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
