reliable see considerably begin useful method see conv l svm k conv bit reliable svm var conv compare efficiency test highest inefficient plot compare envelope reliable reliable envelope mis mis plot plot envelope help reliability prediction good nominal nominal equal conv failure conv equal also reliable conv figure provide figure conv failure concrete conv effective precise conv concrete band envelope tight conv band var dataset mis model conv rank third var conv give ranking summarize row dedicated summarize inter table fourth upper cm fix k var var var conv var concrete var var var var introduce comparison schema nine benchmark contain predictor asymptotically suffer quantile side prediction effective drawback limit conventional quantile regression prediction limit model proportion identically distribute space predict concern model promise idea predictive support machine side prediction model appendix predictive commonly term error risk risk let square error square hence minimize estimator loo see approximately n constant estimator independent distribution fx tend large variance non tolerance interval side level proportion note need side confidence must sided merge obtain confidence denote proportion upper must proportion quantile statement two interval describe q mention assumption interval eq uncertainty equation mean tolerance bias guarantee confidence contain prediction response estimate tolerance desire conditional error center proof proposition definition guarantee conditional predictor present interval build tolerance interval prediction error query predictive give yield interval reliable interval interval analysis science use problem financial forecasting regression variable random different kind technique characteristic square technique design assumption kind estimate response estimate parametric parametric model regression build make currently context interval application finding confidence least desire proportion environmental exposure trajectory prediction security category estimate technique estimation gaussian quantile outlier suffer slow quantile cross least regression technique interval drawback point paragraph interested finding interval confidence point conditional variable tolerance confidence concept interval tolerance model side easily extend sided interval test interval context efficiency obtain variable bias validation local validate predictive interval capacity provide side predictive interval reliability envelope evaluation chapter interval less envelope almost envelope base find guarantee predictor value simultaneous interval state similar high contribution look simultaneous difference local instead test comparison follow tolerance describe state art efficiency interval propose describe tolerance interval find chapter compare square quantile dataset conclude remark sample random find denote underlie regression represent estimated notation training set regression variable predictor regression true variance point response coverage tolerance response content coverage tolerance distribution content predictive distribution predictive prediction quantile degree freedom estimate regression statistical independent pair follow define q mutually variable function usual everywhere idea finding minimize find eq risk estimator detail risk local approximate neighborhood locally neighborhood evaluate fit covariate change regression taylor expansion polynomial write vector weighted kernel estimate tx kernel observation close point big weight bandwidth scalar select scale q author include take distance near neighbor neighbor popular technique choose observation consume common bandwidth plug unknown quantity estimate section plug regression bandwidth weight weight predictor kernel parametric function vector locally problem convert weight interval guarantee confidence specify proportion tolerance mean deviation quantile chi distribution quantile error inter quantile variance use variable dependent yx show estimation estimation tolerance introduce coverage tolerance denote predictor side tolerance content factor express interval regression separately detail see parametric tolerance response regression work address parametric distribution unknown deal finite need inference review square use find besides address method learn explain application drawback paragraph emphasize method method obtain regression tolerance interval build eq certain find interval average deal omit regression interval model instead prediction test square extensive topic square sample tolerance simultaneous study interval quantile square technique x give fold assume unbiased independent finite size statement appendix prediction drawback band tolerance bias interval treat tolerance least square part level mean interval function desire cb create envelope simultaneously part response distribution describe q detailed difference prediction contain least proportion coverage create envelope contain envelope coverage describe quantile side two interval treat separately side sided least sided level far similar tolerance build quantile regression quantile quantile estimation sided interval interval level pair quantile side interval build must side quantile merge interval statement probability combination side square explain overlap side enforce dataset prediction need purpose building plot far compare effectiveness interval note content contain train small medium dataset schema mean inclusion fraction content mis sample deviation prediction sample reliable definition measure content mis mis mis mis mis mis bn mis choose mis normalize mis equal mis value compare reliable mis complicated mis prediction yield mis behind find equivalent predict see interval average obtain correctly response variable gaussian inter quantile q trade sake find final wide envelope mis mean interval prediction visualize big dedicated dataset compare whereas chart compare method across nominal distinct proportion difference compare nominal represent proportion represent represent nominal line value nominal represent reliability obvious failure reliable higher desire obtain respective nominal precise line reliable mean label normalized looking efficiency line inefficient ignore mis represent mis look figure obtain envelope model envelope satisfy constraint normalize mis mis chart chart goal mention response value locate predict interval display contain mis chart mis different gaussian deviation chart display concept predictive interval interval introduce rating measure frequentist interpretation interval sample level tolerance interval quantile mean interval proportion pearson independent content tolerance sample regression confidence induce include therefore confidence interpret chapter frequentist viewpoint true response variable obtain interval confidence interval regression quantile obtain interval true unknown concept query desire proportion long measure predictor interval interval content goal content predictive predictive interval change regression predictor verify entire distinct huge impossible regression make state begin variable approximate normal related definition observation v dataset result deduce formally sufficiently include predictive interval average simultaneous tolerance schema usually approximate level prediction content interval reject quantile reject nan hypothesis predictive significance test simplicity response inside content interval simultaneously proportion build express content content proportion prediction prediction interval training include arbitrary training tuning convert hyper parameter take least method predict give tolerance hyper index denote parameter content divide find also pass result satisfie constraint one fold hyper small mse find small mis space parameter divide tune hyper tune mis model cross give advantage exist package aforementione remain hyper model know content upon sake brevity part interval interval quantile consider construct knn hyper knn confidence coverage interval regression hyper knn svm regression small high guarantee interval begin like decrease interval size long constraint leave influence interval obtain tolerance regression assume mean error locally find properly bias tolerance leave fold error regression bandwidth order interval bandwidth tolerance interval bandwidth variable optimal non consist balance bias vanish sample tolerance interval far find desire proportion find tolerance interval response add add remove response tolerance confidence part context tolerance interval tolerance condition deviation formally let query bandwidth local minimize satisfie condition almost include neighborhood usually regression neighborhood next reader regression satisfie bias variance content tolerance tolerance appendix prediction error variance therefore biased estimator advantage tolerance prediction approximated tolerance error inside represent without note become residual neighbor content tolerance interval compute replace propose take near variable tune depend tolerance cross whole summarize obtain interval ix equation hyper tune interval complexity interval instance evaluation local neighborhood inside appropriately take bandwidth coherent satisfied bandwidth bit great bandwidth behind bandwidth vector begin coverage interval interval interval add iterative uncertainty assumption match yield partially tolerance decrease increase increase contrary interval size choose value coverage interval desire much near number instance tolerance interval min return min kk increase contaminated prediction practice small belong subspace different error serve restrict region usually regression regression neighborhood give small tolerance begin everything calculation phase describe local confidence tolerance obtain predictive provide predictive finding predictive tuning constraint interval interval hyper regression hyper serve hyper respectively interval variable predictive paragraph review application regression method polynomial point near bandwidth refer first interval bandwidth interval instead optimization tune high neighborhood pair great select fold tuning define mean mis neighborhood big tolerance high coverage increase interval little bit begins usually evaluate knowledge tuning first find tune come way find try decrease interval goal size inclusion fix precede decrease inclusion compute local density response dense hyper repeat iteration min max min min mis mis min min fold schema equation tolerance previous compute interval upon capacity content model compare reliability envelope describe variable selection outlier preprocessing organize five part describe part interval discussion nine dataset list find double mention systematically remove dataset distinct dataset name total variable contain concrete compressive strength concrete cpu cpu describe interval programming language specific hyper hyper name linear list method side predictive neighborhood var side explain sided package sided interval regression quantile function type percentile se interval explain quantile hyper cv argument radial set default radial cv non parametric tune cv small mis satisfy radial basis radial basis dataset tuning constraint l svm use sigma sigma conv conventional prediction use near neighbor fold validation training non list must tune var may different mention l svm conv concrete auto cpu attempt tune hyper serve
remove solution c satisfy guarantee iteration remove residual hold satisfied unit sphere nice follow algorithm f iteration iteration lemma remove ia b ic update formula expand term conclude lemma prove therefore q since I c step part directly tensor change ta claim deal difference sum frobenius equal therefore ic ic show true replace true frobenius bind w term product replace leave hand frobenius frobenius normalize version last inequality tensor applying provide vector bind chi q generate tuple guarantee potentially generate bad tuple provide specific output key time without assume case choose work let tensor w I denote hold definition corollary norm bound enough know j suppose initialization correspond tt exploit part prove succeed output center close compute previously find tuple tuple satisfy w jj already tuple condition algorithm rgb corollary criterion derivation observation time bold bold paper provide cp adapt tensor establish third dimension tensor incoherent recover overcomplete also arbitrary initialize vector tensor slice guarantee tensor tight perturbation give overcomplete decomposition popular unsupervised range mixture community certain fourth order sample tensor source separation topic case tensor order exist stochastic guarantee orthogonal solve orthogonal eigen decomposition symmetric second step limitation practice learn especially whiten whiten expensive suffer instability lastly unable overcomplete number orthogonality limit popularity overcomplete many domain decomposition tensor mode alternate method extremely fast calculate global guarantee modify alternate method make update mode update suffer ill whitening approach presence incoherent soft orthogonality incoherent representation extensively literature number context compress incoherent flexible modeling overcomplete signal parameter construct feature tensor incoherent establish guarantee subsequent work various learning cover cp alternate alternate power asymmetric tensor mode project update local incoherent due incoherence update even noiseless set approximation overcomplete incoherence remove run remove approximation consistently overcomplete slice initialization alternating lead global tensor guarantee tensor rank tensor mode overcomplete tensor tensor arise low transform simultaneous prove consider tensor slice update greedy rank perhaps natural cp decomposition lead recovery tensor np obstacle limit tensor incoherent error alternate update decay end random perturbation tight contraction also decomposition tensor square guarantee update orthogonal analyze extend tensor whiten early whiten lead require overcomplete overcomplete tensor challenge identifiable general factor matrix however dimension procedure limitation assume generic time procedure overcomplete tensor procedure recover overcomplete tensor high instance tensor utilize slice mode level additional dimension obtain fourth order slice simultaneous perturbation provide ica slice overcomplete consider overcomplete sparse tucker specifically identifiable tucker decomposition weak guarantee significantly one consider fall minimization provide convergence completion phase code involve notice asymptotic sometimes clarity notation factor convenience identify way pi limit tensor instance mode index arrange row tensor mode mode similarly slice fix tensor slice rd mode r arrange multilinear matrix tm vector eq multilinear multilinear slice write notation represent outer product cp multilinear particular column adjust appropriately denote spectral tensor scale style fill corner sep c initialization svd base power power descent width c line width corner leave algorithm dash corner width fit label draw corner width pt leave alternate guarantee recover alternate coordinate one procedure power asymmetric mode tv w update formula multilinear tensor mode alternate different later relation approximation denote dimensional sphere program multiple local show update optimization vector alternate approach converge initialization iteration unit option option initialization asymmetric multilinear member numerator formula orthogonality orthogonal second power lead incoherence encourage soft orthogonality iteration propose residual discuss tensor initialization true rank random initialization draw unit sphere option right procedure vector initialization initialization know one identify initialization successful cluster procedure appendix standard singular vector update tuple k update starting denote tuple previous power recover algorithm remove residual mainly descent unlike iteration immediately stationary inspire residual analysis step matrix satisfy bind column bound satisfied c update I procedure I advance store multilinear computed multilinear sample method mm apply tensor linearity variable quadratic update orthogonality alternate view least square unnormalize rewrite rao column simultaneously update current vector column efficient complexity show procedure throughout assume perturbation unit vector goal recover rank challenge overcomplete regime dimension loss w deterministic tensor convergence show satisfied uniformly simplicity assume main part also reasonable assume deterministic assumption matrix impose soft w upper ambiguity recover tensor unchanged vector mode sign vector mode vector convex tensor decomposition tensor uniformly appendix perturbation satisfy q weight bind initialization vector addition update formula convergence tensor decomposition algorithm output matrix appendix overcomplete incoherent assumption assume simplicity many derive local also interpret identifiability incoherent frobenius whole residual term two main removal guarantee step convergence tensor result reason provide column guarantee initialization rest regime residual state explicitly hold recover tensor cp propose main change adapt tensor change change slight modification consider tensor tensor power remove order tensor brevity guarantee rd tensor cp recover th mode modify change tensor setting decomposition constant addition step satisfy h number initialization exploit svd propose slice tensor guarantee initialization guarantee mm number w svd initializations unit rank arbitrary tensor setting efficiently decomposition overcomplete arbitrary constant simple svd base argument adapt lead convergence tensor power overcomplete overcomplete mixture moment high low label initialization svd initialize update generalize set overcomplete component overcomplete component addition q theorem proof mode mode arbitrarily overcomplete global employ svd contraction power remove residual break incoherence precise convergence enough estimate contraction incoherence crucial establish part number satisfy close argument analyze dominant bind experiment consider experiment three aggregated initialization generate I initialization vector formula option criterion iteration improvement stop accord size tensor initialization illustrate average depict versus initialization horizontal plot easy column setting linear need initialization almost recover start initialization ahead hard rate initialization ratio average run threshold several eq corresponding estimate relative estimate stop output increase computation recover overcomplete hard descent remove error iteration see initialization nearly rate svd computation expensive desirable theoretical initialization appear experiment additional room improve guarantee c avg avg e acknowledgement discussion alternate early use lemma support microsoft fellowship award award award nsf award nf award nf column remove rao denote norm norm text matrix randomly randomness decomposition unit sphere vector component incoherent incoherence spectral condition norm constant rank tensor bound eq weight factor define particular initialization norm many choice discussion provide brief condition normalize remove issue additionally incoherence setting impose norm tensor w h draw sphere limit provide bind initialization ensure iteration define contraction initialization local tensor spectral seem even condition imply lemma go set matrix satisfy incoherent uniformly unit sphere incoherence high understand matrix tool unit eq hold notice proof incoherence unit incoherence incoherence disk index particular entry inequality use outside last old old proof apply h old application cauchy unit sphere eq unit exploit overcomplete property matrix spectral bound know first triangle cauchy exploit follow independent gaussian eq bound complete union rao step remove rao product norm rao product satisfy idea matrix bernstein prove concentration symmetric
complexity add definition corollary learn computationally protocol efficient agnostic learning accuracy acknowledgement grateful first drawing discussion regard threshold bound interval thank suggestion contract nsf grant google notation convention let type random take follow entropy universe hold equal boolean shannon information sequence leibler entropy jointly recall characterization mutual divergence jointly support say bayes definition exist universe recall principle input uniform computing error communication follow deduce deduce technique distinct refer representation let representation margin boundary setting represent represent representation therefore satisfied ordering value obtain strict function dd mistake tree complete mistake depth inductive leaf choose free property restrict identical threshold interval mistake let mistake replace replace leaf depth depth df z jx v x jt binary min pair ax ax eq impossible send divergence hand identical eq output contradict joint distribution view point occur line intersect formalize pm let hypothesis class quality output follow differentially finish small select independently accuracy call mechanism use affect one privacy consider hypothesis mechanism private independence hx observe multiplicative chernoff similarly fx hold hold hx fx least part work universit paris paris mail analyze private notion privacy introduce provide agnostic study work start main differentially private randomized communication evaluation characterize mistake mistake pure private sample complexity differentially pac open know build communication complexity training individual sensitive medical therefore preserve increasingly mine pac agnostic private guarantee individual observe private formal definition achieve amount computation differential privacy agnostic study show differentially concept denote whether gap class output point bound non proper infinite show either distributional require label domain exist hx private pac representation dimension natural version surprisingly prove characterize differentially private pac c relaxed differential privacy use sample complexity private proper relaxed version leave open question proper resolve establish definition ingredient term function message bit use coin bit public coin share need least definition problem clarity omit c function discretized correspond communication view integer b combine whose dimension yet pac imply result communication sometimes give proof bind relationship private coin additional via prove first study communication complexity line plane complexity class finally pac differential privacy imply privacy separation concept substantially essentially thing know communication notion specific privacy extensive privacy machine relate area survey convert include majority formal differentially private seminal aside already differentially private show learn polynomial point commonly set specific learner learning relaxation know studied denote equal disagreement also refer duality packing cover notion characterize differentially pac agnostic private differentially pac weak differentially privacy imply learner dimension private logarithmic query database synthetic fx sufficient low sample proper agnostic good error principle tell function mistake label label concept subtree remark mistake include leave define mistake tree characterize small mistake online mistake bind relate communication complexity learn class input concept result concept public coin let output h boolean prove minimax principle imply induce every protocol public know know fx protocol event contain hx fx g cf deduce communication also private coin protocol majority communication function hold average one bad fx von deterministic suppose function exist hx fx c strategy player payoff minimax exist coin protocol use private output protocol private learning simplify bt I follow view integer note bt bx communication lower correspond distributional complexity distributional way great use public private coin simplify presentation eq equivalence theorem weak private coin theorem communication private pac learner know use hashing convert learn function result resemble learner provide private protocol probabilistic efficiently learn equivalence pac error notably boost private constant one convert simple repetition protocol back probabilistic show bound complexity differentially concept proof complexity index get get explicitly prove way communication adapt concept complete path root
sampling base strategy simply sampling correspond monte single unlike original need step sample collect bad case mcmc small efficient collecting already early operator experiment empirically whether computationally realistic discuss eq consider factorial differ interpretation early show deep special training naturally lead train contribution conditional combine randomly joint ensemble possible model one fraction step kind uniform sampling subset independently mix precisely choose variable obtain gibbs deep would mix slowly gain consecutive reduce chain provide adopt distribution resample particular visible low transition gradually reduce high mix initial clearly plausible mode step fast visible consecutive digit start another digit slow sampling sample consecutive anneal sample use successive sample realization digit collect investigate propose noise run chain chain increase importantly none chain able however generate sample quantitative instead compute take sampling take get autocorrelation successive starting collect collect autocorrelation speedup factor discount accordingly procedure conclusion really conditional compatible deep multiply computing cost visible markov cost run predictor unless variable trade computation control validate permit trade gradually low step acknowledgment thank frank com op universit neural autoregressive recently show successful modeling multimodal rely variant deeply train unfortunately deep corresponding neural predicting give connection train markov associate way compute sample speedup account consecutive reduce achieve use combine mixing sample thank step low unsupervised representation learn learn great success instance claim art recognition deep convolutional network supervise deep unsupervised counterpart challenge popular model model autoregressive early work model binary network computation connect graphical distribution pixel consequently exact since replace distribution propose yet variant deep neural modify effectively train unsupervised learn autoencoder autoencoder estimator autoencoder continuous unlike aim stationary generating distribution able possible boltzmann approach multi mp instance label special achieve state dataset agnostic cast theoretical explanation allow alternative deep mcmc rather describe description connection sec novel sampling call inspire empirically effect deep deep dimensionality q predefine index index order hide eq weight bias logistic sigmoid nonlinear activation train maximize denote issue original predefine limit capability model ask infer conditional eq visible intractable marginalization build architecture lot share weight parameter layer front predict one sharing unit sum sum plus associate unit available predict procedure train factorial objective intractable involve factorial instead variable regular feedforward firstly accord index set provide input length secondly solve issue original optimize suffer infer lack predefine ordering make regardless depth neural train simply usual training procedure generative network distribution although tractable transition operator chain mcmc whole directly jointly operator argue easier simple mode mixture mcmc powerful parametrization special auto corruption denoise autoencoder parameterize corruption stationary ensure ergodicity word match albeit transition suggest possible observation operator quickly find distribution start random reconstruction encourage burn sampler model explicitly chain find plausible order agnostic procedure sample
absence reliable rarely visit medical typical indicate signature patient stay home infected computer trend positive negative underlie rarely fully subgraph infected node set epidemic epidemic source phase epidemic nature patient hide connect initially patient algorithmic statistical challenge pose consider instant inform subset exact reporting unable report snapshot statistical determine epidemic possibly initial rather infected seek scalable minimal assume neighbourhood need infection epidemic member social network spread share technology medium well possibility medium friend website decision make local furthermore apply negative positive find contribution review extension node decision distinguish alternative arbitrary infected infection accord edge infect infection infect node reporting node represent positive reporting process illustration report infection probability report scenario node fraction ultimately infect probability report false go setting diagnosis work school epidemic distant neighbor independently like epidemic mass medium affect independently illustration epidemic report infection zero infected fraction two truly report denote positive report report easier truly report reporting describe report report report report epidemic report hypothesis visually predictive forward first work focus epidemic estimating topology infection exceed give predict future focus inverse decide question attention transmission mcmc question epidemic source closely model seek completely rely tool topology infection may handle number seed insensitive hundred spread come inherently essentially count node many high level triangle parallel significantly knowledge something compare computational running scale report infect control pose negative statistically resemble second infect large graph play role statistically contact complete two indistinguishable algorithmic regime infection minimal address contribution number reporting require minimal know local performance give epidemic particular succeed infected fraction node infection succeed experiment real world theory easily epidemic denote near indicator indicator radius indicator enough report immediate neighborhood equivalent correspondence indicator evaluate call intuition epidemic infection report infect classify infection cause precisely algorithm epidemic report scenario counter counter epidemic reporting depend near neighbor theorem make choice require positive detail appendix topology analytically hand suggest choose graph information encode contrast report reporting node parsimonious reconstruct knowledge report rather reliable compare similarly memory therefore scalable increase noisy choice topological constraint converge report irrespective truly report positive great every reporting contain report show correctness proceed boundary infect alternative infect infected boundary infect alternatively infect every interior node nod local infect difficulty increase set number false reporting show regime converge truly report succeed epidemic scenario significantly indicator report random graph key technical proof sum appropriately sum node truly report reporting exist kf constant depend problem infection regime succeed negative positive truly infect report detail defer appendix infection infect node condition satisfied reporting goes truly report correctly converge like epidemic report report report report nn reporting epidemic interior non truly report kf key evaluate hoeffding corollary structured correlation decay pattern graph variable indicator independent environment disjoint local also disjoint environment maximal adapt concentration epidemic scenario great interior show appendix corollary explicit include statistical substitute corresponding grid q function difficult solve apply correspond long local contain infect requirement fail mesh ball number ball indeed report mesh case constant even epidemic infection total reporting truly high infected neighboring pair indicator positively correlate uniform reporting indicator I probability set n epidemic apply type ii tend zero kn ii count infect infinity truly report node epidemic correctly source chance environment network model email network internet facebook empirical test focus scalability simulation demonstrate ease real require care extremely set infection infect performance regime false positive infinity test regime rapidly predict theorem report reporting node tend infinitely negative positive converge see remarkably bad diameter considerable random network infected mean false reporting truly report among report zero addition reporting tend false positive epidemic email motivated spread infection negative false positive infection seed algorithm affect radius medium social identification network cascade failure attack security epidemic email comprise number infect small truly report report positive situation error test local neighborhood indeed person incorrect near account network information inter decrease epidemic estimate distance network epidemic report infected buffer negligible report close reporting mistake node ball increase radius distance type topological noise plot distance equally line two node infect positive number positive become increase depend opinion identification process force single reporting map snapshot underlie network analytical wide correctly false negative infinitely positive simulate free graph facebook email chain size practice extremely estimate finally facebook report infected epidemic scenario similarly report report scenario first q tends set corollary zero tend otherwise q equivalently highly first likewise np tend word alternatively converge calculation condition reasoning scaling consider infection noise tend condition particular report correctly alternatively network grid tree noise reporting conclude random sum obey xy xy c replace note xy mx xy mx xy mx hoeffding inequality np xy xy x x xy parameter must infect report infect near infect distinguish two discuss infinite tree start root infected level deep tree infect consider contain infected node ball ball radius interior hence ball case
study simplicity condition strongly consistent dominate x var n normal also account obtain thus asymptotic interval law belong power strongly mean variance denote c ii hold consistent obtain mle however observe size mle shall focus consist iterate log maximize calculate accordance method consider st let simplicity shall notation lk model individual satisfy q notice although lk z l l need follow normalize straightforward notice depend influence maximize log word constraint entire st iteration determine parent iteration observe reach never leave calculate nonetheless include description shall deal case begin stop em k ne em verify log unimodal mle choose determine p criterion denote one unit reduced generation control identical em construct converge step eq n np I prove lk lk l control distribution need verify take normalized maximize subject eq final apply check expectation likelihood function estimator shall result simulate binomial distribution mean rate practice kind evolution presence binomial birth probability subsequent evolution notice distribution consequently consider unity generation take simulate individual material evolution control prove exponentially rate evolution individual solid confidence interval family go observe accordance remark estimate solid line parameter horizontal pdf var pdf evolution estimate right course line approximate dash estimate approximate interval dash line horizontal value situation assume number start per known try reasonable per use illustrate individual plus run convergence occur procedure assess consistency estimate figure show base sample width medium width var right entire solid individual dot line value line kind work information come environment control clearly justify parameter evolution plot observe respective value study unity simplex uniform open em sample insensitive choice initial maxima saddle give law maximization evaluate parameter greatest start discussion observe knowledge kind denote aic consider sample observe aic binomial lead little kind control satisfactory procedure would term model expect cn binomial column need attain procedure mu leave entire family bootstrap dotted line pdf pdf pdf left parameter process generation em sample obtain approximation analogously correspond right variable joint one observe distribute curve give calculate respective preferable assume understand great content former sample interest determine respectively lk z lk max nb associate row element probability row equal product analogously assume let storing store say order determine possible form dimension tree generate consider great lead study supplementary material give complexity indicate one need population available evolution leave sample binomial em iteration store store iteration require sample procedure seed evaluate simulation perform software log package estimation consider nonparametric control observable generalize law practice observe family tree situation observable individual generation generation observable parameter incomplete procedure make algorithm show encounter maxima different high although simulate show provide adequate reasonably store iteration simulate entire tree bootstrappe approximation algorithm contain variability acknowledgement author read comment would thank ia provide de ia de grant immediate base consequently eq fp lk maximize subject negativity kullback verified maximize take maximum make strong law number fix simplicity let k martingale difference account iv verify I sm nz guarantee iv vi ik k I kn last inequality ii consistency take account respectively consistent denote sequence I central sum theorem adapt brief firstly due k cite proof know proof immediate proposition iv shall I I limit denote analogue cauchy consequently pe enough follow q nj prove analogous argument jointly condition vector nu j prove argument consequently material follow binomial distribution variance respectively sample family individual generation robustness em observe maxima choosing unit simplex uniform overcome problem propose likelihood binomial control convolution maximum coefficient degree take em estimate great log start likelihood function versus start seed center width p p pdf pdf width maximize log figure ccccc cc cc p em expect paper convergence figure start seed expect determine value function provide possible maximum notice sample remarkable analogously contribution z b bb determine eq q value possible obtain go go value nan store file take account table relationship r max z max thm thm branching branching evolution size generation interest sample entire investigate secondly practice consider use individual problem accuracy procedure example branching control branching class stochastic characterize random determine overlap subsequent usual framework branching relevance growth biology cancer disease cell link etc structure add branching generation degenerate deterministic control individual practical population possibility affected binomial control would reasonable concern introduction environment introduction specie control branching branching see see state recent therein development applicability model frequentist model theory control square branching sample firstly
switch switching horizon curse dynamic number number investigation switch use recently however priori result point cost switching mode moreover I assign modification characteristic switch function fail solution problem development aim develop cost carry switch network nn weight continuity change satisfy necessary uniform method analyze numerically close literature subject maximum switching need assume state discretize c calculated numerically switch minimize change continuously calculate organized problem detail iv discuss implementation analysis conclusion give vi dynamic mode model nx respective mode active identify schedule system schedule cost compose piecewise cost mode horizon c cost therefore consistency start assume sign definite develop admit reward e time decision make researcher straightforward cost go final fixed depend word e different go observation develop solution switch cost active cause scenario switch previous active controller want current controller want compare see cost switch instant previous active time cost go dependency go consider concept mode operation play switch word mode another go already compatible looking mode active configuration position stick manual transmission car mode mode mode go step operate denote terminology instant calculate form recursive bellman mode give consider concern run select next already switch already active concern address minimization concern address minimize fourth concern however address term confirm go close operation switch approximate cost go linear nn approximated select smooth denote learn algorithm cost go incorporate weight input function subject proceeding training concern nn continuous neuron otherwise belong integer e function continuously versus unless system comprise select continuous used idea incorporation dependency give use function I number weight require train multiplied nn following need prove give continuous approximate set every approximated use capability structure prove main idea adapt go minimize state continuous continuity scalar piecewise identical point open set belong boundary limit continuity every continuity eq follow contradiction side q contradict q hold complete derive example repeat weight training detail cm integer repeat cm cm cm go back repeat guess converge domain interest set step step select pair qx cm go cm load batch backward resemble control cost store final nonlinearity hybrid nature subject utilize find see computational increase conclude note capability argument weight train state eq compare online easily firstly provide robust toward uncertainty secondly program nn local method least fulfil condition result trajectory domain train go valid belong provide flexibility desire behavior switch investigation code matlab request mode give horizon discretization system euler integration select cost switch tradeoff cost state switching dynamic compare convergence fast origin speak pointwise switching basis accuracy capability polynomial sequential iteration result plot dependency go method utilize become less eventually switch happen go mode switching turn switch controller provide argument schedule fig turn go operate entire conduct controller able dynamic lead optimally go controller optimally interesting active mode switching switching stay mode confirm go switch beginning becomes turn go latter case need controller new method switch switch eventually mode active eventually need switch e finally initial active mode depict history mode control system switch switch cc cc objective open half open square height lower low setup upper dynamic mode discretize decrease switching also preference neuron domain z select cost switching machine core ghz matlab simulated solution select job control perfect tracking switch three fig seen effectively switch assign switch decrease switch switching switching resemble utilize switching cost go approximated training threshold switch cost apply another controller switch reason process consider fig study threshold switch versus due fig go turn fig high potentially present switch controller switch analyze function operation cost along switch negative controller reward active assign require train controlling
solution contrary pick minimizer via inverse qp refer operator nothing indicate value actually crucial make satisfactory relate identifiability ensure proof guarantee asymptotically unique exist limit highly unstable expression enable summarize recurrent unbiased gaussian lemma state find instance theorem deduce expression asymptotically worth note initial choice basis besides construction entry problem step close vanish entry rescale acceptable final however original easier asymptotically turn consistent asymptotically limit variance improve one suitably possibly non q must invertible admissible clearly influence minimizer asymptotic variance include optimality argument reach lemma raise problem optimal actually plug performance verify reason suggest favor procedure regularity performance situation devote monte study involve scale begin distribution well result soon estimator replace counterpart scale deal arbitrary application methodology model third consider different distribution integer set transition repetition deal arbitrary space bernoulli rescale digits drawing gap iid distribution markov transition keep way conditionally build step geometric error report relatively remark row consider ht c c p c c c summary size carlo repetition state determine binomial indeed convex state hand observe realization consecutive nevertheless favorable conclusion regard efficiency appear clearly observation geometric deal necessarily q describe two size gap result summarize table http c c simulation matrix size consider square correspond standard deviation error repetition performance seem even dedicated show eventually perform situation allowing draw similar consider asymptotically case simulation confirm theoretical implementation consider naive theoretical optimal property technique investigate markov censor sequence chain observable iid identifiability partially lie markovian exploit set framework studied lie empirical observation close monte carlo various situation situation make imagine support recover support determine necessary condition identifiable optimistic current starting tackle question research work identifiable writing sum p deduce identifiability condition equivalent result point optimal sense know map differentiable converge denote q complete theorem proposition statistical transition presence censor situation sub gap iid gap transition consistent numerical simulation probabilistic analyze huge range field markovian give rise markov hide statistical methodology transition censor simple markov gap jump consecutive problem initial gap kernel identifiable identifiability transition know specific show full regardless element framework markovian exist finding parametric seem tractable find eigen element contribution build lie estimate explicit moreover normality asymptotic variance carlo illustrate situation jump process jump occur consecutive observation grid imagine situation subject application field recognize provide phenomena chemical financial market markov study disease process restrict modern literature contribution sparsity difficult exploit issue address considerably simplify nonetheless spirit paper approach well develop technique model section describe problem identifiability characterize build support study numerical monte carlo simulation proof appendix irreducible transition available jump observation iid markovian generator far gap identifiability transition space dimension choose assume slightly say identifiability identifiable every depend remark never intersection combination problematic
leave tight future research would take number implementation step pass else equal mu v take pass distribution initialization entry procedure standard singular vector alternate complexity theorem require remark row multinomial failure mention multinomial sample efficiently change distribution previous section entry input obtain bound inherently non convex might converge rank completion alternate analyze completion proof similar initialization routine show differ previous work key exist crucially incoherent avoid initialization accord w eq iterate obtain singular good rank minimization hypothesis r tt step th decrease geometrically vanish error hybrid style handle leverage alternate coherent new pass interested store two set web need query ad co occurrence multiply matrix give produce rank final factor intend already column correctness suppose wish calculate rank approximation proceed stage follow intend element alternate iteration accord produce factored remark norm completely parallel matrix element set weight mention fast overall particular ia bt r f multiplication matrix matrix rank however approach compare rank r compute yy application server rank iteration server compute norm norm server entry f k initialization cp server server ij u modern compute million dimension store matrix environment problem however depend computation amount distribute set assume partition row rest store th act cp cp minimize standard interesting row linear compute typical requirement complexity weak depend crucially independently need server row outside critical detailed code simplicity drop iteration correspondingly modify simplification compute server server locally independence initialization initialization compute right tr tr tr similarly communication constant round svd step alternate compute cp server row th computed message server message server size combine pca partition completion server cp number update compute exactly hence algorithm communication bind argue paragraph suggest total communication desirable completely contrast communication communication transfer bit complexity bind ij represent initially bit I f bit hence bit b incoherent coherent projection computationally incoherent coherent matrix line b direct stagewise incoherent algorithm clearly directly error stagewise incoherent stagewise computing approximation low error computing simulation singular incoherent coherent matrix incoherent svd incoherent spread whereas coherent entry vary matrix incoherent matrix correspondingly average iteration sample subroutine generally sample plot projection projection choice hadamard compare algorithm vary plot algorithm incoherent coherent figure b give use algorithm compute low rank set approximation plot incoherent plot coherent dimensional row space dimensional first multiply theorem theorem conjecture edu microsoft microsoft com mail edu randomize approach different exist literature minimization factored intend leverage yet approximation weak frobenius combine aspect generate spectral exist besides spectral approach extension interesting new efficient factor give small alternate instead small pca want low rank matrix array technique modern typically approximation residual run pass approach involve column onto dimensional svd bias factor low rank try approximate crucial ingredient sampling combination distribution element row naturally utilize focus frobenius run approximation norm pass approximation spectral however ratio first singular alternative approach distribute contribution low approximation draw execute algorithm factor intend pass independent problem hadamard similarly reference frobenius briefly review problem computing pass present sample compute give guarantee extend additive norm hadamard project hadamard compute drawback hadamard transform advantage sparsity time frobenius guarantee area comparison heavily low decomposition streaming
propose learn modular refer problem submodular relate know modular maximize popular flow represent problem accommodate find item weight learn repeatedly bring second conceptually simple explore face refer episode item regret quantity interest sublinear world simplify sometimes instead submodular function specifically entry word entry maximum let maximum entry find item decrease broken rule e weight mathematically straightforwardly generalize minimization exist view generalize notion closely ground q vector derive submodular increment eq variable many solve nevertheless solve concept concept involve ground flow network unit flow minimum diverse cover item popularity popular item movie title popularity action movie movie cover movie restrict feasible formally lemma act basis combinatorial nature basis suitable solution maximal basis interpretation recommendation cover basis unknown happen instance diverse popularity perhaps movie maximize modular formalize problem bandit item unknown unit cube assume associated arm contribution feedback solve observation several generalization bandit basis agent motivate specifically minimum observe cost contribute movie movie choose movie bandit choose observation gain observe te te non contribution agent agent equivalently optimistic et e nu u te te w te e te te te te te particular refer compute ucb expect item estimate expect episode episode finally item encourage often episode confidence exploit item continuous simplicity exposition episode episode choose item item regret section sort major episode speak part gain item heavily major contribution exist bound prove bound summarize result key analysis episode basis indexing simplify loss episode episode hardness item item contribute simplicity exposition contribute basis intermediate generate select suboptimal item definition say entry ki rest expect bound subsequent gap restrict ready main expect fraction item observe finally rewrite regret k sum sum item follow contradiction result must contribution item episode stress sequel gap dependent cumulative regret episode key idea select regret bound next gap order increase item et fact quantity far combine item cumulative expect trivially q l ucb algorithm ask tight bandit free notable theorem notion item differ improvement different notion follow let independent item item distribute bandit property gap suboptimal item return item gap tighter prove free bandit bound bandit bernoulli set let independent set item independently bernoulli bandit small solution formalize dependent say episode generality consistency inconsistent algorithm poorly logarithmic instance regret bandit eq kl bernoulli mean first follow gap integer bandit view bandit horizon bandit bandit state adversarial environment bound expect upper proposition constant therefore also factor tight upper bandit bind gap multi armed weight bandit major gap key showing difference expect sum difference gain two therefore contain maximum zero fact entry solution combinatorial bandit evaluate synthetic episode choose contribute world cumulative episode difference compare baseline first baseline basis baseline greedy algorithm episode set probability randomly item perform greedy policy flow show practical experiment dependent tight node capacity link next link source nod one illustrate define source flow consecutive third flow cost parametrize formulate minimize modular function submodular capture note sum indicator draw bernoulli independently function episode three suggest upper second consistent fact surprisingly particular time regret episode observe upper particular surprisingly never ccc greedy policy learning service span goal span lowest expect minimization formulate bandit ground experiment network forest compute edge q record noise range motivated network explain distance unlikely cause high episode second greedy episode cost policy table greedy policy network spanning episode learn quickly edge title movie american toy movie recommendation ten optimal movie decrease popularity movie highlight movie third recommend identify people million rating movie ground movie rate movie movie movie episode user episode user movie rate trend number episode movie bandit maximize modular analysis decomposition apparent combinatorial ucb algorithm solve regret latter ucb tight magnitude perturb geometric online mirror adversarial semi bandit offline variant efficiently optimal guarantee hull problem projection single step order magnitude minimum modular paper paper maximize formulate modular quite popular span gap tight gap upper three learn efficiently leave several match eliminate modify leave thompson perform straightforward thompson replacing believe
category system device resource cloud add category training return refine require intervention relevant automate system set capability sound training extract sound new statistically signal evaluate sound classification coefficient compute pick g irrelevant affect parameter code compute time fourier among related centre mass identify magnitude spectrum fall class various system integrate magnitude band amount express sub energy encode prominent filter purpose instead compute filter bank whereas feature envelope sound thus coarse record channel acoustic scene delay occur sound measure variation channel link spatial system harmonic harmonic correspond occur scene help discriminate feature frame method base extract sequence audio signal firstly frequency representation acoustic time acoustic vector analysis model autoregressive instant previous determine envelope model information employ autoregressive approximation combination whenever basis parametrize function contribute example decompose transform frequency index encode audio occur specific time also extract convolution assume analyse alternatively already unsupervise way machine adaptively train representation brain encode along activation use segment significant acoustic application acoustic salient spectral main use signature event important recognition scene encode probabilistic encode elementary property therefore parameter design extraction comprise operation firstly audio scene spaced pixel obtain extract orientation extract frame histogram signal training event car scene employ within training feature essentially perform acoustic property employ acoustic descriptor classify model audio event language test describe process derive quantity use method enhance discriminative capability feature linear non transform cite transform project subset identify direction property general independent analysis employ subspace score measure feature belong cluster near class feature separable computed frame discrete consecutive identify evolution scene extract stage statistical use category work decision determine likely generate principle basic different obtain category statistic close centroid use discriminative classifier interpret class specific feature model output determine hyperplane class training criterion discriminate allow discriminate discriminate belong class train combination evaluate separate hyperplane combine generative model separate vector former overall acoustic decided vote statistical list highlight technique various aspect statistical moment quantile modal express detailed present baseline benchmark account temporal unfold complex acoustic scene record sound precede sound move three normally encode matrix sound occur correctly unfold indicate probability connect sound occur sound negligible connect wrong sound employ unfold acoustic event dynamical theory recurrence series context measure audio scene output acoustic feed system computation community verification problem sequence vector derive dimensional acoustic probabilistic linear discriminant criterion determine sample decision generally type respective pair decision criterion multi use already audio frame scene accord category majority vote employ vary audio frame contain assign training whereby close according whereby model system mobile device area environment encounter supervised design running instance parallel use result decision rule signal discriminative example meta algorithm multiple copy bootstrappe lee learner category frame acoustic scene technique throughout compression forest output approximate compression normalise compression audio file audio base thought algorithm deal select combine run test optimal majority vote sophisticated weak see range signal processing context allow solve category indicate member universe system statistical classify firstly divide short frame choose frame ms depend signal frame sequence transform reduction aim member yield distinguished far transform clarity processing operator number occur reason belong category let describe note stage illustrate function phase complete apply phase classify return follow combine extraction modelling spectral special case call name whereby word occurrence frame follow ignore feature tb supervise acoustic despite effort benchmark tackle audio acoustic line aim research similar music publicly available difficult previous collaborative include environmental music audio condition repository would suit rigorous fair evaluation available database series general effect sound com series sound constitute challenge provide researcher produce environment record office restaurant disjoint contain long scene publicly researcher back challenge website http uk table list challenge tailor sophisticated recurrence quantification feature scene histogram gradient audio scene acoustic audio extraction frame z lee acoustic selective event detection sound scene machine classifier representation j scene low due matlab toolbox present correct david dissimilarity bank none majority vote descriptor energy frequency near descriptor auto maximum acoustic baseline patch energy moment vote majority vote selective max energy spatial fisher majority vote weight vote moment audio distance summary name statistical fed separate hyperplane reference discriminative frame majority vote weight majority cite method good system comprise phase challenge provide indicate environment test optimisation far fair evaluation private recall training belong depend available represent general problem occur bias produce office ideally complete historical office world bias towards rich office infer office environment incomplete use available label partition phase propose purpose challenge employ fold five independent classification run design allow signal proportion keep signal per avoid calculate yield correctly classify correctly sample th belong classified belong chance perfect confusion classifier whose belong private fold box perform differently definition vote classifier audio refer human accuracy interval display algorithmic comparable human see accuracy algorithm method central dot accuracy average fold interval assume fold whose measure evaluate fold bar ratio fold root fold gaussian interval probability baseline mean group exceed accuracy level significantly box indicate perform red file common method indicate certain independence incorrect classification agree particular label allow relatively meta performance far perfect suggest acoustic confirm confusion figure commonly misclassifie tb confusion obtain highest fit among fold every hold file file basis recall test assign define file correctly classify whose two misclassifie classify return equivalently correct incorrect imply correctly misclassifie performed perform pair test classifier reject fix grey box represent whose significantly dataset ranking bad choose accurate performance range say significantly high majority pair assume x statistically assumption tb distribution classification accuracy accuracy calculate acoustic bottom plot accuracy tail ten result carry understand whether individual evaluate fold signal total category scatter file observe acoustic belong belong greatly vary exception office contain sound multidimensional scaling dimensional pairwise see detail demonstrate achieve tend decision expect mistake aspect decision pairwise number multidimensional approximately distance yield sufficiently suitably place corner private testing plot appear together svm svm human participant ask classify public dataset audio record category office restaurant designing choose evaluate acoustic nothing participant label prior test people allow audio appear ensure file classify work advance test old device high care include participant participant achieve low lack motivation accuracy people outli accuracy box reporting interval participant locate note decide include participant classify accuracy accuracy would lot close median participant public category ask participant total occur participant classify individual classify positive derivative improvement hypothesis participant perform calculate participant right reject great tail fail reject expectation smaller indicate participant find reject hypothesis believe classify individual find tb row confusion insight human confusion commonly misclassifie category estimate belong various addition distribution represent average assess accurately benchmark figure depict public public private disjoint subset informative compare trend human oppose achieve accuracy human appear regular outli whose sophisticated algorithm important factor learn classification abstraction expense detail nonetheless think valuable insight gain result light challenge trend regard statistical infer algorithm extract signal class label moreover whose achieve technique classifier audio discrimination audio signal environment classification boundary discriminate recorded environment algorithm motivation attempt extract encode
ig x k important q denote th block e form q abuse notation knowledge require operation record use equality matrix updating eq let special complexity latter expect average spend synchronization parallel alpha problem proximal discussion concern presence output first accelerate alpha proper proximal accelerate arbitrary proper c yes uniform yes yes yes yes yes yes yes importance alg alg proximal c accelerate modification lemma convex contrast block recursively eq positive clear get z p p z recursive substitution combination deduce positive p conclude k deduce fact sum z z state let sample arbitrary sequence produce recursion expectation reasoning analyze obtain fy eq blocks function hold thus follow descent serial variant without compact span sample unified variant focus unified strongly extend proximal sequence sequence k theorem remark reader refer find classical accord alpha unconstrained attribute uniform hold ad hence reduce output satisfie particular explicitly state apparent approach clear sufficiently accordance alpha accelerate coordinate descent solve proximal choose generate p z uniform sample output corollary distribute coordinate present school focus dimensional minimizing sum regularizer alpha update subset coordinate remarkably flexible randomized method accelerate importance deduce complexity many variant improve grow realize approach problem moderate solution moderate sufficient learn science shift problem truly method reasonably work read describe need able read describe problem information contain type descent semi paper focus seminal early justification convex extend primal dual accelerated coordinate characterize perform operation advantage reduce subproblem theoretically practically accelerate descent latter acceleration parallelism proximal setup accelerate paper unconstraine constrain progress et asynchronous method develop liu deal update coordinate likely coordinate importance recently consider serial arbitrary investigate objective zhang nonsmooth coordinate gradient improve specialized algorithm analysis material optimization close main contribution randomize composite alpha pick value focus minimize zhang possibly nonsmooth appear alpha arbitrary expectation bound cover accelerate variant knowledge complexity randomize arbitrary dependence counter objective admit sampling assumption determine need determine optimize bind serial common conjunction serial lipschitz block derivative situation serial block update need intuitively capture certain smoothness gradient spanned coordinate systematic study exposition focus apply smooth accelerated variant analysis alpha remarkably stepsize alpha reduce focus alpha deterministic gd deterministic new cardinality think distribute computing sampling let choose subset block uniformly take set reduce similarly reduce lead specialized distribute establish robust many one force traditional whether alone often iteration row think put element alpha bind bind upon complexity coincide algorithm depend certain level closely obtain unified variant alpha achieve establish recursion arbitrary alpha differ block convex choose sequence reduce sequence vector equivalent avoid extract relation sequence admit weight say admit denote hold vector formulate observe bind tool design formulate study coordinate method tool generic establish randomized descent establish vector appear directly influence address paper uniform nice particular algorithm refer reader systematic admissible complexity analysis alpha unconstraine cover also different alpha explicit solution reduce sample z fy result alpha alpha proper iterate q particular optimal alpha accelerate variant accelerate proper rest proof theorem alpha general prefer establish proof sample identity notice positive produce algorithm fy line v last inequality recursion inequality convexity side expectation divide side take finally r last assumption flexible modern special reasoning case alpha sequence see modify sequence obtain choice deterministic randomized latter choosing constrain uniform ht characteristic uniform algorithm alpha eq simply norm define likewise euclidean block choice gradient indeed reduce I k follow apply keep reduce positive follow particular special allow arbitrary k fy kx k accelerate coordinate iterate method case direct satisfie serial coincide randomize
large scene height roughly scene feature count none take five see histogram match case bag come reconstruct count estimate case illustrate office scene image overlap extraction create iii annotation row simple view scene likely practice rarely access reliable instead hundred automatically derive infer counting grid different human label output detector image various oppose input window counting grid spatial pattern depend reasoning example image full office fig extend include help overfitte issue count bag discrete counting nevertheless attention counting property relationship bag vision feature combination image significantly simple unsupervised learn discriminative supervision integer continuous lda multinomial spatial relate scene spatial among ignore topic assign co occurrence word model gaussians train counting tie inherent datum bag point count corner window bag define learn region sometimes refer grid help guide reconstruct grid capture region infer feature distribution spatial approximately align relax former represent arrange divide pre specifie assign hand place center arrange flexible spatial configuration generalization level uncertainty image patch build consist pixel generalize small image synthesis transformation translation employ would work sequence represent map multiple independently hand entire single point model however turn mixture bag experimental mainly generative grid mixture bag intersection grid hide simplification bag extract inside feature codebook real value feature sift feature function retain establish thing informative bag count bag organization bag spatial bag bag create window bag boundary image solve arise constraint neighboring window completely determine identity column overlap difference depend neighbor window etc propagate constraint uniquely determined count bag bag constraint constraint likely representation lead bag many window window reconstruct least spatial bag window image g category bag imply help predict count bag extract new category original spatial turn constrain useful way count index grid e count follow grid bag firstly window place sum window count fig letter latent variable network fig give bag count overall mapping sum rhs perform count place mapping location bag th bag represent position probabilistic interesting mapping sample window index grid share share hand variational counterpart window particular compute likelihood datum variable perform exact procedure effect use optimize optimize keep lead simply minimize divergence count bag count appropriate window counting optimize bind involve logarithm summation jensen locations k window index index summation bag distribution think proportion type source window performing add bag q consider distribution feature count know feature map grid additional proportion window normalization reduce count grid optimize variational expand equation optimize full fig dirichlet prior appropriate inclusion influence trivial zero equation location window window represent corner finally location update prior surprising mathematically model consider dramatically cg e grid relatively window g though make parameter tie interpolation capacity mixture grid capacity word able estimate grid distribution aggregate window window position mask one corner zero elsewhere experiment prove normalize eqs constitute iterate summarize alg update z z p return symmetry break align bag local minima important however count accordingly prevent problem boundary need grow count cg cg eq mix express many vision framework choice window size fig count illustrate third column remarkably essentially image bag representation iterative start inconsistent direction lead minima deal image consist bag case region relatively infer bag window formulae bag contribute information bag reduce bag section low break count grid window discrete single index equal equality otherwise likewise discrete former characterize differently make histogram pixel descriptor great performance location alternative count representation hybrid counting conference prove successful also introduce grid focus feature generalization window grid large source highly shift slightly avoid appropriately relationship variation term step reveal grid utilize require convolution operation update sum position show slice compute efficiently cumulative sum panel set histogram count well feature efficiency computation bag window uniformly along compute shift contribute mapping histogram image reconstruction histogram section iv input cg color video classification cluster hundred location discretize feature experiment sift enough grid large feature feature identity difficult simply property count procedure run color patch draw matlab load sample drawing transform map point one color section bag bag separately grid window help grid visualize count location equal z color color weight count attempt reconstruct color information location image aware treat feature remarkably lot spatial dark go green dark discover sense among histogram image counting representation reconstruct scene remarkable left upper low z find reconstruct reconstruction iterate eqs count detailed map bag step patch histogram replace e find count grid eqs extreme individual counting become feature redundancy image extract count image recovery use category large generalization bag sift discretized feature sift spaced way image transform create fair implementation allocation feature unless protocol per assign test low counting grid complexity capacity sample roughly lda windows grid parallelism scene grid even basic spatial think count grid fig window give combination report different capacity report level marker grid allocation prevent hybrid cg comprise consider grid use probably abundance training report discriminative baseline counting grid set hybrid cg generalize mixture lda spatial pyramid annotation report count grid outperform spatial pyramid camera dataset illumination lot foreground object category house office environment location class task place validation bag scenario dirichlet allocation well counting grid limit recover perfectly evident counting grid significantly lda likelihood panel count allocation use information fine limit robustness regime mixture spatial pyramid image window overlap window try scene fit acquire camera largely outperform bar sequence datum represent scene actually recover compose acquire camera descriptor successful method combine concern grid use sift model compute place every use forward observation posterior hmm observation constant significant marginally move camera indeed piece recover single bag fine dirichlet performance inferior equally count grid interpretable counting really fig complexity equally issue window learn scale big copy consider worth collection repeat grid day camera
set embedding reproduce rkh characteristic distance measure rkhs yet large kernel perform distance kernel however ridge herein consistency break problem difficulty guarantee nature make stage result short term even verification condition require care reproduce hilbert question general topological endow work domain euclidean admit density string structured object space suffer embedding intermediate cm rkh kernel introduce discuss borel topology l reproduce xu k rkhs reproduce bound operator algebra product expectation lx lx supervised problem hand randomness wherein generate relation scalar simplicity distribution use regressor model rkh function paper composition result element assume exist regularization sample observable ridge quite new remark access tackle transition ridge arbitrary contrast choice algorithm excess e compare estimation consistency function triplet difficulty triplet set ridge make rather moderately since kk x xy cx short insight assumption concrete example boundedness continuity imply supplementary also choose evaluate embedding yield supplement definite many kernel compact domain old compact topology supplement kernel cauchy separable hilbert hence separability hence verification hilbert space reproduce w latter h old supplement boundedness factorize boundedness triangle page continuity compact universal ccccc cm theorem illustrate concrete consequence convergence result outline idea technical detail supplement take excess pt pt bound upper old lead k tr effective cm h excess capture difficulty class decay eigenvalue dimension smooth definition us class ne normal identical use k argument equal mean gram correspond gram decay matrix source manner theorem cb c b k rate task reduces imply material plug bind material excess bound multiplier discard complexity due index marginal row second dominate word lipschitz since exponent small recent upon keep focused relax capture class open ii obtain acknowledgement nsf grant laboratory department uk em em em detailed proof consistency k follow old I topological borel compact compact space page metric trick eqs identity term second decompose notation exploit consequently notation rewrite present derivation z bind series ab theorem apply trick probability xt series l z fulfil see bound obtain matter rate discard bound get lb b l lb rl nh nh constraint cl lb reduce triplet match convergence match dominant carry summarize first term ii exploit ii e term go bn iii dominate I dominate one bc bc ac bc bc condition bc ii bb ab ii ac ac ac bn bn b b b b b b h iii b b b bc bc b eq ii bc nh ac c bc bc bc bc bc bc thus condition satisfy ii assumption positivity bc b cb bc bc bc bc bc bc nh bc bc h bc bc bc bc bc bc bc bc bc bc bc bc bc cb thus n ab provide numerical ridge experiment serve compare avoid density estimation modern toolbox embed ridge smoothing free goal learn choose entry uniformly matrix select point figure display learn entropy compare root square confirm reason achieve estimation large challenge optical small variability iii ground base consist correspond include bag iii bag baseline expectation maximization achieve follow work train hard sample validation regularization robustness pick different kernel exponential student rational mat ern expression kernel explore increase summarize accord use ensemble kernel improvement study efficiency summarize kernel rmse drop set despite domain fall range fairly precise however kernel poorly boundedness excess em regression problem response little distribution inherent estimate consistency guarantee dimension simple algorithmic reproduce hilbert contribution consistency technique stage endow kernel total observation difficulty answer old consistency classical set al stage response distribution machine side thought side estimation hyperparameter close analytical exist somewhat definition consist distribution meta rather sample l patient might identify measure health indicator measurement one mapping health indicator hope perform mapping candidate denote good derive excess proving rl zero sample triplet appropriately rate establish prior effective input mean smooth large motivation fold parameter gaussian gram briefly focus mild back surprisingly question question
take question representation require query kb schema answer simple topic single kb triple stand language question absence kb triple question answer end supervision avoid manual intervention labeling make supervision indirect supervision question kb triple treating mark meaningful representation question involve triple mostly kb previous embedding good quality scale fine tune simplify separate information extraction question answer language answer highly triple million entity interpret natural initial attempt consider template enough addition triple researcher turn specifically semantic parse flexibility interestingly weak supervision setting indirect attempt tackle answer realistic favor hard cover answer negligible intervention answer broad human annotation vector kb answer embed noisy indirect supervision kb triple complete datum associate answer meaningful kb relationship outperform introduce fine title remove even embedding make lead convergence propose fine embed optimize embed lead rest discusse introduce answering section answer mostly via track question query feed web search subsequently extract top return page approach engineer transform open question answer kb natural huge amount supervise machine language large question robust evolve triple entity via distant indirect supervision connect language supervision actually recently tend require manual intervention effort carefully design kb framework question answer little annotation answer question limited open manner kb trend propose embed answer train supervision get language reach performance many task less recently propose connection extraction work build question answer supervision answer consider answer triple kb triple use remainder kb entity word size respectively consist question pair find answer directly rank rather prediction directly query kb logical question parse aim label indirect supervision automatically rest section create consists automatically create kb set answer give kb triple k relationship entitie broad intervention many triple entity numerous entity triple unclear highly database many coverage despite cover triple hence decide create triple automatically seed question display round triple randomly seed question note triple relation table similar randomly except seed triple noisy create training firstly syntactic human english sentence type triple person relationship name string simply fine many generate rich kb would lead well train intervention choose hand triple kb triple pattern l l c kb triple question connect kb language satisfactory modeling follow learn margin want triple question enforce train however sample triple triple chance member leave element heuristic create triple somewhat positive counterpart scheme g embed carry update word entity initialize triple ensure gradient enforce vector learning sgd update course part replace score two contain conduct sgd example create replace question another switching day core force keep architecture entity embedding around also stay scale sgd appear learn powerful however control properly stop pre epoch solve room able rank fine tune often top embedding dot similarity triple efficiently fix frobenius problem minute bfgs subset example example validation learn embed define dot product identity fine slight triple rank end improvement question test ccc recall gram embedding fine detail label need evaluating create identify question hold likely add question various triple total hand question pair evaluate version provide triple sort whether compute precision rank well full whole kb conduct rank top answer compute precision whole kb candidate embedding fine embedding discuss present various version result essential allow embedding encode rich kb word note provide word alignment try gram bring counter factor gram generate poor gram conduct variant try order fail well perhaps supervision allow fine tune top list grant carefully tune similarity improve score almost initial language gradually iterate come automatically acquire allow flexible variation grant come recall triple among display full kb candidate indicate appear close relationship embed triple plain discard decide filter candidate string candidate string noun phrase noun phrase string augment singular final triple make tractable matching greatly reduce model lot table display example near entity vocabulary tend correspond relationship entity close thank use entity string respectively kb l string match fine tuning match question answer experimental section evaluate learn question another learn set match return entity string answer obtain question model record entity compute question return question matching question performance good system report f design still almost manual annotation dataset besides entity name explain rank evaluation new framework answer supervision embedding indirect supervision previous answering question way embed solve promising challenge remain semantic due supervision work answer even word modeling carry encode question
box predict determine response nothing prediction review base describe incorporate acyclic graph dag hide unit feed membership unit belong define combination nonlinear purpose logistic give radial function combination output class letter letter indicate softmax represent estimate note force bias give accord bayes mx dl need rely dimension well raw suffer likelihood x I model belief take extremely statement quantify incorporate adopt specific ard originally neural cancer ard layer hide part weight normal prior zero control become shrinkage pooling prevent overfitte gamma refer hyper leave value specifically automatically relevant determine whether determine briefly overview carlo please thorough carlo method draw suit high net guide network update value network weight hmc hmc update acquire desire posterior level variance include ard closed form conjugate one draw follow inverse control hmc variance momentum update simulate hamiltonian surface posterior hmc create term energy potential directly log natural posterior variance hmc need momentum formulation momentum mean bad result momentum momentum momentum notation proposal momentum distribution reduce reach distant point leave return accord acceptance probability high becoming period great concern modify probability whose acceptance give condition parameter interpretation modify posterior capable maintain favorable ability ease range acceptable previous evaluation hmc much wise similarly operator wise operation neural network literature propagation posterior feed forward operation gpu likely burden impose bayesian neural conduct environment library code contain ard question ask j relevant framework completely response equivalent variable ard determine relevant nan test variable true simplify calculation additionally approximation posterior state ard effect wish great phrase nan significance familiar operate fully bayesian ard nan ard wish test side iterative derive induction random gamma distribute nn degree value definition chi imply conditional value stage ard network unit input accord start update gibbs iteration however iteration thus update simulation ard ard base induction simplification proceed baseline evaluation mention use use website package repository http project tree influence package degree effect approach test assess test pilot find roughly permutation evaluate method effectiveness hundred prohibitive divided primary analysis section permutation since compare snps two strategy allow popular amount rely perform previously genetic involve contain disease capital letter major respectively cccc aa aa bb bb bb aa aa bb bb bb cccc aa aa bb bb bb symbol table baseline snps allele disease status risk embed causal snps snps combination yield evaluation create language run recorded disease snps take snps power correct recommend perform iteration approach use parameter ard prior gamma ard hyper hmc cutoff bayesian ard testing discard sample inference minute minor allele right allele frequency leave bottom minor allele powerful excellent effect test contrast relatively small never high effect threshold tell combination powerful picture appear well remain genetic may instance evaluate examine permutation size accommodate generate used software specify status I sense determine trait minimal relationship nearly purely effect hard snps contribute trait status method scenario approach purely relationship analyze level range previous h outperform method wide margin may detect snps encourage additionally outperform scenario detect across parameter test mention analysis exhaustive permutation significance conclude method genetic snps scale sized data framework investigation technique study cutoff ard case cutoff nothing cutoff value everything cutoff control tradeoff specificity positive false cutoff previous properly amount rate trade cutoff change cutoff increment record figure size produce roc curve legend display auc ard achieve maintain positive bayesian network design genetic marker tb roughly subject classified currently infect confirm derivative exclude snps subject top cluster membership ard hyper burn follow take top five ard ccc rs rs rs rs snp nd significant rs appear snp snps currently locate gene rs report locate mb report statistically either remove study likely replicate capable contain bayesian network association study approach broad different genetic architecture permutation powerful show performance competitive large conclusion powerful technique association capability availability code implement gpu framework outline bioinformatic university nc bioinformatics department state university department university nc discover causal genetic variant genetic pose difficult assess marker trait status demand presence parametric bayesian analyze genetic accurately involve control status graphic build decrease interaction across broad genetic conclusion framework detect handle collect large number variant ability leave genetic disease incomplete presence gene gene critical account interaction vary computational vast marker typical marker experiment marker examine marker consideration million situation interaction modern genome association million nucleotide require examine half interaction sequence cope little technology advance million several type one last decade extension combinatorial consider select via cross exhaustive suffer previously iteration need rw make
believe corrupted feature hope scheme assumption begin main generalization bind proof build top type poisson write write bound random single available poisson poisson write bernoulli draw sample ingredient establishing imply eq q establish notice generality prove translate proof sum gaussian dropout see define objective term write eq monotone equivalent except write eq possible error note topic excess sub excess make optimal guess know q imply optimal guarantee topic pure represent proof e generalize usual rate dropout datum loss e single example need hoeffding generative define show center write eq equivalently sign assume hypothesis q check hypothesis topic topic generative document multinomial multinomial probability topic topic prop prop lemma prop comment prop science stanford stanford usa stanford cs stanford edu originally design deep dimensional language generative long document dropout empirical dropout achieve like learn perform corrupt preserve induce dimension training increasingly dropout commonly network regression document name entity dropout poisson effectively corruption example hard classifier boost example dropout merely create create training erm word independently generative excess erm multiplying improvement additive word document expect minimizer respectively dropout denote variant logarithmic document text get modular bound erm dropout excess recall classic vc erm number vocabulary excess behind half document whole document tail compare variation document compare stem dropout excess bias bias negligible bind ng exploit generative generative incorrect large bias closely logistic dropout rate range logistic regression naive tradeoff dropout improve generalization factor bayes framework prove decay dropout adaptive improves factor contrast leverage improve minimizer regularizer encourage confident prediction confident rare dropout adaptive dropout language dropout topic base restrict setup count usual function analyze logistic dropout train perturb instead word thin feature binomial time replace independently occurrence probability boundary weight dropout often dropout differ dropout setting coordinate reason choose binomial dropout random remain dropout equivalent throughout poisson depict bernoulli topic vocabulary frequency accord document across topic contain topic simplex mixture allocation although advantage poisson gaussians explain extra arise approximate set bridge generating foundation translate generalization dropout first technical every apply assume feature useful balanced minimizer rule balanced topic topic away random substantially separately overhead separate topic almost dimension center proposition turn feature assumption hold risk dropout generate linearly meta modular standard heuristic previous excess picture address dropout dropout view assumption probability numerous shorter dropout poisson document preserve bayes let topic topic topic dropout rule true class incur classify short pair train intuition affects also discuss difficult simple relationship g score classify play almost drive dropout turn give good rule illustrate tendency spread problem red hard classify open circle hard classify dominate red essentially ignore left classify play less driving dropout fine near gray boundary dropout poisson value poisson intensity remain word neither result dropout intercept clear tradeoff end roughly document appear nearly b examine
deep deep implement understanding help deep neural network standard process standard multi layer perceptron typical infinitely many unit form limit theorem function gaussian result surprisingly put weight positive hide correspondence unit neuron fill red neuron black pt sep width text center hidden black cm circle inner sep text cm cm neuron cm h neuron multiple unit hide architecture layer mapping fix representation unless flexible neural deep c pt cm fill black sep center neuron node neuron minimum neuron minimum neuron h cm cm h random nonparametric black node width fill green neuron width neuron size width hide neuron cm neuron h cm input network whose infinite hidden unit whose weight hide unit parametric activation neural distribute introduce infinitely wide basis architecture show top layer node alternate architecture interpretation substitute eq weight ignore intermediate deep layer direct network deep deep view deep bottom section theoretical characterize deep jacobian derivative ccccc layer parameter control control simply derivative layer construction half normal choose regardless magnitude derivative euler limit gradient approach normal even grow heavy tailed bound small everywhere rare jump behavior next deep draw derivative individual dimension exp jacobian independent ij row jacobian jacobian jacobian composition jacobian composition jacobian jacobian compose deep j jacobian product matrix deep common useful manifold useful argue good argue conversely representation preserve dt node color compute little manifold white robust direction representation tangent preserve layer characterize jacobian jacobian representation vary h layer normalize net deeply large direction manifold dimension cc transformation draw code point successively draw distribution singular spectrum analyze jacobian point examine jacobian identical everywhere net deeply singular dominate effective demonstrate arise produce density locally onto suggest width modeling manifold whose cc identity layer function direction move locally change suitable shape remain increase example circular corner illustrate code deep figure functional graphical connectivity circle minimum neuron fill fill h h neuron draw fill white fill neuron bend neuron cm two different architecture deep connect layer layer connect layer adjacent find visual function ccccc layer layer show layer point singular value singular draw deep compare mean output often connect imply mapping one move put representation depend jacobian recurrence jacobian deep show machine generalization exp interesting non local create periodic first basis rise kind feature principle composition close either fortunately square repeatedly feature exp input prior degenerate degeneracy connect compose feature layer input deep square exp kernel repeat cc input deep kernel draw connect composition draw recurrence function input tail architecture correspond feature implicit kernel enable generally network transform deep derive simply fact rise learn way unlikely useful recently method entail dropping order result neuron maintain overall activation make average dropping neuron dropout procedure examine dropping feature independently dropout weight variable central form apply perform dropout dx model put prior mass understand rate therefore perform dropout input class therefore input depend cccc st nd rd term kernel dropout dimension order consider nearby distant long value model help explain generalization deep network give deep gaussian first use develop analyze relevance
node cycle exact hamming successful nlp instead structure set present analyze structured prediction key finding theoretically error theoretical perspective external tractable implication context grid node rather edge evidence clear improve decode relaxation optimal understand property prediction parse understand score relevant edge proof bad correlate chernoff continue unchanged definition task machine involve simultaneous prediction label maximize sum specific intuitively term formulate classify vertex label ground truth low hamming interesting hamming positive wide graph common theoretically expect toward justification exact edge label consistently bad edge label arbitrarily admit ground truth go graph poor path theoretically theoretically algorithm grid graph important theoretically parse language speech entity set input sentence foreground sentence specify encourage take e encourage neighboring foreground state big feature space condition learn random field structure application quantify discrepancy ground label study hamming hamming inference namely py practice map inference namely advantage computational estimate test worst np inference computational even pairwise understood measure obtain complex perform bad work programming search cut branch obtain accuracy predict test hamming however fairly art unable achieve generative setting high prediction accuracy limitation heuristic indicate instance characterize computationally understand inference good much lose prediction hamming assume version term vision motivated vision grid non often marginal turn analyze hamming expect hamming evidence first use combinatorial worst bind analysis grid graph utility distinguishing ignore knowing sufficiently give rate node quantify observation need make close submodular grid computer foreground thus emphasize grid moreover multiplicative objective setting similar know provide ham good job motivation interesting result computation obtain ham graph relational protein web even constant empirical grid demonstrate recovery structure highly trivial iterate conditional mode poor polynomial obtain constant close goal set unobserved observation various setting important channel e zero augment error deterministic bit send channel version view appear check bit code see channel coding appear notion truth one highlight free input via semi process exact truth quantity every unless extremely small number pairwise inconsistent obvious translate objective allow complete share much motivation technical instead error ground emphasis give truth dense paper paper also time semi numerous variant specify consider significant cc work obvious translate stable seek polynomial time low truth instead paper assumption near objective cluster cluster positive state understanding open level recover ground truth study numerous domain exactly impossible clique detect community notable sort total ordering present work order random graph mainly graph os recovery considerably hamming et explicitly partial oppose recovery degree paper graph infinite addition physics connectivity constant avoiding ground approximate error generally meaningful merely ground truth object classify incorrectly optimization error recover truth relate way ground truth paper ground truth understand theoretic approximate error relationship recovery possible error close high several extension relationship study prediction label observe depict hamming observation nod generative follow observation independently good observe adjacent label bad bad edge another bad vertex whereas emphasize edge bad label node label function label e noisy labeling expectation error two stage expect extend stage maximize subgraph incorrectly label agree yield agreement higher maximal correctly inconsistent truth two statement half bad boundary bad lemma motivate bound give set carefully far error seem difficult analyze directly simple upper induce connect exactly exactly two side side contain side side category component call vertex type component single procedure fs distinct every sided entire empty thus non empty lie imply moreover correctly label type path bottom side incorrectly classify vertex component type bad agree datum half edge minus edge neighbor every error bad ss least half edge probability least event iid th bad probability lemma parameterize square f fu rectangle km km km task bound boundary dual minimal subset property extra correspond face include dual correspondence cycle cardinality part cycle cycle part b cycle choice start part cycle choice final twice orientation set complete intuition computation relevant lemma hamming bad derivation definition lemma size tight statistical physics boundary upper region recall attribute type region expand type f c ic area translation cycle go name self avoid finally show analyze second stage truth label choose via majority second completes start point analysis h cp h kb wrong imply constant misclassifie less well truth prove eq expect marginal truth symmetry error truth truth parameter need away flip coin edge label generate truth function white label white white white chernoff imply white white white give minimize predict white minimize predict white wrong several extension graph graph section theoretically recovery grid beyond robust grid graph fail thin constant analogously boundary exponential sufficient face outer proof computationally theorem graph precise depend structure relational predict protein interaction web page section prove recall analyze two output chernoff trivially bind fact least half misclassifie every lemma overall sufficiently claim discuss recovery noise one procedure compute would constructive efficient observation eq fp chernoff large misclassified call bad let good flip maximize score get b cd hence bad region none share bad observation
potential uninformative heuristic hand infeasible maintain parallel enable possibility ensemble convention effect lie formalize show horizon feedback reward reduce horizon agent must explain different td fast propagation describe exact choose maintain learner f reward adopt terminology al within learn greedy w reward reward shape devise collect vote action shape intuitive voting voting etc ensemble contribute vote maintain happen corner accord policy vote vote schema policy modify act preference interpret give result eventually arrive attention classical car car hill system position velocity position episode end goal shape informative rarely technique ten learn policy uniform time potential progress need first high position energy fig encourage position velocity architecture rank majority voting voting speed helpful likely prefer priori challenge interesting ideally like able two comparable pick well use tune select greedy step run independent run evaluation episode greedy allow reflect refer variant cumulative l variant height alone aid significantly statistically exception scenario performance significantly combination arguably negligible note even table policy framework give sound capable learn learn shaped car scenario well signal former scenario outperform latter able general expect benefit extensive future limitation requirement important expand effectiveness architecture hoc voting combine possible combination fitness r learn challenge happen doubly relate size scalability potentially thousand efficiently context rarely sensible could define potential different scaling combine static throughout roughly mdp go attempt learn well since argue et al potential grant ph grant foundation ac advance guarantee possibility technique reinforcement potential base ensemble induce voting mechanism happen real agent interact markovian agent learn setup allow arguably arising scenario popular potentially temporal greedy implication multiple task share stream sound architecture spirit fast devise ensemble reward reward idea signal instead improve formalism consider scenario latent often setup environment costly failure make imagine exploratory policy note reward part performance purely propagation section rl policy architecture capable sound realistic guarantee limitation general apply brief definition notation policy car discuss environment rl model set denote state reward action markovian depend discrete discount store entry pair maximize mdp dynamics mdp temporal td iteratively method rate td draw trace way al process converge standard approximation large continuous suffice one thought way scale rl difficult rl suffer long however prevent ng show potential ensure original mdp maintain potential auxiliary reward potential towards validate speed uninformative reward augment shape shaped shaped converge policy base process section motivate find suited ensemble reinforcement reward bagging solution ensemble rl extremely combination limit practical usage overhead learn speed lack learner surely reflect ensemble lie contribute reliable fa similar fa disagreement
mixture identifiability necessary requirement usual asymptotic ml condition identifiability covariate provide contaminate identifiable provide positivity avoid contaminate gaussian equality j n classical incomplete source classical component membership govern otherwise specific leverage reference cf denote ij therefore algorithm iterate replace two iteration calculation rl ce ie rv ie ii nj rp eq th calculation r particular algebra nz j nz ij detail maximize perform analysis facilitate code upon request implementation start algorithm constitute respectively select cm contaminate ij nj model probability step operational view thank contaminate observe gaussian consideration base criterion choose gaussian analysis algorithm probability fitting component distribution implement acceleration iteration decide reach e log estimate acceleration q converge analysis r estimate depend q base update write straightforward decrease formula write decrease reduce leverage provide residual effect contaminated contaminate membership leverage component expect arise convergence membership although leverage interested classification ccc leverage thus observation rich information eliminate observation bad outcome detection proportion could th group typical nz nz herein use update somewhat specify proportion outlier advance pre specify point realistic many contaminated characterize far purpose computation criterion well adopt section evaluate contaminate gaussian square mse gaussian datum contaminate regardless b additional choice fit mixture possible switch mse help mse scenario replication concern negligible bias practically finally interesting improve governed value estimator bias negligible result use lead increase estimator contaminate student student economic business although seven illustrative purpose height justified gender scatter labeling line gender fit classification gender gaussian bic lrr contaminate classification display regression line yield six misclassifie six consider misclassification section sensitivity local end perturb set type accordance ht denote perturb contaminate contaminate one contaminate scenario towards point top corner representative entire plot sake local contaminate local contrary scenario two contaminate locally leverage detect leverage contaminate degree contamination group component contaminate ccccc increase group membership regardless point point aim evaluate fitting modify datum uniform square modify denote contaminate lrr gaussian contaminate generally well contaminate term bic sake model ht point leverage detect outlier detect denote bad poor line term maintain misclassifie agreement group accordance contaminate importantly contaminate put gold analysis group approach robust point leverage impossible discriminant fact type supervision provide flexibility compete higher easily work facilitate contaminate interesting directly model state quantify model datum dependent base least mixture contaminate believe robust exception extreme fashion parsimonious imposing exploit development property test coefficient moreover work eigen decompose suitable simplifie contaminate integrate yield divide side contaminate scheme point j complement hyperplane therefore j jj condition identifiable implie imply j integrate yield therefore simplify simply arise identifiability contaminate require five hand derivative partial rv update derivative q nan rv j nz rv j derivatives operator nan nz rv obtain partial ij ij transpose partial eq nan update analogously derivative nz phone variable component response give respect elliptical heavy tailed presence contaminate introduce contaminate proportion outlier control leverage contamination specify contamination crucially approach furthermore fine intra leverage concept primary mixture contaminate maximization outline operational issue monte carlo weight base contaminate continuous constitute flexible powerful often assume original problem compose covariate mixture fail incorporate valid represent mixture see two mixture fix mixture assignment generate value datum mixture paper dependence allow depend member mixture regression weight assume across weighted density covariance vector contaminate observation affect accordingly detection development assume assignment analysis paper contaminate contaminate identifiability give outline operational aspect carlo estimator sensitivity
stream benchmark scope collect ease algorithm incorporate consider entity preliminary introduction feature may significant improvement term plan integrate adopt aim platform optimize stream message twitter media theorem cluster medium identification discussion topic recommendation streaming medium procedure atomic token carry tweet aggregate measure tweet actual concept topic various diffusion tweet variant forget old twitter week systematically able outperform content detection assume network show adopt suited social popularity high message device news discuss reach group interest medium use spam ability pattern great interest classify piece content real platform class appear stream labeling tweet training volume produce tweet well single tweet due brevity modeling help context character user external resource tweet contain content include user user attention tweet follow user exploit tweet topic start character twitter rely conservative discussion twitter content content semantic represent spread person learn potentially overlap leverage entity content bootstrap aggregate entity tweet text exclude entity remainder paper refer initial atomic homogeneous tweet relationship algorithm assign tweet point tweet reconstruct political static although consist group tweet aggregate together tweet represent sophisticated aggregation identify offline static case scenario reality medium stream fashion resource newly operate essential mind pass sort stream broad learning datum point various algorithm income close distance centroid take allow cluster stream content outline base stream summary contribution tweet share test atomic measure stream able require suitable online algorithms variant slide evaluate baseline algorithm solely tweet tweet content plus underlie social network result various stream media stream system direct relationship generate stream twitter google take case twitter platform user follow turn used receive news feed user address directly tweet user visible basic similarity incorporate recently way message big block natural unit aggregate broad tweet entity tweet hash mark leverage activity user user name symbol include link web link resource tweet remove phrase help capture message tweet mention entity way allow tweet use tweet share entity entity stream accomplish real time therefore extraction next various measure broad fig tweet content post tweet projection space feature let preliminary similarity content user similarity cosine tweet respectively similarity represent tf weight word document tweet tweet similarity eq user tweet author tweet involve network similarity cosine set obtain similarity measure weight allow parameter search combination cluster performance similarity measure pairwise two capture base good content reflect maximum formally maximum need importantly previous provide combination next maximum data stream amount incoming datum store memory evolve cluster emphasize represent stream base window slide window generality duration grow point stream use time exponentially decrease high high importance moment contain step adopt model ignore summary slide yet choose processing algorithm simple algorithm online start initial seed assign cluster centroid average member general account stream concept assign overcome suggest check centroid outli also cluster find row column confusion totally measure mutual reflect oppose mining like biased evaluation investigation confirm limitation measure report indistinguishable due negative bias toward tend adopt task event produce suited overlap cluster refer discuss implementation introduce aim assess drive bring social stream compare framework baseline operate baseline outlier handle text tf tweet compute similarity tweet aggregate configuration tweet base current art streaming tweet rely present dataset provide advantage also practical consume stream tweet similarity tf use tf implementation algorithm make algorithm comparison user present cluster time advance slide deviation mean slide duration minute across slide window yield treat ground truth cluster label remove evaluation score window refer period plot cumulative period axis represent hour slide window hour evaluation essential tweet truth therefore tweet explain empty remove old list cluster ignore cluster last separate list assess window perform baseline online time run due topic discussion medium demonstrate baseline statistically significant b measure overlap algorithmic ground truth huge truth several one assign cluster performance detail confusion matrix tweet solution confusion class truth number tweet compute evaluation period display maxima quality performance comparison dark colored value row cluster dark square confusion job capture confusion baseline ground truth ground able discover stream great classic hierarchical access availability full stream set parameter slide window affect overall cluster varying achieve intuitive heterogeneity therefore shorter small time social configuration obtain increment window grow difference understanding stream fails affect day medium production effective figure illustrate gray band day cycle period fluctuation describe reach peak accumulation evident benefit example cluster goal identify measure tweet tweet capture time scenario display whose consistently exhibit low value consideration generally short content produce mostly day hour oppose performance tweet fact perform attribute crucially term generation diffusion cluster social topic stream identification technique emphasize term signature method take temporal feature keyword physical rely well formalize social medium stream present track medium news short act signature identify small create news topic news cycle systematic quality retrieve provide focus news extraction base evolve
topological dag compatible trivially sort dag neighborhood topological sort compatible prove topological generality compatible write x k k kp parameterize impossible logit subject index parent dependent contradict contradiction large mass desire jensen prove function argument neighborhood share compatible argument jj p k triangle k p p dominate term cn maximizer omit pn similar first tr need tend b pn proof consistent therefore convert definition department california ca article penalize data intervention represent logit achieve maximize employ select group variable encode together penalize subject dag biological method competitive word bayesian graphical model encode independence represent acyclic dag recent see popularity medical science infer attribute function dag rely conditional test include goal dag function criterion minimum description et scoring particularly various applicable decompose sequence linear develop penalize dag give penalty theoretical high dag despite development regularization method nontrivial code group penalty coefficient consequently maximize log likelihood become principled sparse dags categorical know bayesian logit propose formulate descent iteratively establish penalize section report type biological section technical proof network dag e j multiple dag equivalence bayesian network possible equivalent dags observational dag joint incorporate experimental detailed causal bayesian please refer therein assume experimental intervention intervention experimental therefore remove point intervention estimate observational empty variable index parent total state px nonzero product multinomial assume level grow parent multi logit discrete development straightforward dag point h dd pd conditional distribution multi logit coefficient logit identifiable particular I ir jx p give total growth assume experimentally indice jx n j logit factorization although availability log apply experimental structure network equivalence order dag estimate see parent wish group subsequently group alternative penalize regular drawback inconsistent selection certain circumstance overcome limitation causal bayesian call bayesian computationally demand logit successful consider j simple minimize quadratic approximation taylor value add approximation log current give suggest hessian small convergence necessary tucker minimize apply approximation meet immediate consequence substantial enforce induce cycle dag cycle minimize induce cycle simultaneously outline algorithm discrete indicate constraint minimization initialize acyclic ir I I ir j j update completely identifiability dag say dag direct set say natural x dag causal path connect assumption world asymptotic interior logit regard theorem p maximizer pn consistent local maximizer pn confirm local unweighted turn construct achieve generalize develop order causal consistently intervention limit network observational node subgraph subgraph asymptotic simulate three simulate chain network simulate regardless parent logit group parameter j otherwise free network generate skeleton edge initially undirecte convert edge direction randomly sort topological sort network dag grid speed adaptive give comparable dag measure false fdr predict estimate dag cd estimate dag low average primarily predict wrong false suggest skeleton accurate three predict false selection tb network p fp markov report problem dag choose pc algorithm availability efficient observational datum complete acyclic take approach pc hereafter produce favorable direction edge intervention direct dag report base obtain solution cd dag edge pc cd pc note chain determine pc alone column correspond consequently count fdr chain column edge average cd algorithm datum flow perturb component simultaneous measurement since three protein observational among measure protein perturb contain measurement level group interaction causal consensus reach interaction figure dag qualitatively consensus cd predict false dag domain good algorithm seem skeleton mcmc dag node scale see
occur use heavy vertex removal remove result plant every boundary define condition short definition degree edge g structural plant th u edge otherwise budget initialization current u ii definition also vertice markov inequality structural du claim size prove claim claim bound x boundary count vertex follow number token send get right hand token send observe token send edge token receive compare red token get mind cover h v h make show property belong eq upper size edge graph apply switch around edge edge short nm dm df du property first fix integer vertex let term equivalently replacement show bernstein two h f subset rewrite imply set g v since balanced least cut balanced cut balanced require half map expectation vertex thus minimum cut cut go equal cut equal cut preserve replace embed net present previous work li du u du ex plus ex budget short span conv var sr sdp cut corollary theorem property section conjecture conjecture semi random plant previously partition cluster edge sample balanced cut plant cut combinatorial arise area science research science dedicate designing analyze make case exploit suggest life instance practitioner real life possible model provably outperform plant stable edge believe capture formation social sciences balanced balanced study bad expansion cut plant condition plant number take disjoint connect probability independent refer edge plant cut give edge sample invariant permutation vertex aside lie inside complex example model fairly large dense consider vertex partition equal permutation informally permutation identity label vertex know formal set cut plant permutation let give recall balanced balanced balanced cut balanced cut give minimize algorithm balanced w fraction edge plant constant plant plant cut state deterministic permutation succeed extensive plant ex model w recover plant cut semi arbitrary sample ex adversary impossible plant arbitrary permutation unknown plant cut theoretically paper high probability planted planted plant graph graph inside plant model except adversary current inside far find planted plant impossible model theoretically instead give plant hold plant model block widely use social science paper relax choice adversarial tractable opinion capture planted cut partition similarity real vertex within impose restriction cut assume cause error opinion model plant edge independently second many tie social tie friend people common interest think superposition tie network extent g tie graph vertex people tie people live region divide group live tie usually tie region live tie necessarily college twitter tie network social tie different formalize permutation invariant choose correspondence identify vertex introduce believe graph combinatorial optimization network equivalent rest vertex v h g r gr adversary arbitrary adversary choose nature graph polynomial exposition attempt additive random edge identically permutation graph identically succeed succeed high overview balanced cut cut sdp cut similar slightly one rao ensure sphere give sdp solution say sake discussion make sdp depend edge furthermore sphere every square euclidean ball small thus long contribute sdp sdp cut discussion remove decrease edge cut edge repeat vector sphere intended progress problem conceptually find radius contain cut cut edge iteration serious edge short edge skeleton skeleton constrain sdp skeleton whole skeleton long special remove skeleton edge skeleton many skeleton structural encoding encoding consist note value reconstruct encoding tell reconstruct encoding encoding less permutation kolmogorov small edge technical accurately overview work skeleton introduce semi partition similar iteratively remove heavy removal analysis structural ensure however geometric notion skeleton structural completely significantly removal quite numerous equal average assume generality iteration relaxation balanced subgraph vector pt height pt height pt solution orthogonal intend satisfie solution graph intend sdp cost intend sdp relaxation cut rao normalization rao work sdp solution subgraph cut edge set nu balanced present cut remove arbitrary lie otherwise vertex edge remove algorithm partition cut piece piece part step store budget budget keep edge remove cut vertex get budget keep sdp graph graph cut long heavy section heavy removal procedure remove edge remove cut long budget extra budget increase active remove loop complete piece use rao component height height hide sdp removal procedure update obtain denote cut algorithm remove pt structural e sum active extra vertex whenever edge word budget edge neither update vertex lemma budget extra allocate piece absolute piece return size separate piece partition already know optimal sdp heavy removal output cost analysis proceeding remove originally equal graph strictly degree r solution leave rely state prove sdp solution divide piece piece cut outline define feasible integral sdp assign vertex orthogonal vertex sdp partition balanced cut procedure cut ensure structural section part plant edge thus edge sketch structural setting satisfy removal heavy removal remove budget active vertex neighbor ball lemma remove control overview du show encode vertex respectively approximately neighbor r random exist uniformly short exist define b ready encode permutation record record restriction complement record number order element precede encoding encoding encoding need bit record record vector record restriction record matter encode encoding encoding permutation q satisfy theorem probability satisfie solution radius around induce embed need scale constant feasible sdp follow condition u sense heavy hence remove control remove remain structural around gm let let vertex assume otherwise structural degree place total initial allocate budget eq finally structural property rather technical speaking v h satisfie edge sdp edge cut increase vertex budget edge extra budget non execution edge cut decrease f suboptimal q since optimal corollary budget long whenever extra never active decrease vertex decrease algorithm use budget pay extra vertex radius removal pick I step heavy vertex find boundary remove removal remove ball union satisfy invariant budget heavy removal remove several invariant independently edge total budget total remove heavy vertex least need sdp pick range
many large coefficient cosine general know datum system consist signal call way stable preserve reduction address solve much e general ill theory estimation isometry isometry vector nonzero entry property rip theorem show matrix satisfying rip noisy keep q obeys depending depend apply treat entire unobserved miss come mean basis basis parent signal thus unlike conventional consequently subject try reconstruct observe complete em q define respect w belong subset find ks call iterate new require maximization maximum step expensive instead maximization subspace iteration iterate claim performance use em em initial check reach minimum close compare reconstruct closeness setup datum reconstruct reconstruct call population conventional residual effect start reconstruct process repeat get closeness naive possible approach closeness assess residual conventional procedure bar work well nice unfortunately attention naive impossible comparison time approach value procedure consideration sampling reduce plot average residual work take work conventional reconstruct high conventional sparse essential ingredient naive algorithm approach treat iid population setup signal may work combination linear combination conventional signal follow shannon compressive paradigm recover reconstruct compressive difficulty subsequently modify compressive comparison simulate huge variety sensor range imaging produce much often store entire sampling
fairly explore kernel aspect multiple reduce idea upon explicitly efficiently construct input classifier positive kernel define avoid instead matrix trick nice impractical harmonic approximate feature specifically shift rbf harmonic eq distribution simplify cosine see rbf correspond decomposition approximate kernel approximate scale projection find key far cf million million lead method induce plug vector multinomial regression logistic regression mainly assignment need address optimal tackle latter construction feature sum investigate report automatic empirical acoustic often use develop major strategy challenge multinomial method sag theoretically empirically descent sgd applicable sag design well suited secondly together partition multinomial logistic parameter combine spirit parallel leave parameter weight converge minimizer size softmax activation extensive empirical study support argument setting model advantage use kernel engineer select basis task hand specify function select popular paradigm latter mkl start kernel identify combine adapt scale mkl novel hadamard product advantageous material use neural network mkl present tractable large scale also function approximated linearity concatenation scale straightforwardly logistic scalable parameter hold fix parameter first general scalability combine highly individual kernel follow construct feature convolution namely need component number kernel additive product argument kernel concrete gaussian rbf layer parameter layer introduce perform pca covariance kernel kernel secondly implement fisher discriminant spirit multinomial probability largely due alone multinomial classifier build readily validate challenging computer vision conduct extensive neural network perform attain material yet equally report finding complementary tune adjust similar effect hyperparameter hide layer rate decay momentum unsupervise pre tune detail describe supplementary specific computational support observation learning counterpart direction understand learn grateful microsoft microsoft work paper california center high http intelligence advanced project department laboratory contract nf reproduce annotation contain herein author interpret policy either express fellowship f support nsf google research award fellowship award nf translation eq variable feature invariant p I ie product kernel p ie new simply use provide image main text divide rbf kernel pairwise select performance unit firstly pre bernoulli restrict rbms sgd algorithm learn momentum learn momentum decrease every epoch mini size stop overfitte augmentation learning epoch noise compare dnn deep net kernel augmentation rate h type gaussian h cccc validation visualization also pre softmax activation first principle mnist network seem spread kernel digit low corner right text like result use achieve feature model show median rate rbm layer three bernoulli intermediate layer cd adopt rbm tune learn rate momentum regularization tune momentum l tune momentum rate epoch early overfitte datum run epoch constant rate epoch momentum epoch momentum schedule epoch trained overfitte observe increase apply additive deviation pixel kernel method dnn dnn overfitte layer deep model c cccc provide comprehensive study automatic speech task model crucial automatic speech recognition acoustic learn predictive assign short speech sensitive acoustic window frame proximity capture b hour hour hold hyperparameter hold acoustic challenging well friend record mobile acoustic environment public place phenomenon background familiar challenging recorded well environment work gaussian frame acoustic feature value dense overlap state million million hold million million hold million report evaluation q attain hold tune often next performance speech recognition inherently proxy goal recognition measure speech recognition posterior probability label interested linguistic truth token token rate error entail perform costly rarely tune token task report train set speech processing area language speech language model currently decide broad comparison system conventional description appear acoustic provide power token use convert raw feature frame show work dnn frame dnn acoustic model contain output softmax nonlinearity correspond dependent cluster dnn connect dnn stage wise layer range criterion minimize descent momentum learn cross design another dnn restrict rbm training layer totally language rbm rbm epoch tune parameter rate momentum back propagation tuned decay momentum strength another epoch acoustic searching factor rbf tune pairwise median well random use range stable text acoustic text use average sag tune loose combination combine one combination train kernel reduction text dimensionality first c dim c dnn train dnn combination kernel multiplicative report hold separate metric across reasonably attain red colored advantageous good different architecture meanwhile acoustic random feature parameter achieve similar accuracy fraction dim million way intuitive convenient instead adapt report measure rbm dnn well perform dnn highlight need reach proximity relationship different certainly plausible space explanation different might combine individual dim dim dim inspire previous learn two preliminary dnn compute pre similarly label pre activation visualize display two scatter represent representation visualize phone label considerably around initial suggest cluster color form spread advantageous tool pursuit conjecture fan em science california em york york edu center ny com em france share question
useful presence miss interest allow slope intercept separate year teacher teacher persistent teacher score model teacher refer persistence parameter one special fix persistence scalable implementation scalable routine teacher model teacher copy cp teacher teacher year gp model product discuss alternative concern identifiability estimate largely structure pattern entry nest diagonal factored nest highly inefficient infeasible large technique newton nr routine square effect dimensionality reduction nest dimensionality grow user specify statement design matrix scale estimate tailor software specify also large model estimation correlate produce failure setting positive estimate accord cholesky root offer parametrization challenge dimension covariance estimation presence miss mixed treating latent miss behind popularity nr depend nr restriction need advantage presence furthermore em additional technique em algorithm nest model correlate effect refer initial maximization iteration compute maximize integration expression observe derive e definition appear generalized persistence gp equation apply g g let conditional remain likewise let correspond j j block gs namely calculation change unbalanced solve sort year size student year although treat table lr student year pattern observation covariance matrix addition denote contain include student let indicator otherwise corresponding observation follow step update unbalanced profile structure score present routine update appear definition appear unchanged derive equal diagonal g ig ig ig yield q although may em definite equation singular current teacher form distinction step definition appear case scalar persistence cp student link teacher year column year score function step solution update require result perhaps covariance relatively fast em produce hessian mle working correction observe score derive value calculate central mle suggest forward also useful r covariate effect year right eq htbp std year year year htbp cp program list figure maximum parameter r year identical variation effect r surprising correlation year good slow maximum lie nr failure cp correlation persistence model student persistence significantly indicate compatible pattern figure future simulation wishart credible correlation include specification distribution compare error predict calculate product em large standard error likely correspond gp interesting interval average standard error prediction teacher effect distribution model teacher residual independent student though violate assign student properly attribute student correct response analyze student status estimate teacher effect miss score miss depend student teacher history student finally relate later correlation usefulness measure teacher depend aspect teacher standardize careful teacher mind valuable improve year teacher student strong later variability valuable year effect student score percentage drop teacher year student assignment estimate progress level unit offer method inversion depend effect total observation produce effect propose availability well sensitivity implementation facilitate provide deal relative effect student case flexibility gp may need able structure readily error predict teacher effect teacher effect include standard distinguish effect although develop well unit contribution health sequentially affect outcome level unit algorithm package membership allow expand situation student teacher careful basic step would remain application pd pd specification step however even usually near space variance long model student correlate much common problem estimate mixed occur pd concern future teacher notice portion pd pd semi matrix pd acknowledgment national science foundation grant finding recommendation material author view national science university individual student flexible literature although structure sequential level unit associate unit random challenge limited availability em compute implement example illustrate efficiency work publication control mechanism reflect publication subsequently mix popular describe unit primary belong belong level level static level child care live family visit multiple worker student membership model complex correlate nested structure particularly datum develop nested case paper algorithm longitudinal association unit covariance exploit speed membership example patient student teacher motivate research come score intend teacher student score teacher expect student potential issue variety different primarily persistence literature gp teacher receive end student therefore belong membership gp add context model teacher predictor teacher teacher score teacher represent teacher teacher heterogeneity causal teacher distinguish current effect would expect student teacher test complete persistence teacher subsequent year student persistence propose teacher year student year complete software teacher student future year though effect perfectly correlate refer persistence teacher effect impact student year multiplicative year multiplier persistence parameter teacher teacher year teacher covariance special gp current effect teacher assume general correlation detailed teacher problem membership iterative carlo chain likewise method estimate school computation require informative covariance teacher avoid need ml practically infeasible point calculate implement software development make calculation model many setting problem teacher student basis patient membership arise network example membership persistence measure membership since similarity former application patient different progress paper organize foundation em algorithm set school structure level response unit response ig ii covariate contain also level unit membership row nonzero value conditionally also observation together form linear consistency asymptotic ml meet addition regularity associate assume identifiable j identifiable school fitting find identifiable student move one next school insufficient allow level unit characterize process valid propose joint indicator accommodate informative teacher teacher score school link student score present specific student year notation depend model student history observation student teacher estimate student year score represent effect teacher teacher current year effect teacher distinguish persistence former teacher student g contain could
strongly see batch hold exist convexity fy list range inspire original linearize loss ms list second bind proof similarly result practical routine aggregation refinement grateful anonymous comment valuable comment cm thm thm universit es paris place paris france new recursive refinement set batch version strongly procedure satisfy optimal rate require achieve deviation weight refinement deviation second regret stochastic iid multiple keyword average individual sequence consider set recursively goal subscript closeness address predictive eq loss learner produce time quantify risk learner aim find measurable eq abuse aggregation good problem ms seminal book instead cumulative risk environment thank use cumulative risk temporal properly time aggregation risk bind risk apply jensen iid lower call ms rate ready procedure problem ms achieve fast aggregation call bernstein define properly procedure ms explicit regular criterion solve quadratic recursive greedy one opposite online achieve rule batch coincide excess extensively study fast rate use refinement describe round exist different none ms refinement crucial refinement thank order refinement distance learner aggregation procedure costly upper risk standard risk previous application bernstein quadratic counterpart instead martingale convention variation bernstein bernstein inequalities armed penalize erm apply theorem estimate successively f tf tf regret equal motivation notion optimality batch necessary appearing batch similar achieve verify center second hold introduce comparable ml ml batch convert cumulative cumulative predictive valid environment extend direction trick rate adapt rate order solve c gradient procedure figure replace linearize refinement q abuse good aggregation regret bound probability online theorem try practice bad case satisfactory introducing parameter learn weight extend tuned procedure order recursive tf f r bound excess learning rate sequentially recursively q classical rate q linearize learner give version range unknown procedure bound theorem second loss article nice consequence batch excess assumption restrictive generality result cumulative online coincide predictive aggregation boundedness small ms come require see restrict loss strongly iid extensive work modify linearized second order also initial theorem fast result online convexity optimality conclude bind well price version batch excess environment next useful bernstein theorem recursive difference classical recursive martingale leibl distance basic variational formula entropy originally obtain measure gibbs identity approach exponential aggregation naturally measure conditionally deterministic provide recursively cumulative respect weight inequality recursive obtain regret r follow classical order aggregation procedure heavily unique property issue consider rate precede sophisticated respect argument multiple cumulative rate let random apply reasoning equivalently inequality identity multiply inequality end corollary simplified favor rate rate bad initial differently adaptive tuned optimally batch obtain individual consider learner expert f notice rate depend exponential multiplicative solve concern modification describe adaptive form rate version obtain order procedure similar convex non cumulative adaptive reasoning theorem recursive argument weight center expression precede basic combine apply probability end choose trick proof effective range linearize tune rate restrict positive learning regret convex respect tuned procedure argument theorem exception adaptive case range linearize error let enough consider adapt reasoning small updating restrict range define correctly range second hold learning tune loss recursive argument proof index belong formula formula prove argument adaptive study adaptive pay variability adaptive loss avoid turn observe recursively thank risk provide introduction reasoning second order order go back moment online aggregation assume sequence see counterpart provide predictive aggregation first note tf tf classical argument adapt recursive conditionally eq fact apply
function hyperparameter call relaxation concave hereafter adopt prior relaxed j get question choose appropriate q target marginal way select consequence evidence type maximum mean probable explain incorporate constraint reconstruction follow ease without convex sequel cost convex nonconvex optimisation concave lasso reweighte jointly convex convex differentiable principle iterative formulate jointly globally w order define explain zero stage optimisation actually also iterative estimate close iterate stop criterion reweighte reweighte sequel heuristic study admm closely admm solve assume dual dual terminate dual stop set relative detail across share consequence call follow approach share new variable derivation simplification share share carry update solve eq k find sharing problem dual arrive satisfied break solve block involve square step collect single lasso problem stop reach predefine iteration penalty soft procedure collect I ip stop break classical biology consider natural frequency coupling strength represent couple topology know exact reconstruct time phase consist identify coupling parameter consider base add column series non natural variance create simulated collect include note reweighted package specify solving call sdp solver solver reliable class algorithm admm admm distribute subproblem implement matlab calculations core intel ghz ram illustration different ratio noise snr range weight reconstruction normalise snr snr show test different implementation distribute algorithm matlab worker time varied computation experiment unit large difficulty report size distribute reweighte processor note computation reweighte small matlab computations parallel core lagrangian parameter iteration reweighte number reweighte distribute reweighted core reweighted distribute reweighte nonlinear series dynamical regression interpretation use normalise classic deal network comprise admm reweighte previously still properly currently establish helpful ac distribute focus nonlinear formulate optimisation sparsity induce concave iterative reweighte optimisation algorithm design multiplier decompose subproblem use reconstruct reconstruction series name nonlinear form problem expand nonlinear model amongst function parsimonious description paper nonlinear use priori candidate system biological chemical electrical time dictionary objective identify explain good cast reconstruction nonconvex optimisation nonlinear solve typically biological e collect big reconstruction handle measurable measure combination build model genetic network capture law degradation mass action hill follow dictionary function I assume dynamic k exponential
model trajectory rise picture brownian bridge e constrain end state phenotype string end impose string dynamic point acquire node node occur lead acquisition trajectory transition start allow probability biological modification signal observer signal ensemble thus string comprise string could responsible compatible wish compute datum calculate compatibility support observe describe dynamic network give rise compatible probability observe specie consideration likelihood concern likelihood ignore simulate trajectory node ensemble particular network compatible high preliminary trajectory broadly uninformative evolutionary dynamic aim evolutionary support describe acquisition trait step evolutionary trajectory build ensemble trajectory probability current amount yield trial calculate take likelihood perturbation transition w obey detailed state simplicity describe quantity consider move proposal propose move propose start investigation confirm posterior converge yield report infer compatible coarse abundance describe possess distinguish decrease bundle bs cell adjacent decrease ratio bs cell volume abundance bs bs independent present order assign datum cluster depict blue algorithm partitioning hierarchical cluster branch partition assign activity activity bs bs illustration phenotype trait yield trajectory evolve accord signal trajectory every possible evolutionary unobserved character yet specie red triangle trait blue triangle triangle represent whose deviation block know presence absence trait absence score cluster partition evolutionary event shown suggest assign trait pair trait remove trait trait trait trait robust trait bundle bs trait link assess modelling evolution start phenotype trait acquire trait next event acquire trait list trait acquire axis multiple trait link acquisition evolution perform compatible different b network variation embed principal axis variation less distinct department sciences united department mathematics college united independently pathway trait unclear evolutionary major trajectory meta analysis specie intermediate landscape experimentally markov predict phenotype appearance determine trait acquisition flexible flexibility trait evolution trait surprisingly camera trait mechanism apply highly involve leaf also find least phenotype explore experimentally ideal mathematical evolution landscape understand evolutionary event generate specie pathway million concentration modification leave leaf biology high repeat unit bundle bs fig bs cell activity alternate generate bs cell co b I I c specie typically classify type trait genetic mechanism evolution gene associate bs involve element act independent co mechanism specificity parallel evolution part convergent however molecular know trait exist cycle one history evolutionary analysis phenotype phenotype intermediate independent characteristic principal component pca perform specie represent transition trait specie represent space phenotype connect phenotype I sub type type data activity five cycle confirm specie trait specie present specie study intermediate specie show representation pathway phenotype pathway specie blue pathway compatible phenotype specie c trait present trait use assign fig bit string trait meta define phenotype combination absence characteristic novel bayesian evolutionary evolutionary trajectory fitness underlie evolutionary dynamic evolution convergent distant reconstruction convergent fundamentally acquisition trait acquisition trait node label phenotype characteristic label characteristic landscape weighted occur constrain inferential model hmms fig simulate chain transition chain evolutionary pathway pass several record network mcmc meta represent dynamic evolution characteristic acquire path compatible specie uninformative order acquisition trait generate mathematical material path generate require impose trait acquire acquire trait nevertheless able trait evolution test positive control evolutionary event pathway trait acquire simultaneously clearly assign acquisition trait link underlie fig simultaneous acquisition trait evolution trait obtain diagonal time four grain artificial artificial specie replicate miss meta generate artificial miss compare absence quantitative phenotype specie abundance white phenotype already grey predict approach indicate phenotype predict phenotype evolutionary trajectory specie calculate phenotype trait outside one trait presence dataset choose datum predict phenotype trait phenotype publish trait neutral strongly trait correct assign neutral highly produce phenotype yet describe quantitative real time experimentally verify abundance successfully infer evolutionary dynamic prediction verification illustrate identify feature evolution resolution evolutionary event probability trait combine evolutionary consistent subset bs specificity occur high insight event well specificity predict evolve prior primary cluster intermediate specie use acquisition trait evolutionary diameter trait acquire evolution bayes acquisition deviation trait order acquire early c bundle bs strong bs c cell increase abundance increase acquisition trait consistent analysis specificity verify robust score absence trait oppose fig hierarchical difference small trait evolutionary trajectory affect produce highly change phenotype subtract trait remove trait predict remain trait appendix increase deviation observe use might affect additional trait trait widely species position despite occurrence trait unclear acquire early importantly trait trait analysis suggest trait likely evolutionary therefore aim trait predict evolve link consider phenotype genome trait contingency subsequent trait increase detect multiple underlie multiple trait pathway connect phenotype evolutionary trajectory indicate evolutionary pathway c trait acquire capable produce phenotype material trait bs acquisition fig multiple pathway trait time investigate I entire infer transition reveal distinct largely infer full comparable represent event generate trait c propose constrain broadly evolutionary pathway probability sub broad difference pathway differ event generate c I differ primarily evolution bs I I convergent I traditional basis difference detect evolution example I density early evolution I specie trait broadly bs acquire pathway generate I trait majority fig consequence evolutionary response non furthermore evolutionary sub restrict phenotype space discussion bayesian infer landscape powerful conceptual transition trait challenge incomplete state report development able evolutionary highly stochastic trait dependent trait record also limit predict infer pathway underlie occur shorter disease cell decrease rate create pressure evolve strategy abundance occur evolutionary majority pathway al change within predict bs cell predict bs specificity acquire early majority bs specificity predict suggest occur recent evidence identify environmental frequency leaf nothing well mechanism key modification leaf evolution bs division appearance evolutionary pathway sub trait evolve traditionally cell specificity I mechanism evolution type therefore different evolutionary leaf c type detect difficult explain early determine phenotype type restrict pathway leaf development biology provide change convergent flexibility evolutionary independent range upon trait upon phenotype specify diverse molecular mechanism series highly complex trait show evolutionary trajectory trait appendix limit small subset recent leaf fitness landscape trajectory pass phenotype specie landscape mechanism generate abundance cell specificity
consider property design element sensitivity subset sensitivity cf estimator base restrict condition usual error independent zero sub row sub independent provide bound role diagonal diagonal diagonal probability least n tw tw tw find mu selector selector thank design formally almost imply imply mu selector selector importantly u tuning p selector attempt term precisely constraint new selector estimator sparse belong selector eq set addition constant assumption admits follow bind establish establishe realize eq lemma imply argument lead imply q next condition view eq literature inspection hold assumption satisfy compare theorem zero mu selector selector probability term mu selector whereas term selector upon original selector whenever additional estimator term selector obtain exploit additional finally go zero maximum finally event cone since due immediately term obtain combine selector follow unknown assume penalty compare selector also infeasible know selector use benchmark ccc ccc rmse pr c rmse pr table provide literature ignore issue lead worse see infeasible easy estimator uncertainty know essentially propose nonetheless fail eq similarly substantial reveal zero simulation constraint exploit optimization potentially constraint ccc ccc rmse bias pr ccc pr parameter seem kept constraint improvement small sometimes test value make perform improve substantially essentially constraint help severe helpful recommend keep case additional appendix auxiliary brevity minimization least use matrix eq selector lemma feasible consequently imply note imply selector probability n least hold fact use together remark view initial least since hold event least complete proof suffice last proposition remark different generate motivated consider assumption explicitly observational propose design applicable require moment observational applicable observation proposal covariate literature comparison convergence model arise error design unknown parameter assume belong set component estimator arise standard selector unstable literature boundedness q denote design set selector selector selector solution level sufficiently offset grow literature independent estimator converge rate motivated bias mu selector minimization problem diagonal accord rate selector selector contrast mu selector design small programming component combination adaptive attain computationally feasible cast mild estimator q sup minimax optimal usually vector know estimator relaxation depend sparsity matching focus subgaussian analogous consistent recovery convergence propose
runtime synthetic confirm simplification give specificity short short path start vertex path source shortest path general acyclic graphs single material structure improve complexity since read would respect describe scale linearly another rule together point inside inside note pattern message pattern extend dot dot see extension general u come interact pair figure locate message therefore define possible p u sg way computation case expression consideration complexity l p assume dependence g dependence introduce rule leave modify algorithm without show weight rule interval satisfy present start calculation message reconstruct find family base px xx need weight parse possibility variable approach likelihood computing turned compute expression replace accordingly turn division compute moreover compute multiplication lead sum polynomial appropriately modify marginal efficiently require difficult sum compute sketch material interaction interact computational synthetic datum solve shortest take depth contain interaction pattern distribute value take varied pattern part dominant varied step value start show dependence global sum crf literature general slow show namely may field chain free processing encode view crf model combine advantage hybrid paper analyze task fast label formulation observation sequence take value appear domain bioinformatics approach field j concentrated subset call pattern word occur define x parameter depend word crf intuitively sequence application natural process syntactic construction secondary structure angle associate configuration sequence suppose model local nature language I probabilistic sentence language gram equivalent represent sentence equal gram know language frequent gram sentence syntactic describe free syntactic structure encode structure equivalently accord give another identify rna sequence certain discretized angle nucleotide sequence label label alphabet local rna secondary model complementary generalization context rna optimally parse interact parse include weight model motivate symbol weighting parameter probability view way define encode long crf problem posteriori minimize polynomial complexity state compute polynomial pattern algorithm appear refined version handwritten character name text optical recognition protein string string extend pattern correspondence probably close represent correlation gram unlike probability successively model slightly mix apply rna secondary former equal show done ignore interested ratio model total hybrid probably intractable conjecture thing argue computational advantage hide require sum minimization subroutine gram correlation unweighted would like combine context language define language parse alternative choose directly complexity admit certain restrict third weight motivate kind interaction albeit roughly speak simultaneously overlap restriction inclusion restriction fourth focus energy depend cost general length pattern general present message standard parse algorithm interaction message compute interaction belong belong partial message message interval variable pattern interval plus change assignment message approach good parse start apply rule need know end fig pattern avoid set e value message argument intersection optimally upper equal count induction compute check step equivalently serve argument message definition compute message assumption via case correctly serve induction optimal optimal first responsible word pattern optimal variable message could segment value q equal message message correctness whether correctly complexity go function proper f lemma complexity dominate step describe interaction successively iteration base parse rule computation relatively update computed parse start divide piece equivalent short short graph correct depth message parse applying since rule parse fx c j option either u u c inclusion patterns intersection set pattern thing interval statement obvious serve k variation preserve conclude k u formula need orient suppose sp suppose parse start rule rule ss ks u rx kx proof order kx parse parsing
whereas selection orthogonal ability deal include operation linear unconstraine focus alternative sometimes constrained optimization bind frank wolfe fast project algorithm cg paper contribution derivation atomic take atomic cg explicit constant proximity arguably pool ball root projection organize atomic derive computational tool regularization proximity section cg project gradient optimization result bold column th upper bold component break arbitrary vector sort non sorting matrix naturally wise increase weight negative ball illustrate fig htb w w low eq chebyshev inequality trivial show negative regularizer set atom compact atomic norm induce convex c atomic atomic atomic norm mathematical attract considerable atomic sign permutation group sign permutation use norm form fig e component strictly theorem atomic see minimal minimal atomic b example case atomic general definition appendix full dimensional polytope decrease sequence minimal atomic atomic inclusion b n unchanged define atomic fundamental linear programming see function use symmetric permutation symbol fact ni x notation norm thus arguably focus proximity derive new simple three simple match v result fact proximity fact immediately sign thus compute eq projection recently show onto cone notice onto monotone pool adjacent summary compute coincide propose mention worth lead operation radius ball compute e previous eq multipli minimizer duality thus lagrange multipli impose function suggest find technique monotonically root adopt van matlab na computation dominate almost lemma omit proximity argument efficient sort clear comment g lead sort operation onto v sg standard formulation regularizer fidelity typically refer three sense solve adjust formulation adjust address efficiently proximal fista proximity direction multiplier alternatively gradient cg frank wolfe appendix b gradient algorithm accelerate project projection onto ball cg projection simpler cg well suited tackle ball free require b norm value coincide factor follow cg address norm f h solve reasoning initialization k kk predefine take trivial benefit duality accuracy iterate stop implement dominate sort total prove denote solution cg q define iteration require optimal like theorem loose follow duality local step duality precisely typical gap subsection accelerate accelerate iterative thresholding bb application solve implement bb backtracking approximate x k stop fast thresholding variant acceleration nesterov fista backtracking know fig fista h cg fista matlab window intel core ghz processor ram similar accord column radius stop iteration total evolution surrogate confirm surrogate gap surrogate duality iteration compare backtrack address formulation accurate tight stopping show backtrack cg sample standard gaussian observe fast fista problem cg h c breast imbalance yield randomly iteration repetition table cv fista fista backtracking fista backtracking cg formulation norm atomic cg atomic dual exploit cg regularize arguably proximity establish pool adjacent efficiently compute experimentally cg accelerate projected gradient show former work application namely logistic briefly basic concept mention fundamental polytope hull set affine hull dimensional state convex polytope theorem degree moreover since sign permutation argument triangular invertible x x rearrange since x permutation
layer deep network differ pattern appear detect reality structure distance advantage must filter unfortunately major capture resource wide feature detector detector hard train likely multiple prevent would train even scale detector capture scales independently share presence absence pattern convolutional network si scale detect pattern scale pool response locally scale invariant convnet architecture differ scale rather scale multiple scale variation mnist dataset multiple scale si scale variation scale complementary convnet make available research incorporate deep feature boltzmann rbms infer high model transform filter large transform densely retain inspire incorporate extremely successful convnet pooling un tie explicitly scale require learn un apply semantic convnet learn sharing detector share propose convnet layers fed scale invariance influential pyramid tie parse layer forward propagation apply connected layer align allow expense increase final restriction capture range contextual interaction scene interested capture invariant feature image unlike pool response subtle effect middle size circle circle architecture scale recognize layer architecture circle two detect circle expense redundant apply parse effectiveness modular incorporate invariance convolutional network feed feature detector layer regressor optimize jointly via gradient computed propagation convnet usually activation spatial pooling idea detector image pixel nearby strong detector extent get convolution operation tie weight location send layer single sub pooling invariance amount detector regardless spatial image invariant convnet si convnet output invariant overall two layer convolution bold box detector figure pyramid scale brevity across convolution scale normalize response pool obtain representation size b scale multiple pyramid come size align map inverse max pool scale spatial pooling multiple serve locally scale allow convolution linear image transform transformation convolution transform detector convnet learn whose scale layer match scale path win bold line pyramid filter analogous filter scale train expressive increase fitting image dataset convnet epoch test seed convolution scale parameter material propose boltzmann rbm protocol fold error deviation si convnet convnet hierarchical convnet hierarchical convnet slightly convnet overfitting convnet convnet convnet introduce pixel exist si convnet learn large rbm fact rbms unsupervise architecture extraction original invariant rbms comparable respectively rbm si convnet si convnet improvements good invariant rbm network convnet paper achieve invariance neuron input unit times neuron response input record robustness neuron arbitrary score neuron ratio report score neuron please step score convnet si convnet layer see pooling response scale si convnet report scale convolution actually si train wide network less demand si filter filter well many pattern h plot consistently outperform detector observe increase test convnet consistently convnet gap si mnist wide variation account si scale scale evaluate scale factor scale vary correspond away mean challenge convnet si convnet convnet worse mean shown verify si convnet outperform convnet end relative lack symmetry around digit human digit robustness keep training factor range si si low rate scale variation show redundant variety si resource h architecture scale representation convolutional net share across multiple scale single detector arbitrary scale feature pool invariance convolution incorporate scale fitting si outperform aspect order align layer write forward matrix encode bilinear interpolation vector multiplication toeplitz propagation size encode transformation response convolution kernel encode toeplitz equation element wise multiplication apply derivative signal stage linear weight way convnet error spatial accumulate layer discuss convolution convolution convolution compute invariant use large scale different scale quadratic geometric sum convolution process subtract range three unless specify si convolution follow another layer feature
element independently distribution desirable property embed approximate embed hamming pair th angle variant distribution easy sake easy bit normalize hamming distance approximately preserve angle unfortunately row make hard analytically understand rand analytical hamming hamming distance generate generate dimensional rand variance rand repeat whole average surprisingly curve variance indistinguishable bit rand intuitive locality sensitive lsh slight still identical significantly hamming variance bit curve distortion distance embed come recent type slightly number projection preserve high also sign dimension preserve distortion one randomize utilize propose learn dependent fashion minimize opt alternate objective bit comparable empirically try row uncorrelated help reduce redundancy code orthogonal vanish rotation distortion orthogonal similar include hard find propose optimize perform input time optimize dft lead element update recover derivation dft project norm preserve formally arithmetic matrix conjugate transpose trace furthermore conjugate optimize becomes decompose minimize polynomial solution hard solution cubic bivariate system minima consider overall guarantee increase alternate optimization run bit optimization kk k understand frequency make domain propose remain heuristic conduct three long internet represent imagenet imagenet represent third dataset imagenet imagenet contain image dimensional unit version embedding bilinear bilinear bilinear opt version bilinear embedding well call code lsh applicable feature long much space show relatively datum high scenario set query recall evaluate query instance define near base experiment generation retrieval experiment fix bilinear order bilinear square full bilinear projection bilinear ghz core indicate generate bit code projection bilinear time respectively compare time fix fast factor due storage availability highly optimize library suitable gpu preliminary test gpu speedup cpu fair comparison bit computational bit opt rand times bilinear opt bilinear rand hundred fast lsh time bit make opt rand fast bilinear opt bilinear rand hundred fast lsh bit bit less bit computational identical opt rand rand hundred time lsh three top show method yield well lsh bilinear code margin compare different rand almost identical lsh hundred opt rand bilinear bilinear rand code save set linear svm code show give randomly degradation lsh bilinear classification task lsh opt opt compare conduct bit dataset shown opt bit gap become much suggest extend incorporate achieve add distance pair supervise frequency optimization update fix q eq supervise version auc imagenet embed long code optimization degradation compare expensive time full high require preserve suffer binary binary project compare dimensionality alternatively minimize extensive approach give provide degradation bit become retrieval massive computer vision biology finance code bit binary approximate retrieval happen directly fall typically hundred thousand linear follow bit eq
view great pac bayes attempt gap development practice propose pac view analyse generating make explicit analysis tight main advantage template approach knowledge rely regret multi classification error assumption pac computable paper see assumption pac encode term study name recently dependent place encode inference impossible illustrate pac enable adopt gaussian treat significantly bayes view agree example first gaussian last centre formulation prior expectation unknown datum estimation finite rest pac svms give multi involve multi svms evaluate bound lie suppose average provide bind bound svm represent space induce function centre pac bayes ready svm bound return average stochastic bayes analysis multi view concatenation p agree identity view view feature example feature though explicitly employ theorem divergence complete contain expectation unable prior satisfy natural namely representation prove margin classifier develop estimation bound error multi semi definite function semi inequality logarithm concave pac bayes involve outer actually determinant equality nonzero eigenvalue inequality independent irrelevant posterior give whose locate still pn divergence moreover inequality omit pac multi pac pac formulation augment feature representation formulate form explicit svms scalar balance weight usefulness evaluated set uci repository include index index feature form view view datum original view obtain partition form form provide svm train view simultaneously svms adopt fold pac multi pac datum normalization feature representation test bayes svms tables multi pac bayes svm multi good far enhance pac trivial sophisticated scheme four pac classifiers pac bayes view promise fill multi view learn possible explain usefulness experimentally possibility pac view learn could far adopt expectation w another motivate pac multi relate support foundation china project service
iv lemma confirm limit note bf model b iii bf strictly bayes bf variable consequence bayes true go consistency identical bf observe orthogonality design give l converge belong l l I hyper prior consistent g dt dx lemma thus n define orthogonality expression derive z dt z dt constant go proof prior affine cccc n k z map act non p ib ap p ap exactly invariant affine problem order monotone function denote normalizing density c j function early stochastically j j strict strictly happen g integral finite density p integrable away zero equal strictly finite proper limit sequence posterior proceed k k pdfs ordinary rhs equality c ab g r k dt true z z lemma e x lemma enough arbitrarily small denominator enough whenever n finite c normalize constant dt f f arbitrarily small claim dt dt b limit complete argument bf r jj ta describe laplace definition appendix behave laplace complete remain laplace integral go go infinity belong must possible group non om om j om along translate happen since nr tm b om om om know one equivalence block om laplace bayes case reasoning converge mean square prior expectation shrinkage dt k k b dt show integral b dt k dt dt algebra make standard lemma expression k z b q hence denominator vanish b proceeding ok ok p necessity tr tr dt strictly coefficient column triangular matrix hyper prior triangular stay yy yy yy dt tr proceed q z b z ok ok necessity constraint tr dt p bayes strictly j k r f imply I j r thus ir explain variation k j j I I fix yy compare yy shrinkage near intercept dt dt dt come result limit bound zero additional corollary section prior traditionally sensible characterize lead thick tailed prior hyper limit prior paper new mixture reveal argue undesirable scale coefficient place hyper avoid provide sample asymptotic normal bayesian method regression central selection conjugate form due ease ease update prior posterior theory conjugate quick factor popular impose analog prior ridge selection prediction consistent estimator desirable concern nonlinear shrinkage place normal regression pareto gamma representation use place discrete model implicitly near hyper placing along investigate selection search hyper particularly mixture retain perform study limit insight substantially wherein inference least prior include suffer mixture prior introduce ordinary prior predictor separately investigate tractable situation fashion group block introduce behavior show poorly criterion hyper theoretical new orthogonal dedicated examine hyper brief discussion supplementary material response traditional include inclusion represent intercept error place prior along retain intercept traditionally transformation provide argument common flat match simple closed expression compare coefficient determination error least value suggest consideration review prior behind variety lead careful result thick tail prior marginalization mix mix estimator least square often primarily prior place proper suggest bayes hyper bf describe several undesirable commonly associate prior reveal take rely information limit hold old summarize anomaly choose hold approach irrespective undesirable hold follow g factor though grow undesirable avoid prior careful mixing provide hyper criterion study limit produce initially see qualitative description formal prior estimator limit limit drive rather mixture exhibit arise asymptotic accept connect behavior limit drive phenomena problem write element linear hold fix variable consequence consider drive produce denote element immediately hence several asymptotic drop subscript result otherwise hyper prior define provide appendix estimate situation conventional zero coefficient leave unchanged appendix hyper suffer call length irrespective least regression big place weight size grow infinitely result tend behavior matter zero toward attribute proper hyper prior posterior define ig material show describe exhibit generalize prior specify situation prior suffer irrespective robust suffer recommend irrespective material arise result predictor must affect similar explain portion mass nonzero coefficient behavior avoid multiple latent place approach concept local shrinkage normal include regression typically avoid limit concentrate hyper prior corollary center limit finite corollary material regression let eq incomplete gamma ordinary prior e hyper eq prior three notion consistency first seven criterion bayesian consistency directly particular apply model model follow consistency hyper prior hyper prior information size block correspond block arbitrary coefficient condition proceeding need notion include block x basic design true establish slight variation prior second bayesian ensure bayes model describe early also strong appendix hyper prior hyper produce consistent concern bayes square tn use form satisfy block hyper consistent proved show prediction regression model satisfy prediction novel prior behavior use scale common argue undesirable shrinkage coefficient presence avoid light bayesian component model comprise consideration prior suffer consistent aspect failure prior replace g hyper share hyper successfully provide derivation development theory raise practical question good select identify predictor measure likely comparable relate explanatory knowledge place correlate existence sometimes indicate previously construct scope investigation establish setting result orthogonality condition relax modify suitable block elsewhere mean dt z increase state g b k b k note finite number k second vanish limit z proceeding denominator b b ok ok appendix material transformation design generality projection represent triangular hyper upper triangular stay yy yy sequence r dt define q proceeding q exist b b z b later k ok ok supplementary material appendix
generalized nonzero order case website thank helpful suggestion p scenario regression pool adjacent operator application decay move away area idea simulated feature regularize parameter solve tune convex large add call result derive application time predict move constraint reasonable determine constraint paper contains order standard order real simulate section auto series traditional criterion degree freedom order work lasso constraint ny setup sense natural convex modify write use component absolute monotonicity strongly encourage interaction solve programming algorithm proximal subsection intercept illustrative purpose cc h elegant obtain obtain adjacent subject solve hence decrease expand operator minimizer compute k order adapt elastic net show order lasso generate plus coefficient colored profile order lasso job recover fluctuation coefficient observation profile colored profile estimate coefficient bottom relax ny subject encourage monotonicity problem derive procedure create extra simply place first outcome predict outcome outcome outcome henceforth continue omit intercept ik ik predict predictor j plausible unit back first write follow block hold block lag choose zero order predictor kp x series different lag x write convert section large detail q block correspond predictor row lag coefficient block kp tr x j time four lag predictor true coefficient order blue plotted job true coefficient average time generate deviation space four predictor leave figure square standard order give mse panel randomly thereby violate noise order coefficient monotone reverse true achieve much mse monotone set eight california divided figure curve measurement day order lasso achieve degree coefficient order order interpretable predictor time lag beyond estimate coefficient wind lag day time beyond time lag h cm predict lag fit time series proposal seem lasso ar derive property suggest autocorrelation lasso example contain represent auto fit validation panel fold behave lasso monotonicity give picture three set htp p
get conclude entry entry eq q otherwise conclude proof key proof purely extension count variation process eq j thompson inequality e e te l f last give conclude martingale use martingale purely martingale moreover mean non intensity bernstein notation introduce suppose hold q bernstein purely consequence case particular let assume process ds hold particular take homogeneous knowledge sub adjoint hence force symmetry z obtain technical proposition p jump entail hold cf successively gets give conclude proof martingale martingale whose quadratic variation analog line point definition case purely martingale replace let z v td cf u tm quadratic row omit subscript get easily eq result n get get optimize z low matrix precise goodness reduce prior connectivity user low filtering induce penalization trace nuclear consider procedure minimization square penalization trace consider process focus derive one stand differentiable subdifferential q give lead way fit h old together use subgradient bind motivate eq h depend intensity variance elsewhere cascade common keyword tag connection message cascade observe cascade counting ct ct user hence depend cascade along cascade reconstruct cascade user motivate intensity cascade decay function decay cascade fit functional model dt end dt matrix martingale write entry ct concentration depend norm algebraic structure lead control market transition finance laboratory universit ed subsection simplify drop index ambiguity aim decompose easy one index entail z whose invertible write computation eq z line obtain eq recall compatible along therefore shall entry j recall use lemma q use finally rgb proposition matrix case purely trace calculus appear statistical system study counting process component new concentration inequality around matrix version chernoff scalar result joint quantum entropy physics base stein extension scalar hoeffding inequality application particular compress simple completion large work concentration inequality see instance concentration however available tool calculus paper bernstein purely see hoeffding probabilistic appear naturally problem system network stand approach social fix connect link edge click etc consider network user action approach development cascade survival organize notation paper purpose essential obtain concentration section bernstein purely whereas study sharp counting quick illustration concentration rank strategy penalization relaxation trace reference sum sake clarity technical f augment nan continuous small value say trajectory x entry assume possible index adapt conditional expectation column equal size context stand square stand stand trace operator denote another notation kk jj cm adjoint denote moreover symbol stand
typical surface output series amount large similarly return occur colored prediction standard deviation uncertain htbp color deviation four asymmetric surface skew contour large relationship contour parallel nonlinear sort skewed quadratic asymmetric transition attempt variance make environment become intuitively except market environment previous asset period volatility section surface grid various band figure correspond section nonlinearity return look pass financial setup make observation short computationally simply learn predict author setting use compare combination setting likelihood n table outperform predictive four inference summarize little expensive financial dataset avg calibrate still expensive implementation inverting covariance hand cost cost gps gp time vary gp space nonlinear present online particle batch method generally financial improvement nonlinear intuitive clear direction relationship volatility price market interest learn pricing derivative speed attractive live tracking accurate limited variance learn overfitte address change gp distribution flexible develop main overfitte offline financial performance often exhibit volatility large return financial frequently period volatility phenomenon volatility univariate capturing autoregressive generalise far inspire variant extension find variant address volatility past negative effect volatility return introduce functional add asymmetric term fundamentally learn likelihood volatility gaussian flexibility unknown variance effect return introduce new parametric volatility flexible place gps furthermore explicitly asymmetric effect volatility evaluate series financial predictive functional automatically asymmetric previous attempt capture main gp work gp focus filter smoothing gp transition dynamic paper inference financial linearly time eq model flexible several limitation term flexible introduce negative return volatility extension asymmetric return asymmetric capture hide hmm assumption limit place state way hmm transition fix inference previous gp develop method filter dynamic dynamic apply em learn gp dynamic use learn similar much call gp real function gp function encode system dynamic return function output I output highly correlate model thick size black node thick connect thick f x connect x connect connect previous enable asymmetric finally variance however gp learn unknown denote challenge task fortunately introduce dependency particle avoid particle adjust chain origin quick filter unknown hyper standard introduce add artificial dynamic forward backward never later consequently distant gp collapse state near input adopt mcmc monte procedure establish framework hide additionally develop sampling markovian learn prior sample particle estimate parameter observe index accord propagate chain forward adjust eq posterior parameter particle part alternatively sample current conditionally slice drawn filter auxiliary filter generate conditional collapse smoothed trajectory alternate particle draw slice particle include learn transition synthetic recover hide gp series measure likelihood predictive finally performance term execution accord equation linear encodes asymmetric volatility covariance give state hyper brevity typical plot hyper figure filter particle show blue generate synthetic gp conduct dataset low daily exchange fx price total eliminate particular eliminate price market return mean standard deviation return training make evaluate
aic even safe drawn method issue make version quite surprising even constrained value see even draw distribution regard training equal covariate shift process goal bayesian probable aic cross validation lead aic typically lead unlike small goal aim predictive concern stress unlike model criterion rather dependence help get thus select comparable highly prediction select predict e frequentist imagine draw within usually predict unseen datum define far vary aic generalization I u throughout additive term divergence interpretation sample estimate independent aic asymptotically select minimizing select close truth kl divergence sequel omit classification two set may interested behaviour give adapt replace capital letter contrary may vary represent slight abuse notation fix logarithm expression square give attain another addition ordinary example linear form correspond error appropriate aic intend bias datum derive evaluate old datum intend aic article structure extra explicitly concentrate remainder focus discuss behaviour aic experiment contain regard prediction conclude proof supplementary supplementary section aic aic point fisher analogously assumption kl divergence trace unbiased minimize term somewhat simple bad aic grow asymptotically aic good misspecification derivation lead generating specifie emphasize aic derivation work apply supervise datum corresponding identically assumption problem aic equal draw unobserved mutually automatically satisfied take assumption randomness learn instance assumption setting assumption regularity material variance use asymptotically unbiased estimator extra error aic new minimize estimator evaluate depend distribution density depend additional evaluate variance one explicitly q equal n extra analogue similarly accordingly believe penalty concrete base formula aic except extra input appropriate extra prediction supervise might replace input computing computing follow aic retrieve special bad distribution recommend course case follow distribution applicable yield analogue nothing point contrary already model criterion implement away input complex discuss section input training extra inconsistent case choose model selection focus desirable input value give focused selection hard focus consideration focus recommend evaluating criterion usually continuous undesirable continuous analogue curve quantity penalty input property apparent exactly characterization follow design linear training set expression aic place possibility aic also show trivial extra mention mutually aic go concern focus aic consider input iid almost invertible final design relate random input extra bias model amount grow aic evident formula term depend data likelihood log likelihood measure independent largely value computation variance great aic compare reduction come price variance selection hold large aic similarly distribution apply estimator affect experimentally small correct version several univariate degree draw iid unknown select compute perform function input intercept variance true fu u additive test input uniform mixture distribution test input report input another label gaussian differ model weight average correspond mean version method aic variant experiment univariate decide standard selection counterpart jeffreys likelihood variable unstable large respect polynomial jeffreys bic attempt probable give predictive probability three like recent focus design good though use unlike focus variance bias global bias available learning function minimize model large correct correct bias test bic cccc spike bic cccc univariate figure square risk table risk weight average variant clearly visible experiment perform expect center select obtain stable center outperform model adaptively input risk multivariate spike overall aic tendency go soon true aic continue matter large sometimes parameter small complex refer well assessment generalization test input achieve risk tendency complex available pick much tendency apparent experiment small cause note vertical axis training risk experiment perform aic difference notable exception error potentially bic try attempt probable give conservative average put select weight away multivariate rarely bad instance spike section select complex model near seem result bias elsewhere one observe switch near complex risk univariate multivariate experiment expect clearly aic occur bayesian predictive nx give model reliable must summarize prediction likely lose instance evaluate loss minimize weighted posterior factor use capture rely consideration find bic affect
four kind fold hidden layer neural operator fold space activation fold consequence collapse subset identify mean region identify offer restrict axis shift encode activation mean bound base network parametrization map lf linear region contain region least preserve perturbation parameter get number particular uniform even small exist volume attain work study piece region investigate number examine behavior mlp fold way activation unit analyze behavior correspond piece piece map piece layer start visualization propose perspective activation unit result example input consider visualization input identify deep mlp analyze deep upon result tight computable deep directly unit maximal overlap ignore remainder unit th subset weight p th select coordinate scalar eq act namely coordinate linear follow interval interval map onto illustrated restrict interval consider vertical line periodic pattern hyperplane composition pre treat compute deep layer layer identify generalize deep hide region neural input th corollary asymptotic assume hide width compute region deep polynomially model improvement em em l reformulate term behaviour efficient maxout feedforward layer unit activation f k jk two maxout unit maxout collection envelope view hyperplane maximizer region envelope show maxout region maxout intersection maxout number structure maxout intersection diagram describe partition maxout region region diagram intersection diagram trivial intersection region behave hyperplane hyperplane mm region maximal maxout look maxout maxout layer twice result maxout layer region turn case maxout whereby positive ray maxout layer maxout grow fast similarly respect certain maxout argument sec complexity feedforward network focus find superior machine piece identifies exponential region compute complicated replication help generalize follow identifie image low input region region correspondence describe adjacency intersection combinatorial simple hyperplane region reference neighborhood identify first layer network equal input distinct linear computed belong different function compute region construction divide layer fold dimensional per unit outline unit choice drop remainder define unit sensitive input right activation fold alternate activation pre argument view sum unit output identify cube unit cube way identify region multiply discuss use sec sufficient fold interval need interval interval x k function total space neighborhood map unit hypercube output th bias hide choose neighborhood hyperplane sec divide equal bind give otherwise complete divide sensitive fed layer activation pass next illustration computed pair fold axis maximal computable unit growth region per expansion number behave behave discussion give imply deep em em deep maximal grow polynomially fast maximal region maxout layer input maxout argument sec maxout unit value linear piece hyperplane boundary input entry hyperplane region go region give intersection hyperplane behave polynomially theorem maxout seed maxout point direction forget hyperplane normal whereby slight rotation output region composition coordinate slight rotation cone maxout open cone identical image shift input neighbourhood maxout width identify region input identify region input rank maxout parallel hyperplane region maxout mention maxout layer whose intersection diagram describe intersection diagram difficult diagram hyperplane understand nice maxout input bias way unit hyperplane intersection hyperplane panel illustrate maxout triplet parallel hyperplane region mention maxout platform feedforward without convolutional unit value array convolution array convolution map convolutional consider convolutional fall difference lie corresponding weight convolutional belong restrict obtain one single layer unit dataset conjugate minimize run misclassifie two layer capture region sec piecewise fully piece piece input map hide unit unnormalize cosine template input map map compute find keep use multiply adapt mlp piecewise activation maxout e g response possible response point large provide mlp face dataset unit last train regularization drop unit column project sigmoid unit regularization sigmoid layer visualize row hide visualize interesting response response activation positive similarly visualize output unit layer show visualization seven fig look difference distinct region invariance four map learn interesting hide map linear normalize attempt behaviour convolutional unit piece wise visualization visualization approximately feedforward neural map minor difference actual implementation approximate simply approximate inverse region train test plot visualize actual map show distinct input three unit face unit corollary theorem universit universit e universit cifar computable feedforward activation linear deep network able sequentially layer way compositional function piece exponentially compositional map contribute new depth family maxout network improve behavior unit keyword
snp every snp predictor gene expression model snp pairs strength evidence every snp calculate regression snp loading substantially suggest identify meaningful association panel specific datum individual color across snp loading type datum first derive sampling conditional represent full conditional integrate collapse sampler full factor w posterior proportion operation element column matrix calculate efficiently obtain straightforwardly replace expectation latent statistical exploratory analysis structure observation couple observation put bayesian hierarchical loading shrinkage factor column shrinkage remove reduce behavior validate compare result apply two study different identify gene co two genetic variant jointly ability guide produce unsupervised factor structure sparsity canonical association analysis attention recently ability exploratory rapidly low representation project gaussian I kk jj diagonal factor latent integrate factorization suggest contribute observation correspond loading traditional exploratory method analysis canonical latent factor model extremely framework load mapping scenario loading statistic loading matrix element sparsity correspond contribution factor effect contribute observe expression loading impose induce prior latent classical shrinkage achieve shrinkage effect substantial mass around tail allow signal spike sparsity induce sparse tractable relevance determination ard induce active focus incorporate loading cca pair feature vector identify cca cca concatenation loading induce sparsity combine estimate couple I result structured statistic wise loading matrix across representation covariance large subset study flexible enable avoid curse develop normal parameter beta shrinkage high property load globally unnecessary induce behavior factor drive allow loading either sparsity induce dense enable correspond feature matrix signal curse discuss sparse bayesian hierarchical prior illustrate signal matrix substantial performance model world section drug versus group gene genetic variant publicly extensively use low model transformation vector assume diagonal isotropic probabilistic diag represent analysis panel bayesian canonical group analysis propose pair canonical cca seeks find canonical project maximize description common latent model distribute definite diagonal dependency among residual within load orthogonal building bayesian cca I k common variation loading vector load originally inter analysis recently cca equation low rank factorization particular observation w p error factor loading mean share factor factor zero relate fix block group factor couple I motivate multi model partition observation concatenation loading block structure loading limited loading loading capture subset subset relaxed covariance observation model vector allow loading factor zero achieve shrinkage effect penalty put prior relevance ard loading share assume give hyperparameter precision posterior achieve wise induce penalty include either element achieve capture share avoid model covariance subset maximally ard penalty encourage element load interpretable meaningful research either bayesian loading penalty still go carefully structure shrinkage prior loading encourage element shrinkage parametric wise selection bayesian wise loading selection method regression little structured prior structure conceptually shrinkage prior shrinkage factor property marginal prior maximum penalty coefficient know double laplace lasso induce distribution heavy substantial canonical spike prior zero flat often model spike elegant interpretability loading exclude include model come many possible loading mixture alternative prior generally mixed term near distribution ard laplace strong heavy prior directly show model separately enable extend normal level induce behavior beta freedom smaller great towards let induce assign scale become shrinkage behavior mode near encourage induce encourage infinity generate put strong shrinkage hierarchical representation flexible representation make ideal induce loading assigning encourage element shrinkage enable load f induce equation depend hierarchy shrinkage loading zero interpret across load observed loading induce specific parameter loading estimate wise strength across local shrinkage sparsity loading level allow column wise give non behavior factor jointly dense shrinkage component dirac variation technical effect g large loading effect equation loading sparsity column wise sparsity modeling loading wise two outcome towards effectively remove column share induce loading factor model factor whether remove put flat loading z w loading loading loading column lead column sparse behavior model factor load column couple h component variance application local hierarchy variance wide couple simplify section regardless develop variational maximization loading column specific estimate jointly fast load addition markov monte mcmc use mcmc update loading parameter start warm encourage robustness variational model orthonormal rotation p produce identical traditional restrict loading triangular carefully right multiply loading multiply structured prior put constraint sparsity regard zero desirable structure loading low practice load sufficient solution address switch identifiability address trivially label switching switch either mcmc estimate sign simulation follow arranged loading match loading sign match switch estimation element sparse loading simulate pair observation simulate compare relate pair simulation small reflect structured observation factor factor table factor element load randomly element factor generate column error specific dense loading generate vector context sparse factor specific last include simulation four sparse four represent dense vector cca ard ard cca cca cca put ard loading encourage wise shrinkage extension ard multiple couple observation run ard cca aim canonical direction correlation maximize loading cca classical cca matrix non singular simulation regularize cca add two matrix accord leave simulated loading I constraint transformation loading choose projection true factor couple cca maximize project original sparsity induce produce encode orthogonal minimized concatenation choose matrix u vertical concatenation transformation frobenius load recover column column switch splitting rotation well quantifie invariant orthogonal rotation switching scale zero extend stability index couple regard loading recover load distinguished value load five loading sparse loading loading load column dense loading affect separation sparse loading matrix treat loading separately recover component final evaluate number dense four run start factor identify number run run correct factor run recover matrix multiply easy loading model panel panel recover loading recover column recover multiply recover random c recover loading model panel recover recover simulated loading inspection among sparse factor ard ard limit sparse ard poor figure loading well covariance unable subset poor identify c well across size ard sparsity figure figure sparse loading ard recover loading figure dense loading factor include figure c reflect figure large well panel across four b comparison loading four size across loading large well recovery small indicate method b across panel loading datum study hour exposure buffer represent gene project quantile apply initial initialization estimate estimation datum recover proportion calculate explain total tr proportion explain buffer sample panel proportion factor type order display panel count scale covariance control share specific control level biological loading illustrate gene cluster run three factor sparse share gene go david buffer term david significant annotation gene treat gene gene annotation annotation annotation david annotation term
specification r cell avoid mark iteratively update step moreover since choice n temporal logic temporal logic mdp decision probably reinforcement logic specification pac keywords pac consider synthesis logic specification unknown environment probability build call model base probably approximately mdp methodology attain probability sample polynomially logic specification horizon maintain initially unknown base logic transition finitely satisfying predefined integrate learning allow complete unknown gradually control effective correct meanwhile obtain paper logic system incomplete knowledge probability planning robot operate affect action differ level robot model movement possibly number approximate actual optimality temporal specification thesis efficiently controller satisfy temporal specification maintain different action approximate exploitation stop either policy information logic specification measure learn logic verification use infer probabilistic deterministic game chain great quantity admissible policy restrict path computationally temporal specification efficient employ inference yet exploitation controller bias apply model may synthesis temporal logic share apply specification guarantee efficient specification improve exploitation probability specification tuple initial transition atomic atomic stability build proposition boolean always represent acceptance tuple run word accept appear infinitely give present quantitative synthesis specification j v ss ff si deterministic policy quantitative logic objective chain v v chain state visit reach unique path follow notation paper path start exact path eventually chain involve denote pair empty define connected edge graph denote state visit state structure give specification specification want end policy end component follow synthesis quantitative objective practice underlie one motion reinforcement learn model use knowledge update eventually policy success specification product horizon h v tf u v v value understand transition neither give expectation eventually step accumulate reward design output value overview full space know visit many time unknown maintain update consequently partition unknown estimation state compute target horizon state state unknown state identify learn result policy informally state satisfy specification less satisfying specification minus small policy encode atomic proposition omit product acceptance condition specification probability follow assumption associate time transition transition probability large variable extend label approximation share action construction approximate learn learn true model estimating policy unknown learn observation error u range second requirement learn achieve close therefore temporal logic influence potentially horizon potential influence suppose optimal policy directly horizon close satisfy eventually horizon mix let u ft exploitation strategy exploration exploitation make knowledge exploitation insufficient zero encourage agent nearly optimal notion confidence interval action probabilistic encourage state aggregate initially unknown product fig figure respectively near optimal product rapid exploration unknown statement visit simplicity gx gx write probability set visit infer xy u tu reach state one corresponding end visit infinitely sufficient become known use logic formula input mix let policy return firstly state polynomial step induce lemma policy attain unknown state visit explore efficiently finite step polynomial mix value obtain achieve policy need unknown specification synthesis product provably efficient learning eliminate detailed discussion elimination tm q ss h h possible state initial explore state tuning run motion planning implementation
wavelet tree widely segmentation denoise document categorization determination wavelet e state mention estimation page gaussian hide jointly wavelet wavelet provide flexible handle latter work well denoise detail exploit lower develop em bayesian state integrate demonstrate conclude matlab code available abstract transform edge parent wavelet conditional modify frequently application assume mean structure model view wavelet child else typically application wavelet state coefficient mass density nature obvious model generic unknown stress dependence consider independent kp finite density index distribution wavelet dependence structure wavelet page property page coefficient neighbor also page variance initial unknown usually variance denote level ps ci appear mixture model use wavelet parent tree log hide level parameter level difference moment wavelet discuss parameter composite consider full parametrization I structure variance recursively conversely parametrize ci ci appendix particular derive e node path assume level induction depend node furthermore em difficulty two marginal child wavelet calculation forward numerical limitation modify replace finite state integration notice gauss quadrature rule quadrature consider involve smooth em estimate composite likelihood wavelet parent child distribution handle sequel likelihood term full child th wavelet composite li assumption estimate denote vector log likelihood apply l rr obtain composite handle mainly conditional marginal distribution likelihood estimate meaningful replace make sense marginal return local composite return indeed satisfactory approximate posterior density w ps complicate handle chain monte carlo end view approximate bayesian transform corresponding show histogram along fit figure fitting vertical panel provide vertical level equally additive noise pixel work orthonormal wavelet preserve denoise wavelet wavelet white eq observation discuss appropriate state model approximate calculate calculate obtain scheme noise ratio image image level frequentist parsimonious image appearance model perform one provide leave image top panel leave panel original see top image posterior median either turn wavelet coefficient homogeneous lie quantify edge thereby coarse tree focus label wavelet label pixel first bayesian compute viterbi third wavelet arise hide first viterbi compute successively step finite maximization multidimensional log matrix additive omit matrix definite third iw ip specify specificity j use pixel associate binary indicate haar mention present scale exclude wavelet transform simply pixel classification positive comparable notice present wavelet direction model variant haar wavelet transform wavelet perform signal determination wavelet note method estimate obtain variational may wavelet omit far variational natural sciences centre geometry advance foundation grateful thank conditioning structure immediately relationship straightforwardly derive appendix moment relation general wavelet consistent estimate wavelet tend consistent equivalent moment strong usual variance homogeneous accordance thereby appendix apply start density density expectation maximum point calculate density product density rule g suited approximate integral quadrature node repeat whereby return together obtain next
equally cluster size experiment aim size observation cluster item correct classification trial mean sc cluster centroid sized sized cluster detailed size centroid part health obvious difference compressive unknown incomplete compressive sense directly b completion follow compare relative completion cluster correct reflect handle part different individual number imagine half bottom half top column mean random rate may handle cluster may offer tradeoff couple offer improve method compare real data data california survey large survey conduct maintain center health phone extensive health health health health health services health health one major difficulty analyze truth come eliminate datum consistent individual replacement miss finally apply completion spectral cluster cluster expect rate decrease monotonically decay regardless reliable outcome even reach group simulate preferable compressive sensing giving recover explicitly take contribution bring mathematics health verify traditionally future performance type prefer aid design health survey miss thereby reduce burden receive university lemma question theorem conjecture remark analyze cluster health compressive sense spectral health datum test low misclassification completion accord advantage compressive sensing near health datum vast cluster technique mathematic refer separation meaningful within group health research unobserve participant response package sensitive local identification model fit also drawback method identify relationship individual measure matrix entry individual compute eigenvector separable dataset sort large eigenvalue entry plot cluster datum randomly large incomplete imply incorporate likelihood analogous multiple x score miss raw estimate compressive fast grow mathematic cs application low rank optimization ij recover completion since np hard due underlying completion complete provably noisy generate remove frobenius frobenius slightly recovery cccc
real triangular factorization implicitly compute factorization refer distribute stable dimension section platform thin svd thin thin q decrease orthogonal thin qr factorization thin svd factorization dimension compute svd svd svd dominant sample implement parallel assumption implement outer key row reduce sum process aggregation practice gaussians processor across affect hull projection datum normalization hull streaming read column disk twice disk normalize column simultaneous norm column norm triangular normalize read idea experiment nmf present section factorization optimize architecture reference restrict architecture intensive several reason dataset eliminate algorithm pass loading disk memory file output significantly reduce cycle many analyze read store disk loading dominant spent roughly measure sort single iteration read normalize show norm combine algorithm value pair matrix may optimal tb tb separable near stanford institute computational engineering gb ram intel ghz svd svd follow greedy gp require representative set approach choice qr svd transformation subsequent matrix million permutation word extreme matrix storage distribute file system function separation rank converge column select extreme extreme select recover separable matrix separation select coefficient identical tb test implementation configuration datum simulation location row coordinate temperature location radius construct near variability responsible additional rank structure nmf select indeed matrix publicly residual separation rank small residual gp quickly heat value residual curve five one extreme different remarkably characteristic case extreme close small illustrate tb heat index extreme look tb fc labeling phenotype individual combination correspond pe pc cd cd represent data interest pairwise cell marker vs marker kronecker pair cell pair marker abundance error quite nearly column figure residual define phenotype marker biological phenotype cell marker researcher omit still recover complete preliminary nature depth multiple similar desirable show coefficient coefficient diagonal compose nearby tb nonnegative need efficacy separation nonnegative date insight massive set structure heat show redundant would analyze scientific test additional practical impose requirement explore explore regime column rough long fit mean machine gb begin dominate solve office stanford fellowship lee office technology stanford fellowship fellowship thank helpful discussion stanford david nonnegative assumption row component improve transformation preserve separability pass suitable streaming multi core architecture efficacy size synthetic world matrix scientific bioinformatics nonnegative factorization nmf value entry general decomposition advantage nmf column pixel coefficient reason nmf broad application discovery hyperspectral property massive hundred million feature bioinformatic many concerned algorithm advantage large scale new technique orthogonal transformation particularly compare community computation implementation easily see begin show correctly recover heat software online remainder review computing issue unfortunately rank finding minimize np assumption separability q index notation live hull extreme index separability tractable nmf near separability live hull extreme algorithm near separable typically base describe determine focus efficiently implement pass severe restrictive algorithm efficiency justification separability exploratory experiment ray ray advantage fact orthogonal factorization decomposition svd top matrix row information extreme restrict column reduce representation separate invertible preserve column transformation transformation rotation preserve technique exact separable transformation briefly describe column preserve transformation preserve select extreme column possess invariance reason orthogonal separation try value rank pick separability gaussian hyperspectral
review finite random exist goal pac distribution denote pac outline compute hoeffding hoeffding without hoeffde tight small worst develop empirical bernstein name bernstein bernstein bound tight version traditional validation develop similar replacement sx j direct computation tight bound binomial tail without tail inversion take pac compute set refer entity identifies match distinguish type batch query match request identify batch knowledge identify query cover actual match know initially bound algorithm independently call bind derive validation actual match match sample select uniformly replacement without replacement match identify classifier start bayes failure sample draw replacement precision failure bind match one random h pr pr pr use substitute rhs rule apply recall classifier classifier recall probability theorem actual match must complete result without desire substitute complete batch bind failure least accord substitute complete precision request match match identify algorithm disagreement identify set actual include select precision q one match match identify actual match otherwise fraction match identify precision node replacement actual compute identify match respectively failure least also uniformly replacement select sample without replacement draw identify include draw without replacement oriented remark need address present validate disagreement actual match classifier collecting collect produce tight bind precision different set match set validate example merge contact profile social set match choose verify merge validation measure query node fail identify match identify match algorithm rate validate rate match validate disagreement example name phone email address location entry contact occur document pair field entity aggregate field matching connect address phone case want email phone set set actual validate structured people set field match multiple multiple home phone uniformly replacement application wish develop develop classifier develop validation subsample subsample separately draw replacement similar without replacement verify validation subsample subsample follow intersection size si uniformly without replacement sample consist independent sample uniformly rigorous validate regardless method network validation validation matching extended match match evidence direction develop guide process iterative match iteration infer match iteration adopt match rely future match confidence independent distribution validation develop could distribution direction research absence verify match use match support place match datum would adjust accommodate might validate match verify identify match validate match identify node whose know know probabilistic able develop assumption soon add recently distribution work proof subsample select without size replacement uniformly without replacement lemma corollary identifying introduce compute probably approximately bound require level match correspond combine network rich understanding person business identify facebook biology different algorithm match similar edge definition set social business strict overlap business match person social represent person business strict rather person high matching node weight matching problem goal pair maximize set similarity neighbor insufficient effective incomplete people name weighted bipartite big prohibitive allowing match must seed match high degree different connection match connection match establish seed match
dense region graph naturally lead class stochastic instance far ellipsoid parameter due limitation guarantee irrespective true still trick robust encode q feasible linear probabilistic normally noise covariance probabilistic k write k cumulative normal practical interest fortunately far knowledge link hard scenario essentially satisfy neighborhood show hypothesis way attempt statistical learn see two rademacher pseudo cover sample draw restriction expectation q definition often unless bound constant within generalization generalization statement eq constant relation empirical rademacher complexity cover use number integral fx fx c upper rademacher complexity argument convex rademacher bound knowledge associate extend operational cost applicable set bound linear constraint f sr minus portion spherical formulae volume integrate sphere bind example decrease influence complexity measure improve generalization know cover multiply complexity rademacher hypothesis p vector upper magnitude expectation first problem sign fix distance scale second generally result lower tight half constraint space vector upper rademacher operation l tx capturing analyze decision extend set motivate define matrix constraint addition constraint manner rx bb parameterized parameterized jk k kp c element small q kk q unlabele give force cover polytope structure point duality rademacher omit assume ellipsoid intersection find rademacher lead desirable namely call ellipsoid pick ellipsoid combination original contain tight ellipsoid family tight ellipsoid quadratic define rademacher quadratic let semidefinite ellipsoid correspond ellipsoid correspondingly magnitude eigenvalue since original region ellipsoid upper tight axis ba ellipsoid program optimal theorem dependence x ta use hand example capture matrix value become quadratic behave bind ellipsoid eigenvalue like lead high nx form depend lower similar ellipsoid regard handle rademach complexity covering describe ellipsoid eigenvalue pick side bind problem involve minimization step singular call ellipsoid multiply diagonal find convex look result come whereas pick qualitatively ellipsoid eigenvalue instance ellipsoid sphere thin ellipsoid tighter although problem stage ellipsoid intersection original theorem obtain bind generalize ellipsoid family ellipsoid minimize side quadratic lead quadratic simultaneously tight follow eq sl kk requirement ellipsoid way intuition linear inequality way result upper arbitrary constraint instance pac concentration statement presence pac though result unlabeled point author introduce notion distribution example quantity impose level hypothesis finite hypothesis q obtain bind approximate disagreement classifier unlabele disagreement zero training assume randomization theorem serve result focus exploit unlabele exploit unlabele restrict work couple rademacher define maximization right great value equal side rademacher thus maximization inside operation dual duality upper us dual eq g similarly prove appear rademacher supremum ellipsoid positive invertible jensen linearity fact orthogonal write define transform become substitute gaussian upper constant bind gaussian upper l relation obtain result give constant function orthonormal li x b concentration function gaussian standard function omit lemma py alternate distribution py py substitute bind substitute upper element let feasible substitute feasible b bound rademacher dependence empirical weak rearrange inequality get term eq expression rearrange term scale empirical rademacher equal sum mean moment need two gaussian use ellipsoid n state n rademacher maximization objective inside max operation empirical write duality maximization equation similarly intuitive reason serve upper get rademacher right desire result upper see duality rademacher rademacher value ignore right lagrangian k z objective maximization z show rearrange complete dual minimize complete term value result remain respect get follow upper upper feasible obtain instead suitable feasible upper feasible give upper give desire use constraint bind know outline side help focus attention give deriving bound beyond traditional paradigm hypothesis study space interesting space quantify describe encode linear side baseline performance rmse root multiple ridge ridge use response type label keep aside number knowledge size incorporate knowledge side multiple construct constraint construct smoothness section sort monotonic accordingly constraint r vector use ease time train knowledge use knowledge test ridge dependence change example summary training obtain across increase impose true rmse value shift difference model side useful rmse legend learn knowledge multiple range bar plot training setup side mit school management institute technology usa supervise side lead tight space quadratic lead hypothesis side knowledge quadratic could potentially domain actually domain nontrivial algorithm use successfully variety learning property constraint nlp language various use work aim beyond sparsity smoothness keep additionally different set example label choose unlabele label hypothesis search motivate expert example learn predict example encounter type constraint linear constraint main contribution linear space arise naturally circumstance unlabeled example connect provide bound linear find section paper constraints family upper provide match novel bounding constraint illustrate arise ball coefficient dimension proposition situation intersection ball ellipsoid theorem set ball second cone helpful circumstance fully side smoothness consider truly low sample complexity improve selection effort true gain benefit motivate erm svm incorporate classification dataset rule generate constraint norm classifier decision tree knowledge incorporation svms review erm demand auto demand method output monotonic demand whose work multi belong class translate svms know regularization augment cloud unlabele quadratic regularize outperform svms overall formulate robust lead classification heart uci repository introduce improvement introduce imputation uci also provide experimental
highly efficient proposal mention care updating describe qr operation rotation fill transformation decomposition investigate maintain update topic derive specialized fuse lasso component node underlie write lasso edge recall arbitrary theory orient incidence simplification algorithm reduce tx cd orient incidence guess linear edge analogous sparse typically find call basic norm strategy apply future place nan readily strategy fortunately fuse onto form solution alternate expression quantity alternate specialize fuse simple norm dc transpose minimum linear system compute z step necessarily section compute difficult maintain norm cover must norm solution linear g work alternate form big term fuse orient incidence linear computed computation hard span efficient orient incidence edge logic onto component solution linear topic science nice solver use laplacian algorithm indirect iterative solver return approximate tolerance system issue explain extremely computer community reference therein direct solver let graph component express block therefore decompose accord fully subgraph exactly follow laplacian matrix graph eq laplacian solve submatrix exclude connect span solution unique prove column orient incidence graph column edge corresponding row repeat show message graph form last column cholesky component finish specialized fuse lasso dual repeatedly compute dl outline connect graph one change whether run first search path specialize fused incidence center first add projection solve w z connect package sparse cholesky decomposition reduce prescribed employ cholesky algorithm see therein unfortunately cholesky admit empirically efficient number linearly subgraph provide solve full dual algorithm specialize lasso implementation fuse orient incidence graph row eq brevity project onto onto underlying row partition g g define edge follow connect coordinate correspond averaging within otherwise component system laplacian connect component dd laplacian subgraph connect component jj discuss linear factored cholesky sake completeness say fuse fuse soft thresholding output fuse lasso lasso long perspective recall presence view dual dx fine generic penalty fast structure retain dx apply usual path result carefully solve step solve solve linear strategy apply solve nd typically give idea calculate solve dense trend fuse prove correctness implementation fuse arbitrary full characterized suitably matrix full give general characterize hand system apply logic rewrite multiplying rank norm dx dx subspace dx x dx rewrite computation compute compute simplification norm system eq offer offer significant hard system td step require involve projection first begin ever reach quantity serve freedom estimate therefore interest stage step last really system qr outline trend st span take consider eq fuse fuse onto orient incidence span connected graph subgraph correspond incidence therefore vector give q remain node connect node write appropriately indicator node section dual fuse reasonably large come report city report date occur spatially aggregated census group census calculate think proportion measurement randomly census block proportion figure task census grouping adjacent census lasso difference neighbor huber block difference optimization appropriate capable produce component total graph first fuse path specialize little computer freedom note roughly region city side city risk city since block incur picture qualitative census city lastly fuse compete measurement counter tendency method create equal sized cluster iteration complexity implementation generalize implementation iteration termination super notable exception fuse termination infeasible undesirable typically regularize visit toward path path vary three fuse cubic trend filtering setting period fuse square bottom third first fuse trend step fuse computation indicate problem hour trend interest section step census group denoise solution hence step one likely drastically algorithm trend filter operator lasso incidence handle well acknowledgement rt support nsf brief review square excellent column form surprisingly qr operation decomposition primarily minimize leave equivalent equation recall triangular look box nonzero entry indicate row substitute procedure square require operation total multiplication qr importantly vector compute decomposition operation permutation span column upper triangular visually look column least criterion admit solution infinitely qr eq first last variable decompose operation let hence least operation compute solution necessarily however qr decomposition need rotation cover key message triangular rotation take compose form operation ok operation write side utilize seek must advantageous problem qr change compute special km find concept except require operation finally total rotation transformation maintain triangular rotation decomposition add qr decomposition notation largely chapter main rotation amount rotation angle orthogonal furthermore simply onto axis inspection note rotation identity except correspond applying affect leave eq rotation apply affect row take row common look example th rd zero begin pre multiplication pattern give triangular structure matrix multiply rotation matrix affect column compute operation logic multiplication appropriately th column look eq rd rotation triangular structure cover rotation qr add remove cover reference decomposition subsequently want qr row motivation qr naive problem separately suppose mm ar triangular therefore rotation n n triangular desire procedure rotation therefore operation qr remove change cover denote th rotation basis let orthogonal qr om add exist one cover rotation row triangular rotation apply rotation upper triangular operation technique qr minimizer subsequently compute norm row actually different qr initial qr rotation matrix triangular avoid rank case g right eq k triangular complete qr appropriate minimum second rank nonzero rotation right side q triangular qr desire rotation require operation alternatively na row rotation qr qr ai decrease zero rotation triangular help nd rotation row rotation j proper qr decomposition operation one obviously qr obtain remove add remove qr decomposition strategy addition removal require clear order initial square problem path correspond p p corollary taylor efficient implementation full rank case trend filter fuse sparse fuse specialized implementation offer numerical solution use repository path qr laplacian computation outcome vector problem choice assume ensure value implementation generalize algorithm compute specialized implementation special fused trend filter implement repository problem early work trend framework brief think fuse row th e contain zero fuse lasso many exhibit subgraphs concept note oriented incidence undirected laplacian realization fuse component successive piecewise across work setup fuse lasso additional refer fuse fundamentally pure describe trend fuse trend filter penalty version fuse term explicitly filter signal estimation statistical begin discuss briefly review aside central convex value parameter solution function former desire number quadratic programming multiplier admm general specialized technique fuse linear string trend specialized admm finally fall category proximal utilize specialized describe algorithm regression also fuse flow track opposite start end primal perspective assume row fuse lasso operate unified allow flexible enough specialized take work give detailed comparison alternative implementation dual path generalize place column name compute help case reflect compute path path level keep track coordinate compute equal lie leave outline minimum first record compute next leave add remove leave record main lie word start set often obtain fully less proceed path second concern solver encounter step broadly speak solver indirect solver solver linear round error perfect computational platform direct exact indirect approximate may preferable system boundary across rely solution sense really stick propose describe dedicated evaluation section consider without contribute iteration update row add qr norm efficiently decomposition save order computational qr decomposition complexity
text take computer intuitively vector close language roughly cluster manner get language identify include sample sample table gram letter histogram guess accuracy incorrect system detection identify gram htb pdf cm table confusion language base letter predict corpus detection language keep track easily accommodate efficacy indexing language solely letter text letter also thousand algebra vector make multiplication addition letter english next advance design mind retrieve arithmetic generality text well scheme way address indexing language indexing identify material simple dimensional signal rise data work memory inherently produce amenable experiment paper acknowledgment discussion feedback computer california berkeley berkeley center theoretical california berkeley random indexing wide application variety accuracy demonstrate encode letter gram method implement require validity task short language achieve accuracy comparable human recognize unknown language course sound especially language say language give language text categorization model similarity various language identify language count letter letter compare profile general store various google recently compact detector profile corpus perfect popular processing word mean statistic semantic context vector ideally word represent vector explain semantic frequency index simple use way al bag locality sensitive lsh differ randomness thousand refer calculate useful paper present detection indexing highly scalable main random indexing project well operation keep high implementation indexing similar language create text latter refer letter frequency text know letter frequency language letter letter idea physics consecutive letter text block appear gram example rise b stand frequency letter block language text language letter plus frequency letter would gram track indexing text vector running create create sequence describe early example block calculate label half half vector sum gram text store language exactly text compare vector cosine unknown define cosine high text language cosine choose
x measure coherent moment learn distinguish continuous function learn possible motivate restriction continuous exist increase old continuous old continuity every another old lemma interval argument argument chernoff probability lemma step arm minimize low choose try estimate risk mean arm explore try exploitation repeat tt algorithm algorithm probability time regret different continuous e match bandit case cover bandit log exponential problem bandit function old demonstrate efficiency eq see front go comparison growth old function sequence example achieve regret follow let consider arm principle arm negligible precisely represent th arm point I estimate arm word ie ensure condition advance arm arm happen sure ie ensure arm algorithm use instead construct knowledge avoid construct small occur arm repeat decision receive time repeat least bound motivate example take take region case since arm hence compute I risk two achieve logarithmic prove bind measure condition logarithmic might even continuous sound follow goal likely problem general open functional class coherent risk could plausible intersect inclusion identification framework pure explore focus notion construct satisfy uniformly previous implication f let every similar minor follow distribution high event union one arm event combine three give focus bound everything derive stop fulfil ac minimize regret multi armed goodness arm present sublinear regret stochastic armed bandit framework clinical formulation one distribution receive arm repeat prescribe regret difference arm small desirable clinical trial armed bandit case arm risk theory learn field learn study expert risk pure propose use variance measure aim latter previous immediately applicability lot risk mean arm risk arm completely bound pac formally state model discuss open possible extension conclude proof main distribution learner
whereas slightly complicated recommend corollary factor besides concrete applicable step discretization rule rate tell perform concave state change might relax impact make tv quantitative characterize readily eq bind get horizon eq tv proof infer initial impact density p small obtained accelerate extent amount definite close h characterize corollary k guarantee strongly concave cf also density log density one former target continuously inequality function satisfied define expect close broad necessarily assume leibler eq pf use formula divergence eq exponential second kullback leibler divergence right claim tv derive follow result p algorithm op comment dependence acceptable get substantially concave dependence able simulate precisely repeat initial distribution op op get case replace properly certainly get replace tight involved formulae mcmc strategy define stop strategy scope applicability spectral inequality employ ergodicity langevin lyapunov since inequality langevin drift ergodic advantageous gap involve dissimilarity explicit dependence exponentially advantage ergodicity even convex langevin diffusion euler discretization section diffusion attain desire lead diffusion hereafter langevin monte carlo h pf accord matrix spectral f approximation theorem defer let direct last provide number prescribe level trivial reader continuous constant outcome also recommend reach desire op pay attention computing exponential versus preferable situation require instance paragraph use worth perform hessian much case first approximate establish guarantee scope remark warm reduce useful replace g density experiment value device rademacher distribution value draw pn gradient precision median employ recommend experiment aforementioned estimator trial clearly large increase although median robust sampling rather posterior median mle put bold frequent winner measure euclidean draw posterior median mean posterior median approximate concave log concave langevin monte regard natural counterpart statistic beyond good theoretical prove computational complexity scale polynomially desire level prove show evaluation evaluation available chi divergence polynomially value evaluation theoretical guarantee computable constant involve work generality result difficult implement get tight constant context metropolis hasting mala concave target important particular investigate compact derive density bound make establish guarantee little interest remarkable prove logarithm power behave dependence dimension warm available bad analyze evaluation build recent connection focus propose sampling come convex hope investigation aim establish recall sake completeness proof second introduce dt lead prove proposition continuously throughout shorthand view global apply subtract proposition hold inequality obvious complete lemma yield simple application schwarz invariant density schwarz give kf relation suitable constant corollary theorem argument diffusion process lipschitz continuity hessian provide analogous gaussian h ss l l complete ease notation proof notation v conjunction sum schwarz distribute last use v go back side conjunction e inequality condition infer inequality hand check pi claim acknowledgment work support various kind pa importance ingredient procedure resort meaningful log concave distribution langevin monte effectiveness process beyond density present log estimator close computing likelihood require impossible provide iterative variety algorithm approximate work gap sampling stand characterize continuously define descent precisely achieve upper evaluation feature logarithmic dependence side somewhat conservative quantity involve expression computable simple stop recursive situation approximately exist study theoretically tune grow maximize density necessarily gap even numerous similarity optimization approximate sampling langevin similar algorithm step recursion often refer center vector matrix identity small product total establish explicit quantity approximation refined version term summarize keep thing translate one large complexity gaussian iteration performing finding work column magnitude number iterate perform error column contain bad complexity complexity one iterate iterate warm set stand norm unless probability proportional markov behind euler discretization langevin diffusion langevin differential equation sde brownian sde strong p call spectral ergodic bound operator value fast behind probably influential probabilistic asymptotic avoid langevin ergodicity choice case langevin diffusion choice result chain influence subsequent metropolis metropolis langevin numerous impose couple continuity chain choose fact differentiable note condition nonnegative expansion view entail point consequence long assessment rate end markov vector upper process precise increment increment iid readily vector h follow drift
image assume soft object dominant specific assign patch relationship quantization number learn distribution sum corresponding patch utilize share compute codebook soft label topic change assignment refine shannon entropy setting codebook remain new refined codebook q estimate class summarize propose patch total infer learn rf correspond train base final real world semi order unlabele image extend paradigm rf feature extend except unlabeled scene scene consist pixel dataset classification evaluate class object dense sift patch size pixel ratio retain rf codebook histogram use dataset table notice though improvement seem narrow soft codebook compare significant final label training good model scene label art despite limited label method effectiveness forest drastically label rf rf limited mind topic generative believe hybrid explain experiment conduct gradually experiment feedback fully setting patch rf unlabeled training rf enhance training feedback mechanism b b effectively soft role visualize patch review effect soft label codebook visualization scene rough original besides normally image rf background patch reduce background patch rf splitting improve rf method iteration rf accelerate soft label feedback framework rf codebook learn image achieve soft codebook codebook reach framework paradigm investigate support university rp center edu united uk china edu understand important due bag codebook essential forest rf tree performance patch paper tackle novel way update codebook codebook base feedback feedback perform patch art rf codebook learn focus patch experiment propose task part base discovery successfully apply unsupervised object categorization scene codebook patch achieve discriminative advantage counterpart g ground forest rf decision tree compare codebook quantization demand rf show codebook object categorization segmentation heavily truth ground belong face however notice patch belong class g red face train rf codebook background patch rf codebook framework novel rf rf weak image patch feedback c codebook arrange development relate visual codebook framework show discussion codebook essential representation find mean tree employ recent research focus codebook use discrimination codebook offer characteristic popular rf feedback mechanism detector create location codebook feedback scheme variant feedback rf learn perform codebook learn classifier set separate feedback node study utilize assignment label rf codebook effective topic codebook popular serve mid group meaningful representation ideally part part represent represent background require image combine enhance rf codebook feedback main contribution rf codebook level superior information patch level rf contrary employ learn rf ground truth label treat build rf codebook secondly train weak codebook label weak rf enhanced train refine
finding iid eq domain assignment exponentially find assignment efficiently add unary potential rbm correspond standard basically I alternatively binary unary potential section feasibility use availability np mrfs efficient minimization propose suitable energy change without change energy intuition perturbation compare put distant training remove false training pairwise potential unary potential remain desirable potential higher visible bipartite interaction need noise bias perturbation potential basically assignment inf new likelihood idea minimization allow landscape perturb reach thresholde fast closely step inference current may deterministic chain mcmc intuitively attempt update parameter unnormalized probability function refer phase cd training phase configuration neighboring state take markov chain train maximum posteriori basic extreme map assignment perturbation perturb ideal approximation use provide upper maximize feasible submodular potential even step could relate basic idea perturb configuration discrete training rbm rbm field n example rating speech model deep neural architecture unit q e normalization constant due bipartite form visible visible rbm rbm local field visible seek maximum log calculation derivative negative require sample current may markov chain regardless form q sufficient rbm calculate term term training markov
discrimination adaboost slightly effort embed machine vision unclear propose repository challenge vision model significant detection haar paper develop briefly review adaboost algorithm explain fail sample adaboost general ti ti tx hx tx indicator variation algorithm arc differ mostly compute adaboost select weak pool take adaboost round error go improve overfitte adaboost margin theory give error direction error size training formulation adaboost tie discriminative whether line pass type red one however illustration eps initial thus sample adaboost weight correctly sample pass usually slightly small discretized classify receive fig round essentially lead back combination keep due adaboost sensitive keep weight pass weak embed switching focus operation way improve design allow case hard feature separate feature often lead fitting adaboost vice versa mix recursively combine cascade lead decision tree weak embed weak describe much long complexity eqn algorithm essentially logistic probability q discriminative upon weight make impact vision million thousand adaboost strong simple use classifier adaboost call however still linear difficulty fig positive classifier failure select subset complexity give detailed description combine operation di tx decrease output straight classifier operation classifier final positive regardless classifier unlike adaboost mis quickly focus solve show illustration eps fig feature weight step weak classifier however positive negative receive weight adaboost create situation weight sample di h tx di role adaboost negative positive therefore decide regardless later weak therefore weak swap positive turn operation complementary decision operation operation htb give di stop output htbp illustration eps cccc adaboost second call classifier weak keep check decision operation naturally embed aspect logic confusion weak pattern fig result operation positive classified margin theory al decide decision much simple cart worth one present happen add major issue concern classifier computer vision eqn desirable produce error low vc good vc margin decide small difference reality enough task none limit since kernel mining also ultimately modern particularly error classifier major scope performance widely generalization al give behavior adaboost margin margin weak combine arc try minimum adaboost experimental arc big error adaboost try find arc indeed arc adaboost decision uci repository breast modify category merge class upon belong sample test trial cancer training compare alternative different weak operation give good similar vision weak eps eps width different number conduct plot weak improve show dataset suggest achieve introduce breast suggest boost decision cart base table adaboost arc decision arc cart adaboost cart decision show arc cart cart tree leaf tree around complexity big massive thousand cart achieve greatly usage classifier currently widely htbp eps width slightly illustrate demonstrate segmentation detection demonstrate testing image label pixel body task classify patch center body negative positive patch haar response filter cascade cascade node select algorithm identical bootstrapping algorithm uci repository improve error nearly cascade well report report cascade adaboost fig among detector haar computer understand
incorrect concentration direction region version incorrect theorem theorem specialize class test ratio quadratic power nan discuss way distributional devote apply autocorrelation series whereas appendix result real vector loss generality identity reduce transformation assume furthermore relaxation general although state stand borel expectation respect shall measure borel say impose possesse everywhere else completely strong need even apparent testing fact distribution line proof theorem problem understand typically identifiability meaningful still beyond moment explicit identifiability assumption randomize measurable space rejection borel together rise region test shall I rejection probability terminology always real transpose span symbol onto complement rank z row orthonormal satisfy vice choice orthogonal denote corresponding eigenvalue symmetric order definite root form euclidean operator denote interior closure w symbol denote lebesgue borel uniform measurable operation composition function r course column eq main invariance w test invariance another invariance remark maximal group denote generally sphere satisfy property invariant obviously additionally borel continuous neighborhood unit sphere moreover choose satisfy claim author define whether r z due assume e remark power every additionally parameter identifiable sense z test far away interest I limit limit confusion stress throughout sample situation I motivate autoregressive order autoregressive power like sufficiently test approach intuition depend employ spatial autoregressive eigenvalue intuition suggest intuition incorrect test significance employ see coherent general structure non mention thus randomize crucially function equal positive model power claim follow exhaustive exhaustive exist neither observe satisfied framework additional remark invariant seem power mt comment notion mt obviously give symbol third symbol eigenvector small eigenvalue invariant validity condition empty mt explicitly statement reader implicitly exclude invariant region strictly precisely complement nan appropriate interpretations cf correct part claim mt discussion mistake generalization correct mt correct incorrect distribution concentrate subspace occur happen infinity crucial effect test appropriate rescaling enforce generalization third mt even mt cover intuition set follow assumption ne normalize converse example underlie weak assumption mt large converge limit exist equal assumption satisfy mt sufficient satisfied discuss converge every accumulation measure coincide measure coincide view claim mt provide end introduce vector say assumption symmetric orthogonal implicitly impose assumption condition arise satisfied hold continuous generally absolutely note family regression convenient avoid indeed weak statement ready present mt stated possibly randomized test invariant third claim mt observe light remark weak mt mt coincide mt sign third specialized rejection region invariant reduce applicable stand indirect region modify test almost everywhere test underlie general note example establish bt region cf see substitute second mt critical n ty condition expression power test ratio quadratic ty substitute assumption remark study invariant rejection rejection rejection assume ty claim say subsequent claim incorrect assumption nd paragraph incorrectly provide claim nan obviously ty clearly rejection region matrix function speak statistic take whenever define rejection region probability certainly family r ii maintain adopt regardless nan turn convenient course rejection nan lead alternative assignment easy understand boundary orthogonal hold constant entire depend obvious actually arise subsequent proposition region form claim provide part proposition slight inclusion region part need hold provide case kb kb provide allow imply mt correlation autoregressive exist root restrict necessarily verify spatial model easier verify root necessarily although independently assumption theory hold claim establish note l arbitrary w even thus condition singleton l accumulation ty te precisely necessarily singleton e odd additionally remark assume entail provide ensure hold appendix formulate equivalent formulation easy spatial ar family density precede mt precede coincide cf precede substitute incorrect mt determine limit accumulation general limit expression explicit accumulation precede decide basis exist apply corollary vanish test totally relation vanish identically show element complement closure rejection conclusion precede verify test statistic satisfied elementary calculation view vanish except b kb alternatively large eigenvalue although belong rule case eigenvalue rejection precede singleton simplify theorem accumulation expression nonzero surely since accumulation accumulation question examine explicit expression observe accumulation open accumulation obtain regularity distributional assumption exclude degenerate case test locally x also assumption give assumption mt sign impose corollary follow assumption would corollary note would furthermore remark kb kb corollary always exclude corollary weak mt cf neither satisfy kb b point occur locally good test literature cite next corollary belong boundary rejection theorem presentation concentrate subsequent corollary rejection neither b satisfied hold view assumption kb multivariate matrix furthermore region iv possess cover equivalent b kb kb b furthermore family b kb limit equal eigenvector accumulation interval u x bc x c e bc non odd accumulation bc bc assumption exclude case b kb empty corollary go matrix bc definite note directly extend version corollary complement part guarantee corollary lem iii obtain appropriate corollary power question critical occur essentially equivalent ask size region size arise least subsequence end rejection invariance quantity infimum size vanish along subsequence limit power proceed narrow discusse appendix unfortunately error discuss applicable structure subsequent see version power equal satisfied test region statistic characterize situation occur level restrict autoregressive order lemma improve lemma relate characterize equal present obtain reject pass subsequent proposition exclude trivial rejection whether suppose origin kb kb kb kb b p kb kb kb b b corollary repeat part satisfied never eigenvector eigenvalue corollary invariant locally guarantee power fall precede tell construct design avoid kb whether property way overcome absolutely follow b kb open possibly satisfie kb proposition concern theorem result precede proposition remark assumption subsequent vector hold neither hence assumption possess w lebesgue everywhere neighborhood origin nan precede neighborhood replace exist assume upper identify power course depend model alone hypothesis indistinguishable test fact invariant function invariant error theorem proposition regard test iv prove ratio matrix I kn nu define group furthermore every whereas mean nan alternative function g invariant symmetric test invariant invariant distinguish alternative invariant less I condition choice differ c I family consequence theorem alternative invariant multiple assumption alternative may invariant give rise corollary l multiple necessarily condition turn matrix non sign hold arise suppose see invariant test cite identification reduce parameter identifiable maximal invariant cf special maximal statistic follow suppose possibly space invariant probability invariant test automatically carry rejection observation part continue requirement apply part possess uniform consequence possess uniform reasoning show distribution weak way everywhere explicit possess continuous explicit difficult give equivalent choose everywhere vanish everywhere possess everywhere continuous everywhere positive iv hence condition always choose almost part part b suppose satisfy atom identity covariance may appendix argument underlie discussion gaussian without apply replace base depend consequently vi discussion positive case correspond mass origin invariant satisfy throughout viewpoint broad setting distribution atom precede rejection family actually probabilitie result paper apart concern invariant impose serious restriction satisfy interested large invariant follow test hold extend covariance accumulation certain assumption hold lag error spatial lag element denote zero absolute algebraic also unique multiplication model random covariance aa cf remark implicitly latter connect hence however implicit typically maintain denote frequent frobenius see e one choose identifiability identifiability immediate nan hypothesis hypothesis disjoint hold without distribution model mild well w absolutely satisfy main e theorem immediately spatial corollary purpose corollary provide fact neither cf maintain assumption possess almost everywhere neighborhood origin possibly nan origin kf f b b n b everywhere hence accumulation iv part next corollary region elliptical assumption elliptical general test give maintain assumption origin symmetric critical b random limit whereas eigenvector consider lead quantity strictly theorem theorem sense limit early power correct proposition critical exclude trivial discussion subsequent always consequence assumption n precisely strictly statement limit b strong proposition statement result maintain rejection neither maintain assumption suppose absolutely neighborhood origin except furthermore quadratic ii p nb lemma test power equal maintain assumption absolutely density neighborhood n b b c automatically satisfied represent less eigenvalue b w empty consider spatial mean random vector identifiable rewrite equation case typically long spirit follow maintain put mass proper e part mt claim incorrect provide mt fit become appropriate version produce turn degeneracy statistic part consequence maximal problem experiment recognize identification occur assume experiment appendix condition condition optimal invariant part proposition immediate contrast result invariant proposition elliptical symmetry proposition sufficient corollary establishes respective proposition precede comment importance weight q first remark precede maintain intercept generality n orthogonal every q together function maintain observe obviously intercept element consequently every onto map mean element say attempt justify invariance spatial lag sense incorrect incorrectly invariant invariance coincide show consider comment error covariance testing negative autocorrelation assume fix distribution depend assumption framework clearly cover gaussian autoregressive readily verify hold assumption show satisfy ii denote fact ii maintain assumption I subject square root critical quadratic x multivariate equal suppose p k corollary expense maintain could nevertheless allow parameterization invariance alternative substantial I discuss expression case notice arise autocorrelation test consider intercept autoregressive power closure interior region numerical contain intercept subsequently indeed intercept limit except degenerate theorem rejection justification one mention extend express ratio note additionally also autocorrelation easily read analysis regressor systematic investigation regressor already mt statement explain mistake satisfie impose page mt simplicity proof degenerate simplify correspond uniformly compact zero everywhere tend degenerate eigenvalue converge obviously tight concentration claim fact opposite happen mass claim claim speak construct mt claim limit power define rejection differ nan family dominate concrete normally without case without q symmetric definite observe rank section arbitrary region invariant r invariance sphere I limit theorem obviously satisfied since absolutely continuous hold mt incorrect start rejection generally covariance spatial rejection region argue somewhat artificial rejection ask mt modify modify boundary subsequent mt impose probability nan nan strict obvious slowly invariant provide limit claim regressor present somewhat b absolutely combine vi trivial power either provide hold comment discuss discuss context restrict concern infimum limiting vanish symbol definition refer hence make usage seem mind test region ty ii remark ty tf tf problematic reason statement region critical region proposition therefore invariant region boundary refer require rejection lead base incorrect implicitly continuity assumption function satisfied value statistic refer size explicitly denote pose absolutely denote correspond stand content statement author typically critical nevertheless case statement however pure read pure limiting power cf irrespective test eigenvector explicit understand assume explicitly statement understand degenerate choice specify test incorrect incorrect discuss lemma would without limit limit case test test reduce reduction argument limit cf necessary clearly establish regard clear precise interpret derive equal test limit operation justification interpret derive course justification however argument perhaps argument version strong statement possible lemma suffer lemma incorrect rigorous infinite dependence without provide necessary reduction precede discussion furthermore statistic case regard test seem point invariant exclude easily test unbiased issue follow event allow argument go paragraph read stochastically also assumption easily relax elliptical symmetry part claim regularity condition contain invariant argument reduce test argument could invariant test strict strict preserve turn regard invariant hold separately elliptical symmetry read also verification display use precisely general refer display hold hold almost concern locally invariant mention turn last degenerate power trivially importantly proposition several incorrect incorrect stand arise equal conclude version probably corollary checked converge form basis order small must equivalently sum I side converge support accumulation sequence subset weak accumulation mass origin weakly say give accumulation certainly subset consequence assertion subsequence weakly almost continuous continuous conclude surely converge continuity symmetric nonnegative square mp ml mu ml mu u proof
output softmax layer model objective consider interaction pair word two two issue individually detail eqn capture intuition language encourage representation language conceptually consist representation word two domain denote word weight sure language embed nearby stacked express q subscript respect source pair derive scale product objective vocabulary argue primary exist due softmax regularization perhaps even importantly strong introduce issue sample objective estimating word regularization word care observe idea optimize et take step representation continuous word use objective architecture specifically language vocabulary sequence bold finally bag represent one bag word representation embedding motivation learn represent target vector embedding maximize sum noise procedure typically entire vocabulary typically hundred word language relate learn language relate eqn proportional alignment frequency step training alignment alignment alignment translation utilize word directly parallel since alignment interpret alignment weight write eqn english word alignment pair sentence level know translation occur sentence make naive align leave advanced english length sample english sentence towards simply bag word eqn parallel corpus trivial embedding regularizer practice document approximate eqn parallel strong proportional eqn eqn really estimate learn cross distribute representation exploit sentence align training evaluate representation task across language also setup goal label language classifier language document vector document utilize alignment instead purely corpus induce cross language induce subset english corpora category market classification document select corpus third remainder size separate development baseline namely word align word language baseline document source al day auto day auto replica cross dimensional embedding directly summarize comparable train english corpus exploit nature english al versus original day demonstrate loss efficient accurate next art train comparable original state art en de en report train english current art also fast embedding task publicly task extract english online google translate service dictionary source individually english pair frequent translate near embed evaluate fraction top return specific method translation baseline et rank base co distributional similarity word construct count word count language dictionary finally word target language c word occurrence induce embedding english translation summarize baseline accurate english improve percentage english word translation percent indicate grain translation raw sentence introduce induce representation require utilize parallel representation advance objective computation training scale sentence thereby enable efficient document art minute evaluate english task relative word university cifar google word computationally train signal raw sentence align datum sample bag language cross embedding outperform art lexical code open language part name entity mostly english technique exist another trivial generalize hard across desirable syntactic semantic feature task interested unsupervised language apply wide language induce representation usually language embed learn training embed sentence induce serve linguistic syntactic nearby embed especially high resource language baseline well name entity recognition translation sentence obtain translation occurrence frequency train regularization frequency pair induce costly traditionally training vocabulary length hundred thousand fast exist evaluate translation pair motivate use believe time limit attack cost head introduce simple induce embedding extension embedding uses align learn training requirement text contribution consider bag avoid induced embedding translation outperform reduce hour several prior approach feature learn cross domain english embedding word phrase similar property use
gate gate boolean surprisingly define truth table bits gate gate define boolean circuit circuit bit output bit decision restrict choice circuit consider input bit circuit computation correspond circuit process circuit specify gate gate wish table row bit ef transpose obtain bit either evaluate classifier require evaluate binary evaluate operation significantly total importantly want computable evaluating circuit consider feature store save operation list logical c five truth gate input output gate product truth gate tensor truth compose complement bit evaluation gate multiple elementary bit million gate evaluate parallelism circuit simple circuit capable obviously simplify optimization produce hyperparameter algorithm circuit second leave optimization global greedy quite simple leaf take coordinate bit input specify greedy leaf bit gate choose truth gate behave circuit decision gate gate reader intractable optimization gate gate nothing gate good gate outcome input gate reasonable mostly gate category choose truth specify bottom take dimension truth know second move tree versus decide success criterion gate gain gate example gate produce ultimately create list maximize information gain set index gate variety range white image datum encode binary depth scale exponentially hyperparameter give internal leave example reduce evaluate small classification circuit hyperparameter algorithm compare net observe repeat configuration section hill leave quite powerful carefully severe quite choose one leaf well back course evaluate array logarithm leave tree hyperparameter need see figure obtain classifier significantly greedy produce tree classify significantly well state art report paradigm preliminary paradigm worth dramatically indicate well feed neural outperform net thing context although preliminary believe investigation need significantly cpu enable fast appear circuit investigate structure initialization leave circuit certainly several dimension eliminate average adding could fitting hill look promise feature select subset simply improve else autoencoder framework net use gradient thing like ham poor substitute future investigate optimistic result thank david thank helpful comment I month finally would thank unconditional experimental supervise state classified bit avoid domain box invariant example example tree update hill hill noise cube bit bit pixel channel bit notice greedy gave depend second recover otherwise describe hill algorithm efficiency experiment core intel core cpu amount trivially experiment code algorithm extremely ask truth preliminary result indicate thousand experiment perform gpu optimistic speedup practically example cube compose pixel white bit cube contain two surface bit cube overlap choose overlap permutation nature pixel data train train train train train train test train train c c train n test k c c test default hyperparameter train run take less minute factor approximately gauss dataset previous list integer bit integer one classify integer receive perform differ plot variance apart round rescaling affect graphical interpret versus l k train versus versus versus test train versus train versus gauss test mnist use dataset distinguish distinguish drive dataset bit pixel range form bit choose bit obtain behave mnist loose presence significant hyperparameter greedy hill l c bit train train train test mnist bit l train train test c c train train convex dataset group hard feedforward net consist image convex set vector rbf feed stack hyperparameter default hill perform c error thorough hyperparameter search expect run time long
difference strong weak evidence bootstrap outline take due bootstrap fold let variance auc train new set training although cross hand side instead difference auc across bootstrap bootstrap bootstrap contain auc difference variability combine set uncertainty train combine net net compare quantitative relationship past technique variety along successful application inspire win recent competition artificial multiple conduct recent dealing report superior structure study relate chemical property chemical mathematical could study desire without actual distribution numerous benefit drug generating chemical mathematical predict property encode chemical structure throughput screen ideal collecting training test generate molecular study encoding descriptor various chemical property interest machine matter prediction competition retrieve use descriptor nevertheless net allow win relative accuracy internal history apply regression neural forest projection pursuit partial machine successfully advantage partial making allow user able assess viewpoint inform assessment uncertainty model important controlling overfitte concern capacity requirement measure importance control layer selection reduce infinite network require cubic number case single issue past decade wide advantage deep network highly successful task vision along researcher train instead network hide unit neural layer thousand million parameter substantial weight unsupervise pre avoid overfitte fully mention differ variety net task operate multiple something literature aware neural net dropout overfitte access feature additionally different neural motivation behind relate task model particularly share task multi feature govern law broadly descriptor task various leverage recent development outline net multi lead baseline model feedforward artificial powerful reduction neural map repeat simple module internal layer feature useful value parameterized eq layer know output optimize standard output vector regression square appropriate train descent sgd momentum minibatch case use network train molecular descriptor activity neural regularize forest order neural architecture vector descriptor input separate unit compound multi neural net backpropagation unit whose compound case descriptor observe towards handle control training go minibatch case case replacement emphasize leverage call profile spirit net difference profile treat compound activity previously compound side descriptor prediction compound activity another net train deep network hide successful numerous notably vision speech recognition complicate capable extract input million thousand net date tend single unit wide deep hardware dominant net overfitte deep network recent network always perform one practitioner depth vice since architecture advance deep neural become capacity well train hide weight regularization wide net avoid model large expressive capable represent dependency descriptor activity regularization broadly subject recent unsupervised training powerful regularizer attention neural early base validation overfitte limited effectiveness little network high help overfitte sophisticated experience dropout network training effect add uncertain hide view dropout numerous neural net produce random net weight powerful avoid overfitte weight l net eight million avoid expert sequential ideally parsimonious evaluation construct function suggest acquisition exploration highly uncertainty job software implement particular version neural optimize range configuration overfitte validation datum neural investigate good statistical error train new perform closely neural net perform single counterpart relate one difference statistically closely net dataset leverage baseline surprising p minor variation particular nearly identical enough merge positive negative relate work nearly gain task single reflect gain task create testing multi net relate dataset multi net exhibit improvement combine second highlight display table single net statistically net since net model ignore capacity irrelevant task much well primary combine even make prediction net prediction treat primary task improve upon net often somewhat task benefit demonstrate sufficiently model still gain multi interesting arise positively correlated net negative r primary allow use prevent enable non weight penalty always dropout well lot include contain reduce dimension drastically although informative thousand net informative feature information gain descriptor typically produce unnecessary drop figure show auc representative descriptor descriptor effect bayesian run layer task second little auc consistent trend multi net deep although depth across neural net r layer c contradict molecular neural network unfortunately neither descriptor public cause discrepancy several likely depth multi task net train regression binary classification bit information effort try descriptor may exist data descriptor open descriptor software package could descriptor add result improve effective leverage potentially inactive decision virtual compound essential plan perform future work version bayesian software package practitioner long sophisticated network since small setting automatically way implement net well rapid progress advance team molecular activity neural minibatch backpropagation gradient objective net parameter objective current minibatch formula strength training list preliminary allow range
belong child node side split essential gps propose partitioning capturing design stationarity likelihood hyper gp regression express straightforward purpose marginal likelihood leaf leave aggregate level tree index start exclude return depth return list root tree associate leaf marginal eq implicitly assume node give factor along path equally importance leaf example decomposition depict inspire work correspond leaf path presentation empirical however straightforward approach infer hyper leaf markov monte leaf approach gps leaf ei acquisition function point ei efficiently gps leave employ iteration propose standard non objective propose comparison optimisation well optimisation approach mat ern gp obtain different benchmark discuss function first figure function exponential hand side whereas hand code please second synthetic precise flat without careful direct modelling paradigm wikipedia article parameter specify mini batch original restrict grid latent structure similarly tune experiment structure setting tolerance performance original dataset community optimisation approach available table exp svm mean deviation end iteration achieve achieve result krige modelling minimal observation krige closely relate acquire describe square south measure gold acquire observation surface make observation measure gold surface exhibit stationarity try highest please comprehensive krige construct optimisation try find historical record approach minimize early run fail advantage latter achieve converge global optimum evaluation exp exhibit fast illustrate shown structured svm converge eventually important lack state mining extraction significantly outperform deal depict figure optimisation flexible leave readily combine approach leave improvement robust range evaluate performance propose competition normal setting yield gain robustness box tuning machine finance mining optimisation optimisation expensive capture function single stationarity consequently stationarity optimisation automatic exploration optimisation minimum generally convex modal whose evaluation objective often via interactive environmental material network carlo experimental reinforcement inspire great automatically algorithm optimum objective optimisation setting whereby th round select return noise belief smoothness observation model describe tt derive acquisition decide next acquisition exploration introduction please refer aforementione tend address process introduce flexible adopt tree properly avoid split parameter observe improve brief maximum acquisition ei remain bayesian optimisation package ei close represent density improvement family one heuristic member project input extend dimension try directly covariance stationarity optimisation non stationarity input cdf
transmission subspace principal equal compare exist system dimensionality reduction initialize observe however distribute excellent close fast estimation htb pt paper distribute wireless network result exist scenario require transmission propose distribute reduce adaptive distribute perform dimensionality follow dimension distribute reduce develop communication overhead improve exist advantage strategy reduction distribute reduce sensor distribute strategy fundamental wireless network grid specific neighbor combine require communication dimensionality order reduction kalman consensus costly processing naturally context reduce technique dimensionality spectrum interference research exploit estimation attribute employ meet reduce estimation joint iterative normalize least rank reduce perform decomposition agent information low neighbor dimensionality reduction dimension outperform compete letter inverse operator complex wireless limited capability topology protocol incremental consensus base neighbor topology fully connect instant measurement accord noise variance measurement wireless fashion operator possible technique diffusion set denote cardinality coefficient eq neighbor reduce processing network strategy alternate technique htb depict depend strategy section reduce unlike auto perform decomposition process costly flexible low cost fast jointly fashion method lagrange consider arrive lagrange frobenius part set describe ki ki ki alternate instant adaptation perform adaptation step instant node start reduce ki ki ki ki neighbor node keep instant adaptation combine rank neighbor estimator conclusion reduce estimator propose htb instant n ki ki ie ki ki ki ki reduce estimator send neighbor node topology ki ie ki update keep ki kl li estimator keep locally final node ki ki ki ki
assimilation formulae framework instance correction say kalman among gain instance replace reduce reflect couple h h h assimilation ms l hereafter cyclic z I system variable divide estimation sub role call term mode plot ms l numerically integrate state collect brevity call integration discard divide estimation assimilation step trajectory synthetic gaussian white state variable integration therefore step assimilation ensemble ensemble state assimilation extra divide achieve treat vice versa parametrization motivation usefulness assimilation combine increase stress running estimation scenario divide assimilation ds joint ds da divide divide ds stand assimilation ds whole da adopt assimilation fig ds da divide start couple split sub slow mode mark act line dot incoming ensemble update counterpart describe ensemble forward assimilation conduct plain introduce covariance localization covariance localization adopt initial square investigate framework analysis update identical experiment ensemble observation repetition std state analysis carry report mainly precision computation depict series scenario ds ds joint da divide ds divide divide da assimilation ensemble scenario extra parametrization ds divide scenario therefore panel framework early assimilation become substantial meanwhile ds divide panel ds da joint mean panel panel panel extra parametrization always panel appear low scenario possible view forecast challenge analytic da joint da ds da ds da adopt left column characterize difference work adopt subtract trajectory da joint da divide divide da joint ds divide divide instant ease visualization box step stand scalar grow final increment set matlab band inside box represent end mark individually fig divide scenario trajectory assimilation box appear time move ds da gradually indicate time remain similar da joint da divide except period short scenario also histogram difference whole assimilation window th trajectory state ds joint divide ds divide da joint ds divide da da divide histogram difference peak support peak interval instead phenomena divide da ds da scenario peak tend tend wide fig seem suggest ds da ds divide ds da da substantially reference localization important auxiliary enkf enkf arise systematic covariance increase explanation extra numerical additive model always bad work introduce dynamical context alternative artificial noise e research assimilation development residual appear sufficient conduct originally covariance localization method conduct localization see localization locate localization localization degree half ms choose divide framework component circumstance localization aforementione average assimilation covariance seem estimation experimental hybrid filter relatively indicate scenario divide estimation framework close joint separately filter performance section framework largely challenge assimilation model illustrate ds divide divide extension possibility ensemble size fast slow mode want gain member mode ensemble size apply filter formulae size address discuss material performance plain factor mode filter tend plain take clear ensemble ensemble mode impact reduce dominate l comparing seem well simply fast mode ensemble fast mode dominant extra error scheme significant fig filter plain plain seem system couple exhibit background window could reasonable assimilation scheme slow due implementation significant interpolation enkf background historical enkf appear attractive divide incorporate largely motivated status challenge operational couple computational resource implementation sophisticated scheme enkf combine assimilation divide mode mean analysis parameter slow forward update eqs use historical ensemble eqs slow parameter numerical fast mode background ensemble propagation ensemble assimilation cycle slow drawing specify whose equal historical assimilation available historical fast covariance historical ensemble mode generate historical ensemble mode step take fig fast mode dominate magnitude fast slow distinction hereafter refer assimilation da divide plain set neither set mode divide localization adopt magnitude trajectory da da interval covariance divide localization divide around relative da da divide incorporate divide assimilation problem couple system tackle formulae consider sub separately bring efficiency assimilation scale assimilation combine option divide sub framework addition possible may couple assimilation background precision extra circumstance assimilation one reasonable current work service datum assimilation couple balance generation additional challenge assimilation extend study include limited complement light mathematical equivalence exist example development tackle aforementione similar divide couple extension interaction domain assimilation conduct scenario kalman formulae complicate adopt divided topic investigate future would constructive comment improve presentation science technology also like realistic research financial step expand matrix side equality line h h combine eqs algebra omit summarize chart ds da divide system eps std eps divide framework single repeat background ensemble change ensemble observation panel value repetition divide framework state variable deviation std eps eps eps eps scenarios panel plot subtract difference due numerical accumulated eventually become c error ds indistinguishable integration trajectory ds da ds da divide eps trajectory ds da histogram da eps eps histogram ds divide da eps histogram difference joint da divide ds da ds divide scenario reference da rmse delta lc vary lc delta eps rmse scenario localization adopt eps l eps l eps eps slow plain inf eps da panel localization apply panel mail edu sa consider assimilation couple interact certain way tackle assimilation sub system ensemble kalman filter enkf assimilation system quantity system
variable clear operator close keep recognize discover relation discover appropriate continue expand linguistic horizon syntactic primary search semantic end language lexical appropriate truly need structure sophisticated understanding seems take theory summarize appendix amenable linguistic aspect seem automated aspect structure aim automatically aside relatively effort topic categorization extremely recursive early syntactic level relatively abstract structure yet clear stage stage proceed aspect reality external world arguably lexical lexical name example lexical specify word magnitude lexical lexical semantic mean lexical thousand learn external object refer word also refer observer purely linguistic say entire singular oppose every time structure require world model word likewise many actor properly semantic capture partition say structure relation direct partition entire devoted decision statement partition sophisticated extract appear already identify thing read sentence apparent deep structure text like build also relation syntactic clean enough abstract unclear language reality act parse external checking reasoning consistency closure say external indicate may particular relation rate observation stop language turn something else something else hard external reasoning external align human construct external visual associated resolve word regularity external name relationship order syntactic create mechanism external world language language overall part aspect linguistic part decade item learn e background demonstrate existence far propose create less final working formalize mathematic remain regard suggest distribution prior treat fairly goal probability evenly minimize broad phenomenon entropy seem theoretical clearly correctly thus room ad hoc cut good coupling mathematical development abstract conceptual depict language general structural entity word entity describe section mutual example frequency unique name require work treat constraint make suggest list short description desire way group actual task grouping grouping distance similar thing grouping entropy cut list item add relation lead discovery proximity another know aspect language armed perhaps become iteration become visible clear know already new deviation candidate relation thus sort compose collection linguistic relationship linguistic entity reasonably compactly pattern observable linguistic property linguistic linguistic construct may find via linguistic construct considerable aid linguistic come language linguistic imply corpus datum alone linguistic relationship linguistic entity purely well language learn seem likely accurately corpus far special describe elsewhere key intelligence pattern token symbol part knowledge part thing represent symbolic token easily pattern component corpus usage linguistic linguistic factor come corpus merely particular usage pattern principle create system loop bias toward particular usage valuable break multiple instance loop focus couple learning loop syntactic linguistic lexical provide high linguistic relationship relationship lexical output result semantic loop feedback regard correctness extract confident result loop confident interpretation loop loop attack sort slow issue dag syntactic associate dag represent sentence raw tree feed input layer sufficiently corpus understand maximization example entropy word fair mutual entropy pair search structure closely resemble dependency show pair mutual mi tree unique unique connect typical modern language million link type viewpoint link link automate discovery something speech pair mi reveal follow mi previously mi obvious word might correctness grouping relatively straight act reinforcement correctness group importantly discovery come rule million unique pair small link connect define logarithm total complexity rule plausible mind maintain million hundred foundation type example noun type mn short name np foundation deal thing discover text relation also theory inherent part language theory appear discuss dictionary list perform span entirely two trivially indicator appear many sentence count exercise often mutual lexical another structure discover subgraph instead somewhat occurrence frequency condition logarithm probability entropy exercise mi appendix formula word precise proper rare linguistic less syntactic group common lexical entry part ask grouping grouping perhaps level previously perhaps perhaps back feedback step early refinement trial recognize direct application observation pair essentially identical flexibility language widely variety situation mutual carry poor english case appear relate term link link apply situation discover limit bit challenge manually maintain dictionary construction english flexible rule certainly pair insufficient case challenge occur primarily construction flexibility human sentence phrase case avoid work child syntactic describe way get restriction change implementation overall core language syntactic context consider triple frequency count mutual particular order link contain come link link parse version syntactic mi word initial word link link word category base usage likely classical use single syntactic large usage link category usage word category create syntactic link remove pair link associate link usage link exist link plus syntactic infer category link modify noun link subject possibly link particle head syntactic sentence infer logarithm chain syntactic type may indicate previous clustering return link goal link category contain one link inefficient language cluster relatively maximize maximize assignment large english hundred type unclear sort variant obtain implement basic purely syntactic integrate loop relationship generate word usage way versus link set syntactic far carry soon carry parse word parse reference corpus rank acyclic dag usually syntactic label different prove parse entropy generate span regard parse actual one probability link agreement cause issue one linkage existence assignment incomplete fail link sentence essence parse parse feedback refined treatment phrase boundary handle embed price list chapter design engineering challenge effort syntactic come semantic relationship close syntactic separate heavily influence heavily system map link semantic prototype consistent semantic network formalism implement code spirit proposal however assume mechanism linguistic content via corpus learning code specifically suggest discovery semantic proceed manner discovery relation describe unsupervised neighbor extract way employ phrase save offer candidate sentence syntactic candidate sentence relation syntactic construction carry comparison make string issue alignment parse establish context subgraphs essence recognize challenge recognize challenge relation understand lambda expression employ place partly trick syntactic construction atom form term algebra probability term notion term network field mathematical formalism graphical range domain generalize distribution evenly prior maximize general drawback abstract dense fall np much simple come lack proper generalize generalize instead get metric semantic similarity np hard maximum rapidly convergent extract construction semantic neither distinguish word relation sentence necessarily justify meaning semantic lack mention neighbor across appear strongly nearby word solve chain appeal affinity word neither language similarity instead strongly grained observable hard detect phrase lexical thus word structure reasonable lexical prove otherwise phrase different measure discover measure text occur time sequence occurrence meaning thus word co occur appearance essence word mutual word expand word occur word slope phrase eventually failure carry standard second usage different occurrence frequent word get rapid sigmoid though semantic variety fuzzy robust structure need one discover similarity mean markovian count likelihood unfortunately formula poor ability discriminate distinguish receiver operating word co trick come embed network completely ignore markovian sigmoid sigmoid serve single importance essence sigmoid say forward discrimination counter probability discrimination learn speed incorrect behavior convert build effort thing useful yes describe build approximate unlikely correction discard handle phrase discriminate semantic apply syntactic approach mind require refinement learn semantic outline corpus syntactic early corpus frequent statistically high mi subgraph occur corpus subgraph word allow subgraph syntactic semantic annotated semantic context combination standard subject sentence form word instance subgraph set base division similarity fuzzy degree fuzzy web involve platform far nonempty determine involve assessment associate intersection find graph association instance corpus syntactic parsing pressure reduce complexity counting summing rule relation class content language exercise entropy away relation never occur leave use express fact semantic corpus category link type point link point sophisticated method assign worth category appear graph symbol recognition loop return step newly semantic noted semantic relationship use syntactic understanding way include part use syntactic formation parse semantic resemble deep level make another link know refine relation build syntactic semantic structure basic derive else build give syntactic semantic bootstrappe need semantic text relationship text gradually instance achieve read corpus build linguistic able together conversely layer guide learn low layer seek associate eventually external persistent object action labeling provide foundation semantics encode semantic reverse later unclear direction demonstrate certainly description nonetheless believe mathematical advanced author idea prototype hundred detailed publish current overall project focus direction explore language language analysis comprehensive text mean text although primarily orient I believe meaning dependency think rule algorithmic implementation lexical core linguistic linguistic semantic dependency primitive atomic thus definition specify role essence entry mean theory mean medium compare properly semantic capture mean term medium medium assignment thus medium distinguish pre mean example medium roughly distinction pre suppose information thing fine specifie style semantic network text speech english concrete object external refer person partitioning topic say medium word increase pt form semantic also syntactic roughly dependency like representation sentence structure capture correspondence translate structure another rule think attempt treat implementation discovery lf rule lexical meaning lf normally lf specify noun lf broad lf narrow scope subject subject noun lexical sampling strongly stop half reason lexical like comprehensive view lexical unlike describe mechanism syntactic description frame aspect mean capture semantic lexical hierarchy raw sure take place style reinforcement must layer syntactic layer somewhat develop semantic layer guide seem appropriate framework must comprehensive describe transform language must lexical pick structure pre viewpoint learn treat learn black able probability associate relation simple work mutual generalization typically linkage type leave right observe distribution observe relation relation fact word certain ambient perhaps nearby word commonly position matter notation observe quite since within less observe word pair almost give give mi associate example subject properly frequent pr large unconditional pair entropy relation obtain simple working relation language information structural relationship notation hard rhs identical appear summation difference singleton likewise singleton mutual scale pattern may rare useful mutual conclusion conjecture exercise fully corpus build approach enable needed generation natural regard explicit code linguistic annotate corpora approach fully satisfactory substantial formalize human annotation principle coded corpus nlp linguistic content operate high abstract parse set address semantic content practice yield full success semantic means discuss scientific community regard linguistic text video ambient idea art automatic third list decade address linguistic issue difficulty arise implementation notable build phrase essentially point equivalent progress mention yet fair say truly progress automate linguistic content corpus useful incorporation system content create code annotate corpora linguistic large text generalization author believe body idea enable corpus gradually decade aspect aspect validate simplicity deal text bring complexity purpose document syntactic handle speech similarly closely finally stress intend conjunction corpus processing unlikely happen syntactic learning guide exactly corpus wikipedia might web certainly devoted provide rather basic rather getting lose important thing class thing step e return correlation almost mutual information strength thing thing hypergraph early iteration thing pair measure accomplish primarily principle word b group word class w hold form take word actual consider complex detailed considerable thing environment able thing pattern probably pattern piece piece final picture match difficulty may pattern input language immediately pattern inter suffice viterbi parse relationship say relationship become essentially combination keep likely really affect minute complex pattern sentence together coherent fashion recursion learning principle previous tendency relationship describe minimization complexity tend increase logarithm logarithmic discover rule confidence phrase book group noun phrase suggest book odd noun incorrectly classify place form deep serve refine correctness rule outline capable lack collection unsupervise fashion remainder devote might outline begin linguistic content characterize ai concept language linguistic approach drastically linguistic assume comparison convert dependency easier verify head word word word image learn syntactic relation normally write abstract fundamentally lexical abstraction abstraction low level parse string word dependency relationship extract case impose structural relation noun point linguistic terminology actual require inherently linguistic discover structure believe outside point system learn assume key list list learn previous abstract view merely back concrete task list link link connect sentence head english link
give exchange action action calculus present causal observational expand structure miss check identify calculus effect observational imputation miss flexible observational causal effect predict causal assume know utilize causal calculus systematically apply standard way statistical available validation fitting observational combine carry summation actual demonstrate dataset use effect strongly nonlinear observational differ causal setup far miss aim structural distribution figure causal effect equation include convenience represent represent differently data eq indicator response completely observe indicator variable cumulative likely value small causal causal index variable indicator individual select sample record mark solid marked circle covariate response indicate missing case joint association covariate associate response observational observational due combination value response path latent child calculus average eq derivation detail mean estimate causal effect estimate observational must address biased design figure datum miss random additive necessarily model suffice causal flexible ready make software exist message work require causal complicated miss external need causal care scientific make present encourage causal package utilize fold tune smoothing estimate perform situation different causal high value around zero replace convenience imputation handle miss causal causal b mechanism form datum causal causal causal causal express association adjustment needed calculate mechanism nonlinear density normal normal skewness contain observation see model directly overcome distribution assume joint allow causal effect decrease panel datum illustration difference observational evident integrate method generalize additive spline smoothing forest layer integration summation forest carry package addition utilize parameter fold parameter smooth hide causal see network fail effect forest design number predictor smooth restriction forest scenario cm major advance take causality still effect estimate observational causal challenging tool model causal imputation estimate causal tool keyword analysis miss nonlinearity structural equation major advance take despite still causality observational causality lead problem consequence effect causal relationship dependency see observational instead causality alone estimation effect suffice whether affect affect effect obtain information straightforward causal easily physics causal calculus systematic way effect concept causality clear practically definition whether specify causal completeness causal prove causal fourth framework deal design bias little attention causal causal use consider causal restrictive flexible framework concentrate narrow theory practical present start causal causal qualitatively nonparametric form
element ols generalize relate ridge dimensional elimination beyond broad create several create sensitive column paper demonstrate square various quantity ols ridge regression full rank term instead study estimation return ridge linear motivate function satisfy full define ols estimator define negative zero soft section dimensional svd orthonormal become orthonormal matrix decomposition multiply incorrect basis multiplying estimator correct contain lasso showed bound fix lasso relate relationship equation proof contain great classical uncertainty I penalty term nan hypothese model cdf package normalize equally variable column length heterogeneous length strength ensure leave right q define element pn jj jj exist inference carry select fitting ol test statistic value define cdf algebraic require matrix reliability p theorems regular penalty objective function control let let minimizer row determine statistically care change high converge theorem hold analogy backward elimination analogy backward g resemble forward selection sparse x sparse around yield moreover even minima long ii compute identifiable distinguish function paper lasso paper connect type ol informative p value suggest classical suggest two statistically backward procedure multiple ol remain neither create multi match become complicated forward classical model weakly sign backward elimination trivially satisfy g elimination procedure consistent procedure become ol incorrect hold define orthonormal orthonormal ol n ol ol element statistic define
logical structure notation object dimension notation quantitative fit qualitative collect metric boolean represent alternative boolean derive degree categorical invariance transformation wide human logical category structure improve domain describe learn structure instead take roughly column category task list first familiar whether particular category symmetry reader appear elsewhere stimulus meaningful slight consider fit metric depict table comparable boolean complexity well boolean comparison exist conclude human conceptual reflect mathematical complexity conceptual difficulty information complexity shannon entropy dimension human paradigm dimension general learner child set mathematical domain reveal connection dimension separable learner able yield learner predict metric experiment learn effectively mathematical predict behavior question cognitive difficulty concept six category call paradigm occur human learner characteristic shape order occur human classify characteristic readily g child categorization identification paradigm mathematically measure logical complexity logical rule order difficulty amount uncertainty remain subset base shannon metric minimal uncertainty predict canonical six type metric uncertainty order type six uncertainty predict well exist complexity canonical learner test sized problem type instrumental evaluation category distinguished rule category type distinguish accord type order particular pattern type learn speed order exist literature series reveal condition encourage formation researcher associate set across wide entirely different specifically separate order separate ordering stimulus theory mistake pairwise comprise analyze distinguish order reach circumstance well classify separable acknowledge reader seem familiar detail shown fully match generalization predict ease unless attention human learner match characteristic order type modify type report child type test difficult significant child accuracy increasingly type child explanation category implement trial trial learning explain paradigm specific order use formal metric logical specific put metric account hand operator shannon subset specify provide paradigm specific learner abstraction attention regard hand correspondingly learner unable logic detail among observable I heuristic calculate theory ability paradigm well section classification boolean successfully predict metric logical shannon provide metric explain component description system system categorization boolean type logical kolmogorov algorithmic short program logical describe categorization boolean begin describe dark circle dark heuristic eliminate remain logical also incorporate selective stand structure category ignore appear consider consider object category natural essence easier similar boolean characterize amount shannon information observer observation shannon explain shannon example fair coin shannon entropy random variable finitely self weighted outcome interpret observation mean suppose therefore interpret uncertainty valid approach probability surprising occur completely uninformative sure happen measure maximize event probable symmetry make event event make likely extreme event event total probability ignore mathematically calculus event maximize coin flip coin outcome tail mention fair coin bit coin bit information coin tail yield information coin less flip unlikely maximally coin whose certain shannon fact hx shannon mention encodes dimension specify uncertainty categorization way aggregate dimension formally classification formulate category function aggregated uncertainty specify consider partition specify partition let example partition second demonstrate subset element stimulus entry particular consider partition fall three uncertainty entry vary determined function amount uncertainty formally function uncertainty entry second q function produce relevant overall uncertainty suggest denote two represent element equally clearly exceed value singleton precisely whole observe uniquely category could categorization category uncertainty completeness metric aggregate measure classification metric boolean type ii depict define type two category imply categorization boolean categorization boolean complexity begin maximally reduce length claim digit digit represent claim claim represent element translate confirm describe type boolean simply compact total evaluate categorization consider ignore bind dimension call proportion consider dimension ignore stimulus category invariant proportion produce proportion proportion least transform structural manifold metric square functional order preserve capture essence case information complexity consider dimension denote dimension say zero stimulus category first maximal therefore uncertainty uncertainty dimension know stimulus category dimension zero third category uncertainty second third uncertainty associate trivial minimized divide subset subset stimulus maximize subset aggregation reveal version reveal paradigm assume yield way divide dimension difference obviously idea good level divide stimulus two dimension min contrast opposite ignore remain object look far though twice information structural reveal search degree operator naturally capture might boolean yield boolean rooted finding complicated category find expression boolean start maximally redundant category redundant could minimum path minimum would sensitive natural analog order setting rely notion play role consider task comparison collect paradigm subsection
feasible connect precede insight prove generalization theorem alternative p building sec strictly course normalize perceptron von connect normalize perceptron duality paper margin determine affine lin coefficient margin key quantity easily intuitively rank lin lin dot matter easily margin zero difficulty feasible see search restrict lin quantity clarity feasible distinction really think unnecessary see often elementary simple unfortunately behaviour unlike conv strictly vector maintain feasibility change amount lin orthogonal lin zero vector lin product lin jump instability margin von relate many know interpretation feasibility center dual cone dual cone cone interpretation conv hold zero closely particular connect margin characterize relationship ball conv exercise return note perceptron yield iterate center origin conv zero sequence end popular learn connection margin scope margin central quantity version radius euclidean center origin mathematically leave lin infinite dimensional space occur dimensional ball hull thing ball span column interior full ball expect conv read large affine ball inside overall matrix brevity highlight conditioning feasibility far reach radius latter singular analogous feasibility ill pose normalize margin perturbation row ill make quantitative alternative negative seem literature derive mathematically spirit precede intuition extremely might proposition seem hope generalization interpretation particular prove definition previous capture proposition similarly must spirit statement equivalently follow either w note one entire often characterize iterate example prove prove multiplier equilibria game name whose follow alternate notice one surprisingly spend simplify proof geometric insight brevity strength measure crucially distribution define substitution see precede omit multiple whose interpretation feasibility govern mistake proof prove angle often insight simplify amenable involve duality primal machine perceptron subgradient mistake track property iteration feasible step find infeasible iterate satisfy interesting feasible margin margin true perceptron variant prove normalization margin subgradient minimize interpret unit unit respect yield argue iterate towards optimum require perceptron step also elementary open surprising singular value similarity communication latter publish go von circle independently identical go geometry von conv say loop von produce establish von though primal like perceptron prove step question special private frank wolfe light problem see connection duality subgradient frank wolfe finding solution converge linearly strict nevertheless perceptron dual mistake machine descent coordinate ascent relevant margin correctness affine relation generalization theorem statement precede tool remarkable theorem explicitly affine margin compare classical iterative turn spend simplify presentation though behind contribution lead final clear clear realize straightforward margin recently universal depend tie take part could round simplify reach remarkably geometric intuition margin duality claim intuition impossible algebraic generalization lastly claim perceptron surprise provide classical seminal theorem rate theorem proof strategy array usefulness lastly classification integral analytical algorithmic idea behind margin choose aa g aa b
inf mh mask blue value show sampler perform poorly fail completely inform improve sampling cluster cell tailor discriminative lead inference proposal probabilistic set future vision identify dependence recently accurate generative allow scene yet trend world idea inform proposal manually readily vision domain sampling reversible jump mcmc method investigate describe manuscript believe computer graphic current efficiency towards inform heuristic technique principle rich generative emphasis aim create scenario basis inversion involve plan computer principle generative model aim adapt proposal exist straight would proposal draw identify time markov accept mh value independent accept use discard replace inform fit exist refer detail technique monte carlo qualitative informed inf still challenge already inform mesh posterior obtain inform sampling inf mh informed indicate red page analysis sampler low sampler individually difference acceptance plain standard update converge high modal take different chain sampler result temperature perform mix acceptance inform inf mh choose combination various mh temperature combination pt inform result mh result poor proposal ar standard deviation four compare plain optimal plain optimal inform inf choose acceptance deviation find proposal show figure select ar value pt coefficient inform inf mh hard variability texture accurately formation belief posterior explain intuitively largely fail difficulty posterior efficient believe usefulness generative vision exist even principled concentrate invert exist graphic engine graphic inform discriminative technology improvement conceptually generative physical formation interest presence position nuisance light generative think deterministic image observation prior inference generative fail variable reference frame bad generative use intuitive fair track record generative vision use heuristic objective function problem design inference therein computer leverage dedicate hardware system generative modelling motivate research inference computer graphic world test stem reason dependency modal forward process prevent exhaustive enumeration believe usefulness generative task argue overcome substantial challenge devise allow different novel scenario want maintain correctness efficiency leverage aid paper markov proposal instance drive method tailor informed sampler vision feature make informed proposal latent accept reject inform implement model sampler incorporate object ill carefully assess sampler investigate probabilistic estimate exist inform sampler library produce inform likewise inform sampler present baseline method experimental inform diverse reasoning estimate body conclude future stand vision graphic build vast computer vision generative mention graphic scene understanding pose pose many infer spirit use segmentation human pose estimation sampler highly domain yet technique believe task idea graphic understand root computer graphic inverse goal category formulate convolution understand deconvolution pose specification generative modelling try also program module program formulate hasting mh appeal graphic plain inform challenge another piece apply bayesian devise show multi invert code mention paper posterior challenge make especially correspond graphic despite apparent observe vision exist distribution interested availability infer accurate offline stage computationally accelerate time metropolis goal particular instance sequence markov repeat step propose accept mh technique differ proposal hasting state image inform stage parametric proposal mh valid call inf move markov inf mh ideally global move local proposal responsible density locally proposal every accept enough acceptance rapidly process density estimation cluster observe unconditional kde choose solve diverse kernel detail kde transition random forest approach map observation simulate fit initialize representation discriminative heuristic invariance nuisance main method across region technique summarize test advantage give need identify efficiently kernel reversible metropolis hasting combine hence reversible ergodic distribution ensure remainder demonstrate three initialize sample except note center metropolis sampler draw dimensional propose move chain modal distribution technique replica exchange chain propose chain work individually high temperature implement adapt initialize fit kde already mixture size valid find sampler inf mh mode sampler proposal inf initially hold dominate mh indicate move slow sampler mode discover mode inf even inform inf due acceptance sampler acceptance sampler h camera also test find variable fast enough median separately camera inform inf inf modal well inf inf mh mh experiment orientation image add bit look image cube location orientation angle label switch color choose resemble leave scene readily solve source previous increase proposal sampler inform propose jointly product distribution single proposal scheme sampler inf square boundary obtain inf discard feature cluster kde observe image kde determine inf empirically inf find inf sampler fail proposal state sampler acceptance around follow separately reasonable guess fail informed inf rate median inf produce localize uncertain show relatively baseline sampler crucial enable heuristic library experiment inform simple baseline sampler inform sampler improve speed baseline produce fit fast estimate single sensor vision human body produce mesh body allow size characterization roughly mesh human pose angle predefine part use parameter mesh component person camera viewpoint orientation pose hold generate mesh representation virtual camera create image person choose take gaussian create depth learn proposal record value height feature normalization feature learn kde cell forest kde discriminative try require reliably forest adaptively uninformative adaptive mean tree score tree minimum leaf kde train time regression proposal place kde cluster example observation overcome curse semi capture explicit parametric linear dependency combine explore visualize angular error ground mesh individually inform analysis sampler include supplementary material test approach standard normal summarize method baseline mh inferior mh low rate inf low acceptance decrease rmse regression inf mh rare
di take choice counting intensity ff ai mark spatially nf turning whereby usual must case may cl absolutely last reference l control mark density take form treat spatial cox marked indicate additionally assign mark auxiliary mark control former process mark connect naturally stochastic forward stationary mark cox intensity mark point small intensity correlation marks cox setup develop affect employ cox dependence mark may idea ground intensity mark note fall category mark discussion mark intensity next turn consider construction spatio look marked intensity look particular define temporal marked cox benchmark locally intensity bound intensity I intensity whereby correlation functional stationary due randomly label next relax slightly specifically process constitute poisson intensity whereby correlation functional g g l al f g mark cox process spatial recall ground locally direct constitute cox conditionally conditionally poisson possibly finite negative locally cox write cox cox whereby cox spatio spatio xt may connect simultaneously way drive random intensity mark translate definition process intensity conditionally mark density extremely development tool kind development recall definition temporal mark order cumulative e construction ground may mark present behaviour n I n ng g assume derivative refer regular write intensity nx x I analogous derive product constitute natural gain whereby component stage f b define c b monotonically locally become respect call mark enough specify accommodate purpose construction martingale intensity adapt process write order relation pd assume derivative coincide except set integral ground l ff sx dx conditional respect tx statement far whole consequently exist l f radius note extension since start recall construct constitute compound process note construction location dependent markovian mark bit mark note also sm simultaneous evolution mark underlie evolution markovian mark pm tt ts give n assume reference measure ng ideal mark mark else point density recall quickly density fundamental exactly specified location see imply sense regular mark give define ng l totally finite treat construction equivalent probability pn measure n multivariate point recall auxiliary imply f f finite intensity al support whereby define f fp mn f g ff I density way current context density class g g l depend measurable interaction markov may conditional intensity intensity exploit formula exact convenience possess spatio g process pt ks markov define mark situation characteristic functional mark functional mark multivariate mark form define spatio I expression play role since sample accommodate yx functional mark reference say brownian motion calculate explicitly density sample factorial moment n yu k derivative absolutely continuous exist assume imply consequence correlation g u g turn intensity expression x df sx f ff df sx well refer marked intensity fashion may mark u l finding point locate outside spatial auxiliary mark conditionally absolutely respect turn absolutely consequently entity lemma turn mark mark wish essentially mark spatio mark point x l nu u grind good account quick could proceed spatially see n l u x dt l sx g sx dt process function kx kx l n u f j respect markov pseudo ground auxiliary suitable conditionally give ft nn mark accord slight abuse notation dt kt dt mark influence outside approach simulate miss datum family ft n exist density nn product spatio sample mark certain situation however argue mark spatio ground process possibly auxiliary mark able evaluation l correspond treat observable time analogous scenario stand birth radius height location retain function reflect nature th remove marked density need mark find find kf xt I xt x l stand time tree acknowledgement author truly grateful idea author grateful technology u discussion education corollary remark gb point spatio temporal mark c mark spatio process indicate sensible connect field spatio boolean mark cox double connection discuss purely intensity tool estimation intensity mark reduce measure pair field spatio temporal functional marked point spatio spatio mark wiener treat arise phenomenon mine naturally table application represent occurrence event death incidence review mid th interest expand point point event volume emphasis spatial process methodology range field distinction area absolute coverage cite reference temporal distributed point name associate model history david cox attention spatio temporal location event origin incidence relationship worth traditionally view principle vice versa instance times though consist zero one represent process point interval see analyse contain mid spatial process spatio regard hence special spatially process lot spatio temporal ad hoc approach spatially naturally binary number event conversely spatially multivariate process common spatially sufficiently justify explicitly spatio mark carry mark variable several variable type mark mark mark interest spatial size shape treat mark concern analysis marked pattern analyse mark conditionally mark mark alternatively treat mark case mark value concept refer process one among know example able density local mark situation mark field field determine whether technique investigate interpretation mark mark conditional variance mark another serve mark another generate process poisson depend mark point history analyse function qualitative quantitative collective tree location stand moreover point community form spatial location local characteristic approach classify discriminate belong clutter belong configuration permit analyse curve depend describe model curve observation number theory theoretical analyse curve analyse develop temperature year analysis none new characteristic process number derive approach spatio temporal mark propose new spatio point mark generality ability accommodate different structure new notion spatio hence provide framework framework mark interpretation connect spatio intensity significant practical aspect structure give thorough section new c functional temporal interpretation motivate spatio section characteristic intensity need development section discuss certain within characteristic cox c mark process mark type spatio version first collection euclidean location value mark I e variable control type random event describe note extra start define continue space underlie inherent spatio temporal filter dx complete borel borel denote identical lebesgue denote measurable dimension lebesgue indicator cardinality context measurable dirac singleton sometimes dirac short understand spatio functional mark shall spatio marked process mark point ground mark turn subset construct process location common compact identifying construct spatio temporal interval point temporal occurrence g birth auxiliary auxiliary may possibly type let auxiliary mark mark euclidean metric consider whereby propose space borel functional marked array thing growth dependence choose allow mark take see space continuous limit function function metric ff ft du borel denote accordance element path process evy empirical f ft bc rd bc f detail c stochastic diffusion path space supremum set supremum supremum b equivalent metric differently temporal point purpose preferable scale may time whenever cross context g forest stand pn xt construction framework application surprisingly ordinary marked spatio temporal x give mark provide similarly I x spatio type mark slightly create mark temporal process constitute marked mark come spatio temporal random xt spatio z xt I nm ti field k belong hilbert g follow model residual mean variance location z jt th tx x jt stationarity xt jt clarity prefer expression follow call describe spatial among across entire scenario one spatio spatio field monitoring location framework constitute consequently framework incorporate deterministic associate section hence obtain exploratory analytic indicator examine pattern relate point regard mark covariance structure live pattern evolve spatio idea local spatio surface spatio surface dimension surface functional mark surface provide spatio structure rise substantial idea extensively mainly context mark process birth death spatial auxiliary holding time distribute conditionally mark ordinary dt gm j l jt individual growth absence individual hx j jt spatial interaction find mention reference application collective stand birth breast growth radial scale add mark eq independent diffusion coefficient would measurement simplify interaction mention wide scope root provide shape first factorial density start set ai density permutation invariant measurable functional partly section region mark dl measure first factorial I definition intensity functional refer note ng find observation probability product exploit assume g n n n np np n whereby n space absolute continuity whereby e find express density give factorial measure regular probability exist intensity functional turn underlie conditional existence whereby mark n stochastic process absolutely continuous I version furthermore auxiliary mark x ff nf valuable dependence spatial see play light q correlation spatio pair g extend measure r r family q functional fix write expectation interpret probability event reduce counterpart relation give fp accordance choice define
conclusion table incorporate trust boost performance recommendation accuracy demonstrate advantage trust aware recommendation social performance significantly discuss beneficial relation relation type relation advantage utilize trust look notice incorporation trust neighborhood base error significant social trust know publicly include trust consistency relation work particular examine correlation metric trust seem insufficient user generalize item works practice preliminary hinge optimization viewpoint smoothness interesting direction gain accuracy corollary network recommender crucial success service due application need preference despite recommender suffer exploit social relation along rating recommender stem significantly influence user web trust list user friend base respectively incorporate information contrast incorporation recommendation potential explicitly incorporate almost paper social incorporate trust relationship quality recommendation trust enhance enhanced counterpart respect thereby demonstrate incorporation recommender huge increasingly cope relevant one recommendation take account history interest network enhance obvious system success online amazon netflix guide goal recommender book news web interest widely broadly cb cf hybrid cb recommendation try give past external item description profile extract analyze recommendation cf recommendation popular method recommender system assumption user express rating past access cb propose cb essence lie neighborhood user user recommendation preference user similar cf recommender recommender representation item user main also cf consider combine model type accurate recommender system heart similarity orient similarity orient cf orient oriented overcome oriented cf usually profile behavior seek orient additional wise also collaborative filter method rating memory collaborative express rating user perform user rating effective rating user rating base correctly decrease recommender major based suffer scalability reason application search identify computationally scalability issue issue overcome limitation observe interpret predict past use predict unseen cf employ datum recommendation learn category aspect bayesian due handle huge method low factorization nonnegative pmf small item rating recommender although recommendation scenario fail rate item instance accord non miss become challenge sparsity problem recommendation prominent tackle problem media application allow interact video page group idea significantly influence connect user share interest life application accumulate would utilize goal recommender recommendation historical user trust user social friend enhance become great increase availability review many review product social trust trust review find online community million comment capable friend site share condition online review social website require trust person user come role product influence confirm exploit recommender long fundamental base recommender recommendation friend probably goal rating trust boost sparsity issue trust among online trust pass one member another social trust recommendation social relation regularization trust trust recommender last user trust relationship relationship also quality later integrate reflect word whose review inaccurate low therefore utilize boost recommender contrast trust great research disadvantage explicitly utilize recently attempt relation recommender system proper incorporation social recommender system prove manner consistent plausible naive modeling trust raise challenge incorporate give challenge involve enhanced recommendation particular trust relationship recommender increase knowledge work model relation trust time intuition behind one interpret relation user preference rating must incorporate deviation latent combine user effectively preliminary experimental demonstrate deviation way incorporate propose facilitate feature user minimum dissimilarity formulation agree item friend friend predefine recommender trust relation enhance friend factorization factorization algorithm incorporation trust system base recommender leverage type propose social network empirical investigation rating particular examine extent align rating exhaustive propose advantage detail trust enhance recommender system put work social recommender system formally incorporate trust discuss include demonstrate generate accurate discuss research recommender system directly enhanced recommendation successful past recommender enhance approach recommendation trust review major enhanced recommender explore user recommendation aggregate trust neighbor social trust aware recommender collaborative inform trust utilize social connection social annotation recommendation construct walk perform ed trust trust trust trust idea depth independent trust two cycle trust remove remove cycle every infer trust trust acyclic walk combine trust item recommendation noisy outperform exist approach walk trust predict rating item similarity great rating predict reliable friend away recommendation approach trust user item systematically user similarity pairwise trust researcher factorization technique user rating among datum divide two regularization method typically social social term friend model laplacian add minimize et build kind semidefinite relation factored rating factorization incorporate graph generate disadvantage user recommender recommendation process reflect make interpretability drawback cause interpretability name social latent user rating combination basic show basic exist trust handle trust recommendation network minimize disadvantage user model incorporate influence make feature incorporation recommendation totally recommendation start recommendation recommender lack set propagation formal trust introduce formal trust trust propagation capable kind trust extend propagation enable feature comprehensive estimation trust metric propagate trust score aggregation use trust trust score trust propagation seminal incorporation information recommender address particular enhance careful incorporation enhance recommender outperform trust counterpart rational algorithm employ address connect two trust accord introduce trust path aforementione predict trust relation social utilize trust combine trust direction examine relation social work factorization effective recommendation l symbol mean latent rating matrix user network user social relation social neighbor similarly provide formal collaborative filtering concern follow literature collaborative filtering assume click item user rate aim rating user rate correspond th item matrix partially recommender system utilize user item factor preference preference effectively recover rate singular method rating applicable fact rating matrix utilize observe formulation e subsection review extend incorporate trust information item user rate goal constitute minimize factorize term matrix respectively control matrix respectively would practical collaborative filter application well trace netflix amount success million rating rely matrix expect user rate important challenge recommender start user start life system rating allow prominent tackle factorization rating recently trust relationship rich side technique ignore trust recommendation usually trust trust trust generate recommendation aggregate user intuition user tend adopt item recommend friend trust positively strongly correlate recommendation trust provide accurate incorporate relation problem keep user user vector close user feature user rating specifically problem subset behind social idea every user similar friend use social friend assume weight relationship user social easy friend simply objective jointly fix one recommendation incorporate trust relationship partially observe present generate computationally gradient descent rating suffer vast majority recommendation trust ignore capable exploit develop recommendation network trust relationship trust recommendation evidence trust certainly propagate social influence basic user close separate apart interest incorporate idea matrix user friend reach desired trust directly replace utilize careful trust chance certainly away regard incorporate another behind stem relation relation consider user agree friend item friend reasonable margin enough friend dissimilarity user friend view viewpoint connectivity feature influence user connect learn map basic feature isolate enhanced social trust relation correspond positive reduce latent edge close distant inherent topology network illustrate behind rating feature friend margin figure example illustrate mention ease exposition consider obey goal prediction rating trust user depict viewpoint user circle latent friend outside dash circle circle safe margin friend impose triplet friend triplet force extract triplet ensure exist friend factorization mention similar enhance existence relation investigation correlation social relation rating include rating people neighbor empirically social support formalize ingredient latent exist social introduce monotonically user user behind I latent triplet function k k two assign utilize assess loss logistic widely loss extract triplet friend becomes learn latent feature consistent consistency reflect problem make hinge assumption write consist output utilize estimate user item compare discuss reveal aim inherent among objective capture recommender system description object gender age potentially description way incorporate meta measure latent feature pair meta datum specifically obtain meta diagonal pairwise similarity graph base meta rating return like emphasize triplet trust link item user add latent item generalize triplet link dissimilarity link accord tag associate tag trust otherwise alternatively trust dissimilarity profile improve recommendation regime popular operate stochastic approximation regime approximation regime large use approximate tradeoff reliable preliminary example gd sgd gd sgd solution gd gradient full discuss gd detailed objective convex minimize iteration fix repeat fix exposition triplet auxiliary except ki jj ij write apply gd indicator take value argument gd computation expensive note large triplet next provide issue descent sgd gd slow batch output gradient update eq problem gd gradient idea fix triplet triplets triplet strategy unbiased make iteration triplet intermediate sgd triplets select triplet gradient decrease enjoy light mini brevity mini sgd triplets triplet gradient set sample triplet unbiased full gd convergence number non smooth sgd much gd exhaustive experiment conduct experiment fundamental question perform incorporate trust social extent user align friend question trust role accuracy recommender system tune exploit social prediction affect recommendation extent lack trust relation trade efficiency subsection begin introduce employ follow discuss choose evaluate aware recommendation customer review website share movie write review rate helpful helpful rating whether worth quality review user addition trust rating present review inaccurate quality explicit negative trust enhance recommender ideal social relation trust full statement conduct come item total rating datum approximately rating demonstrate rating rate overall summarize statement trust well utilize recommendation different create increasingly example randomly select sample predict rating independently fair since hinge well rest stick however loss thank smoothness negligible matlab load machine trust number rating number rating average trust max user min rating user employ mae rmse propose filter enhanced mae offline mae predict compare absolute value prediction mae value predict denote factorization assign item rate user item measure mae larger even rmse valuable netflix competition reward rmse trust implicit relation recall normalize discount ndcg rank friend define relevant friend friend user relevance rank user ap ndcg discount cumulative sum rank user position discount ndcg ideal measure ndcg measure user rank ndcg logarithm ndcg scale normalization logarithm tuning parameter may drastically objective important much incorporate complete partially observe rating user utilize item rating hand dominate reasonable social recommendation grid value two parameter combination good grid parameter achieve validation consider triplet computationally remain perform repeat set algorithm choose recommender system trust relation recommender factorization recommender take factorization trust exploit trust minimize network intuition behind assume corresponding feature quantity user e add problem obtain optimization factorization trust algorithm stand propose literature exploit trust memory recommender predict user combination rating neighbor user already rate idea trust recommender limit neighbor user distinguish trust trust adapt rating relation instead schema trust relation binary social hamming implementation neighborhood information use recommendation predict rating adapt exclude trust integrate spirit strategy use relation trust relation contradict propagation trust exclude implicit trust range rating ndcg ndcg ndcg consistency implicit ndcg c rating ndcg ndcg website allow review product service review allow trust review consistently valuable review consistently inaccurate rational incorporate trust recommendation trust relation investigate rating implicit trust trust relation social important aim empirically investigate user friend rating user assign review rating write neighbor social social could supplementary rating boost recommendation rating review analyze also claim similarity implicit trust use complementary compare implicit explicit implicit investigate relation rating interpret user literature adopt popular refer pearson coefficient respectively relationship behavior user implicit
shift high cause robustness architecture function highly non linear code cnn obtain effect principled subspace dimensionality acknowledge ec axis fp adaptation adapt acquire search leave prediction learn misclassification alternate sub performance aim adapt acquire new target domain several task scenario attention search domain invariant leave stochastic gradient learn world exception think part corpora time subject vision particularly way background motion mention acquisition resolution artificial e filtering train final poor domain adaptation overcome information come target label extensively paradigm invariant searching representation source target aside reduce crucial adaptation induce distribution would perform equally labeling use domain learn jointly reliable source source depend availability annotation semi set provide besides large source method modify formulation train learn source classifier correspondence label sample augmentation case domain part recently propose combine cross domain domain encode kernel challenge unsupervised domain resort minimize sample mmd reproduce use domain nice critical lead poor goal define overcome reconstruction map target promise domain adaptation shift present mostly feature mapping cca couple principal eigenvalue instance space variance preserve distance transfer subspace project alternatively use mmd subspace matrix exploit intermediate idea introduce domain geodesic strategy extend intermediate subspace intuitive alignment sa demonstrate source pass intermediate overall domain invariance less attention dedicate substitute pls maximize unsupervised approach go beyond search subspace exploit source discriminative encoding source distribution shift previous work method fix adaptation adaptive method exploit annotation name rest briefly domain subspace domain shift conclude classification sample unlabele different hold satisfy labeling operate adaptation establish condition source perform demonstrate eq indicate error ideal suppose low low dimensional intrinsic structure source matrix subspace transformation modify subspace target simply measure frobenius minimize obtain transformation matrix unsupervise pls extensively study sa promise cross domain subspace adaptive par keep domain adaptive process separate focus aim domain divergence source domain concentrate sa margin formulation detail target aim representation combination hinge function trade high give importance focus role sa priori c da avg choose see dataset combine office class amazon resolution provide descriptor standardize normalization consider split provide two digit different specifically image normalize gray pixel use annotated image search image per category typically representation adaptation estimate mobile device period contains label collect period unlabeled period location evaluate predict mobile device modify define location set repeat consider target split random subspace domain analysis supervise description turn locality preserve option fair comparison none exploit subspace publicly available implement regularization linear geodesic kernel alignment modify integrate preliminary obtain pls less explain jointly maximize source minimize equal pca baseline also source learn target subspace pca represent target method final tune fold cross three parameter remark annotated option indicate tune baseline source target office exclude extraction modern computer cpu ram compete training slow g slow sa provide optimize besides reduce test phase runtime comparable adaptation model offline issue change high value drop source square trend office source target state target similarity reduce divergence sa respect case lda domain amazon interestingly consistently understand performance target vary source change figure namely amazon appear high stable source target divergence loss appear always red obtain square target analogous trend domain negligible find good domain divergence instance label separate linear final indicate compare domain shift application sa learn exploit vs involve per separate sa produce suggest main feature sa challenging domain want
identity square index submatrix back multiplying maintain simplify definition eliminate minor dpp marginalization formula eigenvalue specifically exponential reduce elementary symmetric order elementary eigenvalue except note elementary polynomial note recall eigenvector denote singleton eigenvalue dominate require reduced consider matrix eigenvector identical eigenvalue get dominate letting eigenvalue update step depend eigenvector eq apply standard rule subscript indicate eigenvector express entire similarly sort expand zero row somewhat relationship body recall background section identity dpp relationship start step long express plug sum elementary substitute change sum assume row derivative efficiently specifically diagonal dominate contain body ccc wishart moment moment initialization moment low set setting set start wishart example draw setting third trial lemma observation single h bar bar bar bar style ne nest marker nest engineering university engineering compactly semi dpp learn entry log likelihood entry thus focus row propose parameterization eigenvector bound expectation maximization world product recommendation gain naive likelihood project gradient ascent example typically choose product domain diverse chance user find define assign quality discover effectively adapt include marginalization arise dpp compactly learn example maximize np gradient projection produce degenerate partial scalar learning make item direction store dpp attractive property lose bayesian unconstrained non method propose differ assume restrictive eigenvector develop maximization style problematic naive maintain projection sometimes nearly interaction lead nature make ascent dpp psd index notice psd non normalization thank intuitively capture quality item diagonal item eigenvalue clearly psd also imply exact marginal convert learn subset section naive project ascent marginal entry eigenvector hide variable apply inequality section ascent low log constraint ensure psd put upper let rule project ascent optimization technique algorithm refer ascent psd eigenvalue notice project guarantee optima poor probably accept though step truncation result still improvement initial near well employ dominate operation projection overall runtime convergence follow em well runtime tb kk p dpp marginalization provide dpp broken eigenvector submatrix index eigenvalue eigenvalue dpp marginal mixture dpp algorithm mixture equation sense intermediate variable em hide introduce auxiliary develop corresponding gradient proper position us eigenvector see appendix step practice optimize take ideally repeatedly various objective exactly expectation size solve perform trial update enforce eigen projection poor optima associate sophisticated technique optimization practice second order hessian use multiplicative em previously assume average runtime come step search test ascent google em require kernel neither start diversity thus explore option naive incorporate statistic initialization wishart degree freedom identity wishart eigenvector unitary output fit correspond dpp place mass wishart tend emphasize unless employ matching normalize single item recall attempt match choose start recommendation task ground comprise product category recommendation account choose popular negative dpp use basic live recommendation build test dpp dataset consist amazon com product amazon product category split toy item filter product least discard category remain sub category test diverse appendix detail number initialization initialization wishart initialization gain advantage truly strong negative create check value gain exhibit investigate far set step near wishart problem close comparable likelihood average gain moment perfect instance poor category
place therein exactly main difference aim nature result therein extension would guarantee direct lipschitz assumption concavity imply combination aim instance nature contrary demand aim necessarily satisfied require need severe use set get remark rate achieve original variation term computational projection solution perform strategy round mention omit sake recall set payoff actually scalar payoff mix opponent eq compute whether indicate actually singleton game hard strategy need compute query convex polytope particular payoff asymptotically statement neither statement opponent determine minimal provide whether cost drawback superior refinement quantify exactly refinement work mention cost aim payoff possible adjust adapt assume exist continuity soon take constraint game linear scalar payoff cost value bound formulation correspond payoff notation distinguish way relaxed propose computationally already simple strategy could well achievable conclude quantify general gain quantify mostly lie material objective fact proof omit main body appendix indeed section path various completeness read understand result main putting optimality prove supremum norm thus induction confident payoff adversary respectively product rewrite eq short cauchy schwarz indicate assumption quantity bound everything induction provide suitable take bound relate time whose conclude sequence induction satisfied assume latter indeed expand hand proof performance self confident proceed force immediate recurrence something order claim von latter convex negative expansion supremum payoff opponent maker get reward product convex expansion aim minimize payoff opponent decision maker value distance supremum negative graphical expansion solid line thin solid assume contradiction achievable imagine choose equal choose possibly stage nature choose next entail stage stage repeat part identically play prove follow computation leading entail substitute supremum component display mean decomposition converse thank start refer absolute combination admit expression leave switching matter payoff regime function indeed round illustrate combination combination absolute indeed see give eq study consist decomposition get help mind main local expectation payoff advance next getting play could play lemma target already theory admissible achievable function infinite admissible show example actually result exist minimal achievable target function totally order subset indeed least element difficulty achievable compact payoff empty empty fix would cover totally addition exist together contain differently achievable existence lemma show target methodology therein strategy maker target choose stage denote mixed maker round payoff receive q negative achieve denote next stage denote action play decision average receive one hand thus equation inequality together symbol play arbitrarily nature round first round play round denote play decision maker round payoff round limit claim argument interval universit paris france paris players want converge try exclude preference relation target possible payoff oppose spirit action player receive vector round approach set require projection scalar perform episode online learn minimization decision maker obtain much perfect average maker quantify g sequential decision cast objective possibly arise field finance resource management many call optimization solution use offline pareto front optimize feasible weakly dominate front choose objective pareto see multi player value payoff player play want vector objective represent admissible state player exclude prescribed start determine task determine hard reveal standard decision online reward priori make game structure every decision reward assume game treat game maker specify approach value reward define goal list multi exist work use efficient maker advantage constraint special present similarly general comparison summarize propose online approach furthermore start game move discuss target family possible base achievable achievable sort individual latter never worse strictly devise achieve goal amount regret modify direction application classical sample path payoff consider theory average payoff asymptotically base know small game repeatedly play two player maker opponent player payoff finitely whose opponent round vector impose restriction lie pick action scenario inform round payoff martingale study conditionally know put differently formulate resort formally maker want average formal latter small target indicate choice take expansion decision maker decide wireless channel maker power throughput channel much ideal work throughput maximal power zero maker look axis throughput power throughput plane value throughput throughput expansion long hausdorff distance one give link consider opponent possibly random payoff round opponent choose rx opponent strategy empty close course meet dual expansion close correspond constant put therein related mathematical mind cost aim player average payoff small expansion constraint prescribe formally matrix payoff vector abuse work opponent maker maker get payoff admissible adapt exposition set pp general satisfy soon follow value fix expansion discuss start target aim intuition target rely parameter denote payoff expansion component infimum continuity lipschitz achievable maker ensure opponent non follow convergence set graph coincide weak useful target function error stage prove resort relaxation see ask hull define supremum convex decomposition I strictly target achievable part prove rewrite eq whenever convex e g denote section strict summarize fact individual achievable second response function theory explicit efficient achieve however calibration intuition target calibrate sense know advance exhibit target call complexity rely property strategy output know advance payoff
q n e mn md replace exactly statement generate projection eq q least conclude distinguish let assume give jensen inequality satisfie see therefore modal universit paris reconstruct entry problem recommender filtering quantum physics work recover sub take correspond recommender necessarily uniform nuclear penalize analyze theoretically bound kullback leibler tackle potentially dimensional attract decade consist random classical noisy entry real observe presence finite alphabet categorical survey survey yes quantum outcome recommender movie rating dataset item survey course incomplete proportion entry matrix completion ill pose particularly popular low constraint completion observation low approximately reference commonly square rank constraint nuclear alphabet among case depth recover probit example observation moreover uniformly recommender rate theoretically nuclear variation norm constrain penalize lagrangian rather first upper kullback leibler distribution general sampling absolute upon previous find last coordinate recently design possibly approximate formulation benefit value scalar order decrease operator give matrix hellinger distance leibl integer link let cardinality finite alphabet logistic rating function entry though multinomial distribution success probability give denote q control estimator logit instance error binomial score link framework uniform index resp resp exist ensure whereas requires sample associate reveal coefficient rademacher material kullback leibler true universal stochastic quantity control ease notation constant convergence bit minimize slow study max constrain maximum rate multinomial logit associate distribute problem consider significantly burden significant impact solution introduce iterative mind operator singular singular potentially computation see vector canonical stand tensor normalize linear real space cone key denote zero singular otherwise obtain conversely auxiliary map map step nonnegative denote dirac k store entry index total allow table give rough execution cpu ram cache evaluate uniformly unitary equal factor uniformly observation logit logit use implementation moreover obtain classical analysis contrary logit completion observe outperform gaussian leibl symmetric rating fold grid gaussian modal unable necessarily value probability probability distribution real calculate kullback leibler truth dataset test
sgd individual architecture connect architecture tradeoff compatible deep exploring procedure begin accord multivariate normal network output prediction derivation position indicate auxiliary attribute j j filter propagate computed channel error fig sum loss detection sum multiply transpose convolutional filter pair fig learn pose expression feature face face exhibit pattern optimize eqn logarithm finding become definite dynamic ignore eqn become f imply value stop task eqn tendency drop period length indicate valuable similarly tendency stop strategy addition need tune threshold decide stop sec model auxiliary attribute readily handle initialize share tune label dense representation pre unit contain layer pool connect commonly conduct overlap fully fourth task stage include failure estimate ground inter failure facilitate choose table facilitate criterion region influence addition divide face category rotation pose face corner task select rest image face challenge conventional dataset specifically pose variation face testing image severe partial face annotate annotate rotation select face depict web face interaction face annotate web annotate densely face face face consist image expression show densely l bag big open gender face head pose profile coordinate different convergence verify train dynamic task slowly stable early coefficient early scheme network em inter task wise stop covariance give relation square root correlation compute absolute five corner correlation attribute determine global pose rotation affect randomly choose attribute attribute visualize attribute normalize one visualize profile leave right exclusive positive face rotation five attribute average absolute intuitive figure head pose heavy positive attractive correlation simply task target term demonstrate effectiveness cm cm cm examine variant along auxiliary global result variant full model attribute simplicity variant cm auxiliary task beneficial task group pose pose global attribute auxiliary mainly capture local information main gain cause attribute fig produce trend accord face corner low face learn share describe corner similarly location localization remarkably face constrain likely pose show demonstrate effectiveness build full publicly image superior cnn legend method cnns cnns implementation cnn layer comparable locally cnn intel cpu cost ms gpu structures table run ms wrong transfer train sparse tuning various pose regression binary auto encoder compare tree jointly head response map face face method detector error image face result annotation report near grey capture follow protocol face set face produce comparison see exhibit capability handle difficult large head rotation representation fig show protocol face face set quantitative algorithm depict achieve detect face cm indicate instead detection detection heterogeneous task appearance expression head pose auxiliary deep share task utilize auxiliary attribute severe pose need cnn gpu techniques future work explore zhang receive b china currently work toward ph department research interest track department chinese university interest deep vision graphic change science research chinese university previously semantic research include vision visual surveillance technology ny ph institute technology research engineering chinese work microsoft interest include vision recognition receive good award conference computer vision conference computer transaction intelligence international computer pt ie edu study improve jointly optimize detection recognition heterogeneous correlate attribute gender appearance since attribute learn address novel inter employ dynamic facilitate learning task extensive learn alignment deal face severe pose ii drastically deep face alignment face semantic corner verification great field detection remain head pose traditionally independent approach include template base coarse fine cascade accuracy compare previous exist system architecture believe number factor govern intrinsic second discover attribute help detect corner accurately also small face large rotation source space alignment divide auxiliary attribute corner specific dataset rich treating constrain detection achieve optimize b average attribute blue face face pose effectively divide investigate main leverage information task head pose gender age attribute multiple appear model allow joint conventional challenging several task face inherently difficulty identify easy determine negative enjoy balanced recognition auxiliary improve procedure poor study optimize correlated signal auxiliary back jointly alignment nonetheless newly address aforementione consider effective deep equally assign weight auxiliary dynamically task show coefficient essential reach alignment heterogeneous task concern correlation exploit well dynamic task correlation learn automatically newly thank effective share auxiliary learning base alignment five readily handle configuration method challenge dataset specifically dynamic coefficient relatively effective heterogeneous objective jointly improve usefulness auxiliary improve technical evaluation conventional category base task task coefficient method difficulty introduce coefficient early network learn dynamic task aim rate convolutional cast detection transform pixel highly nonlinear five tuned label prevent fit general pre procedure former step filter normal face extract detection prediction auxiliary coordinate let face pre imply center corner fine tuning attribute prediction generalize model suppose correspond first additive task model crucial detection correlation therefore accord dm dm tm differently begin training training fitting auxiliary attribute assign coefficient adaptively accord determined sec pointing wise stop beneficial task determine empirical task early dynamically dynamic solution value update summary face estimate filter coefficient posteriori map
ne compatibility chapter part head head head head head compatibility theorem equation em compatibility compatibility compatibility compatibility true false em mu mu mu setting true mu mu mu mu align align end end end align environment use environment style style construct allow allow false tag tag science centre university college science interactive centre college inequality derive rademacher complexity unit dictionary play eigenvalue concentration rademacher complexity standard input pair I complexity lead uniform example member function machine function kernel kernel hilbert kernel feature map norm h interested sequel similar decomposition bound exist follow rademacher replace replace read apply trick lie linguistic specificity sense individual different type linear prediction associate dimensionality dominant bind vector equation w denote large overall example coincide eigenvalue kernel divide bound significant number large tr high radial small type behaviour typical method benefit datum see structured regularizer propose give dictionary reproduce novel include application weak important go concentration rademacher random complexity complexity incur little gain gaussian sometimes convenient simplify certainly exact reference whenever inequality book aware systematic application intend exhaustive appear somewhat derive structured sharing section new give application learn concentration proof elementary covariance proof remark independent variable conclusion replace theorem exercise calculus subtracting point v tr benefit centering raw pixel kernel c use empirical convert bound complexity derive structured norm dictionary hilbert space inner span norm include overlap projection rademacher ix ix ip mx mx mx parameter mx p mx improve tr feature subsequent give value q let rademacher purpose obtain uniform tx value unconstraine tuple real independently become highlight role control dictionary intermediate admissible class expression ti w sequel give another computational order refer eq q delta word extreme point applicable give w ti ti ti special ti tn ti already reverse sharing liu guarantee multivariate extreme point unit k ti ti ti third rademacher token supremum supremum weak ti ti overall depend second weak large sparsity disadvantage sharing penalty outli task norm consider let dictionary orthonormal effective compare bound trick construct finite covering dictionary eq k ti k ti jensen put everything take infimum get compare
variation exponential tail therein et distribution approximately integral carlo sample quantile follow ordinary differential ordinary series convenient recommend take interested find quantile find convolution numerically many light random al cell claim one tail partial partial independence tail heavy marginal iff expectation thus distribution derivation et al comprise joint maximally decrease conservative tailed triangle contain heavy bind result convolution say amongst tail case instance one et ij write tail approximation perform core study involve quantile regression therefore parametric choice quantile regression good quantile choice interesting exercise previously feature company cover convenience two claim amount trend claim development generally development trend skewness mixture implementation development effect skewed symmetric adopt skewed primary conjecture appropriate understanding quantile higher quantile critical derive margin model wide tail development variance al est functions second model set mean detail prefer incorporate year accord development describe variance subsequent whenever c est est est est fix fix fix quantile plot demonstrate quantile fit model indicate specify different trend development depict posterior quantile level respectively nonlinear trend development covariate increase subsequently quantile furthermore trend loss quantile level al benchmark model variance year five triangular heat map heat upper triangle five row study map trend development level indicate light around year year increase loss peak scale rather transformation gb cells upper tail far adopt calculate quantile light tailed claim analyse technical claim expect available cover capital measure calibrate percent projection gradually percentile dramatically percentile c gamma flexibility model popular involve opinion aim approach statistical particular percentile quantify uncertainty uncertainty utilize adjust traditionally admit central margin estimator deviation calibrate approximately percentile total loss moment method suffer drawback influential normality estimator skew utilize model achieve base via var represent total standard iii median adjustment tail traditionally utilize estimate consider risk margin adjustment margin make adjustment quantile statistically denote year capital state uncorrelated presents quantile al percentile applicable margin demonstrate demonstrate amount third party cover million cover period use represent cumulative payment portfolio variance lot drop original skewness data scale drop scale necessity adopt dynamic skewness modelling choice al allow flexibility skewness modelling nonparametric proxy indicate loss adopt propose modelling year reason increase uncertainty appendix h skewness variance skewness fit complexity margin perspective margin estimation c skewness variance year change skewness skewness respectively start year gradually margin ahead risk margin base sound mathematically consistent reasonable margin analysis model quantile reveal quantile finance heavy tail compare parametric five al pp gb provide investigate three regression function generalize estimate margin overall indicate margin offer considerable drawback particularly rare quantile precisely solution aware limitation cm corollary school mathematic statistics university email department statistical science college uk pt develop derive margin capital application utilize entire function quantile regression provide estimation margin capital historical volatility framework nonparametric quantile model quantile model include distribution power pareto case carlo strategy adopt proxy scale mixture facilitate dynamic applied analyze datum discuss interesting regression chain monte margin work assessment claim assess margin inclusion margin issue margin relate inherent practitioner non margin requirement general establish develop risk institute th aim highlight calculation discuss aspect capital article regard challenge capital require plus risk capital furthermore calculation specification recommendation margin several little significant wide technique modelling highlight approach practice assessment percentile traditionally adopt claim central estimate define range outcome however inherent arise statistically robust claim claim differ practice adopt typically eventually cover claim requirement risk model capture uncertainty margin provide claim therefore increase likelihood sufficient require gps regard worth note display heavy margin large stable portfolio margin capital capital method determine margin measure return capital evident capital require initial capital claim estimate return capital alternatively percentile quantile currently takes meet subtract predefine percentile bring percentile ability rigorous manner claim micro environment explain attribute principled manner shall allow argue percentile involve somewhat margin square enable offer mechanism study explanatory distribution center explanatory margin variation propose framework function regression provide apply range economic finance finance explain important quantitative extensively analyze assess monitoring risk regression construct autoregressive risk regression risk var quantile portfolio accurate financial capital level cover portfolio taylor percentile risk margin assumption sophisticated capture shape tail claim stable pearson gb transformation datum result log sensitive et family compound stable extreme structure could claim incorporate covariate multiplicative quantile rather quantile recently alternative skew heavy tail adopt function long outperform conventional gamma generalise gb tailed gamma pareto family provide convenient claim perspective pareto quantile pareto combination difference capture claim incorporation function fundamentally illustrate develop risk forecast perspective distributional assumption markov skew alternative appropriate frequentist asymmetric laplace al asymmetric bayesian quantile et fitting single bayesian information type zhang propose bayesian growth loss form monte carlo fold quantile propose relate quantile risk instead margin rich characterization tail impact distribution merely secondly quantile regression bayesian especially generalize software user specialize knowledge markov monte methodology shape estimate capture claim year heterogeneous apply parametric quantile organize follow section explain framework way use two loss set conclude relevance model novel analytical perform focus asymmetric al distributional family al demonstrate claim triangle assume development claim year year claim incur simplify year number development year triangular predict claim triangle xx l claim triangle quantile structure within predictive quantile estimation cell quantile section component distributional model quantile loss quantile quantile solve ij yu minimization vector parameter vector al give scale clear maximize minimize formulation observe response al coefficient vector study simply link equivalently link start towards alternatively parametric conditional coefficient distributional quantile quantile write cdf standardize incorporate regression scale suggest ij transformation ij ij fy ij parametric regression regard relationship impose distributional naturally framework directly link quantile quantile detail yu zhang realization laplace volatility asset allow al yu regression purpose development context propose residual inverse cdf quantile eq shape affect note skewness shape magnitude distribution skew skew hence skewness moreover risk margin adopt mostly percent rather fairly figure show skewness skewness second parametric pareto function combine pareto main tail pp comprise function u may valid produce power pareto pareto power specification derive u give solve treat location really implicit regression complete pareto parameter pareto plot demonstrate flexible skewness tail feature al modelling data support upon carefully fit interpretability transformation year moreover claim tail particularly tail business class family gb beta modelling express gb include gb link link covariate eq q generalization distribution via ij give beta accord widely utilize gb family relevance context estimation correspond sub family understand flexibility gb gb sub see introduce three shape link hence quantile within family restrict consider adopt model suitable explanatory quantity subsection explain distributional conditional quantile simplify regression possible classify explanatory e trend explanatory distributional consider multiplicative interaction regression aspect well consider scale consider apply pp effectively al addition allow covariate make manuscript differ regression comparison concern distributional relate covariate sub limit specialized explore sub space distributional adopt set year development year claim basis may observe label mean parsimonious specification structure assume trend year behavior popular know level slope trend decomposition level specification give correspond denote year respectively constraint first distribution gb support log al correspond quantile quantile
make datum basis modelling diffusion tree stick prior basically distribution relationship accurately intractable posterior assessment particularly generally tree substantial modify tree prior tree tree child change consideration simplicity generative flexibility generality nonparametric algorithmic towards develop coherent principled inferential form article systematically assess advantage theory abstract invariant limit grow tree model critical condition aim article fold generative model structure mechanism develop goodness simple model tree use goodness fit test class canonical variance consequence technique algorithm pay mention primarily convergence condition process regardless distributional propose goodness base cancer systematic classification intra heterogeneity crucial development problem detect brain heterogeneity group heterogeneity intensity voxel intensity approach summary skewness intensity account structural complexity intensity wherein image hierarchical branch dendrogram tree relate pixel cluster distance branch length carefully dendrogram asymptotically determine conduct look binary density condition model key condition restrict attention relationship purpose estimate condition model goodness approach conduct number test test check group experience short survival comment generalization proof simulation material root order finite amongst parent notion child trees tree non unit branch length vertex describe strict internal vertex child fit article motivate binary benefit construction leaf form homogeneous function add edge inter time choose distribution edge point choose accord length leaf order chosen observe branch branch length form branch branch denote branch term interpret assign mass length binary topology leave root topology branch length order root q arise topology tree exchangeability length leave attractive tree wherein leave detail product act factor make tree density capture total sum branch length th arrival density obtain lead family wherein branch length model sample tree efficient test generate leave characterize arrival characterize edge arrival uniformly length action obtain density homogeneous th arrival consequence speak homogeneous poisson rate function non always convert homogeneous homogeneous intensity process one obtain solely path appropriately modify edge obtain follow determine total suppose generate homogeneous poisson gamma proposition test homogeneous suppose branch percentile branch length generality th percentile testing leave statistic topology test attractive assess detail application wherein branch albeit flexible topological goodness simplicity develop test considerable extend binary condition process lead completely notation finite root vertex eq finite trees difference terminal vertex leave manner integer tree start root give child refer copy vertex two arise probability tree tree ensure address consider tree condition know combinatorial condition condition importantly condition view pick wish accord equivalently tree flexibility modelling perspective useful offer structure order condition success tree trees bin strict binary order unary tree unary tree tree vertex contain tree vertex contain success useful scheme branch condition construct critical condition converge limit modelling purpose strict child proposition demonstrate extend sub super condition model bin bin super behaviour resemble critical finite may source asymptotic branch within branching technique tree reader arise tree grow size banach make tree natural branch lie follow concern critical projection specification brownian need construct tree role structured topological branch carry computation consequence code describe root order uniquely code traversal traversal walk tree manner ease branch positive imagine motion root explore move continuously edge explore come clear twice evolution time root offer intuitive length branch length take walk path length relate vertex length unique vertex formalize proof map order ht span finite branch path branch tree vertex denote root preserve illustrate preserve distance root new original ht v leave vertex pick vertex context reconstruction work purpose choose leave terminal manner remarkably two way strict arise limit carefully tree condition brownian arise limit scale tree first vertex q brownian brownian weak height vertex brownian functional functional randomly subsequent tree condition random leave limit leave branch factor order density tree variance employ homogeneous coincide equation view marginal dimensional use density lemma density dependence manner employ incorporate notation terminal leave subset parameter condition goodness tree condition equip parameterized variance homogeneous process vary length identical topology therefore limit condition child topology consistently propose condition two consistent omit normalised subset leave path length estimator goodness fit test path use clear role tree possible extension condition need retain leave tree choose leave clearly define leave interpretability density exponential go detail upon branch gamma minimal statistic leaf test tree stand purely homogeneous setting model alone position develop see generate variety copy condition applicable article goodness entail generative wherein branch estimator fit test start leave order tree randomly branch asymptotic rejection choose subset leave accord leave denote fix path length hypothese leave loss numerator exceed testing every order follow condition develop inferential statistical perhaps transformation indeed establish probabilistic condition probability pose equivalence class weak consequence invariance experiment counting size law motivation weak prove equivalence regardless variance factor handle condition tree connect pick kb label end move implication uniform tree statistic construction construct look total variable shape principle induce b approach verify translate pick vertex test lead order path choose vertex walk ex seek give meaningful construct depth walk root traversal scale test path degree freedom proposition consistent variance loss numerator exceed denominator testing tree random normalise root normalise distance condition tree efficient simulate large tree expect code value random construction straightforward walk condition necessary vertex description final supplementary examine performance binary tree test topological test statistic identical construct simulation objective critical various check estimate examine tree correspond tree total converge expect path correspond condition binary hierarchical subtle rejection framework vertex trial reasonable second provide carry tree contain vertex interested size theorem critical length sum branch constant estimate compare histogram tree vertex ht cc l first histogram histogram condition tree tree randomly leave solid red shape report two size examine nan sample list exercise trial choose vertex choose leave compute total length permutation base test goodness condition tree bin see grow test valid distribution trial success represent birth death condition tree fact correspond tree ultrametric cluster application concern mr elaborate upon cc cc cone c bin bin permutation condition vertex refer tree death rate f permutation bin alternative normalise randomly value axis statistic vertex verification choose random experimental encourage tree appear large portion contribute estimator tree death tree poorly tree tree cc cone bin condition correspond death rejection goodness permutation tree distribution bin column utility heterogeneity cancer dataset objective relationship hierarchical patient confirm molecular genome database sequence image inversion recovery cancer publicly pre process mr volume follow correction visualization automatically medical intensity appropriate heterogeneity variety genetic molecular occur development classify base intensity increasingly amongst practitioner motivation pixel intensity spatial lie heterogeneity image classify heterogeneity histogram pixel intensity image overview image pixel offer recently histogram factor skewness heterogeneity cell cancer task experience survival tree binary dendrogram individual intensity intensity branch length distance representation heterogeneity image cluster intensity hierarchical patient classify survival time agglomerative cluster intensity group heterogeneity binary examine condition tree finite rejection detection heterogeneity rejection precede power ultrametric trees leave result pairwise tree arise clustering consider survival panel heterogeneity branch tree three tree indicator heterogeneity height branch branch whereas topological short patient survival intuitively patient short survival appear intensity rich varied small branch length scale dendrogram appear observe dendrogram pixel intensity choose cc note linkage small tree perturbation base hausdorff distance ultrametric therefore function agglomerative sensitive linkage distance tree appear linkage test unchanged linkage order prescribe size patient time consistent efficacy test choose nan reject reject reject
exhibit specific variable analytic dependency exploit combine mixed motivation mix analytic consider capability ordinal nominal response datum mixed background economic nominal datum model concern fitting take health cover people study though nearby date water suit contain supplement home collected explore landscape describe area analyze response categorical item ordinal nominal item asset indicator asset car ordinal use ordinal modern power among item datum asset survey derive principal group examine relationship construct asset index principal score observation base previous survey construct asset index upon index routine principal recognize whole collection propose aim factor md md hybrid ordinal data latent nominal combined md ordinal link detailed item ordinal lie observe response underlie conditional latent trait underlie item usually term discrimination parameter link two py view say nominal complicated response set nominal inherent detailed dimensional ordinal response analytic nominal response variable correspond nominal item denote nominal relative cutoff latent analytic mean trait parameter ij analogous discrimination section note binary could ordinal ordinal analytic similarity mix binary nominal denote binary ordinal item loss generality column item use nominal item analytic continuous ordinal nominal item collect set ordinal nominal lie analytic term loading factor ordinal nominal latent parsimonious latent observe mixed loading latent trait vector marginally parsimonious facilitate mixture modeling impose mixture md md model gaussian belong denote loading latent indicator cluster belong augment gd gd ordinal nominal three way define rise particular response detail bayesian parsimonious mixture md provide approach survey unify md novel capability nature nominal datum capability modeling item unified monte mcmc utilize interest membership mix proportion latent trait item md bayesian unknown specify threshold conjugate specify term trait multivariate latent indicator variable follow latent likelihood mcmc marginal mcmc employ latent variable sample exception parameter detailed derivation allocation trait ng gd gd gd gd ig nature corresponding condition detail ordinal truncated row follow mcmc chain prior also gibbs adjacent slow thus overall gibb propose candidate j rr select metropolis iterate md identifiability aspect threshold ordinal constant threshold ordinal item factor analytic many propose literature solution loading triangular diagonal order appropriate md model instead impose loading matrix sample post process reflect closely reference loading trait transform posterior fit cluster loading reference mcmc hoc describe understand landscape md asset vary trait consideration md models md approach within set md well small exist region md trace plot chain achieve satisfactory mixing jeffreys gd absence strong relatively uninformative prior note flat improper mixture assess indicate sensitivity appear thorough switching address criterion md place interesting assess exploratory three uncertainty bayesian paradigm distribution discrepancy employ truncate sum square pearson assess fit context cluster deviation evaluate frequently observe response md intractable evaluate response multidimensional truncation dimension predictive pseudo count set across md consider illustrate quantile median across md plot trait improvement insight well apparent assertion exploratory assess uncertainty plot cluster general confidence low notable increase higher randomly sample latent trait dash uncertainty model residual residual normal latent reasonably focus residual item residual residual correspond membership residual plot model suggest explore fitting component md trait distinct belong divide time allocate cluster pt yes yes yes yes yes yes modal response modal across group response analyze modern group modal likely possess modern contrast poor modal location type less modern status close player cluster small low produce trait phone item ordinal pt detailed picture box binary ordinal nominal binary response code response correspond interpret box plot nominal say item cluster plot ordinal cluster reflect cluster economic status type cluster notably response group plot box plot relate item sample dimension nominal focus mean follow significant high divide item estimate suggest region survey show observe conditional member pt total hellinger total hellinger distance group hellinger hellinger plot modern hellinger hellinger hellinger many pattern evident plot hellinger distance cluster notable hellinger item account hellinger item large hellinger contrast difference highlight cluster notably list table modal differ component group similar component possess modern differ modern possess investigation reveal group group hellinger distance group group see group concern modern convenience interestingly consist essence remain modal differ solution item modal differ group solution component separate asset gray line score colored trait explore landscape base asset survey interest landscape md economic consider code binary principal matrix construct asset raw show standardized principal standardized asset index color two broadly roughly low middle top principal choice rand solution md poor asset score md model preferable base approach type landscape region asset status survey md achieve economic status examine information great benefit various aid decision make development policy result membership interpretation aid future survey survey md model important economic health account study examine cluster landscape exposure landscape statistically principle hoc md provide mixed mix md capability individually usual varied challenge facilitate md formal trait would beneficial paradigm pose difficulty md bic alternative underlie bring difficulty uncertainty incorporate reversible nature cluster parameter md md extended time survey however survey extend appropriately longitudinal move md could insight indicator similar metropolis hasting rank likelihood areas md facilitate consist little md latent latent trait extension achieve unit variance probit covariate could naturally md effect l main yes material modern material yes area room yes house house report none modern report water source report status yes tv report status report player
atom describe layer wavelet per dyadic wavelet level review wavelet first level second atom frame case obtain cross frame relu architecture normalize mask concatenation frame match context dnn refer frame optimize network optimize mse architecture case gpu sir nmf multi layer outperform gain sir high separation discriminative significant improvement discriminative context regressor play well architecture improve resolution representation representation able long context remove uninformative variability relatively finding discriminative improvement separation contrary modeling remain unclear phase architecture believe discriminative promising comparison subject interesting explore good recent study source separation dnn separation improvement speech currently address architecture exploit assess multi role study facebook ai edu decomposition operate feature effectively duration frame modeling frequency multiple temporal pyramid transform convolution modulus first learn widely audio investigate resolution alternative neural fundamental speech processing rely capture different elementary atom become audio negative nmf adopt various audio source recent many separation nmf fail characterize speech increase window dimensionality limitation extension learn code temporal include occurrence kalman integrate rnn recently efficiency train adapt task become inverse completely problem deep range simple frame regressor sophisticate rnn separation music multi benefit separation discriminative role separation take resolution representation pyramid information temporal defines increasingly think generalization apply excellent classification source preserve much possible discriminative pyramid nmf dictionary level selective localize separation regime multi baseline confirm superiority discriminative regime separation work family separation dedicate describe alternative fall category set describe popular nmf discuss aim source representative report concentrate music operate frequency version comprise temporal thought operator typically define magnitude short fourier transform alternative temporal resolution duration non representation success signal shift expense inverting space normally specifically estimate source phase recovery efficiently soft signal resemble wiener filtering demonstrate obtain filter multiplication division define mask impose restriction receive lot attention literature negative activation represent speech ideally measure dissimilarity input channel discriminative compute supervision mild autoencoder generate treat frame ignore work integrate several scope replace capacity minimize fitness truth common space phase recovery study predict forward use dnn fix time context fail temporal dependency speech increase intractable dnn exploit rnn long memory section briefly wavelet conceptually temporal layer band localization wavelet quadrature center filter result bank per define low pass bandwidth wavelet smoothing kernel critical rate preserve locality possible complex modulus nonlinearity critical sampling bank localize trade sensitive local uninformative robustness signal wavelet bank modulus dyadic reduce bank filter discrete every produce non quickly experiment operator desire reach context increasingly produce map low resolution tree pyramid feature source separation pyramid layer transform produce mostly interested layer variability inverse leverage level source sampling carry code test estimate gradient descent approximate phase phase simplify strong
expand percentile student standard help student compare probably ok otherwise compute interval skew advantage narrow interval percentile interval suffer symmetric skewed add coverage coverage skewness matter size z adjust effect coverage correct advantage error formula may large problem poor recommend bootstrap percentile bootstrap short percentile like people bootstrap percentile sample large narrow accurate percentile population percentile perform bias factor allow variation interval avoid allow multiply table population bit red population never exactly multipli population yet continue help long distribution take sensible percentile adjust provide normal population simple adjustment multiply adjust bootstrap q adjust give n table coverage adjust dramatically adjustment skewness uncertain help modify motivated quantile handle skewness adjustment se percentile expand percentile percentile poor common adjust quantile well coverage exponential population mathematically mathematical bootstrap limit assume bootstrap quantile bootstrap q g mirror percentile reach far percentile sample mean percentile interval reverse percentile skewness figure thing discuss mathematical thing strongly skewed depend strongly wrong thing transformation bad everything sample percentile wrong skewed wrong direction figure reverse percentile perform percentile wrong reverse percentile wrong wrong namely say reality say positively positively correlate large positively skewed opposite direction denominator affect skewness show negative skewness figure show skewness base bootstrap generate table reverse bootstrap percentile quantile bootstrap distribution eq quantile vice versa inaccurate table percentile mean percentile percentile interval skewness skewness closely figure overall coverage statistic skew interval test statistic estimate bootstrapping order formula available iterate sample bootstrap bootstrap original total suited location sample mean correlation coefficient transform depend reverse percentile coverage population bootstrap percentile percentile expand percentile coverage side normal population section coverage percentile reverse percentile poorly interval correspond se skewness variability skewness skewness thing expand percentile much still due variability estimate surprising optimize bootstrap extremely sample correction bad ordinary se coverage confidence side population interval non code leave hard side bootstrap interval next skewness correction accurate poor good reverse percentile interval poor population type previous interval axis second linearly side coverage e expand confidence interval skew hard bootstrap interval skewness adjust second accurate require include summarize interval inaccurate interval size bias yes yes yes transformation yes yes yes yes vs partial skewness issue replacement plus another randomly random add extra exactly opposite attention copy another round correction bootstrap finite four thing permutation test discuss bootstrap name permutation test pick replacement label equivalent name permutation test infeasible independence fisher use effect include include report value zero impossible compute side one create great less center measure compute fraction quick use discuss deviation one value may equivalent result mean variance monotone h permutation latter together south gold united united er li china zhang china south south free score two program free score permutation partially dataset slope figure distribution correlation short correlation side add numerator denominator calculate one multiply sided testing conditional one permutation testing test test independent amenable without assumption procedure beyond scope article variance differ pool population positively mean naturally variance differ nan company investigate expensive procedure alternative large live match would appropriate regression shift could perform test wrong slope multiple hypothesis whether coefficient zero side quite narrow tend narrow permutation permutation test variable situation permutation confidence might bootstrap hypothesis relative accurate permutation permutation compare two sample sample way straightforward reject nan interval fail hypothesis calculate value tail permutation nan could recommend skewed impossible categorical keep weight convenient normalizing choose broad bootstrap testing relatively bootstrap bootstrap goal bootstrappe test statistic somewhat deep method show inaccurate provide broad resample article offer benefit concrete abstract concept student tool visualize nan error arise visible permutation student difference work student many finally median odd student formula check bootstrappe role play bootstrappe prediction interval test skewed datum central operate interval accurate sided probability intuition skewed tail outlier observation would allow possibility tail percentile section right tail percentile narrow like computed skewness expand percentile interval percentile wrong thing transformation people large backward percentile poor well interval skewness adjustment formula bootstrap need accurate side criterion bootstrap skewness expand percentile percentile percentile bootstrapping normally sample original datum sample general exception fix sample bootstrap observation conditioning dimension bootstrappe odd independence permutation simulation side fall recommend routine interval handle little coverage se skewness sample correction skewness skewness regularization something smoothly large procedure place zero nonzero skewness software package eventually standard current david mm figure inference abstract article bootstrappe permutation student bias distribution issue practice compare inaccurate latter confirm asymptotic resample alternative percentile cover sample alternative bootstrap permutation randomization test I use bootstrap test student understand concept I mind elsewhere broad application resample motivate include text resample one student behind bootstrap principle guide visual bootstrap sampling thing inference skewness transformation something cause odd bootstrapping statistic also inaccurate procedure skew broad change statistical skewness different discussion simulation asymptotic regression avoid permutation look beyond test finish short test section include r package reading box return may read resample read first permutation broad read skip notation come population correspond often sample sample I say sampling unless deviation r permutation approximation quantile bootstrap percentile expand percentile bootstrap procedure behind bootstrappe student tv channel one come tv pay extended minute half hour tv minute hour channel vs minute perhaps stand random half hour tv actually half hour occur really basic like occur time figure also right labeling give would rare chance difference observe right pool replacement another compare plot histogram exceed numerator multiply sided detail denominator procedure nice visual student look make nan true statistic student learn significance rarely chance student work directly switch like generalize means formula familiar difference formula answer resample bootstrap sample bootstrap draw replacement create bootstrap calculate times bootstrap relate sequel mean figure distribution extend center spread approximately indicate unbiased spread vary bootstrap approximately quick percentile interval basic channel channel percentile interval section distribution tv distribution top extend channel observe bootstrap sampling bootstrap distribution statistic se bias extend elsewhere bootstrap permutation test replacement statistic bootstrap distribution bootstrap percentile bootstrap bootstrap minus draw sample compare size sample bootstrap bootstrap interval include difference variation permutation basic nan center assumption bootstrap center statistic confidence make abstract concrete concept central interval student involve statistic variability deviation student use student confidence interval work statistic variety make statistic also understand student check student know really ignore later hidden formula bootstrap bootstrap bootstrap abstract standard role random familiar tool like plot bootstrap percentile statistic check answer formula bootstrappe permutation student group like student partition nice visualization repeatedly build permutation resample practice country thousand machine count query country query country count add country per query per word ordinary bootstrap generate poisson save instead user come seed generate get count across user try improvement probably user variability across resample lift describe early version design people question familiar see treatment nest people ad subject visit website actual response prediction gender covariate across people prediction person error theoretically difficult derive need instead people survey fold validation resample technique routine imputation backward elimination mixed stability calculate million question among probably behind bootstrap later inferential statistic estimate something sampling distribution principle obtain draw sample sample distribution distribution sample population collect infinite draw sample expensive know sample bootstrap estimate population statistic show world sample population statistic collect center plug something substitute estimate substitute take estimate plug whole raise substitute possibility consist distribution replacement may sample population may parameter substitution tell something discuss important immediately center population bootstrap bootstrap aggregating bootstrap improve matter center add instead people bootstrap think create bootstrap help something nothing bootstrap sample real use accurate regard formula twice standard use quantile bootstrap quantile sampling use estimate bootstrap estimate replacement detail avoid case unique bootstrap infeasible carlo implementation draw simple rule mention fix produce modify question draw sample estimate would actually precision similarly bootstrap testing another nan match modify behind bootstrap behind bootstrap population well claim tell elaborate series thing work well exhaustive bootstrap randomly population bootstrap middle column three five bootstrap center rather shape bit lot bootstrap well parameter instead useful additional fundamentally spread carlo quick estimation interval adequate however tail bootstrap distribution middle small center shape vary shape sample substantially vary monte carlo may apparent narrow go plug fx x x error basic depend severe bias bootstrap course ever department survey result recommend student spread affect confidence sample smoothed sd median bootstrap approximation sample continuous odd bootstrap always original near bootstrap statistic heavily bootstrap distribution turn heavily different bootstrapping work ok shape bootstrap percentile median bad odd percentile fall one value fall percentile right smoothed draw thing though great sampling distribution leave population spread mean middle top two figure like sampling right spread interest binomial distribution depend show bootstrap implication confidence reach long normally middle right distribution sampling parameter sampling chi square depend centrality estimating number mode bootstrappe application bootstrap distribution sampling typically bootstrappe overcome indeed sample well assumption parametric population trust spread datum bootstrap population work poorly depend reduce variability distribution fundamentally look ahead thing accurate allow fact usual interval allow discuss resample little good suggest carlo usual bootstrap distribution sample draw replacement enough confidence criterion computer much slow fast computer easy second random due prefer implementation formula bootstrappe permutation fraction exceed bit add bootstrappe monte bootstrap replicate like bootstrap example se bootstrap percentile interval monte se percentile carlo error package package use quantile side necessary sided chance estimate permutation exhaustive similar percentile confidence quantile fall bootstrap se variation depend z deviation normal approximately calculation suffice round variability percentile bootstrap bootstrap accuracy get routine practice increase carlo want decision implementation coverage coverage smaller sometimes roughly large like coin tail overall coverage way sided value chance estimate within monte percentile interval se routine recommendation decision depend confidence hypothesis skewness population look affect affect functional odd bootstrapping subject low pressure high pressure group likely cc pressure disease risk disease show relative skewed right skewed se risk bottom panel log desirable property invariance monotone transformation invariant confidence transformation equivalent percentile transformation another answer percentile interval risk contrast interval risk take transformation invariant answer zero derive plug principle sampling minus statistic biased summary another r bootstrap square produce bias b generally aware bias double make another kind bootstrap distribution sensitive transformation median bias three cause helpful third bootstrap important nonlinear transformation median bias near strongly denominator e cause optimization bias category high use population quantity give biased bias selection optimize lack bootstrap even quantify fit line obvious curvature section bootstrapping panel resample bootstrap lack bias poor bootstrap estimate functional subtle bootstrap assume solely statistic answer observation twice distribution odd bootstrapping sample functional probability look treat question odd bootstrappe tv observe se bias average conclude biased think say non functional adjusted smoothing procedure regularize se affect calculate solely bootstrap bias observe statistic affect confidence issue skewness skewness skewness york responsible phone service term competitive something wrong responsible suppose quickly customer various different period significance customer slow customer substantially pay test instead service positively skewed odd plot hour period work sd surely discrimination clear aside outli time tend comparable quantile permutation value cutoff test pool test four nan hypothesis solely denominator permutation pool permutation test permutation test absolutely sir fisher originally permutation test computer limitation permutation test computationally computer skewness size permutation biased permutation population close tie value quite wide primarily small center mean se reflect contribution se sample bootstrap skewness answer share question ask skewness solution answer wrong base raw already thing point deviation normality bad sampling alone question statistical effective really work even skewness observation measurable interval percentile limit scale perform permutation pooling draw replacement pool three three pool vary accurately reflect pool variance often side isolate skewness converge correct slowly percentile simulation moderately skew confidence interval test skewness procedure skewness least g bootstrap procedure discuss combination historical momentum simulation note unfortunately much replication prohibitive budget would momentum person software start usual skewed look bootstrap rather twice skewed opposite quantile plot calculate coverage material
behavior dataset good small pls construct regularization component loading obtain assign loading integration center cm cm center seven age irrelevant response point pls en forest concrete next sect principal pls pls select similar sect tuning method sect deviation
lead cite vi establish contribution look community intuitively two study straight optimality analysis exist development provide basis lyapunov present straight forward lead presence vi control also concern versus continuity continuity function address state domain work long entire controller remain valid discuss rarely policy hence apply policy evolve establish respective contribution provide vi vi paper presence investigate contribution boundedness guess control analysis apply evolve analysis straight forward compare available interested reader development author regular rest formulate iii iv vi conclude vi respectively integer dimension control space index continuous semi initial calculation control minimize control asymptotically definite put respective one k hx trajectory control utilize rl boundedness definite condition trivially select assume value assume continuity function require boundedness function lead respective admissible lead unbounded connect contain origin intersection vector set associate contain trajectory utility without origin value satisfie solution mathematically nonlinear utilize idea use either look neural select specific valid entire trajectory remain approximated value pi vi pi initial admissible control converge approximate calculate approximate guess iterate equivalently pi admissible satisfie associate one may compare remain respective control confirm lead require system stability system evolve critical vi scheme arbitrarily prove vi convergence solution guarantee require admissible come disadvantage convergence vi see pi admissible vi also policy guess sake vi give use iteration admissible notational compatibility calculate utilize guess stability theoretical pointwise induction result hand assume compare result consider induction proceed continuous lead issue initial guess vi give admissible control iteration monotonically value equivalent limit lead prove monotonicity h hx hand therefore sequence help continuity result limit may idea proof continuity hold generality applicable admissible contradiction end relation inequality arbitrarily close eq q loop initial continuously note existence one skip result enough index tolerance iteration convergence admissible compact selecting iteration uniformly theorem ref theorem monotonicity lead concern continuous within equal eq limit exists form continuity contradiction one continuity exist open point away margin contradict arbitrarily within continuity versus approach note set hold finally tool conclusion value function induction interest capability sample stage lyapunov compact r ix proof lyapunov continuous control definite eq trajectory asymptotic difference value function invariance close system I origin prove converge origin word time asymptotic stability every stability form find lyapunov another approach system subject stable every trajectory similarly replace left side small per repeat q repeating equation lead hand side bound converge therefore consider trajectory path establish besides address vi guess reason cite either great exist number negative state less present require hold uniformly restrictive detailed analogy establish vi optimal control cost horizon definite horizon minimize subject control horizon time go summation along bellman equation horizon guess vi vi identical converge case proceed consider resemble method control lyapunov utilize terminal respective horizon loop remain iteration converge optimal horizon analogy go cost control rest admissible infinite cf nature also invariant utility lead go origin due q horizon consider evaluate horizon great sequence compare result theorems literature close lyapunov correspond admissible guess vi directly asymptotically control hence restrictive study requirement existence take account control go similar guess simplicity proof oppose difference besides address concern idea establish evolve next x let control control subject value estimation region closed loop inequality next step trajectory hence origin vi hand eq simple parametric rise error read iteration right convergence vi boundedness investigate boundedness worth train control actor approximate give vi control approximation regardless whether value remove calculate minimization side control actor learn training system independent actor control actor hence actor error beyond scope investigate separately boundedness consider comment denote minimizer assume boundedness analysis assume vanishe origin word semi positive hereafter sense vanish I x x respectively value bound exact sequence generate recursive mathematical ix ix ix induction compare note resemble idea programming idea boundedness respective admissible control iteration boundedness actual challenging analysis vi presence monotonicity lemma long boundedness guarantee neighborhood iteration importance system lemma result define admissible word iteration k satisfie initially trajectory result c confirm assume lead complete minimizer lead along show result evaluate side trajectory hand second fx ix x system system similar lyapunov function low upper theorem guarantee parametric show ix ix lemma replace consider approximate vi eqs f conclude vi summation function evaluate state consider one guess ix q leave boundedness lead c ix policy q leave side analysis sign quadratic left satisfie make suitable result non last c desire stability theorem zero assume positive end system trajectory within within continuity lead function error control approximate asymptotically origin theorem vi eq along equivalently evaluate
assume equation fact separately incorporate affine term row later optimality subgradient expand nonnegative must index solution loss dissimilarity incorporate affine rewrite q row substitute constraint dissimilarity parameter beyond row order several follow index assume order index follow denote dissimilarity element dissimilaritie similarly use contradiction select feasible achieve function write involve objective triangle inequality eq write q arbitrary centroid element together obtain contradiction approximate nonlinear ds gaussians notice quite representative nonzero row representative proposition example pt finding center learn health signal pairwise dissimilarity target source set row regularize formulation relaxation regularization parameter show group group deal outlier dissimilarity asymmetric implement alternate direction implementation algorithm real categorization representative series modeling segmentation dissimilarity simultaneous recovery video scene recognition characteristic entire vision refer visualize web video increase interpretability expert describe small requirement work contain original efficiency oppose training help dataset efficient classifier pool extract sift frame video form histogram frame apply ds matrix preserve find lie subspace qr rank try correspond good submatrix find approximate assuming express linear problem rank find lie low low manifold live similarity dissimilarity pair work several ambient high pairwise relationship efficient work high live datum pairwise compute importantly similarity dissimilarity allow suffer initialization find impose relationship try dissimilarity point employ initialization affinity pairwise ap require initialization empirically categorization single point process fix subset set semidefinite diversity impose similarity single specific hard propose unsupervise subset dataset selection solution fall location extensively operation research literature dissimilarity ij right representative dissimilarity problem dissimilarity dissimilarity element optimization recovery formulate regularize minimization put trade closely relationship advantage ap require dissimilarity dissimilarity particularly advantageous pairwise dissimilaritie difficult dissimilarity instance distance dynamical dissimilarity encode base dissimilarity specifically dissimilarity come asymmetric triangle guarantee group show regularization parameter representative propose unlike solver well computationally propose use alternate multiplier admm result show admm reduce world improve art categorization segmentation representative find along element associate representative effectively b red accurately approximate manifold mn dissimilarity well dissimilarity denote collection dissimilarity predefine euclidean representation access dissimilarity social learn dissimilarity use contrast art restrict consist see dissimilarity code error consist even correspond hence dissimilarity kl select representative dissimilarity geodesic set dissimilarity asymmetric triangle inequality contain scene converse dissimilarity bag short sentence converse necessarily efficiently associate dissimilarity denote b euclidean row indices representative row interpret representative represent dissimilarity objective want dissimilarity encode via possible goal consider count convex relaxation nonzero notice program assignment range probability representative appropriate whose representative emphasis select close representative emphasis select value demonstrate select lie absolute dissimilarity illustrate bottom dataset gaussian dissimilarity euclidean distance material compute world source contain section framework effectively dissimilarity find point source notice outli correspond element addition reduce large help reject outlier efficiently element datum may explain efficiently model program enforce often detect allow encode outlier value program encode via outlier equal zero hence encode hence put penalty selection without penalization outlier optimization w weight figure illustrate detect encode notice augment time membership z assignment jointly partition dissimilarity determine would like cluster bi group efficient direction admm compute distance representative element close plot decrease put emphasis encoding approach arbitrarily small nonnegative word element select close show jointly program partition implication illustrate notion joint partitioning follow let correspond index centroid well represent target source solution find target partition correspond say jointly index choose index notion joint prove optimization program select element dissimilarity element index partitioning dissimilarity kk jj k dissimilarity partition regularization determine theoretical kk distance group regularization reveal certain partition optimization x nonempty single dissimilarity identical algorithm like pairwise relationship target impose pairwise relationship theoretical apply theoretical result cluster nontrivial group thresholding dissimilarity fail around dissimilarity centroid dataset accord become emphasis representative form identity representative subset minimize enforce program multipli establish program store build office room consider single finding program optimization program np hard relaxation let soft second row arrive show three cluster parameter notice increase value obtain observe demonstrate effectiveness enforce deal weight multipli formulation evaluate real world improve challenge regard since multiply divide scalar dissimilarity divide unless otherwise obtain result neighbor nn significantly memory demonstrate improve effectiveness scene categorization category class forest make rest testing ap point rand baseline initializations fair spatial pyramid pyramid bins dissimilarity pairwise distance pyramid histogram dataset class distance dissimilarity respectively test result obtain expect rand perform method come ap pass relationship include select ds close training use ds important dissimilarity word dissimilarity group dissimilarity group improve performance rand ap ds confusion classifier ds middle expect increase result confusion important select use training row confusion matrix show store use rand ap ds subject activity motion capture use marker per subject trial comprise walk run stand jump c frame activity sc ap problem generate among segmentation motion framework efficient identification dynamical series trajectory p kk important measurement given regressor define eq series segmentation correspond hybrid system give estimate dynamical form learn collect dissimilarity represent efficiently segmentation membership select long framework argument activity motion capture several marker measurement human body informative instant loss
vector signature attribute signature attribute probabilistic direct add unseen class variant explore gain often work unseen specify attribute signature none attribute unseen see impact stress attribute former focus mid classifier exhibit attribute pac mid level attribute common alternative exploit external adapt unseen class unseen image imagenet hierarchy label spirit label embedding shot category attribute control unseen attribute signature unlike shot generalize shot propose random forest model simultaneously signature label enable shot attribute attribute mid discriminative label however guarantee discover align shoot unseen semantic semantic post hoc discover pseudo idea handling concept explore propagate membership entail shoot accordingly domain differ train label unseen signature propagate signature shoot visual attribute unseen signature attribute unseen training typically association binary real grain unseen signature modal vocabulary discover expressive discriminate unseen zero shot signature attribute unseen time unseen initial attribute classifier introduce zero random signature signature shoot sec attribute shot presence importantly classifier come object need instance unseen category precisely n descriptor bag specify indicate attribute v must descriptor attribute presence per th scaling forest v v v attribute entail operate characteristic operating example instance next introduce zero shot shot decision attribute descriptor vector shoot available apply unseen signature signature train forest signature unseen signature negative build manner learn recursively split signature record signature signature follow training instance recursively randomize signature child indicator thing node split norm signature training forest discriminate grow forest depth branch reach per forest novel test signature test test apply attribute predict posterior attribute zero shot alone train unseen shot random main generalize recursive procedure signature path determine individual predictor expand signature attribute space build appropriate expect error choose split idea extension receiver formation process signature explain signature maintain validation datum sec recursively denote root let record occurrence signature signature split node receiver roc attribute negative negative rate attribute gray attribute threshold sample signature pass leave single signature reach signature estimate attribute stress two thing role sub capture split select split fractional signature child must mean split leave child properly dependencie threshold address miss uncertain building shot choose split information gain signature unseen signal attribute classifier prefer class reliable point remain omit split highly signature find validation positive signature propagate left ji jk explicitly make attribute inherently attribute classifier unseen zero label play important robustness perfectly attribute unseen course forest attribute classifier attribute framework instance attribute signature tackle indicator annotation signature signature fraction forest please attribute signature label signature select split also traditional training gain signature define reflect fact image like actual output require propagation recursively select combined control signature suffice training appearance shot learn forest dataset unseen class scene image material etc solely annotation split select unseen sec include color histogram sift attribute label roc attribute baseline amount report measure trial cross validation negative reach node validation addition state variant baseline train signature attribute shoot literature approach design overcome classifier prediction examine attribute see sec consistently well large plot learnable break minor attribute benefit classifier avoid uninformative attribute yu yu et al name discover next shot apply attribute commonly shoot recognition exact variant furthermore model attribute presence valuable gain especially per attribute less reliable attribute vs table help impact signature propagation full aspect contribute well unseen error attribute combination initial attempt regard include signature unseen negligible compare publish name name attribute decode achieve art shoot simple strength signature bin show image rely solely solely signature shoot baseline method unseen dotted signature shoot towards prior play blue select automatically introduce zero shot attribute classifier prediction unseen challenge dataset indicate fact attribute remain reliably future extension accommodate inter attribute correlation random forest test multi forest unseen contain shoot avoid sec discuss signature sec shot noise sec list class check account inherently avoid completely attribute negative node signature propagate ji jk plug far plug constrain sec summarize deal uncertainty class attribute signature modify soft indicator vector amount perturb copy signature describe equivalent latter perfect signature respectively term signature uncertainty specifically oppose annotate signature run term rh familiar annotation add signature perturb per expand among probability implement uncertainty common reverse association class may per class signature fraction attribute signature zero shoot recognition annotation signature type material surface envelope type closely relate scene instance reason localize belong marked attribute unlikely person category discuss simply shot result similar fig shot shot class trend similar synthetic
another case tight sufficient primary algorithm succeed learner immediately low meanwhile convert low bind learner especially primarily apply separately primarily immediate direction uniformity small although might question paper uniformity uniformity whether far know paper worst well draw confident one program consider learn benefit metric immediate suggest testing coordinate probability see uniformity thin learn within perhaps distribution acknowledgement discussion thank project finally thank estimation introduction survey field section body consider support entry consideration denote distance distance infinity instance treat slightly true specify distance tolerance access sample wish determine sample distribution goal correct call uniformity place relate support immediate exceed already prove want exceed want jensen vector conjugate treat lemma jensen eq back give side maximize exact conclusion inequality side minimize analysis focus coordinate meanwhile q sum sum covariance pair pair case hold triple triple appear one distinct count way sketch give intuitively slightly choose expectation chebyshev chebyshev make regime dominate know advance simplify somewhat prove uniform chebyshev use definition since next part proof u eq meanwhile leave inequality choose inequality side subtracting divide side reduce divide suppose q check remain term lemma plug inequality prove start drop complete rearrange n n failure rearrange drop point iteration least vote incorrect draw theorem q q around integer hold approximately minimize failure uniformity failure far uniformly give oracle output say access uniform family low family indistinguishable usually say usually wrong oracle versa specify random coordinate remain coordinate zero two family toward property n expression chance inequality property inequality show draw number meanwhile oracle observe entirely uniformly family thus condition access access member correctness correctness oracle family uniformity eq immediately unknown careful construction modify exercise bind apparent plan distribution draw draw sample distribute outcome let output eq let analogously line lower prove lemma slightly uniformly follow flip coin construct valid p p aa uniformly choice fix expectation simplify odd case ia ia ia sum ok inner back expectation convert expectation take apparent q precisely mini draw uniformity need distribution coordinate uniform probably never indistinguishable let note sampling say contain argue probability drawing sample similarly correctness draw arithmetic one case uniformity require family distribution possible put coordinate put distribution put let pick let meanwhile length bind drawing inner expectation coordinate otherwise claim probability exactly plug necessary q briefly regime check coordinate draw u outli number sample uniform case except group least every satisfie binomial chernoff bind g follow fall range direction union range suppose chernoff mn substitution substitute suffice recall chernoff group probability threshold suffice suppose least whose chernoff sample least simply chernoff eq slightly theorem failure run draw proving draw run draw lemma draw say sample well sample sufficient logarithmic empirical concentrate formalize lemma state rearrange imply rearrange give th change change change inequality state plug suffice draw sample suffice corollary suffice follow directly bound prove deduce distinguish coin require sample prove formally author distance regardless tight tight recall construct member size pack bind error pack point simplex point exist add simplex contain ball member space factorial simplex set meanwhile ball dimensional set size eq pick numerator next bind q consist coordinate equal minus binomial distribution drop slight optimizer coordinate may entropy determine particular bind probability drop relate success relate guess conditional prove terminology uniformly lemma follow q prove least choice plug get case always take distinguish coin distance distance simply distance bind learn construct follow assume apply construction coordinate coordinate size large every differ sharing distinguish identify coordinate sample construct take choice half precisely point half claim two distinct subset pair coordinate lie half differ coordinate differ first need one last include include include claim test uniformity distribution examine classic learn size sample test uniformity conjugate suffice seem upper support uniformity easier side coin dependence uniformity factor optimal sample uniformity testing meanwhile algorithm metric cl discussion full question broad whether classic whether estimate would except failure study independent practical imagine web company keyword give day motivating require distance uniformity show uniformity support sufficient regime size uniformity testing work choice theoretically uniformity distribution like understand seek address goal survey big sublinear depend support question nice monotonicity query however ask answer say datum suffice primary result general drawing difference depend desire find uniformity sample solve data conclusion something distance must light primary conclusion sample make sake fundamental knowledge coin come might think require support norm measure distribution piece portion distribution find optimize tradeoff immediate application instance use box utilize instance derive beyond draw conclusion less develop deep understanding space test idea address lead simple general sharp broadly vector application stream tool question study norm may refine develop next result describe conceptual uniformity discuss broad prior future proof omit though proof prof low bound uniformity give distribution specify satisfy output except test uniformity distribution failure upper low match constant uniformity intend reference skip key contain factor employ quickly know bound aspect devote conceptually surprising opinion section detail technique uniformity respectively p cc regime neither tight case prove theorem match n consider phrase actual complicated regime chernoff sample algorithm coordinate outli either correctness outli uniform outli chernoff sample group matter count number sample large note large compare uniform put heavy contain outli chernoff probably sufficient prove ix jx output learn satisfy except naive frequency sufficient proven draw sample coordinate elegant general order interesting novel failure suffice number sample note q p x term I let x I contribute contribution x x tight reduce technique suffice learn concentrated expectation well dependence confidence follow must sample draw enough expectation suffice result failure number suffice suffice discrete distribution number
shape simulated size estimate sample true correspond complete use likelihood likelihood aic provide moderately nan model true approximately reason choose aic great aic high general give respectively low could approximation specify low ks r repetition aic provide fraction repetition reject level provides reject ks successfully aic follow misspecification bias section explore branch mis specify case realization realization parametric mle type figure branching article make introduction allow estimation easily useful estimating background branching highlight branching discuss recommend alternative quantification branching ratio observe specification finish provide relevance attempt quantify branching ratio high fluctuation event significantly day em parametric intensity event mini fluctuation heavy explain allow helpful estimate branching level model rich process type cluster process extend marked influence allow easy additionally em introduce cluster associate multiple cluster allow dispersion evaluation enable treat branch parent point algorithm find study introduce quantifie self extend branching branching realize produce along form attract lot combine seminal spatio mark sequence individual review genomic dna neural train brain become frequency fluctuation price location cluster poisson location mle maximum time I identically time distribution dispersion index variance call instance flexibility extra evaluation impossible paper branching next severe simplification make maximum use cluster distribute termination consider event long novel branching derivation complete address already case em miss present process process algorithm minimal datum goodness test present selection branch misspecification discussion relevance define window instantaneous occur value ts intensity process call give mass branching branching inter event become branch construct point generate subsequent process introduce generation define process sum intensity recover intensity realization fig represent correspond generation deterministic intensity pdf necessary intensity function intensity intensity define cumulative distribution intensity distribute inter decay intensity provide exponential thus go realization process introduce dependence location cluster dispersion process simple general likelihood realization maximize intensity intensity evaluate miss relevant section occur last inter factor inter observe density unconditional homogeneous introduce perform mle simplify form use iterative guess parameter estimate parameter expect value complete miss datum f nothing proceed iterate estimate likelihood algorithm identify em give case unobserve process event branch describe matrix diagonal element sub point parent rest split inter event e density square inter relation lag lag lag child inter event lag lag support define index return index distant intensity never vanish account q start neither point include square branching distribute child subsection step step em account branching evaluating complete give estimate irrelevant determination introduce event either one matrix em branching structure write form denote line event derive exploit branching superposition exploit purpose intensity event derive unconditional incomplete intensity weight probability event time probability parameter weight enter iterate time vector clear hadamard vector complement probability discard remain obtain previous compute iteration maximize obtain rather exploit intensity intensity allow process independently parameter form step estimation process complete square event time z I jt event likelihood determination numerical stability become small intensity require branching branching maximize cdf branching expect slightly small total denominator occur parameterized mle parent assume piecewise inter estimation density branching probability inter datum within take approach expect take replacement inter unweighted simple inclusion allow weight natural potentially example smoothness solve variational memory j large within lag reduce take estimation branching clear however tail approximation adaptively em obtain alternative efficient process may estimate implementation branching complete separate value structure concern use specific need rather regard branch numerically part concern useful parametric example positivity lag sec trick combination reduce likelihood evaluate thus must show likelihood aggregate index exclude average likelihood value take description branching ensemble realization likelihood take carlo statistics sec simulate realization set treat follow probability taken select way repeat reach repeat contain possibly repeat index calculate likelihood value treat poisson plug intensity log transform incomplete computed computationally approximation likelihood careful issue encounter average logarithm compare log standard process process process generate poisson observe one transform kolmogorov ks generally ks statistic semi complete h nan statistic unknown vector semi complete monte approximation value discuss consistency sec mis conditional parametrize pdf decay imply exponential hand event weakly become density shape heavy shift index financial ensure markovian calibration outlier tail alternative density typical application account computational exponential function discuss sensitivity start speed complete em require em get optima thus select reasonable starting point understand algorithm speed towards overlap branching overlap detail comprehensive phenomenon insight couple illustrative standard high branching ratio model choose pure initial estimate ii realization low em initial parameter estimate branch ht fig density converge completely density estimation despite poor analysis start dash solid increasingly dark parametric
jointly train challenge learn classical example find appearance address initialize part alternate part thousand part diverse propose still pool use image remove uninformative produce elaborate part comprise initialization joint section part level performance part translate directly test improve art mit cnn contribution part I part informative counter negative initialization extract patch repeat pool discard part fix part filter idea recently approach stage discover approach different score high cover max goal part diversity natural function model another use mapping part contrast part detection share multiple another work visual visual category similar train word terminology negative part concept image information strength pick attribute class collection part entire scene composition scene region scene position pyramid contain location dot product response location treat maximize define image collection response pool pooling region maximize region simplify notation response predict high suggest scene binary often binary classifier foreground score combine classifier predict say foreground foreground intuitively usually response filter binary multiply non score function hand long convex feature capture part part response vs part classifier counter part score another challenge similar object factor contribute positive ambiguity detector head detector class category filter natural class part filter filter weight vector scoring multi select matrix score imply invariant series unlike restriction entry negative part classifier impact norm otherwise therefore classification loss affect propose joint training objective encourage diversity encourage complement substantial train simple multiple example part think vector multi hinge optimize structural svm reduce define line initialize repeat repeat w convergence output joint parameter optimize weight equivalent training represent solve use method involve maximum make minimizer bind sx w j hx sx z j u z I implement minimize optimize function memory joint objective part step find initial part part procedure large pool involve pick regardless image label whiten patch estimate patch background discard discriminant random result comparable method get alone part figure may part select pool entrie th uninformative redundant drive zero regularizer generate number monotonically important mit cnn per part base part maintain pixel extract multiple grid feature value maintain pyramid cnn hybrid hybrid network imagenet fully fc response part pooling pooling arrange mu mu final left get part randomly weight svm pool regularization train show feature mit space pyramid subset part comprise flip selection improve performance large achieve level performance outperform flip dominate compare part cnn mit pixel train obtain performance imagenet improve augmentation tractable term show effect surprising model feature improve cnn extract entire get pca coefficient pca perform part gap increase number part pool part improve select part jointly part significance train part much gain large translate section blue red random blue curve part red curve part part part filter lead initialize initialize correlate positively outline section computationally expensive optimize simultaneously share filter weight secondly part require repeatedly slow mechanism tractable optimize candidate location location pyramid cache cut copy mu require repeatedly image cache subject cache cache hierarchy among cache hard hard global hx possible configuration image x place vice versa z j sx j w I iy yy u old b latent follow w procedure outline benefit intractable solve auxiliary optimization problem start initial cache cache cache remove hard line algorithm input tc old old save cache update cache convergence warm trick call converge cache remain update happen close iteration trick e triplet retain cache update follow work mean c w old w w imply convex strict convexity w therefore since w cache configuration iteration c stop change due step theorem cache stop possible mechanism cache w local implie therefore key idea algorithm visit cache number plane method solve optimization treat quadratic start update direction accordingly cache entry optimize auxiliary objective proceed gradually converge objective price however tractable slack formulation maintain constraint e tuple cache entry constraint training constraint total ie unconstrained problem equivalent qp eq set qp add repeat proportional size qp dual behind let f ib w k relate gradually constraint enough remain rest process early round discard certain consecutive cache vector qp find fast qp optimize pool initialize optimize objective function direction version constant second mu plot result part depend increase adjust part complement provide figure joint subset mit determine category example distinguish illustrate part filter image filter filter filter filter filter part filter image filter filter filter filter part filter filter filter filter filter filter image filter part filter filter part image filter filter filter part part filter filter filter image filter filter part filter filter filter filter filter filter part filter filter part filter part filter filter part filter part filter part image part correspond entire c c scoring cm cm patch highlight green view c cm cm cm part cm window visualization match particular appear part selective color example appear highly weight appear detect row highly strongly sensible apart filter part location patch number class bottleneck test depend affect procedure first standard bottleneck filter nest loop line loop line algorithm cache loop line qp solver number algorithm separately use one cnn dimensionality latent day joint full mit full
edge everywhere else thus encode difference site odd signal express composition problem encourage sparsity odd constant jump odd structure site fdr underlie spatial odd dramatically difficult inspire technique compute ordinary signal spatially constant optimization treat fix separately section describe detail turn action panel raw toy example concentrate heavily true site thick black fdr smoothing grey mean ordinary attempt recover show toy simulate site arise rare elsewhere highly testing come allele dna adjacent site genome figure show simulate reconstruct prior fdr smoothing function solid curve across dash model fdr smoothing show grey favorable stability site estimate though truth mean site estimate average truth smoothing consistently exhibit spatial site adaptation shrinkage raw top discovery leave spatial pattern density locally high score area locally realistic fmri working memory experiment analysis describe detail panel score arise experiment systematically experimental difficult working voxel horizontal obvious cluster shape task study panel score false clearly edge region many spatially isolate spurious panel bottom procedure partition dense area contain significant area locally significant bottom discover fdr region significant signal plausible fdr locally significance region right bottom figure specialized emphasize fdr simply merely raw locally rather signal fundamental address detail fdr two reason nonconvex guarantee find even evidence actually yield reconstruction power likelihood simple augmentation lead maximization negative likelihood convex function log design stationary via conceptually odd e linear plug guess binary variable conditional site give logit sub expand taylor approximation weight least generalized term intermediate solved gradient respect evaluate iterate denote diagonal log separable penalize square work give follow taylor expansion complete computed overall statistic current complete expand taylor thereby form quadratic surrogate problem problem use augment lagrangian describe practice iterate far expensive use date long improved iterate computationally part fdr smoothing repeatedly chain equivalent problem final chain method multiplier original approach denoise express slack could costly step avoid slack unconstrained yield whenever slack eq primal scaled scale augment lagrangian admm update stationary lagrangian describe individually step parameter update value simply separable soft eq soft thresholding operator subscript update must demand euclidean projection w v r objective size underlie change course diagonal linear cholesky reduce front cost system orient incidence therefore symmetric dominant nearly solver system implementation produce iterative provide approximate know solver oppose subroutine affect convergence therefore cholesky hope exploit specialized linear systems dual k greatly affect discuss issue dynamically primal residual calculate drive ensure happen thus next dual variable likewise large value scale meet describe separately amount rough bayes spatial fdr justify appeal stage indistinguishable repeat argument estimate simplify spatial homogeneity odd density formulate fdr distributional describe test problem poorly describe parametric form order produce reasonable variance central shape histogram test near come mostly nan versus empirical fmri comprise proceed construct obtain second taylor scale mean deviation curvature z approach test estimate apply family show fmri analyze assume unknown deconvolution dirichlet use recommend recursion flexible enjoy guarantee choice large right crucial structure top avoid tune hoc fashion adopt often taylor fdr across decrease grid warm find calculate measure small enforce image approach aic maximize generalize change stein involve distinct contiguous prior background remarkable freedom error aware analogous situation plug degree calculate absence number surrogate heuristic seem good freedom automatically improved panel likelihood two panel aic due admm appendix trace surrogate freedom problem much aic place freedom dominate produce bic balance problem recommend additional practical trivial efficiently scale na I spatially separate counting appendix eight cross site signal configuration large gaussian convolution signal spatial discovery rate dirac mixture convolution grey blue eight scenario panel show nonparametric corresponding scenario sense predictive reasonable job deconvolution right signal square ambient simulation four dotted grey correspond convolution curve simulate set nonparametric recursion signal four dash small compare procedure fdr explanation suitably rich trick basis use basis grid baseline group rate eight desire fdr fdr conservative fdr technique fdr multimodal come bayes true power problem mode nan indistinguishable nan fdr come spline fdr conceptual essentially treat handle choose basis implicitly smoothness underlie straightforward smoothing path choose fdr edge happen coincide find require finally benefit involve dense fdr basis sensible fdr fdr greatly interpretable rate fdr fdr region fdr fdr oracle eight error set fdr smoothing true positive scenario consistently two fdr come close slightly fdr small signal fdr modern scientific many analysis exhibit ignore exploit fdr increase control discovery automatically identify strong area improve area first penalty lead slight equivalently toward ordinary adaptive concave problem present important future moreover feature prevent concern fdr smoothing conduct yield fmri three require quickly hardware laplacian gpu programming two approach plan choose model setting entirely principle tuning fourth fdr provide suggest area could fmri literature fmri specialized fmri generally place intend statistical lasso independently body summarize literature reader group analysis fdr effort beyond paper nonetheless r fmri set analyze section acquire process trial grid sequentially ms follow forced series one present presentation hard second block block acquisition versus easy fmri band tr ms te ms flip voxel mm slice orient back ac mm mb scan length fmri develop st motion
cluster robust contaminate estimation need establish global ignore seem remark separation prove consistent computing hold principle property mind justify strong improve weak compare desirable statistical theory statistical method derive example boost view stop overfitte path another fan li fan li prove oracle property besides minimax huber least simultaneous variable evaluate performance begin focus nevertheless reasonable framework asymptotic discuss find subsample reader van van limitation issue seem valuable theorem present selective broad many list number real datum obtain direction property recently yu leverage algorithm square aforementione local algorithm wang liu forward greedy kind estimation problem derive one appeal start li valuable begin challenge fan liu analyze big datum ability expect relate national natural china grant thanks wu grateful key laboratory chinese sciences van square university york prediction van new york fan penalize zhang york york york york annealing algorithm york york company york mit new york zhang h unbiased minimax penalty proposition section section well mathematics sciences ac cn global solution optimization well likely global solution statistical believe call optimization display optimization statistic widely study paper aim establish discuss various several commonly encounter problem subsample see reasonable indeed well property cluster outlier subsample combinatorial estimation separation notable maximize brief rely good statistical extension estimation huber obtain function nonparametric maximize parameter model fit minimize justify fit least square smoothing fan view nonparametric setting minimize good subset norm regularize regularize see involve analysis fisher discrimination sphere projection et al likelihood special besides determinant compute multivariate number design optimize certain factorial design wu design optimize li geometric criterion utilize statistical support machine boost function regularize commonly minimize select thorough indicate modern meanwhile anneal attain global handle serious due time yu difficulty rarely verify solution objective minimization two understood sense seem likely global statistical usually use believe well principle strictly verify problem actually decision ever complex author formally discuss whether fairly dimensional tell sum square possess well screen property asymptotically view solution constrain therefore fitting actually discuss exist experimental design help reason theorem theorem optimization problem possess separation look result maximum good subset optimization optimization perspective refer good inference presence determinant subsample perform outlier contaminate model statistic hold experimental design criterion consider eq variable subset specify design square estimate solution design lead well whose variance therefore justify conclusion criterion design discrepancy seem statistical relate desirable criterion construct minimax act correspond experiment sample thus sequential deal objective sample study space simplicity map measurable field theorem provide two decision estimation second cover variable subsection confusion situation case remark inference lie desirable subset contain solution practice take decision lie another contain bad say strongly separate separation separate property denote almost surely note imply strong distinguish two former generally convenience arbitrary sequence positive sequence number roughly speak decision property describe strong contradiction n strong separate directly imply condition separate value nn e p np avoid problem countable obtain separate statistic p separable subset example always countable dense sense interest optimization need sequence decision contain decision statistical concern lie another almost say separate subset say theorem write separate two condition weak comparison separate separate sequence countable nn respectively value countable separation property sufficient imply confusion depend situation estimation decision space paper omit write despite effective establish many statistical likelihood let density respect problem minimize commonly multiple compute concern lie neighborhood estimator go infinity decision discuss great likelihood view objective fx dx furthermore bx limit segment subsequence number may triangle restrictive require aspect robust inference correctly identify criterion parallel subset subsample exactly usually van e knowledge asymptotic subsample discuss statistical formulate decision integer subsample objective section attain type serve decision asymptotic ia finite measure subsample eq determinant look low determinant minimize underlie assume normal assumption separation denote sufficiently denote x x p fx fx subset consider dx fx fx g I r sufficiently large banach exist condition fx therefore hold matter away indicate outlier class minimize population possess robust discuss see wu subsample new kolmogorov function discuss property separate weak c subsample contaminate far contaminate imply kf unable distinguish strong prove hold similar letting consider na therefore assumption complete immediately far simulation method likelihood let outlier iii generate objective become correspond selector simulation time likelihood end outlier conclusion likelihood subsample iii well well problem base linear estimate high describe I nx I ip r take correspond corresponding estimator mr mr mr assumption denote fix definite n limit rx k covariance hold array satisfy separate na na n n n n lx n n rx n rx rx x rx rx ia x rx x rx h follow conduct model generate matrix matrix search generating size simulation fix repeat time ci selection van nonnegative scad fan li zhang become screening rule reasonable condition well sub satisfactory continue discuss fix linear without full submatrix bic correspond square x know minimize bic lead prove separation I ax definite asymptotic fu strongly separate ax n h ax h pa aa pa aa aa aa complete almost objective rao wu bic special corresponding increase fan screening specify stage coefficient stage possess screening retain fan fan fan et al
originally convert angle w z tangent see say dimensional multiply project specify project pn pn apart mode really similar shape general comment cosine fix distribution unimodal circular mean identity create positive cosine high axis vary flexible shape result distribution near axis increase mode axis bivariate series x ty hmm indicate state indicator time otherwise transition literature circular latent give relax independence circular get fx k bivariate let normal covariance build joint r circular regression formalize circular flexible component circular specify r see circular regression one propose circular circular cl k dependency circular cosine circular circular argue normal density pn pn simplify cl ki ig ig lead f b b marginal posterior mcmc precisely gibbs sampler metropolis conditional full multinomial depend entire k eq mix slow speed try metropolis mcmc decrease dimension marginalization decrease simulate employ step slow toward burden increase marginalization impact simulate simple carry gibbs estimation switching issue class hide state parameter inference output various propose recent tackle decide post chapter regime jump non approach however goal demonstrate cl hmm circular increase aic minimize among evaluate criterion mcmc maximize map bic aic f bic aic generally indicator suggest estimator carry simulation recover empirically unimodal circular ignore linear plan study cover scheme uniform shape circular separate dataset three model diagonal cl circular unimodal circular indicate cl time element consider scheme summarize characterize consider regime circular one circular dependent l scheme variable b circular figure cl circular unimodal ig aic criteria regime frequency cl cl cl respect cl model perform true former latter hand bic excellent exception may expect perform amount dependent circular cl latent high course affect estimate need regime ccccc ccccc ccccc predict scheme cl cl cl cl cl cl c cl cl cl cl cl cl cl briefly summarize simulation estimate cl cl lead suggest whenever cast cl cl circular distribution unimodal interval ci ccc cl cl ci ci ci ci ci ci ci situation randomly accordingly c cl miss rank probability circular circular one measure distance circular identical linear cl cl dealing point view dataset iteration need thin study resource make project education di computational hour hour reach finally cl wind record semi locate km datum record arise environmental study value record profile wind event south north north arise warm air low pressure cell move episode occur pressure interior behind pressure far wind usually associate flow link interpret wind regime aic component suggest decide two ar circular look value choice three regime provide separate interpretable state classification probability display expect transition essentially persistence regime wind probability indicate direct episode unlikely confirm wind event period ci cccc ci ci ci ci ci ci ci cl distribution circular circular circular variable interval ci ci ci regime regime natural velocity concentrate respectively circular regime north episode south circular correlation circular regime hypothesis statistic interval circular condition first regime interval variable circular relation circular cosine thin check tool cl bayesian explicit cl likelihood circular circular component project circular fairly model bayesian arise several posteriori evidence marginalization conditional circular linear make wind new circular interpretation inferential simulation circular concentration derive circular application consider movement modelling drive development include circular circular project interesting propose latter dirichlet
activation nonnegative firing update learn decay epoch back normalize column set example update procedure repeat one practical issue number dictionary assign produce rich stage avoid procedure encode activity mean close across activity correspond activity simulation provide style align center width cm style width encode sn encoding sn sn sn effect example extend alternate pick example ideally example dictionary close truth uniform heuristic attention part choice element apply goodness dictionary element goodness motivate idea large reconstruction ground truth truth select dictionary regard direction reconstruction example critical produce one level put limit observation noise collect happen beneficial measure snr rare salient define location exponentially make channel channel four selector choose goodness selector select example dictionary sort operation epoch simulation possible goodness selector initialize example max epoch good pt nn loop pt dictionary encode example ground quantify epoch minimal span permutation display positive investigate coherence bound coherence regardless high set comprise x dictionary relatively incoherent easy letter rotation sign artificial violate choose epoch activation magnitude sample snr pick set epoch conjunction goodness trend baseline selector across epoch encoding figure column selector stage epoch contrast surprising poor estimate nevertheless good selector soon establish activation exception selector work closely track selector design use map dictionary ht rl ca dictionary cb encoding order epoch distance end leave order robustness algorithm modify ratio db element original good selector across element suggest great advantage selection extremely complete rl gradient descent dictionary indeed dictionary epoch special success strategy contain procedure implicitly rely identify example dictionary sophisticated grouping provably dictionary inter characterize inter whose work generative spatially generation apparent thus case intuitive pick information dictionary sample benefit inference validate snr last epoch figure algorithm pick high snr correlation snr weak suggest factor drive factor contribute spread reveal selector pick select example distribution measure select example histogram tends dash weakly predictive overall suggest snr cb dictionary nd plan instance selection lead empirically paradigm suited stack autoencoder case interesting layer explore question future task font font font edu uniform feature datum active current accelerate inspire sparse activation work code hypothesis hypothesis decade low code idea dictionary starting extensively explain capture efficiently usually uniformly training resource relevant
available construct combination list converge one coincide rely violate open tend kullback leibler weight n n fx I fx primarily focus bayesian procedure aggregate h la impose primary work prior bayesian prior minimax convergence la able minimax aggregation utilize extra da able automatically exist optimality require tune open fall list posterior put proportional reality hope inequality propose novel interest theoretically discrete burden continuous avoid combinatorial potentially computational distribution widely probabilitie rigorous relationship concentration relate mixture posterior effectively extra emphasis study moreover component fix allow unable impose sparsity rest aggregation describe ca simulation theorems technical deferred provide detail aggregation risk base understood design truth possess expect rate estimating case sf f extend precede gain minimax la sparse la ss la arbitrary constraint approximate extend show risk aggregation structure pdf dirichlet commonly simplex priors allocation concentration display dirichlet change moderate small distribution concentrate small capture sparsity htp concentration characterize absolutely exactly sparse relax sparsity consider index sparse concentration property symmetric dirichlet consequence section simplex utilize process fact dp reflect concentration favor suggest uniformly sparse pattern true general method satisfy f put almost mass able concentrate around assume square integrable fx assume convenience theory generalize problem belong aggregated try element mf different produce aggregation dirichlet da symmetric dirichlet favorable concentration property near minimax contraction dimensional shrinkage sparse place dirichlet include power la design write represent training mp aggregation high special parametrization determine correspondence prior prior parametrization example prior induce mind double dirichlet dirichlet equivalently q gamma pdf dirichlet form study focus efficiency aggregation la constant example characterize ca observation specify condition minimal leibler put almost whose towards avoid argument behavior error deviation proof justify setup frequently f situation th make integer jx jx covariance uniformly cauchy bound uniformly condition use part study hellinger gaussian one mean impose without sparse suggest radius proportional ca special procedure iid prior da eq ff non uniqueness quantify la fast constant design since normalize condition condition aggregation linear b assume study consistency boost high regression sparsity also aggregation gain also constraint satisfy spirit identifiability approximate converge therefore sparse sample b q fast among minimizer suggest concentrate achieve explain sparse uniqueness happen suggest depend estimator addition estimating estimator strategy follow divide learner algorithm learner j aggregate sample learner give become ns ns dominate impact equally plot credible ns non credible interval nonzero dirichlet prior change analysis htp cc figure la weight especially model robust result recommend choose conduct aggregation truth I training size first rf neural nn bayesian ba super learner sl sl package implementation base learner burn summarize table root square error rmse replicate cm lasso cart svm sl ba second fit response predictor learner large covariate base use full addition compare ba sl voting learner subset base summarize sl ba apply ba datasets uci repository cart forest regression neural learner discard half burn dataset forest aggregated ba learner dataset performance base learner sl datasets c concrete forest log cart rf sl ba auto divergence accord asymptotic theory iid aggregation problem convergence put mass around section iy concentration become dominate pc imply cc kp problem iy concentration characterize concentration property dirichlet dirichlet concentration cd characterize characterize interest da volume lemma prior concentration property thus part da play role characterize da aggregation property double taking assume property second ensure convergence prior put show dd da probability space sparse approximate la characterize term cover number integer q next complementary utilize stick da eq joint expectation copy distribution misspecification design construct test la design la remark section uniformly bound bound type fix design assumption minimizer kl aggregation space la mean similar need provide da b jensen chebyshev combining yield converge cn lemma constant select sufficiently n n cd combine fact prove construct bn lemma similar help dense accuracy therefore construct satisfied conditioning sparse fa fa let cover net interval net enough ga argument point integer exist md mn rest generality nonzero since j denote gamma q approximation conclusion double allocate application conclusion adapt additional constant similar instead part result apply index satisfie sm imply prove first conclusion fa b lemma nonnegative consider dp unit dp stick representation dp index let combine definition k application markov inequality yield apply I yy df np mp f yield acknowledgment support national institute environmental sciences national health mcmc la idea conduct metropolis hasting update distribution bayesian apply stand old
broadly comparable lasso frequentist method recovery mode give immediate bayesian section bayesian posterior contrast result quantification combine contraction latter asymptotically bernstein new identity crucial separate prior density instance laplace density even prior uninformative contrast parameter bayesian infinity zero small essential bayesian framework organize prior nonzero investigate ability coordinate distributional apply credible show recovery defer section parameter true generic vector ambiguity refer let column prior bayes borel laplace usual situation however setup induce laplace large nonzero undesirable assume value decrease natural distributional posterior hold read precise interpretation regression error unity scaling follow case light sequence condition create prior allow extension precede example unit variance location replicate response regression refer fix input th sample exclude shrinkage situation sufficient turn prior constant dim prior rate decrease reflect coordinate mixture dirac zero laplace briefly comment one replace prior prior model value laplace density even necessarily definition simplify setup discussion concept compatibility compatibility give compare norm predictive consider nontrivial compatibility simplicity value replace denominator numerator comparison norm schwarz replace compatibility restrictive condition concern compatibility p minima number compatibility dimension define small singular impose cauchy impose respect whereas suffice reconstruction matrix division unity submatrix coherence maximum dimension interpret reconstruction norm relax pair compatibility number sparse closely inequality evaluate infimum away zero thus compatibility value certainly zero size multiple e mutual three index previously reverse compatibility mutual coherence possible restrict isometry extensive discussion refer compatibility oracle bound prediction norm contraction spirit contraction coherence analogously estimator supremum coherence way albeit allow norm verification compatibility preferable compatibility valid compatibility maximally design moment situation coherence number true possess exponential compatibility away regression small survey compatibility eigenvalue see column diagonal satisfy example satisfy cover aspect case eigenvalue rate equal final consider vanish block handle index include proof indicate remark misspecification show true dimension interesting constant compatible constant make choose dimension read proof also dominate contraction rate suitable appendix concern center compare estimator concern rate contraction relative compatibility number bound thus rate recovery vector bound zero theorem uniform furthermore every consequence theorem upon choose assertion instance design assertion large selector develop statement compatibility oracle oracle besides give quantify size coordinate put qualitative manner compatibility matter model unnecessary truly nonzero selection nonzero detect nonzero exclude theorem compatibility condition term threshold similarly satisfy true replace large mass sense shall contrast lambda include magnitude bias small lambda regime choice small nonzero lambda lambda regime simplify thus regime regime let submatrix consist column square I correctly possess prior would equivalent von normal distribution dirac measure satisfy sufficiently neighbourhood true lack parameter outside different eq projection subspace decompose x two lebesgue improper mixture correspond weight sg coordinate hand still weight bernstein von lambda regime improper select uniformly subspace dirac note define choose improper enough improve prediction interpretation projection computational auxiliary variable typically laplace mixture implementation unity present problem prior study setting consider recently last year model curse besides time monitoring concentrate low also apparent result make thousand exploit solution graphical genomic view correct move model hundred smooth posterior set coordinate geometric move distant density performance equal exact analytic limited spike decrease model spike likelihood survey setting find reconstruction empirical pseudo bayes approach spike show put present posterior quantity implement computation hardware suffice design correspond ratio support laplace density side bound prove laplace side display occur event bound denote inequality display surely ix integral decay power follow display cauchy possess normal variance I tail normal dimension prior satisfy ty measurable bayes follow old schwarz therefore value right combine display q see infer choose proof posterior support intersection event compatibility q assumption suffice x display calculation theorem tend similar eq first assertion x total distance measure restriction bound constant one hence asymptotically precede restrict measure connect remainder absolutely denominator satisfy second jensen logarithm display set view shall factor right q relation principal submatrix jj assertion event property yx ss inside standard possess freedom apply give ps marginally upon tend theorem imply assertion follow assertion enough sm ss bind ms precede display square dimensional span onto abuse onto shall term start note x x mx mx first cauchy schwarz sx sx j sx together sx normally variance order sx op exceed side probability tend desire positive reasoning precede asymptotically acknowledgement thank grateful helpful discussion supplement supplement state von lambda proof supplement regression prior usual choice regime strength center give von lambda op q approximate normal estimator bias covariance lambda correspond argue lemma assertion abuse notation write right instead c disjoint assertion limit mixture obtain enough coordinate coincide index draw law index law eq thank assertion sum collection contain index write leave multiply element constant statement indeed establish sr ir si last center note guarantee coefficient together lemma nr exponential possess cauchy inequality ball bound conclude small result prior eq vary j orthogonal projection j volume j sd expectation sum contain term sum last jensen hold product normalize denote kullback leibler divergence cs expand quadratic form symmetric term p tv tp tv xt maximal linearly subspace successively union let lebesgue denominator formula translation invariance measure jensen lebesgue weight definition vc vc vc intersect deduce v vc vc euclidean unit ball one deduce enough display upon hold
sect sect equivalence sect sect finding explore extract invariant local detector various incorporate cnns one build invariance architecture yet another mechanism learn rather systematically knowledge work perhaps believe quantify property manner sift cnns function notable invariance representation formally invertible hence existence representation predictor case cnn require input image intrinsic geometric representation capture practice affine image invariance act invariance regard goal establish image object image change study systematically possible representation transformation study invertible satisfie mapping cell direction cell component symmetric permutation one rotation approximately rotation implementation densely sift network sense obtain boundary effect result equivalence must discuss discussion focus deal transformation mapping simple affine g g choice restrictive initially permutation hence permutation empirical datum natural image amounts adapt particularly quite encourage row q reflect component induce exploit convolutional translation locality neighbourhood index triplet neighbourhood site close vx vx measurement site fractional neighbourhood transform site g v combine regression activate neighbourhood location neighbourhood sect histogram hellinger distance layer cnn orient well understand orient cnn train end objective preserve quality ground label train case image set cnns target pre train predictor sect class encourage convolutional say transformation convolutional reason learn implement layer cnn consist map filter neighbourhood sect intuitively purpose channel fall integer coordinate permutation lattice rounding distribute site bilinear interpolation study cnn equivalence orient sect predictor cnns train imagenet transformation loss cnn interpret convolutional representation bank permutation may sect study mapping sect move examine sect mapping apply regression style align leave none legend align legend rr rr fs feature reconstruction cell hellinger learn map different rotation constraint impose structured value neighbourhood size train discriminate cat gradually scale l rr r c second regressor array unless sparsity sc learn fs rotation scale support sect evaluate finally learn validate explore formulation predict feature image histogram validation focus predict small baseline array array avoid boundary rescale restriction later transformation objective square ls ridge rr fs per output fig surprising array f zero error rotation exact sect error fs remain em legend column legend style align none cell align legend fs fs opt fs fs em learn vertical cnn fs task orient reconstruction bottom classifier image good red report evolution flip oriented rotation rotation hard reconstruct suggest learn address boundary effect substantially nothing original mapping cnns conv fs less training orient compare sect substantially less task orient well converge much fs fs significantly small reconstruction depth matching intuition deeply perfectly transform layer transformation vertical next geometric horizontal vertical rescale transformation mapping leave unchanged cnn implicitly invariant vertical rotation however learn substantially generic good cnn sect practice never high invariance replace row evaluated accept relative corresponding channel large approximately result rescale cnn notable invariant conv second conv tend conv possible vertical rotation flip flip c top top top top top top mapping layer flip sc channel r layer top several cnn obtain portion convolutional portion look representation replace sect sect cnn layer train mit place train mit place identical top hybrid learn layer sect fact intuition make feature channel compatible slightly deep specifically conv conv fully conv conv code conv task bad dramatically chance compatible complement investigation far section mapping sect output label fully rewrite advantage demonstrate pose pose object pose cat rotation rotation sample degree rotation line cluster r n contain example gmm train affine mapping cluster center pre image contain fs sect cat version rotation consider scoring report cnn conv conv nearly mapping curve regressor conv conv conv ms report cat head pose direct regressor error degree residual rotation normalise frame method predict h cm rgb rgb legend legend style align legend align leave legend conv conv em curve rotation affine regressor c pdf pdf pdf pdf rotation bottom cat face estimate affine pose represent regression conv within representation several layer deep cnns transform image architecture deep degree specific addition tool regressor elegant science image encode task apart function assign incorporate useful normalise class irrelevant orthogonal feature irrelevant contain encode direct vector reveal object class viewpoint nevertheless orthogonal irrelevant direction achieve invariant linear manually argue task normally different task classifier regressor discard task give image rotation translation invariance alone always ability distinguish differ invariant seek however eliminate factor simplest specify rotation exact invariance closure transformation transformation example closure even pixel minus font legend title style macro despite importance image representation histogram convolutional network mathematical representation invariance transformation effect visual establish empirically introduce layer cnn reveal aspect include cnn certain theoretical structured output regression demonstrate image notable orient bag visual word code super generation convolutional popularity representation remain generally invariance contain r conv tf fs tf tf fs tf fs tf fs tf tf flip tf fs flip flip fs tf fs
improve replacing step conditionally log directly either imply cm conditionally maximize function van combine sense satisfy ascent property share em simple integral pdf thus large replacing give multiple miss em iteration secondary stochastic expectation use carlo em complete single log model structure draw single sample draw carlo sample per imputation estimation pick maximize together map pdf prior penalty general penalize motivated term usually thus solve map modify em framework prominent field prior knowledge prevent algorithm change model em exponential complete variable include poisson pdfs moderate information simple pdfs censor exponential complete data occur observer record datum common trial e medical testing product reliability analysis algorithm replace latent good maximize em space depend generalize nature function derivative numerical complicated coordinate decompose pdfs come population recognition random population proportion discrete pdf pdf pdf sub fy rewrite ease estimate population variable use derive mixture sub population change pdfs population popular area cluster identification map distinct distinct space specifie sample maxima dot contour maxima alternate emission construct emission emission image level chemical scan start array detector around fine emission picture pixel detector simply determine reconstruction set random model population distribution probabilistic detector observe detector detector record geometry sum poisson response individual pixel complete intensity parameter detector emission detector depend detector define go average emission intensity function em lot space use iterative approach incomplete complete fit logistic mm generalize optimization mm specifie function tangent parametrize mm algorithm instead ascent ideally easy em al fit modification constant shift shift depend ascent ascent establish em primary parallel ensure ascent descent property imply curve tangent require observe point precisely another mm use satisfie effectively create local quadratic every produce true tangent estimation argue design less designing complete extra flexibility publish wu optimize objective converge compact close mm global theorem solution interior stationary assumption closure follow ascent property boundedness minimization continuity em example mm satisfy closure broad include closure may slowly information em implementation get complicated converge maximum demonstrate address slow improve rate application chapter demonstrate speed improvement idea expectation speed noiseless show maximum surface positivity corollary em gaussian censor positivity simplify quadratic benefit distribute decrease review formulate behind noise noise improve introduce corollary present variant present theorem positivity theorem law large sr noise amount improvement mutual input early benefit physics biology early description benefit include ice snr improvement brownian particle chemical inspire nonlinear signal estimation describe screen exhibit necessary sr weak signal signal screening help explain observe benefit system benefit pixel monotonic curve non monotonic sr signature curve noise em noise benefit em unlike almost sr benefit apply cauchy model censor benefit condition suggest condition last noise condition em important time shape signature convergence ht mixture deviation normal standard gmm vertical bar noise benefit sr involve threshold sr additive depend independent weak benefit dependent em happen enhanced noise chain proper test converge fast gain benefit examine plausible statistic likelihood appropriate could ball depend ignore working realization alternate specifie experiment high likelihood pay alternative calculate bootstrap favorable likelihood result high condition describe likelihood idea condition sometimes pdf pdfs em positivity condition benefit modify surrogate likelihood maximizer modify likelihood regular likelihood maximize benefit occur noisy value noisy perturbation current noise benefit familiar express pdfs information incurred pdf pseudo noise benefit pdf well theoretic description complete regular incur low difference condition guarantee follow pdf noise differential entropy additive keep average expectation distance pseudo likewise benefit sufficient noise relative assumption finite q integrable integrable permit benefit theorem apply complete positivity user condition generalize noise proof iteration suitably close towards correspond algorithm hold positivity imply noiseless algorithm number benefit noise sequence inequality converge equal imply convergence noise anneal continuity k q fy guarantees exist converge fix inequality theorem dominate condition positivity wise condition pdf positivity condition monotonically pdfs still useful effect benefit model special case corollary corollary population likelihood eq corollary dominate occur gaussian gmm pdfs pdfs ensure dominate population additive noiseless pdfs positive expand q hold conditional similar model pdf noiseless pdfs last prove condition corollary quadratic q figure exhibit benefit standard population mix proportion standard deviation start comparison evolutionary iterative gmm noise count noise similar ht cauchy add bootstrap estimation benefit dispersion equation condition benefit ht gmm dim sample dim sample optimal noiseless mixture normal use normal deviation gmm reduce admissible noise support depend value goal q side hold fall sub population take falls sub population valid fall tend tail cluster sub location gmm extend gmm covariance sub dimensions fy fy gmm vector population gmm estimation multidimensional quadratic positivity geometric description exist fall boundary hyper rectangle centroid eq population determine case multidimensional gmm set interval singleton origin origin illustrate geometry line ht min min dimension specify side length draw geometry towards centroid sub action dimensional sub jointly population jointly population state notation eq product degenerate suppose df pdf sufficient eq hold complete finite model iff sub determinant simplify thus eq give ensure summation conservative analogue condition specialize gmm condition us condition variance condition benefit condition cluster intersection dim case green mm blue overlap mixture boundary product dimension factor component dimension way mm noise geometry noise scheme ease increasingly dimension performance em seem theorem sort superior algorithmic function evolution evolve function come price evolution raise computational sampling step complexity sometimes raw fewer efficient keep noise case still noise benefit ideal positivity amenable inequality corollary complete pdf noise zero random variable decompose product censor unobserve latent gamma give pdf interval gamma density hazard rate describe benefit censor censor speed noiseless inverse gamma distribution censor estimate mean bar predict noise replace additive corollary method modify noiseless average noiseless noisy expectation schema current depend maximum replace em give decay scale iteration decay factor iteration zero need cause closed decay fix even final polynomial factor logarithmic anneal application variant schema generalize maximization simple replace ascent noisy schema allow outside scope involve modification step change step pick subject variant procedure noise change em gmm derive procedure gmm intersection dimensional version generator perturbation fall degenerate another add perturbation perturbation datum schedule difference em bar across add deterministic perturbation locate datum far origin perturbation decay scale perturbation like gmm converge em average roughly noise gmm dynamical interference interference map value gmm use scale fit set square ht algorithm logistic map scale decrease gmm gmm gmm depend apply gmm eq satisfy gmm analyze still singleton go probability noise value decrease condition eq eq maximal sort nest nest empty bound close interval hold fail happen positive produce benefit write intersection second property sub borel fall number converge tight size probability one singleton value identically condition correspond case guarantee em identically limit fall ht identically avoid probability even noise figure model equivalent define event figure event blind differ depend blind use blind blind average blind blind fail blind simple perform ht use anneal converge fast simulate annealing model reduction deviation theorem sparse zero noise benefit improvement entropy decrease benefit data f fx fx pdf show benefit central limit analysis term form random I probable apply population law equal difference arise pdf law number mean expectation positivity state standardize distribution variable noise noise positivity infinity positivity condition fail suggest condition match sample large result noise benefit mean noise benefit available likelihood nonparametric careful convergence sufficient condition benefit signal probable em speed satisfy condition blind noise blind noise condition obey positivity implementation level benefit variability surface likelihood average change control concern optimal noise vary show family cause fast speed noisy noise distribution easily flexibility regular comparative versus interference em deterministic interference three demonstrate important incomplete model cluster feedforward em backpropagation recommendation google news news netflix amazon recommendation often rely centroid classify computationally slow provably speed cluster benefit general include em noise benefit classification improvement classifier compare classifier training use train optimize classify algorithm find pre limit benefit corollary gmm benefit report classifier parameter show agree normalize misclassification fall reach minimum reduce misclassification beyond optimal misclassification ht gaussian special benefit different decaying help supervise competitive algorithm fully theorem generalized version benefit algorithm attempt differ mahalanobis similarity algorithm assign sample centroid close centroid attempt np centroid centroid partition look minimize cluster fuzzy agglomerative cluster probabilistic true population assumption fold mixture bayes population involve estimate parametric estimation benefit derive framework cluster apply population positivity optima suitably em positivity reduce positivity correct centroid additive count square decay constant function give mixture pdfs population mean current gmm posteriori assign parameter eq benefit extend benefit relative em optimal optimal misclassification parameter benefit misclassification free em misclassification procedure satisfy positivity condition reduce condition misclassification decrease optimum follow iteration count eq reduce noise satisfie classifier perform converge less em enhance algorithm dimension benefit figure misclassification procedure partition centroid try centroid euclidean distance indicator arise near classification indicator procedure optima em gmm measure fuzzy much belong population bayes membership cluster membership centroid assignment cluster modify gmm benefit population spherical know proportion membership gmm reduce cluster matrix update equation em centroid em hard change sum hard reduce centroid diagonal proportion knowledge estimate mix proportion update hard occur em sub result probability equivalence confirm predict benefit resemble noise supervision circuit art interaction bottom activation neuron activation input internal representation art substitute supervision field system signal recognition exist fail category within specify match flexible mean art pre specify cluster art update characteristic learn art open question provably benefit competitive pattern adjust weight competition competitive train centroid quantization converge centroid competitive noise system use competitive like imply topology compete neuron fan vector neuron centroid distance approximate dynamic layer center winner topology competition win neuron fan neuron nearest neighbor pick win neuron fan close current move winner centroid little close incoming first win vector equivalence alone benefit second step increment prevent direct em algorithm prevent theorem centroid quantization initialization linearly decay coefficient winner update win quantization update win quantization modify version pseudo alpha rewrite stochastic pick winner closely resemble deterministic competitive neural modeling memory neuron input neuron field sigmoid competitive neuron likewise scalar competition function zero rise winner winner result imply square connection matrix band winner teacher know know fan represent winner fan winner class increment get minus rather sign winner increment context adaptive classifier differential competitive hybrid replace competitive win differential structure law notation activation rather differential neuron learn competitive neuron competition activation reinforcement increment blind law compare simulate competition time derivative neuron approximate competitive simulation competitive cluster simulation white training scale standard allow intensity entire decrease variance distribute three anneal schedule neuron q win quantization perturb version h noise anneal pick winner win quantization anneal schedule noisy winner win quantization four show figure improvement noise speed expectation maximization fairly competitive noisy expectation guarantee extend benefit method also two co co cluster movie movie database cluster semantic use filter document benefit benefit also co generalize chapter hmms probable region space new algorithm train sufficient positivity positivity sufficient hmm figure architecture hmm reduction number take converge likelihood confirm speech hmm observe speech effort neither speech theoretical review hmms em point present boost hmms test training corpus ht variable series speech biology vision hmms speech speech use hmms markov hmm single map state contain density gmm common pdf coefficient mean matrix tuning hmm respective sequence make difficult concavity hmm indicator forward backward function maximize respect lead condition definition pdf em converge positivity enforce simplify gmm positivity positivity additive observation index latent variable noisy positivity eq maximize gmm covariance differ noisy modify noiseless positivity satisfy condition increase h kt st jt st st I jt kt kt n n kn modify perform embed create large hmm sub modify suitably produce positivity variance anneal scale noise iteration decay dataset setup frequency coefficient compute window also second vector energy hmms gmm metric hmm percent improvement percent reduction iteration likelihood marginally positivity number small careful addition average ml hmms give sufficient condition simple constraint gmm per frame hmm develop noise benefit neural mathematical consist neuron transform input signal usually sigmoid connect biological transmission layer neuron feedforward feedforward absence backward self connection feedback neuron distance black inner sep neuron center node neuron edge input layer feedforward network popular many speech translation intelligence computer network biological network acceptable white feedforward borel measurable backpropagation feedforward goal bp feedforward ff example say ff minimize function minimize feedforward network offer insight task lack explanatory algorithm much contribute training error unit architecture backpropagation propagate signal via chain rule local error feedforward expectation consistent miss bp hide match noise positivity condition template noise bp feedforward backpropagation fall backpropagation convergence activation layer neuron noise examine stochastic generalization noise generalization add activation reduce section review detail backpropagation em training discuss simulation bp ml neural parameter layer network deep neuron layer consist neuron neuron activation represent value encoding neuron connect hide target neuron matrix backpropagation follow entropy target backpropagation ascent give gradient ascent log formulation use square observed output error call configuration higher later development backpropagation convergence linear replace value neuron layer value backpropagation assume estimate optimal square square estimate backpropagation partial derivative backpropagation backpropagation epoch eq epoch know entropy expand backpropagation backpropagation show intuition greedy probabilistic hide bernoulli activation formulate ml feedforward parameter e resort carlo sampling converge true follow bayes easy approximation independent identically distribute sample dimensional kronecker importance importance activation output neuron maximize backpropagation perform backpropagation bp fast positivity depend converge assume keep positivity condition form condition activation neuron consider benefit gibbs positivity ml training feedforward neural activation ratio sufficient condition positivity become positivity inequality log activation sufficient noise illustrate hyperplane change output positivity ml feedforward neural sufficient outside sphere network average ht noise sphere use approximation quality substitution quickly neuron backpropagation mnist pixel lie fed pixel neuron use neural layer output digit auto encoder three pixel three layer gibbs activation classification class carlo encoder show backpropagation backpropagation add gaussian mean entropy add mean epoch use monte square backpropagation backpropagation allow develop backpropagation simulation digit recognition square cross nn feedforward learn feedforward backpropagation backpropagation noise chapter application shorter important training take stack neural deep neural benefit present predict estimation describe previous belief bayes heart application spam belief oppose experiment difference two kolmogorov acceptable prior fisher pearson provide free bayes publication involve inference modern individual application estimation highlight frequentist find mle simplicity property intuitive interpretation frequentist bayesian depend frequentist powerful prior rest chapter chapter corruption framework thus previously framework evidence belief build bayes evidence update compete prior belief hypothesis give evidence occur depend converse unconditional disjoint exhaustive follow partition state measure evidence compete hypothesis inform belief hypothesis well compete correspond drop normalize sufficient converse probabilistic maximization inference abstraction hypothesis realization expert ultimately come data chi square kolmogorov part application determine come could expert source accurate pdfs application limit close small pdfs conjugate produce close posterior pdfs posterior come three normal display conjugacy relationship beta likelihood poisson count prior datum htb beta unit interval prior success binomial coin bernoulli shape beta conjugate beta combine binomial produce new beta posterior trial mean square square conjugate still negative replace conjugacy dirichlet multidimensional dirichlet conjugate priors conjugate pdf pdfs chi generalize pdfs sided population mean poisson poisson gamma combine produce real population mean normal normal hierarchical still variable prior sequential previous pdf become pdf conjugacy greatly simplify chain monte posterior pdfs statistical optimal point estimate question belief give new produce pdf measure spread base spread answer incomplete incur wrong decision parameter estimate incur much magnitude function optimality high average estimate calculate subject bayesian optimal bayes opposite formulation decision economic rational choice economic utility bayesian pearson basic testing decision problem determine ideally engineer square loss conceptually minimization loss prior possibly improper maximum likelihood ml uniform constant thus maximization invariant scalar multiplication mle take hold unbounde improper pdfs pdfs integrable ml minimize strategy address estimate conservative estimator typically exist alternate value risk compare admissible rational scenario pp form person game minimax minimax strategy conservative estimate need description specify parameter measure generality around estimate inter variability mode bayes estimate credible familiar frequentist credible subset characteristic confidence interval statistic credible unknown credible interval interval unique bayes belong credible interval credible credible mode chapter frequentist approach also address issue method issue point exposition assume pdfs accurate fail corruption incorrect next chapter model observable datum source confusion randomness datum parameter bayesian inaccurate possibly form corruption raise reliable data approximation address effect analysis posterior pdfs apply bayes pdf fuzzy bayesian express description form algorithm expert linguistic grow function law fuzzy tractable fuzzy carry fuzzy show fuzzy fuzzy scheme likelihood well approximate later hierarchical chapter carry prior model circle draw black distance thick rectangle connect parameter knowledge form rule fuzzy likelihood pdfs pdfs describe fuzzy rule close also adaptively fuzzy rule closed range prior capture expressive linguistic figure tune skewed pdf simple rule fuzzy set fuzzy skew fuzzy fuzzy description occurrence rule rule small small large medium scale cauchy fuzzy tune shift probable set simulation rule quickly learn implicit prior fuzzy reflect evidence inform expert fuzzy reasonably bayesian approximation uniform prior lead fuzzy approximation time fuzzy converge adaptive centroid compute observe probabilistic evidence process little guess expert source source additive fuzzy output uniformly compact learning cluster gradient descent fuzzy allow user add rule fuzzy effort prior likelihood improve review behind fuzzy show fuzzy approximation approximation scheme fuzzy doubly bayesian fuzzy prove uniform fuzzy hence fuzzy system fuzzy rule input interval approximation expense pdfs truncation leave normalization fuzzy output centroid sum scale mx set centroid centroid form ix ip jj rule drop scalar care simulation fuzzy value theory shape cauchy adaptation extensive perform square user fuzzy give corresponding fuzzy practice fuzzy triangle c b b give direct measure inherent uncertainty define current rule cover affect fuzzy affect extent centroid tune parameter depend unconditional mixture density contain rule multidimensional fuzzy suffer general rule turn fuzzy patch tends quickly fill rule extensive high fuzzy approximation extensive fuzzy dimensional iterative chapter represent closed exactly closed strong exactly volume fc ff structure rule bound pdf simulation set function gaussians centroid fuzzy directly available hold bound posterior pdf laws adequate practice train numerical datum use chi kolmogorov pdf figure prior pdf pdf train sufficient simulation approximate train learn prior available use tune part area law convex parameter derivative rule law centroid form law manner expand ht success sample fuzzy approximation time sample prior likelihood approximation tune dispersion parameter triangle cauchy laplace generalized tangent set example six fuzzy learn law law eq law q law law symmetric parameter law eq double generalized set law reverse adapt part maximize fuzzy give law eliminate partial derivative ascent posteriori law six fuzzy pdfs pdfs six law software approximation fuzzy program compute square fuzzy uniform evenly spaced part assign dispersion initial volume centroids pdf location spaced rule law adapt harmonic decay pick numerator approximation representative simulation cauchy set illustration gave simulate six set produce pdfs gamma prior side normal truncated likelihood likelihood narrow support prior fall tail accommodate unlikely setting prior strictly bound prior integral well behave fuzzy cauchy respective squared uniform respective square approximation gamma prior respective learning sample iteration error fuzzy uniform interval training iteration mse mse generalize noiseless noisy pdf histogram sampling pdf systems still pdf random bin equally space pdf beta rescale rescale pdf random central location bin figure second example deviation separate case produce random sample pdfs pdfs pdf gx fuzzy rule iteration figure fuzzy bayesian conjugate pdfs h rule iteration next structure fuzzy generalize centroid long vary posterior expense centroid part form fuzzy accord corollary computationally first special dirac delta arise become law centroid substantially integration delta approximation curve normal cauchy pdfs alpha stable pdfs parameter pdf dispersion go zero characteristic fourier exactly dirac delta equal right hand fail narrow unless maintain unity second system several narrow rule uncertain compare constant pre structure another tractable centroid occur xx gd gd discrete cover system lot linguistic theorem suit scheme approximate approximation free uniform feedforward ff network approximation approximation white ff borel function ff sample back training feedforward procedure fuzzy ff fuzzy ff network linguistic simple case ff ff representation use nonsmooth activation hard ff compact domain equipped multiplication rely approximation generalization compact basis spline bernstein polynomial unstable implement use doubly pdf pdf doubly fuzzy fuzzy fuzzy growth rule lead old iterate ht gx n fuzzy base doubly pdfs rule gaussian approximate likelihood approximate doubly uniformly pdfs mild derive approximation convex fuzzy bind proof require let likelihood dimensional approximate fuzzy approximate sample everywhere f pdf approximations function bound expand error almost everywhere derive parameter measure compact thus totally invoke extreme extreme compact attain maximum maxima minima continuous right attain thus q inequality theorem maxima depend value sample pointwise improve extreme ensure factor finite bind attain minimum assume h denominator error limit guarantee restrictive determine satisfie proof lead uniformly respective nf corollary reveal fuzzy sum produce centroid fuzzy mp jx jx centroid centroid likewise part set sum centroid induce turn produce min centroid centroid centroid even assume minimal centroid centroid centroid doubly fuzzy centroid sum since mp argument thus bound sum bound centroid sensitive centroid divide minimum become centroid bound correspond least informative centroid multidimensional multidimensional component centroid hypercube max r fuzzy fuzzy density descent learn tune might might prior lead fuzzy neural problem conjugate doubly fuzzy general bayesian iterative chapter chapter fuzzy put next increase theoretical complexity conjugate track bayes fuzzy growth conjugacy conjugacy semi conjugacy conjugate size restrictive conjugate nest put prior uncertain appear pdf another demonstrate technique common inverse gamma pdf uncertainty conjugate wishart covariance bayesian pdf uncertain parameter involve make represent circle minimum size mm thick rectangle black label label connect edge connect capture another unconditional pdf conditional describe hyperparameter conditioning prior add integrate remove dimension benefit flexible proxy approach lot marginal approximation fuzzy pdfs conjugate hyperparameter pdf gamma ig conjugacy truncate gx n posterior use fuzzy gamma ht like fuzzy give hierarchical triple build double prove triple scheme pdf input extend call quite depend polynomial function statement require notation pdf denote set qx gx gx xx approximation applie allow prior almost everywhere x f qx prior approximation bound depend expand get assume everywhere finite lebesgue extreme compact attain maximum extreme minima continuous domain set similar ensure maxima depend well arbitrary depend note continuous extreme bayes upper maximum extreme assume gx gx denominator error zero guarantee involve arbitrarily satisfy appropriate integrate nuisance away compact lebesgue eq exist therefore guarantee multidimensional extend fuzzy datum hyperparameter long unobserve uniformly dimensional posterior pdfs hyperparameter function dimensional additive model fuzzy multi fuzzy produce prior hyperparameter representation adaptive approximate tune use law learn eq dispersion law replace parameter law law square fuzzy case fuzzy allow user prior hyper extend bayesian fuzzy inference model prior triple separately pdfs approximate fuzzy conditional separate fuzzy pdf fuzzy conjugate mean deviation beta hyperparameter beta fuzzy approximate pdf approximation conjugacy bc x point fuzzy use fuzzy b gx rule prior point likelihood iteration bayesian update propagate fuzzy exponentially standard conjugacy keep define conjugacy fuzzy shape set corollary preserve conjugacy part fuzzy function corollary conjugacy beta gamma support ht laplace beta semi gamma laplace structure fuzzy bayesian doubly fuzzy rule fuzzy system fuzzy rule j j fuzzy fuzzy product part part volume centroid likelihood fuzzy rule rule represent fuzzy approximate pdf represent pdf fuzzy prior hyper volume centroid qx fuzzy rule doubly fuzzy define fuzzy system shorthand fuzzy system approximate fuzzy system fuzzy likelihood centroid rule weight volume approximate imply fuzzy stage rule fuzzy rule rule linear fuzzy rule iterative inference track involve likelihood prior system rule fuzzy rule rule fuzzy rule fuzzy likewise represent pdf set function centroid rule beta represent four set conjugacy structure doubly fuzzy system kf conjugacy part doubly fuzzy ib functional part form part set product conjugate n f conjugacy doubly fuzzy prior use h gx g gm eq use set hyper function self case eq eq fuzzy show part fuzzy corollary doubly fuzzy conjugacy beta set suppose bm h gx g bm g f g j j special occur conjugate q fuzzy form beta part conjugacy part set suppose gamma poisson set gm gamma q conjugacy laplace suppose semi special shape dispersion semi conjugacy conjugacy semi conjugate set avoid parameter conjugate beta gamma laplace corollary latter center special result parameter fuzzy inference uncertain fuzzy fuzzy uniform substantially extend choice prior close pdfs conjugacy priors user knowledge open exponential fuzzy system device growth algorithmic corrupt record datum also inefficient improper increase daily yet systematic datum corruption lag behind effort corrupt incomplete effect misspecification show improve speed general analyze corrupted rely benefit result affect hmms limit misspecification present show bayes uniform function pdfs produce theorem quality drive approximate interesting know many diverse help reduce intensive application area explore subsection htb c chapter feedforward ng ng hmm extension chapter segmentation zhang al zhang deep refer deep machine rbms perform bipartite structure gibbs energy whole ensemble rbms neuron energy lyapunov rbms chapter show backpropagation feedforward procedure formalism rbms noise rbm rbm visible layer neuron value visible neuron ht sep blue fill red width cm width line visible neuron rbm backward connection matrix draw pt minimum sep fill fill red fill h cm h dot visible energy rbm rbm pdfs hide connection bias visible rbm conditional pdfs logistic cd gradient parameter rbms ascent mean cd instance em backpropagation derive rbm benefit average specify configuration bernoulli positivity hold rbm
increase programming offer still necessary variational propagation first speed subsequently belief insight symmetry work largely choice message bp recently technique notion group solver framework marginal particular lift tree reweighte formulation far present first follow benefit log partition marginal exchangeability immediately lead bound log serve goal analyze symmetry depend appearance probability quality affect suitably symmetric work symmetric explicit compact optimize wolfe frank map appearance effectively algorithm span tree graph benefit polytope inequalities polytope notably random sometimes characterization polytope elegant call compare demonstrate supplementary detail begin reweighte inference serve discrete assume overcomplete probability approximate domain essential view partition polytope seek tractable polytope formally moment entropy subset sufficient statistic moment span tree entropy span belong span polytope denote appearance bind use polytope constraint consider appearance polytope optimize span step span several derive geometric program guarantee relaxation polytope span tree decomposition essential directly tree frank wolfe iteration follow latter optimize appearance direction find maximum span weight build variational introduce construction structure nevertheless preserve key characteristic exponential polytope log entropy exploit tie cell rise tie family partition formulate compact specifically induce restricted polytope supplementary since jensen substitution appearance every edge edge edge entropy node graph edge simple loop fig encode incidence graph follow element assignment ground polytope constraint term connect compute induction length base statement right front base return ground contain let must since since connect path must node argument must one keep disjoint follow log proof demonstrate polytope variable conceptually configuration substituting variable configuration summation configuration compactly similarly constraint pair c k arrive number way sum edge exchangeability condition precisely obtain equality obtain consider arcs model triangle triangle rx rx rx rx rx rx
relatively restrict square real target common alternative normalize presentation concern gradient base feed forward neural target vector non represent compactly pair target naturally equally dimension allow similarly sparse besides target intermediate denote corresponding matrix letter transpose transpose architecture linear activation usual kind amenable backpropagation e g k b activation equally input tie explicitly bias traditionally place I linear possible old trick component hide update descent need vector operation able gradient backpropagation activation layer row non zero need precisely operation write operation part propagation difficulty final incur prohibitive cost priori reason e sigmoid non would fundamentally extremely suppose reasonably sized compute residual incur cost output one update incur prohibitive operation prohibitive maintain matrix update representation order prohibitive intermediate compactly compute respect w complexity prohibitive rewrite maintain much square update prohibitive computational update generally neither difficulty note decompose update yu yu tu new tu u tu u use formula need thank compute one together prohibitive direct keeping date eq date update date update implicitly translate constraint far perform p p yu k operation state precisely update require roughly require propose change correspond factor speedup access whereas update emphasize simply ordinary prohibitive standard separate update equivalent update straightforwardly minibatch yield need careful keep reasonably minibatch resolve minibatch generalize extended function basically function express compute softmax practice something limited class deep neural language embedding predict incur prohibitive updating weight matrix need backpropagation handling base family function softmax backpropagation remarkably without ever yield speedup least order typical part computation dominate kind network deal large representation arise vocabulary user sparse input
improvement text english text remove token document avoid effectively text need pattern dictionary string token effectively like report dataset english collection political publish anonymous corpus interesting claim text claim text study paper write claim researcher tend alone analyze dataset compose text jointly compute describe perform dendrogram binary represent dimension algorithm report leaf document appear evaluation done cluster branch tree e checking text isolate cutting agree text text place cluster result hierarchical obtain compression algorithm document evident close text rise method concept describe quantify string rely notion perform document automatically text text rest close minimize retrieve author adopt minimize gram gram author gram yield former find gram assign incorrectly among base measure outperform obtain rank competition exclude take furthermore subset english test classification competition problem adopt allow corpus belong author classification criterion assign document competition document correctly recognize comparison meaningful per correct author respectively encourage specific task correct make public former violate thesis without properly eventually title media list page derive attempt analyze page fail separate page satisfactory reasonable able text page thesis show text instance picture come page leave cluster justified page refer work happen author consider page author describe close characterize helpful identify text source evaluate similarity text compression propose compression text document report experiment document write english advantage method apart yield justify firstly meaningful contain document full discarding concern employ dictionary number object drive typical maintain keeping promise collection text english propose dictionary effective dictionary heterogeneous style write different come historical period state art outperform traditional task give challenging document interpret machine style therefore machine one author review report widely employ anonymous decade cluster text universal normalize distance general estimate object concept use histogram retrieve carry outperform state support art report compression base superior reduce respect kind analysis meaningful string instance match report improvement traditional usage automate text satisfactory preprocesse document dataset paper compression validate base similarity classification diverse text define represent compressed version string share usually characterize share common well efficiently generality diverse include text application categorization assignment modify version dictionary author apply text analysis character mark subsequently token character account string successively send point successively encode repeat
hand side notational clutter view relation condition complete comment relate alternative agent reveal candidate agent must distinguish turn cost cost scale gap suggest scale detection average iteration asymptotically tend dependence state network characteristic agent spectral centrality let imagine sense want dispersion evidence favor allocate token adversarial aim delay central regularity recall definition centrality regular large element eq informative procedure suppose agent default determine neighborhood centrality gap assume centrality fix idea markov walk replace generalize modify average network exploit optimize modify large eigenvalue min drawing term occur intersection min network agent add improves assign connection introduce correspond remove add self consider communication definite recall apply eigenvalue definite reduce theorem keep tend connection imply roughly behavior elaborate issue finally positive easily impact interesting existence central preserve diameter scaling lie cycle diameter poor communication affect model distinguish rely private signal identifiable consensus learn truth exponentially learn communication adjust fix parameter gap theoretically proposition agent suffer equally impact failure connection discard network bi directional eliminate view decrease amount plot network behavior almost monotone monotone relationship gap roughly signal state analyze study versus counterpart cost turn centrality affect informative optimize spectral speed learn link failure side effect poor discuss round potentially costly alternatively contact informative enough state direction scenario generalize model appendix elementary keep self contain positivity implicit setting equation fashion calculate kronecker discrete linear therefore matrix stationary markov chain observe recall mind break part sample since simplify require eq get q obtain probability take state use simplify thereby complete follow entail jensen inequality right recall ratio schwarz last line jensen I right conclude axiom conclusion conjecture definition exercise theorem remark summary address agent agent receive private underlie state globally identifiable informative signal literature introduce kullback leibler centralized network centrality relative agent informative central fast optimize speed spectral speed provide recent year burden agent regard range network broad class need therefore exchange dispersion adequate big picture consensus protocol gain grow popularity decade early decentralized detection consider fusion center classical centralize collect recently propose framework govern aim recover agent opinion quantity observe stream private signal likelihood condition state informative agent knowledge build learn finite non propose average bayesian opinion tend truth mild dual demonstrate generate weakly exponential weight posterior convergence present distribute convergence entropy individual analysis rate entropy describe dominant factor time finite gain affect work agent equally expert learning compare substantial agent upper bind conditional enough realize agent endow informative interestingly importance design facilitate interaction demonstrate achieve rate chain consistent natural rapid failure less observe spectral translate diameter diameter make propagation match finding paper formal briefly application conclude transpose vector element simplex th inner norm variation eigenvalue I centralize formal agent seek probability govern space assume marginal uniformly reason bind provide let evident state perspective true globally identifiable state identifiable continue triple space correspond agent receive pair throughout report agent neighborhood construct entry th row th column nonnegative normalization stochastic assume path agent assumption unique eigenvalue magnitude centrality negative eq denote centrality take form w entail chain irreducible motivate distributed scheme introduce detection mirror algorithm scenario could centralized agent global state specifically observe signal reveal nature characteristic round expert simplex let depict uniform rate initialize form belief maximize marginal divergence jensen inequality iff therefore fact set observe parametrize directly neighborhood uniform belief initialize let signal belief measure distribute compare counterpart agent opinion expert total cost incur observe
length ambient preserve additive multiplicative error multiplicative mc curve integrate side also manifold preserve geodesic distance geodesic geodesic q briefly map preserve pairwise follow establish dimension provide since geodesic probability q ambient map approximately volume curve manifold discussion manifold linearization mc contain linear theorem normalize cover divide normalize firstly normalize call short shorthand normalize tangent control intrinsic dimension tangent bundle secondly apart euclidean terminology cover decide apart quantify tangent purpose let observation readily inequalitie q net net sphere let suppose let definition q final step tool smooth reach final inequality trivially satisfy dimensional reach cover respect mc cf far computation improve extend subgaussian map remove inherent essentially research compressive like I thesis definition section exercise definition hausdorff mathematics theory euclidean several obtain case improve structure sparse manifold hilbert dimensional various informally curse several I map preserve certain dimensional possess intrinsic dimension dimensionality seek random linear know direction classical show orthogonal distance simple appear author reduction projection attractive subgaussian let entry mean constant denote small formulate compactly manner general dependence optimal random easy component g set meaning require incorporate dimensionality mixture gaussians manifold match processing square regression compressive compressive match rely extension different quantity intrinsic show subspace smooth motivate investigate subgaussian pairwise distance isometry subgaussian stable manner subgaussian e g therein development isometry subgaussian various isometry subgaussian matrix act matrix use substitute isometry unify restrict isometry subgaussian matrix additive counterpart formulate master subgaussian map act new bind focus make non demonstrate extensively extract result concrete known matrix subgaussian class contain efficient subgaussian certain processing application overview include subspace example hilbert variety express polynomial rate fashion type embed union space dimension subspace set measure principal subspace upon recent discussion main deduce reconstruct subgaussian thresholding deduce reduction smooth length preserve uniformly pairwise ambient preserve manifold first result linearization dimension subgaussian error terminology space random center variance subgaussian subgaussian subgaussian call fix norm let metric diameter linear hilbert let cardinality write width semi play role semi I slightly theory generic suppose subgaussian increment metric eq special process metric coincide translate cover small ball call let q finite reverse fail even time analysis empirical early database densely generally readily subgaussian map dimensionality distance preserve restrict isometry normalize preserve multiplicative say corollary radius mc subgaussian particular eq sphere parametrization width refine result obtain make contain sphere easily cover relax map isotropic parametrization set subgaussian expect concrete technical work dimensionality space q often universal say cover unit ball know argument integer cover dimension dimensionality reduction dimension short estimate statement result suppose cover dimension cover subgaussian map ik corollary readily deduce net assumption use arrive illustrate vector rip derive isometry subgaussian result sense let restrict isometry note since subset optimal rip signal sparse linear become count accordingly pn l imply upon explain expect upper lead guarantee recovery subgaussian matrix rip new restrict isometry map property role recovery play sense information define constant covering first subgaussian map tensor rip extend norm metric let isometry notation cover corollary originally subgaussian four elementary net respectively generic necessary signal hilbert signal piecewise overlap prove hilbert parameter natural consider metric subspace angle principal angle define subgaussian apply suitable parametrization h q admissible
irrelevant address p ty classification model intercept induce desirable encode interpretability relate control interpretability relate interpretability specify interpretability iii interpretability training example formulate program ip let objective approximation rate model predictive interpretability attain trade interpretability model attain vice versa convex regularization interpretability produce linear model use interpretability train create state rule classifier coefficient numerical worse induce direct interpretability producing represent example parameter accuracy unit optimal accuracy produce interpretability attain possible value regularization restrict interpretability attain guarantee interpretability set yield equivalence increase replace hinge polynomial ip discrete interpretability interpretability high robustness meaningful training probability estimate case use loss significant digit produce feature order magnitude coefficient attention unit feature coefficient expert kind relationship integer index indice non positive index either define pos j j ip formulation constrain narrow feasible region ip predictive sometimes model weight consider wish yield also training contain drop value adjust use adjustment drop see adjustment ensure value interpretability tie composition grain term number tune composition input alternatively impose include encode complicated encode require leave vast world produce trivial heart classifier predict heart attack handle sensitive positively label adjust respectively generality benefit train weight correctly example attain level accuracy benefit specificity expert sensitivity specificity encode model shoot accurate error label optimization aim positively correctly expense constraint prevent attain high single shot procedure intervention budget find attain high suppose budget action train predict aim positively accurately optimization aim secondary attain intervention budget different kind interpretable produce pair formulation adapt loss switching constraint decomposition scoring allow quick prediction subtract multiply number assess medical outcome simplify value score difficult reproduce system expert le principled scoring eq produce create system restrict interpretability tune restrict value great neither interpretability influence term classifier minimize make add yield coefficient denominator regularization restrict set classifier give value restrict coefficient without function n p j norm variable variable constraint big formulation parameter score misclassifie margin implicit feature interpretability absolute coefficient integer fine grain interpretability user set interpretability penalty coefficient heavily exclusive interpretable iii parameter monotonically train gain require example loss penalty gain accuracy k u r j big identical binary coefficient interpretability constraint coefficient assign constraint interpretability rule classification compose feature convert rule threshold rule model data rule assumption binary categorical exist j jt j threshold feature need threshold place rule rule notation rule regular thus give rule benefit mathematical expression least rule rule fold cv create ip n c constraint big identical ip interpretability penalty penalty scoring version suit outcome medical patient intensive optimize real feature feature selection exhaustive construct fine grain rule classifier feature rule constraint rule ensure binary agree improve interpretability ensure maintain monotonically outcome system follow formulation p p j j constraints big ip section f interpretability include binary value enhance scalability consider generic framework would ip variable constraint formulation however cut scale recent benefit relate stand aggregate setup solver different ip solver individual loss original optimization iterative oracle piecewise approximation iteration piecewise linear aggregate solution computation addition I linear solver repeatedly solve present popular decomposition initialize proxy aggregate cutting support subgradient aggregate clarity add cutting constraint proxy create aggregate plane grey linear loss form lie collection cut loss let function notice piecewise true point feasible function combine lb tolerance gap optimization z lb lb k kk trade accuracy minimize attain definition know control accuracy increase train basic penalty include interpretability train classifier comparison train case attain training range unit maintain difficulty also coefficient train run run ip take decomposition classifier loss feasible severe time summarize experiment setting ghz gb ram logistic runtime compute cut multiplication produce optimal classifier second train loss loss impose classifier train classifier train higher imply classifier impose severe ip solver purpose htbp trade scalability convex ip solver produce formulate even loss decomposition benefit substantial control mean discrete interpretability discrete interpretability loss well suited scale compute cutting cut operation decomposition polynomially exponential cutting performance scalability add cut geometrically center chebyshev center procedure train robust give solve proxy determine example computational provide work filter train proxy objective set classifier proxy datum proxy proxy feasible enough f denote proxy f relate choose filter helpful reducing follow stage reduction proxy compute level set I I stage proxy variant proxy contain force classify obtain exceed upper set contain dataset know optimal original datum provide proxy loss use determine enough loss alternatively condition proxy htbp denote solution original solution train variant force way convex proxy ni n I train proxy large eq appendix satisfy level proxy function p optimizer optimizer proxy proxy follow c z satisfy condition require proxy avoid proxy reduction relaxation ip hull convex relaxation ip proxy enough satisfy let ip optimal ip inequality follow ip feasible solution ip thus use feasible solution ip determine figure demonstrate datum train dataset specifically filter level instance proxy convex relaxation ip feasible correspond value compute user guess e proportion filter increase amount filter high keep mind attain htbp reduction computation include reduction use preliminary procedure ip even conjunction situation proxy reduction proxy similar impose even fundamentally branch particular reduce feasible label branch optimality impose branch effectively exploit linear coefficient g training value margin resolution train large achieve loss equal contain classifier attain procedure attain low optimize directly simultaneously round choose solution produce discretization bound consider use margin margin ki large magnitude proof apply theorem show discretize easily motivate discretized definition discretize solution principle structural minimization guarantee follow important linear classifier obeys proof hoeffding lead motivation discrete loss increase indicate large significant digit notable benefit generalization exclude suboptimal hypothesis discrete minimize function feature principle generalization minimize bound integer n generalization classifier coefficient every minimizer dataset size regularization minimizer obeys translate generalization relate apply objective bound integer vector penalty refine term point classifier obey argument integer relative due classifier improvement rule small left generalization clinical tool demonstrate flexibility real world tailor part laboratory contain patient feature health patient patient upper significant imbalance pr would model clinical list cm high positive maintaining ensure enough explain short relationship factor incidence suggest patient risk appropriate model training requirement parameter limit maximum optimization section process constraint classifier would establish datum fold cv solve hour gb ram summarize table imbalance use varied sensitivity range see explore setting inherently method varied mix dropping requirement violate one fold cv among ptc parameter cart lin rbf value provide flexibility method table ht lr lr ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc lin ptc ptc ptc ptc ptc ptc rbf ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc cart ptc ptc ptc ptc ptc ptc ptc ptc three requirement give base cart rule unable produce method svm lin rbf ridge unable regularization achieve require unable issue satisfy requirement tailor model expect able max net cart monotonicity control net highlight art pt accommodate reasonable crucial mechanism adjust sparsity incorporate constraint poor possible control accommodate reasonable allow control indirect require free extensive find cart max figure tuning consider standard case fold obeys final unfortunately requirement framework encode interpretability ptc instance ptc sign lasso lin cart plot classifier fold right highlight produce cart tune satisfy max two method acceptable sensitivity minimize penalty surrogate hold sensitivity sparsity train figure also performance ridge attain fit small linear integer vs coefficient final unable produce align constraint interpretability benefit coefficient understand relationship easier validate example htbp mml lasso mml htbp index mass point tv solve minute run numerical various popular uci repository comparison method varied nature process mn include include opposite htbp predict breast cancer predict go predict year patient breast cancer patient heart breast cancer predict mail spam summarize setup matlab art baseline package rule suited design interpretability assess via assess interpretability mn allocate minute ip ghz processor ram thus take hour aim run without constraint free hypothesis use restrict roughly rule zero rule contain coefficient coefficient contain coefficient set coefficient dataset attain accuracy htbp tree default tree c default setting lars lars ridge binomial link lar binomial link e net value svm lin rbf scoring np call interpretability different coefficient lasso e mn rule lin leave decision cart rule box svm feature relate interpretability visual table plot figure correspond cv error median fold cv test cv max size variation fold vertical horizontal size ridge net box line coincide path plot figure sparsity mn produced use feature g accurate use set likely coefficient attain training htbp l ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc range ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc size ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc function issue mn rule dataset feasible minute setting far provide optimality train proof necessarily binary mathematical overfitte minute plot unable attain model accurate regularization mn limited dataset correct accuracy possible mn rule measure sparsity restrict mn model mn mn rules particular know arguably happen path provide focused interpretability nice comparison attain perfect accuracy mn cart attain fold cv note able attain omit scoring compare cart lasso highly compare lasso cart structure note round round produce good one round difficult one randomized round require rounding attain perfect rule rule g anti rule create instance point domain unify lr lr htbp lr lr model cv htbp mml point categorical model categorical variable interpretability crucial aspect predictive help practitioner control important show create type design scalability use concept lastly present extensive illustrate art major method avoid achieve interpretability approximation interpretability long dataset integer programming software sometimes thousand integer practitioner allow periodic improvement code pool interpretable include section less point remove ii follow trivially follow proxy iv proof z inequality look rh inequality use get iv rhs inequality incorrect plug lipschitz ii satisfied set invariant define wise difference margin follow case lie margin margin margin trivially margin calculation whenever analogous calculation I I whenever put feasible optimization nz integer thus contain minimizer I htbp produce cut aggregate loss overview decomposition converge ip solver provide feasible oracle second cut aggregate able unable set requirement htbp mml dataset htbp mml mml htbp htbp lift lt lt lift lift lift lift lift lift class lift heart lift college lift class lift age class lift age lt lift sometimes heart abuse lift lift rule lift rule lift lt shift work lift lift lift lt age lt ess ess lt sometimes
whose solve see far increment value one know theoretically important store decide majority often refer majority storing reduce increase size polynomially expressive aware expressive c require compute separation increase result strong exist size imply efficiently except product circuit theoretic use give separation natural simple beyond provide simplified serve technique advanced later section powerful able prove address pose notation standard index row index context communication tool depth b fx negativity apply separate expressive depth define define function compute layer meanwhile easily result interpret easily communication pose whether expressive former require answer technical perturb identity associate equal satisfy obey must layer prove use become efficient depth depth node child efficiently computable state input univariate consist c depth super previously expressive show depth particular know various layer counting boltzmann efficiently capture capture deep boltzmann hierarchy property analogous boltzmann sigmoid belief never prove hold trivial rigorously gain expressive layer make analogous slight multilinear circuit monotone multilinear arithmetic circuit multilinear arithmetic circuit compute original arithmetic circuit construct circuit monotone confirm author circuit serious multilinear arithmetic circuit circuit capability multilinear circuit certain imply thing issue value exploit multilinear arithmetic decomposable value deal hardness certain circuit multilinear establish multilinear polynomial proof theorem give polynomially sized c beyond depth grow multilinear circuit multilinear circuit size depth multilinear function transformation preserve multilinear circuit multilinear circuit slightly arithmetic circuit apply analogous statement turn issue adapt multilinear circuit monotone multilinear circuit circuit small multilinear circuit automatically property equivalence monotone multilinear circuit end size arbitrary compute size give distinguished circuit fan formula restriction expressive answer turn indeed formula input real c compute consist size polynomial slight prove context multilinear circuit value monotone multilinear circuit node multilinear formula compute arithmetic circuit circuit confirm author hold circuit try theorem prove encounter multilinear circuit also apply existence possibly unnormalized imply density theoretic major drawback unlikely efficiently density know tractable counter provably depth notably boolean circuit size simulation capture probabilistic like notably proof theoretic might distribution model like network turn arithmetic circuit efficiently approximate boolean circuit approximate simple simulation efficiently reasonable efficiently polynomial size prop net arithmetic circuit distribution indicator absence label particular take edge present span denote computing formula amount graph span tree otherwise decide span tree check connect exactly efficiently solve boolean circuit trick adjacency solve add neural threshold simulate boolean circuit manner similar depth easily basic computable constant boolean circuit capture small kl divergence boltzmann pr sequence sequence also sigmoid depth circuit surprising remarkable corresponding variable compute possible computing reduce span tree set turn directly reduce derive laplacian argument formalize corresponding value label span main result follow arbitrarily arbitrarily univariate size trivially approximate arbitrarily large denominator purpose simply decompose weak function show output theorem sum weak must exponentially thus weak fraction avoid span polynomial negative weak relatively equal sized suppose negative polynomial multilinear circuit characterization compute exactly compute show arbitrarily analogous theorem univariate different sequence polynomial real value negative tuple satisfying eqn particular must prove show term span weak sum non start simple negativity factor intuitively requirement input think span either value vote always go pair factor reach incorrect despite status span cycle might visible potential cycle produce incorrect yes force essentially characterize conservative voting showing situation span tree receive yes vote real value form describe dx hz particular devote edge red graph vertex triangle one color triangle triangle reason soon clearly triangle simple cycle determine edge triangle impossible neither gets jointly triangle triangle vote whenever edge vote whenever formalize property value constraint triangle rise triangle color graph red total triangle bound lemma triangle choice conclude least triangle graph remain proportion consider span perform walk add walk subsequence sample one formalize suppose span proportion tell proportion span prove however separate span tree complete capture learnable simple tractable separate extend natural way capture care limitation want fundamentally character much statement indeed worth expressive thorough available nature amenable use avoid various exist circuit capable efficiently circuit seem fall open question tractable allow strong statement expressive acknowledgment like thank run helpful regard multilinear circuit support google result transform proceed process child process node compute normalize process node constant divide incoming leaf computing divide compute density product normalize soon child effect sum parent equivalent divide output recursively recursive fail divide normalize leave unchanged multiply incoming processing note income fact denote circuit node product non root node weight linear combination negative polynomial base trivial depth circuit non child scope coincide make child product consist node compute univariate constant node modification tuple modification constant observe add compute replace evaluated underlie circuit node aforementione replace child node conclude complete decomposable modify exist add child dependency theorem multilinear valid multilinear happen depend dependency scope univariate finite choose jx I jx jx polynomial combine positivity fact say term term collecting q use zero polynomial polynomial finitely root see multilinear incorrect evident degeneracy node product expand yield polynomial expression whose term give possible multiply collect sum product positive expansion weighted sum union collect weight degeneracy describe appear statement consider non polynomial sum label compute member scope element induction depth scope suppose member scope member inductive hypothesis mean element part factor element sum node root child induction depth case node overlap least product lemma set say twice thus scope multilinear multilinear circuit hypothesis amount establish multilinear multilinear happen polynomial part least factor member contradict way fail multilinear none scope previous set contradict scope scope distinct scope scope part product case contradict scope identical multilinear multilinear multilinear inductive hypothesis establish multilinear case set multilinear factor might twice contradict way fail multilinear element scope statement hold reverse depth depth multilinear inductive multilinear root syntactic disjoint respective disjoint well multilinear member scope member product form union union equal scope equal scope part multilinear multilinear consist product member lemma union appear exactly similarly scope multilinear node element verify co complete extend informally idea boolean sized output represent satisfie add evaluate output output whenever assignment proceed formal formula associate boolean natural label node count definition input form distinguished give node note accomplish polynomial moreover satisfy construct value validity require rhs validity circuit implement vector node multiplication implement product child component work entry node take node row final working hard produce decomposable function scope process proposition respect transition definition state kind transformation vector elsewhere realize obvious identically weight univariate variable compute claim index index weight sum b fx k iw layer pair obvious univariate sum node compute second layer compute x I proportional rearrange diagonal differ satisfy plug fx I semidefinite root symmetric root nuclear trace unitary matrix root root unitary unitary q eigenvalue equal thus fact multilinear polynomial proceed induction input hold trivially suppose multilinear polynomial multilinear multilinear polynomial eqn inductive proof multilinear circuit product discussion equivalent decomposable decomposable decomposable contradiction depth multilinear arithmetic circuit univariate affine add sum circuit remain multilinear circuit agree contradict multilinear arithmetic circuit size compute decomposable univariate decomposable decomposable suppose compute multilinear arithmetic univariate binary input replace add formula circuit clearly moreover circuit formula multilinear polynomial polynomial agree contradict proceed whose acyclic subgraph proceed label connect edge label vertex span tree span take label disjoint clearly consistent hard spanning construct subgraph edge edge subgraph clearly cause cycle edge span span form eq determinant asymptotic multiplication bad equal tree maximum accomplish replace fan structure scope replace pruning note circuit decomposable complete empty circuit stop circuit polynomial remove express output node remove stage input scope analogously depend completeness clearly consist precisely theorem form non polynomial thus define eqn choose give start root path child size dependency current arrive scope child child scope due child scope never path must become scope let apply polynomial satisfy theorem eqn finitely integer finitely dependency one infinitely often sequence independent dependency subscript subsequence forward whenever replace subsequence I x real dimensional bound sequence bound finitely subsequence replace subsequence converge converge write function resp iy ix complete sequence value resp disjoint converge pointwise subsequence zero replace subsequence subsequence exist zero select finite form convergent subsequence wise remain many infinitely maximize infinitely infinitely often replace subsequence subsequence certainly since wise proof contradiction contain value agree contain constraint triangle span follow contradiction lemma triangle total triangle say triangle upper bounded form take edge form number triangle original triangle original colored attain n triangle arrive triangle tree uniform obeys constraint analyze behavior due sample iterate step vertex current vertex visit edge terminate exposition minor algorithm distribution particularly sake simplicity vertex produce stage stage triple allow call triple pair triple convenience encode triple vertex give distinct triple consecutive encounter triple easy triple visit visit begin visit triple triple correspond constraint triple neither visit condition choice algorithm pick stage constraint triple occur stage choice constraint occur stage upper bounded plug setup theorem circuit sufficient call impose circuit typically intractable generative amount capability understand multilinear circuit question depth various establishe existence capture depth capture generative additional layer distribution capture grow kind hierarchy property never prove contribution include condition sufficient validity various type recently class generative unnormalized density function circuit arithmetic circuit feed connection arithmetic circuit input special valid normalize deep crucial evaluation provably intractable valid practical perspective validity call c impose structural limit kind extent expressive course expressive expressive efficiency emphasize distribution capture super instance latter question paper capture simulate unclear capture exponential tractable computable marginal etc come marginal intractable nonetheless correspond capture deep g indeed hard practice capture efficient general point obvious question density capture deep complexity theoretic establish sense many attention tractable could c analyze depth characteristic exist arithmetic theory gain expressive efficiency c despite expressive also numerous theoretical propose definition connection multilinear arithmetic circuit exploit powerful latter section provide insight relationship validity validity merely also generalize validity co np base use constructive efficiently efficiency give short technique circuit theory go answer open pose leverage multilinear arithmetic circuit prove powerful regard depth expressive extra depth computable thus give strict depth next grow expressive greatly reach recursive learn expressive multilinear circuit efficiently compute existence capture even approximated circuit circuit logic circuit neural network instead operation operation arithmetic formal define type acyclic follow either every known incoming edge arithmetic circuit go child arithmetic circuit value follow rule nod child weight polynomial singular take circuit node define circuit essentially depend node depth path general circuit allow quantity compute computation write compactly give generalized sum take value f f I univariate integral tuple whose arithmetic circuit property monotone arithmetic circuit gain arithmetic circuit node compute polynomial view set tuple define member denote primarily give normalize constant also degree represent subset integration amount summation treat deep density machine partition function density marginal accomplish completeness say let I ia I jx ix jx decode say respect corresponding integral say integral range intractable valid efficiently integrate respect allow f purely make say valid identity fundamental validity strongly multilinear associate trivial essentially think without circuit note reverse example compute count measure multilinear inspection easy strongly valid interesting complete validity purely application paper prove equivalence validity note validity compute way count neither decomposable show choice equivalence validity completeness need degeneracy circuit positive size procedure degenerate arithmetic circuit degenerate preserve structural completeness remove remove node fan except fan sure node node output degeneracy validity consider compute compute condition never degeneracy many non monotone circuit root child monotone arithmetic circuit equal form note circuit polynomial set scope scope degenerate circuit multilinear multilinear utilizing
domain domain control trade solver sgd saddle saddle stochastic time stochastic feed comprise fed factor difference stochastic descent domain minimize loss update learn fortunately layer layer apart meta act subsequent pass precede implement exist object package multiply nothing domain architecture depict result model converge mathematically describe backpropagation identity matrix objective optimize stochastic implement domain predictor use predict derive adaptation define distribution depend particular representation space family enough pick concatenation predictor layer perceptron aim assumption easily train closely relate backpropagation become small cm cm ccc cm ccc sign source mnist source sign mnist mnist source train digits background correspond e correspond domain bind da cover five considerably big portion gap amazon l sa l l last art extensive image office dataset standard vision much data branch include reveal serve assume addition new dataset comparison da boost performance baseline principal target sa activation classifier descriptor learn domain train new adapting compare label performance remain office recent da approach publish cnn general convolutional layer pick exact architecture stick use loss train batch image domain rest comprise domain instead fix gradually schedule experiment optimize detail cnn find visualize distribution digit window digit orientation color variation manually however rather difference structure clutter background propose backpropagation work target sa accuracy adaptation task challenging mnist mnist gap adaptation stay epoch avoid learn anneal direction equally diverse appearance observe separation feed solely mnist probably explain improve mnist scenario see opposite direction adaptation unsupervise mnist unsupervise da capable adaptation sign experiment sign simulate increase evaluate domain additionally split train set part solely evaluation procedure slightly predictor target suggest thorough verification office evaluate office collection three distinct domain amazon unlike previously office rather spread amount available crucial fine cnn pre imagenet recent da exactly architecture domain work transfer task image per unlabeled abundance target domain set art unsupervise adaptation amazon scenario adaptation feed forward architecture amount domain da adaptation alignment accomplish backpropagation approach scalable deep plan usage evaluation supervise constitute work interesting autoencoder deconvolution effectively inspire lead update introduce minimization discrimination loss constitute special domain entropy predictor adversarial loss label gradient domain result rgb architecture experiment mnist inspire single cnn pre train office bottleneck domain branch somewhat adaptation attain architecture momentum anneal eq q progress linearly schedule optimize domain dropout train perform massive amount label datum absence attractive label domain propose architecture datum amount unlabele feature shift augment architecture overall implement deep package perform experiment presence big shift office feed architecture advance wide performance label training set large scale time abundance fully label training approach adaptation suggest approach mapping domain domain compose adaptation mapping target datum either fully annotation semi focus hard although generalize supervise straightforwardly previous work feature combine adaptation training deep adaptation process decision base feature domain distribution feed target shift invariance optimize discriminative classifier label domain optimize optimize minimize classifier domain classifier latter encourage domain feature course green deep backpropagation training proceed minimize gradient indistinguishable result invariant crucially process embed compose feed forward figure use layer loss backpropagation modification g sgd momentum generic add architecture backpropagation component propose rather trivial layer unchanged propagation multiply negative backpropagation adaptation office benchmark considerably previous accuracy source approach select seek map target way reproduce whereas axis accomplish modify representation geometric rather separability deep several approach source among sequence autoencoder source domain approach train domain predictor separate autoencoder feature adaptation unified architecture argue conceptually implementation considerably office benchmark approach domain target context deep feed architecture fine network train require domain quite network measure minimize discrepancy discrepancy finally focus domain adaptation feed network mean may regard seek tight assume work space problem finite generic handle feed handle exist refer
algebraic technique selection use dominant feature improve capability compare classic art scheme technique focus learner approximate classifier generalization base approach one learner apply train excellent introduction method randomize train subset selection scheme boost whole mapping literature forest subsampling model weak learner high author extend ensemble every subset proposal leave future randomization strict simplicity randomization bad accuracy insensitive see classifier virtue propose significance similar propose apply blind randomization learner illustrative inherent interpretation capability explicit classifier non selection apply leave research tp build train decision classifier restrict bag utilize conventional indicate process uniformly utilize object feature two feature next feed accord generate tree collection majority voting apply derive return frequent decision sampling column let nn svd singular column normalize define denote entry singular highlight slow generally strategy randomized score include section experimentally two variant case replace subtle distinction construction utilize randomness split uniformly depend htp publicly handwritten digit people namely case multivariate highly challenge point feature heavily usage benchmark impact leverage result svd rank bagging algorithm default matlab b report htp c accuracy training pair behind requirement superior superior small case leverage less training interpretability limit feature first increase processing time improvement accurate rf computationally depict accuracy moreover well need among predefine time second hour complexity memory maintain feature selection study strategy time efficiency classification indicate tree state scheme interpretability least study effectiveness randomize experimentally feature selection feature experimental evaluation naive forest massive information development scientific ever contradict high feature space proper description easily interpretability curse pose qualitative noise difficulty abstract q learn j j low case report literature classification random due accumulation poorly information removal gain classification fortunately important removal dna influential usually prohibitive storage requirement post snapshot available storage ambient
point fairly raise choice fair dimension kullback kl remain mutual mi remain mi fundamental theoretic via low variant hard easier let kl quantity data mmd mi suggest behavior kl mi stay aforementioned fair kl easy irrespective fair variety distance decay increase alternative characteristic kernel invariant gaussian kx distance choice median always represent biased keep choose power decay bandwidth choice interestingly median univariate laplace variance center choice keep experiment decay bandwidth heuristic maximize choice origin center gaussian covariance say keep number constant dimension verify keep constant encode really try detect accurate choice keep kl fig mmd vs keep calculate aforementioned example b compare polynomial kl mmd actually shrink polynomially bring especially calculation unlike choice direct early aim bandwidth prove look involve taylor simplify affect choice qualitative constant clarity corollary ignore residual population go zero exponentially fig median median heuristic polynomially mmd hope kl bandwidth polynomially mmd smaller demonstrate approximation actually calculate mmd bandwidth population mmd mmd separate previous median maximize mmd present exponentially mmd expression laplace kernel bandwidth accuracy verify use bandwidth median heuristic experimentally verify drop exponentially make small correct bandwidth optimal make bandwidth polynomial mmd vs laplace kernel panel relate aim approximation expression verified verify tr taylor theorem calculation mmd qualitatively vs kernel panel understand various proposal fair alternative bias independence understand popular drop polynomially constant zero zero modern dimension completely behave acknowledgement nsf grant look mmd calculation case characterization mmd invariant kernel laplace kernel translation kernel definition kx iw iw fourier invariant substitute consider change substitute get proof proposition dx example dx dx obtain thereby lead integrate part equality follow manner kernel decompose step substitute equation get bandwidth taylor q tr approximated b median heuristic optimal corollary derivation proposition taylor order get interpretable formulae result demonstrate compare mmd sample mmd mmd observe quite thereby previous unbiased mmd empirically prove decrease bias mmd bias mmd decrease fashion mmd observation machine pa usa nonparametric solution distance behave well high source give rise specifically hardness statistic zero fair hypothesis actually drop dimension fair light bandwidth advance test independence testing sample determine sample algorithm homogeneity draw sample draw distribution marginal r problem parametric class quantity kernel introduce approach parallel testing introduce far summarize subsection identify address normal mean estimate hard whether zero statistic behavior low independent nonzero get hard test fair set decay dimension current mathematical solid completely formally rkh correspond statistic kx kx subscript bias exclude call every statement qualitatively matrix matrix subscript suggest quantity characteristic test provide insight test test elaborate error equivalently power also dimension work well understood type people aforementione hypothesis test outline understand distance claim presentation word claim worse get dimension unfortunately contrary base method introduction lead dimension
combinatorial search proportional manifold dimension ask polynomial complexity demand manifold sometimes ask solver exist barrier recover work theory counter manifold low play role many use treat ambient low dimensional manifold manifold union signal dimensional manifold may make exactly essence completion unknown measurement random low synthesis model signal zero representation group column uniquely recover possible recover manifold however feasible practical technique subgaussian partial measurement number need measurement easily result guarantee recovery turn roughly noise synthesis compress signal low one abstract reconstruction manifold sparsity framework looks apply example vertical horizontal signal becomes represent note hereafter tv g vertical finite define follow addition mod correspondingly thus stack vertical difference analysis model assume characterize denote signal entry denote unknown submatrix row subspace recent reconstruct random position orthogonal row dimensional recover signal ambient thus natural matrix norm count recover np hard gap theory tractable stable recovery combinatorial noiseless utilize exist dimension unless manifold reconstruct dictionary vertical horizontal matter theorems suppose subspace theorem prove combine without huge implicit remarkably efficacy measurement exist reconstruction demonstrate size signal dimension proof technique operator discuss implication theorem hypothesis pack radius maximal eq p piece lemma last admit large packing reconstruct pair pack reduce indistinguishable classical number lemma follow radius pack finite argument begin signal finite q worst suitably trick packing pick estimator side lower minimize highest straightforward relate prove cone packing suppose begin rescale note restrict reliably packing otherwise estimator project word divide side finish metric signal noiseless fail characterize noisy notion idea provide indeed le many problem subspace know proportional geometry state recovery measurement dimension fast pack proposition htb attribute demonstrate look experiment present signal ambient combination interestingly instability indeed dimension recovery heavily soon reconstruct increase htb color attribute square reconstruction additive white standard energy color bottom upper exponentially number decrease become error function correspond minus value synthesis suffice percent correspond measurement behavior quite soon accord increase become become attribute square htb square error manifold connect gray generate image pick pixel start walk assign stop pixel visit result image
penalty value aic penalty number mse blue case mse penalty position similar penalty lot case mis look correct false positive parameter detect lead positive detect choose detect far thought look versus approach value show figure circle could point near also see segmentation segment look sensible illustrate firstly neighbourhood efficiently mis specify something change independent identically analyse let vary mis let slowly mis simulate value simulate segment set distribute constraint set linearly normal mean firstly calculation segment set maximum evident optimisation improvement run neighbourhood datum speed cost without similar general gain red dash neighbourhood bottom mis middle sublinear mis approach efficiently criterion aic detect range range simulate estimate infer within positive positive actual measure proportion detect positive divide positive detect evaluate accuracy segment parameter segment separately true
use ensemble read theorem proposition matrix call subset basically model definite proposition summarize prove dpp kernel dpp dpp condition answer give eq know quadratic eigenvalue positive definite definite q technical conditional positive definite definite matrix hard semi therefore follow definite negative prove positive semi definite positive point process ab corollary ab dpp claim ab b corollary kernel disjoint know q since summarize ab however clearly b product condition dpp definite positive c q process subset claim know ab ab ab cb subset true generalization kernel disjoint follow statement k thing j look covariance dpp briefly graphical process definite give result subset pairwise disjoint subset separate vertex consequence fact markov follow graph disjoint separate markov ensemble ensemble disjoint separate zero place give disjoint separate independent separate kernel
hamiltonian hmc use hmc mainly generate correlate might implement software run discard half sample element sale coarse comparable acceptance across different run take second proposal single tb third concern heterogeneous hierarchical parameter identify package single drug sale drug datum conditional sale sale often binomial poisson per week depend sale heterogeneous contact depend vector intercept population I weakly informative row wishart estimate slightly sale period trend change run baseline double averaging adaptively set hessian implement hmc code gradient playing chain period search recommend posterior initialize trace plot chain iteration panel appear converge autocorrelation progress summarize population final hmc million inference tb tb ht problem bad step reverse regardless hmc resource hmc million evaluation assume sufficient small collect proposal evaluation acceptance however time low mcmc implementation single mac sample collect sample draw hour core reduce minute discuss scalability processing collect parallel give could individually could collect even confirm converged draw inference method density hmc hmc chain density level hmc baseline standard quantile decomposition close convergence parameter align little movement chain thus infer effort tb ability parallel attractive mcmc present favor scalability grow analysis computational compute posterior hessian hessian show achieve hessian pattern independence heterogeneous homogeneous component level add summation grow subsequent number together might hessian namely estimate matrix ideally would code require code gradient would fast yet either evaluation posterior log linearly gradient estimate hessian grow gradient accumulate numerical large hessian estimate ad algorithmic ad refer put ad treat composite practical ad involve code keep track derivative compute log ad generate additional return order package also access option remarkable feature ad scalar five package storage dense format store regardless precision dataset model ram furthermore effort quadratic cubic extent operation mode find computational grow multiply multiply matrix efficiency hessian instead cholesky cubic source scalability sparsity system add since cost triangular system solve triangular would grow cholesky benefit sparsity square hold order one nonzero sparsity next nonzero grow element add average nonzero cholesky linear cholesky decomposition store week period visit single week covariate vary true weakly informative average across replication expect grow matrix compute cholesky multiplying triangular standard triangular tb table factor acceptance say acceptance dataset could influence expect fast confident overall acc data additional method marginal output acceptance consistent easy compute solely importance method pseudo arise hull define demonstrate although effort collect probability give proposal accept therefore equation rearrange available mean observe empirical support proportion estimator regression use truth conduct simulated number number covariate simulate include intercept iid density correspond plus per density number draw hessian proposal exclude proposal present sampling harmonic include mean remarkably improve draw offer negligible note comparable harmonic compute fall input algorithm importance sampling scale sd sd sd acc spend long hour deal mcmc utility alternative algorithm appeal mcmc converge heterogeneous conditionally sparsity construct sampling grow unit attractive practitioner guarantee generate perfect target concern discretization could posterior increase proposal expense computation possibly experience find collect depend density posterior mode mode might determine manual search find proposal little time proposal gradually principle metropolis however advantage begin contrast tune apparent substantial appear collect try many multinomial closed form carlo integration augmentation advance parallelization might make numerical integral year efficient augmentation kind multiple suffer weakly missing could treat implication require involve multimodal posterior find mode posterior multimodal normal idea find mode one mode remain unchanged match recognize guarantee amount practical unimodal care optimizer stop reach optimum package gradient number package language method package rejection take gradient user information author acknowledge suggestion comment date date integrate science hierarchical capture unobserved heterogeneity unit markov outcome develop bayesian parallel chain autocorrelation conditionally make applicable use likelihood little management practically application course picture impact method multiple natural capture heterogeneity customer type constrain source grow salient familiar parameter question idea popularity markov monte carlo sampler involve block unknown theoretically iteration early generate difficult reader hundred reference despite mcmc remain start estimation chain converge hierarchical heterogeneous unit customer preference cycle size data outcome multiple magnitude require yet reason procedure practitioner let day chain answer end processing system technology solution outcome collect core processor also question indeed ensure chain converge hand envelope rejection impractical small inspire mcmc sampling normal mode traditional rejection unnormalize distribution maxima derive conjugacy requirement share effectiveness broad scaling proposal draw algorithm sampling intractable hessian density determinant efficiency fortunately several project scalability heterogeneous bayesian probability observe involve numerically prior note product key issue relevant limitation may require independence assumption scalable nevertheless effort involve joint unnormalized posterior factored index mix serve level likelihood include factor probably kind optimizer trust mode substitution write target posterior restriction least restriction later auxiliary yu simulating candidate satisfy comes write differently involve sampling sampling definition get need estimate simulate proportional kind prior polynomial extremely bernstein polynomial effectively repeatedly proposal cdf cdf draw order proposal become accurate proportional partition segment fall multinomial weight continuous exponential sampling finally draw criterion save sample true rv marginal restriction negligible must accept high high density choice multivariate proposal kind density researcher density negative hessian asymptotic order multiplying covariance valid mode scale three panel potential proposal leave panel log posterior tail middle right panel multiply sample tail however proposal unlikely would algorithm stop try density little inference make believe relative case good application nothing implement manual selection metropolis hasting tail proposal multivariate fail mode quite rejection distribution proposal posterior proposal advantage exact proposal ratio extremely deviation mode acceptance accept discrete exchange non call direct remove concern sample target acceptance sampling bayesian mcmc term resource implementation however direct suffer another even moderately sized large et al consider allow conduct without conjugacy mcmc important feature method several direct focus shape concerned characteristic bernstein draw may proposal ideally already traditional collect parallel issue autocorrelation discussion improvement
break construction equal l evy gamma poisson bound limit matrix independently draw gamma stick sampling marginalization baseline negative task review york times corpora exposition recursive stick break like extension integral give constructive definition discuss gamma far normalize employ model model recent examine negative beta break readily beta beta stick variational though scalability problem disjoint borel subset completely take countable process algebra measure measure measurable kp cp c rate mass respective ensure beta sum finite mass base completely component create atomic measure disjoint normalization measure increment stick break break v break round fraction stick broken piece surely dirichlet stick like construction technique simple process stick break construction mark modify stick break beta gamma poisson process define derive marked process formulae stick break stick break construction beta random stick broken piece weight atom weight along somewhat practice gamma also prove correctness gamma may construct ij poisson probability superposition tell countable poisson measure denote ij c ascent gradient two document corpus loading count document put dirichlet column variational get multiply hyperparameter setup ascent describe first variational nk lower nk corpus initializations rate gamma affect indicator round indicator hyperparameter latent prior kt integrate technique sampling break monte technique g kl integral stick take normalize infeasible evaluate normalize value consider model document count matrix loading count poisson gamma dirichlet sampling element count kn z kn kn relationship multinomial nd condition conditional distribution numerator denominator sample integrate break improper sampling indicator round atom describe poisson posterior c calculate carlo fall draw exactly pc discretize carlo technique multinomial document corpora count document vocabulary model derive compare avoid infer poisson monte affect indicator indicator update prior hereafter ibp put symmetric prior column add variational variable hold time expectation warm gamma stick break atom simulate vocabulary use every measurement measure second minute learn refer likelihood burn though slightly count uci edu ml york corpora count hold truncation update keep representative fig gamma hyperparameter variational improper sampler likelihood initialization dirichlet learn iteration variational attain likelihood dataset hour edge somewhat dataset unlike attain iteration burn convergence second log likelihood truncation among good medium truncation additional competitive small log truncation atom second iteration take dataset fig matrix require update minute hour average find fast medium dataset large dataset measure hold less compare substantially produce novel stick gamma variational
attempt sensitivity generate feature small affine affine hull include affine affine discrimination via affine constraint affine enforce near maximum extend vector challenge unsupervised relaxation maximum margin hyperplane running experiment outperform hyperplane optimal label convex convert definite problem obtain optimisation cut plane speed extend multi hull comparison hull intra hull normally may discrimination contrary near image sensitivity variation balance approach assumption hull model distant combination variation image ns synthesis require remain synthesis must involve synthesis call neighbourhood noisy define point noisy convex affect divide multiple hull notice hull control number divide control extraction reduction set extract convex subset far divide hull variation b conceptual illustration convex extraction hull construct close indicate line connect set blue combination b illustrate sample away set hull cluster individually notice close generate subset point sample area variation solution apply extract however convex nc maximally separate inside point hull eqn nearest svm optimisation combine discrimination eqn eqn margin distant similarly approach local cluster reference cluster convex matching comparison hull match drawback extract individually capture variation hull expensive conceptual illustration reference c separately cluster divide hull grey indicate set contain assign grey close address reference adaptively cluster reference probe accord total reference ie design local sub cluster also adaptive reference affine versus hull distance hull affine hull hull variant eliminate select top image arc wherein reference extract query reference individually way reference query comparable perform approximate set face dataset consist subject pose illumination subject conduct cross select rest normalise histogram choose partly situation face fail test category category various select turn normalise hull reference cluster arc image normalise bar bar arc technique bar correspond consider red combine compare method efficacy choose fix arc select counterpart hull discrimination consistently local capture comparison evaluation vs extraction method number indicate number threshold minimal ptc c c c variant show parameter performance well propose approach helpful variant counterpart indicate cluster cluster performance cluster sensitive drop normalise image achieve propose replace arc arc cost compare distance reference arc evaluate indicate number cluster arc indicate strong set arc argument arc helpful remove data matching nn perform well summary good state normalise use dataset raw method dataset regardless variant average compare two table set convex slow hull extra however significantly lead local novel balance maximum cluster constrain region artificial adaptive arc cluster cluster set hull comparison extraction subspace effect acknowledgement lp department communications centre chen university school university technology point method convex affine artificial noisy sample significant intra variation extract cluster undesirable environmental illumination pose variation two close represent propose enhance near point affine adapt constrain local arc constrain query image dataset show technique information improve discrimination robustness variation pose illumination two method distribution parameter art model attempt linear subspace convex angle similarity single structure select classification row artificial middle third artificial geometric close close point calculate adaptively distance degree variation close discrimination show artificial variation combination distant recent embed representative discriminant patch extract model local acquire local ie exhaustive fix two example assume image person cluster second represent
tool library consider main tool train predict additionally merge finally provide include variety extra tool facilitate basic bootstrap sampling categorical splitting recommend original ratio positive negative subsampling illustrate software keep majority aggregate sophisticated decrease accuracy desirable base number average function set randomly balanced dimension trend training order svms subset subsample kernel overall ensemble fast twice ensemble experiment simple ensemble single marginally fall short train basic aggregation ensemble standard svm ensemble voting tool svms experimental ensemble model significantly maintain frequently train complexity know benchmark may desirable instance benchmark ensemble ensemble scheme currently scheme nonlinear provide quality intuitive svm software date incorporate promising request de de library machine package svm currently offer ensemble storage evaluation support share model experimental result show drastically reduce maintain available online support vector machine bag become practitioner constraint amount become particularly train accurately problem svm nonlinear requires run quadratic set training complexity aggregate small training offer bag unstable bag voting maximize instance binary classifier per instance approach class aggregating ensure evaluation heavy library training moderately system use
domain measure suit connectivity consequently similarity zero node voxel measure voxel construct graph voxel introduce scalability location patch connectivity local patch approximately reduce voxel number dataset use distance voxel tune partition functional connectivity measure patch form connectivity mesh learn connectivity represent local pattern seed patch positively voxel indicate voxel indicate functional k fc functional represent voxel lie bold positively mathematically speak define neighbourhood generate select fc j fc fc index voxel translate voxel em fc functional compute employ close neighbourhood sample operation replace operation near positively correlate operation voxel obtain operation fc exceed fc extraction separate connectivity deviation computation distinction mesh neighbourhood formation mesh seed voxel algorithm suggest neighbourhood connect mesh neighbourhood method identification difficult class distinct distinction recognize voxel cognitive process need within connectivity brain discriminate recognize class state semantic analyze pairwise class represent pairwise voxel unique construct semantic illustrate number relation illustrate connect mesh fc em connectivity em em em fc cognitive eq voxel cognitive advantage use discriminative fc discriminative connectivity figure connectivity compute htbp discriminative connectivity voxel functional neighbourhood fc std kp deviation voxel vary voxel cognitive determined voxel cognitive voxel extraction voxel different cognitive discriminative consider neighbourhood fc even amount class similar divergence exhibit correlation semantic pairwise carry semantic class divergence trivial affect also illustrate divergence voxel functional entropy amount scenario correlation respectively absence last voxel pair semantic high entropy matrix illustrate pair near voxel construct note fc voxel fc threshold nearest fc contrary voxel set fc voxel information fmri list problem period whether probe member old new period period brain activation relate probe activity pattern test whether brain cognitive processing total ten semantic category use activation collect encode retrieval phase train semantic category preprocesse stage pattern processing perform http uk quality procedure assess outlier slice slice global signal correct slice acquisition slice match slice correction interpolation functional datum space affine transformation along basis sample spatially smoothed shift across consistent previous shift account response lag encoding retrieval consist voxel temporal fc generate nn support parameter user always much fc order form practically performance approach compute functional connectivity connectivity introduce threshold user work employ give peak relationship activation scan specific scan percent improve peak correlation namely positively specify discriminative connectivity employ cognitive connectivity toolbox avg htbp avg classical mesh mesh correlation mesh functional discriminative std mesh discriminative recall discriminative cluster connectivity computation employ peak scan generation entropy table connectivity mesh improve classification performance considerably classify voxel mesh performance mesh learning performance nn respectively main near connectivity voxel study mesh classify cognitive neural activation test employ connectivity voxel cognitive brain mesh cognitive state information represent brain focus finding cognitive would insight success cognitive improve mesh drawback mesh select fc brain hierarchy brain pathway rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb edu tr classify cognitive brain acquire fmri form around voxel voxel mesh determine neighbourhood define neighbourhood similarity voxel form voxel mesh voxel mesh linear relationship functional connectivity aware mesh voxel seed voxel voxel mesh relationship seed voxel voxel model arc mesh linear arc aware relational feature fc represent relationship voxel finally fc type cognitive state type information encode retrieve participant study category make recognition neural accordingly machine successfully category belong represent brain memory encoding style circle drop font style style auto b node auto swap auto swap j instant three instant belong cognitive mesh cognitive model individual voxel mesh voxel arcs weight spatially voxel voxel spatial coordinate brain arc indicate voxel time minimize location voxel fmri measurement category minimize employ seed instant mesh p mesh arc instant represent construct memory retrieval stage far cognitive detail
room improvement prefer kernel kernel learner et work component feedforward network et generalize idea sensitivity develop extract classifier manner superior identify pair discriminative act characterize discriminative platform visually sensitivity map rise principal establish introduce introduce let may nonlinear map sensitivity write actual intuitive importance classifier input define definition sensitivity direction denote define allows write classical quantify individual input generalize contribution rise map classical distinction sign direction influence rich sensitivity information eigenvector vector define principal sensitivity grant rich dataset fig unless otherwise recall pca case put q see pca center visually sensitivity exist code covariance map mnist datum sample add common template fig three step bit template picture gaussian intensity truncate intensity sample feed ten dataset dataset conduct gradient adopt final posterior purpose transform represent class standard digital logistic mnist describe way base sensitivity map compare sensitivity map obtain common neural train pixel character region edge st assign opposite whose crucial characterization edge characterization sensitivity rest opposite sensitivity extra sign rich counterpart benefit sub sensitivity st extra information benefit visualization classification problem sensitivity map particular class rd classifier quantify perturbation input look visualize deal binary visualization aid another fig st distinguish rd alone confirm map capture character binary st thus finally sensitivity depict map note pixel assign pixel localize understand discriminative digit dataset hand object orientation mean vary sample make visualization classification brain area necessarily face likewise translate may region digit mnist appropriate later dataset choose familiar summarize standard able character c stands test correspond verify highlight pixel clearly orientation effective template straightforward must template pattern elegant visualization remain decompose space assess artificial visualization least aspect dominate sensitivity classification classifier solve problem sensitivity drive attain visualize may decomposition variety problem medical identify biological region helpful
ultrametric node relation notice imply identity quasi ultrametric imply inequality definition guarantee boundary imply candidate two x show scalar property eq satisfied finally ensure continuity trivially dendrogram increase consequently dendrogram every ultrametric prove identity respectively former quasi ultrametric network arbitrary node ultrametric dendrogram influence either appear node merge single dendrogram arbitrarily identity concept concatenation point point define quasi valid ultrametric link strong triangle chain maximum exceed maximum cost chain xx axiom pick arbitrary node axiom satisfied point transform chain yx reduce dissimilarity chain eq far exceed follow eq q expression axiom separation dissimilarity useful show pair whose exist contradiction satisfy partition find partition exist compose represent node true dissimilarity conclude chain case cost repeat partition dissimilarity recursive correspond satisfie observe construction contradiction assumption incorrect satisfy return main ultrametric showing act admissible I network notice dissimilarity axiom substituting obtain valid ultrametric arbitrary direct chain block u vb b dissimilarity reduce dissimilarity map dissimilarity equal dissimilarity map increase separation since dissimilarity proof hold admissible usual projection say indistinguishable chosen notion distance clear check correspondence finally definition hence triangle correspondence correspondence show pick must element triangle define correspondence furthermore element absolute less absolute noting proof checking imply follow fact correspondence choose way contradict manner must two guarantee exist immediately particular force conclude correspondence write fix yy xx sc influence turn influence thus qp pc support service service sc fr totally influence sc two singleton describe preserve higher state definition dendrogram form node influence resolution influence edge mp rl mp influence cluster resolution service mp join co five five cluster hierarchical fr rl rl main influence singleton keep resolution sc cluster influence quasi partition resolution define define block partition relative importance influence mp except resolution totally resolution comprise mp observe three cluster co mp influence cluster top partition resolution mp red coincide total depict blue vertex height anchor center thick gray gray gray gray specification dendrogram resolution method dendrogram dendrogram component fig b quasi visualization quasi instead ten previous mining construction trade trade inf finance service service care service dendrogram influence resolution resolution dendrogram form singleton formalize dendrogram edge resolution b combine influence remain influence concentrate service whereas qp partition influence business service influence influence mining extraction influence apart influence definition quasi dendrogram reach trade trade service health care form green influence quasi partition include green one resolution green clusters main compose seven span secondary influence singleton influence quasi singleton join one induce correspond quasi interpret ordering resolution min resolution merge together red arbitrary combine red former latter sense edge quasi dendrogram cluster resolution dendrogram dendrogram fig min seem play clear large increase structure quasi merge resolution original dendrogram exclude form influential tendency anchor minimum height width anchor center black fill black fill fill center width pt introduce quasi network preserve quasi ultrametric linkage cluster stability method fulfil quasi study united network dendrogram family partition index arise resolution node group community dissimilarity differ determination difficulty formal whereby admissible respect asymmetric uniqueness former admissible admissible method cluster asymmetric decision derive symmetry dataset stage network relation dendrogram output generalization refer procedure back symmetry equivalence define quasi quasi dendrogram nest hierarchical clustering quasi regular partition contain disjoint include original block proceed respect axiom transformation axiom state quasi transformation dissimilarity quasi quasi cluster axiom linkage equivalence generalize quasi quasi stable establish quasi power algebra united year quasi exploit asymmetric california west result contain supplementary material finite node possible dissimilarity non hierarchical disjoint cover represent induce induce equivalence hierarchical partition nest collection term dendrogram different clustering increase start form node resolution quasi cluster concept node start connect edge consecutive node give dissimilarity link dissimilarity different endowed dissimilarity close asymmetric difference motivate structure partition equivalence relation search asymmetric symmetry binary necessarily relation hold quasi relation term order emphasize equivalence define unweighted self vertex edge satisfied pair b minimum height black height minimum black minimum height cm node blue vertex size minimum minimum cm node height minimum cm height node cm pp height minimum width fill height minimum fill pp height minimum width black vertex minimum height width fill blue vertex node minimum node width black cm node minimum height minimum point height thick bend thick bend thick bend pos edge thick bend right b thick bend pos node block vertex partition influence edge vertex quasi block direction block vice versa latter separate motivate direct influence edge likewise influence influence none influence accordance qp influence relation meaningful lack qp requirement qp qp definition quasi equivalence relation quasi equivalence distinct quasi conversely quasi partition induce quasi relation partition generalization datum far regular partition partition generalization hierarchical dendrogram nest partition nest index recall quasi dendrogram boundary resolution node separate cluster large cluster hierarchy x xx x continuity exist large cluster ever cluster join stay requirement give resolution except merge technical ensure correct definition cf dendrogram imply give quasi dendrogram dendrogram set vary nested recover dendrogram quasi equivalently quasi regard quasi empty quasi motivate cluster quasi vice versa quasi method axiom facilitate asymmetric version exactly node long cycle qp node qp qp qp imply every quasi acyclic dag dag construction quasi theoretic partial partition satisfy identity strong ultrametric regard quasi ultrametric dissimilarity ultrametric follow theorem preserve xx x either construct quasi ultrametric xx theorem dendrogram equivalent quasi ultrametric result allow cf quasi ultrametric apart importance since mathematically indeed paper term quasi quasi illustrated dendrogram ultrametric influence occur belong block conversely ultrametric class quasi ultrametric less contain distinct ultrametric equivalence quasi quasi ultrametric network ultrametric dendrogram dendrogram equivalent bottom first obtain dendrogram component node merge resolution ultrametric cf become resolution node belong vertex conversely depict value ultrametric edge equivalence resolution exist imply moreover one equivalence blue vertex blue bend left node bend leave bend bend leave pos node bend edge bend pos vertex scale blue scale bend bend pos node blue black draw black black encode axiom criterion axiom impose impose quasi arbitrary network axiom direct dissimilarity output direct axiom transformation relation reduce tendency cluster decrease axiom dissimilarity quasi ultrametric justification ultrametric quasi admissible axiom want method admissible axiom direct formally quasi admissible quasi ultrametric axiom turn unique admissible quasi axiom linkage admissible axiom undirected metric general asymmetric network lose axiom recover uniqueness asymmetric linkage clear non uniqueness asymmetric asymmetric dendrogram dendrogram uniqueness linkage cluster asymmetric develop study leverage admissible quasi asymmetric establish cluster space sensitive effect require weak axiom uniqueness aware effect admissible successful space network similar study finite formalize analogue hausdorff denote define metric supplementary material regard position express distance input network nearby yield nearby dissimilarity ensure supplementary non ultrametric outcome dissimilarity function original change dissimilarity quasi dendrogram influence arise original change supplementary material interpret dissimilarity correspond ultrametric quasi search chain infinity construct operation perform power algebra regular sum replace product quasi ultrametric power ultrametric triangle follow argument one furthermore inequality finite quasi ultrametric space building ultrametric network ultrametric operation algebra indeed exist cubic clustering relate instance take input asymmetric complete might modify follow quasi census state plus fraction come supplementary percentage come asymmetric dissimilarity application extensively ultrametric computed ultrametric dendrogram analyze dendrogram dendrogram influence
bound alg pruning step branching step state em branch split evenly subset subproblem ed queue subproblem select implementation subproblem sort subproblem bad branching subproblem select split subproblem element column branch node split node singleton node distribution evenly state node order probable branch b discuss section bound energy yield constraint relaxation conventional sdp sdp large interior sdp solver scalable sdp combine cut sdp constraint arise relaxation sdp integer state per normalization constraint one node negativity constraint become fully negativity relaxation loose submodular marginalization pi pi constraint connect negativity marginalization sdp approach directly marginal polytope constraints cut polytope equivalently polytope extend comprehensive constraint cut polytope polytope triangular consider three binary polytope cubic binary cycle odd inequality express triangular inequality cycle define cycle inequality odd odd constraint polytope define polytope binary inequality separation violate odd adopt graph separate node non represent project pi ph project inequality define cut maximum initialize lower bind primal cutting problem dual meet step round generate discrete combine aforementioned sdp map intersection outer lp several sdp inefficient computational involve polytope connect inequality grow end adapt propose cut plane cope constraint approximately discard case perturb sdp perturb simplify u column constraint condition eq solution large dual relaxation numerical restrict continuously twice gradient solve newton dual gradient step decide simplify feasible dual beneficial stop subproblem sequel far follow yield follow ii search gap x energy branch sdp relaxation minimum value consider lp sdp relaxation result integer relax relaxation sdp drop sdp significantly scalable interior add impractical find add violate cutting plane next redundant impose constraint cut search violate add sdp add constraint violate negativity triangular enumeration violate cycle inequality find separation cycle violate odd separation cut implementation iteration arithmetic scale eigen solver routine robust requirement violate negativity marginalization constraint arithmetic violate cycle save node odd cutting plane omit corresponding memory requirement efficient classic interior need operation sdp tight propose bounding b pre binary state prove solution reduce size apply assign integer persistent sdp bounding procedure sdp sdp computational reduce stop value calculate cast energy computation bounding stop meet see global subproblem low step small warm bound warm cutting procedure cut plane initialize constraint zero dual variable first bound initialize subsequent bound speed bound zero disadvantage cutting size subproblem correspondingly dual increase iteration traditionally never remove prune inactive address issue cutting drop work discard current activate relaxation polytope relaxation fully without branch plane cut plane result lower produce solve approximately impose well low yield cp cp effective iteration sdp connectivity strength performance mrf synthetic problem affect difficulty mrf per unary compare unary increase map mrf example belief propagation experiment mrf pairwise energy sample unary energy I unary versa connectivity link curve concatenation gradient within bound round conduct graph upper low meanwhile impact amongst drop decrease well lower bad amongst synthetic perform mrfs connectivity unary potential submodular denoise mrf use validate whether c experiment whether globally input add truth contain unary submodular pairwise branching relative latter submodular mrfs good lb cp v tr tr cp cp deconvolution instance cp solve instance respect bad six instance within hour six instance respect convolution growth portion size image third demonstrate partial white color gray last column image energy recovery deconvolution cp deconvolution reconstruct convolution formulate unary submodular energy connectivity growth image deconvolution different size deconvolution reduce cp report hamming perform post table result within runtime minute hour give hour omit minute quickly respect size six within hour solution short time exactly recovered v within minute energy cp image perform mrf variable eigen deconvolution core demonstrate cpu hour cp benchmark show inference densely unary good lb l cp bb bb achieve minute hour solution sub category fully unary compare cp bb winner bound method run limit hour hour limit method hour optimal solution three algorithm lb runtime label graph hour instance achieve bound chinese character term cut show achieve likewise comparable within hour solve bad specialized note utilize complicated well method include bound lower particular achieve bound lp validate bind good lb runtime top solve large instance method compare hour graphical maximize modularity six absence unary fully connect lp relaxation odd method modularity six large hour lin kl specialized offer instance solve hour word bound runtime deconvolution inexact detection runtime inexact inexact modularity instance runtime inexact bounding exactly summarize instance deconvolution modularity evaluation present cut map mrf advantage efficient sdp main propose variety incorporate sdp sdp solver cut plane technique branch bind warm dropping inactive unary type solve almost exist bound experiment compare art superior performance lagrangian function u transform serve increase converge firstly know matrix fix trace spherical x problem p p p primal contain duality equal optimal primal problem primal feasible proposition strong duality optimal add optimal objective equivalent bf font bf claim remark definition van university sa centre united branch cut solve mrf core bound sdp cut speed computation exploit warm start constraint method par magnitude unary experiment art non mrfs segmentation reconstruction find posteriori map mrf typically np however solve problem approximately exactly comparative cut mrfs potential exactly cut obtain globally optimal submodular pairwise mrfs portion highly graph weak unary mrfs move swap employ local move swap move energy subproblem solve class popular message max propagation exact structure cycle approximate belief tree reweighte propagation minimum energy approximate prove binary submodular mrfs propose aforementioned lp optimize polytope ordinary max lp relaxation structure generalize cycle exactly achieve tree structure graph dominate cone quadratic standard lp interior inefficient problem exploit lp method descent standard enough map especially usually perform poorly densely unary interest long cycle de se ed loose adopt consistency marginal loose existence violate cluster add search high triplet cycle separation cycle sdp sdp develop np hard accurate approximation primal interior state sdp solver worst require arithmetic operation requirement sdp node exponent even medium significantly relaxation mrf matrix eliminate point solve need many show constraint conventional relaxation consider local consistency mutually neither tight consistency tight sdp high lp sdp able linear branch cut sdp procedure approach either map budget computational art compete main contribution scalable sdp algorithm cut plane minimize intersection outer arise sdp relaxation scheme sdp linearly sharp violate interior still maintain energy sdp bound embed mrf problem optimize bound branching include reduction warm start removal model demonstrate compete dense unary potential line focus metric consider mrf branch mrf generally bb lp relaxation sophisticated include ce stochastic bb approximate inference optimize semidefinite technique work separate dual decomposition sdp subproblem qp guarantee experimental lp sdp well paper integer real value semidefinite aim solver program sdp semidefinite sdp solver interior sdp interior notation list use bold letter column bold low letter cone semidefinite semidefinite scalar statement trace matrix vector consist vector diagonal element frobenius introduce basic section contribution
functional standard less systematic recently machine ml functional principle ridge interact functional free dft produce highly accurate density energy systematically additional ml capable dimensional pattern successful medical stock automate categorization ml apply quantum include fast accurate molecular energy optimize divide new typical challenge new drive dft exact interact reference easy additionally ml approximation thousand million exact dft positivity approximate none issue example accurately approximations ridge system ref various cross validation great euler additionally descent modify euler effect theory background interact spin density spin atomic symbolic energy interact spin external potential potential elsewhere hamiltonian spin eq potential system energy use eigenfunction energy potential tb ref exact ref half half test dft dft variational euler lagrange chemical satisfie constraint density drive density satisfy ground density free dft energy ks density cause functional functional derivative approximation impossible avoid nontrivial von ref poorly mae self bad local add modified expansion little topology representation ref molecular predict energy contrast norm derivative conventional derivative dimensional practice expand finite basis density calculation represent express product result converge truncate fortunately greatly variety parametrized parameter manifold density potential density potential kernel trick structure optimize parametrize form increasingly linearly space linear tb illustrate lie circle transform back belong map assume wish space thought must existence term need compute enable ridge version regression prevent training density determine minimize regularization ij k magnitude prevent overfitte calculation uncertainty identity parameter kernel length find cross sect design e kernel choice reflect characteristic ref choose approximate equivalent notation e norm scale relate ref grid define fig distance tb tb select kernel dash chemical gray hyperparameter randomize fold median small inverse numerically limit force directly center mean fig illustrate example datum unnecessary center kernel reproduce domain domain require center regularization strength e scale standard kernel laplacian wave em radial basis rbf behave broad problem well contour mae set strength see cauchy contour scale neighboring rbf yield poor functional become unity effect contour comparable scale kernel vary nonlinearity region mae cauchy performance middle dash laplacian behave smooth hyperparameter pick value hyperparameter must hyperparameter generalization future never give final model essential selection cross training validation ml set hyperparameter set never determine assess see dot density analyze randomly optimize mae training bin bin bins minimize mae total final hyperparameter validation loo special typically leave simple fold expensive leave intensive good mae test scheme similar mae exist variation occur cross validation generalize kernel dot choice dot global mae relatively flat validation minimum indicate well em wave finally use time optimize functional drive paper ref perform c em n set give error likewise density test give chemical ref systematically absolute mae hand wave increase indicate wave flexible enough note choice need reference energy grid ml depend demonstrate grid cross validate accurately accurately type potential limit energy underlie parameter needs distinguish comparable fine desire large able basis sparse reference greatly tb mean drive density validate hyperparameter mae reduce jump challenge density discussion evaluate density e functional useful must solve yield accurate derivative plot model display inaccurate huge apparent ref dimensionality unable functional information ml direction fig minimization technique interpolation information many orthogonal produce inaccurate derivative dimension produce since exist dimension create fig deviation constrain tb starting show quickly cause density minimization stay euler minimization elsewhere manifold normalization previous long minimize ground give reduce domain avoid confusion call develop attempt reconstruct manifold tb j consistent square function elsewhere implicitly interpolation show accurate red become unstable jj tn approximation manifold aim locally linear principal density manifold weight generalize evaluate locality come distance density density ref smooth weight pca average define density covariance eq eigenvector direction lose keep direction direction tangent first pc projection onto space choose pca approximation square tangent pca project projected approximation density guess density derivative compute project see take trading speed ensure remains weight iterate convergence achieve tolerance max density converge tb take along project direction energy stay density project report error density density factor constrain density data sample constrain magnitude particle density locality density orthogonal individual pca choose pcs mae pc n pcs pca pcs structure see space introduce fail search report constrain giving although generate parameter
lasso valuable rectangle text blue corner width corner biology york ny lasso inherent validate variable illustrate promising synthetic biological field year science engineering throughput drive mathematical variable analyze interest irrelevant lasso popular practice dimensional turn properly adjust aspect practice cross validation tuning validation inefficient lasso lasso lasso tune adjustment design proper simultaneously accurate computationally attractive contribution systematic development square tuning bootstrapping scheme finding response constant exceeds typically ease exposition sequel allow random design support zero entry dimensional regression estimator detail lasso recall tune least criterion parameter regularization influential reasonable choice look bind cf overview proportional satisfy calibration need aspect tail calibration design matrix approach deviation approach recall tuning parameter root similarly lasso determine intensity see square deviation readily locate denominator distinction lasso act inherent deviation make lasso adjust tail address incorporate inherent quantity consistent define accord estimator one one mapping square root contrast establish interesting path omit brevity tucker latter resemble fact estimator formulation equip tackle spectrum estimation prediction variable task bootstrappe fix majority bootstrap bootstrapping scheme bootstrappe rule already practice illustrative sample readily least square perform dual extension pair dual straightforward beyond scope subject currently finite root oracle properly yet type function comprise amenable invoke integer fit amenable tailor convex fitting solution several among effective guarantee fast excellent novel increase non smoothly mcp prove computable example synthetic inspire biological set involve production numerical matlab ghz intel core memory cross validate schmidt approximate parsimonious stopping criterion tolerance iteration evaluate selection synthetic minimal validate mean error cv generate regression inspire simulation sample vector normal error multiply sample row normal normalize scalability repetition thick colored correspond thin bar precisely report runtime plain lasso performance fix function show runtime cubic scalability least comparison scheme reveal fit result setting selection strong hamming consistently supplementary material compare provide excellent cv recently publish biological comprise gene experiment vary profile expression standardize production measure highly predictive production report runtime matlab routine gene list specifically stability coefficient enter runtime single computation approximately select considerably cv corresponding coefficient list majority selection vote list select frequently reveal key insight coefficient lasso cv solution cv rank cv frequency b select plausible locate co express runtime lasso complexity differ considerably often error report cv three error l equal lasso give equal lasso bt intensive omit observe square counterpart
problem maximize eq regime strategy word mean use attain due restrict consider binomial difference conclusion alternative big substantially independent parameter divide english word arbitrarily instance high english model become likelihood likelihood maxima log start per model merge language symbol compute give merge english per english english merging language become recall non english consider entropy accounting eq correction conjugate parameterized function vector clearly distribution document dirichlet pre simple probability zero top topic tend evaluate computed spirit document unseen document hold fraction remain unseen without change topic frequency unseen document run lda improvement let wang medical institute chemical biological engineering department physics database text knowledge require extract document assign document enable search statistical characterization lda state art systematic technique lda yield infer adapt approach community wikipedia reveal big collect store analysis nearly digital keep knowledge challenge language text database gap topic database recommendation digital spam filtering dirichlet allocation topic rely document cover mixture characterize usage address topic document corpus focus might different word primarily word equally application topic mixture crucially rely maximization linearly large known computationally hard landscape gain thorough implement highly specify control theoretical normally standard rough topology landscape exclusive topic enable search landscape document model count mixture topic english topic topic might english two merged vocabulary document symmetric fraction english big variational curve infer algorithm th generative theoretical grey area lda actual model practitioner large almost equally pose serious investigate elementary language corpus helpful provide realistic complex language fully language language document entirely create simple lda use two document step select concentration si language step randomly word document sake simplicity restrict document bag english language language language infer language alternatively merge two count english part likelihood generative model lda fact divide english corpus si increase english document overfitte merge log likelihood per document fraction great likelihood limit likelihood identify model indeed non negative kl critical depend document corpus increase lda si likelihood though great infinite infinitely generative landscape consequence define extremely vocabulary per si conclusion find potentially model regardless likelihood technique highly equally landscape technique yield across accuracy match among fit typically lda represent different language slice language test value overfitte overfitte language line initialization resolve si algorithms require assumption corpus reality need resolve test synthetic corpora corpus ten language comprises belong language class comprise eight language word frequency language validity calculate output similarity infer corpora enough dataset si estimating would lead unable global landscape si show asymmetric implement gibbs result landscape current improve performance build intuition landscape view bipartite network construct word language corpora language component find topic complex comprise unlikely topic use compare co document word dot product similarity distribution depend si significance word filter present corpus identify word exclusive topic si information decide refine asymmetric distribute likelihood actually wikipedia validity lda corpus comprise document web contain publish six economic process document corpus remove list word pre yield topic number topic topic split merge suggest small split topic find corpus journal science frequent bottom topic find big topic journal document assign journal publish infer generative fig perfect standard lda lda comprise si let six infer put paper journal paper publish science likelihood model si systematic different implement choose lda tune topic within corpora extent topic result corpora si test generative size right portion equally overhead guess small easily wikipedia document research limitation model able remarkable validity simple function guess guess obtain correlation topic initial guess propose practical make separability algorithm interestingly improvement degeneracy yield well create synthetic corpus dirichlet probability small document make fraction implement latent topic write topic synthetic generate make lda sized topic equal topic initialization lda describe measure distribution topic use topic permutation label easy quantify similarity compare since topic topic get versus st stand look make average st similarity similar assignment label similarity rand eventually measure model define ht synthetic dataset simplicity document topic topic specify vocabulary simplicity model distribution corpus aspect mixed document topic across word decide drawing probability see word compute bayes corpus choose value easy corpus mostly keep constant document however mostly use since realistic corpus relate code david discussion outline supplementary language language simple set hand find absence sort eventually treat detail lda entropy language sake sec eq since generative asymmetric lda always likelihood document asymmetric information document per likelihood conservative correspond topic argue log optimization cumulative relative difference log generative find log versus match visible accord language support conclusion discuss number topic fit illustrate big english science share music word assume word english write sake language english datum find find language calculation higher lda lda sec fit english use english vocabulary english tell go english merge reason big split lda science see display argue topic make small method first build connect term topic cluster like topic optimize asymmetric likelihood via variational bipartite word number network word dot generic strongly put term relate area filter dot nan weight variable occurrence word corpus rare event approximate poisson average ab randomly occurrence uniformly fix share document poisson precise large dot corpus comprise six document build isolated refine topic isolate word topic build provide belong partition recall assume completely discard time word always generate module word locate since hard module document time topic word optimize describe series move aim aim topic make specific precise move topic select consider independently topic actually come binomial topic small significant increment zero decrease accordingly repeat previous explicit dependency maximize refined lda closely main data document case quickly situation similar describe practice software every iteration well explore filter lda optimizing perform set change measure sec topic need run take run significant document topic might select significant document appear likelihood depend application small topic optimization threshold wikipedia software let remove initial select topic hold fraction corpus algorithm dataset measure hold model obtain method tend actual one lda provide fairly uniform assess entropy topic entropy probable compare topic versus effective topic number show effective dash black line topic select black panel achievable topic topic five method low visualization generative one dedicate measure topic performance lda compare lda synthetic yield achieve provide implement generic generative unknown measure perform poorly performance dataset color allow way get similarly happen language small topic together indicate two across horizontal bar divide proportional document sort prominent sized topic lda actual comparison corner obtain topic sized lda fed topic hard guess information get set reasonably get sometimes give slightly much increase number basic option parameter choose relative difference mean affect lda topic document seed whole dataset though initialization guess mean actual similarly generative get check performance close one define topic lda grow overfitte main decide web science sec lda test difference systematic test step topic provide highly heterogeneous section asymmetric difference probability topic variational language well fig structure dataset model similarly topic topic
impact avoid set expense running find affect speed sensitivity spc generate combination averaged combination overlap length except value also greatest small l l spc clusters entire hierarchical path spc path include number obtained drive explore different still along gap instability select aim spc simple decrease datum log big increase log likelihood sort difference number cluster proportion estimate line implicit behind singleton difference likelihood simulate accord ari relatively ari score demonstrate average ari term ari path ari average show close average range take simulated separate averaged solution large ari score solution path determine whether singleton cluster truly challenging reflect average score table overlap scenario ari score clustered misclassifie contrary select fast appear spc various k scenario cluster well noise spc dataset tf self cell expression tf activity bind tf spc general cluster include difficulty provide spc proportion irrelevant observation search solution adaptively irrelevant spc fine tuning input result set lowest long importantly spc require cluster provide estimate impose two merge stage however eliminate perform split soft thresholding operation discuss practice case split spc assume object gradually split separate cluster later stage receive penalization spc across simulated datum scenario penalization widely apply path surface selection bad feature spc majority exist framework convex sparsity need address singleton outlier need cluster usually high cutoff address future paper solution interesting every fix value cycle minimization reduce thresholding remainder coordinate modify incorporate fix give q recall minimize reduce define see globally sufficient imply note minimizer k ari nm extra date date hyper nm extra proposition remark research fellowship year fellowship support nsf grant dms author amount create discover large scale result problem grouping address methodology introduce regularization difference adaptive provide produce corresponding cluster solution optimization carry block advantage compare simultaneously separate grouping methodology various simulate dataset expression competitive collection discovery information instrumental visualization adequate enable discovery group association deep insight biological cluster phenotype help variety rely usually like hierarchical graph popularity cluster map support name cluster vast survey usually comprehensive general development trend mining approach datum contain amount irrelevant identify recently researcher noisy irrelevant interpretation new popular penalize cluster noise account mean algorithm challenge specification cluster exist solution rule different gain complex penalization fuse lasso method generalize useful large loss compute entire penalize angle spc include irrelevant singleton cluster algorithm solution specify objective couple coordinate adaptive path spc effect spc simple work well work penalize utilize gene spc assume select object author cluster impose center cluster center result converge globally severe cluster procedure handle weight primarily discuss weight cluster unified solving cluster select penalize penalty penalty author parametrize selection pre grid penalty remainder provide path spc several spc spc big direction matrix object cluster achieve pairwise careful arbitrarily minimizer important little cluster naturally separate noisy belong prevent merging simulation adjust plot see section appropriately continuity assignment mcp develop mcp define penalty regularization concavity mcp penalty form mcp concavity compare convex penalty group penalty explicit concavity easily rate bias drive minimax figure illustrate case special minimized minimizing reduce eq plot see difference circular light gray contour penalize contour black minimum force minimizer penalty affect center gray contour figure contour produce dark gray contour objective small value minimize cluster center estimate proper choice correspond penalty surrogate cyclic singleton merge appropriately correspondingly two suppose solution center represent function minimize force merging current step iteration number due concavity obtain q weight adaptive weight penalty weight distance vary whereas change minimizer penalize sufficient step thresholding spc never gradually computation work well necessary could handle thresholding detail soft establish penalty condition meet mcp separable apply objective meet fix mm coincide set consequently eq reach almost iteration controlling amount later degree concavity create simple drive rule parameter cluster start individual form singleton gradually enforce great reduce decreasing depend current warm later decrease initial low concavity decrease concavity behave considerable whether decrease bias ratio cluster behind certain center beyond concavity bias lead bad address evenly lemma simple lemma lemma provide mm minimizer lemma lower serve quantile regard approximate proportion point default need point obtain first denote merge suppose decrease z upper maximum object collection object merge merge case bind decrease algorithm cluster summary involve mm consequently demonstrate affected choice small slightly small avoid unnecessary recommend paper user calculation stand approximate near tight could value dataset well noise force initially merge noisy high dimensional construction describe full spc algorithm initialize assume form singleton warm start require input spc provide kk g spc k weight use adjust ari across ari assignment solution true take partition ari find simulation ari quality compete belong calculate ari suppose estimate assign respective see contingency ari count identify belong estimated sum misclassification sum sensitive identify noise datum noise except table consider plug mean cluster k randomized center input cluster comparison input result compete find stay along user target clustering rely small conversely small noise assign reciprocal region calculate ij categorization cluster categorization identify generate recommend near categorization size small detect perform choose near neighbor cluster convex specification weight neighbor solution path result package performance separate cluster separate overlap equal outside overlap cluster locate simulate demonstrate spc noise also define separate scenario merge output dataset cluster center cluster present true cluster spc solution mis report ari range g spc path note detect scenario scenario spc spc perform similarly well tight outperform separate cluster pre hierarchical cluster remain excellent spherical perfectly match spc scenario initialization c n comparison method scenario average top bar refer cluster find spc report well considerably separate spc merge overlap identify noise spc create big cluster reflect cluster mis specify tendency cluster close together overlap cluster tends produce satisfactory aside add cluster figure assignment consecutive spc separate noise perfectly two method separate one tight figure add large add histogram iteration combine four scenario instance converge decrease ht b simulate cluster datum distribute cluster high detail small merge initial datum cluster consequently though decrease increase along spc cluster misclassifie point cluster satisfactory comparison scenario axis indicate range range spc dimension
g piecewise approximation logarithm scad directly become approximation multiply add appropriate procedure original form scad similarly approximation indicate dc result state theorem except depend way right namely calculation compare close tend poor easy p deep study much context furthermore hold problem via prove norm appropriate vector b b u reformulate penalize penalty solution iff function dc program demonstrate equivalent suitable function iff feasible q feasible solution conversely feasible nu fx nr fx nr nu equivalently special equivalent eq q iff xx fx ft ft fx sect therefore ki li virtue equivalent discuss solution proposition omit ambiguity three suppose right denote problem give existence derivative define prove without generality equivalent equivalently write concave hold concavity r rt dc component k find previous work scad induce dc decomposition k svms I reweighte problem dc program solution dc I k x since update solve algorithm initialize x k form iteratively solve weighted reweighte sparse stage run character reweighte propose justification convergence I reweighte algorithms context linear log z next perturbation third exist reweighted type optimization I I dc program dc iteration dc k k updating initialize compute k I choose dc become expression approximation algorithm reweighte iteration converge cm ir sparse p reconstruction k seem address additional subproblem less constraint dc decomposition suitable dc program dc svm approximations enjoy formulation subproblem subproblem also possess dc program dc quite compute influence property update let ki leave resp k k take derivative variable help exceed know convergence possess kx know uci repository point test description site uci repository htbp feature breast algorithms pc intel ghz gb ram program zero small algorithms accuracy solution training test sparsity solution percentage second experiment propose scheme experiment value dataset five choose suitable cross validation good perform cpu purpose fair run update procedure stop update solve linear last column bold fs fs cpu fs fs fs cpu fs cpu fs cpu comment concern correctness gain three select well fast evaluation criterion update update procedure solution update fold two comparable chosen fold validation smaller confirm analysis subsection become difficult bad contradiction updating updating give global training updating dataset cpu second solution second section experiment parameter follow fold validation point exp lp sf cpu sf cpu sf breast sf sf sf sf cpu approximation considerably number select select good correctness term give train quite cpu study dc programming dc approximation algorithmic consider class dc norm usual induce function consistency minimizer problem original sufficiently large relate solution solve original scad problem sense usual induce formulation common concave scheme approach nonconvex dc approximation piecewise usual concern svm scheme finitely local confirm theoretical identify winner induce dc programming light sparse nonconvex permit establish crucial relation induce elegant way effect dc decomposition perturb algorithm reweighte reweighted specify deep dc model solve world nonconvex large corollary le involve programming consider dc include induce study resp global minimizer resp minimizer use dc approximation namely induce function analyze cover sparse reweighte reweighte well dc tackle implement feature selection empirical comparative various dc function feature zero denote cardinality optimization one refer problem domain finance draw increase attention researcher recent origin nonconvex x q regularization make want solution important learn text micro array analysis feature feature feature preserve discrete classifier classification determining lead identically sample compose explanatory response vector input look relation possibly relate model parameter eq loss take relationship multivariate form composite identically distribute explanatory discriminant onto maximize class I e maximize bs scatter positive class label sample optimal fisher discriminant refer reconstruct dictionary selection available particular amount way among call portfolio portfolio management want investigate sensor digital decade involve divide category convex nonconvex approximation nonconvex belong group replace suitable efficient chapter penalty inconsistent variable bias introduce adaptive nonconvex approximate nonconvex extensively induce penalty exponential fu smoothly deviation scad logarithmic definition use algorithm develop successive gray local stage zhang lasso reweighte reweighte quadratic category name reformulate author context program generally intractable dc program sparse symmetric problem category heuristic tackle greedy orthogonal pursuit etc approach involve regularizer problem nonconvex approximation deep relaxation produce good many local minima approach weak concave bound term show set approximate original nonempty available minima lack mathematical always challenge researcher learn issue cite approach approach programming robust scalable nonconvex nonsmooth continuous contribution firstly dc consistency link minimizer minimizer neighbourhood strongly concave optimal solution secondly depth induce suggest approximation reasonable via identify scad suitable moreover box exact technique interesting sparsity induce dc main show concern dc dc piecewise guarantee permit exploit elegant dc decomposition flexibility dc view perturb reweighte penalize lasso reweighte careful empirical dc programming brief algorithmic minimizer study comparative usual approximation discuss deep problem approach conclude equip canonical euclidean denote identify programming constitute global introduce preliminary extensively le original constraint program exploit deep result elegant approximate nonconvex dc program popularity rich deep rigorous robustness method adaptation problem real nonconvex development programming mainly devoted obviously dc high nonconvex programming dc form equal take dc dc dc convex q dc n wide life operation dc constitute sufficiently reference order leverage powerful dc primal dc also dc eq dc convention function nonempty finite dc program dc play algorithmic optimality subdifferential generalize singleton nothing programming dc condition dc program dc lie distinction local solution global develop dc dc trivially point critical tucker kkt optimality next case quite practice dc program critical local minimizer since differentiable everywhere say critical locally locally inclusion resp resp solve dc program also reference therein local optimality duality programming simple approximate sequence program iteration approximate concave correspond minimize convex generic guess hx note convex far solve important critical terminate iteration bound every critical dc program dc program optimizer dc whole convergence worth involve dc hence dc version dc infinitely dc decomposition implication robustness search dc important dc decomposition tackle scale try either explicit form computation subgradient usual rule calculate subdifferential efficient adapt handle generic sensible study answer structure nonconvex search dc point dc programming nonconvex program field apply science especially learn reference decomposition suitably dc permit recover nonconvex program global algorithm program dc program dc programming use program dc programming reader therein dc dc dc stand precisely h gx yx st idea replace lead dc function function u v get resp resp local
ignore influence form term decomposition extra impose combination basis properly point suggest subsequently brief dictionary environment extensive simulation effectiveness analyze beyond principle completely solve rather target completion liu nsf nsf fa li nsf iii nsf cardinality play role suppose u obey provide ac ie ie ie ie ie ie frobenius sense u ie kronecker definition ie aa inequality obeys constant u lemma lemma u u u I u u I I feasible standard convexity lagrange l gradient svd uv u ia uv ta uv uv convex optimal solution feasible increase x aa denote complement orthonormal cc ta l equality already unless triangle f u validity f last minus height depth computer science nj usa department science nj abstract complete rank establish theoretically may low capture property merely constraint specify accordingly even handle propose low impose restriction datum dictionary properly non uniform lead practical completion experiment generate dataset encourage application need entry general give adopt assumption latent want fairly low suppose ij lie th entry location entry scalable various g notable contribution probably ease shall tell meanwhile I exactly parameter free scalable sum space support besides completeness also completion rank strictly uniformly critical success low constraint low detail structure nonlinear hard point figure e probably reality might uniform datum uniform might fail area vision motion provably advanced rank matrix citation construct learn unlike minimizer fall back regarded generalization properly mathematically uniform provide elementary dictionary devise proper environment dataset encourage summary contribution completion modeling matrix furthermore term version idea replace product concept g regime affect behavior coherence point non capital letter accordingly denote entry particular symbol mu rv shall abuse notation span onto space abuse notation space support complement identity function norm norm I large singular frobenius square denote I sum singular letter coherence besides extra structure coherence hence promise direction non influence parameter follow prove detailed noiseless svd svd subspace numerical confirm dictionary ask imply equality elementary learn h e one everything else dictionary unit coherence recover incomplete noiseless reality true observation contaminate entry completion accurately perform modify noisy completion follow theorem svd ac mn give recovery theorem potential kind framework like firstly utilize computation solve equivalent provide contaminate gaussian moderately support location solve optimize rank solve svd normalize column I denote optimize summarize whole encourage well sufficient facilitate dictionary unit theory backward whenever already successful recover successful effectiveness algorithm randomly datum index create rank vary step step fraction result trial matrix successful recovery successful pair mn sense denote compare learn work area success white significance dictionary real motion incomplete sequence database dataset dimension vision
acceleration use unit neuron time architecture cnn particular slightly architecture cnn architecture stable behavior hence cnn architecture subsequent cross cnn deconvolution aid behavior cnn positive ct image cnn module sensitivity candidate label away positive cnn subset rate positive patch balanced beneficial cnn balance channel patch patch mm physical ct window times translate scale take run extraction ct volume take train cnn prediction candidate parameter receiver operating amount quickly improve fp per volume fp auc point fp perform cnn candidate demonstrate task effective fp building reduce initial scale sampling rotation around prevent overfitte increase cnn exhibit range fp art sensitivity recent fp obtain fp assume candidate generation note available moment material publicly convenient improvement joint coherent literature alternative classification aggregation fusion indicate quality prediction work investigate fusion cnn show variety high orientation cnn clinical center publication automate detection clinical diagnostic structure distribute state sensitivity positive fp shot haar paper operate preliminary generation towards sensitivity fp level patient volume interest consequently orthogonal view via rotation respect coordinate train deep convolutional classifier cnn employ probability final validate ct volume patient fp respectively drastically art segmentation node play many cancer become e computed image small diameter short axis ct slice fig play assess follow classify coefficient manual processing consume clinical system ct base feature pool haar strong classifier availability intrinsic structure particularly appearance patient fp moderately sensitivity achieve fp range part region clinical employ high focus reduce shot via multi fusion voxel prediction multiple descriptor candidate topic sensitivity fp image red map assign slice volume channel slice simplify jointly three individual separately cnns slice prediction image ct scale edge length number voxel order increase variation analogous datum augmentation translate translate orient random permit neural net classifying unseen average per candidate classification one image patch purpose decompose channel combine slice cnn directly burden curse
proof axiom bind name axiom type interface already release since library include release tracking treat new logical axiom take name bind body canonical definition name type axiom name type infer body definition prove progress proof type name strictly step environment add exactly internal proven implement kind environment tracking walk type body construct tracking note previous proof main parse language expansion phase store collect try auto contain language try level else third typically basic atomic dependency tracking available implement protocol user dependency message progress make dependency obtain look define algebra anti convert logic z bi proof machine necessary dependency apply evaluated set theory logic available concept divide implementation class symbol meta mode separate object long type system treat proposition proof type name theorem name name term name without check type sufficient evaluate proof average advanced pattern combination modify combine harmonic classification method implement tool external near fact conjecture theorem naive assumption conjecture fact naive proving feature formula independence assumption say occurrence occurrence feature predict relevance ng filter keep track conjecture perform iteratively fact fact score roughly irrelevant occur fact perfect score fact feature feature cover coverage proof dependency suggestion cover cover dependency precision suggestion call need whole large enough position dependency order relevance roc rank close rank use rank iff evaluation sort fact try predict dependency learn topological sort fact ai interactive proving proof theorems performance nn naive comparable library correspond symbol proposition system heuristic dependency need recommend proof auc k ensemble far overall encourage library prove conjecture hard system include technique could implementation work directly logic thank help terminology research support proof dependency obtain dependency formula proof come dependency comparable large corpus last decade interactive various system system light reach previous proof already available library system important relevant theorem library actually separately early automate ai
evolution universe lead formation without shape shape information human understanding complex distribution galaxy formation evolution universe shape view point must perform way free shape galaxy image approximate perform analyse dataset galaxy show general idea sparse code approximate combination predefine approximate dictionary adapt basis well predefine galaxy solve frobenius package choose image must eliminate image small standardized band method galaxy low dimensional image intensity paragraph dictionary call dictionary collection collection suppose pool dataset fit dictionary vector suggest mmd statistic mmd select two dataset see image generate randomly n jk jk jk cv ht approximate galaxy dimension galaxy help distinguish distribution future work constraint galaxy shape shape elliptical one galaxy pa
square exponential periodic define kronecker delta kind symbol define sx replace sigmoid simple application start equal expression propose search operator parameter optimize scoring search greedy search operator base c acc sp cp se sp mkl se lin material report maintain recommend read first review discuss analysis demonstrate temperature exactly call centre demonstrate many highly structured white automatic regression explore open discover explanation set treat consequence trend compositional language allow term rich domain statistical field rely machine researcher simple automate package little interpretable intelligence modeling automatically paper conjecture ai statistic work incorporate open expressive enough many composition world phenomena search efficiently span language evaluate complexity fit automatically explain visualize choose quantify improve part describe note know call automatic covariance discovery define gaussian process information evaluate compositional develop language show automatically report interpretability term find art series consist learn expressive complex nonparametric smoothness process jointly practice mean equivalently addition rich structure trend composition way particular incorporate figure expand amenable automatic extend table list common model e operator include material descent bayesian criterion q implement describe generate natural description language convert expression simplify sum product kernel different multiply multiply original rule write kernel different independently product separately describe contribution noun description justify multiplication remove long since decrease increase linear cause vary linearly multiplication multiplication multiply periodic formally ht l phrase smoothly periodic linearly amplitude polynomially act table product form noun phrase noun noun phrase uncorrelated smooth polynomial number way description head noun interpretability description qualitatively rapidly description include periodic period description extra linearly report noun choose head noun choice area attempt present first add component fold q convert q head noun description uncorrelated smooth correspond finally third describe periodic period period demonstrate ability discover material year cycle rest rare identify description summary see identify component component third term trend well slowly vary trend next model component accurately describe periodic white express description system learn constant replace capture offset product approximately periodic component heavily approximated span kernel significantly interpretability discuss rational gaussian express infinitely capture short one rational quadratic material component capture medium term trend short visually describe contrast separate medium term deviation mat ern reason ht xshift xshift yshift box right find periodic great zero semidefinite similarly product become qualitatively anti credible xshift yshift box xshift xshift yshift yshift credible interval anti manually composite kernel supplementary include dataset procedure choose similar paper whose fit composite interpret automatically manually model spectrum delta kernel spectral kernel table body construct rich mkl polynomial time express language form specify time covariance function search decomposition graph space define select criterion model differ automatically statistical natural output automate ht accuracy building evaluate performance list time library report supplementary material six spectral trend produce interpretable predictive include brevity experiment default express restriction restriction mkl trend spectral greedy procedure marginal optimisation advanced use method move average construct class series interesting future bic parameter nest produce
become huge distortion distortion equal gap couple run distortion effect distortion contour source decrease curve source plot figure increase increase correlation se section corner terminal observe wave start boundary proceed center recover gradually particular create wave boundary depict experiment keep couple wave wave stop proceed recover result spatially couple see result error gradually increase contrary spatially case wave proceed variable stop sharp transition measurement rate transition depict sc wave wave space variable write finitely q binary nonzero index unless similarly linearly deal case terminal joint try single terminal matrix two check message fix appearance om variable check message different therefore get obtain eq negligible asymptotically fact tend similar giving replace obtain similarly equation intuitive justification terminal amp matrix word additive consist amp follow asymptotic get theorem row similarly gaussian converge compatible similar derivation tt argument obtain imply give equation ks mmse observation similarly variable covariance elsewhere matrix whose provide mean let conjecture remark amp initially compress sense matrix extend multi terminal show terminal behavior characterize se distortion terminal source spatially couple rate distortion curve phase approximately distortion fully measurement distribute match se pass amp measurement coupling terminal pass enyi dimension interested sufficiently many matrix respectively take separately recovery terminal hoc environmental signal temperature etc one imagine sensor terminal fusion center sensor communication low processing joint terminal process fusion recover usually exploit redundancy particular scenario sensor device turn increase kind temporal temporal correlation result slow temperature signal correlation sensor densely environment precise result redundant energy resource desirable sensor environmental densely distribute sensor assign measurement case require distortion address terminal signal terminal correlation sample study terminal make connection distribute source please refer study prove regularity condition encoder negligible enyi decoder prove block enyi necessary exploit hadamard capture spatially rigorously prove rate source source feasible rigorously analyze terminal term code replace also develop multi terminal variant compress well behave probability signal define independent reconstruct source behavior se distortion distortion spatially couple distortion low eq x dx mmse eq q mmse limit eq lebesgue theorem decompose continuous part e enyi singular well weight restrict singular space brief overview linearly conditional realization terminal vector whose component joint se depend state choice mmse estimator point se point simply check increase hypothesis result case already traditional rate require rate situation code threshold result pass different threshold spatial necessary briefly describe structure spatially couple measurement band weight roughly e band diagonal denote row column indice terminal ratio n measurement rate terminal variable explain terminal whose index variable mmse spatially couple accord index take belong block column belong block terminal spatially couple output equation obtain asymptotically infinity base terminal case give follow terminal terminal q ensemble couple measurement separately recover negligible rate code role discrete code corner achievable terminal distortion multi terminal term mmse use se dominate converge converge step single terminal zero gaussian use linearly similarly possible achievable dominant face measurement achieve corner copy ensemble negligible region rate dominant distortion
use way gaussian different kernel object similarity generate case interpret kernel ratio show problem class mkl rt formulation applicable like orthonormal pls procedure guarantee problem experimentally mkl rt select reduction modal retrieval mkl rt well mkl dr formulated optimization discuss ratio present mkl rt conclude indicator transpose denote appropriate denote absolute two ratio trace datum transformation use prevent overfitte popular pls make solution correspond pair kl transform x early computer vision trace evaluation mkl rt label correspond representation use cross modal retrieval map modality dimensional concept modality highly correlate modality feature ratio trace modality xx pair two modality common latent suppose provide label modality directly extension label sample class way implement replicate without implementation ratio modality xx n z far mkl parametrize combination learn mkl ratio trace formulate nevertheless symmetric rank eigenvalue eigenvector let define optimal please supplementary material iteration compute successively restrict maximize verify optimum individually unconstrained program system eq h compute solve I update summarize algorithm ratio transform representation mx summarize rt summarize new linearly transform mx mkl rt trace explain covering application discriminative modal retrieval wikipedia modal rt trace define solve trace use correspondence trace mkl approach dataset use svm nn rule reduction compare mkl rt nn best svm nn nn kernels eq reduction use use number recognition category kernel various image available online omit different split average use per regularization rate split follow approach produce pt discriminative constraint mkl dr figure mkl dr training sample multiclass image per come predefined split moreover author brevity omit matrix mx mx predefine average regularization set various approach svm report use standard figure weight mkl rt dr third split rt rt rt mkl rt mkl svm svm rt rt rt mkl mkl rt mkl mkl dr experiment modality modal wikipedia article retrieval text design training test group broad art etc text linear allocation provide histogram orient consist histogram pyramid rbf sift descriptor order descriptor sift descriptor pyramid histogram rbf matrix pt histogram rotation correspond pyramid kernel simple generate pyramid generate rbf descriptor record pool score see mkl rt give dr perform poorly rt rt mkl rt respectively show mkl mkl mkl rt mkl select show cross modal pt image rt rt rt mkl dr rt wikipedia convex different modality common distance wikipedia dataset number average score text mkl rt give poorly rt rt rt figure mkl mkl rt whereas mkl dr kernels text rt rt rt mkl rt rt wikipedia non mkl wikipedia select object image test text linear tag rank provide various kernel rt retrieval performance mkl dr approach perform poorly mkl rt consider rt mkl mkl rt select kernel perform result much present query rt rt rt rt c dr rt paper show formulated ratio trace include like orthonormal pls provide propose ratio problem demonstrate discriminative cross modal retrieval rt non mkl dr plan trace line mkl mkl plan rt pt pt automatic mkl attention various research community mkl context machine explore ratio trace formulate popular procedure converge global experimentally mkl mkl rt successfully use modal rt perform well recently propose mkl many vision application often transform initial new representation application consideration transform different interested image query may want representation correlate plan linear transformation many trace whose solution extensively computer vision popular algorithm ratio trace semi supervise fisher locality
maximization equation require chain hope dependency detailed indirect identify conjecture conjecture maximizer conjecture give easy identification testing purpose chain period allow coordinate least test conjecture random create distribution rbf arise choice g adjust adaptively decrease systematically varied along unit dimensional attain bottom conjecture argue justify heuristic indicate global possible start configuration adaptation optimum computation maximize drive coordinate adaptation progress cd consecutive progress provide optimally wise sufficient coordinate algorithm wise become ensure condition rate stationary approximation progress algorithm strictly decrease equilibrium quasi deviation additive progress deviation term leave side accord thus small enough towards equilibrium equivalent assumption indeed fulfil practice approximation cd extremely step argument gain validate test clear fulfil proxy deviation result progress usually flat optimizer careful variance variant discuss software problem cd class analogy stop linear linear algorithm soon tucker drop establish value even compute problem logistic component dual drop straightforward indicator always reliable indicator coordinate correspond roughly zero lasso cost cd varie svm extension library solve iteration solver pick derivative set evaluation list separate parameter varied grid apply shrink carry class comparison similar shrink solver well considerably time wrong shrink decision shrink well drop shrink uniform instance news benchmark multi svm experiment experimental baseline value r iteration second class svm subspace descent parameter vary within implement logistic solver analog linear dual logistic regression shrink applicable apply l c fold cv iteration second well speed indicate mark stop day news center good cd bit slow case improve significant relevant namely result training time logistic cd order cd tend meet coordinate overhead adaptation value regularization run extremely soon start pay sometimes dominate outperform svm shrink shrink priori contrast generic speedup specific technique shrink result introduce coordinate coordinate coordinate cd aim maximize minimize cd coordinate obvious feature cd guess inform particularly coordinate inactive perspective coordinate maximize characterize progress coordinate property true provide well simplify assumption towards equilibrium understand markov chain primary goal application notable shrink set compare art implementation four new show systematically outperform fall rare case adaptation descent universit cd method support lasso logistic general cd coordinate fix coordinate frequency remove need estimate requirement usefulness arise offer speed art becoming increasingly solve task svms lasso regularize cd pseudo newton stochastic contrast cd gradient iteration particularly vector problem step factor point stochastic plain equipped computational e cd obvious uniform cd non consider probability interestingly recommendation nature e run keep constant simple quadratic objective difficult propose realistic optimization scenario proposal adapt progress therein need adjust run often run similarly trust online parameter local characteristic adaptation cd technique extend inspire coordinate close coordinate coordinate direction step maintain adaptation inefficient adaptation general turn technique refer follow basic coordinate selection summarize review machine method algorithm coordinate selection probability new problem variable simple value want allow index let minimized course handle technique cd present cd partial derivative compute computation partial coordinate solver optimally e search e analysis analysis find cd variation highlight selection coordinate prominent cyclic epoch loop loop coordinate coordinate visit always epoch predefine coordinate avoid equal attention view dependency structure epoch pick select simplest distinguished index random generator nesterov draw often iteration greedy manner knowledge inefficient notable linear svms working heuristic drop svm cd refer extensive cd old apply factorization linear prohibitive cd distinguished coordinate often machine choose uniform advantageous uniformity question often solve address nesterov derive runtime bound constant minimization optimization offer problematic coordinate problem priori lipschitz derivative coordinate tighter costly continuously importance run reason outline concept implementation machine exception shrink svm cd cd suit problem one advantage quickly often speed insight heart shrink svm often result regularization absolute selection operator alternatively dual result svms demonstrate outperform four serve course sparse start output label unconstraine either instance simple form et propose coordinate result empirical term restrict coordinate time step newton derivative distinction need newton step end zero costly step denote number zero compose input input find follow situation linear solve dual cd key keep track I program dimensional newton derivative cd read eq interval densely step arrive applie remove originally remove normalize technique common use online note coordinate matter robust result costly follow warm adaptation justify svm importance coordinate course outli move maximal fix variable drop become relative upper helpful online trivial svm different extension binary exist arguably reduce two multiple turn lee al piecewise originally svms perspective classification denote subspace solve either qp solver style default box treat attractive problem solve arbitrary derivative computation sub logistic regression smooth hinge solve efficiently cd logarithmic share property allow solution cd iterative implement shrink technique applicable represent subject simplex review technique adaptation value type technique drive extreme trivial aspect parameter online adaptation essentially case performance increase tune efficiently online fashion break parameter differ gradient poor machine descent unbiased carry sake minimization inefficient solve online adaptation al model function online available adjust give outperform sgd deep reference therein propagation backpropagation constitute maintain wise roughly gradient sign derivative adjust multiplication agree simple scheme refine understand fix grid adjust characteristic problem probably make evolution class randomize last evolutionary highly respective sample recent es matrix denote turn extremely poor parameter fix decay roughly online e classic modern es treat whole free adapt essentially filter maximum generate block despite satisfactory theoretic understand adaptation newton case suffice adaptation powerful optimization raise date apply coordinate ask indicate beneficial direction adapt cd optimum optimum course condition fashion property seem coincide progress objective value become independent tool namely maximize decide answer straightforward quantity average progress coordinate progress make progress coordinate small roughly monotonically decrease soon progress answer monitor progress optimum relatively little progress equation nearly progress formally speak add beyond standard cd algorithm progress turn case product step majority progress acquire avoid rely old sample coordinate record progress order progress average fw fw quantitative update many possible form fine right reason adapt unnormalized preference track define coordinate step progress formal preference adaptation record progress default parameter initialize inform available e coordinate p p ip l default ad hoc tune insensitive simple possibility ideally like complexity despite variable index list output coordinate coordinate next inclusion guarantee argument theorem alternatively terminology algorithm enjoy scheme g cyclic formalize section chain first distribution formalize mathematical show quadratic eq exactly finitely coordinate rule exact trivial relevant chain divide unconstrained quadratic relevant understanding cd sake twice optimum
bias sigmoid linearity branch model bag translation highlight share connection sentence bag sentence translate language representation align translate representation achieve regular autoencoder propose reconstruction sentence representation language vocabulary representation sentence encoder able language reconstruction reconstruct language x z k z notice share encoder language given reconstruct reconstruct experiment equally weight investigate promising investigation exploit task corpus mention word train simultaneously encourage frequently align similar network language whether look representation phrase mention useful linear separately skip language propose train neural network learn phrase segment phrase base model learn setup test corpus importantly label document quality word language embedding overall early stop training document word language selection classifier test set word document linear compare use step embedding english english pair vocabulary document embedding either version choice normalize result autoencoder worse report original might preprocesse datum category word picture visualization show visualization frequent language confirm autoencoder learn embedding language en test embedding train gr en en gr al english pair right near english meaningful without rely autoencoder word preliminary extension bag autoencoder bag model thank probabilistic assign act useful acknowledgement help also provide dataset big thank universit universit work rely translate align word language autoencoder word representation autoencoder reconstruct sentence encode extract translation english compare exploit prove nlp meaningful representation syntactic similarity small generalize vocabulary start look align across representation machine translation common approach word alignment translate sentence relate embedding without word alignment corpus align sentence usual model learn relate paired bag sentence word want language document language level initial reconstruct autoencoder set work bag vocabulary bag word correspond order sentence word autoencoder bag word representation embedding train reproduce word meaningful encourage original bag choose decoder representation care choice reconstruction decoder efficient design reconstruct document associate reconstruct decoder treat word multinomial q must ensure compute efficiently
problem show k ij inner inner unbounded conclude k equivalent define exploit common imply tight go iterate appear redundant restrict lie feasibility h every closed definition eq minimizer follow separability write eigen decomposition fix minimizer define kb imply return argument prox prox exist singular decomposition form prox typically hold parameter rapidly set discuss exist convergence penalty experiment increase penalty admm I n ik I I see process stationary k f k block non less serious surface precision stage optimization stationary briefly need covariance ss rs problem covariance optimization search range appendix parameter close form minimize show objective stage equal square exponential unimodal true univariate minimization iteration stationary ss optimization e block estimate compute block prediction precision provide simulated real allocate parameter prediction th denote algorithm th standard parameter also define square se covariance range replication standard replicate covariance however deviation replicate value set prediction obtain value admm rs rs scheme split block test ss rs domain ss rs scheme show model fitting cccc replicate ss rs estimate appear unbiased deviation considerably point though violate point contrast seem reasonable domain rs scheme per rs well setup result range rs sensitive ss moderate high scheme come deviation replicate ss rs test different review code code method generate isotropic square se value method describe input rectangular mesh select boundary freedom knot initial replicate hence provide therefore evaluate fair randomly initial replicate solve first initial solution learn standard replication htbp see estimate covariance degree substantial considerably alternative carry intel ghz gb cpu require stage fast method fast fast unable unbiased range crucial considerably bad air total measurement website analyze cloud collect imaging website matlab read format input software website use covariance deviation table htbp covariance rs cf rs deviation quite magnitude magnitude confirm alternative precision selection method provide considerable scale parallelization stage optimization alternate direction approximation distance frobenius segmentation rs capability method turn square compete surface block since second line investigation numerically matrice isotropic near corresponding work analysis asymptotic covariance problem realization fix investigate estimation simplify range correspond direction connect range pairwise block inner vector ij rd ij ji eq multipliers replace four nonnegative minimum solution optimization one line parameter field via optimize function solve point computational ml inefficient procedure process solve multiplier parameter magnitude approach precise alternative handle enable without approach stationary convex gaussian markov selection gaussian field include traditional rely maximum likelihood mle difficulty yield furthermore mle operation routine inefficient call overcome fitting realization sparse inverse likelihood use regularization precision matrix parameterized constitute stage solve alternate method admm covariance square problem consistent solve region fit result let realization additive usual without mean countable joint I mean point krige variance rely correct unknown simplified family c main lead matrix ty j f j I respectively correspond marginal require solve contain family e local minima evidence stage work one basically deal precision global small block feasible precision resource note estimation big six function spectral approximate covariance usually determinant expansion reduce computational process technique achieve split estimate formulation predict full relate addition class approximation covariance propose localize rest present method solve efficiently line numerically prediction method literature conclude provide remark propose four isotropic c comes study sparsity property vector propose symmetric pd matrix denote cardinality operator envelope growth interest decade optimization include work exactly markovian way line work include regular random conditional assume neighboring lattice index lattice prevent exactly elaborate markovian markovian precision equivalence advantage approximation e behind lattice let determine conditional independence set employ retrieve implicitly precision make computationally hard require motivation spatial utilize precision indeed function exponentially precision utilize likelihood h compose convex likelihood program inverting fit least perform propose k kn note predict block follow scheme consider
utilize n use theorem thm computer science engineering university decomposition randomize randomly sample form compute statistical leverage refer incoherence simplify incoherence bound incoherence row limitation full assume besides row sample randomly randomly compare advantage perfectly include entry addition rank even finally adaptive randomly db bb ab uniformly random entry index goal optimization problem recover notation rank singular vector svd decomposition column incoherence incoherence measure projection norm matrix let assume satisfie subspace span column ii function need analysis lemma us theorem assume accord theorem directly directly perfectly observation need note incoherence assume incoherence row assumption sample hence convexity e let define canonical theorem minimum row end use define nm I k thus skew skewed constant q next incoherence measure numerical rank incoherence constant easy utilize replace eq
evenly precision equivalence definition near ff node different j vector first continue unit assumption repetition follow distinguish algorithm communication bound bit bit end algorithm atom atom claim different output atom similar argument node select bit constrain simplex extend case diameter factor match dependency communication bad case two body loss locate alternate direction multiplier dual communication cost function total section none method rely gd sdca need iteration converge illustrate method partition q regularizer across iteratively subproblem prediction node current converge converge within iteration lasso dense iteration slow add hand feature create tradeoff number communication admm tradeoff study select way fw communication scheme readily match pursuit enjoy series experiment strategy tradeoff distribute approximate first baseline strategy select atom method node select subset random strategy fw sized objective subset batch compare communication decrease admm section evaluate distribute computationally carefully particular kernel receive computational implement connect cpu core label versus rest rbf kernel average point runtime different synchronization issue across exact see illustrate assign share large center atom balance I reduce runtime unbalanced training negligible way cost randomly drop suboptimal iterate show average node asynchronous properly bit slow drop version set future element focus communication overhead frank wolfe favorable communication quality experiment confirm support nsf grant grant fa award microsoft research fellowship contract nf reproduce purpose annotation herein necessarily represent policy prove execute ik execute moreover execute information part iteration require constant value claim frank wolfe I come moreover step frank wolfe stop k g communication fact use wolfe update communication case change objective bind extend two node apart put communication claim carry university edu frequent machine element locate address balancing end propose distribute frank wolfe theoretical cost combine bind construct validate synthetic world baseline compete method study relaxed fairly execute computer year become increasingly machine mobile device wireless sensor naive interesting fundamental perspective study tradeoff attract interest view lot therein practical problem sparse combination spread support vector adaboost formally broadly dictionary basis etc weight weight atom g continuously frank wolfe adaptation centralize counterpart able communication atom introduce balance distribute machine term depend family prove low deterministic construct implement parameter problem practical lasso distribute baseline atom criterion sparse compete iii real world centralize communication drop update review frank centralized propose variant practical example match review conclude scalar k k simplex j j frank wolfe fw constrain problem compact space say fw move linearization current iterate stop surrogate product let optimal theorem fw find curvature terminate satisfie fw solve minimization extreme exploit subproblem norm find combine show find turn bad derivation matching improvement desirable interest subproblem ensure iterate nonzero entry problem result distribute overhead f sum criterion k iv compute atom k connect edge simplify assume simplify atom matrix wise local set index efficient set first identify absolute node ii corresponding subsequent iteration iii fw update round nod termination node optimality atom entry show frank wolfe execute far allow communication dependence total appealing scale I strong theoretical linearly atom suffer overhead due atom cluster classic greedy repeatedly center run intuitively atom nearby center select lead gradually center essentially following result local index opt opt g gap radius node exist atom claim second claim follow practical pick proportional another variant highly unbalanced problem efficiently include code learn regression aim approximate sparse feature respectively subset training distribute logistic perhaps interestingly lasso come source multi view encode categorical n dy psd iy augment notice lie version machine frank wolfe direct svm h j adaboost formulation tune thus straightforwardly classifier wolfe update add base classifier define point currently misclassifie potentially family classifier learner
homogeneity x although since correspond unit norm polytope finite accord program convex depend magnitude attention sort hull belong none combination depict complete vertex sign configuration notice choice mention previous paragraph zero large sort large n denote obtain take derivation prove let moreover allow write group value split respect proposition obtain group group obtain denote operation averaging finally satisfy generalize permutation proximity efficiently solve fista alternate direction admm aforementioned algorithm operator state short decrease weighted sort fundamental namely email lx lx family regularizers sort generalize recently norm instance argument sort non paper derive sort proximal splitting algorithm introduction recent year devote sparsity regularizer variable grouping unlike tie negative simultaneously encourage sort multiplication q sorting replace penalize regularizer write sort relationship different regularizer comparison regularizer convex relaxation induce encourage induce regularizer convexity lasso consider special regularizer sort negative sequence notable give entry include follow notable case non increase sequence group form focus term dual regularizer manuscript aware formally motivate result paper work dual optimality square denote wise sort tie
find optimizer later optimizer distribution dl consequently draw eq speed loss optimal estimate denote draw another probability distribution distribution dd rejection help condition stage algorithm return mm odd learner oracle depend passive addition learner active learning require linear dependence open lemma odd p j show first relate suppose probability empirical average odd last inequality statement union prove appendix provide ready definition alg inequality lemma note cr nc require lemma noting follow cd active imply observe achieve learner n px p p contrast passive receive label estimate mean variance mp passive learner many open active whether plain ridge static carlo estimation allocation constant open type active stem slight condition lead require theorem side constant since convenient derivation follow label example standard completeness sample md xu lemma jensen inequality definition france propose parametric set improve passive setting passive nonetheless characterize optimal learner risk regression linear square error design draw I cost useful costly obtain essentially learner guarantee regression passive excess unlike finite passive study simply square relationship wise predictor rate allow adapt underlie technique common integration increasingly refined globally learn parametric active specify use design normality approximately noise design possibility adapt propose adaptation query potential consistent pool focus learn formal setting notion state approach passive support strictly label example without subscript marginal distribution denote denote predictor predict dy dl optimal dl dl dd drop class throughout integer round effect negligible expression universal whose crucially simulate cost select sampling regression explore mostly asymptotic regime bind finite hold use label cost example rigorously derive lebesgue integration follow l dd r np passive learner solution eq constraint lemma proof complexity reduction ratio highly symmetric general active access conditional learner knowledge approach disjoint subset class I
deduce learn understanding rnn help utilize recurrent recently research recurrent institute technology chinese ac cn com xu com edu wang microsoft research microsoft com microsoft microsoft com bin wang chinese sciences ac liu microsoft research microsoft com click fundamental click search ad yield high behave past ad ignore spend page ad observation recurrent neural sequential behavior click click engine approach click major business modern web account google yahoo search generate keep accord click user maximize engine crucial click ad click click extract historical ad element ad ad work click ad recent point achieve ad query conduct study type user behavior behavior explicit click ad ad view ad fairly query ad finding motivate advance art click prediction temporal click although kind model still hard identify explicitly ability various kind dependency event click user query ad thus natural model dependency behavior series trend periodic ad capable recent study long span massive language neural language rnn speech recognition much improvement feedforward capability specific leverage dependency click consider user intrinsic hide previously accumulate hide embed recurrent large search reveal click state art dependency fold ad relationship use network user sequential dependency experiment validate rnn verify dependency might affect click rnn click experimental study last future understand sequential click discuss effect collect click enter ad page stay click user order along effect click click ad previous click click obvious long user likely click next click second ad click quick observe give rise click user experience user behave quick click long month quick click click interval quick back click figure along click significantly stay certain tend gradually pass study effect system ad automatically query also order bind user query query topic click ad topic cause strong dynamic long enhance click big challenge manually necessary kind widely recurrent output previous try sample previous effort devote towards long context back rnn overall rnn base click view deep recurrent share identical input sequential dependency denote unfold network output click ad weight layer gradient layer hide layer represent wise error hide weight recurrent weight slight contrast rnn test sample feedforward recurrent weight testing process user feedforward sample current make prediction store current hide last matter unfold validate propose really enhance conduct click engine month th engine event whole traffic ad week click prediction week datum dataset ad ad train click ad testing dataset record click click follow click employ metric investigate effectiveness compare performance rnn include lr network nn study click rnn framework able comparison rnn leverage improve click auc lr fair comparison every model achieve include epoch nn unfold rnn good epoch hide auc three rnn click particular nn relative system click prediction increment overall model conduct help click click rate varie often refer analyze rnn ad position evaluation position lr measure position rnn traffic come click rnn achieve position ad ignore user click rare drastically lr nevertheless well still inference rnn far importance utilize historical remove nn ignore dependency auc severe drop information recurrent structure significantly part check history collect sample fed accumulation period continue sequence auc user long feed accumulation maintain long turn model perform good setting accumulation period
constrain hypothesis class contain regardless theory predictor bound fix would correspond lipschitz square notable exception smooth tight contrast implicitly take prediction excess manner magnitude learn tend norm predictor value remain fix lower universal return logarithmic predictor r forecaster give good algorithmic return bm imply even trivial dependence specify common distributional upper dimensional symmetric dimension lead third optimal dimensional bound distinguish distribution excess risk bind respect eq predictor either uniformly attain lower bind theorem leibl instance target invoke kullback divergence plug excess lower pick use construction type quantify standard dependent careful constant returning later uniformly index calculation notation inequality deterministic kullback leibler compose md moreover get kl divergence eq fact back excess eq value pick least pick expression case thm prove thm take thm otherwise helpful short predictor agnostic distributional performance machine statistic well linear standard parameter existence mm example consist pair instance respect hypothesis possible focus excess randomness problem uniformly despite unable result include one agnostic nothing boundedness rely mean work
u uniformly leave find asymptotic get pe pe u asymptotic choose way component sphere vector uniformly sphere e u remark get integrable note q euler last rgb rgb cm assumption asymptotic aggregate elliptical universit france universit de france science business economics university abstract asymptotic tail sum elliptical motivated extreme risk finance result rare calculation numerical illustrate order key word aggregation asymptotic elliptical modeling risk management pricing standard common log risk g despite derive behaviour sum risk appear contribution al normal underlie threshold parametrize importance vector one natural aggregated elliptical paper asymptotic elliptical index contribution precise quantification claim approximation positively bivariate obvious numerical derivation asymptotic univariate log risk elliptical risk b asymptotic impose restriction elliptical risk dimensional normal random singular diagonal equal speed decrease probability imply rest section positive variable function uniformly sequel satisfy far hold calculation accordance finding hold scaling see von fx e et al since radius satisfy far j u c u u u locally numerical mean correlation practical second asymptotic replace term equation approximation use monte numerical study first tables monte carlo monte approximations column first provide heuristic measure quality approximation calculate inequality fulfil improve still order hope asymptotic order asymptotic significantly asymptotic quite display sufficiently c mc ratio main result sequel positive constant fr give equal every exist
set method bag bag paragraph bag learn third paragraph maximize record time paragraph error produce desirable triplet paragraph vector baseline tf weighting perform raw result tf paragraph paragraph significantly outperform bag suggest capture average bag word bag bag paragraph vector table far dm dm alone table dm often well sum dm achieve order information validate guess many varying parallel compute paragraph set distribute representation paradigm language nlp parse translation phrase recent receive attention style model phrase sentence typically parse sentence obvious extend paragraph contrast paragraph vision know fisher kernel vector generative paragraph unsupervise piece text predict sample paragraph classification stanford sentiment art demonstrate paragraph paragraph bag text text parsing machine require text common bag despite popularity bag two ignore paris distant unsupervised learn length piece predict algorithm paragraph outperform sentiment g spam heart machine require represent length text gram often however lose representation use though bag consider word order short suffer gram semantic distance word paris distant strong paris learn text variable sentence name vector emphasize method text phrase sentence predict paragraph paragraph word paragraph vector unique vector share paragraph vector infer word vector train paragraph neural network vector average use neural language use concatenation input neural network try next close researcher try word level sophisticated approach word sentence operation average word word vector show sentence rely parse paragraph capable construct sequence variable text length document present dataset paragraph sentiment analysis text give discuss previous paragraph introduce word word every word map unique prediction word give word multiclass softmax q compute softmax concatenation extract softmax prefer softmax fast structure tree short assign frequent good speedup common binary word descent model commonly implementation code google com converge word map close powerful paris distant difference vector carry mean word answer analogy algebra translate word phrase natural processing understand statistical extraction paragraph word ask prediction word despite semantic indirect prediction vector paragraph ask contribute task context paragraph paragraph figure paragraph unique paragraph word combine change paragraph token act miss current context paragraph call distribute memory paragraph dm slide paragraph paragraph context paragraph powerful stochastic descent obtain backpropagation descent context paragraph compute gradient use prediction paragraph vector descent rest softmax suppose paragraph map word dimension exclude though update paragraph token model concatenation context fourth paragraph context act paragraph length sentiment dataset stanford sentiment length al consist sentence also task document query subsequently extend sentiment sentence movie review site consist sentence sentence amazon dataset achieve stanford label human label label dataset http nlp stanford classification positive task axis whether label sentence sentence label full al apply dataset movie review play set follow protocol independent representation sentence feed logistic regression rating sentence representation feed logistic movie experiment validate window window concatenation vector dm paragraph character paragraph less pre special model grain bayes svms na neural matrix rnn neural paragraph report table highlight table bag gram perform poorly average word fashion bag sentence compose g word recognize sophisticated linguistic phenomenon recursive require parse take perform recursive parsing grain classification term translate relative sentence recursive parse unclear parse many demonstrate et sentiment analysis review key review sentence movie review divide dataset unlabele balance http ai stanford edu sentiment paragraph paragraph vector label feed predict paragraph sentiment review paragraph previous task validate window word present concatenation vector dm dm paragraph vector special character treat less word paragraph see model perform combine approximately come variation work considerable go barrier
minimize kl divergence minimize descent fixing variable form update execute meet feed output bias softmax illustrate correspondence bias vector tb forward iteration bias give bottom layer chain schedule determine b update schedule update bias drop grey height update point special feed network bias tie equal pairwise mrf pairwise restriction feed forward viewpoint nothing stop restriction restriction mean field output relaxation discuss aim beneficial field specific obtain grow expect layer layer different thing getting converge connection pass fast add flow usually helpful layer allow layer connection create relaxation crf potential aim crf test equivalently output budget compute potential target train output kl compute back develop feed forward network pt eq gradient chain use output discussion inference discriminative model factorial layer via minimize output sx sx tf part loss indicator form step layer step discriminative hinge usually straight integrated paradigm relaxation different relate pass step train graphical fashion empirical risk back propagation optimize graphical compare feed network see restriction straight forward derive enable restriction like tie another briefly connection binary mrf paper compatibility inference inconsistent approximate long align algorithm time problematic compatible neural side people try neural intractable belief therein connection field propagation paper approximate limited share spirit background intensity white intensity english foreground noise pixel intensity add pixel noisy pixel two foreground consider crf posterior output image pixel unary pixel vector unary potential potential one horizontal inference initialize take unary maximize conditional approximated use marginal initialize except unary mf improve inference crf parameter train layer layer baseline test mf mf number divergence constant log kl improve significantly well mf iteration train denoise directly start three tie weight baseline mf achieve mf learn hinge field baseline
vision application interest heterogeneous sensor span classify image view directly learn view transform newly canonical correlation cca pls sample correlation pls consider variation besides mutual theoretic encoding narrow inter sketch bridge view discrepancy view label method discriminant instance discriminative canonical propose local learn discriminant extraction large approach recently jointly view transform view extensively encourage transform capture share view however agree indicate transform extract deep attempt learn via vision stack deep structure building include restrict encoder stack auto effectiveness denoise domain speech etc know canonical also widely learn method much flexible learn process set inspire deep natural build view totally modality suffer representation capacity view make infeasible two couple seem couple auto building stack auto encoder margin couple input denoise encoder modify maximum criterion kind etc counterpart add margin naturally layer network illustration see multimodal auto deep canonical tend layer canonical constrain deep representation great separate well view handle insufficient organize solution couple efficacy conclusion basic part discriminative encoder stack network deep fig circle pixel connection middle part whole project separability layer build stack couple couple layer insufficient stacking layer whole compactly represent set transform network gradually narrow gap discriminative capacity enhance discriminative couple train maximum margin incorporate training maximize learn discriminant couple two formally discriminative encoder criterion describe threshold maximum sample note representation attempt nonlinear transform project two discriminant respectively neighborhood separability preserve preserve denoise learn representation discrimination modify denoise margin criterion intra inter nonlinear trade preserve separability denoise auto error formulate follow version specify transform decoder calculate decoder representation tangent operation consist intra similarly counterpart view add rather characterize meanwhile class separability sample class formulate follow belong nearest neighbor satisfy condition function project common term show red work intra class penalty adjacent inter sample couple nonlinear transform transform view training process discriminant gap eliminate representation consistency world single complex real insufficient stack subsection network compactly significantly large transform gradually narrow gap ability enhance training couple nonlinear transform achieve canonical wise precise couple feed input feature stack layer gradually adopt lagrangian multipli first term constraint call utilize decrease far help prevent balance parameter local empirical separability call usually set employ l bfgs often nonlinear optimization problem large requirement utilize calculate differential objective achieve fast section sketch evaluate pose neutral illumination choose pose two subject subject dataset image dataset subject sketch subject train rest subject testing image without baseline art cca deep seven cross view jointly transform view utilize report reduction default dimensionality cca pls tune report cca adjust good cca strictly tune inferior reason cca training datum besides pls vary hide gradually layer pt dataset mean stack space limitation accuracy explicitly illustrate learn conduct dataset project learn common principal cca principal direction method attempt merge fail convert view diagram gradually layer describe come view respectively stack compare cross face acquire seven pose multi set experiment result show pose probe rate method supervise method significantly superior deep cca significantly superior
generate comparable good coverage generator mine fig original generate produce split train name test use auc fold average b b datum important build original test opinion indicator close generate would substitute original comparable generator try work generator less successful evaluation examine generator scale empirical evaluation repository great variability attribute class interface uci assume artificial original instance load evaluation limit original instance generator author htbp anneal cancer breast cancer breast cancer screening band heart heart disease vote diabetes primary heart new generator attribute produce generate instance parameter describe sect skewness per attribute exclude skewness comparison ks reject value attribute compare value hellinger response exclude similarity generator report average hellinger exclude ari exhibit report ari repetition generate use classification illustrate forest robust performance various come default tree set cross validate set htbp test match hellinger ari rand train test comparison applicable ari breast breast band heart house diabetes heart encode attribute generator represent nevertheless measurement core intel cpu ghz time instance report column label equal mostly happen primary tumor cancer generator contain unit activation consequently attribute gaussian unit column label average difference attribute individual ks test hellinger attribute label percentage k compare attribute ks difference average hellinger set distance considerable tool ari rand considerable high ari set low validate forest model train generate test trend original well set datum lose confirm generate mostly nevertheless satisfactory substitute mining namely build original test original large case overall conclusion generator semi reasonable substitute mining sect encode single make value report test statistical binary encoding nominal nominal include ari annealing screen band heart heart house primary tumor heart mostly large binary encoding case generator time create need compare significance binary encoding produce hellinger ari indicate original finding generator instances activate form instance overfitte estimate replace dataset anneal breast breast cancer screening disease house vote diabetes tumor heart disease compare nominal encoding nominal attribute pair proportion difference mean attribute difference original recommend safe try rbf test success rbf rbf compare forest forest successful auc package default forest produce accuracy accuracy side auc skip htbp anneal balance breast breast heart house vote diabetes post primary heart disease forest rbf networks rbf significantly tried identify success datum generator success difference model original difference factor difference accuracy rbf rf generator gaussian collect correlation pearson coefficient difference indicator rbf classifier generator turn rbf test scalability framework public available preprocessing already effort additional effort source turn work provide different characteristic instance proportion practically artificial successfully big generator exploit property generate tool useful adaptation use datum randomization ensure privacy simulation test scenario huge tool generator datum set uci able original generator success classifier generator rbf successful versa unable successful generator intend generator classification turn future extend generator module rejection base acknowledgment thank interesting discussion uci author support research definition university si expensive generator semi artificial similar original enable development simulation without generator learn generative generate structural similarity technique generator uci set challenge well know bring attention application opposite reason inherently rare business privacy record expensive require significant human interest long reliable performance development specialized tuning solve original similar great development specialized problem yet easy background weather context aside small purpose aware extract overfitte exist generator low mostly review sect approach problem construct rbf prediction consist instance generative overcome space attribute categorical paper organize generate datum rbf actual handle nominal generate statistical section present try determine working condition cloud generator data generation cover method problem group generate r support uniform normal beta cauchy multinomial generator package need mass provide parameter several generator distribution less effective generator multivariate simulate decomposition symmetric contain covariance decompose normally datum sect normal distribution normally successfully generate proportion requirement multivariate desire propose datum replace population iteration desire intermediate space capture limited datum clearly kernel estimation population make data approach frequently gaussian intend space copulas copula copula describe write univariate describe dependence datum copula knowledge number careful copula family copula type limit rbf radial function tool continue see contain consist unit class describe dimensional instance unit rbf probability multiply radial function center function kernel away rbf architecture layer must start avoid manual set center weight standard deviation solution rbf adjustment build encode process dynamically algorithm comparable rbf present hide rbf add training fully hide consist possible encode class winner take output unit determine illustrate fig threshold bind activation train good separability class achieve inner circle center idea extract empirically notable ability eq multidimensional give definite decompose r package pseudo generator encoding equal e would encode binary attribute binary encoding require line line datum unnormalize attribute generator generate training specify size function create ic instance generate var zero g sigma fix attribute jt span min l data consist gaussian attribute recall also specify generate control kernel width start create generate kernel list weight kernel ic ki width spread generate around line width line covariance diagonal diag width instance particular spread dimension generate take number instance generate diagonal exploit kernel check line interval attribute retain generate line transform nominal form transform generator dimensional grid attribute center red blue generator eight class illustrate generator fig location center kernel rd simple example well width scatter color generator instance fig considerable pair scatter aware generator capable exist attribute exist generator evaluate skewness indicator insufficient attribute attribute thereby overall account difficult also datum mining difficulty incorporate deviation skewness attribute generate hellinger kolmogorov ks whole process illustrate normalize attribute attribute sensible generate statistic especially attribute attribute
handle column nonzero determinant determined get another dimension add consider body euclidean space fr almost surely exist map let require k singular iid distribution laplacian observe equip state form readily moreover arithmetic fr unique secondly non condition satisfied number context interest index readily satisfy verify outside coordinate straightforwardly derivative express second derivative finally diagonal matrix v ki ij ij vanish indeed covariance give state appendix conjecture axiom grant air force scientific research functional project international resource lin suggestion year become summarize assume structural relationship difference network group familiar strategy object method apply challenge geometry high dimensional geometric fr motivate result functional univariate mean understand modern working say consist signal collection two dimension voxel build various form high representation employ traditionally naturally increasingly denote often consideration vertex image notion tensor association representative structural connectivity brain hand fmri association think connectivity together count clinical prominent research brain towards database compose collection database answer network nominal collection gender contribute finally say change network question network estimation testing fundamental practice dataset fact combinatorial object simply edge nevertheless certain natural euclidean combination tool geometry manifold principle practical framework analogy classical tool undirected laplacian matrix denote loop edge correspondence space subset euclidean either corner subset complicate nontrivial structural constraint geometry nice allow goal certain fr define borel fr mean similarly realization fr thus geometry define manifold able theory average test require nevertheless advance researcher mass field motivate illustrated sharing costly consume conduct discovery paradigm system release set access throughput scale lead finding functional subject locate center participant year year old old scan minute strength varied across center voxel plane slice center project medical school datum fmri automate labeling template voxel series region result two consider prove literature respect researcher impact connectivity human genomic profile brain compare specific observe research consider edge investigate focus specific global consider network extend characterize space sequel evaluate difference organization research question characterization inferential framework network explore report propose discuss entry database percentile address summarize compare principled device available summary comparison operation e symmetric difference broadly various hamming univariate conduct edge adjust fail draw whole multiple difference necessarily lead globally treat point mathematical formal equipped understand geometric topological underlie desire manifold particular shape fairly history back seminal work shape notion average nevertheless little seem study geometry derive fr also relate object formal characterization associate fall cone semidefinite psd lot explore notion choice geometry choice adopt motivated shape analysis play key space psd cone furthermore cone immediately latter necessarily discuss eigenvalue analogous riemannian formal date establish certainly none impact establish canonical g etc aware work characterize subspace psd correspond sharing rank crucial mathematical embedding involve embed smoothly matrix embed lee seem compare via g embedding useful g hausdorff literature embed onto technique particular reduction isometry geometry domain precise manifold employ describe exist embed space manifold preserve information average geometric variation affect geometry space probabilistic sampling equality I loop associate laplacian diagonal e far connect positive correspondence graph therefore correspond admit affine sum appendix practical importance usual notion curvature edge edge distinguish purely say correlation choose thresholded version theorem include corollary possesse manifold corner text smooth manifold lee convex euclidean entry non positive corner dimension convex space proof provide importantly indicate real distance space concept fr analogue well networks topology fundamental complex tend possess mark structural characteristic example edge heavy tailed suggest appropriate formally depend implication extend case graph connect component positive sum entry column proof intuitively graph community increase graph maintain characterized network inferential framework select derive average construction number correspond combinatorial identically might image necessarily definite definite place statistical power increase simulate base experimental design process rely generation network mutual matrix type order second second randomly firstly group whereas proportional grow secondly specify small world construct regular topology proportional edge diagonal small family topology simulate base label diagonal give adjacency simulation scenario thereby produce ratio distinguish type simulation result definite consequently definite default brain investigate main simulating stem absence produce pattern sequence draw process first scenario sequence realization random group scenario sequence time use identical restrict autoregressive autoregressive autoregressive provide subject network matrix interest either guarantee positive semi choice subject mutual respective mutual combinatorial association give every subject aa ds target combinatorial follow laplacian group matrix ds moment modify covariance estimation bar indicate proxy measure effect compute frobenius distance population network vertex group condition topology simulation size representative subject find study secondly fine region practice size allow power decrease power test effect frobenius two population mean varied thereby result difference frobenius discussion bar standard mean figure figure network roughly increase power proxy noise comparable material covariance behave topological subject poorly high likelihood b mutual define small size poor greatly albeit slightly resulted type consider fail hypothesis small result suggest estimation preferred compare subsample test pattern five subject exclude subject analyze york provide unique extract template connectivity laplacian subsample subsample nan reference hypothesis reject high partitioning figure highlight introduction influence brain reject database group accord age equal subject subject respectively hypothesis state draw nan use hypothesis voxel voxel mass univariate despite good effort may connectivity sample focus site international brain imaging york subject stating reject univariate univariate test independently entrie age panel denote significant significant small indeed paper usually subject last size order produce might sample conclusion hypothesis find subject whether mean reference different age subject subject site extract likely population univariate laplacian subject even highlight advantage context univariate fail framework average importantly mass collection purpose exposition summarize collect allow development analysis produce theory offer quite broadly direction briefly confirm sample beyond rate see sample control analysis condition subject current global therefore applicable however subset e analysis single low sample alternative difference employ challenging laplacian inversion matrix different facilitate modern modification sample covariance force wish use behave able theory however year although possess structural relatively common heterogeneous distribution subgraph establish importantly formally impose choice network geometry embed inside riemannian lead nontrivial matrix need apart simple manifold psd natural euclidean moreover measure psd fr mean course impose risk relation psd hence
rapid auxiliary intermediate previous current intermediate solution construct ff ft mild adopt way previous surprisingly explain fitting kernel svms solver n regularization search scalar use alg method trace regularizer corresponding regularizers g proximity regularizer variable u j g scalar tt alg close let htbp rapid dimension per variable normal start code toolbox rapid please toolbox manual norm implement fig comparison identical rapid empirical guarantee rapid big sometimes use rapid svms use optimization matrix sample svm formulate entry wise vector predefine rapid solve alg contain line ignore update variable alg alg fix fista test rapid contain dimension per randomly select rest solve rapid compare solver check f check rapid begin check value add future rapid result well rapid rapid ii present result rapid fista property alg arbitrary clearly definition alg alg satisfy lemma rewrite rewrite alg satisfie let eq strategy prove accelerate rapid speed introduce way construct auxiliary intermediate current small upper bound gap algorithm I summary rapid converge algorithm sophisticated edu proximal simple proximal variable intermediate solution gradient step upper current objective method fista converge accelerate attention area process convex convex non differentiable proximal gradient variable proximity operator proximity convex minimizer feasible return original classic alg size fista prove speak bottleneck follow aspect gradient could consume recent inexact allow approximate proximal method gradient fashion gradient decompose locally achieve proximal alg unfortunately line search decrease instance tune step gradually search lasso order convergence auxiliary intermediate solution iteration prove arbitrary consistently imply precision probably empirically apply lasso machine svms correctness fast case algorithm sophisticated solver include line search construct svms demonstrate empirical solver theoretical prove conclude proximal
uniformly dx cumulative integer parameterize hash u equivalent presentation step scale hash furthermore grid move present optimal depend ratio threshold threshold interested top therefore desirable hash threshold inner similarity contradiction let monotonically show simple lsh eq hash random mapping lsh every lsh hashing eq monotonically increase lsh desirable achievable hashing quality compare hash desire also hash bind tuned hash quality give dominate lsh well optimize recall hash setting high code hash function netflix procedure rating entry zero unobserved entry rank top netflix define presentation movie user user movie high hash hash code length movie selection sort movie ham hash movie hash break tie randomly precision curve average precision randomly recall item hash code netflix dataset suggest setting unfortunately whenever similarity show strong negative fortunately normalize pair transformation always cs mapping ensure h x monotonicity also modification search measure characterization asymmetric lsh possible inner symmetric lsh query bound vector universal lsh symmetric lsh lsh symmetric lsh motivate asymmetric lsh query database two actually third set asymmetric hash indeed suggest asymmetric even second set specific universal important emphasize asymmetric hash asymmetric hash one must normalize query database hash hash strictly identify acknowledgment partially nsf award advantage asymmetric lsh use two mapping use hash hash theoretical observation valid mapping alphabet universal similarity establish contradiction query universal c cs monotonicity max row I mf p indicator conclude max jensen rr complexity sign matrix margin remark claim theorem design locality lsh similarity argue lsh problem lsh symmetric lsh enjoy lsh variant setting asymmetric lsh follow maximum inner product search collection datum maximize inner query matrix svm score efficiently approximate locality sensitive hash lsh locality hash lsh object alphabet lsh hash word hamming distance word recent explore lsh different mapping approximate similarity similarity hash may enable obtain lsh lsh obtain well lsh tree search tree method impractical regime lsh superior vice versa yet lsh tree lsh considering argue inner similarity distinct query asymmetric lsh entire lsh thus show succeed lsh enjoy guarantee performance require motivated understanding obtain simple lsh conduct lsh crucial issue lsh entire lsh lsh normalize bound normalize symmetric lsh asymmetric lsh also lsh also enjoy well hash recently structure recommender study lsh property alphabet function hash distribution family study study two however assumption want l able assumption database lsh subspace lsh modification lsh inner similarity locality lsh hash lsh say efficient neighbor lsh object quantity lsh minimum hash symmetric require truly asymmetric space formally hash asymmetric deterministic make lsh asymmetric locality hashing asymmetric say lsh lsh lsh finding lsh lsh asymmetric help also hash inner similarity assume contradiction exist similarity consider sequence define zero triangular also set
sampler describe require sum computationally prohibitive efficiently key channel force initially compute pass indeed simulate conditionally employ normalize forward message top message column reverse draw define perform sample calculate message chain work message k instead change furthermore resample weight secondly interested calculate proportional function take capacity practice bias negligible dominate resample lead stability logarithm multiplier subtracting improve stability resample must add sequential modify propose explain sequel refer tree compare run algorithm time bar run capacity sampler give capacity channel bar approximately iteration error run burn scale size subsequently display error experiment tree sampler log oppose example tree poorly slow mix enumeration sampler apply compare width either plot versus perform magnitude tree seem gain noiseless capacity upon order furthermore obtain particle sampler improvement modern significant day rate source support contract suggest derive carlo capacity channel capacity run channel idea generally yield improvement computation ever capacity page orient storage constraint help amongst interference analyze theoretic channel storage numerically capacity channel utilize sample auxiliary target exactly focus propose generalization propose capacity problem constrain fundamentally sequentially state backward art algorithm propose continuous goal imply two adjacent bit lattice probabilistic underlying lattice graphical interaction x mass product encoding pairwise depth exposition refer reader constrain channel square lattice graphical wise configuration cardinality support capacity hence capacity channel unfortunately calculate intractable particular know noiseless capacity agree eight digit finite tight bound calculate noiseless necessary use thorough see adapt mean propagation minimizing adapt significantly reduce tractable describe undirected chain see specifically normalize sampler however target sampler approximate collection particle column point approximation kronecker delta particle mention adapt resample particle auxiliary square graphical well constant subsequent proceed hand simulating give particle decide particle generate particle drawing resample variable index particle correspond resample emphasis
parent boolean graph hamiltonian describe indicate clique consist whose second variable far assume dag maximum parent however situation desirable absence search hardware limitation prevent arc need accounting wish arc generality exclude free variable arcs remain substitution utilize reduction degree penalty hamiltonian purpose increase energy ground hamiltonian energy constraint sufficient weight necessary make necessity tight low may exist meet ground duration necessary theorem penalize quantum less remain property penalty arc concerned parent arc bit penalty great arc third absence bit formally penalty least achieve show justification appendix briefly quantity h ji associate quantity define fact monotonicity monotonicity removal arc energy iteratively iteratively I h penalty weight degree hamiltonian construct bit reduce locality general conjunction sufficiently conjunction done bit contain arc heuristic reduce bit penalty compute appropriately use penalty consistency high cycle less cycle encode cycle minimal consistency ng encodes cycle whose strictly k contain contain cycle dag dag minimal cycle penalty weight ij ng encode ground overall ij nh state maximize dag ensure dag parent however latter dag interest per se enable former amongst resource logical device embedding done often value exact great heuristic sa use present embed hardware wave deal nevertheless quantum state quickly anneal future advanced anneal code describe grant recommendation necessarily author acknowledge advanced support foundation award grateful david useful discussion decompose argument simplify trivially regardless value still eq calculation equation right reasonable weight necessary true ji ji ji construction min claim h I I h cycle contain prove show cycle h l graph cycle imply existence l complete arbitrarily assume r l ij ij furthermore claim modify bit q contain direct cycle h direct triangle triangle switch direct triangle j h r r h l claim let finitely triangle construct l dag proof contain consider cycle h h claim n quantum scoring equivalent enforce propose prove weight penalty logical mapping appeal instance give network equivalently factor distribution broad class mode learning specifically structure diverse discovery produce reasonable formal practice require heuristic quantum heuristic certain exponentially computer however exist complete provable speedup one believe speedup availability quantum anneal device wave determination whether generation quantum efficiently formalism mathematically ise interaction physical annealing device mapping develop lattice planning scheduling diagnosis training classifier computation number encode indicate arc pseudo boolean function encode necessary necessarily degree add pseudo result variable embed physical appealing penalty physical fix strong logical compressed problematic scale compress inherently strength prevent sufficient resolution logical energy utility mapping anneal method motivate anneal highly simulate limitation body interaction device respect simulated anneal gap present annealing still arc interact directly one penalty strong tend produce local optima annealing directly exploitation topology energy landscape make heuristic unlike undesirable solution utility scoring sub probable optimum inherently run produce low energy structure utilize perform averaging done average formalism quantum anneal find provide weight discuss useful bn would require ground encode optimal implement implement reasonable overhead construction encode specifically hamiltonian embed minor minor another disjoint individually subgraph edge edge whose map adjacent edge hardware call logical physical embed two logical physical describe physical hamiltonian logical distribute couple physical logical act minor minor practice use bit arc direct vertex indicate whose adjacency graph graph encode consist construction arc score structure direct cycle parent encode structure vertex parent numerical logarithm equation likelihood let score eq minimize wish ji q pseudo boolean multinomial head encode loss simplify parent I note include reduce require many therefore limit parent q value parent otherwise slack node define convenience generality take
variance weight constant one difference train exactly experiment template drop template accuracy template conclude scheme indeed small pose scheme specify necessary ease similar operator operator operator relu max capability powerful generalization interesting architecture apply operator follow view artificial neuron feature space perceptron output block locality generalization machine measure similarity rise generalize rise specification include kernel level convolutional express property statistical initialization initialization arise employ unsupervised manner include acceleration benchmark convnet single require around convnet architecture incorporate weight input convolutional incorporate weight unweighted similarity rise kernel building respectively hand realize svm thm building block carry highlight unweighted similarity rise superiority display linearly say template build block go hypothesis respect ability incorporate layer pooling locality template instance patch locality constraint support process spatially sharing correspond template apply patch locality sharing compatibility patch partitioning patch constant assign patch pool classify z learn fact coefficient add coefficient add character character extend accordingly mapping x v v v power v accordance correspond hilbert space v h dd dd h v v v r I rule eq interpret locality outside pool pool character constraint eqn enforce pool entry pool entry template index conclude express instance option learn locality eqn classifier reduce train instance character index subject locality eqn connection demonstrate share pooling svm illustrate locality sharing illustrate translate associate locality sharing play constraint proposition example university deep neural similarity family convolution max operator operator relu pool additional capability architecture input special machine gaussian basic abstraction iii use experiment capability achieve comparable much largely large visual task convnet architecture include thousand employ convnet capacity layer convolutional windows assumption image control form success still fall reach human level recognition merely size convnet obtain network increase abstraction motivated convnet change since early create architecture arguably success year ever compute power contribution advance secondary importance attempt initialize observed scheme little advantage carefully initialization initialization scaling capacity therefore architecture give natural initialization convnet paradigm completely take developed kernel suit flat make convnet architecture architecture body machines convnet introduce network lift convnet architecture something carry old learning decade abstraction layer potentially provide third architecture endow potential determine channel generate architecture operator generalize special role addition capabilitie cifar layer layer specialize comprise experiment preliminary capacity experiment deep extensive code apply scale operator generalize soft min replace convnet relu max pooling layer input weight stand negative similarity form mapping inner I pp architecture similarity height index patch template z channel width step patch horizontal vertical dimension linear mapping convolutional whereas mahalanobis template every pixel weight datum globally unweighted architecture template unsupervise initialization responsible spatially necessary operator follow alternative expectation I smooth illustration divide possibly overlap map run serve later convnet relu activation input layer block area omit l obtain process allow flexibility layer correspond conventional form layer omit block wide layer special possibility particular subsection multi perceptron addition process patch involve locality pooling operation convolution something consequence make consist unit apply template straightforward weight maxout attempt generalize notably unit pp maxout operation create fix unweighted p inner high conclude feature prove assertion express z indicate k lin x extension vector I neuron feature unit view neuron mapping turn straightforward extension include unit mlp signal set hidden similarity similarity hide associate label index activation operator follow produce classification rule classification template dependent operator combine template attract template template value mlp fed line derivation carry mlp work unweighted unweighted hold n rl gaussian reduce multiclass whereas classical multiclass svm generality offset svm summarize mlp layer rise reduce svm kernel replace unweighted order rise underlie special laplacian similarity rise similarity neuron feature abstraction category input consider r index template index readily condition first linear half space intersection half union unweighte l union conclude mlp unweighte qualitatively equivalent convnet set exponential kernel abstraction level consider fix tell governed view space region surface polytope shape region unweighted abstraction induce linear convolutional govern non separate hyper surface divide plane piece wise separate boundary unweighted divide shift cause less allocate template thereby around template weight setting expect weight abstraction convolutional plane panel template unweighted divide equally template weight portion plane allocate template locality share process patch locality template weight share create stack template map pool layer predict label design patch base locality realize conventional divide possibly similarity channel match template template local patch value ij li j l output layer implement map coordinate pooling normally coordinate window layer layer take max implement node run coordinate pool template coordinate pool layer node offset coordinate fig output employ locality sharing conventional make case kernel input first associate identity p w l l equality operator eqn template e patch pool kernel eqn denote ij mapping consider eqn concatenation structure pool detail proof give chain pool basic building similarity support svm realize form unweighted weighted weight applicable provide rich experiment sec validate weighting show merely build design decision determine learn datum manually chain new give rise encounter idea switch layer play pooling role finish pooling interpretation layer majority vote form final decision paradigm enforce constrain result rule b l range template constrain apply identical rule eqn estimate manually find outperform rule j scheme network initialization take select initialization initialization scheme date hardness typically design large represent true latter overfitte prominent one master properly scheme scheme effective local minima thereby reduce support small computationally validate show unsupervised scheme improve initialization similarity focus z application unlabele data template shape stand coordinate stand stand mean coordinate layer weight follow template order would output hold probabilistic heat estimating shape mixture unlabele patch follow learn linear patch come patch prior make template likely likely appear corner global initialization patch estimation shape calculate output location probabilistic heat statistic certain template unlikely template heat map aforementione region correspond fig correspond convnet correspond illustrate network illustrate c similarity reach competition implement patch currently report later cifar image process pixel template z use template make softmax descent sgd include momentum sgd momentum decrease epoch epoch decay template equal validation compare architecture convnet follow follow illustration comparison convnet learn parameter outcome reach convnet considerably comparison layer study depth cifar art template single pass sum svm produce refer triangle template mean produce c scheme initialization validation report achieves convnet et al superiority acceleration enable large deep meaningful benchmark deep convnet implement use toolbox sgd use batch decrease every dense layer convnet image array whiten accordance whiten
easy frequently life reference common non network structure address edge function edge community experiment follow detect assess available normalize mutual unless specify weight scenario detection propose iterative nmf nmf network visualize network site bi different color node assign community lot split artificial consensus community red entry observe bi select create solution around bottom solution provide c preference pt consensus consensus solution bi copy horizontal create new colored propose consensus notice base treat generator modularity provide consistently perform produce consensus bold number truth community stand normalize mutual modularity cluster consensus mod b mod mod b b mod mod b happen information network match play organize play division ground community community network recover blue team match play division bit manually figure red community modify row logical mistake consensus pt blue pt pt c seed pt thick pt pt sc sc represent blue division green node th mod stand modularity respectively allow community analyze aspect g modality evolve facebook dataset facebook observe attribute gender minor build modality different gender link student otherwise assign share major minor share share independently run modality approach allow coherent something multimodal without need consensus gender modality assign gender consensus modality exhibit inference might build perturb copy copy result perturb still good non chance network long balance perturb solution hence perturb network trial original one perturb c multiple corrupt connect bi context object might intersect describe refer sense describe model outlier intuitive zero set use threshold measurement noise goal ill pose standard implicitly impose recover pair tractable design example least formally perspective minimum number uniquely circle cloud element model define trial describe apply sequentially model impose find idea sampling seek like shift outlier rejection heuristic parametrization high element degenerate level conceptual model object cluster model propose modeling exploit spurious produce throughout iteration object element traditionally analyze row cluster vector art call j linkage agglomerative agglomerative linkage j linkage use merging process cluster object sample application address relationship object need find cluster I already bi section conceptual advantage object allow situation line intersect share object translate block diagonal technique assign instance compose exceed far one analysis discovery technique bi benefit efficient meaningful method discard bad contain object object interested consensus simplify belong appendix consensus false mention form ccc linkage assignment wrong assignment detect tune notice final process leave corner bottom green linkage yield return wrong decay cluster finally propose vanish detection application use segment element intersect plane intersection segment help helps reliably detect vanish point could consider length sophisticated vanish candidate might lead pt bi description detect bi cluster overlap rich characterization datum detect parametric concentration robust give element simple every pass central consensus many create completely consistent formulation bi since similar circle lie along line concentrate miss constraint along address formulation employ greatly ccc configuration lr dispersion threshold number dependency bi common parametric transform share clear properly value many present reasonably intuitive concern simply select application learn outli detection necessary uniquely characterize detail trivial quantity actually try discover relate establish sound relation know research practically aspect nature chinese brief discussion drive principle form image make use namely deviation randomness occur principle atomic image assume subset object share common orientation position etc stand follow uniformly distribute object observe finally ask probable play principle state assume uniformly multiple control proxy occurrence visual exploit twice preference adjust row accept bi configuration framework repeatedly arise dimension nonetheless play role assess configuration bring sampling would fact appendix mn stand observation might develop simple present connection geometrically actually possess mathematical characteristic result configuration bi intuitive nature bi mean configuration summation bi configuration reproduce low stability framework configuration mathematically formulate pr need validate intuitive conceptual reach consensus grouping grouping detection parametric pose group bi conceptually rich modeling powerful tune though instance framework particular multiple parametric pose bi highlight explain investigate whether hard actually instead work value object provide quantitative theory visual show research work fully perspective fundamental problem explore test sharp coarse avoid huge clutter bi would alternative could extend wide preference th would explore depth like stress limit present address formulate solve lagrangian ij multiplier descent successively fix recent value I update multiplier step svd object identically law present tight carefully probabilistic specific form already useful demonstrate capability framework need pass point line equation area bounding bound box lie band width define circle circle eq q lie band circle aa point plane plane pass plane plane write bound lie band around approximate line use segment define pass equation detailed segment root band width aa area bound reader description community merging algorithms bi perspective reach dataset formally pose highlight equivalence connection seek inspire event bi tune handle community bi paper noise outlier element describe outlier application group group general characterization example address traditionally broad perspective overview dataset see comprehensive obtain parametric segmentation name pool universe candidate group run grouping modality pool candidate grouping would discard prove task even modularity cut select pool combine pool attempt consensus problem say group pool subset mutually quality maximum clique extend pick candidate candidate new issue need principle field assess classify develop unfortunately community example exploit group candidate family typically goal consensus partition consensus involve recall hierarchical thorough way try decomposition relaxed matrix factorization negativity community address consensus within thresholded adjacency new step iteratively make build mention aggregation individual pairwise relation relation large group lose partition might poor quality involve prohibitive node reach consensus bi advantage keep relation small tractable dataset bi stress goal find function grouping obtain good dataset pose bi problem highlight attention behind finally formal visual insight approach section show community multiple estimation diverse experimental finally provide remark candidate universe represent element th group preference fig present simplify object consist form star object consist point four case group visualization take form uniform weight simply incorporate element object good object actual fig pattern discovery need bi formally connect consensus algorithm contribution address problem grouping preference intuitive belong analyze classical consensus work grouping problem estimation commonly go back analyze base need overlap cluster formulation base mistake translate characteristic common consensus way detect mistake penalize mainly due conceptual algorithmic interpretable association iterate criterion meet bi element indicator presence row suitable correctly de via use find extremely challenging specifically tune value motivate bi matrix positivity intuition negative sparse entirely suited analyzing result adapt solve work algorithmic loop norm correctly detect bi conceptually bi subtract enforce set successive orthogonality non negativity maintain bi share element simple control bi cluster row discuss encode minimum element bi encode bi cluster theoretical value show tune part experiment bi intersect enable quickly eliminate spurious parametric bi intersect set posteriori strict high value use decide posteriori clarity homogeneity could bi compression encoder select yield compression loss fit I break ij symmetric matrix approximation encodes method parameter discover bi allow construction orthogonality present benefit double act pool frobenius outlier compute robust median preference carry need ingredient consensus candidate feed consensus time bad bad group phenomenon extremely pattern assume general candidate nature group parametric employ simple testing eliminate vast candidate currently investigate non candidate negligible consensus grouping mistake grouping algorithm uncorrelate possible approach ideally mistake cause algorithmic decision systematically appear candidate group procedure network gmm c c cccc preference bi preference ccc ground assignment assignment em ccc assignment subset intra one key exploratory bioinformatics application grouping pool preference therefore fashion weight assess ground truth standard evaluate express f figure synthetic gaussian cluster pre shift size consensus algorithm approximate qualitatively visual inspection compare figure rank first visualize difference nmf nmf iteration average preference whole active bi candidate I approximate single nmf fit group active bi depict preference l help bi cluster ground tuning method first consensus correct self argument get parameter valid although really j linkage popular
efficient approximation possibly graphical across advanced flexibility view toolbox discuss take particular mixture conjunction simulate kernel costly warm branching tree computational assume work tree practice several decomposition present systematic self ask explore address strategy mix decomposition mixture interact run sir direct argument joint run hence detail leaf extension balance unbalanced introduction expense contain resample tree index live may simulate run write count inclusion transition sir particle recursive particle distribution sample additional indicate recall straightforwardly division simply correspond component numerator way convenient upon respect derivative identification implicit notational sufficient counting denote importance particle product h particle unnormalize consequently q turn consistency particle normalize binary denote capture argument henceforth dm finite everywhere dm relax simplify residual perform particle I extension simple set cover definition outer coincide moreover approximation semi cover disjoint finite algebra approximate q assume apply simple converge everywhere apply outside bad q everywhere imply px equal leave internal induction weight ht induction imply population lemma measurable without indicator equation f f lemma pick cb compose union cx therefore next resample perform dm resample particle unweighted particle plug quantity eq dm induction use ci c accord simulate ic n refer particle use reversible markov z z result ise text repeat show estimate mix leave correspond particle flip mh numerical inverse ise temperature result give figure sampler box flip axis dash run box mcmc iteration flip mh axes region toy consist field periodic boundary respectively potential variable py distribution sampler distribution batch apply sampling precision difficulty arise interaction note tend whenever relative pose difficulty site mh shall setting sampler use random manual note small ising log constant site fails converge posterior expect mcmc superior sampler attribute c sampler approach simulate multimodal account draw mode particle sufficiently c sampler suit difficulty population site gradually take account step application effort axis box plot axis posterior sampler site correspond run clear sampler much x correspond data york city year indicate meet level correctly meet school code student student school extract school remove character school support student cognitive delay school extract repository check example obtain four baseline particle correctness argument experiment create detail leaf otherwise metropolis propose proposal unit first within package collect time implementation turn sampler adaptively select marginalization kalman shape adaptation single bootstrap sub forest build precisely fix order traversal traversal forest parameter sir internal propose child describe text package show range top level plot support approximation row dc note std similarly configuration intel core processor ghz detailed architecture dc approximately magnitude burn take run exclude attribute efficiency method locality array non linearity run theoretical time particle implementation std related particle std particle forest std tree tp use challenging demonstrating environment emphasize computer core fast several computer connect advantageous resample communication less example alternative contrast computation communication main cost recursion location tree simplify exposition introduce set root recursively depth depth vertex assign vertex computer architecture connect depth particle particle strategy decentralize fashion library perform decentralize decentralize package discover automatically rough node specify value fail still york mathematics varied particle compute consist ghz processor x node note parallelism either level population speedup distribute scale pt remark university correspondence laboratory email sequential divide auxiliary structured turn inferential collection recursively applicable loop employ particle merge empirically outperform accuracy expectation likelihood approximation novel implementation option possibility hierarchical sequential carlo simulate collection particle weight particle generate sequentially particle generate iteration depend see therein generally much distribution arise shape graphical auxiliary decomposition decomposition contribution propose divide sampler tool bayesian variable set suitable easy parallel result merge discrepancy approximate exact sampling divide methodology repeat overall weight merge costly mcmc via particle methodology broad construct auxiliary create small connect sub assume interest shape indeed demonstrate methodology obvious structure artificial figure either exploit conclude remainder provide work build methodology property slight abuse respect anonymous equip borel density z point wise two concerned integral correspond observed computing expectation problem often arise formalism encode dependency probabilistic graphical structure model summarize conditional graph often random field abstract factor formalism reader convert graph write factor graph space convenient factor bipartite vertex depend throughout convention take sequential monte carlo algorithms section precisely z evaluate wise normalizing computationally sequentially sequence provide estimate constant smc population population weak expectation sufficiently regular expectation test weak establish example resample depth slightly recursive propose recursively easy problem interested sequential nature procedure return empty convention hence detail unweighted resample copy particle multinomial distribution n resample informally particle part state preserve resample weight weight particle sophisticated resample sampling line base unnormalized density unbiased practice particle degeneracy severe monitoring ess ess small execution arise recursive call show correspond illustrate dependency computational chain largely structure chain factorize sir employ importantly fact possible original subset common distribution local mcmc form suitable auxiliary substantial define admit thus original structure backward formally restriction reversible weight selection kernel present methodology appear key approach belief importance sampling respectively provide sense propose approximate model loop method component rely practical sense population particle full rather employ system interact smc particle system attempt degeneracy markov inexact technique numerous author purpose generally employ useful auxiliary convergence provide concrete aforementioned algorithm strategy apply direct undirected modelling scenario include cycle method extension classical framework chain section simulate subsequent chain describe execution organize necessarily specifically auxiliary structured distribution distribution first coincide target recursively incremental towards leave tree recursive execution detail subsequent root tree point computational nevertheless relate easily understand consider target intuition get hierarchical panel latent put prior node auxiliary use prior filter ultimately weight upon course affect efficiency finally far illustrate left topology exclude observe turn carlo target purpose present thought analogue approximate particle maintain particle merge indice iteration bottom auxiliary repeat resample closely mirror specify operation recursive definition equation constant equally weighted ic recursive population support particle obtain measure child particle population equally weight merging implemented resample description population merge extension methodology resample effective resample severe resampling amongst result require filter degeneracy particle employ lag setting extension degeneracy scheme improve challenging setting step target independent multinomial sampling replacement lead particle exploit capture variable simple see replace step simulate basic event step case problematic mixture possible introduce significantly resample possible reduce branching introduce tree merging gradually variable strategy something distribution proposal instance transition detail clarity sequential previous resample unweighted c straightforwardly sampler c ti particle system reversible argument complete particularly mixture use enable efficient choosing warm annealing anneal simulate costly sample seminal demonstrate use approximation proposal particle algorithm extend demonstrate particle mcmc essentially smc year resample sometimes resample adaptively analyze formally mcmc location appeal advantage concentration effort sampling adaptation intermediate subproblem start value anneal size comprise adaptation subproblem effectively significantly throughout population fewer simple represent adaptively adjust remain direction investigation useful markov illustrate lattice ise spin configuration x graphical near interaction periodic boundary original lattice simplicity continue recursively see decomposition c operate leave initialize independent population merge successively leaf define basic procedure three mixture section edge connect correspond merged method section edge connect anneal schedule severe later stage use mixture sampling flip metropolis hasting ess mix ann warm anneal threshold sampler adaptive annealing flip implement matlab grid list particle exception closely flip sampler iteration discard burn value result bottom flip mh box plot increase flip mix inferior ann ann mix exclude method ann mix ann flip ann mix ann comparable whether section anneal take sampler run site number ann mix ann mixture mcmc take ann turn half standard mcmc expensive cost automatically illustrate value anneal process warm c mix level computational edge add merge leaf anneal less figure able warm effectively anneal need mix ann particle report numerical another square lattice multimodal posterior result elementary preprocesse acquisition appendix path root leaf follow root school school come across five standard distribute
aims influence output tune hand uncertainty simple reference variable belief turn total variance hoeffde functional induce uncertainty consider importance sensitivity index write estimate sense index order hundred thousand output monte carlo carlo approach science community include pick index obtain hold sample replication combine input large practice hence dimensional index claim applicable context parameter property want efficiently sparse context method would input describe relate recovery exact decade therein estimation goal draw bridge knowledge draw dimensional model contribution describe index use give new exact thresholded coherence prove proof preliminary prove remain appendix appendix rather thresholded frame input know output want index present index number index remain small observe estimate eq identity lead q high index evaluation generate realization expensive costly inefficient require time good difficulty new estimation scheme let method stem generate index pick subsection define hypothesis one subset choose compute use encode estimation error thereby extensively compressed regularization lar appropriate one key often choice pick bernoulli method monte estimation deduce sized use binary parameter gaussian vector satisfy necessary high property design matrix context observed remark proof thresholded function correspond thresholded version address issue done lasso use clear randomize rademacher summarize estimation sized obtain obtain consider state suppose large real satisfies property pick estimation bernoulli rademacher matrix page test lasso ie plot figure design monte require rademacher two pick design perform fast bernoulli accordance perfectly active ordering evaluation estimate index identify classical estimation index interval k correction length hence show limited new relaxed note apply approach design proof sake give proof exact recovery thresholded adjacency rademacher great invoke get probability great probability constant appear invoke observe read thresholded coefficient elementary analysis fail applicable analysis lead thresholde greatly sup error rescale rewrite expectation thank besides inequality proceed define q denote random conditionally n standard tail statement prove subtract correction minor vector clear absolutely give finish q hoeffding union notice satisfied statement r side observe side conclude assume convex program enjoy assume n yield deduce recovery thresholded thresholded assume denote column convex program enjoy order optimality program r moreover entry devote proof thresholded adjacency bi set exist otherwise regular e exactly per unbalanced define unbalanced unbalanced graph parameter expansion satisfy property unbalanced degree quantity sake magnitude partition observe sub row cauchy expansion namely
data laplacian slide regularization inverse problem depict curve inverse regularization test newton inexact insensitive parameter state number solve inverse size perfectly cg degree freedom inversion report iteration cg newton cg total linearize solve adjoint newton gradient residual newton drop tolerance norm inverse inexact independent dimension surface velocity refined mesh slide refine horizontal also evaluate optimum optimum maximum curvature criterion instability field depict slide parameter surface ice slide field vary nine red value low slide ice flow see slide successful observe surface velocity year dark blue ice red bottom image surface velocity reconstruct ice infer slide inspection velocity field agree region pose nature noise field nevertheless inversion slide ice model datum critical region ice ht l pa ht fitting ultimately deal pose equip answer turn systematic quantify uncertainty bayesian ice tackle quantify previous keep discussion begin present inverse problem rank approximation hessian scale dimension compute hessian computation uncertainty adjoint solve ice space uncertain physics problem assign member rise discretized scale problem freedom parameter field combine parameter explicitly rise pdf note analog pdfs discretization usually impose rough component observable pde operator allow build solver operator variance prior ensures bound well pose infinite actual observation velocity measurement represent vector ice flow extraction ice velocity theorem n covariance take role formulation clear prior posterior nonlinearity pose challenge typical dimension surface solution numerical matrix completely question chain sample statistic expensive govern nonlinear ice flow slide number million discussion execute ice lead log gaussian posterior posterior know posteriori amount deterministic appropriately weight inverse hessian give evaluate adjoint involve derivative posterior expression accurate nonlinear direction inform approximately well space inform linearization tackle massive govern ice flow method prohibitive compare compare posterior ice hessian stand alone intractable difficulty large covariance map forward pde forward ice flow vector linearize pde problem adjoint like vector form linearize adjoint pde ill discretization compact zero understand mode strongly influence present spectrum ill pose numerous mode highly one effect effectively mesh decay illustrate rapid spectral hessian ice demonstrating mode slide parameter mesh exploit action rearrange alternatively symmetric inner approximation generalize q generalize generalize rapidly low approximation eigenvalue eigenvector eigenvalue linearize nonlinear solve newton linearize solve inexact outer cg product cg outer newton hessian linearize solve incremental adjoint tolerance find point total linearize solve approximate map low representation employ hence cost incremental forward adjoint linearize solve measure solve quantify magnitude less depict figure show slide gaussian bottom pdf gain slide across west observe large infer surface slide slow velocity velocity slide field intermediate interest uncertainty ice uncertainty subject fast ice flow region west variability infer slide unknown slide field uncertainty quantify make ready propagate uncertain slide parameter ice flow yield interest associate ultimately ice decade model ice body couple prediction ice steady ice describe scalable formally solve uncertain sliding result large mode quantify uncertainty slide expense carlo prohibitive millions solve curse solution map point q forward ice flow use slide operator give linearize prediction jacobian evaluate map action direction adjoint enable scalability uncertainty problem jacobian prediction determine field steady state ice net ice mass gradient respect forward slide solve adjoint adjoint stress ice infer adjoint resemble adjoint regularize functional adjoint solver purpose turn linearize velocity adjoint find expression without dynamically ice expression find describe section adjoint incremental solve cg iteration rank equation algebra product product straightforward ice uncertainty surface velocity uncertainty slide parameter ice prediction univariate low hessian adjoint solve forward solve uncertainty propagation map influence quantity identify inform observational require quantity interest adopt denote hessian linearize map g give influential direction prediction map q influential simplify mode depict bottom ice east sensitive difference row uncertain vice versa imply role mode sensitivity respect slide right bottom east uncertainty mode show sensitivity influence ht visualization plot ice east column ice mass present solve flow ice level observational infer parameter uncertainty parameter propagate uncertain quantify uncertainty section solve parameter state processor well exploitation infer space cg converge newton estimate inverse datum admit rank require forward solve also parameter uncertainty ice flow quantify pdf form hessian turn oppose parameter approximations pdf interest ultimately relax approximation result scalability remain subject work u air office scientific program office advance compute scientific advanced award er nsf innovation office de dpp empty rgb rgb rgb research science give broad question observational prediction scalable uncertainty infer propagate uncertain prediction quantify uncertainty end context ice effect ice observational surface ice flow velocity uncertain slide heterogeneous ice present day ice require forward ice parameter dimension processor despite size sparse exploit linearize observable randomized adjoint quantification problem approximation adjoint hessian nonlinear inexact newton ice flow ice year warm decade indicate acceleration great project flow ice current rate drive even ice potential accelerate cause ice drive conservative rise economic million risk projection ice play ice considerable uncertainty ice leave panel change th assessment state uncertainty ice level rise dominate uncertainty ice temperature ice model severe mathematical challenge significant improve ice include aspect geometry nonlinear algebraic discretization result widely vary slide range thousand phenomena great mathematical challenge lie quantify uncertainty prediction characterize field describe slide slide ice heat observe ice lead pose quantify uncertainty inference ice accomplish inference discretization take pdf belief express candidate datum pdf challenge evaluate pdf forward ice statistic use art associate ice confidence prediction nonlinear quantify uncertainty infer prediction ice flow execute intractable field quantify ice play significant future paper scalable measure linearize adjoint processor core posterior result infer ice velocity uncertain field ice yield prediction map ice flow slide satisfactory prohibitive ice equal dimension difficulty map inherently limited direction influence limited measure ice scale idea achieve low svd result framework ice adjoint covariance scalability adjoint ice adjoint ice scale number processor ensure uncertainty quantification operation term fix ice solver need scalability prediction predict day ice infer slide ice quantify uncertain slide ice flow model yield prediction ice mass quantify uncertainty nonlinear ice flow solver fundamental repeatedly solver scale brief summary ice ice balance mass momentum denote ice velocity pressure ice acceleration relate tensor invariant tensor exponent temperature determine approximately equation accumulation use heat pressure surface accumulation heat come boundary ice impose flow direction slide onto relate phenomenon depend ice water ice section discuss inverse estimate observational boundary specify stress must pressure water boundary neither interest deterministic description ice ice ice surface refined mesh element ice domain describe ice mesh refinement est mesh constrain profile vertical hybrid refinement quality mesh control element refinement refinement refinement accuracy locally refine precise velocity pressure finite pair tensor velocity pressure inexact arise upon linearization control minimized velocity pressure iterate iterate correspond linearization evaluate exploit tensor nonlinear linearization tensor identical adjoint adjoint iterative efficient critical scalability inverse algebraic approximation give matrix algebraic grid expensive jk add cost bottleneck scalability therefore construct element presence element problem low order discretization ice scalability base coarse mesh successively finer also slide slide describe velocity intend parallel sequence successively refine ice demonstrate algorithmic despite severe nonlinearity coefficient mesh refined mesh system iteration iteration count bottleneck scalability scalable refinement scalable nonlinear solver tackle uncertainty thousand solve section km refine discretization maximum aspect create successive element respectively slide boundary p temperature exponent c c core solve p scale ice algorithmic parallel p pose successively fine tolerance run freedom cpu core core iteration multiplication solve block incomplete robust poisson solve linear poisson pose ic algorithmic ice iteration linear definite homogeneous efficient scalability core count setup phase increase parallel inverse infer slide ice ice adjoint surface velocity slide parameter ice use derive gradient vertical negligible vertical gradient horizontal horizontal compare scale ice inversion ice equation fidelity ice quantify solution ice propagate prediction remainder inverse formulation brief overview expression adjoint ice flow inverse give observational ice surface velocity slide coefficient boundary ice fit square optimization velocity slide parameter observation ice field surface vary normalize prevent division regularization slide ice restrict field reference slide regularization smoothness typically add invertible normal definite need stem small sliding inverse pose sensitive reference inverse value classical reflect prior slide high domain three dimension inexact cg
heart show filter setting delay extraction challenge one device receive heart like design separate capture record add also external human dealing correct person heart record person utilize record heart case heart task heart body close obviously utilize filter device process correspond paragraph extract child body solution obviously heart go heart follow heart explain extract child heart eight try fail organize describe setting filter conclude everything show signal like extract consider denoise filter going minimize least mean square mixed child delay last assign element subtract mixed shown section find approximation lr correlation notable l measure size later converge fast slow parameter store frequency hz sequence test child available home page consider lr experiment filter first change mean refinement signal table several threshold cccc delay e e inf e e inf inf inf inf inf table achieve child l act versa I analysis finish discrete child different dft domain
normal particle filtering learn posterior query probabilistic approach limitation belong similarity distance kernel primarily inequality task case another measure benefit say utilize knowledge fundamentally experiment modularity separate modeling retrieval handle respective explore aim query multiple together dirichlet conceptually article also observation enable part corrupt describe create come cluster center retrieve irrelevant depend share map prior objective reasonably compare retrieval rank generate randomly experiment database query experiment likelihood prior choose dl q compare consistently ordinary notice posterior descriptor show form show variation performance two feature average posterior sample observe well store reduce posterior store weighted likelihood posterior maintain retrieval performance store objective improve retrieval precision approach informative training representative present elaborate weight preserve rank experiment cluster investigate experiment split grind ground evaluate number store posterior perform equally analogous generalizing beyond posterior gray correspond partition database show store toward corner contour sparsity retrieval demonstrate approach restaurant multi label map fig train probit regression dataset gamma posterior sample dataset query consist detecting experiment highly like split class separate sufficient due reason retain computer miss measure propose preserve ranking observe sample able preserve however store since consist customer categorical sense drop sample decrease collect essential rank experiment query state retrieve matching categorical annotation argue actual relation outcome learn measurement relation retrieval compute likelihood query learn measurement exist towards highly extend include rigorously model yet beneficial able sensible storing often express analytic store select informative present acknowledgment support centre coin energy calculation resource school science science project available web acknowledgement blind review institute technology computer centre statistics college institute technology university taylor ac measurement outcome retrieve compare annotation valuable incorporate retrieval employing utilize measurement argue metric sensible inclusion analytical resort store sample therefore informative load retrieval demonstrate efficacy simple procedure comprise independent outcome example wide explore association trait control variable genetic variable specie cell outcome microarray traditionally retrieve qualitative assessment document explicitly handle experimental datum researcher throughout public compare manual annotation suffer terminology annotation effort beyond annotation compare experiment toward acquire annotation information term assume researcher model measurement study utilize metric idea query experiment metric efficiently likelihood posterior issue store evaluate computationally demand paper deal select storage requirement maintain retrieval achieve individual likelihood preserve suitable weight compute reduce storage burden illustrate define collection outcome di di rank database relevance query experiment suggest eq previously retrieval capture keyword generate document marginal notice certainly query kk store grey dot performance pool discriminative computational requirement dot trade therefore contour dot grey dot define binary constraint formalize highly realization entry aspect matrix assign switch actually maintain balance e dd mm dd second third first figure triplet retrieve p q sign arbitrarily matrix block diagonal block gray solve label help likelihood normalize use library solve logistic interesting sample dl treat one I map ratio observe consistently sample come observe store high noise positive negativity naturally
probabilistic two logical predicate logical reasoning system predict attribute gender education location relation user probabilistic logical reasoning result effectiveness course never explicitly gold standard really promise perspective medium directly offer gold facebook contain preference movie book comprehensive different type information offer pt propose logical answer question twitter user new york fan building reason attribute user location gender network friend inference friend I likely new york distant supervision attribute education preference text proposition feed investigate show logical improve also baseline extract online social political public opinion evidence preference come preference like friend like popular help collaborative describe preference movie social combine share attribute relation may latent result user preference source user preference tie like twitter knowledge attribute infer preference twitter relational reasoning framework markov logic probabilistic relational logical combine inference like people work device associate perform rule probabilistic attribute preference stage extract user attribute knowledge describe although medium facebook sparse profile site twitter medium describe activity education system combine supervision structured profile text message able construct comprehensive list attribute mention dramatically increase extract explicitly attribute investigate possible coverage extract profile attribute mention logical feed framework logic infer rule user attribute predict evaluate preference attribute like system describe predict interest behavior medium life twitter technique general contribution summarize logical reasoning estimate attribute medium combine relation relation preference present relation introduce probabilistic logical framework stream extract relation preference logical represent two kind predicate mapping object another return object predicate whether object friend twitter predicate give predicate graph twitter discard twitter publish tweet million next describe extract predicate attribute education gender focus detail extraction goal unite infer publish tweet base tweet specific entity drop publish tweet publish location united entity match name job education attribute extract probabilistic user obtain feed google profile publicly education major name twitter account google account adopted take percent least friend google circle twitter account person percent job education google account name job entity publish twitter content education job entity tweet require user mention education job publish percent job education many framework devote gender twitter study whether high feature help absence predictive without name rather extent logical social implement national social security contain record annotate gender birth name highly gender assign gender gender least gender user user twitter people friend follow straightforwardly twitter al publish return indicate likely confidence project straightforwardly extraction approach user preference specifically predicate sentiment analysis social medium e extract sentiment extract object sentiment resemble sentiment use base manually manually collect tweet among massive people discuss tweet form collect semi distant supervision extraction new new new begin pattern seed like think entity twitter package tweet tweet distinguish tweet tweet model like token entity feature tag dictionary crf crf package context window tag entity tag pos tag word use model iteratively distant supervision distant supervision label draw external sort come treat relation sense hold publish express supervise heavily seed influence seed add example distant help training overview combined distant cm begin tweet label tweet newly entity add training cm preference stop optimum stop tweet entity tweet positive match negative tweet label label dataset view parameter tuning score algorithm c tweet model tweet l tweet entity tweet tweet tweet tweet label purpose without distant supervision naturally constitute employ decide token entity tweet predicate construction exist assumption different like entity category would connect belong category treat predicate priori predicate evidence instead node number branch let system clique major challenge miss example situation mention entity deal user preference preference report entity illustration express summing variable system optimize entity denote user specific entity would mention setting address infer attribute multiple joint along partly retrieve mcmc probable explanation infer attribute able size greedy inspire attribute logic attribute value consider iteratively estimate attribute friend standard confident prediction relation would decision round expect yield former benefit gold detector user attribute relation preference base dataset section value gender testing attribute preference global logical entire network improve baseline use state user tweet state evaluation approach gold standard location report prediction location precise set predict user attribute baseline include assign attribute unify assign usa california svm classifiers feature predict extract attribute feature encode presence absence entity job education friend value attribute presence simplify relation simplify rely cccc acc acc unify naive performance outperform detect network consider evidence logic capable yield bayes correspond people north half draw gold gender assign high precision security inform svm logic relation preference c svm even incorporate design directly gender entity write style gender nonetheless inference achieve accuracy network table give gender infer prefer prefer form select specific distribution positive relation make pr co occurrence classifier global baseline user view location assign location assign proportion value recall svms relation prediction leverage interaction pre ability preference complex gold twitter user towards entity actually know like ability detect say opinion proceed predict opinion task useful structure alone solve could combine sentiment technique begin scenario entity try attribute text experiment sentiment type user g friend sentiment toward entity distinguished york e prediction pr pr gold guess evaluation term prediction employ classifier feature individual attribute value location gender friend along network cf account popular recommendation recommend similar construct user cosine attribute entity base entity describe entity indicate whether entity see outperform cf believe try sentiment difficult entity entity construct total distinct baseline employ include popularity whole decision naive classifier decide specific express entity attribute detail performance evaluate precision table like mention extremely task tweet percentage entity predict one difficult require much kind access
fidelity parameter inverse sense additive noise structure measurement task assume sparse paper group limit include segment block group connect element address regularizer review regularizer recent year attention non zero sparsity structure lasso make group encourage sparsity group propose regularizer lasso regularizer group select make encourage sparse formalize goal generic fuse regression curve model group lasso firstly regularizer net former induce corner latter strictly create grouping effect regularizer regularizer fuse lasso compose total encourage successive similar able equality absolute pair variant sparsity regularizer fuse fuse pair extend fused grouping model net grouping neither fuse elastic regularizer ability outperform grouping fuse grouping ability net inferior focus group paper nonnegative constant control term become lasso regularizer ball divide two locate axis four fig depict solution contour induce contour lasso specification fuse strong ability operator proximity term give q denote stack column operator function convex minimizer split fista admm sbm detail experimentally fast result aforementione fast step criterion satisfy type regularizer bias common say support magnitude phase estimate produce conjugate matlab pc intel core processor ram algorithm assess estimate nd f nd error time matrix negative fig h l l cm mae yes yes yes fista sbm fista sbm stop mae recover quantitative report solve accurately sparse fast obtain conclusion regularizer group
achieve uninformative depict dash mmse plant show matrix see intuitive finite dictionary available large amp relatively towards mse uninformative illustrate plot amp depict diagram calibration static identifiable amp asymptotically dash recover subsection illustrate learn happen noisy sensing become phase amp signal surface sharp transition mse compare fig relate relatively pca evolution qualitatively section fig zero smallest sparse mmse counting bind blue line fig amp line depict line amp mmse separation correspond length signal long sensor case signal reconstruct also correspond mmse error remain neither rank blind early appear phase observation theoretically achievable system perspective development successful lead algorithmic concentrate theoretical performance version however way match predict worth discuss algorithmic conclusion parallel evolution see recent study issue compress optimal correspond belong avoid issue basically sequential pass small correctly size part function helpful explain compressed sensing however implementation explicitly pca able prediction moderate proper incoming bp message income bp incoming incoming se realization cavity cavity se se se notation equation acknowledgment receive research european union fp grant financial core program equilibrium dynamic soft matter department intelligence technology sup universit universit es paris bt france france correspondence shall send fr analyse factorization measurement product two original arise matrix calibration principal principal component cavity replica analyze bayes achievable computational efficient message performance term extremely promising motivate development algorithm give wise measurement factorization requirement like negativity appear many application completion computationally tractable understand understanding limit randomly result fix existence phase formalism amp physics cavity method context replica cavity widely present rigorously describe closely amp amp place derivation phase diagram study framework via amenable diagram establish principle amp find treat separately problem dictionary think kind interestingly two unify formalism measure information denote q probability factorization assume case various various identical know provide parameter blind calibration case measurement normalization partition infinity degeneracy probable prior define index symmetry symmetry break etc usually write explicitly general simplify necessarily expectation maximization multiply arbitrary change derivation paper mean deal happen assume evolution concern square mse original marginal measure square mmse explicit formula mmse achievable element non treat mse achievable message pass mmse amp need notation input output amp uninformative another random take initialization symmetry nontrivial mmse initialize iteration close result agree amp mmse amp mmse entropy section formula mmse example whereas within physics cavity replica provide situation partly establish rigorously recently important receive lot recently factorization analyse data compression compress devoted basis basis datum decompose paper analyse teacher student scenario learn iid elements zero gauss non mean noise zero variance delta goal infer noiseless observed want provide eq assumption likely hold analyze benchmark develop application satisfy spirit pass algorithm mean measurement kind typically look basis compressed mind measurement regime multiply row learning column degeneracy optimization formulation normalization information supervise use hence measure perform reconstruction blind blind calibration dictionary eqs work obtain follow variable variance zero element quantify know measurement provide compressed explicitly factorization rank know small fraction element one know element function precise arbitrary long hence large limit analyse paper keeps work work analysis apply low completion need order reconstruct negligible counting know element reconstruction generate construct keep knowledge principal well decomposition keep svd approximation however tractable variant relevant require rank component sparse approximate product teacher analyse eq still comparable region dictionary learning way hence work zero counting need eq blind channel measurement blind typically number channel sensor sensor unless variant close interpretation create analyse largely non noise element require looking regime interested another robust pca zero measurement noise counting satisfy completion counting element position analysis infer fa representative methodology mean fa assume generate term entire x fa factorization form characteristic site dependence variance normally obey distribution compact manner py z dx ff ff mf mechanism determine minimize certain determining employ fa coefficient signal dictionary negative easily impose student teacher scenario element algorithmic give exhaustive concern paper work paper message phase transition limit dictionary learning identify visual extensively overcomplete many svd assumption g view redundant closely problem principal component blind source guarantee another differ exist mention difference work concentrate theoretical guarantee work work analyze case arguably bad practical application case provide tight literature error signal diagram message direction find processing work base give many variety prove rigorous performance less often consider algorithm stand core performance bayes offer total derive course heuristic mmse rank note amp derive rigorously zeros dimension whereas literature consider amp use zeros result mmse fraction derivation message state amp analyze two amp replica agree give prediction factorization phase diagram blind separation completion summarize conclusion sec identity many calculation physics system identity basically identitie equilibrium configuration signal know identity type average double trial similarly double average dx df xx ff identity configuration average see nothing last step identity problem generation limit introduce paragraph hold realization size f f average useful useful self limit x remarkable concern left measure overlap right self two independent sample symmetry average matrix relation straightforwardly elementary expectation crucial certain strictly explain amp investigation greatly nine equation impose leave evaluate mmse reach uninformative initialization plant bethe mmse bethe entropy exponent normalization bethe furthermore uninformative optimal mmse hard algorithm mmse physics instrumental transition two mmse one zero mmse noise coincide poor mmse simply enough recovery tell happen generally case state barrier prevent attain mmse suggest limit system amp derive mmse subtle low blind informative uninformative initialization evolution may divide one uninformative fix free hard fix phase hard phase mmse achievable phase mmse uninformative initialization denote amp transition useful minimal achievable mmse mse achievable mse mse regime region make mmse amp eqs view complementary cavity replica cavity method amp experience three rigorous proof follow cavity belief propagation correctly bp randomly evolution iteration bp term large cavity asymptotically derivation fail replica break fortunately bayes optimal exclude analysis replica replica normalization logarithm saddle assume saddle class replica cavity replica lead mmse apart claim result exact cavity method amenable rigorous analysis describe assumption make obtain belief propagation equation factor fig income eqs message obviously locally tree assumption rigorously show however resemble compressed prove rigorously expect matrix bp asymptotically sense equation start order phase transition need iteration planted reach justify contribute assumption replica calculation self average concentration basically assume instance exchange top interested evaluate bayes mmse unfortunately prove basic replica easy factorization cavity alternative replica physics strategy conjecture evaluate mmse factorization path cavity approach eventually conjecture mention namely sense matrix apply extensive need prior match fundamentally simple major replica break physics computational many problem spin via pn overlap define measure symmetry permutation replica replica delta peak limit limit continuous symmetry peak simply multiply symmetry configuration respect reference compute free take field purpose physic self transition whereas hence delta plant identity converge delta allow inference decay sufficiently prove measure truly replica break happen trivial even result trivial instance compress prior practice even know often full variance precise amplitude learning straightforwardly include algorithmic detail implementation leave statistical maximization maximize normalization measure message turn update impose compress satisfied empirical variance channel time perform order condition straightforwardly within compressed nice pass back line maximization improve quite opinion bayes amount marginal resort approximation approach combine belief propagation bp reconstruction approximate message pass subsequently ht il factor depict fig bp iterative equation message argument graph bp trees normalization product bp exact incoming happen know rigorously success message yet system point propagation equation sense lead understand theoretic factorization transition bp iterative integral intractable definition scaling shall show bp simplify write involve parameter bayes optimal provide careful loose term limit rewrite square expand order definition without square q message introduce variable integral expand exponential integration function definition term keep term way distribution input auxiliary instrumental notice analogously use message scale quantity hand lt ip recall general belief simplify closed message means define iterate definition notice simplification reduce approximated probability define analogously message amp derive message exploit always close message physics equation spin limit asymptotically literature present factorization clear independent equation avoid might negligible one careful might spin reaction define order equation keep track order equation expansion iteration factorization variance familiar recognize equation compress three algorithm dependent amp similarly find dependent amp index iteration evolution miss big amp expression serious understanding eqs derive factorization principle iterate show way improve wide application proxy describe analogy call identity hold generate correct meaning square equal mean difference identity hold conditional incoming simplify play amp expression rely definition parameter analog mean reasoning quantity hence simplify far simplify eqs stress independent contrary variance depend index sense propagation amp normalization logarithm normalization bethe logarithm normalization energy physics bethe five contribution derivative bp equation practical mathematically tractable amp message pass equation express eqs last keep expression write form put piece evaluate amp bethe log decide amp bethe free transform fix identity verify bp message bethe free entropy belief point stationary bethe amp factorization bethe free entropy allow achieve bethe order derive form bethe amp check derivative derivative note nothing quantity l satisfy equation deriving conclude point free fixed rely iterate compressed entropy current quality sense trick useful bethe entropy simply eqs variational expression kullback leibler divergence prior distribution eqs simpler compressed expression generalize bi white depend channel entropy point channel last obtain simplify term entropy large expression fix point hence maximization expression algorithmic amp compress expression increase adaptively iteration increase bethe free read check il z il I bethe exact approximation field derive factorization element iid matrix iid random generate bayes distinguish output channel element aim recover term order pair index index adjust characterize channel instrumental hz w quantity equation product write average expand double belief exactly indice quantity realization expression proceed conditional incoming bp equation obtain many assumption gaussian variable mean distribution mean incoming message z zero correlation argument lead quantity asymptotic uncorrelated incoming message separately new second realization behave assume case true bayes equation similarly see non mean gaussian analogously integration analogously together equation limit variable satisfy absolutely essential convenience formalism satisfy prior match true simplify justify statement need hold state simplify gaussian integral choose come expression line distribution take initialization evolution symmetry initialize evolution ise equilibrium correct converge absolute value general phase satisfy hence lie line e satisfy reflect factorization evolution equal aim equal denote q analogously exactly get condition result get analogously satisfied write product perform part measure equation integration respect q thank integral integral mind remain analogous integral dy dp np however recall self average randomness average use factorization problem replica know rigorous approach inference factorization problem result replica fully evolution expression constitute statistical mechanic generally examine statistically current expect entropy converge unity expect relevant central systematically carry replica evaluate th dy utilizing assume form generality equal hold come trivial
always unitary require continue triangle zero remain eliminate desirable impose negativity example constrain remain variable symmetry break although problem eigenvalue avoid wish firstly optimal position latent prediction survival use standard p accomplish optimisation make prediction optimisation show log global correspond perform analysis top exist attempt start initial search examine behaviour different generate datum firstly kernel value gaussian repeat possibly value c te independent censor individual censor radial helpful compare retrieve true value purpose plot allow quantitative sample circle origin origin radial belong circle similarly angle circle circle angular point line total square finally note coefficient determination take property secondly rescale ability accurately extract dataset time describe patient induce project latent observe lie one manifold space split equally sized basis structure observe whereas manifold clear separation illustrate extract reveal survival intrinsic show example determine alternative offer explanation third variable explain three non offer survival simulation study include survival retrieval dimensional effect extract insight sophisticated survival hazard burden european fp ec agreement partial derivative optimisation partial derivative likelihood identity purpose drop clarity partial define second partial derivative define derivative begin simplify q kernel eq second q zero otherwise detail sum eliminate perform appropriately matrix outside loop hessian compute partial likelihood define q bind manually function partial derivative become second require practice q partial section convert high survival challenge primarily spurious relationship infer fail representation survival proportional flexible extract intrinsic combine source simulation study overfitte intrinsic increasingly research current experimental measurement hundred snp nucleotide automate software thousand high outcome available fail test datum noise great dimension covariate outcome proportional problematic coefficient uniquely covariate broad selection aim either covariate cox forest elastic net survival suitable association survival attempt redundancy redundancy parsimonious prediction reduce covariate survival outcome interpret overview survival one contain popular survival flexible probabilistic non reduction gp choose kernel type specify space variable infer consist covariate maximum posteriori principal component pca intuitively probabilistic subsequently recent regression successfully detailed overview find dimensionality explain combine simultaneously term easily extend account extension information discriminative class incorporate extract include output advance possibly outcome capture dimensional covariate survival outcome hope capture covariate survival dimensional freedom overfitte recently recognition face term image angle analogy think latent different addition survival outcome underlie yet share dimensional complicated likelihood involve optimisation choice via overfitte extract low dimensional brief section dataset column assume call individual latent space hyperparameter th product map I paper control characteristic length vary individual st primary censor primary indicator primary occur indicate censor hazard base need infer combine hazard latent variable denote survival integrate base hazard rate dimensional outcome use make data latent eq likelihood scale
possibly dimensional hilbert drive geometric model another one generalization near ambient hilbert space drive carry pca subspace factor literature encourage report room constraint underlie one lack snr another limitation subspace usefulness highly nonlinear subspace handle dimension art nonlinear objective regard aforementioned kernel subspace nonlinear paper geometric term mc union ambient distinguish force formulate mc metric operate deal third carry mc extension consider term mc map lie subspace feature additional close regard formulate avoid addition derive novel iterative mc learn complete final synthetic justify main geometry presence geometry investigate result confirm superiority art discussion point literature specifically treat lead alternatively subspace experiment confirm vector denote identity resp submatrix row resp index corresponding row column index transpose represent organize constrain union subspace mc mathematically sec present presence miss mc feature sec conclude remark mathematically formulate learning example problem subspace ambient begin characterization mc canonical model ambient low dimensional subspace simplify ss formally capture distance constrained u paper distance hausdorff specifically orthonormal basis respectively subspace easy orthonormal basis note orthonormal basis principle angle order mc model manifold exist ambient formal characterization collection likely n definition pose goal learn I iy force approximation setup closeness beyond scope cross variant training geometry unobserve miss pose learn synthetic real mc learn true manifold ambient mc long learn mc theory still program iy practice solve intractable dimensionality space involve transform evaluation kernel semidefinite function trick union subspace mc mc scenario scenario remain unobserved quantify mc average test conclude point lead find pre datum address describe mc nonlinear begin available begin center define simplification membership different subspace notice orthonormal collection basis minimize likely infeasible alternate minimize term fix alternate minimization orthonormal fix carry amount matrix optimize large address update keep updating show solve define collection reduce find close subspace reduce pca appropriate pd form eigen sa subspace basis I l orthonormal subspace mc point convergence however indeed require knowledge subspace generalization require loose collection basis point carry update iterative fashion unlike redundant subspace subspace assignment involve removal training subspace iterate subspace pruning update merging subspace merge pair mathematically merging involve find pair subspace define symmetric td find subspace merge subspace great greedy loose pd p l initialize basis merge move subspace estimate dimension subspace effort g incomplete focus level mle first use denoise project onto write average every e k order basis l describe study mc location set explicitly author sp mc propose comprise subspace assignment update subspace carry lc manifold note ta respect component step specific function short direction gradient geodesic direction respect step term subspace orthonormal basis stop w I I ta pd tu l algorithm highly trick mc function solve image mc discussion g ty ng ng centering map pre stage mean map map data nn tn product membership let write discuss algorithm solve trick far sn orthonormal satisfy mc involve kernel trick write ny iy inner n c center inter subspace kernel center number randomly n c detail alternate begin alternate strategy early represent independent map since linearly independent method refer easy subspace kernel ng w l te u p sa move assignment equivalently solve subspace write update careful belong assignment initialize basis ready subproblem e intuitive reduce subspace reduce follow set eigenvector associate c c mc case train sec assume index observe uniformly case clear mc mc complete data inner propose incomplete mathematically proxy I proxy underlie type begin result ij span ij z z corollary plugging replace distance iy kernel skip proof elementary special case find estimator z describe deviation inner establish high relationship correspond j odd inequality trivially counterpart proxy entry therefore guarantee definite mc eigen u predefine parameter mc feature mc kernel conclude call far space complete trick noise processing task feature space give visualize ambient onto many term mc address mathematically reconstruction state need z pre begin calculate square distance q describe p express nn unique odd reconstruct entry see kernel pre regard value estimate value solution z note separately define entry experimental demonstrating propose denoise test sample geometric structure effectiveness mc algorithms sparse dictionary ssc sparse clustering component use ssc variation choose case data ssc robustness every belong carlo simulation datum power synthetic yield denoise subspace dimension pca note every pca experiment ambient orthonormal basis random w control distance generate stack norm strategy white experiment range repeat realization correspond average carlo htbp htbp synthetic level svd ssc ssc collection sample union subspace stack learn orthonormal basis experiment complete miss experiment respectively choose performance ground truth basis pair avg avg l sd avg performance mc regard error te te reconstruction noiseless part simply te te avg see turn fig learn well increase range complete close omit synthetic examine goal know experiment exclude step term time trial fail contrary give subspace next pca evaluate fig method outperform sometimes high effectiveness propose real city image shown generate clean extract signal way add form experiment range patch mc learn parameter trial fair state reason return therefore ssc output coincide estimate
explain table phrase sample encoder phrase show five accordingly rnn decoder propose phrase look phrase overlap phrase encourage whole rnn encoder decoder propose encoder decoder task translation look language use embedding rnn project vector expect fig embed word matrix learn rnn encoder european project document rnn encoder decoder dimensional word phrase embed use visualize word encoder consist hide unit sigmoid wise multiplication bias fix phrase phrase start encoder word encodes source decoder learn phrase word phrase representation fig p universit de ca university university universit du fr universit find I network call decoder rnn rnn encodes fix representation symbol encoder decoder translation empirically conditional computed encoder decoder additional exist qualitatively meaningful linguistic phrase great recognition see furthermore many neural successfully use language process nlp limited language model extraction promise summarize successful feedforward phrase line neural focus neural system architecture network act encoder map decoder back variable sequence jointly maximize conditional hidden ease decoder empirically english english phrase part scoring phrase phrase reveal phrase encoder improve translation qualitatively rnn encoder decoder rnn decoder capture quantitative improvement translation rnn encoder learn phrase semantic syntactic phrase recurrent rnn rnn function sigmoid term lstm case multinomial code softmax activation weight combine use learn novel neural length vector length perspective sequence condition input length differ rnn read symbol rnn change end marked symbol whole decoder another predict rnn describe also condition input hence computed symbol activation valid probability softmax architecture component train maximize log output algorithm encoder sequence output hide motivate lstm graphical describe unit gate th gate activation unit formulation gate hide force hidden state current allow update gate control act similarly memory network rnn variant unit scale short dependency active preliminary crucial whether previous hide eqs detailed statistical goal source sentence first side see practice weight normalization constant often optimize introduce factorize translation probability phrase phrase well probability additional log accordingly propose recently neural translate sentence sentence input rnn decoder phrase score train rnn encoder decoder frequency phrase original corpora expense phrase table simply occurrence one exist translation frequency pair capacity decoder ensure capacity focus toward linguistic distinguish plausible manifold concentration plausible train add phrase phrase enter tune overhead computation encoder decoder rnn decoder generate target phrase phrase phrase present similar rnn use neural short phrase phrase phrase length phrase increase apply sequence handle propose rnn encoder suit feedforward predict phrase improvement priori train author phrase embed distance score phrase pair feedforward phrase phrase closely similar use representation encoder decoder network propose representation sentence one rnn decoder target phrase account decoder aforementione ignore close encoder recurrent translation convolutional hybrid decoder list gold english translation build corpus include news corpus side word count concatenation datum necessarily lead result handle relevant give task subset model selection word translation training network decoder source vocabulary english vocabulary token system build default system achieve test c rnn rnn encoder hide approximated matrix approximated learn decoder intermediate maxout decoder initialize deviation except recurrent recurrent white gaussian left vector train rnn encoder phrase three explain depth supplementary end de la es de la la la pour pour la pour la pour pour la united et es et des des les one un des des un plus des ca phrase communication communication communication communication communication ne des send es les world du es du r du les day le des les les des et la se et le le le cb character effectiveness scoring phrase encoder decoder try use target especially comparison phrase score part system add redundant train gram target corpus project dimensional fed uniformly train achieve random partial score computational buffer gram stack decoder buffer stack gram perform matrix source rnn decoder de pour la la united et dans les un un une des ca frequent phrase sample decoder communication communication le la world du des du past les le cb top phrase sort decoder try em baseline baseline rnn word penalty expect neural consistently baseline
rescale w q check requirement polynomial px px hence lemma dy dy r rr dy find enough enough prove immediately explicit introduce density inner product clearly z give norm density probability sphere du enough r follow fellowship theory research part google fellowship author valuable comment theorem counter proposition open present r classifier approximation ratio technique regression technique learner access require classifier whose namely good run general improper hypothesis extremely practical application extensively machine computer unfortunately bad perspective algorithm proper agnostic know hard learning return improper agnostic agnostic ratio show assumption agnostic various restriction natural restriction marginal uniform even distribution hard regression efficient show namely uniform hardness exact naturally fit expensive biology make experiment useful complexity detailed interpolation approximation bind exact run almost match question obvious question extend product concave problem intersection opposed return build combine algorithmic previously learn approximation outline technique present course tolerance technical exposition algorithm input efficient hard running grow hypothesis xx x w w rw description hypothesis return introduce terminology df df df fx error good motivation enable convex enough function efficiently number time collection function contain good find spirit studied minimize dm px classifier overcome ds dp polynomial naturally approximate proximity define projection polynomial order find find output less access distribution return return detailed appropriate depending run time outline must show assumption w lemma w xt th handle th first tp suffice find polynomial sign except area origin say invoke say sign regime find map resp move accuracy maps resp function x second find polynomial sign area use tail complement section dimensional show aspect case erm linear error agnostic good approximation naive exponential distributional uniform know concave present run margin except instance boundary time agnostic assumption uniform hardness hardness formula imply hardness every hardness agnostic hardness proper ratio concrete argument uniform first w dimensional orthogonal x rw x vx vx vx accord follow lemma lemma explain approximation every lastly complexity complexity proof step dd tt r ready
closure inspection limitation theoretical issue accurately reproduce probably inferior overall fit specific validity specify order goodness sometimes research know set solution accept aspect consider goodness fit selecting reject reject neither finally report criterion package gap tool limitation aic observe statistical case detail aic goodness sometimes want could observed tie alone could generate tie dyadic know explain tie describe aic former approach relate would two available desirable goodness fit goodness structural would property propose make explicitly direct propose assess observe fit choose make assumption parametric simulate r package network adjacency link node consider undirecte laplacian sum zero elsewhere order brevity eigenvalue spectrum refer eigenvalue edge great spectral sometimes associate extensively characterize structure complex recognize scope basic intuition structure note reflect zero magnitude tie cut network divide connectivity network magnitude relative illustrate imagine comprise totally spectrum contain connect rather eigenvalue eigenvalue add successively large eigenvalue successively fine network sub magnitude eigenvalue one correspondence weight series cut large spectrum describe total strength relative strength definition goodness fit normalize unity give let likewise calculate normalizing q change multiply change size shape result tie increase tie size shape change structural spectra b double bar notion spectra however otherwise spectra spectra say calculate distributional statistic context sum error euclidean propose goodness necessary natural density also know enyi network remainder adopt note place nan observe mean survey important degree model nan model generate goodness fit simply divide mean euclidean fit one q additionally network imply calculate percentile goodness nan define zero percentile useful model extend great narrow structure explain percent randomness distant spectrum percent fit like likewise improvement finally model distant nan r enyi random approximation fit consideration fit incorrectly structure ordinary sensible specification highlight strength exist ever formation rather plausible quantitative sometimes play role case consideration reject negligible nan tie meanwhile substantial spurious high nan even account include fourth tendency tendency toward second tendency star star star simulate accordingly perfect fit aic dramatically star nan clearly fit complementary fit structural dyadic hand assess aic structural dyadic explain case star illustrate aic adapt add health survey go modeling effect final differ type create test make significant improvement aic star certainly likewise aic indicate superior model star however generate alone model measure fit measure absolute fit tie finally exponential indeed aic generative generate compare aic qualitative gain fit visualization spectra euclidean observe spectrum fit spectrum second fit spectrum color spectrum lie spectra fit improved spectrum light green error model plot opposite spectrum explain spectrum indicate red turning see fit differ spectrum observe spectrum contrast distant nan observe visualization produce red area net rgb green red fit sometimes intrinsic algorithm theoretically plausible algorithmic network form initialize tie walk theoretically generate skewed distribution assess fit find implement add second exist member network network combination pair well add random walk tool identify rgb fit goodness spectral distance possible two sample hypothesis purpose certain favor construct material separate graph direct focus establish graph follow remark network differently adjacency treat transition markov chain vector leave eigenvector corresponding distribution strongly connect zero way finally direct feature view consider certain condition semi calculate error rather e eigenvalue clear priori fit certain study present statistical analytically conclusion size network mean calculate biased number likewise bias median quantile exploratory simulation recommend examine simulated establish seek conclusion spectral goodness spectrum laplacian implement indicator complementary table summarize percent explain measure relative say absolute fit htb absolute yes measure yes sensitive yes fit yes sensitive yes yes aic specific absolute network ultimately however exist specific structural tendency structural aic assess allow hope ability compare facilitate prior cite end
trial mathematically nk k stable method average dictionary k digits mnist digit remain split enyi site among sample centralized cloud level detail give paragraph dictionary class x cd average detection experiment see cloud local svd use local representation considerably low detection achieve highlight variation model term cloud collaborative approximate massive across consensus converge centralized efficacy extensive datum rely guarantee estimate obtain site centralize distribute remain end consensus sum absolute initial obtain consensus fix z long iteration make desire iteration centralize power power site similarly denote centralized mix consensus iteration eq nm th express site consensus inequality bind notice also therefore eq remain notice plug finally iii plugging expression note obtain claim accumulation power finite consensus iteration fact help keep total argue begin centralized method next need show purpose q furth elementary invoke induction eq geometric main lemma therefore recursively result argument follow code however code accumulate dictionary atom differently dictionary atom overview know bound know j k j r recall challenge source eigenvector k b distribute combine first bound difference dictionary bound sparse code iteration know derive error bind remain satisfied bind eigenvector r fix theorem nc k te f dictionary mean get n mainly get k f ready know atom also consensus p nc prove first decompose atom centralize denote principal r notice consensus iteration theorem eq I algebraic lemma replace follow consensus satisfied k coding eq reality interested te td get centralize sec notice next apply n ref write appendix notice proposition follow definition bind perform consensus nc c n c b due fact submatrix c I follow note algebraic support centralized main case b q k fixing hence thing need cloud svd centralize dictionary j j I thereby prove fix assumption c b te tc q induction argument fix assume follow carry one prove claim conclude apply error theorem accord theorem atom satisfied nc nc must nc show however exercise algebra brevity nc appendix collect support perturb assume singular reverse triangular sparse perturb version sample jj note also p proposition eigenvector let define comprise eigenvector denote vector u u conjecture remark author electrical engineering edu representation big assume number site massive contrast union cloud svd learn overcomplete distribute cloud require communication individual analysis deviation dictionary learn site centralize global numerically efficacy svd synthetic high ambient lie near dimensional knowledge success task knowledge great much often focus centralized available effort work work distribute entity responsible assumption lie near extensive communication entity ignore detail topology paper site massive geometric datum public network term raw among site constraint big justify local concern age justify graph prohibitive internal topology main paper work structure svd base nonlinear generalization receive community interest drive overcomplete approximated atom dictionary learn compare recognition cloud name popular classical estimation average collaborative main deviation dictionary site measure early rely consensus heart cloud initializations dictionary k arbitrarily svd long consensus involve synthetic efficacy cloud k usefulness local distribute processing date nearly decade kalman recent example localization relatively little drive geometric include consensus geometric explicit consensus topology development carry average computing extensive communication distribute closely relate study site oppose site setup fundamentally learn dictionary cloud svd atom collaborative helps rigorously cloud centralize analysis finally cloud cloud provide cloud similarity unlike perfect consensus leave analyze cloud also analysis carry iterative difference consensus average carry work nature matrix operator nonzero operation entry denote product denote submatrix column norm max cloud algorithm cloud svd algorithm remark section main theorem site possible sense system server collection robot mathematically site denote connection site next assume site massive express site distribute ss denote fundamental opposed structure global collaborative learning outperform representation suboptimal level fraction outlier etc uniform site collaborative fundamental structure study learn column specifically centralize overcomplete dictionary non although convex alone alternate solve centralized datum location site impractical communication big individual learn dictionary ever sample concern rigorous establishe arbitrarily basis collaborative iterative decomposition svd enable exploitation distribute purpose svd presentation term learn iterate stage involve denote state combinatorial complexity pursuit sparse stage move k svd lie carry dictionary iterate individually atom atom computation define ts ts location easy find u v leave singular update coefficient svd atom move stage continue alternate stop criterion prescribe learn svd understand stage end site proceed propose compute solving site coding greatly simplify justified long dictionary remain close establish next challenge collaborative arise stage singular reduce error r resolve propose k r k ti matrix locally eigenvector site analysis cloud stop algorithm learn consensus svd stop comment communication cost cloud cloud total give thing site function available site cloud svd big normalization redundant retain consensus averaging understand dictionary return cloud k obtain centralized address understand cloud include code update closeness centralized solution property datum begin cloud svd state term centralize svd thing need deviation cloud svd dictionary centralize sparse solve practice analysis following focus sparse cloud centralize omp assume code accomplish q centralize svd identically centralize dictionary cloud svd centralize initial cloud k svd dictionary cause cloud dictionary centralize iteration dictionary centralize property lasso analysis among collection large among collection define hold singular te kt comment property centralize svd unique code algorithm compute bad result svd centralize centralize svd denote centralized site iteration ready mix doubly weight eq inverse gap denote modulus time dictionary cloud k centralize identically addition cloud k carry iteration k p centralized method initialize eigenvector long method mix comment major implication establishe arbitrarily centralized show happen iteration certain manner call multiplication need significant big datum enter relation fashion finally consensus distribute dictionary learn remain close centralized require consensus averaging provide brief heuristic dictionary identical initialization k svd dictionary begin centralized step perturbation happen consensus happen perturbation therefore cloud svd start perturb deviation code cloud svd add source perturbation update summarize consensus average cloud top eigenvector estimate centralize site cause error r r e k cloud svd cause deviation dictionary centralize mainly two source result need theorem error dominant eigenvector symmetric obtain site consensus site cloud step stability distribute dominant eigenvector
discover network increase user create tweet new movement external connect news event interested event connect learn dynamic comprise activity question piece develop quantify occurrence intuition user connect exposure target link quantify post compare similarity new predict make gain event property paper structure empirically cascade propose whether piece briefly conclude section twitter removal effect tweet propagate month english speak user month overall subgraph million user investigate grained dynamic create cascade post propagate user reconstruct subgraph examine evolution amongst million user million million relation begin month remove twitter user edge month twitter highly think grow evolve add edge twitter event highly amount twitter network average user month follow degree note even gain month twitter skew receive month dynamic twitter examine information diffusion mechanism user scale thick thick thick tweet thick thick figure process user user user tweet happen post get exposure tweet decide follow tweet way create diffusion new form newly cause somewhat local current tweet frequency tweet cause dominate mind investigate phenomenon twitter examine activity tweet dynamic scale partially tweet frequently high network plot even clear relationship dynamic user user similarly user tweet tweet tend degree month loose flow dynamic user flow event think graph steady flow perturbation steady refer user tweet around hour receive number hour negligible increase consistently receive gain intuition temporal dynamic twitter several time plot arrival rate per hour tweet plot gain tweet month notice hour correspond daily activity twitter steady twitter hour receive new diffusion perturbation arrival follow occur hour arrival activity still receive nothing next two steady arrival understand information diffusion identify steady period user receive compare expect identify perturbation henceforth periodic fluctuation arrival hour day remove employ transform abundance prove problematic instead treat arrival month user hour month clarity describe exact interval increase hour hour month hour day decay hour user demonstrate however occur remove employ method actual second method find follow activity hour pointing primarily likely hour day receive remove detect proximity event change arrival new coincide cascade generate tweet tweet follow time standard deviation occur hour hour much normally tweet tweet occur hour explain user tweet tweet weakly connect component examine least identify million instance follow arrival frequent detect reliable focus exclude occurrence contribute evolution ask user similarity pair information tweet aggregate tweet month document tweet tf user document although robust aggregated tweet document account small tweet largely tweet similarity whether similar content similarity hour interval day normalize exactly across user show average significant user follow similarity tweet tweet course month become gain versus gain diffusion high tweet even tweet tweet cause similar tweet compare entire month spurious cause increase tweet separate run randomized action preserve user user would randomly select tweet user decrease observe spurious follow tweet similarity interest increase user homogeneous cause jump apart user tweet user tf tweet measure precede normalize tweet increase tweet tweet tweet become examine neighborhood weakly number subgraph discover user month cause increase tweet increase proceed case connect day community analyze network density potential value follow edge largely grow node follow tweet decrease fast decrease tweet day around type steady decrease behavior tweet density change user decrease connect tweet become related user likely discover b network address ask certain type content cause new iterate tweet create token occur tweet measure token increase decrease tweet filter less identify token violate follow probability pearson tweet cause token significant whether tweet cause quantify token tweet tweet create movement movement token ratio associate movement token table additionally tweet token time cause tweet token event cause spam token increase star datum switch team would lead huge old team would team create link far cascade explain predict cause spike process applicable closure create user exist one short follow edge result closure light user come follow refer user neighborhood predict occurrence follow need neighborhood neighborhood thick thick thick thick thick thick thick thick thick dash dash thick green thick thick rectangle blue thick rectangle circle circle circle circle green fill white fill blue fill white node white node fill white define steady instead potential discover diffusion user user tweet interest unlikely know user interest tweet content via intuition interest previous tend tweet follow tweet effective compatible interest compatible expect edge increase tweet user cnn news follow wide range user stay informed event low tweet increase tweet similarity tweet intuition tweet notice variability tweet user order reliably distribution tweet follow normal tweet quantify distribution tweet choose equal quantify way normalize day confirm plot say form follow edge normalize tweet moreover also explain user tweet new current follow way number follow neighborhood fact condition occurrence quantifie arrival new steady occur high steady follow user tweet compatibility reach reach set normally reach follow time tweet quantify well predict occurrence appearance event potentially twitter need order quantify predict follow simple experiment randomly fit likely guess high guess calculated area performance compare model baseline baseline likely precision baseline user rank neighbor tweet follow rank sort model outperform baseline score score perform baseline user l poor performance surprising user raw user receive predict current highly experience large compatible tweet degree user new could potentially follow circumstance follow user also order improvement baseline due follow phenomena number successful might lead main ideal occurrence user compatible tweet instead steady arrival user portion nearby yet discover focus work various useful continue
df statistic df df df df df df show association layer layer contain number subject survival identify layer primary result association survival identify sc c test statistic df min df association sc sc compare patient differentially region cancer datum apply identify identification section choose variance identify report mean identify identify cancer also associate identify sc identify sc tends identify variance contain cancer patient sc less run ht exact composition min var var na var var min powerful study hierarchical case sc compete simulate sc several likely set indeed even necessarily observation arbitrary difference method maximize note also detection although sc may improve apply scope particularly high dimensional big sc provide nan determine difficult however demonstrate frequently true return exist criterion sc accurate criteria future limitation beta require another unlikely interest correlation tend even feature particular violate sc spurious correlation preferable weight interesting variance identify small homogeneous heterogeneous merely advantage mail likely detect sc identify variance area research chen department university north hill nc email north hill nc email university hill email liu north hill nc mail live chen candidate hill nc mail live edu distinguished north hill nc mail hill nc mail email edu grant de grant grant grant situation feature represent homogeneous patient disease cancer example feature identify variance world publish predictive time mean unsupervise exploratory play analysis high sample express matrix correspond powerful reference overall many subset useful situation aim identify subset patient exist subset differ cancer aid cancer sub contain feature influence observation cluster one partitioning however solve may identify identify respect identify feature identify heterogeneous exist approach study set maximize individual equally important giving produce inaccurate problem propose novel sparse maximize represent observation iterative initially appropriate maximize respect unweighted soft justification imply sufficiently subset subset sparse cluster tuning force small observation differ contain possible observation identify list approach experience nonzero weight propose identify belong first soft perform nonzero motivation apply denote statistic hypothesis heat map artificial quantile plot versus weight expected second remain weight explanation weight kolmogorov step nan hypothesis intuitively choose line return feature either small recommend sc verify give belong increase least exist belong assume law wish datum identify one denote element repeat identify fail reject hypothesis know weight exist approximated repeat step time approximate procedure expensive computationally develop fortunately modify exact calculate assumption write sum square modify optimal imply difference observation imply imply nan kolmogorov distribution easily numerically nan weight subsequent unless sparse produce mean obvious choice differ feature however situation desirable useful procedure design differ wish high low example analyze dna exhibit variance reveal region genome differ initially cluster repeat observation randomly partition variance assign cluster half small initially cluster preliminary approach latter subsequent feature replace version initially respect apply procedure appropriate letting secondary one denote denote standard note alternatively chi distribution manuscript slow variation procedure cluster applying identify well variety approach apply improvement search approximate sum search submatrix reduction sum assume iterative search overlap computational fail valid define consist identify compare simulated signal generate normal focused identify comprise row represent standard rectangular layer background simulation structure fail identify simulated describe illustration heat scale primary rectangular middle panel sc white region matrix approximation overlap moment simulate set comprised entry entry follow overlap shape construct manner background layer layer background layer background primary expect partition evaluate heat scale rectangular block panel sc two comprise layer contain final two layer contain show one simulation scenario illustration single simulation panel show heat map bottom corner one panel region approximation exist remain assume approximately dimension although reasonable situation fail violate spherical dimension linkage job spherical see reasonable expect sc linkage compete spherical sc iid partition versus fourth simulation panel method sparse belong limitation capable detect point high datum describe heterogeneous ability sc simulate set overlap generate way assess illustration heat scale variance corner first identify white overall plot scenario table prediction valid table simulation stop rule simulation sc perform misclassifie across misclassifie low sc also except misclassification second simulation scenario sc normality violate low exception sc produce fail produce fail detect simulation identify comparable sc simulation scenario sc perfect spurious entry performance identify false negative misclassifie excellent sc exist first variance high accuracy usually although negative poorly design generally fast sc slow sc simulation pre rank time simulation incorrectly th identify th consistently comparison return return simulation scenario present simulation determine present simulation simulation sc sec min min min na min sec l primary identification misclassification
intermediate array dim dim double dim cell dim across k multiply dim dim j I double dim template numerical vector multiplication operator array class template place header file appear evaluate pattern pass object deviation interior produce sampler code quantile collect class member hold supplement first multiply evaluate value e analog array response development evaluate treat name pass across np appear pseudo weight supplement look false array array beta dim array double dim master double new dim double double dim I dim dim double dim std integration std dim j dim std std std stream create array element correspond array transform normal process without choose quantity pass master receive master dimension collective product process array process hold name private member feature pass master average process approximation available supplement array object array double pz pf array dim parallel pz pf pz pf beta dim pz double pz pf std dim std std std std array object generate differ object normal might arise create array safe location provide master parallel region assignment np cl np seq context seed force mc np across double np function worker package manner stream initialization list program approximation hold expect program produce change change stream carry monte ratio serial parallel times table five processor program package expect execute program ccccc uniformly spaced point quadrature product require scheme package stream parallel use share minimal use stream correctly accomplish array employ uniquely determine access context option create seed master memory latter inter processor demonstrate package build demonstrate within acknowledgement computation carry grateful anonymous lead software create parallel environment effectively program via way generator package example monte application appear final publication http simulation parallel sense without inter processor case occur practice however division multiple processor stream processor behave sense fortunately stream effect stream dependence see random seed x cx seed along distribution stream derive default generator stream construct apart component permit normal mt although stream situation arise message remain review parallel initialize spaced position generator assigning randomly generate seed overlap contiguous cycle sequence segment parametrization associate generator varied overlap stream processor seed small period mt provide long period feature availability mt generation repeat generator ensure result generator addition division come stream example generator guarantee overlap number generator exhibit modulus generator split may undesirable g quantify nonlinear generator show stream split generator generator period generator recursive generators cycle division roughly multiple generator large period generator combine inherent recursive generator good generator well provide generator good cite mt option efficiently generator advance generator stream make generator easy cycle division package context generator jump mt stream roughly time slow package produce process parametrization generator package generator fall stream several reason package quality stream practical tie library quite provide orient uniform overall incorporate property cluster provide toward parallel generation generator available generator capability software setting package belief contain fundamental foundation safe evolve practice integration item trait describe basic generator section connect similarity generator generator recursion seed via produce number length roughly less use parallel divide overlap stream key stream transformation initial initial seed generator carry multiplication prototype none double multiplication result array purpose take vector allocate contain anonymous name manually stream long I function support header appear methodology false none h long seed start np seed world std random double std std else long receive seed master world double world idea straightforward master conjunction determine stream seed initialize communication like result program alternative array formulation reduce amount expense usage generator example generator set align seed use generator generator element seed seed cat random six seed require integer seed store code however treat integer sign complement representation must add integer seed use generator example seed seed although language another access package instance generator stream return generator request specific option generator use random seed replicate stream previous generator package computing package base suggest parallel capability function package illustrate cluster four random parallel related confirm generator seed advance correctly window machine replicate produce seed stream expect call frame line mc core important specify desire additional core master core must window gain valuable option multiple frame label k
mcmc distribution modeling dynamic inversion mcmc different evaluation use function typically intensive g many query situation computational burden evaluate computationally surrogate inverse expansion projection reduce underlie applicable type surrogate forward solve require evaluate full space span represent basis quality rely crucially compute employ projection base innovation select sample snapshot integrate simultaneously process numerical adaptively reduce couple together accelerate approximate induce reduce order latter biased h collect posterior exploration build reduce concentrated space reduce build offline accuracy space cover typically drive approach increase information dimensional contour show sample compute approach region drive basis reason drive scalability offline orient develop pde constrain optimization simultaneously process reduce parameter trajectory remainder section outline inverse efficiency section drive reduction drive reduce delay acceptance speed ergodicity drive order construct explore provide conclude brief overview detail find discuss computationally intensive pn bayesian construct model likelihood function represent knowledge denote uncertainty without normalize distribution metropolis hasting use rejection sufficient expectation distribution carlo integration unbiased dependent integrate detailed wish improve cpu mcmc adaptation learn posterior online proposal stochastic newton adapt local geometry mcmc give forward cpu intensive major model effort mcmc require enable fast approximation algorithms transition delay acceptance couple ensure full acceptable error explore section drive adaptive framework simultaneously construct explore describe nonlinear differential finite difference discretize discretized system discretize discretize nonlinear pde parameter e realization observable model solve reduce span projection greatly full define output must efficient solution presence parametric low necessarily solve reduce evaluate matrix residual onto result element expensive unless dependence miss empirical term order selective adaptively tailor focus evaluate sampling localization multiple structure approximate replace order give approximate posterior normalizing drive adaptively select snapshot scale output current whiten compute deviation evaluation dual indicator exploration employ delay reduce construct initial step simulation step candidate full density value delay acceptance employ decrease correlation fast rejection stage delay rely approximate inaccurate order potentially stage rejection delay algorithm accurate acceptance introduce delay posterior state snapshot snapshot exceed reduce drive provide select reduction approach sample full full algorithm full length error threshold reduce define acceptance q mx n full posterior mx x gram schmidt chain step ensure post last posterior control length second acceptance spend simulate proposal reject dynamically evaluate indicator infinity describe adaptation control must update scale exceed exceed adaptation definition precisely reduce model become adaptation stop step model least coverage defer use adaptation online reduce adaptation stop delay prevent reduce adaptation continue simulate model large range adaptive variant newton computational evaluating also throughout reduce order lipschitz since constant continuity continuous ergodicity proposal adaptive condition ergodicity adaptation delay carry continue exceed reduce furthermore adaptation stop finite adaptation reduce ergodicity algorithm regardless adaptively analyze induce reduce bias monte error approximate cpu adaptation terminate reduced basis provide hellinger full hellinger translate directly analyze full construct respect feasible posterior feasible posterior complement feasible reduce hellinger specify region formalize proposition theorem follow feasible set assumption exist constant set z z lipschitz exist full induce z x x x apply constant certain pointwise model reduce feasible decay eq hellinger characterized adaptively update drive reduce posterior user threshold practice check mcmc heuristic motivate adaptation definition invariant step reduce decay refinement basis refinement hold basis exceed terminate recommend choose delay time adaptation model ergodicity ess number evaluation consider reduced indicator provide reliable reduce reliability refinement indicator evaluate acceptance update exceed approximation algorithm full sample approach approximate even monte estimator acceptable monte small amount metropolis algorithm remainder provide detail analyze carlo adaptation discard evaluation error exceed mean use line indicator model threshold order reasonable acceptance correction less adaptation stop reduce full accept reject directly use line reduce basis finite adaptation full evaluation proposal reach prescribe potentially guarantee length threshold threshold reduce order acceptance full distribution accept reject accord delay acceptance x acceptance reject adaptation stop acceptance accept reject analyze estimator posterior first finite effective produce monte ess eq cpu sampling effective speedup factor depend expense reduce interested situation rearrange reveal ess produce suggest single stage mcmc mcmc satisfies mse monte carlo estimator bias hellinger result condition square bias reliable approximate way specific govern rate reduce var mh mse amount regime reduce dominate hence factor avoid expensive mh var mh accurate efficient propose steady denote spatial pressure head represent pressure head field govern superposition gaussian standard center prescribe normal problem impose boundary equation condition nine experiment various aspect spatially project radial inference radial apply endowed proposal distribution covariance proposal run offer h radial radial basis weight independent log normal field use pressure head evenly grid signal carry efficiency target iteration evaluation order start begin simulation adaptation single stage proposal burn match reduce evaluation algorithm discard remainder algorithm algorithm build prior impact step define complement ex ex ex ex ex ex ex ex ex ex ex ex ex summarize evaluation reduce cpu ess reference set acceptance target show produce value first metropolis acceptance decision simulate enhance reduce evaluate dominate thus speedup choice approximately reasonably situation could simulate iteration full target produce dimension basis reference full inspection parameter contour distribution black reference algorithm blue red accurate simulation show visually plot suggest contour various cause assess accuracy approximate second amount situation large speedup advantageous full reference algorithm monte posterior complement reduce construction reduce estimate suggest hellinger finite dimension target draw sample affect achieve desirable posterior measure conduct lead additional basis add result report consistently adjacent construction asymptotically decrease basis situation minor increase computational load small full compare reduce reduce construct reduce proper plot
draw seed proportion seed geometrically decrease multiplication refinement cluster principle select manner post laplacian cut energy use factorization walk principle another factorization aim provide total variation attempt find optimization technique initial initialize algorithm random partition factorization seed increment seed algorithm large quickly seed algorithm converge quickly allow exploration high clustering reflect speed generally slow accurate algorithm dataset experimental text handwritten text remove stop remove occurrence occurrence nn cosine tf variety graph unweighted graph mnist extract unweighted raw preprocessing source c news mnist reach comparison report either news speed cluster calculate accord ir ground computed count represent run obtain l refinement mnist accuracy refine refinement refinement get graph quite implementation neighborhood benchmark c rand spam k k news exhaustive exclude ad matrix entry perform similarity main body author word similarity text dataset prefer truth matrix obtain slow news unweighted raw document less time neighbor cosine tf unweighted similarity datum extract page classify keep library appear less remove similarity tf source obtain simply cosine unweighted similarity word count raw document library stop appear near neighbor cosine format publication describe value indicate absence corresponding dictionary mnist matrix image nn weighted similarity compute euclidean raw vector contain raw near von propose basic seed thresholding seed randomly thresholded state cluster benchmark run achieve accuracy refine approach remove still unsupervised task automatically graph point vertice graph whose encode point popular use cluster vast literature graph area resample base partitioning graph contain reasonably algorithm intuitive implement scale number well experimentally benchmark algorithm order multiscale refine experimentally combination refinement maintain expensive direct heavily optimize one arise unlikely leave quickly provide work guess come label label come extent seed vertex partitioning clustering cluster seed algorithm combine approach cluster comparable size low assign vertex might good propagation assignment label context overcome seed vertex assign near cycle gradually draw seed throughout refer formalize idea weight undirected encode degree randomize iterative initial rv rr rv number seed successive iteration update current similarity increment rf mf f w fm variety choice grow circumstance implement strategy initialize subset vertex column outline routine seed small increment cost prove lie cluster generate ff f simple grow routine choice diffusion circumstance utilize give size extent usually indicator grow amount measure another burden grow routine terminate produce diameter allow handle indicator experiment yet modify useful negligible cost experiment combination turn outline discuss upon idea notion label point form heat heat outline grow precisely propagation first unsupervised label propagation seed work localize vertex partition instead algorithm iteratively alternate appear presence step incorporate manner resemble principle inspire upon vs clustering relate interpret maximization minimization kernel power weight depend exactly replace weight directly appear utilize generate embed graph cluster random main cluster primary incremental process process grow quite
private use bind differentially private must show essentially follow provide tb p work upper mirror nn nk dependence curvature bound come free give differentially private erm output incur generalization error loss incur privacy asymptotically think privacy come free generalization many generalization lipschitz dominate ball loss loss dominate privacy risk r internal randomness private mechanism private property risk function respect work differentially contain analyze differentially precise tailor differentially set norm nuclear view assume convexity far objective perturbation dependent less restriction function respect norm use frank wolfe curvature precise polytope private algorithm linear eq q code bind bind nearly tight private q table summarize bound combine tight tb op l nk n generalize weak condition smoothness ignore form distinguish two lipschitz brevity lipschitz call aware risk dimensionality erm width mirror build aware work make rsc privacy risk depend meaningful tight strong hold practice require dependent generalize provide notion complexity complexity closely formalize width show constraint measurement need underlie analyze descent convex norm get sgd noise necessarily polynomial dependence depend width noisy gradient attract asynchronous work geometry applicable differentially private mechanism context answer database reason differential privacy provide arbitrary auxiliary privacy quantify beyond scope randomize differentially neighboring record equivalently hamming distance review exposition algorithm closed define strongly within norm convex bregman divergence always strongly convex within follow duality pair dual say establish mirror design closely geometry early set precisely private mirror depend width instead noisy express expect drop proof private mirror take input set refer bregman differentiable sub md differentially private mirror descent loss b privacy differentially privacy guarantee composition reader utility useful body strongly theorem depend standard mirror counter role one convexity guarantee suppose convex set gaussian diameter strongly norm notice reduce time scale ease parameterization refer begin jensen excess sequence tt suffice next used step triangle third step expectation choice gaussian proceed term q assumption strongly imply sum round zero sampling theorem obtain maximum exposition convex diameter respectively r algorithm lipschitz randomness literature mean mention dependence remove convex section twice continuously main perturbation perturbation smoothness privacy analysis necessary towards affect sub optimal sample tight guarantee strong full dimensional think simplex set differentiable strongly hold special theorem particular correspondingly ball htb l dd p guarantee diameter function continuously norm utility gaussian suppose twice continuously differentiable suppose convex lipschitz ease j n definition algorithm true due condition q algorithms lipschitz task linear e ij op private mirror loose case effective frank wolfe frank wolfe algorithm move towards frank wolfe algorithm remark curvature q v quadratic frank wolfe frank wolfe gradient major first reduce solve minimization vertex secondly step frank take boundary intermediate inside sometimes outcome combination boundary vertex outcome example polytope correspond reason frank wolfe many machine useful polytope exponential version frank wolfe achieve privacy replace private polytope polynomially vertice much small frank wolfe algorithm high compressive apply able gaussian appendix frank wolfe polytope I hull corner algorithm basic convex hull vertex apply mechanism differentially exponential cover polynomial perturbation differentially frank polytope set corner arbitrary privacy guarantee differentially privacy straight strong composition wolfe convex polytope c algorithm curvature ease represent utility invoke utility frank wolfe private theorem corner probability plug immediately get frank wolfe private wolfe tight define loss minimize preserve problem variant regularization start sparse success lasso private frank wolfe check bound far programming x apply fw work assumption might realistic setting consider probability ensure bind nearly private frank wolfe ball differentially private exist differentially private algorithm prove code argument similar implicit theorem code remove integer addition produce dimensional agree row write construct database pair th row argument mutually consider px eq crucial small consensus subset three proof contradiction denote use w contradiction must lemma theorem private markov agree privacy hence complete mm lem lem lem lem lem lem lem claim part research perform microsoft li zhang microsoft research empirical erm minimize function consist natural differentially erm line private erm output loss fairly well understand private mirror descent lead error loss dimensionality denote width constraint improvement strongly private wolfe bound common regression lasso optimize match supervise learn select point availability sensitive guarantee individual rigorous differentially private algorithm risk minimization suppose reasonable empirical model overfitte algorithm stable constrain ball private
ever additionally demand hold def sigmoid log layer predictive seek detail observe hold slowly approximately second modern baseline dataset move improve stack gamma lead performance finally function find gamma outperform distribute normal train deeply poorly def embed double row replace figure click px item click indicate paper collection movie movie rating star user document fit two def size regularize mf test introduce previous section recommendation test user computational reason subsample observation mf report measure un truncate ndcg outperform mf table highlight compare deep activity ndcg architecture regime practical hierarchy deep acting prior compare hierarchical capture semantics interpretable collaborative filter appendix field tb model initialize randomly parallel pz l p l variational family poisson bernoulli necessary score function poisson function shape score function approximation eq normal variational parameterization q score enforce positivity avoid gradient unconstraine score softmax ascent scalar reciprocal root historical gradient show topic choose three group concept super gamma achieve fix poisson use iteration def shape rate def run converge hidden network family perform recent box variational inference combine pairwise recommendation datum go layer prediction interesting exploratory predictive performance model develop exponential flexible distribution reflect behind unsupervised cascade layer latent inner fine advantage probabilistic family many kind representation network graphical model variable def count product share document layer activation via turn condition super topic term super via inner product level article york style though conditional word article define cascade layer topic layer branch leave center def recover something deep factorization context computer vision fall feed forward differ away addition weight value music binary sigmoid multinomial choosing amount choose finally build traditional research literature def pairwise double def user item representation low representation exponential family property context deep network collaborative def exploratory well generally rich landscape modern article word family family exponential family important sufficient log different sufficient base sufficient statistic construct chain family together layer control parameter z n weight k top layer inner model call depict conditional dimension vector variable hierarchy control product draw layer px separate def model pairwise focus count likelihood count entry next return document put form topic similarly layer weight concept topic topic document depict compositional sharing neural family discuss family connect natural specify link moment statistic exponential moment completely family link via identity deep exponential transformation source linearity exponential poisson level log correspond conditional layer modeling represent super property early next layer function derivative equal link require positivity factorize distribution relation layer case softmax function softmax preserve relation mean product well similar sigmoid allow extension sigmoid observation number feature turn summarize gamma r kn graphical model neural distinguished history highlight result deep broad focus sigmoid exist stochastic feed forward sigmoid network dependency undirecte infer compositional boltzmann rbms rbms layer undirecte tie direct away independent become dependent make hard rbms force parsimonious conditional exponential certain infinite def tie equivalent tie family represent broad family rbms variable relate model family computational partition computation sigmoid belief network kind develop variational algorithm seek kl approximate share across observation approximate variational running observation factorize component parameter expectation general avoid analytic noisy unbiased gradient respect approximation monte compute carlo gradient able evaluate markov detail function primary computation latent top q gradient similarly gradient apply gradient sum normalize
accord latent depict except difficult part reconstruct em cluster detail rand fig leave reflect parent child root channel flip medium technique fairly continue ica give global guarantee find optima example grow latent attempt reconstruct theoretical method technique paper order structure uncorrelated variable input connectivity show top start connectivity ten describe end visually trait largely reflect type experience design various intend trait survey statement neutral slightly ten belong trait answer learn show question big trait recover perfect try reproduce confusion advantage none recover type exactly interestingly ica close behind among contrast search among minimize ica trait predict true take consist describe nucleotide rand ari substantially perfect match gender layer east china dataset different message analyze typical feature signal principle word usage recent unsupervise hierarchical linguistic ten frequent token datum latent level tree normalize mutual parent extent post report fraction coincide expect portion intuitive relationship relate cluster word perfectly structure user send encode match precision short token perfect match g elaborate signature identifiable variable perfect precision sale predictor discussion discussion match version basic measure appear context interpretation less connection share describing provide extra optimization present bottleneck extension bottleneck behind small typically label maintain compression relevance redundant store ignore transform optimize use optimization lagrange multiplier reduce drop clutter fix totally py py actually next proceed fashion detail j formal define py py py px partition calculate sum term imagine py py py py py linear combination weight consist requirement sum replace py px py estimate marginal term marginal optimization correspond guarantee elsewhere sec appear exactly enforce exponent instead contribute non objective involve share common variable begin desirable allow learn structure value force amount correlation smoothly magnitude fall reach set base synthetic bottom algorithm value label sample reflect approximation practice nearly maximum well fig help dna gender visual root connect reasoning learn structure channel integer deal categorical data form outcome miss fact dna snps miss sample look redundant miss information briefly comparison cluster gaussian affinity neighbor affinity neighbor bi transpose note clustering nmf try boltzmann single cluster input variable neuron dimensionality either nearest clustered look component none build version dataset unsupervised combine normally turn heuristic look single end file line obviously many signature format result strongly perfect soon collection letter character I frequent occur average occur word use reflect word train top start resolution smaller apply fine grain representation discard layer unit minus plus minus plus minus ex ex plus minus ex sciences institute california hierarchy successively objective search make meaningful structure source include dna automatically learn uncorrelated really view univariate cause cause responsible generate hidden cause reconstruct propose principle principled search condition factor minimize multivariate word simple explanation account correlation building foundation paradigm tractable provide rich insight begin technique detect structure succeed perfectly reverse dna perfect predictor independent relate gender text recover hierarchical theoretical connection future notation capital instance ambiguity cardinality take always subset index high entropy way total multivariate correlation write group conditioning explain correlation symmetric argument zero distribution explain encoding group specify dag appear redundant carry connection see explain let search solution correlation datum draw worse surprisingly difficulty perfectly generally expect correlation therefore correlation impose additional single group objective bind regardless give iid output give search explain end begin
indicate suggest provide focus range ridge regression optimize correlation highly less configuration consider preliminary suggest provide beneficial minus ex ex microsoft suitable store core file pass processing framework g strategy provide excellent iterative canonical correlation cca multidimensional nystr om find machine familiar application include semi representation locality involve unlabeled partially datum vast motivating need find common view subject form give cca projection kkt multivariate equation strategy moderate sized directly reveal qr decomposition qr decomposition technique solution possible analog power iteration block variable multiplication via furthermore require pass b proposal outline good nonetheless iteration attractive cca e exact ultimately utility simultaneous translate document extract european sentence level process sentence bag compose preserve hashing bn embed hashing projection find spectrum pass excellent approximation exhibit ultimately decrease comparable dash pass b ex ex ex ex ex ex find hyperparameter vary set free sufficient precision identity covariance run
box base say something significance order side information loss confidence click include observe increase case order overlap include click summary find hyper combine side alternate parameter think model complexity parameter strength sequential alternate would particularly yet separate bad hence properly landscape review combine collaborative information provide model elsewhere rigorously day click rate manner confirm albeit difference popularity medium digital finance service product ad attention click website specify page ad determine place potential bid pay ad page user page bid act key evaluating calculate ratio predict face context click empirical poorly click scope dyadic label whose binary label give introduction transaction record shall elaborate record display click click dimension construct record click number click click click view always unobserved log hence formulation click naturally able entry prediction confident view probably enough pair entity predictor model information collaborative filter netflix movie rate rating unobserve filter binary outcome object domain entity collaborative filtering build latent think try offer applicability reproduce extensively investigate web search content retrieval logistic technique maximum operate user reveal query feature click model framework knowledge namely ad direct query ad base much feature model explore area well combine explicit finding combine advantage weak recommender effect prediction collaborative dyadic demonstrate model side inspire collaborative filtering superior notably dyadic single per classify data page factorization page henceforth refer ij mf dx formulate regular logistic distribute linearly continuous index click involve click distinct small total click thousand click particularly click summation significantly jointly vice versa exclude minima help control overfitte suggest penalty factor latent batch bfgs regularize special take newton set really learn framework require regularizer skew non click click capture baseline effect row case page hence latent actually explicit attribute name etc page ad server learn overfitte discuss intercept alternatively want page bias page index encode dyadic prediction introduce dyadic introduce tensor follow factorization p si p ij optimize bad local minima section odd input rewrite hence feature fix learn model find insufficient obtain good performance fitting model hold fix leave work unique click regardless quantity initially potentially involve alone section click obvious predict click empirical number predict rate involve either factor feature experiment dataset extract transaction international technology due nature consecutive day auc logistic hold e period total day report label click observation feature include encode include stre weak user country visit day site top country user never mid mention thereby result feature overcomplete experience sampling intercept correction performance investigate particular usefulness explicit feature logistic regression side alone strength log use correspond logistic practice feature include feature practice side lr hyper tuning highly weight bias search tune hyper dataset reasonable consume instead success follow heuristic tuning logistic regression alone weight suitable run experiment different well hyper verify experiment setting including find along train first day worth initialize test feature dimension annotation mark except low order grid plot peak advantageous increase advantageous run far decrease regularization experiment bias b performance add dimension hard us conclusion preferable thus report day strength top auc intensity loss mark specific configuration day confirm trend peak performance auc agree calibrate auc seem confirm logistic rate intercept could beneficial alternate latent necessary confirm ensure performance claim serve general pick mark little experiment daily basis day production warm start run epoch fitting
pure model advanced could instability mathematic computer university cascade neural network architecture investigate thin e international conference artificial intelligence artificial intelligence volume intelligence title cascade investigate surface propagation page publish self policy department engineering department mathematics surface interface attract circuit device cascade nn develop greatly reduce architecture density interface coupling surface enhance device optimisation band chemical sense enhance paper literature propagation optical mode call offer control assess propagation phenomenon well establish material conversely wave due poor understanding propose novel neural separate medium currently determination suitable nn sensitivity novel weight term share parallel hardware amount put proper nn perform software characteristic field grow rapid due behaviour light recently investigate light surface outcome investigation capture cell main interest wide range frequency decay direction coupling field wave flat interface half space real fulfil wave interface medium adopt interface air fig simple order purpose paper investigate relation proper nn affected structure concern equation material isotropic isotropic frequency interface propagate optical frequency take finite free connect length propagation field wave wave geometry separate media package match layer external surface wave range many varying obtaining interface decay visible range correct cascade different dedicated computation say propose novel paradigm run comprehensive cascade separate training novel topology feed comprehensive cascade feed whereby stage new stage depend stage prediction phase output validate stage deviation spectrum behaviour novel topology gaussian fourier analysis ad act transmission purpose peak window training phase optimum perform gaussian window hide number window fourier module act connect layer neuron module fully consist module variation introduce weight connection neuron neuron respectively signal output send last validation module perform transform window module dynamically allocate buffer implement latter signal perform admit plan validate send input module allow validation relative finally global consist neuron neural module implementation asynchronous validation barrier mechanism overhead result barrier process therefore version overhead avoid much join construct produce reason use share memory communication overhead process avoid however require handle concern overhead propose parallel solution process care phase cascade core processor memory support nn cascade predict value input vector evaluate kind consider epoch output see activity execute cascade training intermediate third activity activity fast transform predict signal result pattern datum processing perform module act appropriate merge merge give positive neuron activity start execution new training stop soon epoch involve though performance error vertical execute
fix qr factorization orthonormal projection skeleton uniform sampling approximation column approximation column well bound onto span theorem apply term error
ghz go ram perform nmf quick remainder good exact exact nmf linear slack start strategy nmf observation develop initialization slack cell ms ms ms initialization strategy strategy entry poorly new effective random fact issue random interval generate usually zero feasible contain sparse start exploration away entry position four initialization resp resp single sparse explain initialization exact nmf nmf scheme order assess performance compute mu accelerate alternate optimize block compare real document interested nmf matrix identify nmf subroutine perform number nmf poorly nmf poorly entry modify subsequence discussion c g nmf remainder heuristic widely use simulated annealing framework briefly multi explore neighborhood initial fashion hope solution subset generate locally temperature solution next iterate keep important happen go temperature temperature allow solution final accept procedure crucial nmf slow j u tw e truncated frobenius step locally nmf nmf class slack h h k consist combine heuristic example instead computed refinement word initialize hybrid run stop exact find run see h c ms point start heuristic rather poorly compute hybrid hybrid compute fix hybrid practice recommend many fast slack fail hybrid find nmf nmf become increase complexity depend illustrate exact report slack matrix g size slack regular none heuristic find important question difficult likely newly hull generate use procedure whose circle arc part middle distribute angle point run hybrid run table maximum nmf ten h extension complexity slack matrix equality hold another exact find ten might complexity leave issue far rank interpretation nmf nmf rank equivalently nest slack vertex inner edge polytope dimension scenario outer coincide slack outer inner outer nonnegative corresponding conjecture take role inner outer hence rank small slack reader detail hull one correlation polytope ts slack exist instance index q submatrix slack hybrid run conjecture slack correlation polytope nonnegative rank would exact outperform simple strategy nonnegative relevant initialization suitable exact research include development heuristic especially difficult compute nmf good approximate document hyperspectral would interesting fine tune heuristic test would library heuristic library factorization still speak compute useful develop transform example done manually stress heuristic exact heuristic conservative g practice would easily parameter exact nmf nmf table sa initialization strategy sparse good initialization sa exact initialization difficulty exact c value temperature end nmf computational g value j j j performance strategy good sa nmf initialization fail c sparse observe get poorly reason tend similar rank explore c c c g g strategy note sensitive sa able situation hybrid strategy sa expensive g g mm mm mm corollary ac factorization factorization heuristic anneal namely linear slack randomly demonstrate superiority strategy heuristic combine heuristic insight exact nmf particular conjecture generic conjecture value submatrix slack polytope exact heuristic slack matrix extension finding give nonnegative nonnegative reduction successfully variety mining include image nonnegative nmf look despite nmf successfully many practical dedicate nonlinear try minima optimization improve initialize much aim well nmf tackle minima finding nonnegative compute nonnegative problem factorization nmf contaminate subgraph subgraph bc exact correspond nonnegative union rank nmf provide upper conversely g reference polytope lift exists possibly grow crucial optimization whether ideally linear equivalent formulation appear call polytope slack slack polytope worth nmf recently extension know reference show perfect answer open question bind nonnegative counter belief euclidean matrix standard linear conjecture motivate strong along nmf nonnegative closely nonnegative computation probability matrix check arithmetic polynomial nevertheless check nonnegative admit nmf polynomial I fix rely exact problem matrix show complexity later rely elimination translate deal unable either solver run number dedicated elimination perform exact nmf scale tackle organize class benchmark description correspond present compare initialization select nmf locally strategie dedicate nmf sa along compare strategy perform remarkably nmf class matrix strategy discuss problem finally discuss heuristic well ii slack regular generic submatrix slack polytope heuristics nmf outperform heuristic algorithm previously nmf nmf counterpart heuristic relevant first time nmf algorithm generate matrix concrete use heuristic nonnegative datum set use make hope promising show motivate nmf compare however linear fact distinct linear allow non trivial nmf slack matrix polytope vertice nonnegative entry slack th polytope ai introduction nonnegative slack equal slack matrix several class correspond nonnegative low pattern interesting slack therein also appear cover class build sparsity clearly admit nmf table slack slack slack slack slack slack slack slack randomly symbol value well bind heuristic extensively value rank nice nonnegative uniformly matrix precisely location uniformly ensure specify table fact nmf compare heuristic nevertheless comparison illustrate present heuristic explore strategy main heuristic initialization main e apply strategy nmf multi strategy locally final refinement exact also
common illustrate fourier series n n n polynomial prior differentiable derivative universal contraction poor contraction stem poor rate polynomial tool fourier series spline choose choice fouri negativity coefficient expansion appropriately target restriction moment technique wavelet wavelet series correct wavelet expansion n rate adaptation noise multivariate spline situation consider tensor spline number univariate function product apply theorem sm always present deal class clear remove optimality establish sharp precise loss situation logarithmic remove contraction latter allow mcmc moment conjugate coefficient frequentist rate contraction log spline contraction unknown adaptive contraction possibly consider induce function spline spline choose spline appendix hellinger distance hence bound small hellinger distance hellinger distance bound square root lipschitz lipschitz constant assertion n n posterior hellinger use identity density condition univariate relation spline true level different direction old smoothness kf integer great direction product jk nk n n hellinger mixture density restrict maintain rate put density observation stand k take nonzero interval involve step posterior variance sum index redundant bin weight series prior view develop similar product dirichlet correspond part restrict coefficient gaussian error contraction tensor basis additive ease consider covariate outline suppose satisfie nf ip constant compute get covariate bound satisfie entropy contraction parameter exponential regression positivity boundedness distance contraction spline view choose link computation b ib result contraction model functional contraction base give assume h discuss covariate response integrable formulate coefficient treat away satisfie posterior write ik dt expectation schwarz argument apply distance longitudinal covariate suppose hence contraction logarithmic calculate evaluating term get geometric restricted ensure truncation use default dirichlet unit interval gp normal though summarize calculated replication computation estimation estimation case perform mixture low evaluate become compute term survey result term pointwise credible band nominal von establishing interval dm beta gp e htb dash band solid right functional data http spectra sample objective chemical predict consist training channel spectrum functional use gamma insensitive yield generally result rmse analysis brief introduction spline continuous space always nonnegative scale spline function b follow show approximation b coefficient approximation adapt smoothness spline assertion corollary universal depend ct j infimum attain c know f b kk tensor spline less f j kk maintain assertion assertion product spline I ct univariate multivariate value multivariate spline tensor product univariate basis uniformly bound basis maximum spline give use respective isotropic value restrict integer treat multivariate decay part beyond use prior function coefficient contraction smoothness model contraction statistical nonparametric regression interesting canonical coefficient mcmc accuracy estimator comparable apply functional contraction series rate proposition contraction rate contraction estimate typically practice investigate contraction logarithmic hold adaptive estimation give secondly regardless result establish estimation mixture construct rescaled range widely spatial drive besides common prior put corresponding series contraction series recently univariate basis estimation use class general basis function posterior contraction normal put component paper contribution obtain contraction problem univariate setting arbitrary basis show basis posterior carry conjugacy without general property conjugacy abstract many accommodate support obtain abstract depend coefficient term metric compute rate relate way series expansion consist eigenfunction prior flexible alternative contraction establish use relatively elementary finite computation process procedure knot approximate give prior b spline conjugacy like represent posterior analytically size small moment sample term derive contraction j degenerate point indicator open old derivative pack respect hellinger leibler kl divergences kp p p triangular basis stage make explicit notation j jj hc respectively dirichlet conclusion belong satisfy every unify
pixel update representation regularization add expression prior pixel probability analytic update formula distinct pixel also step formula involve find material crucial good accordance learn parsimonious start later idea model shift version model residual difference train good contrast bottom grouping base composition start elementary structure part joint covariance training inference experiment confirm representation distribution htb two show template composition letter character use come character letter initialize initialize example converge vertical bar sample realistic cover principal first figure character motion necessary max minus min autoencoder boltzmann machine synthetic pixel grid divide activate activate black probability non activate task recover part minus converge visualize obtain template model denoise autoencoder restrict boltzmann machine run rate tune tune corruption compose template max minus sum ground visualize material fair comparison leave transformation template supplementary material contrast dictionary learning st learn th minus min entropy cross truth composition compact representation learn composition propose test alternative competitive interaction rule create affect update focus interaction value consider rule mean achieve intensity layer via multiple minus layer science il interpretable solve review way expert binary interaction suit learn propose composition vote vote attractive novel procedure correction year network drastically improve task use cascade learn lee upon effective typically class compact solve natural representation letter term horizontal bar consequently efficiently six orientation work learn representation little example apart intuitively appeal low description scene know compositional use traditionally location size partition image grid overlap window size point detector local restrict certain spatial hard part split stable deep consist expert learn large compositional usually knowledge area expert priori deep realistic base model expert describe structure representation explicit task locate strong key challenge network learn robust discover part correspond large achieve rate whole image patch restrict key define composition new particularly well describe composition process correction handwritten conditionally template composition template rule vote order create compose template counterpart activation feed formally composition apply composition eq opinion word image black state white pixel underlie able odd rule composition pixel responsible procedure show plot see part opinion state asymmetric symmetric template intuitive composition simple average compose template interpret template note composition impossible opinion restrict manually composition odd type composition boltzmann sigmoid belief networks link pca trait coding factorization opinion odd probability vote odd arbitrarily adopt opinion pressure adopt opinion time part likelihood competitive rule uncertainty rule however computational tractable propose redundancy among encourage vote composition possibility increase place multiple time use opinion hand normalize lead subscript negative composition extreme opinion low symmetric summation composition like odd max min log consider create white image pixel probability attempt
brief investigation mrf turn learn extend preliminary result future crf ising mrf discover several red throughput connection include copy expression connection level commonly trait model connection connection level available level sequence process ii level patient merge mutation indicate copy leave patient expression gene patient yield level converse variable order level field rna sequence wise poisson mrf crf crf sub crf sub mrf family relax positive wise describe stability determine regularization estimate figure blue gene identify type red circle include link indicate well novel several connection mutation link expression expression level breast cancer sub previously mutation link growth breast cancer latter mutation link expression sub estimate novel connection validate mutation link tp know tumor gene long breast link expand breast breast cancer mutation link expression know affect tumor marker positive sub applicability graphical generalize mrf special case field flexible dag mix show fairly knowledge density permit rich implication analysis involve mixed expand ise model model paper broad big beyond image national internet economic work provide fit understand datum discuss need proper post inferential impose relaxed condition practice mrfs additionally work extend variable priori knowledge assumption unknown variable remain still learn mixed estimate dependency theoretical build need several dag little know mrfs direct undirected area broad domain p equality side eq q exponential family expression hand side q similarly furthermore plugging generally consider triplet reasoning product construction respect connect decomposable disjoint respect means trivially extend proof shorthand suppose partition show unnormalized mass suppose c fx complete mrf develop crf crf infinite linear crf cm infinite one ty even simply result condition mix mrf useful restrictive mrfs depend response dramatically relax example poisson study mrf neither hold mixed mrf formulation relaxed block need satisfied statement crf poisson unbounded statement crf poisson mrf crf component label ise mixed mrf crf gaussian crf mrf poisson ise ise crf ising mrf crf mrf crf mrf block binary continuous poisson z generate model sample lattice edge estimate mrf mixed mrf find connection surprise dependent mrf dominate influence influence meaningful assumption l department computer college comprise heterogeneous variable skew continuous area image national internet effort computationally amenable multivariate direct dependency paper direct markov univariate conditional directed acyclic directed field markov condition instance scalable conditional simulation sequencing expression mutation datum storing become big varied consist refer mixed variable sample belong binary categorical ordinal skew among big comprise measure genomic imaging single subject collect varied call coordinate message tweet history internet history among effort collect internet history update history online among type variable dependent motivate throughput detail recent molecular mixed copy variation binary categorical functional comprise measure rna sequencing count genomic closely relate belong complex multivariate graphical task range important understand molecular disease type develop class distribution expression well influence expression genomic dependency mixed address rich dependence mix term mixed popular multivariate relate multivariate multi especially popular trait seek link gene genomic associate response mix type machine measure probabilistic model copula potentially non loss efficiency parametric regime forest handle modeling approach popular especially spatial statistic propose mixed count latent gaussian latent guarantee possibly mixed tractable statistical seminal early review next background random mrfs case discrete continuous include sufficiently expressive rich propose markov random field serve heterogeneous mixed organize simple class heterogeneous group group turn chain exponential family mix mrf call exponential family family exponential family mrfs seminal work include show weak preliminary mrfs direct acyclic class model direct undirected edge general mixed statistical model regime study simulation well throughput cancer field develop distribution model heterogeneous condition set covariate suppose locally neighbor x suppose condition sufficient statistic form consistent graphical similar covariate global neighborhood conditional notice provide ultimately tb heterogeneous variable partition heterogeneous set group setting could cause variable could interest dependency condition dependency among clique clique suppose direct undirecte among node solely edge show armed notation propose natural follow exponential crf elementary field intuition distribution assumption mixed graph mixed mrf distribution introduce analyze restriction several example mix mrf counterpart graphical literature specify mix undirected within arise edge mixed markov marginal undirected edge response conditional mixed purely undirected drop additional restriction undirecte ask classical specifically restriction answer direct node markov clique depend respect entail specify neighborhood investigate implication global property direct edge edge factored set mix mrf write noting covariate similar covariate mixed mrf distribution differ mrfs hence covariate write primarily consider form mrfs eq covariate mrf distribution covariate normalization consequence next distribution restriction characterize refer require ensure density ensure finitely integrable mrf conditional mrf crf mrf covariate n impose mrf shorthand partition mix pairwise follow normalization overall term mixed normalization identical affect class necessarily impose mrf log form form assertion mrfs general demonstrate several counter next illustrate implication broad model ise graphical provide formulation binary domain give conditional give respectively two binary primary specify formulation distinct modeling model log normalization graphical mixed ising follow discussion highlight focus mrfs linear flexible permit relationship mrf give another advantage mix datum distribution mrf crf give let node partition exhaustive direct undirected purely set higher exist literature order block edge within block recursive mixed induce direct acyclic direct conversely dag graph graph thus partial correspondingly elementary chain understand elementary vertex lead seek define construction direct field however theoretic indexing parent clique subgraph theoretic define markov q specify crf detail substitute joint distribution arrive result different block distinct mrf consist note mrfs correspond distribution arguably could cg chain previously must ensure notation within specify covariate remain restriction joint assumption vertex mix edge undirecte mixed clique respect markov solely neighborhood discuss give well mixed distributional valid crf individual mixed crf weak mixed mrfs analogous extension flexible graphical ise multivariate variable domain node conditional block statistic finally block specify denote extend node node illustrate joint mrf gaussian crf follow poisson crf covariate take form mrf crf crf crf exist permit dependency vector three model combinatorial way crf gaussian crf fairly variable type rich class recursive non dependency specify block determine edge block seem restrictive variable measure block block clearly analogously natural language obvious arise setting throughput snps binary rna sequencing via sequence genetic count marker continuous marker genomic variable interact fix marker sequence dna thus influence expression expect mutation marker point type conditional dependency undirecte precisely input partial mixed crf neighborhood setting variable fold unknown mixed graph task graphical problem mrfs normalization close form belong maximize typically intractable secondly heterogeneous varied lastly structure mix underlie undirected variable connect even solely direct graphical unless know accordingly dag priori assumption relevant area throughput assume partial dag mix completely mixed crf overall mixed outline x require therefore mixed linear function correspond clique parameter pair clique parameterize overall crf reduce estimate mixed graph accord independently follow allow side partition function neighborhood seek zero estimate crf univariate node intra block
bias unchanged relative envelope change envelope correspond change providing except determined binomial n pn probability except last risk z z provide bias line contains easily calculate analytical fig function get z z compare empirical complexity denote estimate empirical histogram classifier substitution risk plot vc case continuous feature maximal risk histogram analyze maximize risk fix minimize risk change maximal bias histogram probability stay increase decrease stop change risk second bayesian misclassification feature easily hypercube probabilistic need assign belong piece area hypercube volume supplement hypercube volume call varie vary distribution maximal risk histogram sake define level tree distribution distribution great bias examine distribution interval one estimate confidence equivalent estimating must hold q build superposition play role base interval estimate analytical infimum probabilistic empirical desirable minimal finite believe unable dedicated figure empirical terminal equal sample dimensionality hypercube py x ng define equal misclassification tree consider construct misclassification bias find formula establish bad distribution impulse similar distribution modeling decision suggest bias risk attain histogram classifier particular build estimation maximize risk derive risk histogram carry detailed estimate empirical area machine arise disadvantage complex size order choose need notion reliably training sample estimation carry interval well reliable latter require form estimation among devoted refinement risk approach approach already exist reason work paper equal nearly chapter vc make vc empirical classification shall estimation however primarily orient toward bad classification analytically construct worst realize practice quickly practical although however contrary bad estimate probability misclassification scenario key subscript usage notation decision risk simple misclassification n drop notation risk empirical leave validation etc sample eq given leave remove nearly unbiased taken leave unbiased get expect misclassification estimate risk v rv method fc fc fc possible dependency
molecular often size abstraction develop infer specie leave think molecular sequence correspond first ultrametric b form ultrametric species tree topology eq reconstruct dissimilarity ultrametric distance sake agglomerative ultrametric mutation population represent diameter correct sequence prove tell succeed molecular sequence suffice reliable use hamming modification crucial branch tree possibly mutation first define ultrametric recover specie usually distance clear arguably paper dissimilarity condition I restrict leave information dissimilarity form additive leave hold surprising molecular topology result appendix reconstruct dissimilarity define base algorithm like dissimilarity recall mutation diameter succeed reconstruct replace tree employ enough argument appendix tell like tree reconstruct root reliably note assume mutation gene mutation change work irrespective gene accord specie branch big none branch argument formalize able reliably reliable shift fact px kullback divergence even specie q mention achievable irrespective achievable provably attain require raise tradeoff quantity topology separately specie segment tree notational clutter write leave effective mutation rate correspond exponential check condition denote common also depict condition correspond leave ac use numerator next observe stochastically dominate exp observe distribution ab f cd substitute respectively let figure see condition let denote fact condition eq hold ac ab cd observe fig observe hold early eq tell reconstruction molecular recall dissimilarity call like neighbor return dissimilarity result algorithm reconstruct f side approach make exist leave union term inside summation second say ac q proceed follow clarity us ab ab definition use condition mp ab ab particular realization ab ab first end process hold claim tell apply leave pick conclude observe begin follow lipschitz similarly condition ab inequality condition claim q variable stochastically conclude theorem remark discovery department mathematics university requirement inference problem evolutionary set species specie gene gene specie tree analysis far molecular specie method account estimation gene devise previous regime sorting distance molecular history gene lead population thereby tree topology specie try reconstruction develop reference rely roughly generate illustrate explain little accuracy focus consistency access tree specie gene convergence denote branch length specie show agglomerative dissimilarity specie need agglomerative instead gene reality gene finite molecular sequence estimation level quantify estimation progress towards perform fold gene correctly therefore light require modification reconstruct particular overall sample regime secondly restrictive molecular mutation rate population branch specie extend previous beyond specie call algorithm distance molecular begin introduce paper heart evolutionary isolate isolate leaf tree assume branch assign branch time small branch length role interested vertex unique path connect length tree branch ultrametric leave restrict species branch associate mutation small mutation produce different genome specie figure thick tree specie standard leave tree resp parent branch draw gene gene copy gene describe evolutionary history isolate population represent branch draw exp interact give random gene correspond length branch accord independently time draw accord notice evolutionary history agree incomplete sort road tree refer reader model completeness proceed record fact exponential exp density tree focus species tree gene gene enter branch enter
trade bandit provably develop combinatorial semi bandit consider bayes least adversarial combinatorial bandit combinatorial bandit combinatorial bandit large problem weight approximately exploiting model independent achieve linear across assume agent know lie transpose refer literature bandit coherent emphasize agnostic well agnostic learning uniquely moreover coherent choose episode episode episode randomization necessary randomly episode learn combinatorial thompson motivated combinatorial combinatorial two control specifically close small exploration reduce hand control decrease slow quickly converge optimal kalman te e input combinatorial structure dt n te episode randomly distribution second compute base specify kalman point episode distribution satisfy obviously case agnostic three regularization decrease rate matrix specifically converge sub coefficient insufficient exploration slow matrix ta te episode step confidence ucb compute specify oracle update kalman filter bind bound detail bind algorithm coherent appendix regret logarithm second hold start episode bayesian conditioning conditioning sample combinatorial independently conditionally simplify exposition q inner conditioning two key briefly upper thompson focus adversarial reasonable since indicate hope hence due remark word work oracle offline speak proceed confidence set self normalize develop regret bad event worst associate base detailed proof regret low factor modify achieve still hold construct real world evaluate thompson ucb problem demonstrate suggest likely generalization agnostic outperform art serve baseline solve two episode expect cumulative return divide rather illustrative evaluate path grid path grid corner corner coherent generalization experiment figure trend learn per return episode return episode remarkable poorly insufficient respect episode time imply enough discriminate people music likely music website specifically grind recommendation user fm music tag dataset tag assignment tag assign tag na classifier respect person optimal algorithm algorithm similarly per return episode discriminate propose stochastic bandit linear establish bound item variety show scalable robust exploit generalization contextual either process adversary state item pair feature contextual combinatorial open several question open bound agnostic open combinatorial bandit generalization believe combinatorial monotone leave future prove I sample parameter immediately outline conditioning independently even fix conditioning also furthermore conditionally conditionally ga ga constant notice function follow tb full use cumulative function cdf base follow c naive notice conditioning inequality follow implicitly point v dd constraint uniquely consequently consequently fact x rhs choose prove specifically provide ta q arise specifically notice equation imply imply logarithm notice hx hx hence previous subsection write reader proceed follow construct normalize develop confidence analysis start useful specifically martingale bound gaussian far define self normalize notice base cauchy schwarz moreover know obviously immediately imply exactly complement event lemma state q last naive realize hence bad case conditioning event recall eq first inversion plug equation induction far note suitably regret scale eq solution scale subset oracle theorem assumption last realization conditioning eq lemma put together corollary combinatorial semi bandit problem choose subject combinatorial observe item receive combinatorial semi bandit learn call computationally efficient long offline combinatorial establish statistically efficient assumption develop sublinear thousand experiment robust choice baseline combinatorial combinatorial modular many short weight bipartite matching optimize need bandit bandit bandit adversarial combinatorial establish combinatorial estimate weight website view million user product internet formulate impractical expect movie decision condition item use bandit extend thompson ucb semi generalization combinatorial thompson combinatorial ucb efficient offline combinatorial solve major establish sublinear major problem dataset evaluate thompson usually ucb practice scalable briefly linear differ paper
location output location single forward full replace convolutional map size output forward pass consideration need influence matching fail assumption within violate prefer adaptively collect pixel comprise pixel begin construct cross intensity position spatial construct analogously arm know horizontal vertical arm union arm suggest support region right support region number average refine matching enforce image define energy q function cost term add neighboring pixel differ penalty neighbor minimize image np energy effect direction minimize energy many direction recurrence second prevent effect match proceed zero deviation hyperparameter final car drive around city resolution transform object vertical right camera dataset predict pixel percentage pixel translate depth tolerance object camera object camera train cross epoch decrease example million class subtracting divide standard pixel intensity method method rate dataset mc sf ss anonymous visually map method select example prediction bottom pt implementation gpu take hour predict evident prediction pass matching aggregation everything seem good big learn also beneficial suitable application robot improve runtime university york extract information image convolutional two patch aggregation error method refer horizontal location position know object camera subproblem reconstruction dense frame cost optimization refinement plane propose convolutional neural pair obtain match pair patch intensity aggregation smoothness consistency check eliminate perform final depict method contribution convolutional previous typical begin match cost position consideration absolute difference intensity rectangular denote interpret associated patch image patch image good bad match publicly attempt solve match supervise learn convolutional small patch match comprise image center true example center randomly randomly method aggregation good match near size architecture eight convolutional
observation generate normal follow apply typical strategy propose anomalous anomalous anomaly measure normal anomalous drawback generally apply quantitative ordinal transform group dense belong anomaly detection curse dimensionality suitable curse commonly anomaly exception work perform subspace cluster dimension subspace belong contain anomaly subspace similarly anomaly dimension subspace find drawback kind find subspace ignore could lie method intuition anomaly neighbor normal anomaly score outli instance local near radius center near call multi deviation record deviation local neighbor use micro anomalous record improve avoid unnecessary calculation calculate micro detect usually thus big suffer curse mention training generate anomalous anomaly decision fit focus salient desirable characteristic deal anomaly adjust threshold network tree network determine survey decade anomaly previous divide trend subspace periodic curve anomaly detection outli periodic modify algorithm representative new anomaly time close centroid poorly massive method alignment restrict periodic limit scope outlier light star end correlation light light curve unfortunately operational large cluster occur find light calculate operational cost unfortunately method light imply observational periodic separate star galaxy characteristic anomalous galaxy close develop one former mixed factorization unsupervise explore subspace also normal dimensional basis quite opposite anomaly outside subspace basis factorization approximation additive mixture differently user consequently heuristic anomalie probabilistic model generative mechanism score main drawback inference stage consider restrict dataset survey observation make subset anomaly anomaly score anomaly combine many observe variable equally disadvantage detect outlier space variable approach candidate set train forest class proximity value proportion forest terminal proximity measure create outli score decide suffer density detection method expensive slow big datum evaluate instance set train decision label decision classify main principle follow divide generate robust combinatorial create rf tree forest describe data bag bootstrap sample bag replacement use pick optimize principle individual tree create diversity classifier feature contribute improve rf furthermore subject minimum size allow classify new tree rule node reach rf tree vote proportion bayesian direct acyclic graph encodes among represent variable application bn joint pdf bn determine simplify probability bn describe simplify bn decompose product small factor challenge bn word bn structure probability direct acyclic modelling represent class greedy possible structure explore structure find parent per node checking create network actual number parent input evaluate probability factorization impose assign probability vote distribute rf vote multimodal consequently multinomial get rough capture divide set every fall interval use probability parent variable probability determine value parent combination parent outcome consequently follow expression q parent example posteriori calculate take purpose overfitte estimation stay predefine value previously situation conjugate calculation posterior multinomial conjugate dirichlet parameter act find parameter variable detail methodology illustration stage panel follow rf probability bn discovery pass bn previously mention manner analyze classifier confusion immediately model start descriptor light class descriptor rf label construct bag tree bag tree train vector class ij v use bin discretization perform end dataset rf vote belong variability want decide unlabele rf vote vote rf object store outli joint bn recall product small model vote object rf vote already rf associate already bn calculate necessity possible case choose calculate hope uniformly discretization bin parent three reasonable empirically different massive observed produce million star observe small cloud reader find description nm present table light comprise study long period collect composition object star rr period variable set exclude never see class star rf present leave visualize voting rf scale vote class classify rr color vertical bn rr child voting high star node expect completed perform train rf vector learn bn find figure show class object value top panel performance outlier top ideal result place see class bottom b magnitude rr accuracy rf bn run million light list candidate bn fraction easily period day bottom panel bottom day figure outlier obtain characterize day period year probably observational pattern intrinsic anomalous kind obviously light appear many remove spurious outli candidate day variable non light curve object candidate list consider spurious remove list visually candidate group obviously spurious example shape add outlier explain repeat expect step visually candidate list move outlier candidate candidate candidate either noise snr match know collection know example period variable collection long period contain ray far outlier identify nature object summarize match number appear number rare kind incorrectly classify many automatically error human involve bias present final outlier class day curve appear identify know uncertainty result consequently outli deal uncertainty topic snr actual amplitude indistinguishable variable signal light uncertain reason attribute lack perfect method classification heavily ideal quality feature start rare type never train discover recover object post cm c newton source rr classification cat candidate anomalous motion star examine candidate list type advantage population fourth group set light curve interesting rare class paper survey drop cause star system mainly magnitude distance valuable look outlier light curve cv unified characteristic include candidate list magnitude become approximately recurrent surface year quasi recurrent subject research interest show like blue unified light training variability star disk emission characterize star disk appear candidate fall example light curve location member source newton source confirm ray binary counterpart ray source optical periodic non ray emission w contact ray type ray particularly ray either star black together star ray emission cause study ray process e representative ray outlier star rich extremely cause result drop magnitude light class comment run candidate visually object show individual outlier notice belong locate field therefore star confirm reject hypothesis peak variation variation average since year reject class diagram outli high outlier score box period c class period day snr class class class outli individual right right right left outli survey develop automate star characterize detect anomaly classifier discover forest network exist method work process computational resource nevertheless curve analysis explore anomaly periodic star project identify belong error available find rare previously perform follow characterize identify object isolate construct furthermore aim survey help plan release software tool service na ia cat grant ic institute cat institute university usa surveys lead massive need
text dynamic pooling sentence operation limit max pooling extension type unit last distinguish turn useful later variable sized vs bag gram document gram svm gram infeasible also bag gram one share contrast learn text region intend task convolution turn neuron large never confirm convolution simple layer deep layer explore parallel text improve pooling region representation one produce region layer take concatenation input topic sentiment detailed internet activation descent word appear seq region word vector avoid cnn typically input case scale multiplying test connect neural vector connect net dropout classification frequency scale unit always binary traditional vector component corresponds test nb lm later modification exceed generate bag gram vector gram training hyper net hold choose hyper movie k character review consist amazon review choose movie follow test set one half review review text text review set internet corpus news article describe category hierarchy document categorization thresholding strategy algorithm model like categorization month period early size concatenation except regard nd level nd show rate thing note cnn demonstrate effectiveness look detail first cnn convolution seq cnn table sentiment choose region max pooling cnn seq pooling phrase strong sentiment great movie receive high irrespective rest sentiment classification categorization configuration cnn size pooling pool vector classification short entire pooling location predictive text pooling point seq outperform outperform seq cnn indicate region turn parallel seq gram frequent nb lm paragraph lm unlabele seq k seq method nb lm report learn unlabele supervise due use resource cnn categorization cnn outperform even tf micro macro micro average macro average categorization study train cnn report word train cnn training unable type need tune dimensionality word vector modification sentence modeling notably word convolution region cnn find resource demand task various memory hour gpu cf svm excellent performance sentence classification short sentence categorization time minute horizontal baseline linear take minute core intel gpu figure effective explain cnn look learn comparison gram gram great perfect perfectly bi gram gram gram gram svm heavily gram tend gram nb bi show learn cnn convolution region embed produce dim component th top positive value predictive note embed list large embed vector tend proximity relation target positive sentiment effect help layer c return totally bad fail easy I p super example predictive bag gram training cnn gram long good ever ever ever satisfied overall region contribute entirely gram show test partially bi gram yet embed heavily predictive thus certain pattern entirely overall satisfied need entirely ever adjacent gram seq successfully exact gram g positive cnn effectively bag gram fail show cnn mechanism effective text categorization embed parallel combine complement art classification classification achieve anonymous author nsf nsf ex plus minus em ex ex ex zhang nj usa convolutional neural neural internal structure study categorization namely word text instead low done directly embed text straightforward adaptation text employ word convolution layer explore demonstrate text categorization automatically assign type categorization different categorization detect spam sentiment classification determine typically review standard categorization preserve classification loss cause problematic sentiment bi gram gram gram text categorization general topic categorization add gram reference benefit order text categorization neural network categorization structure word successful image classification win imagenet challenge token pos cnn search sentence notably cnn learn additional elsewhere cnn aid vector train arise purely supervise really categorization essence text region size later sense convolution bi gram gram directly cnn text go sparse efficiently handle infeasible dimensional gpu turn simplify tune cnn text publicly internet effectiveness cnn categorization explain cnn cnn test seq cnn straightforward adaptation cnn employ seq versa winner conventional bag previous cnn text particular work improve extension combine convolution layer embed improvement gram method document review task seq image compute linear share unit feed convolution layer illustrate top perform unit small concatenation pixel channel red blue conceptually region region center set region distinguish layer weight sharing region compute wise activation component bias train learn irrespective location appear regard convolution output consider pixel neuron convolution dim vector region convolution layer pass essentially merge abstract pooling region merge average compute text vocabulary
neighbor rough appropriate smoothing theoretically common point learn learn irrespective know regularity minimax instance loss incur ball achieve regularity general introduction still want hyper attain logarithmic regularity use study contraction extend line reasoning multiplicative ahead latter argument prove desire describe together discuss mathematical conclude copy poisson cube underlie integrable every counting set stress expectation involve notation respectively consider intensity equivalently observation priori q gp index sigmoid section model gp hyper endowed choice hyper inverse spectral decrease inner product denote respectively parameter density constant instance common tail prior satisfy fulfilled gamma increase infinitely smooth condition fulfil precede quantity construction result concern intensity prior formula g describe around intensity generate data contraction depend smoothness quantify belong belong old contraction square root intensity hellinger f sufficiently large intensity hellinger convergence smoothness intensity fulfil sigmoid length gamma extend cf adaptation case write fs fs n away suppose b form commonly encounter contraction rate put intensity datum remain mass together require prior concentrate term grow posterior subsequent subsection theorem result derive like extend adapt deal intensity section define smooth hence smooth compact away vary hence q factor assumption second constant low proof easily complete gp background notion unit determine subsection since second moment give sequence conditioning pg pg bind eq term rewrite view precede remain approach enjoy favorable result show stationary process link room multiplicative contraction lead believe sub impose strictly speak matter open necessary generalization instance generalization prior analytic generalization work believe example definition cox process approach learn poisson
exist fuse fuse give former base dynamic programming programming specialize admm utilize optimize routine q specialize admm update run admm approach neither dynamic beyond knowledge solve high order admm far specialize admm outperform simulated wave three trivial smoothness homogeneous examine spaced order space q small penalty polynomial instance admm record achieve across scale case particular show arbitrarily qualitative behavior clearly routine dominate optimum qualitatively simulation parameter specialize standard specialized term first rough interpretation utilize subroutine difficult subproblem linear progress towards minimize criterion reasoning admm hard subproblem concrete explanation come admm invert trend filter specialized admm ignore admm think perform alternate minimized lagrangian parametrization difference correlate bottom scalar block regression triangular make progress update direction align think contour contour form filtering may update admm allow step overall special admm run start mean without share warm large second large warm absolute illustrated specialized admm axis iteration noisy right warm start advantage middle representative therefore warm worth discuss augment lagrangian parameter use associate filtering rather introduce admm admm optimum however rate stability set numerically stable across update appearance soft programming level intuitively make try adaptively heuristic stable recall paper selection optimization default state otherwise specialized admm write author put also implementation specialized implementation actually number barrier barrier barrier backtrack line choose former size interior point close original linear trend setting try vary lead performance take early time admm comparison achieve mind criterion versus time repetition combination except size repetition complexity offset scale versus large scale admm iteration time fast specialized display display relative absolute regime algorithm n high regime admm large admm statistically curve piecewise fitting figure convergence barrier backtrack meanwhile admm stable across hundred routine small trend desirable fit near specialized outperform admm fit converge regime trend encounter converge winner robustness case defer brevity discuss specialized show size moderate barrier backtracking present suggest case poorly condition core take admm suffer system important denote admm iteration form eq augment important buffer make possibly poor conditioning meanwhile drive across optimality complementary zero dual lie strictly inside buffer condition instability issue solve iteration many dual strictly mean input extension specialized algorithm highly tool filtering much generic evenly space change trend space recursively begin admm input panel fit location slow converge admm slow alternative admm design aside change admm routine general lie augment arbitrary input use motivate admm evenly routine routine try match practice adjust make progress strength solve readily adapt modification trend extension novel manuscript exhaustive list stage trend filter trend tool across input parameter calculation specialized q still operation per example htb suppose extension order act order penalty specialize admm naturally extend trend estimate detect component adaptively detect outlier routine operation iteration encode suggest update fit enforce monotonicity x specialize update modify dynamic fuse take time specialized admm trend leverage strength solver fuse lasso order higher regime interior superior accelerated descent admm finally major strength propose extension basic trend filter software specialized package around c package package associate helpful reading manuscript discrete vary straight polynomially size moderate problem roughly speak flat htbp nk like moderate computation linear trend tune level regularization trend filtering path track computation quickly intractable panel figure point algorithms problem descent primal solution run proximal gradient accelerate proximal accordingly iteration operation multiplication operator truncation onto ball select large still intend top leave fit curve piecewise acceleration acceleration clearly basis matrix solution lasso proximal iteration time computation dense map see acceleration output close corner bottom leave explore non order apply formulation admm iteration descent operation full multiplication exact part capture piecewise although visually perfect piecewise panel reader illustrate iteration special visually indistinguishable exact fact iteration specialize admm operation per actually solve admm begin show size order magnitude specialize fast converge serious difficulty specialize admm steady within important prediction trend relevant proposal package implement feature prediction describe filter nx fit evaluation factorial filter k trend underlying calculate factorial algorithm corollary department fast trend develop nonparametric trend achieve minimax optimal way lack scalable fitting paper efficient specialized currently robust interesting filter software relatively point across trend k derivative operator write difference trend fuse variation th order trend order input knot generally knot reader jump ahead next future need evenly spaced operator structure recursive basically broadly speak motivation argue filter strength spline regression spline polynomial highly minimax optimal locally adaptive spline relatively inefficient computation trend filter minimax comparable spline focuses derive explicitly selection choose concern computational want trend filtering mean course still helpful th trend fuse already use dynamic equally special two direct issue affect classical
integrable check density belong check large class small versa present result smooth variance represent easy define choice stein estimation depend contrast stein estimator long knowledge mild shrinkage applicability datum drive show estimator kernel define kx x nn kx kx exist prove bernstein separable space rearrange therefore space theorem eq depend kx kx since depend kx use easy h n dt e dt follow theorem consistent kx n kernel bound verify kernel hold kx stein obviously kernel unbounded exist carry chebyshev inequality cost slow explore stein set deal estimate view mean restrict df x know replace assume stein seminal shrinkage estimator dominate principle classical notably estimator distribute like see kernel provide shrinkage connection estimate minimizer functional formulation ask question note regularize pose regression consider regularize minimizer lie span g term gram depend set minimizer eq shrinkage nice shrinkage wherein shrink towards require ill since estimator propose estimator constant yield rate n verify alternate choice refer correspond shrinkage measure quantify differ wherein instead omit observation deviation omit observation show shrinkage analytically minimize minimizer give q define nn kx df yield easy show derivative positive latter show u kx see statistic counterpart relatively theorem stem evaluate score guarantee satisfie n kx follow kx kx proceed remark show proof remark carry schmidt norm smooth difficult show x consequently q unlike take refer let q compact operator effect contribution expand eigenfunction empirical operator basis difference shrinkage contain particular account allow coordinate direction value wherein small coordinate large reveal wherein filter generalize alternate representation relate smooth operator formulation formulation x n use leave side xx xx minimizer jk remark suppose bound linear xx I ix bernstein q xx make observation universal bf along hilbert schmidt follow universal universal consistent guarantee knowledge xx cn g gx universal n universal achieve technique convergence shrinkage r leave shrinkage na expensive provide alternate expression n n proposition formula eq proposition aa x n I aa n unlike consistency comparison comparison outperform empirical mean dataset estimator obtain empirical regularize parameter whose obtain use different estimate copy j polynomial degree rbf lin rbf analytic form kx kernel dd begin simplest mean shrinkage comparison shrinkage origin gain small underlie follow generative represent wishart quite dd depict rbf realization uniform allow depend shrinkage determined figure important substantial appropriate knowledge incorporate close origin factor probability improvement iteration improvement estimator copy end conduct experiment shrinkage rbf addition whose loss figure illustrate difference improvement fraction proportion suggest amount improvement however trade become amount decrease intuition part compute shrinkage validation b choose leave cross shrinkage estimator increase shrinkage note non eigenvalue find outperform tendency actually support suggest one validation parameter improvement risk percentage calculate vary size tend outperform performance depend eigen dimensionality figure intuitive improvement size surprising especially function substantial small paradigm r choose classification window zero implement powerful non learn class positive negative window cccc statistically close rkhs kernel mean resp expect window employ counterpart uci repository bandwidth cross employ vote multi experiment test report error classifier consistently kernel wherein fit density unlike experiment goal different shrinkage estimator well initialization return pair significance via large log degree kernel highlight involve discriminative probability representation embedding approximate non investigate shrinkage end categorization support measure anomaly physics rbf fold cross setting report area roc mean shrinkage estimator performance compare summary r competitive estimate rl cccc cccc r segment tp cc classical stein phenomenon improve square show propose estimator shrinkage learn data satisfie also shrinkage wherein exploit spectral outperform importantly accurate performance focus mainly covariance cross tensor rkhs rkhs covariance pca preliminary present numerical xx yy unique center respectively covariance rewrite equivalent idea call trick tensor compare reconstruction rbf kernel dataset different shrinkage center center r omit detail perform center give shrinkage covariance generalize shrinkage use hadamard clearly sense mean considerably significantly affect summary encourage application method feature thereby center write nx nx nx center test thus kernel rgb xx edu university usa ac institute com university college house ar united bs institute reproduce hilbert central kernel algorithm also step modern embed stein family demonstrate especially small paradigm covariance shrinkage stein improve reproduce rkh measurable integral endowed guarantee existence integral kx estimate commonly key show improve gain advantage account preserve operation carry product rkh lastly require homogeneity sample curse dimensionality hilbert embed many mmd embed hmms kernel rule rely component principal component discriminant heavily operator kernel cluster basis early mean certain well nothing construct extent stein seminal show though stein square stein square interestingly stein outperform ultimately result usual relevant parameter definition ultimately look together remarkable counter stein phenomenon receive stein follow estimator space stein shrinkage rely work fundamentally stein seminal radial rbf see throughout paper draw independently identically measurable kernel subscript notation resp quality notation q kx r r mean shrinkage toward
plot crcr color forget crcr solid forget table sep crcr blue forget crcr width height jump ylabel title datum color mark mark option forget plot row crcr nan nan nan nan nan marks forget sep crcr width axis xlabel ylabel title color mark solid forget crcr color green forget crcr green marks mark forget sep mark option forget sep crcr blue forget sep crcr r ex ex sentence fill circle draw black pt ex draw black fundamental special map classify label equivalence set measure finite empty offer joint abstraction multi cluster maximally probable related estimating infer maximally measure program solve np relation order practical three maximally probable jointly mixed program mathematical learning given decide every instance classified element choose precisely every example decision pairwise cluster feasible relation equivalence introduce two pair albeit conditionally constrain feasible maximally estimate infer maximally relation rank relation mix problem mathematical programming appendix rectangle rectangle set subset relation structure associate pair secondly independent albeit conditionally principle observe maximize invariant optimum disadvantage require maximally probable know mathematic art solve branch set non priori probable would discuss equivalence closely equivalence segmentation measure maximally probable know np branch property polytope heuristic order word linguistic assess solution linear explicitly concentrate vector l finite every measure unique approximation form contrast nonconvex deep relation I set every depict random variable variable random realization hence characteristic realization random relation namely realization independence relation separate likelihood equal feasible infeasible measure probability define distribution family alternative respect ab ab ab ab ab form generality one finitely class characterize zero unconstrained introduce fix precisely exhibit pair secondly maximally optimization technique implement open notably estimate probable relation complexity write convex optimal solution time infer form classification problem estimate finite non element non empty firstly logic secondly characteristic arithmetic uniqueness ab ab ab obviously establish rest appendix vector measure separate optimization cluster pairwise subset characterize relation corresponding equivalence consist whose belong aa aa cluster relation logic integer arithmetic symmetry equivalence relation inference set satisfy correlation exactly branch cut exploit lin terminate estimate relation total first logic order general instance exactly branch exploit order feasible find heuristic chapter formalism map equivalence report computation core intel cpu ghz computation unbounded jump log true xlabel ylabel relative anchor north fill legend align format cd precision white red crcr nan red solid crcr nan nan nan nan green table e green table row crcr green table sep crcr e sep crcr blue sep crcr e color blue dash sep crcr white blue solid crcr blue solid crcr firstly classify image handwritten raw multilinear misclassifie linear thank multilinear error reduce random multilinear feature bad misclassifie image fall deep encourage work multilinear width axis xlabel ylabel format precision cd white forget sep crcr solid forget plot crcr white green dash forget sep crcr color dash table row sep crcr white solid forget plot table sep crcr forget plot row crcr forget crcr color forget crcr color blue forget crcr blue forget plot crcr xlabel ylabel test format fix precision cd red forget crcr color green dash forget plot table forget row crcr crcr solid row crcr consider handwritten digits mnist image contain digit show digit pair vector digit set randomly mnist test partition problem misclassifie pair test multilinear parameter infer equivalence cut software loop separate resort cut instance initialize lin solution rand variation information see lin unlike branch mnist solution image incorrectly elementary sentence estimate word occur word english replacement sentence english wikipedia sentence every occurrence feature vector index word occurrence less second sentence test word optimal use branch loop inequality metric distance sentence sentence statistic appropriately account sentence horizontal indicate normalize metric sentence see estimate sentence sensitive abstraction linear estimate maximally probable related problem infer maximally relation np np instance distinction motivate distinction partial evidence estimate broad research state jointly integer toward aside np order understand continuous relaxation necessarily state integer exchange idea learn community define definite convex eq partial quadratic semi thus convex eq example one correspondence establish define correspondence establish trivial respect estimate maximize eq solve impractical distribution approximate multi form polynomial variate variate linear irrelevant secondly iff let j f b x ab iff solution say equivalence pair define ideally otherwise property every feature invariant multilinear polynomial multilinear form width jump scale axis xlabel ylabel cd cd color marks option forget plot sep crcr green mark forget crcr mark forget sep crcr mark mark forget crcr blue mark solid crcr jump scale xlabel ylabel time title cd mark mark option forget crcr nan nan color marks mark mark option forget crcr
divergence rao read family etc density lee lee minimum distance method robustness inherently possess goal simultaneously minimum g several power divergence technique model divergence smooth later et measure divergence small divergence family outli family divergence family al family allow divergence divergence density base divergence connect read divergence smoothly end numerical illustrate estimator discrete natural unknown frequency bandwidth simple need article develop minimum estimator continuous set minimum hellinger estimator along use avoid smoothed model density obtain estimator divergence prove normality minimum interestingly minimum rest minimum estimator divergence continuous approach section minimum estimator detail present influence general divergence support suitable simulation study interesting discussion divergence divergence divergence limit give family family family sense density measure argument distribution f ss discrete technique estimating xt sg whenever influence estimate contaminated mixture divergence represent influence simplify let density lebesgue assume respect lebesgue sample model discrete choose density respect divergence immediate discrete model continuous distance simply frequency nonparametric density generating suggest construction appropriate density kernel density usually assumption rest process substantial derivation normality property smooth various complicated impose kernel survey take propose integrate smoothing justify minimize data need condition kernel vanish asymptotically play procedure play get consistent even hold work smoothed version entry derivative minimize datum routine estimate minimum minimizing behind substitution family smoothed lead prove advantage ordinary data scale get scale normal get bandwidth phenomenon prominent little dependence issue pseudo sample selected analyze mean value bandwidth choice r remarkably variability sake brevity little effect substantially impact bandwidth power originally divergence family divergence minimum kernel fact empirical equation unbiased kernel minimum model smooth us estimator simplify q spirit becomes estimate asymptotic result show estimate become produce identical estimator consider minimum minimum relation influence derive gx degenerate contamination suppose contaminate smoothed g x derivative present estimator give true belong e turn interesting follow integration prove divergence correspond define smoothed equation section assume rigorous minimum general definition property say density parametric independent say compatible support integrate p satisfied place assumption identifiable definition integrate distribution x parameter dominate minimum condition also asymptotic discrete define lemma see boundedness without provide us eq limit eq dominate dct hence markov finite distribution next contamination distribution shift degree contamination heavy freedom brevity present contamination shift mle get contamination estimator positive bad mle estimator ignore contamination heavy member moderately ignore contamination ideally location minimum divergence robust contamination significantly affect contamination variance contamination imply symmetric heavy tail divergence estimator similar contamination ii contamination small consistent pattern recommendation tuning would estimator contamination finding overall divergence estimator second contamination except minimum density short determination angle surface angle mean raw pattern use outli maximum likelihood likelihood remove outli previous section table minimum estimate positive positive outli minimum divergence continuous similar minimum triangular low corner affect finding light datum table unimodal outlier previously many suitable fitting usual see minimum obtained ht automatically discount unlike maximum measure region instability corner table develop minimum similar fully term derive estimator continuous section power case smoothed model exist divergence estimating kernel coincide kernel minimum ensure special justify function corollary true belong model e condition divergence become equation prove corollary side get represent derivative th taking expectation respect side condition first integral complete smoothing mean thus minimum coincide kernel family asymptotic become remark although calculation normal letting equation see crucially although discover role similar limitation classical influence estimator discrete influence divergence minimum divergence examine role robustness contamination approximation bias provide second taylor predict get scalar extension straightforward scalar influence effect special structure measure model influence potentially unbounded cloud one zero close preferred counter balance effect first illustration mean calculation yield bound imply constant unless decrease predict approximation counter balancing effect zero combine property empirical finding present member solely divergence divergence discrete particular efficiency decrease increase however efficiency estimator member tune logic combine asymptotic couple moderately highly loss tuning suggestion window substantial variation performance tuning applicability propose surely enhance situation literature like estimate select tuning consider briefly
reverse process reverse copy start interaction action interact environment forward reverse artificial intelligence offer encounter offline state art control ease burden user perform primary suggest intelligence also feedback part biological contain operation natural forecasting partly encode hardware biology therefore artificial intelligence use take device interpret way result user way improve intelligence system contribute preliminary machine learn prediction action generate electrical load potentially environment user biological copy computational human robot feedback platform arm extra subject ax di data acquisition signal modify send computer interpret send robot angular velocity temperature load design four term locate person fig platform therefore feedback find common device subject ask give inform accordance contain stationary ensure arm position movement work outcome subspace arm leave right ask separate control ask reach load threshold maximum threshold consider arm away addition provide source prediction subject feedback subject music throughout task volume music increase level arm also turn right alternate fashion contact feedback current robot reach effectiveness examine final task participant sound isolated electrical load robot system train acquire prediction task predict load advance load change load threshold datum determine level threshold main study incremental previous make world system divide increase decrease resolution angular position distinct dependent far divided counter vector indicate position direction feature contain unit store prediction robot arm standard technique update accord instantaneous load load product temporal abstraction q step abstraction discount electrical load second cycle roughly hz retrieve predictive reporting threshold acquire update learn weight principle continue assessment experimental giving find reduce load case load trial predictive feedback robot subject feedback subject observe experience drift experiment subject contact approximately arm contact uniformly visible arm contact difference contact leave predictive feedback robot spend angular position bin approximately bin feedback user spend middle angular fraction bin feedback case bin b robot significantly angular load load bin aggregate aggregate five fig subject load subject plot bin bin feedback robot spend fraction angular position record electrical arm motion fig aspect control note work area approach form feedback expect might never sensitive threshold minimal impact consistently feedback comes delay observation demonstrate feedback case load device bias user another desire operational logical load indicate improve feedback though cumulative load matter load already reaction reaction operation load load varied trial indicate device see status device particular highlight difference feedback area bin bin bin cumulative load measure feedback successful device spend two beyond bin feedback less frequently load spend impact condition device feedback great load note sensitive boundary indicate advance much feedback setting qualitatively observe subject contact level contact artificial parameter adjust device achieve objective learn feedback behaviour feedback converse provide feedback load sensitive load mathematical position load purely clarity intelligence acquire update period machine set period fig subject use feedback learn slight shift result user side correct machine study technical algorithmic prediction operational offline expectation computationally nature learn suited environment prediction subject requirement domain suit change persistent real pressure texture temperature body research wherein interpret largely grow attention match proxy location biological substitution device thought device take action prevent separate choice substitution information device hardware device encode prediction hardware helpful operation device natural thing substitution feedback training note et need perhaps minimize modern interpret device act e g response however think term substitution intend window research intelligence salient internal biological device device decision domain suit substitution match room area use user load distance appropriate specific intelligence intelligence learn take user help
long integer choose vanishing appear hold driven choice price pay translate degradation low since similar obtain appropriate minimax rate change inside construction near neighbor paragraph theorem improve algorithm observation accord density formally n number spatial link tail statistical correspond near sequence slice interpret spatial bandwidth small stress term omit make minimax tail drive several previous general assumption classification satisfie satisfy balance balance generalization neighbor drive result clarity obtain low enough next assume balance use suitable jensen inequality normalize denote large meaningful illustrate result small enhance discussion nearest investigate margin rate whose cumulative value translation argument study illustrate reach near tail margin k tail illustrate power law neighbor low q classification rate thin tail close purpose phenomenon tail previous classifier margin near neighbor represent degradation decrease performance neighbor underlie distribution theoretically several law distribution provide reach rule counterpart improvement compactly support point tune neighbor neighbor kernel base process preliminary estimator include fast neighbor j nx n q law location location still strategy nearest training c c gauss cauchy power theorem rule increase growth meaning procedure tried modify theorem become increase empirical subject study address balance risk leave future resp resp proof present inspire minimax testing refer comprehensive deriving measure ix bx mb opposite use density ball later lebesgue measure leave circle cm cm circle cm cm circle cm cm leave compactly fulfil fulfil sake convenience law couple argument soon briefly observe contrary study deduce lipschitz q study one satisfie involve inequality contain ball case tail fulfil soon meaningful soon denote constraint constraint optimize n lead end compactly point paragraph tail recover fulfil tail necessary show long compactly support ball keep q possible arbitrary hence distribution exist hold classifier consistent tail satisfy violate idea ensure tail assumption new whose measure proceed way paragraph satisfied satisfied negligible soon dx soon sequel great support long paragraph obtain large make slow rate value apply decompose assumption yield consider example thus j lead j side use keep notation satisfy equation equilibria previous optimize respect choice conclude proof want minimal lead excess natural balance slice proof j tail thank eq proposition lead sum n see bind term value conjunction balance equilibrium reach conclude compare classifier concerned use hoeffding n state upper error design nx sign nx follow apply proposition clearly misclassification term belong bias concentration write use belong attention sometimes quantity bx bx bx soon version variable bx eq n conclude proposition attention sample resp give incoming set discriminant particular classical measurable prove define provide analogy assumption equivalent keep study smoothness margin assumption quite eq minimal principle neighbor introduce two near decision minimax excess lower bound immediately minimax model knowledge model smooth never appear analysis reasonable think optimal discriminant conditionally spatial aggregate label difference section around subsection make satisfy excess logarithmic remove inequality associate asymptotically random soon remove logarithmic paragraph introduce tail derive near neighbor association argument assumption step major conditionally long appear proposition become yield plug last upper bound discriminant exist e eliminate introduce variable build part pick realization recall aim deviation introduce poisson deduce x nx nx nx nx nx nx study term n hoeffding q nx chernoff poisson nx n follow eq conclude q nx x nx nx nx nx nx x k slightly modify write n tail yield q section give vector law long supervise give near popular flexible machine community ability devote neighbor situation attention margin consistency derive sharp near end core numerous contribution literature recent continue view feature interest paper provide x retrieve law technical appear assign label conditionally provide exhaustive associate among divide family erm select minimize candidate context boost depth maximize decade excellent performance rely recursive dyadic partition state introduce improve refer theoretically bayes overview within motivation plug property overview neighbor correspond plug attract decade seminal integer training also seminal contribution receive even core general identify importance concern influence excess neighbor notion penalize plug entropy mainly asymptotic therein see away compactly support provide exist associate appear law boundary order paper compactly low contribution break first reach excess couple illustrate show classification result binomial one margin second step investigate near w secondary encounter vanish non compactly additional involved allow position marginal near bound classification different tail uniform see link establish even situation consistency open describe attention near neighbor devote prove reach convergence excess mild support location include discriminant model poisson model particular case throughout couple admit lebesgue hereafter measure resp real supervise classification complete predict decision interesting associate possible well q decision rule low unfortunately function benchmark hence possible excess also appear primary importance paradigm minimax define infimum classifier classifier minimal mass near distance build distance context near neighbor near neighbor measurable term estimator regression particular neighbor worth amount neighbor large process satisfy consistency detailed compactly observation core worth fall density strong assumption admit lebesgue density satisfy eq soon bound strong minimal assumption proposition away mass assumption support low conversely include regularity q possible involve involve omit relationship minimax rate consequence theorem exist conclude minimax rate support already high increase curse low tool nonparametric primary importance sake refer size study decade discriminant alternative binomial assume independent sample income predict correspond come classification draw completely conditionally long discriminant conditionally neighbor rule rule standard provide near neighbor discriminant complete detail neighbor near binomial binomial rely sample cope yet discriminant regard rate discriminant set difference smooth discriminant ok situation binomial weak log margin obtain fast seem margin finally applies bound compactly supervise compactly improper setting situation
enyi pairwise comparison suffice consistency retrieve rank relative consistency eigenvector os pair retrieve see similarity simplify laplacian sd preference specifically comparison rank matrix q obtain normalize drop positive multiplicative affect eigenvector result bound perturbation local perturbation rank rank retrieve large conjecture still assumption rely inequality eigenvalue translate seek inequality perturbation corrupt let fix could seek jk jk jk jk bernstein get bound bernstein get deduce n c ik jk equation p use independence time perturbation anti symmetry part diagonal indeed bound rest hold seek high probability p construction laplacian unnormalized close get unnormalized laplacian provide robustness unnormalize worse unnormalized regime perturbation proposition expression laplacian want eq get k I similarly k kn notice sx ks laplacian define odd eigenvalue know eigenvalue laplacian experiment show grow towards symmetric perturb get perturbation relate perturbation perturbation simplify notation l hence deduce constant absolute ensure large hence symmetric cf rank sort going goes order preserve make result compare result true simulation able suboptimal factor proposition perturbation translate start eq fix mean use union jk f c jk jk j first term begin bound hence apply n main comparison sample probability least notice f l lf sf f sf writing triangle deduce f term separately deduce f use therefore use remain notice cn numerator norm perturbation rank perturbation pairwise retrieve argue apart indeed far apart quantify far notice probability use odd negligible deduce similarly w desire real classical synthetic dataset pairwise comparison miss give item similarity nearby rank retrieve percentage corrupt comparison scale deviation interest consist comparison pairwise extract competition pair maintain rating update competition benchmark performance figure percentage transformation refine metric consider participant appear ccc corrupt miss blue centrality green line vary corrupted left proportion top synthetic dataset right ccc top various l city city city city city united united united united united west west west west hull west west hull hull west west hull set enforce retrieve e ranking comparison outcome home away game pair top roughly rank satisfied rank team spectral enforce rank correspond last figure ranking missing comparison index miss define monotonic technique similarity matrix similarity compute miss score eq whereas still show finally single corrupted outside unchanged together retrieve indeed strict strict matrix desired extend index index miss remain proceed except shift give comparison item rank comparison index either corrupt condition remain shift divide comparison unnormalize ccc corrupt miss rank centrality dash proportion corrupt vary parameter asymptotically eigenvalue laplacian characterize eigenfunction limit differential equation enable eigenvector eigenvector characterize x md kx fy dy fy dy fx twice expression solution differential root degree nonnegative calculation constant unitary numerically digit normalize proposition eigenvalue laplacian enable index uniformly calculation comparison rank eigenvector form pairwise valid detail extend input comparison option power robustness describe pairwise comparison assign compare construct pairwise recover observe rank still comparison corrupt miss spectral rank bind observe benefit formulation real rank competitive superior compare classical comparison item back seek ranking comparison order pairwise comparison two science player play player mark formulation music preference term one e rank fourth practice large noisy comparison incorrectly inconsistent classical vary website web page web link express link preference measure adaptively iteratively extract parametric likelihood metric algorithmic derive arc problem pair player understand point preferred item preference information reciprocal provide simple scoring difference example pairwise preference maximum usually fix technique rank classical item order item decrease distance chain reconstruct underlie ordering pairwise exactly solve noiseless case serial ordering small practice perform similarity matrix correct order item organize rank provable robustness formulation supervise equivalent additional impose focused minimum linear excellent guarantee case albeit summarize pairwise recover apply either corrupt incorrectly classical scoring method comparison os enyi independently suffice suffice retrieve consistency eigenvector os enyi effect order retrieve result produce supervise organize rank rank noiseless solution exact subset analyze os finally illustrate synthetic dataset base pairwise item item item close place item decrease formalize say impose coefficient row column without monotonicity permutation put call identity reverse permutation permutation strict strict permutation consist permutation matrix permutation reverse matrix strict order pairwise strict ranking rank item decrease able true comparison vector strictly monotonic monotonic nontrivial maximal contradict strict distinct first suppose monotonic reverse permutation addition identity locally reverse order row since strictly section spectral produce rank computing call comparison totally assume tie sort matrix recover item strict combine deduce repeat hence permutation strict ranking order rank candidate ranking apply enough comparison item item pairwise comparison rank sort either decrease goal item first rank refine also cccc row strict r corrupt keep strict recover right inside enforce strict noisy comparison cause spectral recover numerous first corrupted comparison multiple corrupt provide perfect recovery begin definition row minus comparison item rank index remain item write write simplify similarity impact corrupt score whereas incorrect comparison induce tie tie write get
connect induce asymmetric flow agent strongly network triangular desire primitive network sometimes matrix likewise contain internal appear edge sub contain network corner network receive sub upper combination show h examine steady behavior examine denote block example stochastic primitive upper triangular block sub network network limit eigenvector connect eq start establish norm subset long entry therefore establish contradiction assume frobenius attain denote ensure eigenvector happen attain strictly conclusion entry identity conclude consequently primitive power matrix tends fact claim useful hand side denote total verify gr gs gs gr gr stochastic group reach token represent scale internal information within similarly subsequent involve power pareto solution characterize fully explicit size agent likewise vector denote scale eq accord pareto cost towards conclusion network collect pareto solution across sub limiting extend agent uniform within individual limit total agent collection point network limit point sub argument follow identify transformation show within recall sub happen limit say within converge readily group collect limit key relating eigenvalue non right eigenvector equal establish establish ss ss sr sr obviously block kronecker product claim next observe follow equation condition uniqueness another possibly subtract equation however claim generic agent discussion whether agent limit right collect within sub group agent hold substitute find dynamic network evolve accord statement function assume noise size hold q triangular vector independent group stochastic recursion study argument correspond sub group need evolves instead introduce canonical except extend quantity side recursion appendix start agent group besides moment strongly agent bottom drive hyperplane wrong run logistic topology weakly sometimes elliptical separate data class concentrate outside outlier belong locate away obviously weakly connected agent group outlier advantageous suffer represent connect green weakly influence agent employ term coordinate approximate weakly connect strongly connect strongly large curve weakly compare boundary curve curve infer concern limit point associate streaming datum via white weakly topology assume agent agent obvious get agent simulation illustrate profile noise agent db likewise theoretical agent sub theoretical value agent sub find simulated cm examine learn agent weakly relationship converge agent limit reveal agent topology beneficial reduce impact zero variable step b norm sufficiently condition substitute give introduce proceed spectral purpose norm last next verify establish eigenvalue compose eigenvalue associate eigenvalue eigenvalue large large eigenvalue algebraic must occur block algebraic geometric small matrix follow size eq choose claim replace rr let scalar expectation next error proceed verify substitute return bind use jensen inequality introduce combine use use introduce scalar step follow scalar lastly arrive expansion eliminate since last I sr sr rr sr b sr I expand stochastic emphasize jensen conclude simplified lastly consider recall recall square arrive simplify subtracting block identity agent define block product verify equivalently operation operation stack examine simplifie canonical recall primitive exception equal correspond conclude canonical decomposition strictly eigenvector expression limit decomposition relation proof order q since dominate small substitute start q one moreover early identity th clear agent performance agent group arrive equivalently lyapunov satisfie obtain agent need compute diagonal return step introduce exploit block height department electrical engineering university mechanism agent reveal interesting flow topology result exchange mask certain totally agent determine arise example due attack failure critical work connectivity topology adaptation combination outli connect graph exchange pareto solution cost global minimizer useful developed diffusion strongly path self loop technical minimizer large size learn ratio agent main article examine behavior affect topology necessarily shall examine effect flow consist one back setting important arise attack neighbor inaccurate feed back behavior asymmetric information exchange model second example context interaction regardless regime flow medium third twitter small subset bias asymmetric information exchange reasonably connect connect reveal relationship agent outside influence scenario arise attack asymmetric result failure critical contribution help strong topology combination weak connectivity beneficial namely reduce plain letter letter trace radius matrix besides denote derivation consist nonnegative edge connect agent scalar datum receive zero network say agent direction say strongly flow agent possess loop strongly emphasis agent neighborhood agent denote assume default include neighbor coefficient condition stochastic connectivity network imply primitive property primitive matrix eigenvalue lie circle right normalize entry add eigenvector associate denote variable collaborative combine diffusion positive step iterate adaptive advance far ahead diffusion g individual use convex aggregate unique agent mean
outli dense eventually subsection turn truth problem coordinate descent subproblem column optimize convex q second subproblem base give alternative descent algorithm descent initialization convergent sign b I production stop first show converge point would actually denote part could convex combination optimal global aim arrive bernoulli complementary matrix submatrix simplify outlier appear outli successful index outlier index measurement reflect acceptable determine entry apply usual formula set equally insight advance estimation measurement show suffer bernoulli accurate fortunately inspire way combine remove entry lastly apply succeed would let consider recover contaminate rank factorization base start derivation thing contaminate corrupt appropriately derivation th column appropriate fit regularization eventually public dataset dark line ground line line achieve high efficiency ground efficiency top dr precision poor f specific tuning assign dr pre measure dimension fast implementation inexact alm alm speed previous improve non increase matrix type simulation low gaussian noise matrix matrix gaussian matrix indicate bernoulli matrix generate random matrix dr measure q correctly detect indicate corrupt detect precise three present tune observe high accurate meanwhile cost conduct two video lot activity foreground frame activity illumination equally left scenario frame row box last row run video simulation strongly validate robustness regression detect outlier contribute achieve rank outperform art look lemma one form eq entry except obvious contradict feasible obvious eq chinese china ie edu wu technology china wu ac outli additionally outlier bernoulli help approximately solve high popular algorithm extend factorization recover component extensive like aspect realize accurate estimate computationally intensive certain prevent robust estimate paper aim estimate regression denote denote signal denote fact assume noise noise consider outlier totally occurrence assume bernoulli operation eventually bernoulli accurate fortunately find certain level estimate efficiency extend apply massive modeling recognition rank contamination recover component equal understand
mu selector appropriate alternative non knowledge bind entry show depend report propose yet pursuit need focus subgaussian show omp analogous consistent error variable setting satisfy remain practical enabling although suggest subgaussian less independent component remains minimax state however answer situation two question assumption order cone attain close contrary knowledge compute focus subgaussian matrix appear extend suitable deviation property check eigenvalue solve convex devoted selector programming course compare selector furthermore bound fast uniformly subgaussian subgaussian subgaussian product subgaussian deterministic random subgaussian random subgaussian furthermore assume condition selector gram shall form consequence latter indice complement cardinality subset cone restrict constant selector start follow meaningful term prove lasso prove addition sensitivity fast regressor appendix base computationally section bound knowledge appear theory take efficiently polynomial feasible set appendix main assume parameter assume risk admit appear assumption simultaneously coherence see show subgaussian row well bound extend fix assumption probability although theorem result spirit minimax give bound feasible inspection reveal valid provide bound selector next state somewhat probability constant base selector prediction design slow convergence solve issue specific obviously reduce however shall additional even program simple programming algorithmic therefore detail brevity write set note problem q tuning constant program justify penalty parameter separate h ccc bias rmse pr rmse l ccc rmse pr report selector outperform option estimator high bias benchmark mu selector selector nonetheless feasible reason case align property error report qualitatively display respect model selection separate zero h ccc c pr rmse ccc rmse pr high dimensional regression mu selector programming estimator time practice use theoretical namely optimal somewhat bound nevertheless reduce linear helpful support gene grant appendix give appear diagonal square denote zero follow prove n tw w n tw tw tw pp eq constant give let random subgaussian together independence random subgaussian union subgaussian random analogously use cauchy subgaussian exponential imply union yield b bound preliminary feasible problem two theorem event lemma lemma feasible consequently note event belong let feasible together eq exceed since intersect probability least corresponding finally lemma initial event probability plug throughout proof probability hold imply cone definition sensitivity recall q proof definition finally q collect property restrict constant coherence constant cardinality provide useful assumption q relate ss trivially collection convention solution solution solution minimum solution obtain divergence omit gaussian proceed constant include denote hamming zero see denote component equal constant obtain next condition example school es mod universit et paris centre en paris linear sample new estimator selector turn noisy regressor introduce sense efficiently programming estimator numerically model design noise matrix estimate example reduce linear regression error investigate size show presence severe procedure particular
concentrate small hence efficiency run chain mean give coherent selective spurious couple nlp property construction truncation motivate principle flexibility facilitate fully develop remarkable parsimonious achieve accurate validate pre covariate sample serial correlation capture multi modality may desirable remark perhaps readily adopt develop strategy general adapt generalize graphical em state k direct prior multiply divide cauchy continuous differentiable ac pg n arbitrarily continuity integral make arbitrarily implication direct slight respect finite dominate state n mle consistency either strictly mle density limit constant case generate kp note identifiable model long singleton nlp definition imply lemma linear nlp nlp take form rd cg n prove result penalty sufficient hence write let x z k dominate obtain eq show choice g k k give explicit product integrate dominate algebraic integrate prove continuous grow dominate adjust divide multiply r x expression density rearrange conclude l kx bn n n n mapping k x sufficient nm ng n algebra k ng k theorem combine k k informally point need trace converge satisfied grow converge k contradiction e increase hence monotone improper complete indicating without suppose second derivative hessian rearrange obtain approximation convex maxima occur maxima mode occur jj j kl incorporate assumption mle solve jj pn pn intersection contour length shrink evaluate generality jj converge n far pn mode denote generating argument lie add approximation kt nm third bound finite term leibler law number kt pe n largely mode ti factor case manner probability note pm pm pm follow denominator apply pm km pm te pn ie pm pn pe pm k whole regard non proposition desire iii adjust section exposition indicator e np ip n prior straightforward algebra eq il state eq proposition ii p p n orthogonality give q normal v ji laplace approximation around pn pn pn pn express minimize numerator decrease mass denominator go apply q survival equal discuss min md md min side finish notice mp continuous notice min side md min h min mh md mmd min md constant marginal multivariate function decrease integral plugging dominate apply I apply I md I sample I gibbs non inverse z sampling follow straightforward denote truncate ji z l jk adapt algorithm address exclude sampling write union fortunately increasingly set cdf monotonicity convenient log determine denote q monotonicity z monotone evaluate inverse strategy guess continuity conduct interpolation dominate hence drop equal z z z determine interpolation update either continue guess often quite research remark university department department achieve possess appeal dimensional use estimate spurious quasi depend spurious differ mle theoretical constructive practitioner perspective enable extend notable high benchmark hyper validate magnitude remarkably pre contribute estimation selection actually dimensional develop good challenge appeal discard spurious fast extra important consequence density dimension denote fix set nlp density zero univariate prior pm km nm nm tf minimize leibler kl pn contain spurious hence truly ratio shall show lp nlp regression fully probability decrease show induce asymptotic particular light yet relate strategy fast role pm consistency consistency prior manuscript characterize imply address practical justification add modality performance simulation expression datum grey top truncate bottom intuition vs possibly prior estimation grey line preserve set significance combine truncate exclude line assign truncate coherence detect prior mass concentrate around decrease truncate cauchy express marginal integrate cauchy go nlp mixture assign roughly induce shrinkage nlp k often always induce integrate likelihood nlp lp k dd nm kl assume singleton kk k tm integrate dispersion term converge require mle see include identifiable l n converge typically asymptotic proposition grow rd k n x kx bn x kl generate n n cc strongly estimate satisfy condition minimize kl divergence ki ki ki ki cn k ki pn pn ki pn pn diagonal ii quasi bayes mle generating prior ti pn priors eq residual ti pn pn set spurious become interestingly term affect spurious asymptotically correspondence simplicity omit truncation nlp manner give let truncation define nlp p nlp convenient later truncation define nlp truncation gamma inverse gamma obviously affect p nlp truncation representation p p p additional greatly sample component univariate prior product freedom set function normal truncation trivial cdf survival quickly computation univariate right nlp form behavior depend nlp h min min min bayes depend penalty give small show nlp h p multiple truncation tail functional depend solely whenever recommend significance give analogous predictive yet possibility unit regard draw analytical threshold section posterior linear simple nlp truncation highly straightforward algebra truncation may challenge apply truncate draw truncation representation provide adapt unknown dispersion set hyper multivariate rectangular efficient algorithm serial implement sampling package important property truncation negligible assign often negligible separately posterior eq square represent eq chi algebra lose variable require multivariate assume appendix mh gibbs p efficiently combine search token require os posterior corollary h h require algorithm k algorithm yet multi induce benchmark scad default assign beta binomial whenever adapted never visit benchmark prior parameter beta binomial lasso scad penalization cross respectively supplementary material attain method help covariate contour top middle bottom e e simulate bivariate normal first compute possible integrate function package full model rounding draw obtain mass shift resemble elliptical table auto correlation reflect
parameter algorithm notice sampler effectively transition stochastic initialize sampler kt kt ft ft kt kt synthetic latent ft synthetic recover short audio physical time play simultaneously different audio fourier transform ms overlap yield matrix hyperparameter nmf figure variational proxy component right great see learn structure activate implicitly reflect pattern activation performance sampler audio report run twice sir sampler partially due discover last interference assumption ideally regular method conjugacy directly mean field variational degenerate delta effectively map implementation detail number sir structured field inference beta process optima reasonably hide blind task par exact hyperparameter channel mask activation particularly leave factor desirable capture mask prior encourage mask mask binomial model binomial efficient incorporate structured field non factorization gaussian study literature nmf process conjugacy variational relax conjugacy approximate result synthetic propose reasonably hidden approximately matrix refer activation application domain music recommender system hyperparameter latent offer put component activation allow focus model negativity impose address process nmf binary approximation variational inference gaussian computationally intensive computational burden field variational process strong dependency among latent local develop attempt utilize address first nmf inherently derive framework blind nmf model kullback plug approximation beta kl hadamard properly spectra latent activation binary mask formulated set large easier introduce auxiliary random making auxiliary enjoy conditional conjugacy helpful algorithm inference divide variable structure distribution approximate activation factorize form completely mask correspond local gradient global parameter intractable gradient collapse make forward quantity define ft h jt burn per distribute ft ft ft kt kt kt
function cv cv outline use interior demonstrate every iteration give ip optima costly optimize extension convex hull method apply machine possibility relate rgb rgb rgb false pt false title r thm thm thm thm thm thm convention exercise question develop selection save practitioner tune manually particular implement test fold extend complex compare extensively ill generalize datum crucial regularization smoothing svms characterize follow convex program reduce l general purpose regularization generalize description aic criterion recent also discover develop homotopy svms tune selection classical regularization scheme constant choose minimize subject offer outline practical tool tune manually bring automated examine rather characterize approach simultaneously purpose write ridge approach apply benefit minima approach rigorous see solve value value think speed computation svd solution search decompose low provide constraint parametrize moreover form hull easy verify set hence true characterize entirely polytope together follow relationship relaxation solution counterpart prove bounded pass construction hence ip kkt fix response notation machine optimization set problem optimal among kkt simply solution set relaxation immediate comparison require grid step define high qp latter
landscape subsample anneal landscape factor langevin effect subsample anneal speedup simulate subsample although many linearly choose moment probability energy simplify boolean generate unknown probability generating ball ball hyperparameter ball conjugacy since indistinguishable project blue ball posterior multimodal beta prefer segregation time variational jeffreys bias ball ball effect subsample subsample langevin intrinsic limit red single pde drift depend intrinsic schedule fast mix schedule model total thus absence datum energy mode subsample linearly shall linearity preserve energy inference behave like subsample annealing sized colored stepsize generalize model naive mcmc size anneal behave anneal speedup toy around two mode limit slight limit energy infer rather inverse barrier assume within mode entire mode separate barrier proportional appendix proof interest time arbitrary energy necessary anneal schedule proof thus anneal exponential feature already subsample equivalently anneal difficulty significance learn combination easy infer anneal relate anneal subsampling dynamic equation extra fix subsample annealing exponentially cross categorization partition r categorization categorical mixture feature partition learn hyperparameter partition categorical feature inverse grid wide range hyperparameter partition hasting sensitive significantly value us census dataset dataset include value categorical train log score compare empirically subsample anneal well fewer bad partition subsample anneal consistent learn mcmc sequential initialization initialization toy empirically sequential initialization deep hyperparameter poor initialize state subsample anneal address slowly subsample subsample allow tradeoff speed stochastic schedule inaccurate accurate slow much simulate simulate wherein landscape anneal landscape vary temperature subsample anneal portion yield posterior subsample size employ van multiscale monte avoid site inference hierarchy compression phenomenon liu generalize wide subsample annealing anneal technique suitably apply problem proposal clustering hasting carlo particle streaming dataset addition grow subsample minibatch approach variational inference practitioner increasingly subsample annealing cope principled heuristic demonstrate improve real subsample anneal speedup subsample annealing improve cross categorization provide improvement landscape section subsampling offer energy gap moderate sufficiently sufficiently relative barrier still observable subsample annealing may rare anneal poor hope subsample subsample annealing provide anneal help dramatically intuitively hypothesis grateful red ball resp leave red blue removal symmetry intrinsic define moment scale inspection subsample anneal schedule anneal approximately scale version dynamic effective length anneal lemma system sigmoid mix state barrier height dynamic low inference homogeneous transform case state transform anneal schedule increase equation sufficiently bind simulated clustering learn subsample datum portion inverse schedule gibbs temperature e subsample final state jump energy anneal subsample annealing subsample improve million census rating categorization simulate landscape anneal anneal recently model categorization infinite model rapid structured result stochastic sgd towards inference scalable inference method find site implement applicable discrete contribution extension sampler easy implement yet model subsample subsample grow subsample extra schedule terminology simulate anneal treat subsample indeed deep mathematical connection indicate anneal approximately simulate anneal stepsize langevin prior like subsample move towards increasingly long schedule assume linear anneal section accurate sample draw restrict exchangeability index assignment assignment part point possibly index set immediately classic finally nonparametric increase crp polynomially become grow subsample schedule gibbs sampler respect strategy initialize assign sequentially sampler
profile player particular game player profile gain player time strategy adopt profile nash equilibrium ne change change formally speak three question arise ne actually ne iii ne reach completeness presentation interested reader ne remain player strategy decision game ne may pure ne finite reach optima agent moreover search local polynomial pls polynomial game interpret shift gain favorable strategy player optima detection vertex ensure nash equilibrium community function represent vertex modularity gain community gain ensure detail towards nash equilibrium confirm let ne furthermore ne reach possibility bipartite direct conduct experiment hand traditional hand four produce instability compute modularity higher produce local nash sound community modularity vertex reach ne modularity modularity except appropriate interestingly stability good result benchmark checking goal partition event accord known identify group bipartite top edge associate event eventually highlight red blue author three beyond present overlap overlap namely modularity event partition community produce could member early entity nearly author appear overlap event cause compare author blue green community design vary community compatible nash figure nash reach step display tendency prefer association another unstable modularity become community author detect community assign community community number first value column unstable horizontal observe horizontal modularity third nash reach one facebook account tag different tag tag person name link obtain overlap tag individual tag ignore individual display community pass understand individual life period link life g community member friend life particular contain individual first except person community include constitute partition would one community isolate friend community thank present limitation conversely thank individual finer find unstable community suggest reach begin modularity nash amount increase facebook quite visualize address facebook overlap figure situation reach equilibrium display situation separate color person color nash community equilibrium community community lot lot person potential situation equilibrium community community whole person modularity increase lot nash low automatically spread goal vertex nash equilibrium reach second computational bipartite scientific sized bipartite benchmark dataset describe display company capital member hold vote find whereas interesting overlap community modularity equal value scalability test relationship order library http www scientific paper paper extract second unstable vertex demonstrate unstable one achieve objective knowledge appear suboptimal several criterion define partition sound nash equilibrium yield remain still heuristic offer modularity entropy nash difficulty obvious seem visualization dataset like community linguistic hard distinguish regard visualization raise challenging hypergraph diagram visualization high community read display initial overlap need bipartite facebook must narrow display row vertex vertex constraint predefine number traditional like knn modularity modularity use aware article detection proceeding assignment situation situation take every problem great introduce quantitative criterion example number community limited part may address simplest hypergraph many mention particular complementary bipartite profile distribution community else distribution community team company community evolution quite body field interesting since highlight social appealing temporal evolution cause network facebook evolution scenario dependent service political community partitioning vote overlap network nash equilibrium less detection method contribution display detection stable solution nash equilibrium community unstable membership find modularity vertex collective research enhance herein limitation present take community study unstable situation detect community contribution provide external within would service reality operate usa social focus method aim optimize call modularity maximize community inter completeness problem heuristic optimum paper introduce optimum necessary condition nash equilibrium function simultaneously partition visualization interesting experiment either graph medium sized social field survey topic detail algorithm world overlap focus mainly previous article extract overlap bipartite use bipartite oriented partition call benchmark author represent partition address yield partition community modularity optimum potential clearly closely take vertex community stability condition equilibrium enhance nash equilibrium acceptable powerful semantic address partitioning calculation mathematical modularity connection community minimum link external community extract partition weighted graph comprehensive art report adapt formulae modularity bipartite classical spectral genetic analysis strategy often extraction use design combine spin interaction annealing optimize local treat dual involve weighted link rare focus community method extension clique representation remain bipartite orient graph overlap community property membership decide apply iteratively modularity function merely stop overlap community change stable result article accord strategy community detection problem vertex assign nash modularity bipartite recently several modularity bipartite analogy modularity yet modularity modularity hereafter let adjacency diagonal also modularity community edge kronecker equal hereafter consider graph weight transformation show modularity bipartite graph column ij formulation introduce edge belong modularity yet detailed node community characterize apply partitioning result without edge modularity modularity graph numerous researcher remarkable build modularity community replace vertex represent modularity optimum corpora dedicate overlap vertex several community
strongly conclusion theorem sdp ability include write mc c eq q w fail outside block large sdp weakly constant slightly define weak replace respectively q translate imply hence sdp consistent proof appear relaxation mle apply sdp tight relaxation coincide sufficient translate read interestingly establish sdp specialized result predict success sdp analog mention somewhat base adjacency call eigenvalue truncation impose degree though report outperform let collect general regard zero element sdp proceed solution use conclusion second unique generalize optimization maximize consequence statement order cf sdp sdp construct ij k k j extend ks ks ij ij j moreover bm bm dominate block consistency generality low concentration discard sdp sdp variant summarize sdp adjoint kkt like obtain cluster matrix primal imply psd equivalent condition unique e ks mb I e mu primal solution optimality condition triplet satisfied linear eq accordance rewrite cs degrees community subgraph row column word vector sum take p also ks b k choose feasibility translate detail remark involve let projection feasible least hold feasibility use verify assumption ks ks u sdp next hence infeasible carefully increase block kp construction work balanced plant start k cn mp bernstein around simplicity effect diagonal I n defer assume enough turn imply q e sdp nb operator accordance k k case equivalent act k expansion block later read equivalently b ks u note analogue indicator table satisfie strict verify dual feasibility j ds proof I chain definition distribute assertion assertion sufficiently right inequality sum hold hence take satisfy enough imply eq need drop corollary low imply complete get form replace adapt first method sdp reasonably implementation admm start language affine symmetric th everywhere else element equal everywhere else variation affine subspace derive implement project sdp admm easily projection onto eigenvalue sdp balanced ideally suit require estimate mean truncation ambiguity fix abuse compute sdp introduction enforcing tendency sized block ideal sdp enforce flexible flexibility disadvantage histogram estimate block estimator ni ei q eps eps eps eps eps eps eps row plot experimental sdp sdp sdp amount spectral cluster choose graph operate tuning parameter value drive hard decrease separate behavior four community recover output sdp relaxation scatter row lead eigenvector clearly provide sdp recover exact sdp geometrically organize figure space point superior eps eps agreement information monte carlo replication relaxation sdp dominate outperform eps eps eps eps eps eps sdp sdp go block block prediction theorem conjecture regard sdp always community strong fall however remain weakly replication sdp sdp boundary difficulty recover rd sdp see show f reconstruct sdp severe behave sdp exact difficulty reconstruct see error result close see eps finally show sdp adjacency estimator row order addition sdp block nearly equal result sdp sdp sdp true unbalanced equality one poor representation htbp eps eps eps eps eps eps eps eps eps k eps eps eps eps eps eps eps eps eps eps summary good block histogram flexible unbalanced block desirable balanced sdp sdp nonnegative relationship treat relaxation balance plant various define tight analyze show block respectively sdp strongly class also conjecture outside weakly remain sdp mixed community direction simpler plant sdp obtain adjacency around turn growth new rely sdp sdp hard dependence latter tuning lagrange duality make general empirically outperform adjacency spectral call reflect guarantee dependence seem inherently noise doubly cone equality constraint outlier alternative relaxation far another hybrid replace function explore connection unbalanced sdp relaxation zero surely mp mp c mean bernstein eq rest tool detection involve infeasible problem new programming problem sbm relaxation sdp relaxation sdp guarantee carry however show sdp recover community wider relax consistency condition sdp thus class applicable derive primal construction sdp suggest sdp sdp evidence cluster real relaxation tendency balanced sized ideal tool histogram popularity literature throughout relaxation connection community attract physics sbm widely community analytical connection challenge assignment art accuracy rely start method popular spectral difficulty help even accuracy semidefinite sdp sbm optimization hope like cluster likelihood way optimization easier analyze also relaxation make noise outlier see drawback sdp sdp solver sdp continuous advance relaxation sdp tight relaxation relaxation unify connection empirically derive admm keep reasonable side throughout equal obtain sufficient condition exact sdp sdp wide sbm previously literature current sdp relaxation implicitly strong cf whereas sbm requirement success previous sdp relaxations sdp sdp limitation success sdp primal construction suggest relaxation flexibility complexity successfully recovery see sdp relaxation instance work inspire trivial extension general complex sdp doubly nonnegative also sdp strongly sbm divide sdp relaxation strongly weakly sbm purely mix sdp focus additional assumption equal class sbm simplifie since q recall depend q albeit adjacency mle obtain sbm desirable consistency sense optimality hard relaxed computationally restrict otherwise induce alternatively derive relaxation relaxation admissible kronecker convenience node recall feasible note relax positive psd column compactly since propose sdp relaxation recently first xu slightly remark relaxation via since psd addition affine thus main focus relaxation replace sdp directly tight constraint separate affine break though
high shall derivative property value conjugate mapping use follow hermitian taylor scalar order note jt kt expand order shall equivalence hermitian operator transpose evident hermitian subsequently second term eq term expand term expansion augment presence augment important consequence augment hessian numerical hessian minimization hessian analytic numerical solution parameter problem method order expensive store view c jt kt rewrite inversion h q newton eq substantial simplification complement remove operate reader subsection derive calculus apply find minimize weight gradient become substitute vector see calculus also rule complex original base difference arise formulate give error minimize setting reduce system equation way generalize express rule hessian become error vanish equivalent real calculus hessian real calculus counterpart descent derive perform computation field product greatly simplify optimisation procedure serve complex optimization solution apart address newton acknowledgment dr take discussion calculus wise definition weight calculate traditional nature substitute eq proposition example real array practical solution often calculation analytic address issue propose novel calculus derivation optimization transform domain practice calculus chain correspondence hessian counterpart usefulness calculus simplify derivation generic corresponding complex design calculus analytic newton numerous physics graphic processing communication reduction procedure accord procedure typically rewrite real take variable treat analytic framework often calculate show save burden expression calculus field conjugate algebra solve novel calculus rule value enable derivation error carry field transform problem elegant derivative calculus instrumental basic counterpart invertible first operate augment propose obtain operate conclude enable technique traditional q q difficulty calculus chain rule derivative j rule derivative intuitive rule chain calculate leave derivative focus lot consistent systematic component calculation pseudo make derivation equivalent elegant approach calculus respective applicable gradient base term gradient prove follow also comprise novel eq matrix n jacobian conjugate jacobian convention convention j approach q
filtering minimize optimality sketch doubly random walk general jump cubic second boundary equal inequality interpolation constraint segment change leads estimate trend whose neighboring neighboring coincide neighboring prescribe fuse trend trend resemble detailed consistent recovery change change regime tend infinity keep bound magnitude slope show location specific change perhaps change elsewhere point interpret boundary within detection result slope lead order interpolation therefore choose alternate slowly boundary outside neighborhood variability lead note walk grow boundary away random recover discussion sec natural consistency sign idea change segment detect point therefore filtering detect segment trend filter new proceeding go recovery situation successfully neighborhood location spurious appear actually estimate change due proximity one blue line correspond estimate mean nearly dash solid denote signal goes stay slope purely expect presence trend change point segment change point change remove segment move close end remove presence estimate point trend filtering show far monitor scheme remove point trend piecewise trend noisy provide intuitive succeed build interpretation corollary trend total tv mean tv suffer detect objective paper suffer interpretation integrate walk avoid fused point trend dataset generate non stationary piecewise piecewise way via variation tv counting measure method impose penalty least maximum trend convex tv piecewise without penalty impose translate regularization balance sum simplicity univariate trend filter generalize spline detect piecewise method dimensional denoise fuse lasso filtering fuse rigorously
artificial trajectory cost expectation probability respectively cost go illustrate scalar estimate expansion follow simultaneous value parameter rademacher random rademach cost give sf gaussian sf policy respectively condition accelerate approximation scheme iterate gradient base rademacher go instant component variable stay within requirement descent direction along sf independent hessian invert hessian state matrix ct bt tx identity would denote identity dynamic policy constant w u measure distance th denote radius contain least finally expect build trajectory lemma bound bias least describe difficulty establish asymptotic difficulty bias contribute recursion equivalent update towards critical establishing assume negligible particular ordinary differential ode discretization converge equilibrium ensure remains propose none stack none gray plot area legend forget plot none follow dynamic define linearly parameterize policy discount truncation trajectory artificial trajectory expect carlo go minimize xlabel iterations ylabel col comma exp ylabel sf pos north east index blue table xlabel ylabel legend entry pos north east thick red table index col comma exp thick col comma exp smooth order grid run sequence algorithm run project project run curve sf approach benchmark variance sum discount cost go recent direction actor notable unlike carlo policy resort value expect sum constrain discount expectation sensitive mdp constrain formulate technique x denote solve enhanced variance cost follow ascent dual multipli cost classical estimate lagrangian primal ascent lagrange multiplier q project risk criterion artificial criterion bound lagrangian operate project search batch simultaneous hessian order sf simultaneous scheme difficulty establish bias evaluation future plan condition horizon introduce follow give preliminary state cx u cx fx w follow trivial continuity assume u continuity use lipschitz definition continuity plug iterate proof give artificial trajectory affect transition expect return give x I equation u cx cx I continuity l dy iterating bounding truncation add x bound b dy l u dy l l end lemma w w reformulate w n expectation w end triangle lead contain width c p n observe exist differentiable evolve continuously markov policy irreducible bias rl algorithm visit number horizon impose ensure negligible proceed return finite denote ordinary q proof asymptotically eq projection operator ensure ode stay set equilibrium regard govern theorem prove correctness rademacher easy rademacher easy true analyse set taylor easy vanish discretization ode ode lyapunov follow claim pp set converge asymptotically equilibria ode asymptotically stable equilibrium govern govern hessian tx jt mt tx employ taylor expansion derivation refer proposition lemma theorem arrive update true see discretization ode govern converge ode albeit similar use sf perturbation I j proof proposition rl action monte policy search policy order newton incorporate hessian cost simple continuous paper stand field control infinite discount decision address batch set trajectory access simulator formally tuple state action policy objective develop policy control attempt policy cumulative discount govern develop descent cost go obtain batch discount set advantage henceforth refer state action space go parameter use gradient possess bias well know simultaneous two first second popular simultaneous perturbation simultaneous descent sf estimate parameter sf usefulness stochastic action set metric space policy mainly reinforcement gradient rl least policy extend optimal batch mode rl ensemble rl see aim scheme approximate gradient perturbation scheme minima observable via irrespective simultaneous perturbation method perturbation functional differ simultaneous perturbation scheme sf hessian sf illustrated operate perturbation step cost go perturbation see value
mathematic apply mathematics south road south centre road cb extend divide arbitrary simple analysis deal case pair coordinate apply covariance scale expect change width matrix normally effectively employ fisher generalise straightforwardly include error carlo data include cast software matrix fisher become widely statistical low limit er rao estimate maximum given jointly determine fisher much compute distribution posterior describe distribution sophisticated experimental sophisticated forecast surface space implementation ti fisher useful proposal survey survey scale dark european space purpose basic fisher matrix formalism case uninformative taylor expand constant irrelevant discussion constraint likelihood third curvature fisher case deal variable straight ad hoc axis ultimately combination average fit axis slow new error extract population formalism formalism treat discussion straight line fit bayesian example remainder follow describe arbitrary describe formalism particular conclusion replace equation throughout paper formalism taylor derive generalise principle correlation formalism cover represent extra may measurement simple wish observe amount depend assume expand condition expand integrate parent gaussian main limit consider datum limit text affect concerned bias slope coordinate bias unless prior taylor essentially assume integrate z nz ii form covariance include element intrinsic scatter matrix give final define collect algebra marginal algebra calculation simplify look covariance compute use standard formula find eqn replace standard variable derivative model uncorrelated find correlation recover key observation usual matrix uncorrelated width propagation variance increase main interpretation make likelihood surface generalize straightforwardly replace covariance example illustration diagram apparent length correction colour act around include instrumental dark matter plus correction colour relate interest dark energy hierarchical description principled galaxy may large error colour correction galaxy mis galaxy couple error apparent investigate survey find correlation theoretical could couple arise make maintain error divide error error across pair term generalise modulus matter chain technique simple distribution range modulus look contour generalise good accurately bivariate good agreement orientation actual offset accordance depend coordinate analysis general
meet game theoretic learning instance engineering fluctuation thus wiener lipschitz strength player observation regularize admit strong mind derive evolution special decomposable convex support remain govern k open dynamic quite interpretation recover learn study variability player impact vanish sufficiently correction follow stochastic dynamic eq denote population specie environment fitness coefficient impact weather population evolution account account jump besides fundamental difference term drastically highlight contrast evolution reinforcement important first evolution map correction player also summation form drift determine wiener process substitute prescribed compare eq conclude constant dense continuous show trivially aa necessarily solution interior logit nash equilibria depict payoff vertex wiener reinforcement converge nash equilibrium analysis player payoff environment evolve player involve consider player action reward context compare player payoff payoff could nature advance player integrable stream payoff equivalently originally context agent past action overview reference main seek reinforcement notation focus integrable stream payoff wiener process noise assume player payoff extend consistent payoff arbitrarily assumption generating evolves induce carry primal dual denote terminology reflect negative strictly provide see primal primal bregman divergence express regret term grow consequence iterate logarithm whenever ft z hand wiener obviously w ft iterate logarithm coupling benchmark process begin formula therefore player play proceed sublinear hx hx directly lemma recall suitably restrict face span strongly readily yield conclude decrease control rate choice pick regret law logarithm identify section discuss describe remark specifically payoff player opponent cf play opponent response instead correspondence stochastically perturb play time analogue fundamental process elimination suboptimal dominate formally give kp strategy obviously mind play say pure strategy context dominate strategy regularize dominate strategy eliminate dynamic aggregate surprisingly condition dynamic exponential show variant dominate irrespective elimination dominate basic proof dominant drift coefficient away dynamic study suppose solution substitute recall obtain rhs infinity become virtue finally iteratively induction dominate show vanish dominate regularize decomposable dominate decomposable form eq dt complementary solution player sm problem interior decrease denote cf independence yield kt expand complementary around establishe run diffusion affect probability observe mean elimination let formula reinforcement equilibrium end recall equilibrium obviously pure correspond strategy vertex noiseless dynamic exhibit follow equilibria solution nash lyapunov also strict nash equilibria turn generalization noise interior nash equilibria sure rest reinforcement ordinary definition differential lyapunov asymptotic let say exist stochastically neighborhood whenever evolutionary show strict nash equilibria stochastically asymptotically across strategy show irrespective noise rely heavily logit generator thing approach seem xt x lyapunov nash nash equilibrium stochastically asymptotically contrary direct rely result regard stable point brownian admit solution equilibrium nash must kx v kx k event contain nash one dimensional wiener fairly provide measure respect derivative eq brownian see theorem follow note proposition strict nash equilibria stochastically proposition tolerance strict neighborhood mind sufficiently consequence fair also ks end change wiener fact conclude kt finite w kt kt conditioning conditionally adjust aggregate player empirical average principle converge nash equilibrium version arbitrary regularize deterministic setting show average aggregate modify cf remark average principle extend even arbitrarily large payoff error game dynamic score difference grow distribution play surely nash case sublinear growth martingale multilinear dividing yield kx xt xt xt proposition always assumption player game satisfy addition almost surely equilibria interior kp kt kt claim identically interior term sublinear finally sublinear claim otherwise strong kt correction growth require vanish alternate cover kx k kx denote regularize response player average solution chain arbitrarily break arbitrarily jump invariant proper development subsequently average deterministic reinforcement importantly show response let game deterministic player game player solve q vanish lie vanish xt x xt xt xt x xt track perturb dynamic thank conclusion full empirical nash average nash equilibria game constant game nash equilibria follow converge nash equilibria games logit response match fig nash equilibria display also take evolution fig trajectory horizon tune average nash collect coupling strongly interior induce penalty convexity constant fp py n well claim sake subsequence necessary hx contradict compact get let strong combine rearrange claim let face contain py hx tp xt lie interior sided derivative show compact visit infinitely imply fp claim n subsequence necessary assume pass fine need two thus let absolute eventually k fp nk hx fp coupling capture function dy definition third theorem conjecture example game noisy robustness stochastically perturb cl national france france fr tucker stochastic differential chain theoretic payoff noise provide unified extend game dynamic irrespective noise player strategy become nash asymptotically independently perturbation magnitude finally player nash equilibrium sum game acceptable state equilibrium dominate strategy dynamic widely action payoff probability score payoff score cumulative strategy reveal evolution govern population attract well lyapunov stable nash strict nash equilibria stable
within actually distinction crucial concern spread disease node status suit capture precisely core contribution include name core identify core network weight core decomposition uncertain probability exist biological model instance see core decomposition scale insight worth decomposition would rigorous os enyi center every style fill sep n scale rectangle scale auto center every style circle inner sep rectangle n core core high core empty center fill sep pt n short fold interpretable core theoretical property simulation study rely algorithm develop decade contribution core wide range application core find generate graph core guarantee graph fit self loop remainder manuscript let graph node context subgraph vertex often core algorithmic vertex figure isolate right every style fill n n n n auto node circle fill sep n n contain core thus statistic core map vertice negative integer clique vertex natural information second interest unlabele graph summarize histogram whose vertex symbol example respectively illustrate instance vertex imply observe represent function advantage theory family rewrite term compactly parametrization normalize pg model define list increase empty every arbitrary eq rewrite provide count node simple vertex copy copy copy vertex index vertex index center circle black inner sep scale auto center auto style black sep rectangle center black inner sized usually maximum mle one resort testing solve generality scope short study question often value necessary fix value address remainder consideration conclude interior hull implement initialize vertex remain empty process condition vertex order yield pre sort index increment condition vertex small precisely argument vertex algorithm yield ii increment vertex hope add know new could get unable happen problem modify since potential equivalent make adjacent zero adjacent remove give sequence sort impossible definition initialize initialize edge construct graph condition option produce positive comment conclude simulation randomly construct node unlabele evidence example discover call resort monte proposal propose accept pg swap however chains choose generally mix bad behavior remove markov hand reach value size record group network code b high quantify observe like simulate graph similar goal accord proposition mle mle scope instead illustrative impose prior comment skew degenerate distribution lead concentrated clique behavior node compare probability hand whereas distribution exchangeable large label produce skewed towards account balance effect graph summary distribution triangle centrality allow correspond observe value formal goodness heuristic evaluate well goodness general use markov usual plot ensure sufficient convergence plot consideration figure histogram compare notice histogram triangle capture effect quite model centrality core histogram centrality large capture edge much expect fact tend densely connect consider visit mode distribution include mode truncate result
highly backpropagation generative forward propagation approximate latent variable undirecte graphical restrict boltzmann rbms boltzmann numerous represent unnormalized potential summation integration chain mixing pose problem mcmc belief contain undirected layer difficulty undirected alternative approximate matching require specify interesting layer unnormalize denoise auto autoencoder match rbms employ generative discriminative discriminate distribution dramatically correct desire approach machine prominent extend generalize denoise define parameterized chain generative chain framework adversarial loop unbounded activation loop generative recent work bayes backpropagation learn generator distribution noise perceptron rather simultaneously train word value function eq present adversarial net enough formal explanation optimize computationally prohibitive overfitting optimize maintain optimal analogous maintain step markov part learn procedure poor reject confidence minimize function cm blue horizontal domain domain impose non transform region inner train discriminate sample converge classify unable gm p momentum generator like capacity infinite study probability show optimum net mnist database activation sigmoid activation net maxout apply framework permit dropout generator test set window parameter introduce various report somewhat variance perform advance generative directly motivate far c mnist stack deep net figure generator claim well competitive generative highlight direct deep undirected autoencoder inference need partition tradeoff generator approximate variational mcmc base inference markov difficulty intractable approximated approximated explicitly approximated design nearly extreme property differentiable function framework advantage framework primarily explicit much updating avoid enough negative chain boltzmann keep date gradient inference function summarize adversarial net generative aforementioned advantage primarily adversarial generator generator another adversarial sharp chain somewhat admit many generative predict net advantage inference generator conditional index training net implement extension mp net improve determine adversarial framework useful acknowledgment like acknowledge helpful code would fr ed need support would cifar provide support google learning like thank les corollary xu david op universit generative two generative model procedure maximize mistake player game arbitrary recover everywhere entire train backpropagation markov network generation demonstrate qualitative quantitative deep discover represent kind encounter intelligence natural speech symbol
straightforwardly correspond learn formalism boltzmann generate notice random joint estimation instead w log introduce log r j reduce infer maximize may employ give gradient result estimation assume increase increase coefficient zero method deal parameter crucial problem employ put zero fraction small work suffer detail ability difficulty pseudo fraction step value interpret refer infer stage maximization assign ratio stop belong pseudo log put log likelihood present pseudo maximize note decrease keep one inequality likelihood drastically pseudo lot early stage close wrong quantity vanish take stop student true ratio pair ability maximize estimation toward give descent guess go step infer shown drastically put whereas threshold show comparative estimation put choose minimize example test remarkable increase error value bar put quantity uncertainty put bar conclude good detect infer emphasize ordinary boltzmann kind ability second also process stop pseudo call maximum instance optimal case lead curve sharp fig point pseudo pl max pl detect due drastically pseudo result existence must decide preliminary addition number number ht likelihood coincide vertical line vertical large datum infer lambda determine zero infer advantage remarkable formulate item student boltzmann degree test characterize difference ability algorithm base show remain outperform pseudo function determine function suitable function decide terminate experiment desire author thank discussion perform grant work education student communication inverse ise method theory describe correlate development part science complex assume boltzmann form define ise pairwise complexity system boltzmann likelihood training coincide well bias interaction number sometimes structure could enable week vast expect deal present conjunction reduce training put difficulty assess keep applicability response specialized type test kind although set item student detect method correspond student boltzmann give brief introduction third good existence student infer theory various difficulty well specify ability express resolve problem accord problem logistic form express answer ability problem define express th answer pl item extend simplicity pl
play brain noisy make online adaptive describe nine within open software parameter adaptation performance test repetition variability decrease less suffice reduce across variable delay signal state implementation user calibration plug accuracy scenario source implementation sophisticated dedicated matrix inter variability promise less stability covariance riemannian three minor modification single convenient software development besides riemannian framework bring since rely know spatial variation metric draw wavelet basis adapt show limitation increase channel condition riemannian regularize big mean covariance decrease degree france work ph brain centre national laboratory france research include riemannian brain minor university post institute france france dr research centre national laboratory dr grant human eeg time tool blind journal geometry riemannian geometry good subject method online adapt information geometry far brain interface phase precede actual depend regardless necessity calibration drastically appeal orient cognitive patient limit plug operation consider requirement device besides training discard calibration inefficient proceed pose plug achieve completely generic parameter derive previous continuously experiment possess namely albeit user trivial filter bad work level high show lot solve geometry regular temporal signal allow geometry introduce way experimentally riemannian geometry enjoy addition rigorous elegant conceptually easy constraint thank present able calibration less minor modification type present new dedicated p dataset compare section attempt plug purpose filter unsupervised adaptation inter make property adaptation build efficient see none art requirement classification essence prototype comparison obviously relate hand definition appropriate trivial accomplished nature problem process able metric support nevertheless generally predefine loose generalization approach relevant extract approach process fulfil quality subspace separation overcome difficulty matrix svm classify adaptive implementation adapt eeg useful selection generalization subject separation filtering thank geometry field riemannian manifold field establish increase imaging strength geometry natural lead eeg mean high filter common eeg model statistic eeg application geometry zero directly eeg eeg riemannian definite matrix information invariance invariant inversion invertible latter consequence signal remarkable formulate effect spatial source etc essence working mean denote fr square distance expression manifold find toolbox geometric geodesic short matrix manifold geodesic riemannian metric geodesic could matrix interpolation compare eeg trial prove useful eeg relevant feature spatial rather source separation variance matrix task difference riemannian class adaptation subject subject intra one database interpolation geodesic non evolve riemannian equivalent combine replacement mean covariance filter adaptation class limitation interface p symbol covariance matrix show illustrated potential dataset record game brain inspire classical paradigm e probable single classification score switch begin previous repetition level repetition motivate signal filter th hz filter name filter method spatially filter factor aggregate build use method name stage size classified lda selection regular linear discriminant reduce epoch frequency laboratory pz hz subject compose use offline canonical test paradigm calibrate record area report subject ht performance difference vs contrary pair vs overall usually evaluate trial result converge need reach efficiency contrary reduce fast spend calibration phase accuracy repetition correct potential occur always stimulus delay hardware software well stress classification method delay delay ms performances auc zero subject delay effect test delay draw ms loss selection sensitive delay performance super average p use experiment show cross subject pair vs particularly matrix experiment come source kind come show calibration intra estimate covariance performance subject accord leave l cs cs cs
walk represent agent go adjacent node remain sufficiently mobile choose neighboring connect irreducible vertex add one index home elsewhere control small chance return home base node ensure h run target motion walk sequence cumulative tracking target cost edge short path current normalize state steady plot versus bar evolution versus show minus stationary grow cost strategy good baseline stationary run baseline regret adaptive thus give h initialize start total cost minus see policy realization sequence combine aspect control online theory several mdps mdps cost regret ergodic stationary policy yu et online mdps believe regret space et achieve involve efficiency open address attain online mdps mdps bandit setting learn current cost realistic online mdps state construct regret whether promise far apparent duality control special equation certain dynamic state cf set correspond plan introduction lyapunov criterion ergodicity entry f nx e nx irreducible state right eigenvector fp uniqueness solve irreducible strictly well irreducible frobenius say exist unique show uniqueness essentially function inductive markov proceeding substituting turn imply x xt markov property w prove immediately idea programming map express unchanged add show fix assume item prove proposition guarantee term g establish pick explicitly complete show nb xx e hold begin measure calculation hoeffding purpose state substitute involve prove follow strategy proposition proposition always know fp side last equality fp j fp set reach passive dynamic mh mx p bind proposition second prove proposition rearrange get form expectation follow hoeffding write simplifying use due rgb rgb corollary problem involve perform space action action kullback leibl agent next aspect fact learn construction computationally efficient strategy mild along simulate process kl control markov sequential decision make dynamic environment time agent observe system interest choose system possibly vary admissible pair policy basic assumed function transition probability offline forward effect past action practical degree advance neither reinforcement rl learning variant learn policy online rl operate stochastically expect environmental need ensure agent eventually framework seminal widely sequential effect model cost step reveal minimize incur single action contrast mdps necessarily backward looking incur past observe bandit construct minimize strategy agent cost reveal unbiased full fed strategy minimize regret information reader recent discuss al combine mdp framework mdps mdp observe current choose fix like framework function reveal take minimize relative horizon interest aspect policy brief statement idea later notation motivated machine artificial intelligence merely memory emphasis desirable formulation recently transition feedback law resp kernel underlie one deviation action fix default passive reader paper graphical model mdp action simplex markov system transition govern probability law cost consist cost give prescribe corresponding motivate situation implement free low actually desirable active perturbation prescribe attempt balance tendency cost strong inspire leibl divergence next prescribe property automatically online version detailed state arbitrarily cost determine stationary policy since usual mdps mdp state state feedback law cardinality mapping range subset simplex account cost case state state control represent map law contrast agent choose finite simplex freedom choose measure quantify leibl free external control kullback widely desirable secondly purpose control shape joint relevant system law correspond dynamical derivation controller programming similar nominal canonical bayesian filter variational entail space motivate sort track multiple passive specifie motion target quantify target location target possibility target cost attempt track passive motion rapidly visit prior e target tendency exploration tendency potentially target exploitation another example set brain interface position device passive dynamic natural dynamic absence assume prescribed minimum device execute intend want intend individual cost deviation run well nominal dynamic potentially include rational etc tendency operate nominal mode offline circumstance offer meaningful class boundedness agent ergodicity passive moreover computationally divide increase length apply average cost function precede take yu mdps action space advantage time horizon comment far sequel wide simplex subset hence exist applicable extend regularity yu underlie satisfie uniform ergodicity need et strong significant simultaneous exponentially markov chain space law verify determine mdp ergodicity verify automatically ergodicity function correspond recurrent possibly e matrix denote x ergodicity show supremum consider agent perform walk environment proceed l draw select knowledge incur cost incur agent suitable agent collection mapping f knowledge function cost q define regret gap could achieve use walk lack sequence regret markov transition make follow every yu chain adopt standard terminology consistent w law process certainly complete may outperform stationary stationary indeed truly online strategy alternatively interpret consistency horizon achievable term achievable cost reveal time passive passive irreducible former latter great ergodicity ergodicity coefficient everywhere equivalent frequently ergodic impose satisfie exist point actually et show make ergodicity kernel every policy ready consist r contraction precede construction strategy mdps section overview several recall general mdps action feedback r chain control state construction optimal equation q h function paper process solve describe informally simplex euclidean topology transition correspondence policy passive specify absence shorthand average eq average transition stay prescribe form side form see obviously quantity zero uniquely relative write multiplicative construct policy compute instance solve due boltzmann boltzmann various context physics deviation involve gibbs affine term term indeed minimize hand sequel cost whenever etc policy list explicitly assumption irreducible invariant f relative solution additive solve fix subset cone relative function subset exist smoothly exist map basic steady optimality similar yu et behind partition contiguous phase duration policy match reveal precede phase steady within yet short policy use successive phase phase phase duration give need growth phase phase comment form h mp mx mx end policy throughout phase evolution induce describe frobenius problem begin phase obtain determine stationary follow solve algorithm compute radius nonnegative irreducible perform outperform experimental general major step notion steady
entry kernel provide inner transform include bandwidth turn derivation gram center counterpart problem scope center keep two machine pattern use fold formulation seek axis well variance project lagrangian multiplier axis eigenvector variance eigenvalue I account variation note total datum get lie mean allow onto represent eigenvector I normalization scale along axis fix scale estimation apply mathematical machine learn enyi form analysis detail probability sample estimator q entry expression enyi therefore expression composition motivation eigenvector small term contribute pca eigenvalue contribute emphasize center quadratic study detail issue center way one mode singular center svd reveal gram matrix hard take carry outer turn approach inner scope complete extension center gram product obtain center examine need link center projection projection onto span te complement projection interested projection eq consider onto subspace center give property use gram counterpart equality reveal center subtract column add center understand trace measure due center counterpart correspond trace verify verify straightforward next explore eigenvalue eigenvalue sum proceeding central literature theorem let singular gram counterpart apply separation dr conclude gram proportion divide trace center eigenvalue behave coarse low state theorem theory eigenvalue increase also diagonal entry direct separately connection decomposition gram spectral decomposition product orthonormal show eigenvector eigenvector eigenvalue next apply eigenvalue low I large describe equality due apply observe characterization beyond relation trace latter equality bound furthermore worth state term block far gram follow zero eigenvalue equality zero eigenvalue dual j matrix center wise j j j supremum inequality combine theorem gram constraint one drive normalization box eigenvector relevant principal component analysis axis get axis axis axis examine outer matrix covariance moment dd expression eigenvector satisfy section provide essential eigenvector I hand simplify since correspond equality eigenvector get build entropy easy illustrate theorem simplify therefore diagonal entry apply analogy matrix consequence state associate first investigate consider proof large arbitrary replace side simplify denominator upper cauchy schwarz combine result conclude immediate result cosine angle eigenvector straightforward product namely give relation eigenvector large center counterpart non center theorem eigenvector obtain counterpart significantly simple likewise simple property comprehensive conventional issue scope pca datum investigate weighted consider derive neighbor reduce centroid detail machine weight data become map orthogonal need get analysis analysis long raise orthogonality projection proceeding relation light follow substitute expression become become unchanged weighted expression easy verify eigenvector non eigenvalue matrix complicated due expression matrix weighted mean several special turn section great machine variation optimization genetic machine method lose generality tn presents see hard problem solution investigation evolutionary provide elegant free principle adaptation distribution relevant region solution covariance latter promise fitness progress update matrix zero gaussian present insight firstly trace derivation covariance easily step update simplify angle diversity multidimensional seek preserve pairwise distance dissimilarity expand get inner equivalently entry column vector entry respect translation inner center double center relevant axis describe sample center gram positive definite next study statement paper thorough description conditionally positive investigate worth positive bias associate issue mathematical derive analogy diagonal provide new insight former apply show show variability I give end inequality one latter tight decomposition valid give describe large eigenvalue consider retain define gram conclude interesting completes describe j former normalization projection axis scale feature approach extend establish easily verify repository extensively recognition since seminal divide attribute gram derive show illustrate datum center datum shape use parameter kernel center center contour feature row principal relate mean obtain center experiment c c c c cumulative great gaussian shaped c c center center shaped raw center bridge gap center center explore pca gram center gram nonparametric several beyond conventional include embed address impact function center impact center input oppose center space connection theory receive engineering degree engineering ph security france associate system laboratory technology france interest analysis representation machine learn wireless sensor network paper award theorem france recognition rely empirical moment principal analysis recently researcher work kernel theoretic even center order bridge gap design machine machine conduct product center center datum several explore outer provide extension beyond conventional centering shift rank update multidimensional illustrate relevance gram machine recognition analysis machine singular svd seek axis give component prominent feature axis large amount variance multidimensional partial pls cca fisher discriminant elegant nonlinear rely concept initially introduce fold regression write substitute inner without significant cost perform matrix space property reveal version pls survey machines cca origin algorithmic center center way deal center moment centroid available relate second central issue center propose
gd gd sum correctly lag np lag scale double range lag scale least double point gradient store np long iteration epoch epoch else I lag lag add entry date amount lag lag k lag np pt double frame university team sup paris france sup paris france call spirit sag sdca recently sag composite proximal use sdca strongly strong convexity effectiveness remarkably advance provably fast expectation machine problem requirement strong convexity likewise satisfied form strongly proximal operation incremental contribution incremental prove rate strongly sag sdca rate composite additionally sag method applicable modification establish convergence fast incremental stem start derivative derivative structure inspired sag discuss make take store unchanged proximal size composite requirement hold expense geometric exclude prove supplementary step strong give theorem size incremental convex via small additional avoid explore relationship fast incremental unified brief property consider figure composite list times one sag sdca convex prox storage simple sc summary property question experimentally apply amount trick sag gd decrease convergence unlike gradient reference therein sgd constant get rate present call mention sag reduce relate reduction one estimator convex x yx zero sag past store sag update non bias explain sag proximal update able use sag relate bias every appear inside outer start outer iteration sag pick whereas update gd number iteration henceforth make trade gradient usage sag predictor gradient vector class prefer method iteration loop near practical tune prior non composite intermediate quantity nf unchanged quantity explicitly store recover describe size simplify discussion introduce interpret update consider change expectation identical advantage operator require store prove proximal big pass access order access empirical speed storage sdca sdca transformation sdca work closely pick index compute kf zhang require operation simply use search conjugate simple primal equivalent line search sdca variant line evaluate datum instead full store weighting storage requirement sdca class store heuristic average use sag state slow derivative update suggest sag ensure update practice strong quadratic amongst evenly implementation step scale scale standard trick problem supplementary code implementation expectation respect condition start constant equation lyapunov lyapunov expand quadratic value shall k combine fraction yield size note square verify together ensure constant set round ensure expectation constant explicitly expression index optimality effectiveness test mnist test suggest fast test epoch basis expensive per epoch basis evaluation epoch double sdca sag slow begin sag adaptive confirm discuss duality sdca bind include convexity low k x convexity f apparent compare k vector instead definition proximal sdca experimentally proximal say sdca among slow disadvantage sdca handle although use objective proximal disadvantage sdca additional let strongly continuous gradient gx substitute let forward fy f follow argument tight key trick convex choice available use similar theorem lyapunov different define k x true property begin use prove quantity property full
output less since report worker put payment effort never worker function value observe expert modify regularize ridge regression square include depend two mechanism know possibly minimization ii also unknown expect beneficial point advance still optimally worker problem accommodate mild modification separate bias variance term matter estimate minimization use unbiased estimator payment extra worker reason behavior know term utility depend worker reason although objective payment mechanism accommodate many variant modify minimization reflect unique dominant strategy achieve list modify problem accordingly generalization payment accommodate sum worker square budget payment update effort budget another replace total payment increase square plus impose modify square effort exceed function combine behave modify mit mit computer berkeley berkeley remark assumption rgb propose estimator quality low estimation range include also generalize include estimation subject besides concrete problem mechanism design worker label reality essence business science entity crowdsource popular phenomenon unlike develop related crowdsourcing recently treat subsection interaction rational agent pursuit solve item draw prior sophisticated keep area come back datum aspect reality employ context one quality crucially poor quality worker effort level effort worker center worker lack worker knowledge participant include create mechanism protocol contract worker worker act effort rescale effort worker nonnegative otherwise th payment protocol contract value never must depend know quality worker mechanism may seem optimistic go follow worker worker regression worker determined expectation define opt opt economic opt bind cost opt precise optimum small minus effort effort mechanism light seem quite surprisingly linear example polynomial construct way minimize extract important consideration end assumption concept discussion turn extremely worker dominant optimize payment minus else course design space mechanism design attain opt mechanism worker depend datum seem mechanism work entirely relate oppose mathematical justification root create accuracy worker statistical optimally mechanism include regression advance assign worker even broad include ridge extension besides accommodate objective loss plus go beyond mechanism crowdsource pay crowdsource optimum crowdsource contract expert crowd expert perform scheduling mechanism use crowd accord paper keep assignment treat bandit budget optimal reward optimally participant address concern treat regret use crowdsource mechanism crowdsource worker pay create competition worker enhance nature design individually rational agent produce bias outcome subset datum paper proper scoring rule agent ask level effort contrast final decide design mechanism want maximum effort agent consist effort allocate increase label allocate task besides fact care reward use agent whose signal report effort look except quality sample decide mechanism equilibrium could worker surprising contract chapter contract worker effort contract worker effort observable worker worker utility formulate worker effort appropriate design worker observable two technique difference apply apply worker payment effort observable still instead mean effort employ worker take randomness output ie produce worker payment able effort worker achieve need expect point decide assign discuss yield much apply mean stick development technique minimal modification objective accommodate minimize square subject budget minimize sum square arbitrary discuss eliminate use objective discuss able predict effort worker result payment decision decision worker worker familiar comprising subset worker effort eventually worker outcome able evaluate agent behave game concept game whose close behavior rational player prominent equilibrium nash equilibria randomize guarantee dominant equilibrium comprise worker payment unique dominant strategy equilibrium iff expectation everything word matter effort choose unique effort level worker dominant equilibrium game fairly trivial agent decide induce equilibrium general rare design satisfie constraint objective finally note prediction worker worker capture follow comprise payment effort satisfie restrict induce game worker dominant satisfie might requirement solution requirement individual contribution establish induce dominant equilibrium later several familiar iff well regression regression behave accord optimal behaved definition relax necessary discuss remove optimally solve behave well dominant strategy satisfy strategy equilibrium optimal achieve objective worker effort worker equilibrium induce dominant strategy evaluate individual individual worker combine would achieve effort property always induce dominant individual objective behavior worker achieve even main unique satisfie value equal quantity argue choose assign payment worker worker payment worker choose constant induce unique equality worker worker effort ie estimator worker rational maximize expect payment minus effort level response find maximization ensure effort decreasing make sure unique game worker regardless
hundred md home collect configuration gb molecular activation mechanism analyze reversible atom position experimentally determine inactive hmms achieve interpretability choose monitoring relaxation reversible pathway mechanism onto structural transformation inactive unfold highlight rotation highlight interaction portion overall protein fluctuation freedom largely process degree grey simplicity dataset find activation pathway field drug design effect protein common entire behavior tumor well function intermediate likely unique target protein b state project loop red detail activation pathway framework analyze dataset propose reversible hmms identification state interpretability physical switch without theoretical discretization integral formally control transfer enable quantification error theoretical guarantee yet exist reversible long markovian reason believe regularize represent analyze md challenge hyperparameter aspect reduce required amount manual adapt bayesian unsupervise linear may facilitate goal massive reduce complex thousand degree statistical hmm turn raw molecular function protein extract biology rational drug thank md trajectories r support acknowledge gm machine modeling protein approach hide protein via molecular motivated massive necessity provide biology drug criterion physical contrast improve implement apply home dynamic activation mechanism protein challenge relevant disease cancer characterization pathway protein fold energy surface protein biology molecular problem genome phenotype furthermore dynamic protein design molecular md atomic resolution forward potential quantum reproduce moderately sized million freedom integrated burden md simulation central challenge achieve independent purpose accelerate home computer utilize cycle google production science dataset major contrast problem goal merely md datasets scientific insight protein model physics physical paradigm chemical understand configuration often fluctuation dominant understand move state paradigm motivate base location unknown latent hide hmms thus mechanic symmetry respect version law call equal essential detailed balance probabilistic knowledge protein study describe long see therein substantial protein via shift together motivate term deviation amongst uninformative freedom furthermore pairwise transition state differ along reduce formulation reversible hmm introduction scalable fit standard framework md physical interpretability follow describe associated learn potential protein indicate direction machine protein discover protein protein fundamentally system offer insight concern computational study protein md primarily quantitative approach include movie protein structural number pre specify degree characterize critical biological quantitative method capture rich temporal dynamic model chain cluster md fully hmms recently hmms emission distribution employ notable lack characterize manual purely state lack introduction order inefficient sized regularize reversible generative multivariate continuous series series simulation ji converse ij ik ik ji ji transition derivative term bfgs aic bic selection criterion discretization support recall choose become correlate increase discard criterion propagation state eigenvector stochastic eq eigenvector collective dynamic process equation describe central molecular modeling perspective visible protein perturbation relaxation enough long change discretization gpu cpu across spend exp architecture parallelism gpu fine grain array forward respectively fully utilize gpu parallelism update specialized write kernel trajectory matrix accumulate rest speedup optimize standard implementation intel bridge achieve multiple hmm dynamic hmm unlike feature hmm diffusion govern brownian differential reduce diffusion constant process double well euler produce ten simulation trajectory two fusion state surface learn display sensitivity discretization accurate relaxation long unable accurately lag time succeed identify identify state primarily loop region axis fail state post process show protein protein intersection pathway protein human link would obtain dataset md million protein compose perform extract configuration hmms choose monitor relaxation penalty hmms correctly ease comparison
prove prior lead consistency reduce regression consistency lasso expression completion prior spirit spike selection section lead describe eq generate summarize finally calculation bit principle array common gamma row inverse derive joint therefore gibbs appropriate gibbs large variable hasting high dimensional make quickly bayes expensive bad large certain distribution optimality work iteratively update skip elementary simulation prior describe first optimal factor necessarily whose denote iteratively formula entry th connection bayesian certain penalty posteriori mode recover provide insight prior correspond popular easy interpret easy map penalization nuclear see page rewrite link penalization close essential number hand penalization gamma case integrate respect lin group map prior proof contrary scale problem hand give nice gamma column toy generate square ie corrupt study grow second sample quick illustrate figure take lag case iteration remove period gamma report hyperparameter prior fix gamma gamma four prior result line consistency grow result explain section discrete hyperparameter discrete stable result bad rmse well reach gamma distribution prior automatically note poor slow heavy bayesian available http dataset challenge vb instead vb netflix challenge model rmse quite end approximation hyperparameter rmse gs vb gamma vb inverse inverse inverse discrete gamma vb consume test review propose two gamma discrete real life tensor spirit proposition come end acknowledgement would like comment thm thm incomplete recently several recommender netflix behaviour behaviour gamma decomposition conjugate conjugacy interestingly nuclear prior classical netflix netflix science statistical community increase movie column become netflix reasonably pattern movie problem http first recommendation norm preferred computationally ground result ensure noisy observation recovery exact recent general trace popular multi task derive reconstruction basically quadratic well bayesian context prior learn completion computational rely dataset netflix know rating must keep mind application observe entry long completion netflix movie less sensible define must posterior
one cell memory ignore cell unit unit learn lstm per time relatively memory cell store temporal computationally alternative architecture architecture learn lstm architecture connect layer connect cell output layer layer non output n r I unit recurrent recurrent note effectively equivalent projection layer unit compute unit gate gate sigmoid gate gate gate activation output activation wise cell lstm recurrent recurrent recurrent unit height dash height em minimum width thick circle thick thick scale name xshift name input name name unit name node cell bend auto west node dot cell bend center east cell cell west xshift yshift recurrent xshift projection right xshift output cm north north cm leave west projection west east xshift west east leave west east near north east cm north east north output east dot cell block mm memory block implement architecture core gpu cpu machines network bottleneck operation eigen library implementation operation activation parallelization technique gradient multi operate computational multiplication rather time effectively sequence batch process backpropagation update use step e propagate activation propagate original propagate activation activation frame error propagate gradient entropy criterion error finally parameter weight accumulate state subsequence different shorter could reach next new dnn rnn architecture vocabulary million hour google dataset represent frame compute every cd network map ci initialize try architecture result phone acoustic hold frame system rate report million sgd minibatch graphics gpu softmax represent phone state stack window frame either frame denote lstm partition propagate backward propagate truncate rnns rnns recurrent unit tangent activation cell unit sigmoid forget gate recurrent activation rnn log energy window frames future frames decision frame frame ht frame name contain architecture state cell rnns recurrent rnn dnn configuration name layer low projection layer evaluate rnn significantly rnns training limit activation lstm rnns converge fast project architecture well lstm rnn architecture projection generally lstm recurrent layer ht converge speech lstm recognition state embed mobile phone relatively output figure architecture obtain accuracy depth compare application large vocabulary speech scalability effective use lstm architecture introduce projection recurrent recurrent flexibility architecture improve lstm architecture performance recognition lstm large gpu cpu implementation long memory lstm recurrent address conventional rnn unlike feedforward neural rnns cyclic model use task label acoustic contrast rnns recognition phone scale task lstm architecture make acoustic vocabulary recognition dnn parameter configuration lstm quickly give speech recognition relatively lstm recurrent neural rnn speech feedforward rnns cycle activation network make store provide contrast fix contextual windows input rnn history capability rnn modeling rnn direction make current input labeling recognition
variation identify often body method illustrate box problematic obviously detection guess location unknown common stage informative hard training identify initial annotation higher informative training key accuracy final issue characteristic automatically discover discriminative correspond wrong background e water occur image configuration particular configuration object occurrence patch positively label image patch discriminative covering discover patch part cover formulation cover density would strongly cover formulate independence submodular subject case part effectively take viewpoint demonstrate configuration observe combination patch produce accurate occur interest box informative negative short contribution frequent discriminative visual inclusion discriminative detect method tight bounding box annotation object reduce train detector binary presence effort prominent object image recent challenging multiple intra category variation achieve rarely negative convolutional discriminative patch contrary contain full merely piece full end mutually object far select negative object discover high object mid level patch foreground grouping e patch contour texture recent weakly supervise discover discriminative relate formulate submodular formulation alternative discriminative less scalable use geometric occur visual pattern improve visual recognition performance occur represent star shape among inspire supervision feature dataset work relate object phrase truth annotation full box annotation discriminative positively configuration address patch discriminative occur efficient configuration patch easily retain configuration identify similar positively box selective regardless label remain one discard neighborhood within image identify small representative construct copy near near neighbor label cover bipartite monotone submodular aim configuration must informative patch merely redundant frequently occur often modification treat identical submodular covering case candidate densely identical patch thereby configuration independence may pick together redundant cover ever identify whose neighborhood overlap overlap patch identical diversity phrase optimization problem constraint express neighborhood greedy bb disjoint first visit high list immediate greedy bad intersection axiom exchange insight intersection ground set k ce kk color may node say adjacent k check set patch representative patch different occur head person top visible consist relative practice preference maximize occurrence count configuration amount write find frequent inspire supervision least occur patch operation translation viewpoint bin b I I fall bin share location via ji position edge occur edge characteristic configuration enough occurrence frequent also determine localization small configuration discover frequent configuration well localization estimate hard negative let configuration part object foreground include foreground overlap foreground rectangular region box negative specifically rectangular bounding box foreground hard negative l f foreground result negative foreground introduce undesirable false negative detector adjust coincide foreground finally rectangular overlap foreground box negative uninformative overlap cover foreground select negative region configuration likely foreground object discover lead count compare stage frequent configuration discover patch find positive discover detector foreground derive configuration correspond region detector selective search retain detector section discover configuration impact discover hard negative employ fc feature proposal discriminative discovery transform shift vertical px px px px visually pair loose handle detail space figure qualitatively illustrate configuration box box combination upper frame failure object configuration bottom protocol use art weakly level annotation supervision annotation table baseline improve detection majority class note improvement person
connect unit connect ram roughly reach attention digit translation invariant experiment search big object aspect classify presence clutter operate full clutter learn invariant attention learn clutter focus image experiment task call translate mnist place mnist digit add mnist digit random classify translate error fc layer fc layer layer ram ram scales ram ram ram core unit perform suitable hyper train million good video attention track ball demonstrate ability learn policy introduce attention neural internal focus control signal environment unified architecture method appeal ram control ignore clutter image center ram architecture comparable classification extension terminate make classification take confident allow object fix extra action train policy procedure encourage video google google com convolutional linearly extract video adaptively select select high convolutional neural amount control independently differentiable learn evaluate convolutional neural network explicit neural architecture recently great success challenge classification object come typically currently reduce object second run gpu follow slide paradigm literature classifier object box independently thousand window computation come map entire least one scene human focus attention information representation scene future decision focus resource scene pixel process substantially object interest place irrelevant visual environment clutter outside naturally ignore role cognitive scene bottom play specific see review novel attention task consider scene general video module play recurrent location video build scene bounding box past number amount control independently pixel train maximize decision backpropagation train gradient learn effective strategy look result attention clutter image receive attention vision instance dedicate slide window focus primarily reduce cascade e e g propose window likely contain substantial may approach add cnn root window exploit past processing way class approach computer detector approach process salient identify contrast detector capture human typically propertie ignore semantic task observer model framework setup restrictive work attempt rnn integrate visual decide sequential rely architecture interact environment attention decision direct interact visual observe e full extract narrow band agent control affect true integrate determine act receive scalar execute delayed agent maximize sum reward diverse detection static image play visible game engine sensor operate frame frame game reflect detection static state environment environmental action would decision reflect resolution image extract location map independent another combine produce core take action internal state action recurrent fig sensor choose sensor receive observation environment agent rather focus band bandwidth sensor encode region resolution far refer resolution vector extract history past agent instrumental act sensor neural external feature perform via environment state choose stochastically parameterize formulate softmax dynamic formulation environment agent receive signal reward reward distant less delay otherwise rl process unobserve need map subject case outline choice agent network interact agent combination dynamic playing induce interaction maximize ps ps ps involve unknown dynamic technique sequence episode rule running sample sequence adjust log produce gradient define computed backpropagation provide unbiased estimate gradient reward obtain action expectation log action small type baseline reduce well unknown priori image total episode try good bad know detection output optimize correct achieve maximize truth observation gradient core network evaluate several describe common center location successive twice corner layer e nonlinearity network train location component network core core environment core unit attention softmax select search reward classify reward
distribution often ignore gibbs general therein lie sensible inefficient particularly true allocation recommendation hundred thousand em issue optimization dependence step augmentation marginal simple gibbs replicate state additional state marginal cause approach optima make temperature compete competitive fast symbolic hold problem define joint tie latent eq power original optima optimum peak often subscript write perhaps obvious add considerable parallelism factor method us order magnitude speedup sampler limitation sampler discuss factor hardware present experimental expectation graphical compute expectation compute work likelihood expectation require optimize equation em locally attempt kl estimate usually unlike em estimate common factor approximate factor simplifie make constraint vb nevertheless factor describe gibbs sampler perform joint slice sampler gibbs sampler difficult slow especially dimension slow variance hybrid expect em suffer may likelihood propagation class variational message conjugate family vb coordinate factor although estimation independently infer aggregate initialize z sampler standard sampler group ignore normalizing product conditional product product member represent normalizing imply closed adjust anneal lda sample rather taking multinomial among conditional parallelism sample replace fully capture sample fast completely integer long code increment lda word source considerable independent approach similar coordinate factor use distribution lda process lda parameter previous variational lda line dominant count step line eliminate evaluate factor implement approach vb acceleration source system system al vb implementation lda et collapse sampling lda parallel al lda implementation art gpu parallel fastest lda date fast implementation system evaluate pc equip single core cpu intel gpu gpu cluster gpu come gpu report gpu yahoo york word token corpus million k token million word convergence gibbs marginally beyond start beyond flat mini number per show validate pass pass gibbs vb online vb dataset vb converge pass pass vb converge pass three within usually take pass dataset per reach pass begin converge gibbs move runtime different fix gs runtime system benchmark time c runtime illustrate sampler take cpu hour gpu implementation improvement vb second performance art implementation use million article machine overall construct repeat news article run iteration find news dataset comparable take k gpu process k simply second thus gpu accelerate gs sample give benefit parallelism study schedule logarithmic scheduling max max tm max max total sample size configuration identify anneal fast number c runtime iteration paper hardware accelerate estimation accelerate pass introduce parallelism show hardware gpu accelerate gs sequential fast code com gs applicable explore inference factor conjunction full approximation
phenomenon whenever cause importance construct wise boundedness publish theorem thm hypothesis moment particle weight ensure expectation particle converge filtering extend derive require boundedness boundedness moment boundedness convergence boundedness fourth moment distribution moment hold boundedness leave use also perform unbounded importance moment approximate filter carlo approximate measure estimation inference system measurement modeling dynamic measurement density important particle exist g reference therein convergence unnormalize wise particle filter wise boundedness assumption particle derive moment importance modify applicable square importance bound spirit assumption main filter construction filter construction equation borel probability bayesian bayesian regular require equation model approximate particle approximate solution mean empirical convergence importance boundedness qx I unnormalize dirac delta iw aim induction combine assume assumption q need bound second combine tn use complete proof lemma n together imply generalize assumption unnormalize qx hold lemma assumption lemma remark proceed inequality assumption proceed measure assumption markov borel argument cox priori reflect brownian brownian intensity density respect lebesgue measure respect require include select gamma parameter importance particle importance lebesgue eventually numerator nonzero accord particle guarantee empirical use combine recall argument integer even negative square borel measure converge cox process right
lagrange time meet tradeoff iteratively regard ranking score code sparse regard dictionary popular meanwhile database ranking score important irrelevant yes explore paper joint sparse explore sparse bridge code neighborhood sparse rank construct objective sparse dictionary ranking learn ranking reconstruct combination element linear combination coefficient zero coefficient sparse code dictionary code meanwhile near base content retrieval rank accord refer ranking distribution point performance neighbor search firstly code code rank code use rank ask question relationship yes explore boost ranking code code jointly explore sparse coding however consider sparse rank distribution approximate code consider reconstruction error unified function code dictionary score code rank score internal explore algorithm optimize regard sparse code dictionary alternative introduce unified assume data th one query query learn rank rank top rank ranking score point nf th code learn sparse function learn rank coding aim learn reconstruct dl code minimization error point norm sparsity measure use code local sparse neighboring approximate score learn propose score complex local function tradeoff please rank score force close problem tradeoff please score predictor regularize code ranking connection solve optimization score code role repeat ranking fix gs rewrite ik kf local code please compose objective point objective minimize minimize regard regularization consider sum objective point rewrite th rewrite diag
ex berkeley edu computer california berkeley rp approximation program approximate essential since program quite cubic addition random projection useful usage optimization ratio constraint broad random hadamard transform project tangent cone constraint dimension illustrate consequence include unconstrained vector implication sensitive connection denoise compressed sensing optimize fundamental mathematic program solve prohibitive many may prohibitive dimension million concern sophisticated cone program program rigorous approximate program scheme perform projection constraint cone program program case interesting statistical problem formulate side belong dimensional include matrix set combinatorial shrinkage relaxation principle deriving method extensively decade ambient dimension context program solve g therein result generalize provide broad program addition analytic exploit analytical banach sketch probabilistic unified sequel sensing case program addition storage useful modern financial record medical concern store small solving preserve interesting problem trade set mutual set statistical optimization organize precise main corollary concrete close section devote appear conference international problem turn goal simpler via problem natural geometric tangent cone denote optimality problem feasible decrease move belong cone transform cone transform define banach main relation vector satisfy I matrix matrix bernoulli rescale sphere say sub projection universal example width scale statistical freedom project preserve sensitive program possible sketch retain mutual rescale entry quantity privacy sensitive discuss generic draw random many guarantee per vanish problem type vanish symbol straightforward combination show condition mutual symbol eq substituting claim disadvantage vector multiplication multiplication main apply matrix order randomize begin orthonormal ij hadamard matrix multiplication basis random matrix rademacher base choose hadamard fourier product scale linearly involve corollary many gaussian width rademacher randomize orthonormal system draw orthonormal size approximate dimension general pre example corollary follow potentially offset form product main convex constraint consequence theorem way simple lead square provide manner accuracy quantity small corollary confirm intuition guarantee least unconstraine corollary estimate hold paper overview qr approximate least square substantially qr require consequence theorem sub result refined argument investigate corollary simulation problem fix form problem generate datum datum give hadamard project perform randomly predict scale trial consistent regardless projection approximation become geometry tangent statistic quadratic unique number zero program unique cardinality eigenvalue sketch low solution optimal improve establish dimension order requirement linearly constrain solve another popular use solution iteration algorithm obtain let cone take sign triangle upper tail imply applicable set turning lower involve low corollary application third follow specialized standard size form curve ft solve radius hadamard matrix predict approximation ratio control increase plot qualitative note corollary one simply correspond function perform wise operation noiseless approximation guarantee recovery compressed sense realistic sensing imply well moreover closely precise summarize denoise projection dimension recovery conclusion hold sketch q version rademacher generalize randomize atomic type provide atomic classification represent collection label specify vector label formulation serve least amenable let b paper take correspond vector svm case coefficient lie machine version lead scale sketch machine omit corollary linearly conclusion ft trial place equal program obtain support either rademacher randomize repeat perform trial bundle curve sketch involve simplex portfolio return subject return let semidefinite typically give pair allocation give factorize whenever expect return constraint tangent cone portfolio turn operator dd general estimation primary relatively constraint typically norm illustrative wish dimension frobenius non negative norm obtained long accordingly replace program dimension iy rank solution provide sketch low rank condition bound probability sketch bound guarantee likely substantial version however leave b nuclear eq block duality nuclear result norm matrix follow example enforce group notion sparsity collection subset index note special group reduce usual norm generally group enforce group analogy group restrict size bound solution generalization ordinary indeed reduce similar maxima upper detail turn randomized system high depend central vector arbitrary quantity fix randomized significance ratio triangle q consequently use convex optimality add optimality right appropriately basic need bind long change universal prove et let sub universal subset theorem proposition particular inequality universal involve shorthand q eq apply bind imply three term calculation give denote orthogonal write consequently v follow substitute establish eq supremum decomposition set put piece rescaling appropriately begin state randomize define width universal least universal give complete lemma c claim rescale suitable immediate introduce q proposition unit inclusion thereby base inspection put together piece q projection along numerical fix diagonal randomness h event rademacher complement turn truncation level
cone element side exist prove reverse converge obviously kn sn paper tangent derive definition normalization use verification mention use definition theorem follow existence analytic value path easily modify prove analytic curve put analytic curve involve probably since exploit formula imply formula singular reverse state normal cone imply cone n also know structure tangent cone calculate turn moreover carry fact generate fx fx kk orthogonality estimate square rank large singular value nonzero remarkable priori statement critical semidefinite illustration relative minimizer make tell impossible since finish matrix representation involve sparse second rank huge never form rank well unlikely event fix feasible elegant backtracking projection hence less gradient completion setup matlab choose six ghz cores gb index generate generate normal kn least uniformly start guess choose approximation start start zero exact line smooth relative error visible inferior latter plot think plot perhaps explain fast entry relative error sample unable figure th confirm approach search elegant riemannian instance synthesis increase strategy grow inferior cg miss solid full error index nk explicitly point consideration difficulty arise unbounded curvature optimization real grow treat dynamical important subspace rank tucker hierarchical tucker rank take intersection low otherwise nothing also generality hold n accumulation pick prove limit put hold induction inequality project use matrix closure gradient relate cone project arise unbounded pointwise analytic k know result estimate curvature fact justification assume point method rank riemannian descent concern low thus know riemannian simple projection take tangent plane become tool low several equation lyapunov completion newton search search sufficiently alternative project discretized flow satisfy dirac variational integration ode euler method size also relate dynamical admit lyapunov analysis manifold ambient manifold break boundary might happen need lead allow size serious difficult priori statement regularization certainly convenient analyze closure satisfied principal search method direction tangent instance explicitly project easy corollary need map tangent choice aim attract optimization seem proved n sequence sequence possess satisfie possible strong one critical information rate advance notable class real analytic use wolfe step selection flow consider descent riemannian manifold local search like convergence regularity discrete projected integrate ode existence ensure convergence via second taylor would unnecessary term gets iterate bind constant one limit plan via involved formally reason establish lie order estimate insight repeat necessary impossible regular considered detect idea summarize singular likely generate real differ search thereby establish convergence successively turn tangent theoretical explanation outline part search analytic highlight result descent method line algorithm select backtracking notion tailor direction need identity sequence neighborhood analytic must critical short derivation compare tangent simple priori critical section consider concrete direction matrix result constant provable rate black box limitation work completion cg unless something else euclidean optimality problem far close cone since general metric projection may uniquely denote always cone equal necessary optimality relative optimality paper complement everything fundamentally assume cluster satisfie project hold original unconstraine parametrization basically real analytic analytic open open induced least neighborhood terminology exist subset derivative analytic map derivative cone without assume prof since may depend know convergence statement generic happen version positive hessian second hessian definite clear concrete line shall consider hessian likely treat constructive remain meta intend shorthand enough assumption imply convergence point limit replace assumption prove technical satisfied gradient add estimate behind manifold context norm distance reduce complexity minimization case low rank change besides give actually simple sufficient sense e unfortunately full rank later force smoothness know project gradient flow manifold mean plane property map respect however smooth manifold make tangent cone call implication algebraic variety exist arc implie make cone decrease much importance analyze search upper impose serious practically always remark define cf proposition mind guarantee search call angle satisfy equivalent euclidean good angle moreover n f pick size small point descent choice subsequent importance principle use backtrack eq hold minimum ratio converge follow eq since continuous adjust chance restriction need calculate formalize proposition choose iterate assume constant exist property follow obviously necessary step impose point assume whole generally estimate apply two assume otherwise x lead subsequence inferior notational convenience rearrange hold may disjoint eq finitely contradiction wolfe linear mean
module partition contiguous duration book compact unitary say gx g k k detail template book matter say group invariant close come specific hypothesis see specifically training video show theorem temporal simulation template consideration consider temporal association like think clutter experiment unconstraine face decide person task differ contain face position visual fold contain video transformation module template face label possible fully thresholde dot product section determine chance pool representation performance clutter present association drop clutter without pca model individual row experiment individual template book background low hierarchical feedforward module signature vector module architecture concern traditional single hierarchy face datum oppose commonly distinguish face face rarely clutter positive high response template activate face become severe face spatial pooling resolution selective template clutter act gate prevent face representation model use present module leave low template second class specific template plane rotation layer temporal association video unified architecture operating domain plane temporal video face theory consider plausible module template book arise naturally consequence face specific resemble recognition hierarchy cell face resemble face use viewpoint explain study purpose prevent investigate face categorization distinguish train video fashion training patch layer template video template generate template video face video video may thing face third speed video frame average complex cell overlap complex cell complex domain depend place video purpose none cell final biased long video video complex equally evenly video simple video complex video simple cc c acc acc acc align sift mrf al align observer far concentrate recognition thought depend thought learn natural video depth categorization angle invariance class specific follow temporal strategy explore apply categorization figure video also per category template frame performance drop pool part think object perform use face amount like regard contribution establish possible minimal supervision representation enough height width network apply image colored video frame randomly central plane rotation horizontal flip single patch frame aspect ratio pyramid scale ratio scale pyramid try level feature filtering size pca project eigenvector x image low scale pyramid pyramid scale pyramid scale layer template e pyramid separately convolutional convolutional network three scale template layer dot cnns dot dot product nonlinear cell pooling result pyramid template pyramid rotation stage pool location plane transformation layer concatenation encode second store template simple face feature scale location normalize dot store layer training template e cell adjacent frame frames cell dot product perform input cell final concatenation response perspective spatial domain domain detail image randomly sample patch rotation patch call single video preserve pyramid ratio pyramid frame layer concatenation face first template use template internet x pooling step dimensionality template term refer try modeling mostly explore cnns pyramid get scale pyramid template pyramid template separately cell convolution pyramid template pyramid rotation pool location rotation angle output video consist template column project template window eigenvector fast dot product reduce dimensional space adopt plausible test versus frame pool pt natural video tend remain tends rapidly effort temporal enable useful representation demonstrate e video visual representation perform computer vision benchmark big million example advance aim understand plausible feedforward hierarchy representation video operation perform normalize dot product pooling
robust noise ij adopt project update ordinary project newly projection compute reconstruct compute top singular svd suffer optimal helpful multiple cauchy pca recover simulated usage real corruption evaluate robustness cauchy pattern magnitude try recover sample entry call corruption magnitude corrupt pattern student pca gaussian pca intrinsic matrix cauchy tune trade large recovery true result pca show corruption row display high third pca pca corruption display reason quickly great rate great work noise become pca perfectly pca increase analysis suitable sparse kind small cauchy comparable third row cauchy small corruption large instance average cauchy suffer student pca similar pca row student nearly pca large second student bad conjecture reason em student thereby face recognition image severe randomly percentage pixel corrupt recognition adopt use pca recover matrix training project basis face face subspace assign near testing corruption define recognize face extend individual individual randomly pixel replace integer take replication normalize unit pca vector tradeoff laplace pca recover rank figure corruption see robust dense noise exceed pca stable laplace pca drop pca drop pca achieve recognition bad laplace pca face image heavily corrupt reconstruct severe laplace result reconstruct recognize reconstruction even original appearance fourth wrong successfully appearance recognize student show cauchy outperform pca gaussian pca possess comparable laplace real theoretical future solver cauchy cs edu principal component pca application machine text mining computer vision pca noise laplace assumption propose cauchy pca simple utilize cauchy derive pca constraint matrix regardless sparse robustness pca robust statistic view present singular experimental simulated demonstrate component analysis relate subspace factorization widely reduction compression image extraction visualization pca pca magnitude effect gaussian quadratic method probabilistic paradigm accord noise item fitness try covariance remove corrupt probabilistic paradigm difficult possible desirable treatment pca mixture factorization facilitate probabilistic technique robust student student heavy tail magnitude suffer new laplace pca dense laplace induce thereby dense avoid drawback distribution try probabilistic student opinion roughly abundance noise limited noise dense neither pca suffice reality factorization illumination optical tracking quite tag image attribute considerable negative popularity low mobile device million video publish share capture capture video contaminate video move noise corrupt contaminate large component pursuit corrupt wise magnitude infeasible application probabilistic cauchy derive unable handle dense formulate entirely corrupted assume corruption entry another pca student distribution observe generate student student infinite vary expectation maximization infer learn parameter empirically noise well pca laplace pca intuition noise compare cauchy value parameterize family distribution transform parameter shifted estimate maximize negative laplace pca general specify laplace respectively curve gaussian cauchy enable aligned location peak motivation put mode give sense heavy away center drop quickly laplace cauchy density heavy tail far reasonably heavy since certain amount thereby cauchy pca naturally possess deal location
predict persistence remarkably achieve curve operate characteristic week student persistence last week build modular framework iteratively rest organize organization feature make present technique present present present finding previously focus circuit raw click stream learner page visit server side object comment store note view infer click stream passive view infer stream example database contain correctness infer click stream include release raw receive include learner scale organized schema result schema design capture thereby utilize standardized schema report scope intensive raw database significantly disk gb gb normalization crucial entire ram enable snapshot schema slice require explanatory express explanatory due basis week module regular modular slice slice week week ht regardless nature take active interaction etc assignment course access course page stop slice exercise illustrate assignment module week attempt definition learner consistently course assignment week learner stop show week ever course learner stop week never never analysis another learner week week course learner week learner range never predict week week week ht predict week predictive label represent many historical lag ahead value diagram careful stop week word stop learner point include stop easy stop illustrate realistic platform could week week end week predict exist discriminative form covariate learner attempt week lead ht treat treat learner decide divide rough surrogate variable choose course specifically divided learner page four learner passive learner name passive view learner learner learner assign chart size learner use build engineering length sake brevity pt height htp stop duration spend length distinct problem number distinct number distinct number per duration per ratio total spend distinct time problem week event max duration duration time spend spend book duration total spend resource terminology attempt could therefore htp covariate response response percentile student week percent maximum week week week student week week student past percent percentage total correct average problem due date week logistic commonly covariate input z shape note range ht range function weights logit function rather arbitrary linear logistic coefficient suit fit covariate covariate predict training example datum training iteratively likelihood random accordingly call represent final step evaluate comprise covariate evaluate logistic apply label datum point decision label confusion thus obtain operating evaluate multiple classifier heat present problem different heat roc predict hard become historical enable prediction week change relatively understand feature useful assess treatment randomize logistic regression hour ht student early maintain every lead combination set outline chapter involve fold train fold dataset put follow determine auc evaluate validation auc roc figures receiver operate lead logistic high experiment predict week collaborative accuracy result auc diagonal represent experiment lead capable week fairly week across model high accuracy passive far result week compute ht deeply try persistence week week practically enable predict student finish week potentially reason content become student early sign intervention model successful capture student persistence remarkably generate auc reach passive include ability reach student stop become give rough student week hold true course content thing firstly remarkably sign persistence secondly student would perform size increase student student student course power include collaborative predictive predictive detail refer user week yield fairly auc student persistent counterpart week instead thereby four perhaps passive student attempt lag week week attempt week auc collaborative week week week equip reasonably prediction significant week achieve high suggest prediction finish course small week accurately train student summarize reflect may persistence randomized methodology briefly four importance summarize finding regression datum service call ever group mit multi optimize attempt find armed bandit search balance fashion confusion validation training run lag since create choose lag combination regression predict lead lag lag file pass manner give good test roc auc auc attain similar hmm model varied include stochastic support lead indicate power predictive note vary self self crowd change lead conclude focus significantly lead combination neighbor great deal size evidence relation identify dropout analysis context distance education relevant literature table list list axis intend purpose axis identify whether student dropout model completion reason excellent study actual course insight record single survey collect module however build contrast take would lag interval historical predicting interval first good knowledge systematically prediction week course excellent study concern could intervention error persistence receiver operate auc metric measuring efficacy error aim provide choose intervention receiver curve categorization capture behavior student behavioral student age financial behavioral self model neither depend behavioral age play especially far allow transfer behavioral common behavioral education use college study student behavioral student resource performance tackle challenge capture student platform argue enable identification attribute knowledge detailed orient variable derive exploit vary summary per minute per process g scoring week invariant time important factored behavioral study variable summary usually aggregated course aspect close course different survey play survey collect specific motivation reference describe student test common among collect student among mistake manual survey student response survey survey ask question fails accurately trace include trace fine grain consider build interpretation student htp contain comprehensive overview number year follow finding summarize study find htp interest completion study goal primarily identify research study steady progress identify source influence identify student performance student behavior entire course week form longitudinal study take longitudinal variable later success formation language economic explanatory final integrated behavioral strongly persistence three close week ahead paper predictive throughout emphasize click stream feature explain student success variable auc week ahead passive focus form frequently capture learner interaction resource work student derive learner interaction available subset third learner generate different lag build require detailed description employ operate characteristic advance difficult predict student end week extra effort trend rather important successful variety consistent yield superior consistent across make exception less notably crowd familiar propose highly effort crowd research suggest education inform crowd realistic effort make overall incorporate student problem result arguably student relate student trend predictive count involve class strict frequency collaborative e power project reveal modeling choice variety way model fed model numerous challenge turn high systematically thorough exploration never know successfully aspect quick conditioning create time manual manually etc ready think extract flexible way utilize crowd rich discretize additionally number model discriminative include enable scope building especially include framework possible run hundred framework resource investigate mind create scalable methodology source would apply due shared schema attention scalability support wide applicability multiple would review produce microsoft template web template camera copy page template format space acknowledgment inside complete please detail item entirely web process available conference web paper universal review reach consider publication abstract anonymous facilitate blind review identify appear title page format another publish apply substantially 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simply accept argument file include author review publish author refer phrase reveal e work show remove name exception choose copy mind supplementary accept final camera copy camera ready title name appear point bold type mail address author whereas email author address leave author author file package final address abstract bold type abstract leave hand hand margin space abstract self limit paragraph six seven reader understand pt bold type leave subsection pt bold leave please level within subsection line paragraph without relevant appear want
random possibility bivariate fit separately kolmogorov distance fit distribution base distribution computational address component follow bivariate mle recommend asymptotic mle interval adopt computational approach handle statistical inference satisfactory acknowledgement author constructive suggestion b b row represent base represent approach c united united united rapid rapid pt pt department university university deal strength component stress asymptotic testing asymptotic discuss confidence test carry example give complexity wave wave wave induce wave reliability life role density cumulative survival vector joint independent random survival stress variable reliability take exceed stress interval due point reliability stress strength attract estimation normally consider estimate determined bivariate type application stress collect logistic laplace beta inference reliability stress ml system consider pareto strength twice system study white count stress derive bivariate exponentially distribute stochastically independent distribution estimating likelihood pareto bivariate main random mle give different confidence interval step computational provide result illustrate stress survival function easily let sample binomial obtaining determine also observation observation mle size interval percentile bootstrap compute bootstrap estimate approximate bootstrap reliability approach cat use estimate sample goal suitable express obtain equation cat restricted mle denote artificial iid mle testing calculate alternatively calculate either great cat construct reliability take point cat curve suitable suitable smooth finally solution interval intend computational present behavior various parameter numerical confidence bias size parameter replication mle bias size satisfactory increase decrease property average length increase coverage small
choose make derivation simple derivation original square number observation large exploit datum often low low solve root lasso fast modification effort find sketch sketch take solve approximate observation singular retain validation fast traditional approach employ phase extensive body algorithmic include power random sampling nystr provide flexibility work problem widely different regularization parameter popular include warm algorithm strong incremental training employ technique learn develop multiple generic solver present dimension complexity conclude assume robust lasso bad perturbation frobenius rewrite elastic net directly design employ low rank approximation algorithm approximate leave analysis design leave elastic problem dual u term since onto give optimal unbounded unbounded dual let matrix original bad exploit complexity grow present eq without replace rank robust root application attractive nevertheless emphasis care take involve show root approximation tuning parameter approximation lose insight absence sparsity square root exist uniquely suffice show cardinality solution generality discard column provide feasibility optimality dependent j zero alternative formulate problem constraint simple case lasso author order barrier generic primal dual specialized specific paper develop specific root involve barrier function original root inverse hessian q w therefore rearrange b log function rearrange q ap p hessian therefore barrier plus iteration invert hessian inversion lemma cost original root synthetic real life set set scale table gb safe robust dense sim dense random dense experiment complexity model time run repeat time reduction focus classical show run fold digit second second perform leave become impractical even carry validation report life second binary sparse corpus pixel data evaluation testing require cpu need second set second framework imaging image analogous spirit leave one analysis imaging remove lasso explore topic query datum share answer query robust text corpora papers york table query computational query topic computer political research data query image error image student
outcome row marginal contingency table choose odd risk logit function obtain association measure natural logarithm cumulative follow formula cumulative ratio pn response q age age I k k k k otherwise paper support indicator n order logistic model jointly response covariate odd ratio become rich model ordinary sometimes parsimonious fit response configuration proportional odd fisher scoring algorithm either fail association approach penalty difference effect demand ordinary curve surface penalize limit study proposal application patient play role order categorical datum marginal base bivariate order completely solve multivariate strict appealing constraint contrary like discretized version due lack subject matter restriction strict ordering constraint appropriate helpful possible range penalization consider surprisingly penalization ridge ordinal computationally longitudinal association refer form smoothing spline response regression parameter score
remain difference minimize divergence evidence lower solve maximize especially linearity trace form extension term generate create initialize comment variational determinant operation multiplication weight equation assess vb glm fit process interaction canonical intensity center intensity parameter strict definition usually parameterize simulate range maximal packing limit intensity replace lead additional prior flat correct tight around wrong state expert practice experiment point configuration current gibbs package legend correspond row respectively error estimate simulation vb standard r show vb prior induce extra performance tight value concentrated prior high likewise intensity large fully intensity order expert knowledge consist cell whether trend component intensity fourth vertical figure pointwise trend intercept posterior trend trend clearly separate especially model figure question trend derive repeatedly simulate point variable execution pointwise trend call priori give function converge gaussian exclude self whenever conditional intensity back situation impose prior short interaction conditional connect intermediate interaction weight thick step smoothed gray line mean red interaction estimate line depict motivated atomic molecular interaction model show window intensity acceptable characteristic infer maxima mean range characteristic range extend extension polynomial depict basis function characteristic range resolution width correspond smoothness beyond exposition account intensity process decade communication superior use poisson pseudo connection currently estimate recall surprisingly little approximate use feasibility scalability value example trend pattern consume model involve simulation describe provide computation computational bayesian counterpart simulation bivariate replace community orient emphasis might numerical connection advantage author support foundation rgb rgb rgb attractive logistic method exponential spatial combine variational technique technique demonstrate gibbs point model interact location inherently costly normalizing constant likelihood technique act become pattern popularity amongst likely mcmc design costly loop probability simulation provide mcmc refer depict flexible approximated framework software describe variational
layer plot learn cifar cifar activation cifar decrease introduce novel activation function neuron compute parameter activation function along parameter demonstrate lead significant deep diverse suggest activation fit suboptimal acknowledgment fellowship also acknowledge gpu thanks uci edu research com science university california usa uci theorem artificial typically neuron piecewise independently descent activation neural network compose achieve benchmark mode artificial rapid engineering imagenet science searching component linear sufficiently arbitrarily nonlinearity deep network fast accurate deep network active easy train vanish another innovation maxout achieve benchmark maxout compute activation replace impact function previous effort largely attempt set strategy powerful parametrize piecewise learn neuron descent input experiment piecewise activation unit hinge shape piecewise activation hyperparameter advance learn unit total typical maxout piecewise equation assume theorem reconstruct constrain condition may input function eliminate segment unit slope boundary point special term slope elsewhere element summation slope elsewhere last term elsewhere almost verify thus activation maxout unit unit maxout tune train impractical maxout network maxout unit expressive maxout allow maxout reproduce coefficient one maxout unit maxout tie implement unit maxout would maxout expressive penalty improve file solver tree cifar cifar color subtracting value cifar convolutional pooling pooling pool fully apply pooling drop layer pooling layer drop fully layer softmax cifar pool use activation cifar cifar almost case cifar try improve dataset also try cifar cifar image zero image knowledge report augmentation activation relu relu value cifar cifar cifar unit consistently outperform relu initialization report deviation bold cifar augmentation cnn relu maxout cnn relu relu relu supervision relu data cnn maxout cnn maxout maxout selective relu cnn units relu relu high energy event characterize energy supervised physical decay distribution area operate
x txt header x serial saddle ylabel error font pos domain title header result serial header serial txt header serial txt xlabel ylabel error font legend north restrict header index txt header table header index comma bundle risk like smooth well specialized solver regularize admm present dataset use detail table term minimize experiment svm comparable outperform demonstrate serial quickly poor probably processor problem optimum converge solution comparable trend logistic dataset time publicly dataset recognition million entire sub sampling reduce room improve performance derive random r implement communication machine run eight core sgd also reduce accelerate dual initialize machines degeneracy svm restrict tt epoch update useful recognize order order processor update st epoch processor update partitioning eq suffice serial convergence f end theorem dt convexity concavity let inequality order regularizer rearrange sum derive eq get additionally get tool update q geometric term bound conclude q instead constant term theorem remark definition axiom claim massive optimization efficiently propose remarkably linearly processor verify empirical evaluation batch minimization risk machine give furthermore regularizer brevity recover machine svms regularizer change loss general minimize risk regularizer smooth algorithm bfgs hand regularizers bundle popular solver alternate direction multiplier belong batch algorithm update iteration well computationally expensive point fact empirical risk decompose empirically accept effective regularize stochastically replace gradient step compute datum fast gradient descent second batch method bfgs usually computation speed rarely execute component processor parallel somewhat hoc paper fundamentally risk parallel minimization saddle solve prove processor verify empirical svms binary regularize risk saddle point rewrite introduce multiplier eliminate likewise component duality switch maximization minimization conjugate yield formulation minimize eliminate obtain moreover minimize saddle f equivalent problem coordinate coordinate ij denote training cardinality remarkably component optimization define interested saddle stochastic step step surprisingly regular implicitly replace therefore approach stochastic algorithm key would stacking optimize partition depicted processor fraction red processor dark rectangular active area depict processor exchange variable figure processor processor epoch active processor either active processor key carry processor intermediate processor partition begin processor coordinate point work memory distribute memory hybrid architecture fourth across machine redundant storage linearly fact local rectangle red rectangle rectangle node red blue node rectangle green rectangle color area processor dark color leave fix partition pi j epoch inner inner communication processor execute st detailed stochastic saddle prove uniformly random distribute certain order would recover produce serial convergence believe technique independent interest differ convergence parallel point respectively epoch duality please understand implication theorem processor partitioning note perform update time subset inner per finish epoch theorem number obtain conclude linearly eventually dominate effective risk receive significant limitation unfortunately partial consequently execute serial focused working limitation computing gradient parallel update processor popular share memory latter
cnn single image lead suggest alone effectively table baseline understand iterate triplet lead consistent drop equation refer word sentence dependency consistent drop dependency advantage end gradient raw pixel additional insufficient amount training cccc search ranking devise rank recall good sentence retrieval level score sentence apparent retrieve top mention blue bottom right imagenet powerful complex attribute object generalize novel class learn sentence triplet triplet high box generalize grain ball limitation failure model sentence bag relation relation noun align people phrase play count moreover maximum people inside person spatial hard distinct spurious person many example compound become separate white two relation black rise careful sentence progress address learn modal inter modal reasoning fine formulate alignment traditional ranking improve retrieval previous interpretable future counting reasoning spatial move beyond li department computer stanford usa cs stanford introduce sentence modal embed unlike map common work sentence addition previous add alignment learn associate modality extensive reasoning sentence fine improve sentence retrieval additionally interpretable prediction infer inter modal explicit ability image automate image conversely ability retrieve query immediate particular set language description rank fix sentence vice challenging require understanding modal correspondence query water retrieve corresponding entity relationship sentence complex primary deep language multimodal sentence image object dependency tree relation common embed explicitly reason inter modal allow image sentence k dataset publicly available grow mapping sentence write automatically closely naturally allow bi align scalable quadratic limited sentence triplet scene field relationship relation fall modal probabilistic represent sentence boltzmann bilinear autoencoder closely relate et introduce embed common embed rank adopt describe put entire correspondence sized top convolutional network object complex scene neural connect raw neural representation domain computer vision state detection propose gram representation sentence representation paragraph document representation retrieve image query conversely sentence query training image neural score compatible pair feed otherwise evaluate sentence sentence sort score location list core insight complex interact entity intuition break propose object tree relation true inner interpret margin sentence score strong alignment compose aforementioned objective cnn green map embed relation embed box score box compute extract visually identifiable entity child attribute dependency similarity ground higher learn objective hyperparameter validate objective detail blue box score box image mention blue violate way triplet visually identifiable triplet detect lastly visual nonetheless incomplete alignment sentence interpret visual sentence incomplete define together otherwise intuitively encourage red along green objective red box member bag score accumulate objective dense play attempt infer sentence triplet bag precise mi minimize define set bag sentence return belong inequality states sentence objective solve heuristic score bag global rank sentence truth sentence thresholded score set range member helpful add smoothing cross validate image make dot product thresholding descent sgd batch momentum validate epoch initialization epochs cnn fix switch keep overfitte concern run batch retrieval image annotate amazon sentence normalize sentence simplicity stanford compute sentence hundred overfitte concern consideration remove occur reduce implementation imagenet detection gpu second discard prediction imagenet activation connect immediately detect training image sentence sentence image compute fraction top follow
dynamic top spectrum amongst differently overall represent compactly gain great field natural language speech dictionary recognition construct phone hypothesis hypothesis actually understand sentence store lattice lattice target tuple node alphabet element encode lattice encode every character string notation simplicity path define lattice node initial consider lattice string b b string lattice representation string share among element common edge instead twice lattice share common common save data complex viterbi viterbi pruning magnitude lattice input string alphabet objective lattice task number correspond computation lattice minimal size moreover graphical inference construct lattice minimized think transition factor minimize suppose character stop correspond character merging node construct thought share merging make powerful heuristic utilize graphical structure naturally lattice encode particularly frame lattice lattice vertex transition vertex correspond character encode string lattice graphical structure determine v pg j p iv n jt dl pg n edge stay vertex source deterministic state e pg p please depend edge traversal however take jump unlikely speed please note gm stream like go rather lattice benefit prune speed underlie gm compress certain datum discriminative convert within certain around peak lattice alphabet integer time overhead query show model lattice behave gm feed track number frame max value max decrease specify share rest lattice rest theoretically lower pr cb negligible show contain z fed vertex intensity act affect gm compress encode instance small compare instance intuitive illustration lattice contain disjoint path path separately lattice simple lattice merge share get would size grow task mass hundred within redundant may prune algorithm inference fit perfectly prune strategy prune frame space state original various single candidate pruning strategy still spectrum pruning candidate apply pruning space end candidate lattice jump depict jump good matching small rarely lattice allow pruning pruning tree pruning jump mention gain acceleration amenable tool cause well boost overall evidence let generative find ps wish maximize maximization much discriminative wherein simultaneously parameterize set hypothesis candidate within mass criterion define denote maximize respect approximate converge obvious dx I p possible candidate denominator make objective compute efficiently use stochastic gradient ascent ascent calculate regard vector correspondingly previous detailed practice begin generative move maximize denominator encourage improvement numerator incorrect denominator influence ns discriminative execute denominator calculate possible infeasible represent e hardness constrain peak graphical theoretical since unit theoretical peak value matter sure mass mass perfectly solve scalability lattice together strategy discriminative lattice feasible lattice general hypothesis dynamic denominator lattice even graphical capable encoding amenable lattice achievable consist spectra consist high spectra regard database find available supplementary order score engine set rather protein score choose specify error low resolution mass tolerance th whereas set search peak whereas search provide two benchmark neutral peak peak model absence therefore follow spectra work target identify fdr identify significance monotonic score instead define minimum fdr incorrect plot dataset margin though two method ii gap training representative high quality currently incorporation entire believe critical score arbitrary worth ms equal evaluate regardless flexible result build dataset peak alphabet effectiveness varie mass peak candidate window increase reduction effectiveness improve time lattice pruning pruning setting describe prune wider early prune moreover prune space absolute efficiency give original speed respectively record engine experiment ghz cpu cpu report test utilize run expense high confidence capability hand place unit along access figure low arguably important note prune seven influence choose intensity spectrum peak peak train mean mean well scoring model dramatically fold ability compactly entire framework gain also greatly allow future investigate way exact computed increase training train state effort encounter plan high simplify process throughput technology quantify produce assign observe responsible generating spectrum recently network rapid achieve variety return valuable regard work significant improvement widely use process score give spectrum database thereby allow share candidate demonstrate across datum introduce variant entropy rather maximum enable spectrum discovery datum protein separate micro mass primary thousand spectra ideally single arguably responsible generating spectrum identification problem database genome review spectrum database scoring spectrum identification dynamic correspond model viterbi decode align candidate peak spectrum spectrum peak peak axis training model observe spectra without axis current introduce improvement word make processing represent context lattice compactly collection mass spectrum viterbi allow lattice reduction expense range examine sharing among candidate dynamic programming spectrum exponential score compute ms via denote potential tractable sequence basic determine amongst last frame expand depict figure r spectrum frame spectrum
histogram reasonably criterion estimate mae suggest reconstruction besides mae rmse observe large h error norm mae rmse unobserved calculate come population mae seem display indicate obtained suffer unobserved mae star nan difference star reject indicate significance uniformity er von approach section result poisson describe marginal reliability observe panel reconstruction whereas display marginal diagram fluctuation value fluctuation b b b reconstruction different clearly table well even good c mae figure display power histogram seem value difference reconstruction reconstruction clear interpolation scheme display partition employ make tail adjust h table reconstruct mae error little reconstruction reconstruction density cc cc mae rmse unobserved display mae rmse clearly display reconstruction mae ks uniformity er present loss e confidence serve reconstruction confidence reconstruction table simulate set additionally absolute var reconstruction var increase list calculate replacement datum cc error ccc cc table var reconstruction belong empirical confidence table aggregate nevertheless apply allow compute individual probability laplace contain help determine uniformity reconstruct large datum loss cumulative cumulative difference reject critical statistic distribution asymptotically summary reject level respectively test ks difference problematic sure sure test kolmogorov test version ks point case cumulative cumulative ad von statistic maximum integrate distance sample variance distribution ks statistic uniformity great er von reject statistic behave von goodness tail al fs normal make powerful rely uniformity besides test provide autoregressive possibly lr formulate function associate autoregressive convenience nan hypothesis hypothesis respectively supplement normality ensure standard skewness second third central square freedom normal nan hypothesis follow normal reject nan affected skewness modification utilize absolute deviation skewness number total loss robust degree nan hypothesis reject confidence respectively test ks ad whether integral help detection reconstruction dependence also power skewness sometimes box test overall randomness lag et plot serve fit close additionally overfitte fit lie line curve diagram cumulative go tool marginally estimation graphical device versus cumulative loss calibration quantile comment business iii de present compound empirically fractional method reconstruction obtain variety criterion good estimation var important management loss loss side compound entropy analytic transform available observe determine distribution loss sum frequency probabilistic inverse know compound describe accumulate loss towards advanced capital determine work shall frequency loss compound distribute technique exist propose fall transform actually try transform determined analytically axis frequency poisson loss compound may compound numerical happen many regard loss follow transform laplace transform carry begin describe loss frequently amount business cause tail possibility large claim correspond loss company bank implication determination laplace calculate simulate loss infer fractional begin think identity order relate mass change paper knowledge historical frequency loss available possible loss could order remainder paper organize recall additionally quick overview robustness determine reference devote computation two measure could interesting operational loss capital devote role conclude remark appendix test method fractional variational inverse consist find constraint care natural requirement point maximize concave functional entropy minimum problem actually moment explicitly generic minimize denote scalar obviously another technique auxiliary determined numerically factor add normalization equation positivity determine seem improvement dimensionality view continue positivity constraint borel point coordinate reference search measure first restriction upon hull generate purpose respect positivity introduce probability close whenever finite routine setup generic entropy define version duality achieve case idea value reference product poisson measure unit dirac delta certainly integer notice minimize forget matrix recall determine interpolation necessary inherently explore make set add detail exploratory comparison tool like calibration agreement reliability serve determine quality proximity quantile close loss cumulative empirical minor fluctuation zero estimation see among bin histogram bin I data disadvantage bins rmse distribution versus calculate distribution observe al measure fit bin due possibility density transform back et transform integral interest sample deviation uniformity indicate reconstruction fail aspect uniformity visual inspection histogram autocorrelation plot ks er von al inverse joint normality independence combine normality use evaluate quality reconstruction comparison make simulated compound overfitte well help ability perform unobserved example compound process frequency period poisson distribute analytically deviation simulate
fx hx x element state control denote rl function compact continuity function concern scheduling note cost constant time become infinite run multiply cost step geometric utilize option utilize continuously versus vanish origin definite guarantee state bellman equation read policy give horizon end unknown vi consider several question arise relation guess guarantee I sequence converge continuous author vi iteration go compare convergence question simplify horizon extend answer proof though continuity necessarily concern address uniform convergence advance utilize state instead vi reason horizon infinity final horizon actually vi regard vi address three dynamic system perfectly iteration vi approximate analyse need set vector besides origin sense running function zero need reason trajectory stay dynamic iteration lipschitz lipschitz recursion one vx vx hold stability lyapunov stay stability follow negative invariance lead value approximation matter fact converge ic x next provide stability iteration give continuous ic eq vx asymptotically system eq inequality stability low boundedness guarantee lyapunov continuity continuity error approximation rich helpful boundedness function place offline scheduling eq toward utilize separate rl control learn dependent learning infinite let take q vi select guess learning call however scheme follow replace action fact require independent stage eliminate learning require also matter learn car action let approximated possible second initial select measure action dependent go entirely need identify e g worth fit rl condition conventional decision algorithm chance behavior system call online decision motivate rl conduct note still select never chance learn never exploitation concern concern still decision exploitation possibly result concern utilize guess work ref investigate stability present make switch hand result theorem extend basis due page constant access denote dependent vx step long inclusion iteration recursive relation admissible call analysis policy address already control controller simple admissible policy obtain respectively admissible within select eq prove action dependent dependent iteration converge action select ix result action iteration system however apply step fix stability system similar conventional require stability form converge finding function lyapunov policy however policy subject adaptation origin origin region evolve policy r policy subject iteration analysis stability learn present new sensor controller together respective utilize stability capability monitor switch boundary stability scheme respective discretized sampling fourth conduct least take computer intel ghz single control condition simulate real world force open loop fashion calculate assume result comparison purpose history open loop fashion plot call performance deal incorporate stochastic nature policy utilize drop chance force simulation fig show capability loss transmission try controller successful another important simulate performance free dynamic van feedback linearization discretize policy element h calculated guess act additive apply case plot communication entire online learn use respective exploration exploitation choose scheduling black plot history respective also see end fraction bandwidth compare policy designing policy provide control approach load call take several leave particularly online lead consider limit function equal hence prove integer since x w term non continuous continuous finite continuous characteristic ref establish feature proof result dependent induction result use therefore completes consider finite cost step history horizon remain finite early nature consider limit function action decision decision evaluate unbounded per set cost go horizon cost cost consider value infinite horizon great value fix prove continuity result theorem function v dependent continuous function continuous guess switch lyapunov initial admissible continuous definite hence induction every proof continuous v ix consider invariance theorem close ix kx close origin monotonicity establish let kx k hand since definite drop inequality hand non entire trajectory contain v kx v definition v kx r prove trajectory inside school rapid city sd phone email edu problem allocation unlike discuss develop infinite include zero investigate development extend model present optimal analysis unlike conventional system loop controller require measurement task load maintain example generator point common spatially generator power different control engine etc sensor spatially throughout unified cost facilitate monitoring capability change literature develop approach design decrease reduce load design consequence induce quantization error digital datum effective approach loss delay periodic typically monitor current system scheduling available hold measurement generalize controller design assumption cite paper literature optimally rarely investigate study aim extend application dynamic rl horizon simplify control feedback receive investigate switch behind advanced case delay scheme horizon stability lead load scheme controller sharing scheduling state respective conduct law policy adjust transmission continuously though approach case previously receive state controller one finite horizon design scheduling function minimize store last receive measurement store time state measurement network control looking dependency consider system suppose characterize
transition assumption continuously lipschitz main assumption big theorem take ts ts sequence dynamic achieve select policy regret treat trajectory include layer ts repeat ts ts ts ts ts ts ts ts ts get side notice martingale hoeffding inequality almost surely probability least union simultaneously least union fix let l tc l eq inequality lemma sorting ts ai optimistic function optimistic optimistic policy obtain proof efficiently maximization confidence possible transition complexity number transition side provide algorithm aware precise beneficial enjoy regard ts parameter achieve rich far superior free state concern policy I td control method mdps reward transition change arbitrarily set see difficult version reveal learner select policy expect dynamic regret pool sublinear interesting future mdp make assumption learn adversary future plan ahead markovian trivial combination idea principle sequence construct achieve setting cm cm study finite markov decision mdp clinical trial recommendation objective dynamic take process mdp problem clinical patient response past outcome collect side simple make would treatment patient model patient current regret mdp alternatively mdp problem change kernel efficient paper problem markov decision transition depend new previous side test option apply patient decision transition influence patient formulate decision principled utilize reward goal notation markov decision mdp characterize previous policy lp consider free transition reward influence transition choose give sigmoid vector reward parametrize feature episode rise nearly dynamic policy account achievable expect algorithm begin action space construct equation l algorithm principle employ online
scene composition texture rest supplementary request figure relation indicate decomposition trivial square provide powerful hope various scientific show computer vision code large image acknowledgement berkeley france berkeley program research centre theorem axiom department california berkeley chen berkeley edu unsupervise call code gain lot though interpretable efficient implementation publicly important scientific bring fast scheme active demonstrate computer task codebook visualization unsupervise technique widely used automatically discover underlying structure serve several purpose look exhibit interpretable example neuron activation population similar topic text collection vision unsupervise model mixture gaussian yield descriptor unsupervise visual recognition probably task purpose visualization call interpret provide prediction analysis discover unlike factor force point association association centroid centroid among interestingly popular nmf independently around time approximate factorization lot analysis address develop optimization active scalable exist implementation believe application bioinformatics processing perform code recognition second analysis database image analysis section vision conclude let represent factorial look vector approximated combination close coefficient simplex replace factorization challenge non relate briefly negative seek component negative analysis fix norm sparsity induce produce sparse formulation negativity aside main vector encourage become combination useful entry input variant code datum use decomposition element interpret anchor represent automatically differ propose variant huber often replacement scalar since grow cost section huber natural notice solve qp update rewrite residual coordinate optimization g tb input initialize initialize x x decomposition matrix strategy way efficiently quadratic simplex huber robust reformulate per fix optimize program vector optimize block guarantee converge stationary point tb input initialize I carry carry line various code matlab toolbox performance software package implement publicly toolbox qp solver package level implement original analysis intend method unfortunately software package severe report computational intel report result package order magnitude slow experiment qp solver study scalability package mnist potentially main limitation limitation share classical regard slow converge right mnist far image patch small pixel encode descriptor sift codebook pattern call image finally histogram occurrence yield powerful task typical method sift descriptor encode max yield simple bag benchmark dataset ultimately svm sparse code conduct image classification replace code categorization demonstrate learn codebook sparse code similar slightly involve recognition report perform well use class rest testing lc classification testing randomize even rule digit class test classify good eq q normalize want thus hull near mnist remarkable aa
panel seen estimate right panel single u rl filtering sequence similar previous complexity evolution outperform vector eigenvalue improve structure n nu summarize fig depict outperform competition twice n gaussian also regard three compact yield single stream filtering meet united union present focus algorithm future pass type pass enjoy convergence value product present ridge space minimizer j weighted definition rgb novel iterative projection reproduce hilbert nonlinear nonlinear component multiple permit propose meet hyperplane certain efficacy reproduce hilbert filtering reproduce nonlinear adaptive filtering investigate reproduce author kernel approach multiple subsequently propose multiple component ii high adequate amount unknown limited time vary adequate system investigate norm situation multiple compact representation efficient name affine seek functional gradient hilbert space rkhs build unknown implie datum selective new criterion coherence criterion introduce raise issue regard enter discard contribution though adjust coefficient dictionary algorithm systematically enforce algorithm considerable dissimilarity project hyperplane instantaneous operate e rkhs operate filter present reveal significant interest stream meet bl bl bl bl bl bl bl bl bl bl bl bl bl bl bl bl product bl bl formulation article filtering projection fig characterize superposition lie infinitely way cause avoid particular trivially I share cover important mean sum direct sum space derive rkh uniqueness decomposition sum hilbert imply derivation sum key turn another simultaneously nest structure cover case intersect trivially nonzero intractable product hand close formulate formula selective numerical example effective nonlinear low apply efficacy rest space show particular theorem complexity example conclude nonnegative matrix bold face identity denote function sequentially output focused case component nonlinear etc describe rkh associate denote element unique indicate sum rkh reproduce apply recursively kernel real hilbert w reproduce ridge easy handle appendix processing product close fortunately inner build adaptive kernel span n atom initial initial filter assume element section useful due unique reduce correspondence tuple hilbert equip example positive typical know nonempty rkh associate gaussian nonzero assume nonempty manually gaussian kernel within contain devoted corollary mind present possible nonempty see j elsewhere n c kernel grow pruning adequate grow start n criterion modification novel n time instant new measurement strategy accept normalize project zero instantaneous relaxed therein assume search size affine dd hold hold computation involve p lemma rkh definite jj normal eq gram matrix entry indicate inversion invertible kernel size determine unnecessary orthonormal case reduce complexity selective update kernel geometrically coherence p form approximate n coherence justify example subspace dictionary element follow compute subspace propose state although proposition rather slightly kernel arbitrary satisfy l l hence obtain w light share common product article rkh straightforward nonempty interior exploit characterization prove another appear theorem strategy case analogy adopt criterion use space unknown contain system element enter virtue argument translate derivation product space fortunately translate even therefore formulation follow emphasize write case hyperplane selective gaussian kernel appropriately nr build project vector euclidean obtain call algorithm except individual dictionary multiplication dictionary gaussian complexity dictionary size subset denote submatrix suppose use selective update coefficient matrix inverse partition inversion addition inversion need update demand number coefficient computationally demanding coefficient n n reason
considerably efficiency unified complementary capability advantage base approximation residual sparse residual impose conditional preserve efficiency remark paper argue strong refine residual exploiting perhaps scalability work utilize unify describe approximation relax conditional large section improve predictive leveraging advantage reduce residual latter kullback leibler subject consequently trade size gp rank parameter achieve comparable accurately represent markov spectrum approximation advantage sparse parallelization core great scalability perform traffic implement cluster speedup empirically evaluate real represent input realize value output variable unobserved finite gaussian regression provide predictive u dx limitation practical poor inverting incur propose scalability sx approximation sx input realize unobserve reduced covariance section representation include unified approach approximate matrix contrast refined covariance set input partition scheme evenly disjoint correlate key block matrix b impose process comprise illustrate ease block block recursive series reduce block diagonal e large rank residual covariance matrix approximate v v r five block band block nb specify band rank approximation offer interpretation impose far bb fall outside outside block approximate v generalize vary markov utilize u predictive uncertainty directly invert regression issue leverage associate block n use block band follow specifically spirit impose residual independent give assumption relax importantly assumption equivalently achieve proof though utilize representation unify residual matrix impose strong assume reveal dr closely kullback kl minimum r proof appendix exploit derive formulation amenable parallelization core construct tuple tuple construct summation master master construct tuple u receive tuple uncertainty u local recursive discuss parallelization show mention scalability b b centralize incur cubic increase core reduce centralized respectively speedup parallel centralize increase improved increase overhead incur cubic cost achieve desire toy local exhibit prediction partitioning evaluate predictive scalability art real dataset dynamic degree freedom robot traffic km segment road peak hour road comprise five traffic dataset relational structure segment road topology traffic dataset gps whose prior define ix length scale kronecker delta estimation support dataset change large spectral table platform via memory core core respectively compute storing respectively subset metric use evaluate error b incurred rmse test parallel core incur parallel average core predictive likewise data independence incur expect scale incur incur minute incur time parallel achieve comparable predictive process remark oppose report table incur cause dominate incur incur time remain stable structural result parallel counterpart instance vary core incur centralized centralized increase centralize incur centralize respectively hour centralized incur almost possible setting huge centralized block e expect incur huge entail scale operation huge cache highlight sufficiently support single cache incur speedup increase explain speedup appear increase core primarily cache incur cccc datum size core gray rmse right fig markov use incur order vice versa respective second second incur indicate increase predictive performance latter cholesky easily incur I second use second see core less machine scalability w input denote month day incremental day start day output mean level pressure pa setup platform node run ghz gb ccccc incur core vary size incur parallel set parallel insufficient share memory parallel issue incur hour summary experimental significantly scalable achieve comparable fast achieve incur centralize achieve performance considerably markov early trading support markov incur reliable alternative increase achieve huge cause cholesky factorization insufficient share dataset describe computational markov assumption utilize residual centralized plan automatically variant stochastic plan release http code google com acknowledgment mit technology observation block band observation r follow lemma necessary derive sparsity cholesky u mn upper mn mm b mr proof directly cholesky q lemma fourth last last fourth definition
challenge agnostic general hypothesis requirement disagreement base active open key contribution independent interest connection active rate allow error rate construct label query agnostic contribution rate guarantee classification low extend predictor agnostic setting rate general active consistent complexity show bound label lie label space example marginal denote access oracle label input vc respect h h ss data oracle oracle query oracle possible say agnostic active frequently use disagreement two hypothesis assign formally b r h hypothesis sense significantly finally set hypothesis connection rate rated allow consider predictor hypothesis rate predictor label ensure risk predict predict rate predictor measure performance proceed epoch epoch achieve maintain contain epoch select rate predictor epoch run label adjust excess generalization error oracle vc rate excess call generate rate predictor induce example set distribution excess confidence get minimizer label class rate predictor excess minimizer label risk recall precise explain oracle oracle rate predictor target confidence label still maintain rate guarantee predictor candidate hypothesis minimizer example simplicity distinct output k query epoch excess passive factor agnostic key achieve large excess fraction rate get generally disagreement active input oracle hypothesis confidence rate predictor particular return satisfie agnostic state adaptively find query lemma suppose excess h j succeed hypothesis set example target excess confidence constant algorithm succeed use give non oracle hypothesis class rate labeling draw query label erm follow confidence rate guarantee rate error predictor optimal interest receive likely contain guarantee predictor predict constraint expect disagreement assign goal fraction key program disagreement maximize equation solve program rate confidence rate predict wrong predictor version px set rate coverage rate predictor much coverage essential true label observe rate guarantee subroutine consistency oracle rate predictor target confidence bind subroutine set rate disagreement formally x h db learning label oracle confidence rate predictor exist agnostic disagreement active characterize region label disagreement bind simplify require q label disagreement base active learning contrast replace disagreement noise noise label class condition condition oracle hypothesis rate target constant eq provide c analysis dc comparable consider concave much lead log establish area demonstrate active rated potentially label thank nsf helpful thank wang introduce problem selective vc corollary vc bind pick vc due pick label consistent version pick copy joint induce rate confidence rate predictor datum satisfy elsewhere x combine lemma hold succeed example immediate consequence dense labeling target j assume h j hence stop equation j equation third exist turn make iteration ensure crucial noise necessarily hold namely meet lemma dependence query suppose exist algorithm event succeed approximate true risk first h jj combine output event iteration j n cn cn cn cn last fact cn cn cn j j bc j j suppose classifier set minimizer happen lemma second follow cn equation suppose exist iid let combine equation cn equation triangle c plugging equation get prove thus happen induction clearly show inductive know succeed eq get lemma triangle dx dx dx dx kx ix dx divide q induction event thus k equation q follow happen induction clearly q succeed target combine combine equation px px x px assign I exist violate generate could eq event equation lemma run follow assumption h yx third succeed last second fact b h dx h c yx dx yx second equation c yx yx yx yx c yx yx dx follow lemma triangle yx dx yx h inequality follow divide get succeed dc total immediate item concave homogeneous classifier exist exist follow algebra label excess follow example excess confidence v happen succeed h combine eq call input oracle confidence rate excess succeed input example eq algebra begin observe combine thus equation moreover equation rate x two get eq algorithm consider solution lp z z show feasible coverage h n union bind probability follow hold program k satisfy weight iid copy least proof fairly vc u n n jensen third fourth hypothesis
sparse regressor leave prevent overfitte consensus message come form indicate algorithm belief presence pixel job square probabilistic segmentation regressor perform effectively count stable inference stage train improve accuracy latent fig insufficient demonstrate inference capability message pass normal image leave product n pl pn pn pl pr pn circle south fill red red south south realistic face primary motivation recognition pose approach normal pixel fig formation infinitely distant prior normal light side vector side approximately face line code model rapid although model formation similar successfully reliable usefulness simply true formation accurately column sep row mp mp mp visualization infer map fig consensus type describe fig predict contextual message predictor information message predictor guess estimate image directly forest create behaviour contextual message median max patch around contextual imagine regressor performance experiment dataset contain illumination remove entirely leave around normal proxy code qualitatively assess inference inference map map message pass match closely produce reference strong region mp pass inaccurate illumination cast inference produce arguably improve cast cast suggest future cast fig ability infer task recognition estimate rmse strongly compare close reflect choose result experiment take cast light error cosine angle distance estimate variational pass mp perform poorly produce inference horizontal line consensus pass fine light pass fig mp bad demonstrate use oppose direct prediction help message well presence mis make east column sep row mp forest stem kind decade intuitive rational see message consensus form pass intuitive exist proposal lead speedup parameter long early dedicated inference pass define predictor jointly train system produce long distinction predictor within pass regressor random forest message work concerned message attempt reduce accurate technique message recognize see rough success depend forest completely take work generic broad building contextual like framework computer vision make consideration cast face develop understand application broadly major scalable model interpretation increasingly heterogeneous appeal complexity difficulty barrier goal barrier anonymous microsoft research wish map message challenge message special care methodology give incoming review forest approximate pass message represent represent message different concatenation call parametrization value leaf label previously unseen construction likely contain similar example leaf message multivariate term contextual message leave regressor greedy manner node incoming split correspond j set residual incoming capture contextual message root leave forest might predict tree however sensitive choose moment moment show map normal choose show illumination condition produce close illumination extend synthetic image image use superior baseline quantitative use cosine posterior superior baseline synthetic image h light normal anchor column sep sep mp forest forest mp consensus normalize row cast infer normal map node anchor sep cm cm observe forest forest variance consensus estimate variance pass consensus pass style circle width inner sep cm minimum height width fill black minimum height red generative model reason imaging reason conditional traditionally domain purpose message pass expectation ep message simple vision introduce modification message learn message guide variety significantly ep generative probabilistic applicable wide language computer vision graphical incorporate surface normal approximate symmetry make counterpart perhaps mix bad effort pass many difficulty cost purpose message pass pass simple model attribute influential pass meaningful observe top additionally variable property learn early experimental variety efficiency standard message implication add tool toolbox improve bottleneck restrict exploit vision aforementioned illustration layer factor notation variable case experiment vision normal variable intensity reasoning message desirable purpose could send possess message inter layer factor practice access oracle message inductive argument regressor sec layer regressor predict variable layer inference global loop graphical due global consensus send contextual message e cl cl south pt south fill south circle circle circle fill north north north south cl north south cl north south north cl south cl north south north cl cl south cl north cl north cl north matrix cm replacement cl cl x pt circle pt south pt fill south south circle fill fill circle fill north message pass aim inference message point pass would reach useful good message pass message oracle predictor way message guide point marginal except latent instead message point target would label even strategy experiment consensus challenge message distribution need take fact supplementary material review forest illustrate diagnostic square use improve challenging face predictor train second sample significantly preserving pass experiment use default tree leave pa x sum pc pa pa c red south begin behaviour standard message gauss message initialization significant speed demonstrate circle wish coordinate circle radius graphical translate latent finally observation model express net circle layer presence take iteration converge circle marginal iteration repeat fig dash black figure marginal contain
arrive retain retain execution sampling retain particle signal retain particle state child retain branch loop entry previously retain execution retain yet able align retained particle next probabilistic strength improve approximately version inference evidence operate usage yield improvement program use posterior correctness carlo benchmark distribution experiment large sequential emission distribution crp mixture class probabilistic implement particle language probabilistic system implement approach run metropolis choice execution simultaneous metropolis engine run particle core cloud amazon ec run intel processor implementation gibbs engine generate good order engine implement particle gibbs sampling particle run repeatedly draw contrast infinity recommend kl sample posterior fair reasonably band cover median mark slope monte distribution complete particle amount sampler effectively immediately converge individual sample produce fast probabilistic engine much operating system call os compare core across count ec intel processor tb frame l forward monte particle programming language standardized system share exploit intermediate language machine link operate library yield efficient target new hardware optimizing source transformation phrase intermediate language library intermediate representation language intermediate probabilistic normally operate system library parallel programming language leave readily resource writing program computer architecture illustrate forward employ operating program program via monte implement execution trace operate posteriori trace program reflect data language language model purely forward generative process program execution trace site mh requirement operate primitive also level use inference albeit include var predict mu return output semantic include transition static double initial static emission double program state mean predict course execution implicitly define interpret probabilistic programming capability library execution datum mark expression posterior make choice log pass expression library include macro another loop nonetheless show gaussian return log particular program function predict posterior value underlie emission mean nonparametric generative program mixture gaussians chinese restaurant crp normal gamma point double mu var draw sample double variance observation draw alpha alpha invoke mu var return proceed draw trace entire virtual memory address machine probabilistic operating system construct os create identical execution identical continue correspond choice program forward program report match sequential smc resample building complex particle smc intractable p identity construct q program program execution particle unnormalized importance normalize set execution trace program continue trace correspond lead bad concentrated single trace execution trace trace index execution trace execution trace tb particle barrier unnormalize serial serial serial serial continue execution parallel trace terminate barrier l serial effective eq statement form barrier execution current unnormalized execution trace arrive barrier take particle reach current unique execution effective number store memory reach barrier retrieve child child execution terminate execution normal outline step execute parallel barrier desire additional mcmc hasting particle sequential mh propose set sample inner substantially smc tb particle program serial serial serial serial continue program trace terminate barrier serial serial current
marginal exchangeable partition group inference beta introduce exchangeable probability describe exchangeable collapse nonparametric one convergence good modeling prior increment impossible transform problem marginalization dirichlet random chinese restaurant structure chinese exchangeable lead collapse prior despite significant progress decade modeling usually limited point beyond interest group point exchangeable group number hierarchical dirichlet hdp popular wide gamma process beta gamma none group marginal hdp chinese restaurant derive collapse fully collapse neither unified mixture modeling membership law partition derive partition share marginal group count describe column random count mixed membership stochastic important contribution several additional simulate exchangeable group collapse topic beta update straightforward implement produce representation size exchangeable know constraint express addition addition allow one dependent count column membership propose dependent group beta random product finite continuous measure define evy large one hence define analyze sampling th binomial binomial jk nr pf rp truncation slice recent binomial describe binomial ibp ice relate ibp binary ibp different paper focus count generalize develop truncation free collapse nature value binomial poisson poisson potential bridge count assign count disjoint borel multinomial unclear exactly random us model cluster derive exchangeable probability govern partition later derive unit mass assign borel size membership amenable marginalization analytically coefficient categorical jk n provides obtain describe partition group group arrive matrix th nonzero permutation count detail matrix appear direct calculation j identically dirichlet pmf n jk r rr generate poisson partition count follow lemma denote summation govern fairly complicated derive prediction group govern exclude contribution membership rule select popularity cluster whose govern simulated gibbs run sampling exchangeable partition different setting critical role row kn infer fix data point group sum represent nonempty ji term rewrite product ji j hdp mechanism fully collapse assignment collapse globally derive collapsed hdp chinese book keep link tb topic iteration lda b analogous plot corpus curve correspond www edu ss toolbox http www cs edu corpora restrict occur count corpus document corpus document total count evaluate lda bayesian word one order collect j jk r p n j hdp per word jk jk sm final used matlab cpu collapse sampler topic take per infer topic sampler hdp comparable complexity infer topic infer large considerably speed first mix collapse sampler topic trace plot column slowly reach right quickly reach hdp small quickly quickly smoothing left column lead middle commonly corpus evident hdp corpora corpus middle small topic often hdp topic multiplicative control topic whereas shrinkage lda tend lda comparable topic able predictive hdp corpus setting sample moderate support moderate usually prefer could hdp lda three suggest topic achieve hdp lda view count model differently variance topic collapse gibbs sampler gibbs sampler truncation heuristic hdp collapse collapse comparison parameter posterior topic topic smoothing topic display scale omit membership develop value binomial exchangeable govern influence group dispersion construct nonparametric exist one intuitive interpret fully collapse sampler converge art representation method group unique interest investigate derive mixed membership modeling value gamma poisson gamma binomial transform representation beta transform
nx v nh fu nh obtain complete admissible sm sm combine desire lemma remark proposition type consistency entropy consistency learn function entropy consistency consistency coincide surprising provide infinity illustrate play analysis minimum entropy enyi use concept entropy divergence substitute covariance inspire series minimum error later blind source comprehensive survey advance unobserved power decade theoretical complexity empirical perspective early utilize bandwidth motivation minimize require term bandwidth parameter imply unfortunately simple yes complicated try full establish relationship entropy measure power statistic model statistical mean output measure two setting produce goodness approximation entropy sequel need denote h set enyi rf rf b z entropy involve make look estimator dependent summation involve u asymptotic converge q adjustment measure constant probability entropy power power imply clearly metric consistency good approximation regression serve contribution I model entropy necessary coincide consistency firstly tend consistency consistency result show empirical bandwidth choose large lastly special consistency bandwidth give make analysis regression throughout regularity density function usual tailed minimizer uniqueness obvious trivial remark simplify statement learnable constant cover h h first expect unbounded second ensure learnable happen impose theory fulfil target target adopt relaxed situation admissible verify least rf z literature implementation e choice rate minimum somewhat version later respect regression instead consistency complicated situation otherwise say state model f n regression corollary entropy consistency relationship entropy state theorem entropy consistency complicated illustrate example consistency fail error consistency show imply consistency coincide model prove bandwidth form z look surprising minimize entropy approximate consistency infinity view empirical motivation consistency theorem adjustment adjustment ib main bandwidth positive situation order case denote integrable recall transform crucial univariate monotonically decrease unimodal nonnegative define set assumption noise family look symmetric distribution say evy transform cauchy median choose fourier distribution refer statement exist proposition combine two prove consistency independent throughout notation obviously maximize prove translate prove excess quantity last equality part take expression nonnegative f fu f x fu u transform ensure nonzero interval identically iv corollaries ii previous prove error entropy counter denote specify subscript dx r f error functional denote minimizer measurable function bound entropy generality lemma uniform fu follow integer translate orthogonal f f measurable corresponding minimum equal write e e f x f u last x x x see I u b u fx fu x fu u fu fu fu fx fu minimize condition equivalent minimized minimum f x x fu fx fu fu fu f fu fu v fx fx fu impose need fu fu fx fu x fx fu fu combine minimize give take f rf conclusion iv minimum value regression choose tend suitable depend consequence proposition notice h role empirical use algorithm tend least special subsection unimodal let integrable unimodal unimodal variable belong unimodal consistency immediately state second hold appendix recall fu dx find fu tell minimizer notice take fx fu f fx f fu fx fu f e ef dc hc bounded noise w h fact follow xu virtue
hold several application interval instead extend omit aforementioned contribution besides growth apply theorem seminal mention stationary process task sample section concentrate case center begin preliminary specify proof omit df scale mutually wu u df hold indeed constant argument n u j integer u large r hence establish therein satisfy argument x hold theorem hence x nt limit side follow center stationary process proceed four u probability u u constant imply thus proof covariance stationary gaussian grid hold convergence rv u u almost index let z vector vector distribute unit sphere satisfie may denote df u satisfy n shall asymptotic interval appear define hereafter indicator jt ij p ns argument lemma side e u rest line tt nu nu k hence hold x e asymptotic supremum process methodology seminal paper dealing process check technical assumption process otherwise extension many finding mutually generic rather stationary process minimum statistic result order li lemma statistic gaussian component li show goal li maximum nn statistics dx define maximum order statistic ij n b bn il thus complete minimum il il ij nn il u il I lt bivariate bivariate standardize u z n n grateful careful reading suggestion greatly improve rgb remark proposition grant mathematical pl department university copy process constant define brain mapping interest empty impose condition tend asymptotic supremum process limit stationary version surely sample mutually independent copy interest th central interest fix conjunction time interest empty relate least prominent concerned imaging fmri establish seminal therein euler characteristic high smooth discuss result non field derive exact obviously phenomena skewness orient field engineer environmental study emission collect concern brain model since calculation general contribution approximation result empty df let satisfying assume random point distribution constant integer suppose simple c validity stationary process show validity copy asymptotic generalized notational th condition concern process derive proof establishes li vector stationary
object side gmm massive cluster sharp right gmm draw undesirable classification prefer guarantee gaussian super object number super position function super dense gmm pick pick decision boundary discard continue realization figure sample figure bin adaptively use correlation map count x ray galaxy build map place one pixel divide subtract q superposition delta eq attribute infinitely delta way make realization cross spectra make make realization cross power spectrum cross correlation red point show cross spectrum green band cross map simulation classifier galaxy readily scalable lot determination observe galaxy without dimension modification position object another us depth coverage add obtain statistically characterize generate improvement manual neighbor replicate sample feature space drawback observe random nn always discriminative sensitive ignore internal decision object algorithm properly paper survey galaxy consider among possibility angular ray space mass density q overall normalize unity sum theoretical realistic possible monte analyse complex selection explicitly limit application expand include example find physical principle promise direction near future method effective thank point solve discussion machine thank discussion like whose comment improve package computation calculate take reference namely rgb rgb blue problem thorough usually realistic complicated process galaxy specific mass simulation dark matter obtain kind purpose put together call pick physical property determine deal combine observation task rule derive exist object observe express rule behind machine learn learn classify simulated dark like synthetic galaxy select ray ray measure target galaxy detect eliminate dimension feature ray cluster dark matter simulation look observe aspect introduction machine detailed terminology application recent classify survey image galaxy detect large survey accurately etc support vector dark book paper classifier simulation also galaxie statistical sample classification goal distinguish train outli available boundary target target chance class property widely density boundary commonly reasonably sample size estimation determination well constrain demand compute paper boundary dash white surface sample supervise surface show class simulate black member target red construction svms family dimension use function adaptively class decision outli classify attempt mathematical formulation svm example function whose determine target separate order make separate linear surface space need use introduce space choose problem kernel dot kernel radial rbf determine decision return minimal elsewhere separate region zero region translate region end need solve follow dual optimization need kernel upper regard belong target example vector margin task uniquely give offset svm figure rbf decision separate outli define property uncertainty parameter set boundary right look outlier everything else right black successfully separate population panel figure increase modify distribution quickly inaccurate surface green boundary use show style sharp boundary uncertainty surface classification gmm combination distribution covariance datum cross open machine library model one svm dark sub observe make beyond threshold generative gmm mass ray ray cluster combine survey reduce rare subsample cluster galaxy scale procedure convert mass see detail develop parallel patch structure formation form matter peak measured predict position filtering individually solely initial match much simulation carlo construction galaxy formation phase simulate whether body semi method peak light core body particle train polynomial separate part observable region boundary boundary corner plot nearby massive object object far observe select surface top sample massive always decision cross randomly contain decision measure call rejection setup score specific change small outli surface derive boundary boundary use simulation x ray target though sub statistically object gmm parameter determine criterion
next arbitrarily ergodic period thus observer become become predict various drop result stochastic merge relevant purpose weak focus closeness predictive distribution obtain merge example merge matter strong merge unnecessary context decision discount period usually law stochastic elementary object seminal theorem de representation exchangeable ergodic decomposition exchangeability temporal parameter maker ergodic posterior weakly decision belief concept difference see outcome tail generate dirac infinite belief concentrated realization outcome highlight posterior belief little agent represent way make sensible learnable prediction event become connection ergodic long stationary canonical decomposition however ergodic learnable weak meaningful sense trace cover therein horizon proof rely technique literature update prediction derive maker observe start observe space realization generic product element way uncertainty capture bayesian view process stage outcome represent extreme dirac dirac assign capture intuition fundamental dirac copy trivial discriminate admit well know convex topology ergodic belief stationary admit unique parameter set ergodic belief ergodic stationary belief block realization equal process equal frequency ergodic every define extend recover ergodic finite process special decomposition exchangeable distribution outcome thus observe consecutive number outcome configuration good outcome equally give equip coin represent outcome give first outcome ahead cover finite horizon introduce merging setup outcome period merging let belief realization horizon learnable case strong every weakly rare explicitly merge merging establish potentially bayesian belief say period connect every take outcome average period strategy nf course agent weak learning say weakly play sufficiently patient horizon period payoff discount sufficiently discount let weakly period period motivation learn calibration idea calibration realize empirical show weakly forecast pass calibration weak characterization idea concern quality near horizon common think consistency think estimator reasonable bayesian weakly converge dirac measure reference hold property realization future dirac coin observe outcome belief uniform agree indeed insight future learnable main weakly learnable implication underlie change unobserved period period hide remain change period outcome represent decomposition ergodic belief topology concentrated parameter prediction complicated prediction merge weakly general history period outcome consistency estimator give agent observation block assessment probability appear wrong still agent horizon event process predict coin predict frequency economic weak modification setup bad period last period outcome unknown parameter formally define ergodic borel belief markov last occur value stationary observe time agent keep track parameter outcome probability agent know randomize deduce observe consecutive period point prediction agent predict decomposition learnable decomposition learnable time index hereafter algebra subset measure space exist unique equality measurable unique measurable dirac decomposition sigma algebra borel ergodic induce algebra borel interesting let shift learnable sigma ergodic consider trivial agent observe prediction distribution accord finite history give history therefore stationarity shift invariant therefore limit maker generalization ergodic cover simultaneously let let n formalize intuition belief event far I periodic mixing condition finer learnable necessarily stationary every sufficient establish equivalence decomposition fine ergodic decomposition decomposition gap proposition tail tail sigma algebra distribution trivial property prove tail weakly imply ergodic stationary stationary weakly learnable lemma past learnable tail shift stationary set outcome equal induce learnable rely outcome comment admit process asymptotically reverse tail decomposition show learnable process asymptotic reverse asymptotically reverse contain dirac atomic extent theorem tool merge weak merging extend say belief extend tail infinite equip borel give every fair coin entire tail tail dirac decomposition however
sure strong result unfortunately couple argument however contraction contraction convergence contraction operator value need exact variant satisfy establish weak asymptotic give explicit optimal value classical operator variant assumption satisfy bound satisfied contraction weak follow argument derive bound fix give give short reader refer proof construct chain stochastically dominate rate value get construct chain structure markov zero show stochastically dominate chain concentrate establish zero mix convergence sample asynchronous version even asynchronous consider visit full action deterministic shorthand denote asynchronous operator operator value leave unchanged respectively visit least contraction visit full slightly asynchronous probabilistic checking progress sequence update turn cycle random introduce pick compute bound apply result pair visit asynchronous q online fast though guarantee convergence show result compare mdp cost iteration state pair equation code since step figure rate fast reach relative take speedup iteration mdps establish limit unlike classical scheme mdps actor analysis incremental preliminary experimental state action update would pick pick sure sense result case infinite even continuous action partially mdp method rkh average reward reward dynamic programming operator reward mdp contraction mapping however provably convergent reward mdps ode approach interesting mdps reward criterion part fellowship department technology support office nsf award proof series lemma control markov show strategy stationary define r k markov depend expect value coupling strategy k mdp two ss remark abuse common copy let w j analogous choose choice lemma get extend control homogeneous completeness sequence state chain state law k dropping chain probability control corresponding limit empty start govern contradict stationary irreducible statement non stationary pair mdp simulate arbitrary k k p q proof conjecture notation corollary example remark propose optimal discount cost process classical algorithm mdps converge surely mdps sure approximation rate preliminary fact asynchronous popular empirical c h algorithm popular use dynamic programming actor td recursive observe payoff slow underlie evolve effect average ensure scale design property incremental therefore evolve scale usual conditional averaging underlie control obvious advantage expect work rigorous simulation ball indeed reality theoretically construct coupling simulation finite yield look backward guess terminal reinforcement programming view refer classic present asynchronous extension present possibility mdp action let ps ps action mdp control control px ps objective admissible infinite discount cx infinite irreducible integer bellman q bellman tv tv find arbitrary iteration calculate contraction mapping banach though iteration extremely develop mdps value q exact transition generality mdp drive function uniformly algorithm empirical distribute iteration maximum iteration counter sample stop main result proof every rule step general convergence incremental noise horizon obtain look guess terminal cost precisely value guess look showing backward iterate immediate contraction unknown convergence formalize equation overcome difficulty define rigorously approach backward converge almost quantity underlie space forward control chain c time couple chain f backward notion proceed side infinite sequence negative integer sequence n transition ease independence give represent drop whenever value define strategy map action ks ks resp resp ks write condition mdp implicit use notation whenever drop since markov k prove couple z k two simulated strategy proof consider technique thought function initial offline simulation initialize compute stop else simulate composition start backward composition forward simulation composition successively transition generate state collection choose contiguous familiar backward strategy q kernel important iterate let kk induction note definition get k
bandwidth smooth factor estimator estimator rule package study step estimator unlike case function case f point tend around observation equality f property conditional asymptotically application bandwidth selection datum tend large tend happen often usually rule rough order narrow develop trick depend impose way particular bandwidth rule bandwidth clear left section investigate simulation confidence quantile compare end bandwidth package finally equally distant grid estimate grid quantile xx rule multiply ex ex ex c ex ex generate normal variable quantile quantile namely univariate grid heterogeneous take coverage cc nonparametric quantile perform especially result simultaneous band derive asymptotic hand homogeneous heterogeneous asymptotic mention cc probability yield bootstrap general nominal wider getting find sensitive bootstrap specification upper show coverage cc probability simulation surface asymptotic nominal coverage see volume ex c ex volume cover derive expansion somewhat confidence theory worth note portion uniformity grid homogeneous heterogeneous specification level usually univariate notation sense median point complete probability proportion finding firstly improve volume band increase heterogeneous compare proportion tend curse dimensionality bivariate bootstrap part coverage coverage though cc volume cc much table similar bootstrap size method regression nevertheless obvious stochastically dominate scenario improve dramatically great word low growth growth induce great picture stochastic testing treatment effect quantile consist treatment observation control group group year year common describe unconditional density distribution validate unconditional density treatment treatment tail concentrate two unconditional slight deviation distribution get unconditional group er von reject empirical lrr statistic kolmogorov treatment first year sample lie age lie avoid boundary effect sample year validate density package result treatment group validate validate handle level repetition regression quantile level quantile estimate lie particularly group exceed quantile group hand drop age tendency risk reduction old individual benefit treatment heterogeneous age weak quantile group level h turn suggest treatment risk education nonetheless group rise group certain level curve associate group cc tend potential growth spend school note heterogeneous treatment although heterogeneity education age analyse separately condition age covariate setting quantile cc group overlap extensively sufficient find treatment large surface inside cc upper boundary hence tend improve low growth program high growth reduce negative conclusion surface age treatment effect heterogeneous age interesting case occur year conditioning correspond school increment boundary control old material detailed theorem lemma intermediate contain incorporate positive continuously differentiable xx continuously moreover function continuously differentiable xx h continuously xx old continuous old eq b b assumption frequently ec sequence characterize convergence hold dimension smoothness ec relevant tail nd email em em definition proposition construction suitable interest series inequality regression demonstrate band coverage finally national article material bootstrap goodness fit quantile treatment effect nonparametric c analysis inference curve center covariate though even curve finance management tail event thing event conditional covariate traditional way tail curve extreme alternatively moment tail description general regression confidence parametric available corresponding view inference turn observation necessity form check different kind explicit estimate pre kolmogorov type employ deviation smooth display cc band consider technique construct extreme sup center quantile predictor classical histogram one univariate band derivative year growth literature spirit quantile curve integrate simultaneous cover estimation wavelet adaptive estimation bootstrap poor bootstrap density quantile progress confidence band multivariate expansion somewhat extreme portion small classical uniformity set multivariate band go study aspect cover quantile minimum procedure improvement asymptotic generalize goodness quantile treatment quantile work randomize datum participant treatment beneficial individual year treatment word negative evident old spend school unconditional heterogeneity quantile treatment devote investigate simulation numerical propose assumption theory list discuss reference supplement theorems section bootstrap construct devoted issue discuss reference independent random nonparametric quantile quantile xt xx immediate context local curve cc regression theorem constant xx purely suitable curse regressor via linear dimensional think extension offer modeling parametric clearly band influence subsection property point trivial task challenge technique nonparametric add give xx kernel however investigate seem nuisance parameter consider error replace xx function heterogeneity equivalent sample issue residual ig cumulative chapter convergence develop subsection difficulty residual estimator residual base obtain true residual residual bias less residual estimator conditional conditioning condition v nh n nh price pay converge plug suitably speed coverage error asymptotic
difference return completely balanced hardness design characteristic hardness measure easily goal repository provide meta ready meta one store set server name experiment db relational database curve integrate implementation incorporate access without learn schema store store instance result preprocessing store preprocessing integrate current white purpose discard experiment compare implement correctly store machine algorithm gain provide access learn store comprehensive storing well previous result important sense database easy access desire meta ready access researcher help meta specific database database store prediction aggregate instance diversity choose choose non trivial meta deal select set hyper previous although research focus focus machine specific domain lack meta amount resource differ slight implementation thus meta learn study aid problem uci repository refer set meta snapshot underlie user update experiment keep comparison meta set typical meta meta learning instance level study effect generally level important ensemble diversity classifier important create characterize misclassifie work also prediction meta learning unsupervise meta algorithm score meta set recommend behave treat individually training base level meta weight create repository machine information hope bridge prominent experiment database report purpose database experiment store result learn unfortunately curve inherent storing complexity difficult beneficial potential user additionally acknowledge maintain database add offer store meta learn provide set comparison provide repository meta datum ease meta currently algorithm algorithm database section detail give use access result experiment learn instance meta understanding complex least training describe information three file run allow compare example name seed hyperparameter seed seed hyperparameter default parameter practice classifier bp momentum tree differ distinguish hyperparameter setting backpropagation hyperparameter implementation include case unknown meta mapping ccccc parameter l h momentum separate file fold run include column test file unknown represent represent filter instance unweighted instance value represent instance split represent tool fold sense prediction datum meta hyperparameter tool hyperparameter table accuracy acc fold e different seed partition fold provide hyperparameter single ccc ccc ds acc access researcher practitioner meta snapshot history provide even database store feature algorithm traditional database allow expand database store new schema create schema database piece store database collection value pair collection represent experimental la value document information instance store respective collection visualize show set hold collection document output file name number correspond seed document store collection include contain snapshot allow learning evolve machine experimental file modify future meta include repository store meta commonly meta al easily future meta et examine meta represent affect algorithm performance use deal attribute algorithm incomplete attribute ratio variance small measure set identify set value fisher discriminant attribute discriminant attribute class expand instance belong feature overlap tail overlap bound return discriminative attribute region return attribute return previous remove return separability training separable class value extent training linearly separable boundary tree entire return span belong instance nearest instance neighbor neighbor class set geometry topology set create interpolation return create classifier center measure number provide clustered structure attribute modification et accuracy meta include list create
incoherence use mathematically recover informally underlie sufficiently spread low recover sample incoherence follow relaxed form consider rank name decomposition rank cardinality program name q nuclear norm convex surrogate prove probability spatial moreover handle noise incorporate compressive equality constraint relaxed serve nuclear give rx leave right singular name thresholding specific two proximal pg alm applicable pg likelihood comprises nonsmooth pg often nesterov accelerate use pg alm alternate multiplier mc alm subsection eq usually nonsmooth name nuclear theorem pg pg convex continuous gradient repeatedly easy simply gradient nesterov accelerate another modification please initialize k kt pg method solve via solve update denote soft pg fix also technique accelerate implementation augment direction lagrangian alm classical tool lagrangian read alm ascent primal fix marginal turn nuclear norm proximal efficiently solve soft thresholding repeat convergence variable inexact augment special alternate multiplier summarize initialize k alm equality alm apply augment lagrangian proximal alm factorization completion recently nuclear surrogate minimization typical turn nuclear consider nonconvex surrogate minimize model low decompose matrix therefore small rank finally low matrix factorization rank matrix recovery method dictionary summary basically minimize fixing turn least extensively study computer vision literature recovery completion adopt alternate follow efficiently additionally scheme accelerate nonconvex empirical accurately meanwhile theoretical show completion computer adopt high alternating alternate newton update via well similar algorithm reader introduction vision incomplete ill collaborative address square norm name factorization idea use ridge stability follow equality establish indicate minimization study optimization alternate constrain optimize manifold solve denote form name iteratively matrix square theoretical method incoherence gradient follow estimate square develop name optimize framework matrix three type factorization invariance solution underlie nature class trust space factorization structure perform work form manifold conjugate gradient algorithm name completion manifold toolbox name develop rank handle regard outlier traditional biased address alternate minimization carry linear iterative reweighte similar alternate minimization example robust factorization direction instead generalized norm improve optimization alm group sparsity establish address online process incremental tracking online incomplete corrupted cost observed solve square update introduce extend handle outlier robust replace optimization framework matrix architecture large name incremental scheme nearly proportional number adopt strategy implementation first matrix combine subproblem version shall treat pca latent classical pca linearly prior mle give eigenvector covariance large span pca dimension advantage helpful automatically choose inference imply consequently automatically automatic relevance determination ard machine similar treat factorization method handle miss consider likelihood set representative work treat following probability derivation posteriori map estimate turn interpretation correspond impose modeling regularization predefine determine introduce later full method pmf markov chain change factorization laplacian model prior laplacian large term connection propose observation give moreover hyperparameter probabilistic include etc probabilistic factorization determine serve model hyperparameter play automatic determination extremely drive category rank constraint optimization often greedy project constraint conceptually name propose use scheme project intermediate result theorem similar denote os turn os exist decrease small convex alm iteration notice factorization fast pass method vb competitive parameter curve shape remain high os unchanged factorization shift indicate attribute directly depend case existence factorization relative attribute relaxation introduce remove consequently especially prove figure decrease os os curve rate influence besides recovery ratio proportion rank completion recovery program achieve noiseless know stable version test minimization mention subsection variational overall require perform drop surface dynamic contour shape track move object intuitively low underlie hence recover low many recognition face person show intensity image face recognition characterize face matrix input image face person image low face could remove correct face component face boost dataset surveillance include perform frame background video detecting stand background underlie video camera unchanged illumination variation matrix compose image foreground clean traditional presence foreground illustrate image foreground background include moving reconstruct low show appeal capability modeling spatially contiguous foreground use smoothing segment trajectory foreground cause camera motion lrr subspace dimensional subspace whose affinity combination neighbor estimate eq encourage wise orthogonal point affinity learning dictionary dictionary represent combination claim intuition signal block column order rank minimize transform transformation estimate align difference compare represent single assumption character reconstruct texture camera character etc widely analyze track video seminal observation track camera e perspective perspective coordinate motion object across point frame motion addition possible low solve frame frame frame pose adopt art method shape reconstruct limited basis give measurement constraint tracking feature across frame motion segmentation track multiple feature track group track belong discuss track rank therefore segmentation formulate subspace divide track detailed please completion desire reconstruct lose text approximately corrupt formulate illustration completion image randomly sophisticated texture recover model sparse unknown detect corrupt adopt minimization nuclear norm video video group group share finally completion patch group low coherence image noise removal medical analysis denoise mr usually frame image diffusion image suppose significant component classical achieve denoise importantly threshold theoretically stein bring great original shape statistical candidate shape shape denote consist vector describe training datum candidate often shape shape model moreover pca shape model active appearance shape method build alternative make segmentation similarity shape nuclear minimization rank modeling imaging coherence image mr imaging concept separability ps mr image spatial component correspondingly tu notation coherent space reconstruct sequence basic integrate specific wavelet total meanwhile temporal periodic model dependent locally ps nuclear dynamic imaging imaging modeling multi image modality ct emission example rank model detection parse fusion image paper concept rank review representative additional reading reader factorization low approximation convex programming noiseless case exactly true signal noise shrinkage try nonconvex relaxation requirement repeat svd make computation svd approximate widely real recommender mostly computational convenience factor moreover cost function variable online process technique work probabilistic great real probabilistic real knowledge purpose extract remove etc recent sparse powerful framework technique rank expect acknowledgment manuscript image yu refer interest rank achieve success processing bioinformatic convex programming completion apply collaborative topic recent advance overview concept rank model challenging advantage limitation application context dimensionality great document natural customer recommender bioinformatic fortunately high mathematically big translate correspond signal intensity array reader real raw low recover conventional approach ij respectively minimization interpret analytically correspond vector
tail sum base devise label example parameterize compute discrepancy true label vector predict approach capture widely adopt many multi label form constant prove show determine rademacher complexity tight rademacher motivate objective eq large value control get multi multi label rank correlation label regard minimization may incorrectly zero contour unknown trace range propose constrain derive singular successfully discover norm fail start regularization approach far reformulate linearize regard approximation term problem iterative ignore trace solve singular thresholding handle let svd diagonal ml base set evaluate evaluate average rank large value error jx auc area roc accuracy auc compare label increase label appropriate achieve predictor important trace structure compare three implicitly design structure ratio summarize improve nearly rank discover norm low rank multi predictor encode multiple predictor obtain increase number local complexity tight generalization unseen example behavior dataset confirm conditional singular solve principle guide new erm algorithms discover sum multi predictor inspire complexity rademacher tail tight experimental label complexity risk minimization erm base trace norm instead tail use minimize singular predictor play exploit validate effectiveness rademacher example one conventional many categorization annotation gene straightforward decompose series binary label poor number different classifier margin multi learning dependency removal learning theory justify successful label dependent measure dimension cover number rademacher erm trace explanation effectiveness multi hand implicitly exploit correlation rademacher favorable subset drawback rademacher consider result rademacher complexity rademacher complexity seek rademacher complexity erm multi label sum predictor motivate tail value predictor rather I trace advantage multi predictor predictor exploit learn result function efficiently newly thresholde real validate effectiveness new training distribution function loss global rademacher measure complexity class global error estimation pick subset instead rademacher complexity reasonable intersection center function rademacher global rademacher describe error give every variance lie error theorem rademacher complexity analyze complexity multi illustrate motivation develop multi model
computing reweighted weight minimization exact method share determine automatically yield exact equivalent instead heuristic particular weak truncate nan subproblem solve combine bfgs cg problem solve hybrid bfgs cg effective penalty propose zhang quadratic numerical decomposition solve whose vector arrange give open neighborhood center section verify minimization eq programming constraint variational characterization original increase statement hold locally solution locally locally open vx associate arbitrary feasible vx e solution globally unique solution neighborhood x sign locally conversely locally optimal hold part cm equivalence solve solve penalty penalty develop paper study penalty linear program give optimal say sequence sequence imply disjoint q moreover inequality hold e e e nonempty globally solution single problem solution coincide need since prove contradiction present arrange conclusion clearly hold lemma proposition follow hold I use arbitrary last rx r I complete cm solve unknown advance problem nonconvex solution fix motivate propose tolerance choose cm otherwise stop go go solution nonempty introduction feasible feasible converge bl ba ix b yx v cm termination result terminate subproblem algorithm know q terminate terminate cm result end write nan satisfy vector satisfie condition step nan argument yield next induction satisfy note equality inequality show nan use nan use successive vector equal together iteration nan space condition devote subproblem involve nonnegative one order cone software however consume suitable motivated recent partial proximal follow partial approximately first subproblem via denote function r respect gap approximate solving yield ty kx k v problem minimization summarize favorable continuously q solution map jacobian satisfie hx x note follow continuously continuously gap primal result expression replace cm know form cl I bfgs yield scale continuously differentiable need bring newton find root nonsmooth scale problem conjugate newton cg given go next step cg linear seek integer hold set step positive direction always direction convergence reader may approximate experiment form remark diagonal otherwise subproblem bfgs cg alone subproblem bfgs good feasibility newton meet involve develop subproblem solve subproblem solve bfgs step r penalty decomposition method large suitable bfgs ty end k start use k cl solution account convex surrogate iterate feasibility sequence bfgs good find turn sequence unless involve store choose employ bfgs solution step l bfgs minimization involve algorithm testing slow terminate l advance turn subproblem unless default search subspace matlab windows operating system ghz cpu gb verify effectiveness compare return solution zero denote component iterate yield solver remove entry entry small nonzero test algorithm problem table magnitude lie problem realistic include six solution limit problem magnitude htbp result solver e e e e solver product involve relative recover record four solver bad incorrect set subsection sense number gaussian whose distribute qr element independently hadamard column dct large six zero normally law decay decaying whose matrix store explicitly matrix type store unless state sequel signal successfully relative original htbp htbp take influence four solver type took test solver four solver solver signal little well testing display six kind see type among solver take illustrate solver signal number curve figure average successful much high less solver comparable even six type comparable even signal noise test signal noise ax identically constant choose algorithm htbp take example test recovery error solver consider take randomly curve relative solver vary type signal type algorithm desirable solver little yield residual attain whereas average residual yield word yield type compute high three signal type compare solver type took randomly test recovery solver vary number signal little together superior computing subsection collection report desire feasibility less require almost problem yield problem desirable also yield good feasibility zero norm yield numerical comparison solver collection solver e e e e e e htbp c c e e smooth numerical subsection conclude even superiority collection particular require time much bad
proportional approximate method quantification approximate assess denote random target extensive next functional neutral dirichlet process simulation study consider survival moment explicitly evaluate compare exploit consider survival illustrative purpose coincide specifically beta set jt end section importance section obtain high exploiting moment true one shape approximated distinguished finally fit truncated normal density conclude instant numerically integrate measure beta exploit average nonetheless apparent incremental gain moment respectively numerical instability good accuracy ht combine characterization together gamma expression characterization independent gamma prior survival estimate equally spaced interval distribution us survival median survival interval principle functional step draw summarize hyperparameter posterior moment investigation reveal second sampler describe admissible latent set l l approximate n st sampler every exploit remarkable survival cdf c subscript devise cf side sum depend nonetheless crucial sufficiently estimation functional meaningful median st I mode pt credible divide part survival focus real median credible four observe observation survival plot investigate credible region true survival concentrate investigate performance methodology grow summarize credible credible around length interval reduce close h ci involving time patient time patient treatment star detail censor mechanism adapt right censor observation estimate credible figure plot estimate posterior different behavior mean optimistic posterior median worth must large censor different depend specification nonetheless show capture posterior rely sampler refer panel compare estimate underlie completely gibbs sampler detect credible significantly credible moment credible interval avoid credible survival great panel figure plot effectiveness treatment credible two credible plot credible line comparison black credible european provide integral eq gamma denote conditional adapt right censor observation notation posterior change censor censor time accord censor rewrite observe censor carry case right censor replace jump component occur function result corollary straightforward inferential markov variety suffer limitation functional goal present methodology extend hazard order inference survival limited include remarkable credible approach rely moment polynomial inferential performance methodology mean tailor survival adapt hazard approximation survival inferential rely characterize marginalization element define variable henceforth refer besides identification markov method approximate evaluation form typically predictive mixture dirichlet estimation hazard become well establish practitioner implementation stress relevance worth point suffer drawback easily posterior marginal suitably endow credible choose choice absolute median mode preferable median nice issue provide focus trajectory truncate stick present paper aim propose combine close method approximation posterior develop estimation survival hazard rate function cumulative hazard fs fs ft area soon devote class accommodate rigorous bayesian inferential survival mention neutral cdf conjugacy drawing inference also benefit conjugacy propose full specify hazard popular gamma originally generalize random hazard mixture recent allow quantity statistical marginalization identify see quite implement hazard rate survival mean sample rich complete understand exchangeable survival I pt random suitable instant along posterior allow straightforwardly moment indeed use integrate approximate evaluation almost turn one survival st set gamma continuous kind gamma finally posterior kernel ease exact extension case censor straightforward latent joint transform among display result let coincide hazard evy jump jump display jump coincide jump insight see obtain value conditionally thus point introduce point introduction combine representation alternative aim trajectory involve distribution trajectory approximately since illustration survival achievable trivial issue evy conditional latter address augmentation scheme suitable recall realization jump section marginal goal bayesian effort minimal compute posterior evaluation yield expression moment conditionally technique next hold kernel monotone hazard rise hazard property generality notational display integrate end describe explicit knowledge moment receive great motivating application therein interest motivate determine density distribution
index rx x functional dependency affect connect configuration unobserve superposition unobserve whose column unobserve configuration capture residual product loading identification sparse mean vertex affect zero I seek introduce lagrangian element infer network np combinatorial approximated method novel degree freedom invertible determine minimization full expect svd tp sense sparse use outside support relevant know exclude unnecessary explanatory variable base compute problem infer compute eq constraint introduce degenerate reduce problem still combinatorial heuristic enhance solution propose approach I define index matrix index compute approximate right small perturbation singular give small angle vector iteratively remove converge equivalently large index singular unit sphere sphere unit ellipsoid ellipsoid singular solving require move origin choose index choose small ellipsoid selecting eliminate remove component move small vector removal max ij j compute vector max max single one apply thus replicate choice inferred generally go least solve matrix absolute pearson recover less stability reflect connectivity vertex topology great practice direction require update kk usually work increase solution close old notable memory step compute previous computed inversion take impose variable latent dense demonstrate theoretically test numerically well absence scenario structure simulated random bipartite network encode unobserved variable simulation adjacency infer close versa two limitation permutation many instead choose learn small mean degree avoid algorithm recover test easy slow rest less law uniform advantage probabilistic rather response factor accuracy network observe unobserved fig configuration fig vertex increase poisson choose mean wide h degree case compare improve iteration improve fig scale algorithm hold speed range use analyze unobserved simulate infer assume element close truly unnecessary valuable physical corresponding yet correspondence outline practically situation state unobserve situation modification versus observe assume state measure number index enough determining follow pearson infer likely correspond physical u similar enhance interpretability physical outline rather element significance fraction know come quantify observe alone hyper geometric know provide excellent rigorous evaluation rigorous biological activity limitation throughput scale state identify apply follow platform unobserved directly adjacency overlap tf h identify common gene third column observe chance list go infer correspond network many detect limited experiment perform gold partially topology algorithm interpret penalize edge predict new paper comparison allow identify experimentally crucially dynamic logic gene paper aim infer sparse infer degradation g salient distinguish inclusion ultimately computationally compete sparse acknowledgment thank discussion extensive constructive david david work research engineering technology rgb rgb rgb develop case observe fa decomposition underlie novel accuracy noise scale efficiency method k svd component recover exactly unobserved decrease range noise increase analysis fa decomposition useful systematic allow interpretation variable fa problem
transition consecutive point secondly since noise scalar degenerate cope difficulty drive variable dt equivalently express markovian latent motivation latent result significant mix gibbs dependency pair remain improper prior infect figure half batch write bootstrap kernel current law forward system mean exclude hx hx establish induction equally draw weight markov particle b f b b b line since bound induction sl bounded statistics california berkeley technology tool inference sequential present analyst highly enable mix particle reduce computational burden typically conceptually backward suited dependencie markovian nonparametric thing systematic sequential carlo analysis dynamical wide scientific linearity originally indeed decade process markovian model various either marginalization discuss tool useful however combine sampler make kernel rely markovian stochastic relatively area finance build construct sampler trajectory trajectory effect reference leave target regardless particle however suffer serious drawback mix kernel path degeneracy underlie unfortunately degeneracy high problem applicability address generic add simulation sampler yielding denote considerably much problematic model markovian trajectory backward pass degeneracy modify thereby achieve sampling explicit kernel publish illustrate use problem theoretical validity ergodicity markovian illustrate example applicable severe indeed section direction work unnormalize tx draw enable carlo draw sequential nature suggest use filter start standard consecutive meaning let particle system weight proposal complete notational dependence particle resample particle define particle proposal weight give initialize assign x sampler summarize draw w generate explicitly I natural turn construction trajectory generate thus state review algorithm introduction serve trajectory informally think simulated particle pass sample particle implicitly event reference trajectory retain pass coincide reference trajectory sampler invariance precisely particle return property hold note keep reference result leave recognize degeneracy fundamental turn new idea improvement considerable implement trajectory range current history connect particle encode connect path assign refer form trajectory understand importance formal invariance outline shall modification mixing map stochastically family index trajectory set set w draw ix w k argue mix illustrate numerical option pricing assume observation give smoothing employ particle simulate path mix report reveal significantly well view poor rate degeneracy sampler process pg reference trajectory system clarity illustration respectively blue retain throughout trajectory red extent identical perhaps much insensitive understand system generate thick line particle index line effect informally reference trajectory broken piece point prevent path degeneracy particle degenerate something red probability substantially enable update blue line dot line particle line degeneracy particle grey dot panel trajectory piece degenerate toward something different view state show invariance violate sampling kernel invariant apparent establish particle expectation possible however treat auxiliary thus avoid intractable integration view trajectory index recursively variable function density refer intend extended factor important marginal invariant kernel kernel partially collapse meaning basically refer process variable gibbs sampler invariance algorithm implement partially collapse instrumental draw collapse conditionally construction respectively propagation expression use plug expression numerator consequently step analogously leave conclude procedure commonly within simplify carry correct dependency procedure collapse conditional full clear variable never subsequent sufficient furthermore collapse gibbs leave law procedure lemma recall x b procedure conditionally law particle matter reference position imply give desire ergodicity target obvious modification basically cover support ergodicity establish argument apply irreducible ergodicity boundedness assumption basically slightly exist ergodic condition ergodicity normalization discard equally analogously make integration important illustrate nonlinear gaussian eq wish latent prior seek simulate sampling posterior recognize poor autocorrelation block invariance draw n also maximum q maximize maximize auxiliary integral intractable involve particularly auxiliary make detail smooth intractable general recognize distribution auxiliary simulated particle summarize draw run pf non standard nontrivial density x understand bayes likelihood also recognize step highlight use pass particle replace line extract trajectory backward implicit denote turn case joint bootstrap internal particle proposition build bootstrap backward simulator appendix handle trajectory particle filter establish equivalence outside class sampler variable model eq share conditional independence property history specifically markovian implementation truncation motivate area classical replace processes markovian include regression non nonlinear successfully marginalization part rao result useful instance drive sampling transition many simulator transition exist illustrate markovian statistical among probabilistic model markov employ backward need bottleneck markovian hasting know step retain sufficient leave invariant kernel propose move simulate eq sample accept set keep weight whenever moderately large variable recommend construct ensuring would operation force move mh force prohibitive markovian decay past markovian weight assume sl divergence compute quite useful inferential truncation truncation increase converge experimental tv level exponentially decay remove requirement introduce design easier small rapid change well mix evolve take truncation stop evaluation early accurate approximation dimensional right dotted respectively overlap leave vertical show result scheme way replace computational simplicity truncation another use proposal mh suggest accept mh evaluating implement property linear mixing consider example markovian depend illustrate inference degenerate reformulate markovian initial unknown superior parameter sampler simulate system time step discard sampler sequence row hold sampler require drop much hand robust comparable ideal sampler consist clearly difference sampler degeneracy result agreement discussion leave top row ideal admit density problematic backward sampling degenerate backward first equivalently degenerate state measurable markovian simulate single joint smoothing non markovian possible backward truncation
precise previously element query observe query former process identical query support amount single minimal low pair pair mass preserve detail result effectively intuitively modification unless match formalize support obtain crucially significant aspect difficulty bind obtain regard technique decision appear conceptual turn bound family distribution query tuple sample identically last analyze query actual size intersect one front indistinguishable uniform set adaptive essentially give big query prove return query indistinguishable obtained consider separately indistinguishable uniform spirit follow guess reference upper obtain double exponential refined new idea supplement effect dependence know therefore yield low size al rest proof shall cover uniformity test upper support reader independently logarithm probability countable ss sx sx testing set reproduce conditional oracle choose oracle element si oracle deal choice situation always include put outside support hereafter et uniformly use tailor let call either output follow definition straightforwardly independent goal property deal precisely resp distribution allow possible focus draw get relation make sample et al show invariant convert core core tailor setting form atom denote generate capture something like summarize label information obtain ta call jk sample summarize aforementione argue algorithms access configuration albeit algorithm act internal randomness configuration query obtain j I query refer query specification draw random whose atom element require previous fully besides already decide belong atom put differently belong one stage stage either ever access label know intersection actual identity contain chapter principle deterministic draw randomly apply instead work analyze work importantly randomness external behave core randomness sample external therefore stress external way via element random affect ease observe intersection q apply intersection prove attention intersection show indistinguishable simply pick element implie indistinguishable like correlated concentrate around expect consequence expect invoke eq intersection simultaneously concentrate selection element ns ss satisfy q ns adaptive break recall query intersect receive thus establish support describe analyze shall help support enough probability reference verify inside support last check small significantly big actual query output exist access input either subsection stop compare keep reference doubly guess support meaning repeat call j j rest formalize rigorously conditioning indeed pass reach step bind return subroutine subroutine return claim query query call j th cost query similarly call overall query claim hereafter output meet requirement fact ensure significant fix nx nx minimum suffice rearrange work get enough issue indeed element care support want case support perform therefore use increase detect work give indeed estimate heavy end fine since support failure time vote j kt jt jt multiplicative chernoff call specify happen break analysis happen get call routine comparable e probability factor unless support mark discussion return conversely mass less point mark overall fail probability good call guarantee repeat majority vote success step setting repetition overall subroutine fairly enough uniformly conditional mass mass h access ks fall behave return latter least claim complexity therefore turn subroutine derive random intersect r enough repeat sufficiently whether indeed roughly repetition apart return vote call ms step outer repeat vote repeat overall query sketch derive upper adapt level perform precede identify great guess al find return estimate return intuitively time pick set element non adaptively increment call correctness outside support remark lem prop theorem question acknowledgement dot ref ref sublinear support cl property sublinear sampling low enable allow condition domain explicitly support size contrast require whether hold establish bind equivalence testing recently et whether dependence domain explicitly pose sublinear answer conditional interestingly qualitative turning investigate size doubly generalize weak logarithmic lower bind carry uniformity author necessary adaptive end probably far seem technique like conceptual nice without discussion author decide plan perhaps discussion loose end fundamental seminal al computer science particularly set decade explore well often complete property sample optimal testing recent year situation specify subset formal need thereby sample prove testing extremely namely distribute outcome develop closely uniformity testing hereafter oracle access oracle decide far decide oracle decide factor impossible trivial generalize least show complexity strong uniformity testing provide allow make give oracle access distinguish uniformity understand uniformity sufficient tight uniformity additional suffice namely task require polynomially focus query show even query lower open logarithmic answer uniformity question define access unknown equivalence study decade equivalence upper need open possibility uniformity admit constant equivalence bind equivalence weak conditional oracle restrict pose sublinear answer possibility show sufficient approximate et al obtaining require almost subsequent question establish bind multiplicative distinguishing knowledge getting factor version yield intrinsic equivalence testing estimation uniformity framework
eq fact eqs j dropout study weight drop unit unit probability drop weight iii drop input layer hidden layer hide activation function many use activation assumption full connect type I denote eqs networks network hide f element draw I I show complexity drop weight give complexity apply result interest miss rs cauchy rs ne n hold jensen rs ne rs ne inequality layer rademacher complexity irrelevant unit dimension eq k number layer easy hold network rs ne lipschitz give term empirical rademacher another layer respect nr rs hold layer prove combine b j similar complete real implementation develop aspect deep far dropout effective strategy reduce intuition adaptation detector work study rademacher complexity type result none rademacher polynomial whereas neural lead rademacher fundamental type current light way claim algorithmic implementation aspect deep interesting usefulness motivated intuition detector type dropout theoretical network dropout deep neural deep network wave recognition speech recognition many however theoretical aspect far complicated million risk even control overfitte long research various weight early bayesian success main randomly omit hide execute certain remain overfitte though encouraging phase dropout able improve reduce theoretical dropout clear influence rademacher dropout hide rademacher complexity polynomial deep rademacher complexity relate design dropout dropout whereas unit hide dropout dropout et regular inverse generalization dropout analyze present pac et al derive rademacher keep element dropout generalization dropout reduce complexity neural neural dimension complexity polynomial complexity parameter worth function complexity influence complexity weight irrelevant unit follow presents general rademacher usefulness different classification throughout restrict make neural respect depend network dropout unit necessary introduce different dropout drop weight network dropout give objective introduce performance output entropy goal minimize risk r draw rs risk try rademacher classical chosen rademacher develop datum task notational simplicity w I hadamard generalization mostly affect thus rademacher dropout however generalization thus generalize rs rademacher define easy rademacher rademacher generalization dropout follow sample dropout easily find dropout every n
product subspace similarity angle subspace cosine approach database product aim focus compact e represent objective include three aspect accurately query code evaluation compact code quickly approximate compositional summation investigate compact code inner approximation code compositional database short code propose approach exploit inner operation compositional operation compute handle vector approximation product approximation approximation property inequality vector query difference product inner related meaning euclidean query query approximation potential basic compositional combination quantization aim database application mean jointly optimize basic code compositional produce form call separately combination code source combination vector combination size write combination combination concatenation nm furthermore extend scheme element set selection selection produce composition differently element form compositional length keep represent code zero consumption computing become large scale increase increase compositional dictionary sparse problem optimize transform quadratic w algorithm paper zero acceleration separately subproblem dictionary subproblem minimize hard propose greedy dictionary determine source selection dictionary good previous issue vector similarity database approximate code nm construct dictionary search handling compute relatively compare search sift search sift similarity selection formulate equation constraint ni nj nc formulation extra constraint equivalent mean case summarize compositional transform one successively regard mean combination guarantee validate solution feasible selection dictionary compositional four another generally compositional dictionary summarize combination case kk small inner follow complexity iteration updating multiplication involve code whole respect code sift reach intel cpu hz also benefit parallel acceptable sift supplementary material quantization correspond source source dictionary lie comparison permutation order closeness vector thus regard divide sparsity impose approach sparsity sift netflix lm query conduct sift sift linear lm netflix sift sift descriptor database vector sift descriptor contain sift vector lm dataset around linear query image engine sample feature query rating give movie aim inner rating user give movie recommend pca description quality fraction retrieve among inner evaluating return result perform subsequent item neighbor inner compare database raw compositional code three search neighbor fix source encode number bit selection observe result search compositional group compact include product quantization eigenvalue allocation equivalent quantization random projection locality lsh design cosine see superior nn improvement close show sift consistent improvement achieve bit use indicate achieve hash hashing lm inner product aim find query fit mean relevant retrieve view soft attribute apply search large provide fast flat hierarchical tree recently see netflix rate user apply rate perform much well bit code little come code perform second classifier classification dimensional signature compression ram raw large memory pc raw thus closeness contain recognition conclusion table achieve svm kernel inner product table inner symmetric symmetric sgd algorithm really product addition preserve similarity evaluate search average bit cc bit score product compositional accurate evaluating approximate computationally future generalize euclidean rewrite present show approximation absolute product dictionary quantization rewrite center entry extend quantization space rotation optimize product approximation equivalent analysis product view source dictionary case subspace rank near closeness accord towards distinguished anchor order closeness contrast find partial permutation rather representation compute aim coding find basis code fix sparsity group code approach divide sparsity main paper dictionary code group update time update close operation multiplication multiplication matrix matrix transform om dictionary select selection dictionary contain vector thus time four dataset average approximation composition production error approximate near table observe vector inner error another achieve c c sift netflix cc c sift netflix cc product cosine similarity database norm approach without way maintain extend spherical clustering learn accuracy product addition hold cost evaluate euclidean code produce little subspace si equation show give database thus table item c inner orthogonal map corollary microsoft china address near
third order vector respectively application learn et al liu algebra b especially rapid technology tensor become increasingly multi image video work datum arise due memory call curse dimensionality underlie though due major govern relatively tool curse tensor order tensor small commonly take tucker tucker cp find tucker decomposition tensor decomposition approximately reconstruct hope tensor three challenge selection tucker base give rank involve tensor liu problem decomposition attention problem video reconstruction matrix tensor show capable advantage provide completion nuclear addition theoretical development guarantee partial reasonable condition al progress recovery trace decomposition trace require addition improve scalability relationship trace tensor tensor cast convex regularize high orthogonal iteration scale direction multiplier admm solve fast insensitive robust extensive notation detail th high mode vector ni index map ni two sized tensor respectively denote tensor j ji popular cp decomposition well cp compute high tensor tucker tensor core multiply along core much small tensor g tucker decomposition tensor storage decomposition significantly rank unfold particularly tucker decomposition analysis width tucker factor tucker orthogonal al iteration b good rank general well possible tensor goal norm difference progress cast minimization trace value regularization handle satisfy trace due trace difficult introduce equivalent parallel direction method solve minimize augment lagrange parallel updating multiplier parallelization modify variant parallel proximal solution eq et al solution proximal unfold proximal term show description algorithm use block variable n verify n nn nu n k let n converge solution rank tensor problem tucker employ solve fix imagine optimize consider well optimal value repeat alternate solution clear sized matrix k nr norm computational complexity nr convex iteration outline hence attractive admm adaptively strategy lin al initialize validate rank converge update multipli n algorithm operator multiplication proximal operator operator ni tensor synthetic real world datum experiment core tm truth tensor tucker decompose art et al et normally tensor regularization method tensor datum vary three four much accurate solution outperform efficiency size rank time color kind algorithm important relative regard depict plot tensor increment increment trial perform htb liu al approximately singular large time trial attain relative fast convex addition tensor analyze low cast trace solve experimental regularize method noise outlier tensor decomposition handle useful comment support grant microsoft thresholding processing reduction multilinear parallelization alternate multiplier n via recovery condition explanatory modal splitting lagrangian structured inequality tensor multilinear b approximation lin liu linearize alternate estimate missing value visual minimization rank decomposition h li trace recovery j l j spectral decomposition decomposition via tucker three truncation singular zhang u convergence behavior behavior residual k k iteration provide residual drop much fast iteration relative residual drop quickly estimate rank experimental size rank estimate rank unfold tensor ht plot datum tensor rank varied increment increment generate choose uniformly give tensor size iii u accord property nu u nu
idea tv place somewhat difficult employ place derivation vb go similar fashion tv fact r ni mode formula vb isotropic tv detail mention model issue pixel index variable issue solve numerically value handle deal issue lasso approximation laplace heavy tailed origin similar approximation positive tv functional gradient optimisation method tackle thresholde hierarchical student prior freedom since density hyperparameter present smoothness smooth distribution derivation tv sum write regression invert vb vb problem block structure covariance inverting domain use approximation could similar vb domain matrix nice iterative conjugate drawback step system precede author could vb ignore vb appear corresponding mode density variant quite formula usually hyperparameters done style solve em problem brevity laplace difference prior aspect could laplace discuss freedom gaussian noise image reconstruction present divide comparison penalty deterministic tv method transform quadratic moderate image mask reconstruction hierarchical provide mostly equal slightly optimisation whose multimodal maxima partly improper good test quite near study optimisation determine penalty formulate prior mixture property formulation tv parameter assessed mean posteriori derive method less flexible slightly test tv tv tv tv work edge preserve comparison reconstruction cm vb sampling compute intensive vb region develop vb solve iterative satisfactory algorithm might become improper improper prior even hyperparameter dominate dirac delta study simplifie although numerically comprehensive study hierarchical implement test although separate typical scenario method give also method tailed image relate fact generalise denote parameter unimodal skew moment use formula certain asymptotic formula reciprocal give gamma gamma define laplace distribution q agree parameter see convenience exponential curve exponential square root multidimensional inspire laplace proof similarly degree freedom x tv vb cm mcmc vb sampler opt deterministic optimisation tv image tv model tv tv laplace prior f tv tv tv laplace prior tv tv theorem laplace gaussian prior total variation approximation find bayes method alternate compute automatic preserve promising result difficulty encounter future model variation subject classification variation initially alternative non useful want copy total example differentiable origin bayesian tv information result assess inverse study study paper penalty interpret laplace tv set laplace compressive study variational use hierarchical also tv fast optimisation g solve tv tv propose fix optimisation auxiliary trial estimate inverse scale isotropic also inverse posteriori although fully method present laplace mixture encourage try mix penalty know tv prior tail useful edge preserve scale inverse gamma gaussian interesting laplace multidimensional tv prior model paper familiar hierarchical formula vb section technical remark work discuss appear classical linear discrete solve measure formulate nontrivial solution even singular numerical try introduce term ill pose penalty optimisation simplify formulation deterministic approach proper choose dominate term ill optimisation assess notation image column stack special notational indexing presentation stack array pixel notation pixel tv discrete total functional tv boundary boundary could tv allow tv version might easier deal origin version rotation invariant isotropic tv isotropic tv tv isotropic tv penalty link study pixel bayesian optimisation tv detail idea variable g fix ordinary character denote distribution information prior x usually estimate also depend hyperparameter nuisance think hierarchical involve example specify consider tv jump connected control conjugacy parameter principle available value specification appear choose could improper integrate improper improper posterior nevertheless improper later kind emphasize minimum trivial tv challenge discrete tv conditionally laplace random variable normal place adjacent pixel hyperparameter priori easy see formula r convention root take component wise direction consist difference discrete operator naturally basically allow satisfied acyclic show mainly tv generality derivation follow tv tr show indeed notation tv alternate maximize keep variable loop value derivative optimisation present formula simplify use value solve compute sx derive approximate posterior conditional density sampler sensible analytic solution slow typically base subject converge compare approximate posterior pdf partition factorize eq parameter
des universit propose translation traditional neural aim jointly tune maximize machine encoder sentence decoder generate translation bottleneck improve architecture allow part sentence relevant target word explicitly approach achieve performance comparable phrase english qualitative reveal find agree intuition newly unlike sub tune translation build translation encoder decoder language neural read length decoder output translation encoder decoder language correct sentence issue sentence make neural cope long show rapidly length sentence increase address issue decoder align translate concentrated predict vector target distinguish basic encoder attempt encode sentence length encode adaptively translation fix length cope paper translate achieve improve basic encoder apparent long sentence sentence english translation translation conventional phrase qualitative reveal sentence sentence perspective translation equivalent finding maximize source translation parallel learn translation search recently paper directly encode sentence rnn source sentence length already report neural machine translation term art phrase machine english translation add exist phrase candidate allow art propose upon build novel architecture align translate simultaneously read input sentence beneficial hide predict predict word probability conditional potentially rnn architecture convolutional translation rnn search r architecture encoder approach context word annotation map input word detail annotation annotation eq weight annotation alignment position state alignment feedforward jointly unlike traditional machine directly alignment whole model understand weighted annotation computing expectation probability translate word reflect state intuitively implement attention decoder sentence attention decoder encoder burden throughout annotation retrieve decoder accordingly usual read last annotation word summarize also follow propose successfully speech recognition backward rnn read h xx result backward h xx summarie tendency input annotation focus sequence annotation decoder compute eqs illustration evaluate english translation corpora rnn encoder model english corpus word reduce combine although possible encoder news usual frequent word map token apply ji arbitrary select among length type decoder sentence sentence decoder encoder consist neural rnn single maxout hide word minibatch descent together train update minibatch sentence train day translation approximately neural architecture c list translation importantly conventional sentence word consider propose basic limitation may encoder dramatically drop sentence length sentence performance sentence superiority model basic decoder fact intuitive generate row weight annotation position consider english monotonic differently english fig see translate phrase european economic align area european economic back phrase alignment oppose evident source phrase translate translate le naturally look see able correctly translate synthesis ask character coefficient alignment alignment predict increase weight annotation machine need translation drawback translation applicability scheme probabilistic neural however network largely translation system exist feedforward network phrase additional phrase machine translation recently report network translation traditionally target although approach objective translation consider work work generate translation sentence conventional neural translation approach length problematic recent architecture address basic computed target encode source sentence focus target neural translation long unlike traditional piece towards produce propose english translation reveal outperform decoder regardless sentence length source able align relevant annotation sentence perhaps importantly comparable statistical translation consider architecture whole believe architecture promise toward well translation understanding language general challenge leave future match art machine translation context acknowledgment thank acknowledge thanks hill van framework recurrent alignment describe experiment activation hide conventional simple short unit share backward easily vanish possible lstm new employ eq element multiplication output update represent omit maintain much sigmoid use hide maxout normalize one need sentence layer w nu h pre minimize detail code output sentence vector language respectively length source sentence recurrent w mu embed number hide unit logistic sigmoid rnn matrix backward annotation eq decoder annotation encoder language weight word embed dimensionality initial nc alignment annotation v nu weight matrix encoder decoder decoder maxout model word embed dimensionality maxout unit hour gpu matrix rw initialize weight initialize sgd automatically explicitly predefine minibatch update require time sentence minibatch every retrieve length split table statistic model experiment htp p cm right admit patient medical centre carry diagnosis status health care worker le le en un patient dans un un centre un diagnostic un un est le de un patient un centre un diagnostic un diagnostic en est un un patient un un centre pour un diagnostic une
hold side apply get upper moment q conclude remain technical ready uniformly lead identically need set assume supremum first follow depend excess statement surely last going subsequently employ hold great great great inequality minimizer set step auxiliary assumption great combine analogue centre department university university sampling replacement broad learn study theory excess localize convergence localize prominent case class localize empirical class ingredient empirical interest role theory theory sup heart advance localization chapter yield excess base theory implicitly many machine many case I violate test different point typical computational biology area paper example sample replacement population sample replacement unlabeled predict label naturally appear text recommender system effectively constraint inherently realize system world image categorization area recognition manual inspection label unlabele e image several bound still remain good knowledge rate bayes assumption restrictive contain localize hold hypothesis view analogue common risk inductive setting error take function thus yield excess achieve prove process go concentration tool instance arguably prominent without cross fold replacement pool help non asymptotic cross procedure investigation novel inequality outside scope paper lie protocol inductive draw space choose predictor fix hx nonnegative use setting sample without replacement output remain unlabele label denote respectively l u mh empirical datum reason regard n play role quantity invariant partition test use define hypothesis obtain tight bound paper another commonly obtain generalization minimization note excess obtain risk bound bound introduce deviation side relate sup analyze fundamental obtain probability variable replacement particular concentration sup author general present implicit error certain belong u also attain several pac learner algorithmic roughly mention concentration yield rate present without replacement result closely concentration base first suggest notation finite replacement countable countable refer page page sampling without replacement use random remarkable type version due refer understand lack inequality present without replacement also hold great replacement hold appearance might want expectation however lemma order control could prefer worth novel begin obtain setup theorem write inequality mn replacement say theorem material frequently bind compare bind constant thing provide drawback preferable use theorem question whether least significantly tight theorem far present appendix briefly outline proof supplementary material consequence although result instead concentration refer proof inequality reference define partition finite set set inequality mathematical learn validation factorization procedure learn localize yield complexity bound apply concentration obtain risk set nh mh u mh nh nh nh nh play f nh h hx nh mh sum note random center random learning boundedness immediately calculus material sup naturally inductive upper process quantity mh follow theorem repeat proof follow corollary define corollary convergence corollary us relate rademacher contraction inequality section result excess yield fast reduction loose extent localize bound modulus associate tool end sub concentration theorem convenient loss class satisfy discuss common satisfied instance n hx nh nh nh uniformly bound relaxed theorem unbounde unbounded depend whether use notion nonnegative positive let satisfied root point emphasize appear well one appear share instead appear bind satisfied sub modulus continuity empirical replacement large briefly outline part theorem section rescale center class ff variance obtain slice able also sub root definition obtain finally use great replace excess great minimizer repeat error intermediate obtain detail present key apply develop inductive though apart tight multiplicative constant point regime potentially bind fact leave excess satisfied variable ij n uk depend result direct section decay question state assume reflect gram evolve grow generate think possible adapt relate asymptotic counterpart broad nonnegative excess risk learn localize class excess risk
trait value show ari us group c ari parsimonious membership party membership table selection bic component trait party misclassifie mainly mainly response individual group exception issue slightly vs voting versus l response individual one issue group response variable variable concern aid group education b concern budget anti test aid mx vs b plot group well select contain two four question contain contain pearson test applicable count check fit show count attribute htb count f texture plot group separate htb trait slope variational parameter model draw sharing enable plot latter plot give term estimate mean latent representation covariance place useful structure application herein cluster covariance wish investigate alternative nominal mixture logit acknowledgement author acknowledge helpful comment anonymous review grant aid development grant university research grateful access trait recent binary latent extend incorporate common accordingly low incorporation block consider approximation exploit determine demonstrate simulate extremely classified whether circumstance record something think like table trait term ht categorical latent analysis trait categorical profile latent principled cluster mixture combination component dominate cluster wolfe focused mixture elliptical lin lee base receive little trait table binary anneal chance gauss require ht type likelihood logit annealing logit gauss quadrature logit variational probit order categorical integration logit order categorical wherein identify trait accommodate dependency approximation likelihood implement quickly approximation trait come latent similar structure trait probit numerical heterogeneous discuss use gauss quadrature distributional parameter mixture item trait quadrature integration accordingly analyze underlying propose trait cluster variable exclusive compose latent categorical parameter considerably slope visual representation model accordingly identification trait integral use variable provide low likelihood variational parameter demonstrate vote cluster propose trait slope present real suggestion future approach assume trait slope ng datum reduce trait low assume assume trait p ng ng ng ng p ng ng z ng ng ng categorical group ng ng ng datum cluster cluster arise design repeatedly homogeneous accordingly likely variable outcome specific explain dependency inter variability response equation ij write model closely relate mixture item assume latent level discrete block one effect multivariate trait parameter reduce take follow parametrization matrix density eigenvector normalize eigenvalue determine orientation rise four corresponding covariance respectively assume namely parsimonious make ht l free g g dd dm dd dd dm dd dm g gd dm dd dm dm g dm gd dm g dm dm list parsimonious group need component grow almost ex finite model variable geometric contour principal thus function ng response variable plot effect block statistical common characteristic categorical employ load analogous trait covariance analogous function distribution course mix equivalent dimensional difficulty version analyze binary latent trait class range key trait provide also dual closely trait obtain framework series expansion point bind likelihood obtain maximize fitting integral intractable variational em obtain latent ng ng ng ng ng ng improvement take ng ng ng ng ng ng ng ng likelihood log adopt lack stable application voting facilitate comparison determine acceleration eq detail gauss quadrature expression b md eq likelihood schwarz criterion observation component low preferable likelihood purpose calculate maximize log use gauss quadrature convergence attain common small check goodness adjust rand index rand ari chance rand ari trait discuss give detailed trait constraint determine free parameter identifiability hold rank error maximize likelihood identifiability study categorical assigning n g initialize approximation matrix ten initialization low variational exactly
classification significant modelling field model clustering give partition disjoint batch gp gp govern deviation behaviour model may layer hierarchy construct model e use dirichlet prior gp group inspire cluster time fail previously inference procedure sample agglomerative regular span development limited resolution subject technical biological variation occur clinical grouping relate specie take inference require gibbs need fast dataset resource avoid model key likelihood construct reduce make derive link conjugacy manifold gradient optimization method apply expression series several differential process expression proposal lead propose presentation conceptually gaussian poisson intensity attract lot gene incorporate time potential dp like make gp gp assign function observation noise gene differ far describe gps expert aim cluster propose gp rich replicate infer aforementioned considerably widely gibb adopt agglomerative suffer scalability gene collapse collapse share see yet collapse relate apply collapse variational collapse unable correct overlap mixture gp objective track reduce remove gp dp use free variational show process introduce gps structure introduce perhaps particularly publication space one zero everywhere covariance publication exponential kt length publication separate collect write critical vector eq covariance construct function version write conjugate interpretation gaussian covariance directly construct gaussian consider group group experimental draw draw gp development normalise log gene frame posterior subsequent frame biological group replicate solid area covariance amongst depend compound function j application assignment well popular involve infer mixture widely treat assignment estimate treat approximate posterior parameter mixture achieve prior approximate assignment vb find mix proportion variational avoid selecting though dirichlet allow density proportion cluster concentration break mapping dp concentration stick break length break proportion distribution dp gp provide though empirically procedure variational bayes use gp dp vary atom function hierarchical hierarchy high level know dp gp construct series break atomic proportion atomic draw gp atomic reporting perhaps infer occur probabilistic assumption lower serve assumption factor turn recently collapse specific variable analytically equivalent scheme show riemannian conjugate direction serve dp mixture model collapse break stick length variational similar parameter aside cluster allocation simplification gradient propose gradient move merge split cluster ascent direction relate information quantity empirically free infinite unitary element shall select truncation merge split metropolis hasting collapse gibbs collapse dp gp gene necessarily simplify exposition somewhat let collect stick break vector wish simplify illustrated figure mixture group occur usual hyperparameter omit brevity point vb graphical infinite connect gps dp dot process cluster represent cluster graphical hierarchical collapse standard gp separation analytically symbol description collection observe stick length dp breaking length mix stick stick assign component component collapse select collapse observed wish examine graphical representation would separate variational truncation valid softmax computation natural gradient shall analytically jensen likelihood trivially tractable conjugacy without integral separate complete square integral straightforward study collapse break integral first trivially stick break length beta beyond trivially substitute integral similarity collapse stick break et break length make lead variational tractable expression n g softmax problematic suggest write though avoid first inversion size since symmetry softmax gradient nk nk g nk divide many avoid compute divide obtain expression natural eq expression natural multinomial softmax correct optimization collapse posterior field exactly kl correct bind step length coordinate coordinate natural correct field bind recover unit kl correct correct simple deal compact optimization gradient mean field bind round update perhaps kl correct enable conjugate gradient computation conjugate fail recover merge suggest mcmc current one cluster collapse nature correct particularly helpful merge split deal natural move bind split select examine mass new cluster accept move empirically merge appropriately order move bind collapse contain deal arbitrarily log increase hyper scale span datum account account variational unless model synthetic set randomly function around randomly per select correlated offset add present development aside across specie replicate measurement every pool gp replicate account correlate replicate use hierarchy gene eliminate gene expression hour dark cycle gene periodic projection dp gp nature rbf periodic far discover structure show reflect establish gene provide synthetic dp construct hierarchical dirichlet infer concentration parameter cluster infer diagram synthetic indicate cluster grey square represent gp find correct confusion gp dp account structure model ground truth allocation infer structure amongst introduce component fail signal unable infer method variational trial condition parameter create standard log riemannian conjugate parameter merge routine reflect case variance correctly place formation
instance distortion training impose objective minimize coefficient value size update rate update exponential decay decay well decay iteration dataset consider throughout produce good preliminary randomly describe loose treatment similar slack separable impose remove simply regularization generality train must term adjust score inner although type sigmoid give range sigmoid equation empirically effect variant formulation supervise feature scale equation verify compute derivative become hold likewise convex irrespective form convexity depend upon sigmoid scale irrelevant finite although sgd empirically function parameter tune adjust range require variant equation convex respect sigmoid class sigmoid second derivative compute order derivative proof scale regarding among product appear inside thereby reduce number train name scaling convex feature train pass online dataset dataset dataset evaluate classification real purpose uci learn repository detail dataset attribute instance heart diabetes compare feature implement train baseline vector update instance encounter theoretically weight fast unsupervised train binary logistic scaling feature unsupervise feature describe weight vector train unsupervise dynamic feature section method describe classification passive binary predict pa passive pa average pa linear passive pa binary weight parameter avg avg fs avg fs fs avg fs fs avg fs fs avg pa avg pa avg pa pa avg accuracy good sgd avg fs fs avg fs fs avg fs avg fs fs avg pa pa pa avg pa pa avg algorithm sgd avg fs fs avg fs avg fs fs avg fs avg avg pa avg pa avg mention q three e number negative test instance diabetes dataset passive purpose produce high accuracy next fix parameter table suggestion instance train initialization see scale joint supervise scale pa average unsupervised dynamic averaging accuracy dynamic method unsupervise dynamic deviation instance unsupervise likely demonstrate accord pair effectiveness among report performance pass training become critical compare set compare method experimental keep plan dynamic method compare diabetes method average version outperform compare diabetes ability perform situation cumulative current pass online train classifier method obtain error cumulative total encounter misclassifie misclassifie avoid show dynamic stand low scale pass dynamically popular dataset experimental unsupervised significantly outperform approach improve several supervise scaling evaluated explore learn range conduct preprocesse problematic reason stage change effective dynamically adapt scaling complex several classification classification machine represent often range supervise train frequent extremely relative informative value feature algorithm typically scale range use approach feature reason manner label document task scale possible scaling preprocesse pass online instance extremely stream call scale third compute value instance text stream might factor old dynamically term refer oppose processing focus specific multi classifier main approach assign scaling algorithm pass online scale store stream training maximize memory evaluate online three dataset binary interestingly much unsupervised dynamic scaling consistently algorithms compare include online necessity engine online learning impossible fact new continuously situation train tweet case instance predict trend stream note iteration example gradually criteria example another moreover whether training manner unseen ever stream web pass convergence confidence develop online feature scaling supervise
combine pd lda share among gram pd across n gram phrase post lda topic construct performing pattern mining phrase four heuristic permutation test phrase phrase extraction apply social service twitter twitter specific topic candidate phrase extension topology twitter extend corpora corpus frequent enhance topic investigate objective overall quality phrase concept place lda investigate markov probability independence assumption sentence extra generative hierarchy assign topic sentence output extract rank list phrase phrase operate partition document method comparison interpretability scalability six collect publish paper token area artificial intelligence database retrieval language contain word token article token k word token review review token address phrase remove english stop discovery comparable pd art approach topic use bi add specific phrase post permutation merge term lda extraction speed frequent frequent phrase phrase demonstrate effectiveness framework detection option randomly phrase one top phrase ask select indicate unable task separate ask answer question evaluation second motivated extract quality phrase interpretable visualization evaluate phrase visualize list phrase sort expert science score phrase qualitative coherence homogeneity phrase list homogeneity ask expert coherence phrase list ask standardized h discussion demonstrate quality stem frequent mining phrase inspection suggest key phrase notion may aid believe quality occurrence phrase many hyperparameter pd two intuitive interpretation topic rigorous permutation employ phrase addition phrase induce phrase high belong evaluate predict hold corpus evaluate review demonstrate lda demonstrates validate phrase high lie addition see phrase quality model incorporating phrase analyze decompose framework contiguous take bag phrase output separately demonstrate phrase see scale increase dataset addition one phrase portion negligible topic model scalability framework runtime hardware dataset various art compete computational lead intractable requirement whenever author implementation special gibbs user hyperparameter optimization ensure fair phrase model runtime model pd intractable magnitude runtime runtime intensive permutation n gram method able full dataset scalability large pattern scheme make large long intractable short entire phrase word table phrase method text probable well probable phrase automatic perform post visualize phrase interpret phrase naturally news phrase coherent phrase believe may phrase good great phrase display sentiment emphasis poor cm cm day day day na runtime calculate runtime topic runtime great runtime language mining solve classification character association sentence propose stream genetic large gram genetic programming language mining optimization speech machine language object orient stream natural machine evolutionary feature selection execution series run sign mining recognition orient programming spatio objective program gram drug nuclear house aid environmental health year medical west test chemical disease gram department health care environmental west bank house medical nuclear organization united united house nuclear anti report drug abuse drug united capital patient pay control house member heart member test h gram room store good ice stay roll place restaurant great area great gram ice lot store front great hash great price chinese room dim sum pool area center great good price reasonable mac systematically allow driven estimate topic another scalability currently decrease stem efficient phrase mining investigate another focus pruning strategy similar phrase discrete structure top phrase count merging phrase well properly notice occur due filter principled enhance mining framework arbitrary phrase phrase phrase first frequent phrase mining efficiently aggregate score objective bottom phrase termination phrase phrase assignment phrase phrase computational phrase principled construct phrase post processing demonstrate scalability interpretability phrase nf office agreement nf national science foundation multimodal synthesis foundation fellowship nsf collapse derivation find second department laboratory md edu microsoft com mail algorithm model corpora interpretation rely inherent discover length either utilize phrase suffer scalability moderately approach effective solution combine novel mining segment word operates induce document high phrase extra bag word publication review news recent topic modeling become discover abstract typically model multinomial multinomial retrieval search engine support extraction document query question answering topic topic human interpretation exploration within text corpora qualitative list probable topic yet topic probable provide intuitively description term phrase clear organization exploration collection difficulty scalability attempt make phrase simultaneously create mechanism phrase assignment gram pd appealing incorporate phrase element overall word phrase topic guarantee model propose new compare phrase method interpretability phrase mean word phrase phrase mean lost insight phrase systematically motivate perform phrase scalability issue phrase significant phrase use frequent phrase mining text candidate phrase second phrase effective within phrase assign phrase word frequent mining significance frequent generation frequent current status future share advantage phrase mining algorithm phrase aggregate domain specific linguistic purely drive candidate phrase title frequent phrase determine whether frequent phrase title segmentation phrase incorporate need reduced phrase maintain analyze phrase review conclude input document document sequence token convenience token vocabulary throughout refer word vocabulary corpus corpus visualize phrase statistically characterize example research high speech characterization advantageous statistical interpretability depend member fashion phrase phrase help define terminology sequence contiguous token concatenation proximity restriction phrase phrase despite word phrase mean motivate representation flexibility segment concatenation phrase title token represent outline property mining list phrase demonstrate phrase valid human interpretable phrase principled manner efficient comparable specify phrase designing phrase phrase phrase construction naturally validate candidate phrase constitute human interpretability phrase important regard within frequent topic formulation list probable lda probable phrase give topic corpus refer token frequency chance commonly appear frequency yet english language consider frequency informative insight motivate necessity analyze phrase mine frequent mining satisfy yet intuitive phrase human interpretable phrase phrase phrase divide phrase mining constrain transform bag quality frequent phrase segment agglomerative phrase induce topic phrase phrase phrase phrase phrase phrase partition topic bag phrase input traditional token propose partition phrase collapse token phrase phrase mining corpus tokens interpretable phrase operate contiguous pattern meaningful phrase break corpus frequent candidate phrase aggregate develop technique quickly collect without phrase step great subsection frequent phrase collect contiguous draw upon mining frequent closure phrase phrase frequent document frequent phrase length frequent phrase length closure first mining frequent exploit property mining potential merging termination induce upon original create bag contiguous token h significance left phrase place contiguous token significance key left phrase phrase put track construction agglomerative merging operate merge two contiguous phrase merging significance algorithm follow consider newly merge phrase consider newly merge phrase single merge free phrase compare occurrence occurrence merge phrase phrase aggregate necessary calculate significance merge proper datum structure contiguous pair significance merge document terminate next meet merge meet significance termination natural bag phrase partition remain algorithm requirement phrase completeness corpus actual phrase hypothesis phrase phrase reason phrase consider independent hypothesis absence phrase corpus random number occurrence token corpus assume fairly reasonably normal count phrase corpus trial probability phrase phrase compose phrase phrase population minimum phrase significance quantitative consecutive phrase merging compare expect occurrence equation deviation away number occurrence aggregate candidate phrase efficiently frequent phrase algorithm significance generalization statistic identify check merge contiguous phrase merge phrase effectively free phrase address concern significance score rely naive testing guide phrase merge phrase merge overall corpus phrase frequent contiguous mining aggregate bottom agglomerative merging segmentation frequent contiguous mining differ transaction pattern mining bad search pattern contiguous reduce exploit text splitting searching phrase rarely algorithm frequent contiguous mining corpus pruning datum heuristic reduction lead runtime termination phrase merge phase segmentation phrase possess complexity experimentally verify section new I often chance token share latent brief review dirichlet propose gibbs develop optimization phrase serve collection cm topic phrase token token phrase token multinomial token phrase token phrase multinomial distribution word token assign k token k
compute digital purpose need describe ground essentially quite consume investigate one look shape cloud cross road hundred consume research matter need misclassification difficulty type ground cover automate entirely require manually tree great success manually classify evaluate ground cover need review procedure slice random rather unlikely approach thing point datum set big disadvantage course chance class assign class moderately million make enough sample mean require ensure survey human sample rare representative population rare guarantee numerous e population draw method use weight examine described section go setup subset area select point visualization contain give overview size ht lr estimate variability cart framework remainder datum sample cart outline classification use metric point misclassification rare influence feature accurately classify rare total misclassification set q marginal proportion metric section classification conduct series result size sample large size misclassification classification notable simple metric reverse picture poorly among provide cart prior composition turn great agreement post present remarkable misclassification working finding great metric per right sample sample quality metric put correctness bootstrapping result class quality achieve interestingly case cart provide precisely specify post lead composition improve cart become ever survey integrate cart rgb end improve handle big usefulness sense class ground aim supervise various size result project cover area handle usefulness sense cover experimental setup discuss conclude suggestion character change dramatically decade availability ground provide cover code massive amount cover vast handling
limitation scalable linearly pixel since modern second minute produce system oppose invariant meaning scale reflect level recognition task require way size typical pixel instance scalability searching location scale limit coarse image achieved achieve desire outcome contain interest image recognition neural train patch image test compose scale pool bank cascade operation promise location attempt building paradigm visual domain task propose learn candidate look level class predict next location feed another object across integrate geometric treat perform search well empirical able competitive result mnist handwritten digit drastically cut maintain improve accuracy part unlikely work relate notice densely processing may fully image resolution feed hide bias time dimensional vector vector softmax linearity produce since component vector sample image level output fed predict normalize coordinate resolution generality sigmoid training specify look next resolution generally predict width patch patch specify diameter fix feed patch sake neural network network call give resolution outline system distribution eq show weighting sum weighting yield uniform work produce subject training geometric resolution multiplying probability predictor agree object predict take un yield system fairly robust increase instead look give location look propagate describe solution incremental restrict predict latent upon depend compound parameter resolution location predict patch image positive first low assign resolution network term similarly train step possible current perform minimization perturbation nearby around center predict minimize descent fixing variable log correct minimization rather retain empirically confident class incorrect location force location improve location predict correct location nearby train hold resolution generally hold fix work location location reduce preliminary certain slightly far order achieve prediction effectively put around previously visit encourage different position never unless last effectively parameter fit several extract high leverage context resolution interaction take location extract interaction predict simple parallel proceed location extension summarize fix fix n fine tune adjust last minimize alternate find loss confident wrong see update descent relate prior map although analyze informative task rely method drive across however train discriminative method grid image inspire psd variable psd resolution drastically minimize local descent include update design function meaning reconstruct internal represent share module predict make depend location recurrent fed self oppose propose cascade shall depend input pattern classify resolution dataset dataset pixel place location pixel zero pixel digit random set quality training sake original method well convolutional time l c test resolution fine tune input fed pixel consider scale original take patch pixel grid spaced patch etc fairly hyper run stochastic momentum report rate require well connect baseline hide unit convolutional classification convolutional stage pool filter neighborhood add third yield marginal improvement moreover fine reduce system match performance however amount computation magnitude low convolutional row cost threshold classify resolution classify digit second final rejection regardless overall bad sample time use fully resolution method resolution term diversity share apart experimental validate effectiveness rate location look remove loss function cross right dramatically increase demonstrate low nearby perform output assess nature suggest would well trade computational efficiency accuracy compose exploration experiment prediction nearby suggest term tune reach report tie last different qualitative
sequel ingredient modulus ingredient affect boost restrict attention gradient depend smooth sag particularly minimize explore singular consideration however difficulty application might add pre dense computation tackle major question order loss boost efficiently exploit different newton inverse hessian address turn construct construct facilitate first value continuity scalar difficult loss logistic loss obey notable entire I bound positive constant rate motivate follow easily show I become matter far construct transform worth note similar whiten transformation transform whiten transformation modulus change discuss method elaborate number ingredient namely lipschitz modulus ingredient namely upper bound ingredient summarize smooth ingredient derive I bind discussion upper diag rank two eigenvalue decay c probability derive bind individual quantify incoherence incoherence incoherence theory correlate canonical two incoherence since bind state number condition previous condition ensure reduce number l hand maximum original loss function unknown value condition loss let g square assumption smooth loss original datum covariance indicate due mean decay easy logistic balance always check calculate unknown trial achieve comment theoretically r practice especially size sgd begin unstable ingredient follow functional ingredient naive boost convergence verify example sag solve least similar carry sag suffer much study indeed proceed third question need cost complexity expensive attractive efficient key construct subset denote q define computing construct carry follow theorem datum scale condition improve vary exhibit understand ingredient I ingredient decrease aware increase would pose strong effect datum sample construct first theory inherent e numerical synthetic datum matrix vector constant around eigenvalue polynomial decay decay construct value vs eigenvalue convexity modulus similar vs large number cifar namely smoothness individual inner suggest sag like unless specify optimize performance pre initialization size zero either suggest generate describe least variable sign curve sag boost justify straightforward limit supplement boost validate generate plot constructing sufficient regression cifar experimental four regression predict year song year stock return financial tf task pixel classify image predict forest cover experiment report tune sag epoch cifar use datum plot e datum sampling could yield could finally comment original per convergence problem method minimization characterize experimental validate theory condition lemma thm assumption lin science department computer science engineering management science exist boost I ratio modulus effect method minimize good yield bottleneck provide loss minimize practically random validate supervise regularize eq denote decision convex least square become light computation reduce full
show prove compact path ball radius enough ball cover say ball center suffice ball great equation wise thus equivalent c b ia vector path construct connect path connect path statement proposition correct density immediate x k c px statement follow hold proposition intersection conditional independence cx statistical independence hold direct additive causal relationship challenge science decade causal statement discovery assumption relate joint say acyclic dag imply correspond reverse statement class correct identifiable skeleton method restrict linear noise order acyclic intersection property know hold sufficient necessary condition intersection corollary interest weak strict positivity mention model require intersection identification positivity characterization replace strict positivity variable connect identifiability lead intersection property graph identifiable correspondence noise path aware method infer dag inference technique generic mention formally possibly metric space introduce regard existence part absolutely mass continuous contain path see throughout conditional independence independent conditional bx property eq intersection hold intersection property condition intersection assume intersection give lebesgue example apart existence necessity eq make arbitrarily summarize clearly bx ax mu proof important connected equivalent within component consider mu predict intersection formalize intersection minimum cm circle draw circle draw dag circle b area dark gray function value ten minus ten correspond important implication inference causal graph lemma hold positivity characterize intersection density condition corollary notion become important space closed set component wise connect equivalence union member denote treat figure path connect correspond another contain three equivalence formally variable take definition able direct consequence density weak intersection property intersection property set joint distribution characterization property intersection identifiability additive noise distribution structural q function parent direct acyclic simplify identify node eq require intersection since provide weak obtain result structural density density thus disjoint variable
later another generative choose versus use bayesian user specify false determine change precisely change occur change present network difficulty detect synthetic real evolve network digital recover generalize hierarchical attractive change naturally capture structure interpretable dendrogram binary tree thereby interpretability quantifying vary point connection probability quantify uncertainty generative compose vertex edge vertex nest relationship dendrogram vertex connect low density vertex distribution produce eliminate possibility allow compact illustrate email communication connection edge direct dendrogram approach set choice provide little room increase consider connection maximum drop outcome set quantify prevent become convenience employ beta distribution hyperparameter binomial analytically tree requirement spectrum hierarchical end spectrum contain single internal connect enyi tree add model binary leave tree reconstruction amenable classic technique instead tree costly reconstruction solve majority sample tree consensus leave occur majority binary tree contain consensus internal connect mcmc derive probability remain thus prior empirically observe count connection produce observation become uncertainty model close root far structure implicit prevent structural improve infer noise piece network change point determine whether change slide occur fit ratio framework factor ratio two different alternative hypothesis time rather likelihood likelihood update hyperparameter restrict consideration window network occur denote window change hypothesis set hyperparameter window conservative network let window say exceed literature choice must occur result block suggest technical numerically way exactly rather possibly hypothesis see desire calculate count proportion ratio high test choose say change pg w pg pg g bar bar offset time positive negative unknown change systematically different network control circumstance variety change network provide since hard characterize two add one formation p p single parameter control switch distinct state merge change merge single community edge comprise formation change community simple version convert scalar mean degree geodesic show change slightly time slide window quantify detect change formation detect later false among method positive match false alarm rate negative widely even across four size fluctuation merge notice around rapidly change experiment change enyi point rate different change type magnitude change evolve network mit proximity email external evolve human interaction interaction quantify term detection delay estimate delta otherwise actual proportion proportion event delay change mit network comprise proximity student record continuously phone raw edge denote physical proximity week dataset external public period detection three use result figure term well baseline approach close detect reveal external geodesic distance begin cluster activity majority cluster detect exception begin seem inconsistent event along week week agree well week involve typically shift work dramatically seek meet project goal additionally find point fall examine change infer fall fall establish pattern largely highlight additional interpretable within evolve email show detect bar follow statistic colored bar comprise mostly management company energy investigation company apply simple long window window size highly large suggest window operate resolution window resolution bottom window size bottom mit network poorly perform precision examining external event identify share fluctuation examine large structural occur formation take evolve change detection principle non stationary utilize test principle detect fashion large scale change network significantly change equally datum reliably furthermore community merge internal connection accurately split many community formation question technique eliminate whether add weight make easy say point like cluster yield rate structural poor performance result discard utilize apply evolve social good recover external network measure computational inference work ref propose change accuracy scalability believe yield good generative place model graph work detection
pick term entry observation construct constant etc basis ease presentation pick coordinate coordinate remain family space incoherence condition notice sign uniquely whose last uniquely free consequently solution symmetry column observe plug approximation analysis phase fairly straightforward one reliable estimate column measurement show norm translate inequality provide invertible sampling apply proposition observation q observation replacement unbiased apply rectangular bernstein inequality triangle inequality cauchy schwarz plug turn quite soon equality follow linearity apply matrix second arrive norm large assumption state collect scalar bernstein scalar vector bernstein adjoint dimension rectangular relate task procedure adaptive allow eliminate passive project algorithm rank coherence completely row enjoy exist improvement adaptive necessary complexity singular compute nearly coherence eliminate decrease significantly scientific application fail keep study network involve inference individual challenging word amount generate complexity statistical network modern statistical result inference extremely compressive paradigm presence acquisition severe setting aware adaptive outperform passive scheme matrix completion exactly recover fraction recovery aim low precise low observe sequentially feedback drive manner thesis sampling assumption analysis spread passive uniform sample suffice algorithm problem uniformity monitor anomalous recommendation popular item highly active user fail show contribution completion column completion term place simple complement low row space passive must demonstrate completion approximation approximate appropriately rescale good approximation coherence outperform collect definition section defer proceed interested approximate column denote column r r capital symbol refer orthonormal basis subspace subspace versa orthogonal complement refer projection deal subsample denote list coordinate vector form rescale dimensional subsampling orthonormal row column uniquely define depend nevertheless onto subsampling operation exact require rank mean exactly error relax rank approximate effectively interested find matrix satisfy excess risk approximation bottom require svd limited divide energy apart observation coherence coherence spread fairly column loss matrix analog see parameter sample uniformly informally mean capture salient sample scale incoherence translate uniformity decompose incoherent result stochastic uniformity aim line remove stochastic alternative rank behave quantity usual incoherence rank space vast attempt idea relate well well norm uniform sufficient exactly recover involve imply subspace must incoherent strong relaxed well weak prominent work space incoherent notion consider essentially thesis matrix goal preserve property main completion observe relax incoherence scheme completion input constant requirement essentially optimize objective series paper span column approach column form unfortunately unobserved approximate approximation also aware strategy recovery possibly structure passive method difference view iteratively discard irrelevant remainder rely extension rank signal community adaptive effort approximate structure related recover binary recover ultrametric matrix idea similarity impose reason useful develop manuscript suffice rank coherence complement passive adaptive excess uniformly replacement tu display stream direction maintain processing norm new algorithm add lie ingredient onto orthogonal follow x lead side identically deviation allow subtle critical element importantly ensure randomness logarithmic dependence test reconstruction proof defer rank coherence time guarantee vast incoherent column space restrictive exactly number row incoherence remove column match incoherent principal translate assume assumption mention paper weak polynomially coherence weak super dependence interesting consideration operate pass store coefficient represent column lead computational dimension dependence run matrix allow take read input standard algorithm alternate iterative coherent recovery low estimator function let set lastly incoherence account adaptive passive strategy entire exception entry believe bound success probability minimax risk concrete whenever passive sample complexity passive completion fairly form incoherence non apply incoherence recover relax incoherence depend coherence sampling take measurement absence incoherence universal achieve relax incoherence finally argument even require express polynomial right system since must many impossible nearly passive pass observation random replacement x te tr obtain pass pass frobenius second pass additional place rescale preliminary computed top show pass column square column norm motivate pass algorithm non measurement approximation although case incoherent assumption follow assume satisfie complexity defer serve compute run dominate cost dependence mild dependence translation bind bind bind well set tight mention assumption q give matrix improves imply weak relaxation similarly completion recover recovery incoherent draw variance appropriate dimensional signal ratio constant term ignore spectral gaussian r bind f positive recover long use close frobenius number sample bad fix term bind et frobenius square root equivalent significantly thus particularly suited uniform apart sampling soft versus singular value choice regularization amount capture replace thresholde soft thresholding first soft ensure rank approximation second guarantee translate unless quite thresholde frobenius norm guarantee completion therefore consistently sample notion boundedness incoherence agree much uniformity lastly emphasize adaptive uniformity miss incoherence give uniformity enjoy significantly sample uniformity similarly rank uniform column sample plot per rescale simulation figure behavior binary construct span incoherent row collection want various algorithm fraction sample exact recovery demonstrate fix number column constant per column plot rescale rescale versus matrix vary show fraction successful show figure similarly confirm incoherence rank appropriate capture correct plot algorithm adaptive show govern norm figure record maximally simulation coherent space pattern algorithm relative error target axis display set figure column space span vector whose also log normally length construct via clear column plot relative equation function fraction next dependence rank relative error decrease rapidly need increase qualitatively suggest different matrix matrix column sampling length coherence norm standard plot size plot decay plot rescale curve phenomenon proposition sampling threshold approximation column norm dramatically outperform confirm lead distribute provide theorem concentration measure reproduce
report automatic surveillance traditional confident diagnosis perform surveillance combine advantage medium replicate traditional limitation generally individual may result predict actual second surveillance train fundamentally detect strong pattern instead top measure address new method generalize disease traditional surveillance detect even user system user long number mining approach near user five many twitter describe collection twitter tweet focus additionally anomalie behavior social meta receive information health service individuals privacy know month participant mostly student expect collection university twitter receive account account tweet friend user month keep tweet twitter account multiple time hour look day collect profile query limit collect parallel tweet account account account recently update account friend total collect tweet account tweet follow diagnosis content tweet predict tweet divide month user tweet tweet c odd occurrence keyword set keyword name addition serve expert fisher occurrence user six seven keyword table select keyword first find rank information choose top list space character stop perform convert text naive classify user month tweet message shoot bag technique classifier rating tweet time tweet study tweet human extract text rating machine human rare accuracy health table content individual tweet affect tweet dimensional anomaly detection follow tweet month study discard tweet avoid user twitter rate month q mean standard user month user z significant kolmogorov highly biased cutoff roc currently give health status twitter user account follow account user friend consider friend analysis analyze normalize count character friend activity friend detect user friend stream far analyse strength measure roc variance examine source friend number keyword classifier follow user well tweet account twitter user news rarely account htp cdf cf detect twitter analysis tweet reason would detect aggregate meta show adaboost boost voting start classifier roc boost j decision boost voting evaluate leave cross roc adaboost high boost medium high aid
annotation take various image level indicate count bound box seed problem input crowdsource low box document may tag document preferable bias annotation difficulty category object thing annotation category I label number researcher recognize importance annotation semantic label propagate annotation training paper weakly annotate employ weak annotation structural learn incorporate loss consistent annotation different one impact fully annotate since less kind challenge include former involve satisfie annotation latter involve consistent show solve algorithm closely sequential annotation define perform loss augment carefully initialize iterate conditional mode train annotate consistent solution unlike train specific function simultaneously fully label allow cut base augment finally different weak annotation use inference decompose relate supervised area pass inference three high bound local convexity utility minimize annotation establish annotation apply annotation type label bound box seed efficient function mapping mapping express maximization discriminant depend call learn vector variable supervise output learn appropriate compatible paper follow margin formulation call respect instance commonly distance imply maximization discriminant function decompose r problem cut plane replace approximate polytope violate constraint determine run fully weakly annotated subset individual annotation segmentation bounding box segment intensity component segment seed make weakly annotated simultaneously utility formulation seen degenerate weak annotation therefore equivalent slack balance ignore show order loss augment weak augment inference side use convex approximately semantic represent group co similar represent pair node node feature connect correspond discriminant unary potential restrict pairwise nonnegative attractive potential maximize exist expansion weight hamming loss label number pixel weak annotation need annotation bottleneck combine annotation level label box seed arbitrary truth annotation ground truth image plain hamming full argument annotation ground ground area derive tight label wrong well full feasible full need make change per miss significant augment inference decomposable unary pairwise maximization inference cost accounting label box annotation consist box image annotation image bound additionally road certain define level bounding box type category bounding box bound box satisfy certain outside bound unary potential adapt class unary yet infinite guarantee least contrast unclear neither heuristic significantly well initially bound opposite guarantee segmentation annotation seed annotation particular case pixel label annotation locate weak annotation seed centrality infer seed neighbourhood seed bring loss inner take pixel central pixel whenever estimate eq seed loss decomposable factor sift training image label sift dataset database label category crowd original unary sift word build use dictionary histogram color uniform normalize joint approximate dimensionality triple pairwise share strength code unary appearance vector context triple distance case histogram segmentation accuracy recall correctly label per recall correctly divide total category exclude pixel rare see exclude rare sift one target sign car use define label category uncertain model boundary would label follow least pixel cm acc local strong strong ccc c il bb os possibly image hasting sampling try distribution count approximate training show accuracy recall various comparison scenario label weakly provide stable recall annotation full need discover dependency strong sift supervise weakly label rest label recall label substantially complicated randomized hashing connect image may get local label annotation train additional annotation specific tight bounding box object description thing category category divide list background water road include category image building background enhance tight bounding box seed available seed segment summarize seed box annotation give significant box notably thing category numerous overall bounding box inferior label accuracy per object annotation give box contribution support image ex compare dataset foreground kind believe easy background foreground latent objective label bound box seed weakly annotate foreground partial infer annotate necessary look label train consistent annotation train similar outer
estimate label correct construct base refined prior information solve count policy expect accuracy call attain supremum mab problem correspond decision instance arm identically distribute collect reward maximize problem collect reward involve final although intermediate decompose technique lead mdp since note optimize stopping stop wise attain supremum proof present stage reward interpretation accord expect wise take stage th collect label instance receive correspond remain use get reward maximization tuple stage possible reach element next expect reward define technique space budget action take state equal justification mdp induction bellman illustration dp calculate label uniform one correspond last stage table instance label second c although dp find policy intractable since grow exponentially accord develop computationally approximate policy need uniform choose policy decompose horizon mab horizon mab discount infinite horizon index rule cost require complexity decompose reward show discount reward horizon index heuristic hand side stop instance large note state reward instance reduce calibration space exact method require time index policy attractive knowledge ahead next reward policy labeling chance refer break refer many mdp fail assume integer let consistently budget go proposition provide label instance infinity address proposition behave sampling case undesirable subsection propose allocation policy consistent input parameter budget ta ii I output reward instance index optimistic opt opt optimistic next opt optimistic reward reward obtain obtain optimistic maker process opt opt accuracy e almost prove label obtain opt detail b opt framework budget allocation labeling base conditional adopting measure two small probability x q x equal problem opt policy could select large experience limited sake opt opt computationally budget policy space connection optimistic ucb particular ucb select reward side upper confidence optimistic reward arm policy directly fact opt exploitation policy utilize exploration exploitation take optimistic reward play problem also outcome instance consistently budget discuss presentation beta parameter practice easily incorporate beta distribution allow easy skewed address adopt w cc bc labels negative bayesian put hyper parameter hyper proceed parameter hyper prior beta please apply reliability next worker reliability correspond reliable worker poorly inform worker crowdsource worker maker could assign worker worker reliability introduce extra get reliable worker worker th worker pair worker reliability get negative label label provide worker implicit worker single label instance increase reliability get reliable worker reliable worker worker underlie soft label poorly inform always assign wrong worker reliability beta j decision next worker notational word j decision outcome vs decompose wise reward posterior sophisticated long marginal long beta make reward apply opt large scale approximate technique worker budget worker posterior set parameter opt need posterior distribution close monte since inference variational matching technique close highly compute omit discussion adopt approximate q match analytical appendix still new collected distribution stage get worker present opt allocation heterogeneous establish opt heterogeneous apply crowdsource worker section worker fully reliable share unknown reliability maker label pool mdp address crowd labeling briefly discuss two opt policy sake presentation extension noiseless homogeneous set heterogeneous section contextual contextual assume budget allocate among soft average receive label observation budget level budget explore less receive necessarily receive budget sufficiently instance receive receive infer instance assign soft label worker worker simulation reliable vary run budget receive instance opt simulate four opt red opt generate blue comparison different opt highly opt opt robust underlying highly belief reliability worker informed situation prior average different generate compare opt red line quite generate compare worker include index solve mab discount discount labeling horizon since computation instance vary budget report independently last figure outperform regardless opt inf opt improve sample heterogeneous worker budget note homogeneous inf heterogeneous worker fail reliability simulate instance setting set three report use figure regardless choice different section instance pair different worker homogeneous noiseless set diversity worker prior set decide randomly dataset policy times opt inf although inf perform solve linear could expensive scale opt much quick require comparison opt inf cpu level inf opt opt inf worker reliability incorporate belief worker perform lead accuracy opt policy small accuracy using label time little bit particular partially observe opt go experience phenomenon worker opt heterogeneous worker opt worker beneficial worker reliability address crowd label markov decision propose computationally optimistic mdp binary contextual contextual crowd labeling crowdsource several direction work great opt performance instance worker equally crowd pricing motivate work pricing crowd interesting dynamic third label worker worker useful opt policy interesting decision liu share constructive comment quality appendix proof final maximize expect condition write maximize therefore positive set take b beta ia ia ia ia ba ba b second equality ia b ba ba b b proof follow decompose incremental vs determinant second conditional gain labeling th last change next reward formulate proposition deterministic reward integer lemma present therefore ia ia ia ia ia ia integer corollary reward ba ba way proposition accord instance I expect label I break first instance policy expect randomize getting randomized stage instance randomly instance consistency opt first show exact eq plot integer lemma symmetric decrease monotonically symmetry prove symmetry decrease r bb fourth odd r ba r accord property opt many go label go infinity need assume instance never label r accord opt label lead contradiction therefore take label th label ss independent identically distribute law conditioning h path event k q recall select integer utilize basic summarize property monotonically monotonically visualization algebra lemma integer instance stage instance label accord consistently stage select single incorporate reliability worker text approximate beta assume posterior take p b ic ij ic z ij assume independence ij form approximate technique ij make hold assume stage worker far place heterogeneous worker extension utilize contextual feature budget allocation sigmoid ts ti ty log tn laplace method newton follow matrix calculate reward also dirac mean therefore place summation omit need possibility use bayesian logistic multi categorization use conditional expect accuracy tc side ic h ic stage value write q way entry I rs convert integration cx cdf gamma perform dimensional could monte accelerate theorem corollary em minus popularity crowdsourcing task worker internet services amazon crowdsource worker label amount budget labeling desirable budget worker instance label consider reliability task aggregate simultaneously allocation policy obtain dynamic dp dp quickly propose computationally optimistic budget usually collect digital tag picture landscape datum labeling group provide become inefficient costly thank crowdsource service amazon unlabele crowdsource big crowd label availability crowd raise many usually non expert crowd suffer crowdsource service resort redundancy reduce collect multiple worker particular crowd label unlabeled crowd worker ask aggregate collect raw chance true label raw label come pay worker pre reward usually correctness website total raw collect raise central crowd labeling decide budget decide knowledge vast provide avoid easy raw bring fall near boundary worker inconsistent worth collect boost maximize budget simply put highly aside save instance ambiguity reliable worker despite budget ambiguity reliability unknown beginning raw online fashion dynamic conduct optimal budget ambiguity worker reliability formulate horizon bayesian decision process distribution bayesian necessary bayesian mdp budget allocation however dp computationally intractable level policy gradient performance policy opt dynamically choose worker optimistic marginal general opt measure opt achieve superior guarantee well opt start task instance amazon galaxy general worker worker assign entire worker worker worker reliable noiseless ambiguity budget crowd bayesian opt prove converge mdp worker annotation team microsoft allocate worker worker finish instance fully reliable worker parameter reliability next optimistic knowledge flexible extend information web consist fold budget crowd mdp characterize computationally optimistic gradient propose address budget crowdsource organize first process label task fully worker motivate section via dp computationally policy opt heterogeneous worker reliability incorporate contextual multi dataset follow allocation homogeneous labeling simplification incorporation extension category model instance true label goal infer assume pool note worker true incorrect due ambiguity far latent soft percentage crowd crowd reliable worker receive crowd large characterize concrete ask person person old person worker regard person infer raw positive labeling definition soft follow sense
approach outline rather seek boolean identification unique causal remove estimated entire causal iterative bottom order however causal structure bottom approach algorithm find subsection table variable list mm mm compute measure output mutual adapt fit high possibility input mm output h note depict fig bottom represent sort independence sort equivalent fact practically frequency incomplete computable obtain probability less follow mutual sort represents sort available table possibility every begin compute follow proposition list predefine truth step frequency output entire table thus memory accord requirement note loop find measure aggregate thus loop find function conditional element truth function repeat time accordingly variable moderate show later also favorable efficient choose potential parent choose care obtain truth table simply generic artificial way generate respective assumption successively derive value obtain time various combination index evaluation adjacency represent parent child relationship generate estimate wise operation obtain performance causal order true truth value algorithm explain combination uniformly random choice generate summation accuracy hand greater higher sensitive affected strongly reduce indicate causal approach thousand initial stage estimation even amount give various accurately probability heavily proposition require extra strong dependency density true est direct true est direct true est est form experiment previous level pc frequency relationship true direct miss direct non edge count double penalty incorrect causal failure global number pc approach distribution accuracy advantageous obtain pc identify child exposure nuclear binary conventional college school study aim variable causality yes college high conventional candidate select status retain individual background maintain transform binary threshold high produce fig intelligence affect college influence college algorithm five pe external noise directly check simply skewness external operation generic boolean generate hand accordingly model show check mention subsection portion incomplete miss introduce completion issue applicability variable promise way overcome may combine base approach discuss approach toward issue structural noise derive identifiable skewness distribution external computational promising real datum base continuous discrete perspective causal accordingly mutually fact set complement accordingly equivalent rewrite h variable except every upper since pe pe exclude exclude assumption imply one hold exclude variable obtain take relation accordingly one iii mutually accordingly imply acknowledgment discrete project aid scientific thank dr valuable comment causal discovery skew discover intelligence discover causal model paper novel causal datum new causal binary experimental excellent causal structure artificial intelligence derive candidate causal model set assume narrow scoring multiple causal equivalence linear acyclic unique acyclic equation drive causal require objective extend unique bivariate post apply derive order applicability continuous function identify causal multivariate bivariate address identification unique causal multivariate contrast domain computer bioinformatic maintain recently accumulate stochastic structural set knowledge address principle practically unique follow give regard objective discover feasible next briefly review issue third binary exclusive skew term represent estimation fourth section novel causal base characterization causal network develop efficiently focus causal structure within search space second datum search possible dag variable issue acyclic identifiable relation bivariate identifiability base identifiability aforementione derive structure applicability additive model constitute need acyclic relation algebra addition need bernoulli distribution adapt independence order identification estimation characteristic present concern introduce acyclic among jointly boolean function define every external boolean algebraic formulae skew cover generality kx essential identification analogue
uncertain input define include variation fluctuation value design pose comparable density virtue unique need regularization make minimum specification improve control modify relationship variable far ideal system get close specification metric function scalable ensure familiar norm integrable assume bound ignore choose integration interval discretize differentiable approximate replace approximate compute approximate design argument eq element define write oriented row put need function gradient uncertainty available finite evaluate matrix product select bandwidth density use ensure differentiable infinitely bandwidth spurious exist select bandwidth optimal gaussian pdf estimate formula sample optimizer numerical matlab plug minimize compare skewness estimate indicate kernel point row beta histogram four reveal pdfs generally represent indicate apparent skewness relatively small error cc cc distribution sequential optimizer every choose search second subproblem solve appropriate compute forward approximate hessian bfgs feasible line iteration response bfgs gradient hessian fall dimension estimate need interested find joint pdf response interest match bivariate cast extension quadrature uncertain sufficiently may response pdfs surface pdfs quantification include employ place dynamic solver surrogate sample surrogate surrogate polynomial variable optimizer depend design point q eq mean similar bayesian inverse maximum posteriori optimization chain monte consider representing determine minimize distance pdf eq thus pdf distribution deviation optimization seek fig blue pdf vary yield thereby minimize distance quadrature fourth place nearly number observe analytical match quadrature distance quadrature kde example attack computations euler stanford design upper surface design amplitude smoothly amplitude show location select analogy flow solver mesh lift store design objective include pdfs choose number surface produce moment sample speed four simulation geometry begin fold pareto design inverse mean write eq represent optimization value mutation mutation yield call make optimizer early carry plot clarity plot x axis ratio include individual design marker skew dark positively skewed dark highlight low moment pdf tail variance happen yield value however always design attempt two moment take use explicit adjoint method put significant disadvantage solution still design suitable mean inspection necessary especially satisfied design uncertainty ideal select initial design optimization carry pdf numerical computation optimization matlab programming tolerance met optimize previously discretize estimate pdf quadrature quadrature value visible quadrature rule compute polynomial gaussian carry product design adjoint computation surface sensitivity lift perturb mesh gradient rule computation carry result lagrange polynomial surrogate adjoint plot sample form summarize obtain problem require call lift adjoint adjoint examine four distribution pareto front match exactly target lie target exactly optimization carry design optimization comment target pareto front front infeasible pareto front principle robust design require total evaluation yield initial optimization trajectory take iteration call optimizer see second second roughly obtain nearly variance skewness contour attribute arise adjoint surface spirit plot take evaluation take second call yield extremely target target positive skewness skewness magnitude design close obtain routine design pareto pareto front b second design target third difference take iteration cost attribute positively skewed target skewed target find closely target match solely well average order less seek distance performance pdfs smooth differentiable approach illustrate concept demonstrate effectiveness design low tail acknowledgement award physical sciences uk author acknowledge center stanford college author aspect reference reliability typically use even prohibitive limitation statistical moment mean high order issue computational uncertainty specification find probability density pdf matches density robust design quantification stochastic critical modern design strategy range systematically availability computer trend engine environment device operate static uncertainty uncertainty physical
compression exist provide triangular consider involve form entry inequality bilinear arbitrary knowledge concentration inequality literature symmetric treat provide defer devoted selection operator norm validation operator norm loss computational criterion unbiased frobenius general principles stein risk sure begin frobenius bias unbiased unbiased omit follow straightforwardly estimate frobenius risk give could similar bandwidth analysis bandwidth asymptotically need optimal operator estimation encourage selection large weight term bandwidth moreover class entry large albeit high contribution estimate exponentially decay estimate interval simulation first bad secondly operator dominate sure tune sure tune except one sure variability error study uv n ix therefore distribution equivalently similar derivation combine lead q bind op eq establish rewrite derive result incorrect therein quick eq prove derive eq joint proposition follow remark yield q let joint equation equality uv eq great eq proof matrix put net moreover select thank estimator suggestion paper name section lemma remark estimation norm matrix improve addition sure estimator simulation estimator attract largely inconsistent estimator low obtain nature ensure cholesky regularization besides examine matrix mention visit rate refer covariance covariance matrix define unclear rate optimal date minimax stein unbiased sure aim operator norm block thresholding simulation inferior performance estimator motivate establish norm exist theoretic gap novel inspire stein sure outperform bandwidth conduct propose compete detailed result show norm notation denote inequality constant constant show estimator exist operator lead contribution take close argument shall fact see equality triangle op high show proposition
speed provide implementation facilitate predict svms linear inner test equation support vector theorem vector label expand rbf kernel product induce large slow prediction exponential inner instance appendix enable product exponential per factor front summation inner replace equation eq weight svd need compute rbf kernel least square formulation approximation dimension induce yield appendix guarantee compare eqs translate inequality combine assess approximated check observe prediction extra cost prior case norm upper rbf yield form polynomial polynomial eq effect kernel rbf relate rbf kernel must contrast polynomial fixing approximate rbf relative second rbf support act factor polynomial equivalent add approximated exact overall decision equation benchmark exact evaluation make predict computation differ algebra library consequence ii predict approximation dominate follow use loop matrix modern heavily use algebra software platform algebra library approximate evaluate simple operation exploit multiple datum support gain speed speed problem investigate differ exact approximated gain accuracy list report exact percentage differ exact difference report discuss briefly summarize verification set available website make without extra preprocessing set predict various information two class instance handwritten digit recognition contain versus competition feature testing instance instance contain training benchmark list key set dimension differently exact approximate list g illustrate even though combination inaccurate error overall remain always high ignore guarantee regard assess exponentially bind experiment get acceptable approximation occur schwarz bad upper grow approach schwarz conservative input mnist default somewhat optimize much run intel support extension operation prediction model evident time number number dimension approach ratio ratio optimal time minute impact library column naive implementation gain result file benchmark ann model enable ann range factor grow complex unit straightforward propose amount prediction model consist scalar dense small significantly counterpart table illustrate approximated datum compression ratio would approximate square svm even kb kb mb mb kb mb kb gb mb finally subtle approximated sensitive support instance model approximate consist combination vector reverse currently consider rbf series svms wide variety gain benchmark list approximation application benefit generalize regard normalization establish remain approximate svm approach fast versus iii prediction neural network always quadratic validity approximation illustrate absolute second absolute enough frank use order series inner support evaluation speed relate quadratic number approximated prediction memory optimize gain set additionally acceptable bound task ability use trick trick operate transform result attractive trick favor less cost rbf kernel situation evaluation limited span vision denoise radial rbf know parameter q prominent take
normal spherical scalar g call elliptical distribution transformation spherical elliptical elliptical xx ec ec cc elliptical characteristic guarantee specify elliptical mb continuous elliptical k symmetric function fourier elliptical fourier transform entire partly grant aid scientific b definition section property claim advanced institute mathematics nonparametric kernel bayesian incorporation certain nonparametric call rule rule current bayesian deal novel base mb exploit distribution incorporation mb nonparametric bayesian flexible inference nonparametric combination mb filtering model observation transition mb additive extend general conjugate model develop hilbert rkhs exploit reproduce kernel mean expectation feature respect distribution map associate kernel kernel map distinguished kernel guarantee specify estimation machine application estimation hypothesis classification chain rule rule kernel refer kernel sum rule kernel combination rule bayesian entirely study inference propose filter represent sample develop capture relation restrict nonparametric probabilistic robot vision robot localization hide mobile capture robot position estimate image specific model encode probabilistic desirable model mb exploit include conditional aim refer mb exploit tractable incorporation mb rule model focus mb develop filtering inference use mb middle probabilistic additive sample simply non twice mb mb respectively introduce mb estimator represent sum feature mean difference mb burden g addition bias error mb regression determine knowledge reflect describe mb mb sect focus additive mb case describe systematic consider conjugate comprise rule non mb space observation summarize mb mb incorporate mb thereby yield inference model mb additive filtering algorithm process kernel transition handle arbitrary e sect provide definite whereas filter whereas particle filter observation propose mc filtering space combine nonparametric transition dynamic simple present explicit expression exploit necessary filtering transition noise mc method analogous kalman require paper next preliminary mb mb propose sect ground robot sect review unless state positive definite arbitrary nonempty kx xx semidefinite gram matrix definite shift definite hilbert rkhs nonempty hilbert function exist unique product triplet definite kernel rkhs measurable measurable generate rkh study mean property use p w iw n follow consistent consistent rkhs consistent kernel rule space marginal let p p marginal deal py e computation x conditional non I ij ng li operation deal q kernel rule employ qx py give introduce mb define combination mb infer sect define mb sect mb example show sect describe mb sect base include relation obtain mb deal k mb consider conditional manner additive noise compute mb additive gx gaussian vector constant even rx additive gaussian gaussian noise gaussian additive gaussian conditional density kernel fx fx substituting expression mb compute case output mean often require mb additive gaussian computed systematic focus mb estimator differ estimate combination feature whereas mb input conditional simply mb give regularization smoothness whereas mb tune determined reflect knowledge type mb combine let factor comprise marginal chain fig middle model mean mb operation ng probabilistic distribution non operation matrix g ij computing evaluation n g l I z k I previously consistency mb sect kernel express weighted feature mean rkhs r g lm iw mb section dynamic model arbitrary misspecification mb mb coefficient fix horizontal exact leave mb value indicate horizontal correspond sensitivity misspecification mb perform misspecification set determined eqs analytical rkhs x gx pz xx ia q analytical describe error estimator show type mb mb mb iii mb mb decrease reflect knowledge mb bottom time variant state filter algorithm dynamic bm u dynamic gaussian kernel regularization parameter base grid search estimate trajectory trajectory sequence green dot sample color color top learn dynamic process know kalman kalman nonlinearity report accurate mse due mb result dynamic additive obtain appendix example dynamic exist fig phase demonstrate deal bias error sample thereby demand vision robot position mobile capture robot state robot comprise permit domain definite localization contain mobile environment building angle represent internal e comprise employ mean model motion part database comprise trajectory predict current current measurement argument learn dataset spatial pyramid sift descriptor image gaussian kernel bandwidth tune base filter maximize kernel test experiment size na I dataset maximize kernel pyramid markov property near nonparametric near image training determine demonstrate incorporate size perform property method yield combine estimation conditional distribution additive gaussian mb non mb demonstrate consistency mb show mb filtering nonparametric observation additive transition mb elliptical comprise contrast mb contain mean dynamic mb inference partly allow kernel appendix describe model mb kernel mean infinitely distribution probabilistic map convolution indicate convolution positive convolution definite gaussian generally stable closed definite light degree distribution definite kernel gamma density positive laplace close convolution note kernel mb additive noise stable case let index skewness stable rkhs x function generalization dimensional stable measure unit sphere denote stable aa fx stable comprise sub location denote stable gaussian case stable assume additive stable rkh stable density gaussian elliptical conceptually general elliptical elliptical elliptical dispersion characteristic generator elliptical elliptical nonnegative distribution stable respectively additive elliptical function elliptical rkh elliptical ec ec follow independent elliptical vector elliptical normal mixture elliptical give scale characteristic generator give h generalize normal elliptical elliptical elliptical elliptical lx xx generate laplace lx x laplace laplace laplace density laplace rkhs exponential direct computation omit derivation conditional kernel rkhs
sparse capture outlier tensor model multilinear interaction hierarchical tensor hierarchical student hyperparameter independently fully bayesian treatment linearly works implicitly tune discover adapt prior various tradeoff maximum extensive many art completion determination condition pixel pose illumination show latent past decade g recognition processing framework tucker cp know different e partially tensor miss cp miss cp probabilistic exploit exist manually severe emphasize surprisingly limited straightforward give tensor np fact determine even bound ard framework solution point applicable incomplete tensor bayesian incomplete tensor propose cp infer multiplicative handle heuristic way generally show convex gain considerable completion seek optimize apply nuclear framework completion low multilinear approximation auxiliary exploit completion suitable worth nuclear since standard optimize apply straightforwardly determination still challenge outlier frequently robust factorization use tucker outlier nuclear norm lead limitation quite tune evaluate unknown imply exist impractical obtain therefore automatic appealing limitation robust tensor mainly overfitte especially prediction unify model group additive modeling outlier partially represent tensor rank specify induce share automatic determination hierarchical student element hyperparameter learn evidence vary outlier fully bayesian framework derive anomaly automatic demonstrate term robustness allow rest discusse introduce preliminary multilinear specification factorization summarize show robust beta exploit however jeffreys outlier laplace treatment employ model approach crucial high pca recently tensor tensor error knowledge deal within g capital letter tensor denote letter g ni tensor element wise product instance hadamard product denote hadamard rao reverse except denote incomplete denote index define tensor otherwise measurement sparse cp express outer vector shorthand factor interpret tensor small integer representation factor denote row wise factorization denote correspond multiple n affect factorization latent vector multilinear intrinsic structural dimensionality unknown tuning parameter challenge seek infer partially datum attempt minimize individual hyperparameter denote share mode gamma maximum dimensionality concentrate tend yield minimum induce place hyperparameter place q individual correspond element framework laplacian student commonly apply enforce preference hierarchical student student control improper marginal laplacian lie encourage keep fully conjugate possibility fully bayesian treatment setting specification place illustrate simplicity notation factor matrix map sec principle provide treatment infer predictive miss infer resort approximate inference latent involve advantage employ present derivation proof approximate evidence constant kl imply mean field posterior explicitly derive virtue conjugate fig mode message likelihood incorporate parent express shown factorize subset entry mode index update evaluate straightforwardly introduce sec I rao sum index imply interact intuitive denoting fitness word fitness information current firstly coefficient scale fitness matrix incorporate posterior gamma q rd eq evaluate I far update obtain c thus strongly zero eventually several explain sparsity component predictive miss posterior yield student eq parameter uncertainty cost denote much cost scale polynomially automatic component rapidly present sparse correspondingly fully observe previously computation posterior matrix need denote row independent appendix accord efficiently simplified incomplete hyperparameter cp approximation efficiently result random sec computational addition simplify present treatment gain method characterize automatically tensor require predefine completion need discover various outlier elegant characteristic tradeoff rank evidence estimation take regard solution deterministic algorithm empirically computational firstly rank I first component possess third component mode draw distribution realistic setting subsequently randomly utilize relative evaluate recovery match cp cp ard factorization carefully truth generally cp tune tune penalty contrast cp ard tuning fully vary type signal ard miss entry percentage magnitude tensor ard ard poorly within recover tensor gaussian percentage confirm capability adapt true outlier cp ard observe ard rank value multiple datum term high note base applicable situation demonstrate robust partially observe tensor completion outlier compete performance quite stable vary robustness miss confirm accurately estimate multiple tune good run tensor completion runtime different miss ratio efficiency anomaly real surveillance video separate foreground highly tensor foreground move conduct popular sequence extract frame background alm perform video necessary optimal separate foreground frame separate person stand capture person foreground except clearly obtain auxiliary contrast factorization handle robustness robust capture tb property anomaly conduct experiment drop pixel compare alm illustrate fig alm recover background presence pixel effect severe robustness miss factorization color background whose consume fashion image tensor completion face multilinear first use pose illumination impulse gaussian noise removal poisson ratio tensor cp tucker ard carefully ground tucker ard require initialize cp perform multilinear tucker ard show image people pose robust satisfactory visual quality cp tucker ard removal detail quantitative efficiency runtime significantly note tune parameter ground truth outperform performance tuning computation tucker another automatic selection superiority non gaussian achieve determination tucker ard cp cp characteristic tucker ard robust characteristic framework overfitte determination
compositional regression compositional consider compositional start application become greater compositional specify compositional proportion heterogeneous method analyze assumption normal see compositional since compositional modelling consider compositional compositional compositional correlation analyze compositional consider simplex real distribution transform ratio compositional transformation regression response compositional give ij g ny interval j j percentile compositional methodology unbalanced compositional unbalanced bernoulli moreover generate size number follow software perform square error confidence proportion attack serve game win team follow attack serve understand compositional adequate kind since multivariate sake usual analysis analyze separately present proportion attack covariate transformation cc cc proportion confidence overlap proportion component attack serve different regard belong win outcome analyze result impact analyze compositional contribution winner team attack serve opposite team individually multivariate interesting covariate serve competition need simulation cp discover cp stable near coverage real pointing consider datum sp mail competitive physical international player carry element paper new compositional methodology estimation compositional illustrate datum compositional transformation origin change rule evolution player interesting field question competition development technical strategy factor player decision way team attack opponent among serve point opponent analysis mainly motivation compositional attack serve opponent compositional appropriate indicator objective procedure example team assess home advantage effect period country water verify home advantage way follow evaluate specific difference among home advantage home advantage et investigate play home indicator score home loss play home home relate country indicator assess home away quantify variable block attack opponent involve game home game game apply identify analyze play team multidimensional statistic attack attack attack predictor evaluate serve receiver multinomial obtain occurrence et al effect match e attack efficacy select action classify match status indicator action serve attack opponent lead team possibility win competition initially normality homogeneity kolmogorov quantify among independent comparison loss point break set also test work examine technical factor identify statistically game kolmogorov win match point difference win match factor service context compositional error team
condition number upper theorem sufficient sufficient affected greatly analyze asymptotic complexity result bound recovery section recovery measurement correlate optimization suggest significant improvement investigate iterative reweighted noiseless iteratively step except iteration suitably close regular constant iteration reweighte estimate lasso place weight identify justify property solve reweighte penalty encouraging justify much well reweighted minimum penalty make get undesirable minima choose reweighte bind ml expect reweighte achievable bind successively still efficient simulation reweighte lasso closely vertical interestingly reweighte nearly fail snr comparable require large furthermore gap infinity imply sublinear regime vs compute see recovery snr correlation achievable bind reweighte lasso db variant omp omp noisy miss sense omp vs information theoretic noisy variance support omp perform noisy variance theoretic recovery much level achievable conclusion reach omp recovery omp affect correlation degree recovery observation matrix information central discrete support conjunction tractable nevertheless identify gap exist fundamental sample computable consideration compressive miss show understand parameter snr correlation measurement set show vanishing also identify complexity gap lasso theoretic bound gap get notation conditional variable distinguish variable variable w set collection size hold theorem slightly weak clarity difference binomial line refer reader separately fact u summation sum specify condition equal sum chain rule entropy follow random outcome follow conditioning conditioning follow second depend index size contain prove even though idea general discrete generalization variable exposition fix I denote joint variable ix ix nn cumulative density assume continuous random quantization value joint ml decode indexing variable strategy quantization level level since decoder bind upper bind quantization convenience furthermore boundary equally e x j last theorem calculus also increase small upper furthermore boundary space py third variable assume measure lead lead eq derive easy independent across gaussian reduce inside plug integrate leave long expectation exponent exponent p independence integral leave write integral integral integrate r I analogous incorporate bound p integral last replace lemma necessary q q readily condition bind straightforward manner go infinity ii I sufficient exact theorem become element incorporate equivalent asymptotically satisfied choose choose necessity mutual necessary note jensen therefore necessary hold show clarity consider initially third take give result probability difference exponent integration term also replace w finally outline prove condition relax condition w incorporate scaling go definition salient recovery sample restrict non mutual linear performance gap consider frequently arise number scenario dimensional compressive low formulate observation set salient aim characterize arbitrarily probability parameter signal snr illustrative correspond realization markov depend combination element give nuisance theoretic salient see testing processing identification formulate namely code overlap share h namely severe related overcome limitation iid relate recovery instead general inequality bit represent set represent uncertainty output quantify observation exceed furthermore sharp exponential identify salient linear sparse characterize snr sharing markovian sparse regression sense cs probit sparsity variant particular analysis extensive deal model square variable list contribution markovian formulation follow repeat sake exposition literature specialized tailor instance relaxed testing quantization descent noisy form penalization sparse conceptually unclear come markovian viewpoint sparse observation pattern recover rely sparse thresholded introduce unnecessary distinction estimation discovery well support easy reliable conceptual tool capacity tool infer message indeed resort one tool necessity pursuit etc cover require impose extra reduce discrete natural object discrete pattern design ensemble rip key set herein purpose sense incoherence largely model herein tight ml decoder tight sufficient condition compare information theoretic bind practical omp gap gap room improve solve recovery recovery analyze easily analyze recovery observation variable obtain bound expression exponent identification code extend iid latent observation conditionally non identify gap practical regime explicitly row indexing row indexing use logarithm base latent markovian sparse exposition variable iid variable arbitrarily indice random randomness error recover salient salient salient label index iid latent elaborate mapping variable incorporate assume elaborate conditional example eq extend different nonlinear boolean indicator item result certain e denote set conditional conditionally iid joint coupling appear restrictive describe arise across see correlate condition account several possibility meta iid identically j nevertheless note iid exchangeable de bind analyze choose ml decoder decoder assume decoder upper decoder average methodology deal set true I exist analysis characterization lead sufficient necessity recovery select decoder disjoint iid hold observation sample condition arbitrarily sample di condition hold arbitrary constant ix mutual information salient total mutual denominator condition numerator bit denominator information subset control account error necessity support change condition support x mild mutual satisfied
size cluster encourage motivate encourage law cluster goal principled infer complexity generalize chinese restaurant objective small value law incorporate formulation result spectral long inspire objective derive optimize cut algorithm guarantee converge optima view precisely particular gaussian extensive segmentation asymptotic yield line normalize cluster approach mixture asymptotic fail relate base adapt modify small analysis power cluster discuss segmentation prior domain cut cluster undirected vertex similarity entry represent cut within cluster several objective cut edge normalize minimize relative complete spectral eigenvalue normalize cut seek minimize objective fall generalize cut objective extend original datum objective degree surprising fact establish mathematically follow normalized cut definite equal degree node effectively objective kernel monotonically minimize kernel objective appropriate weight objective equivalent mean goal graph objective bayesian chinese restaurant yield cut cut objective nonparametric chinese crp decay crp customer enter restaurant infinite table customer subsequent customer customer proportional new extension crp power modify crp customer exist table table increase probability start explicitly crp formula exchangeable indicator cluster crp express original crp cluster distribution law cluster distribute treat assignment negative log objective normalize desire clustering result give preserved size tradeoff standard cluster objective propose normalize relax cluster optimize globally technique law incorporate regularization maximization turn impossible instead normalized kernel adapt mean problem derive regularize weighted standard section normalize weight monotonic law treatment equally applicable cut association objective indicator justify cluster representative cluster objective e regularizer update weighted indicator minimize mean usual mean step make less fairly objective exist new cluster effectively correction arrive assign em cluster go increase restaurant table computing get rich new cluster start immediately specification analogous convergence easily show monotonically regularize equivalence cut weighted equivalence vector cut objective simply replace term exactly cut diagonal whose kernel matrix use space node regularize distance unchanged expand write weighted kernel problem scalable application utilize objective trade repeat suppose singleton cluster regularize w distance cd eq points nonparametric refer chinese joint maximize yield precisely mean equivalence cut inference author law datum add function objective incorporate encourage follow author exceed trivial datum minimize objective law experiment demonstrate utility approach mean law standard cut method cluster mutual truth cluster block specifically first assignment block random graph create cluster process adjacency leave figure stochastic model stochastic gaussian entry law cut algorithm compare cut big nearly ccc mean ccc grind truth upper next compare perform cluster power law distribute select uci dataset law class ground truth randomly split feature validate algorithm fair setting average perform k dataset dataset except mean bad power whole validate mean split large ccc r part convert normalize cut graph cut obtain adjacency similarity vector adjacency weight section truth cluster cut true cluster cut power cut cut normalize cut finally qualitative image berkeley adopt affinity matrix cut normalize cut generate image cut tend
careful constraint program strictly sake efficiency scenario impose crucial allow fraction predict unobserve via inspire success ground sparse matrix yield loose uninformative suit broad class pairwise partial angular interior second unable practical first optimization admm program empirically amount produce desire round procedure make matrix entry wise ensure round detail admm round solve original may return fractional rounding maps proceed round map denote compute rr estimate ji ji match repeat next recover set ground even vanish portion succeed minimal soon randomize universe herein generate point independently coincide independently incorrect matrix uniformly impose primarily simplify presentation remark significantly need generate mean theoretical appendix universe equivalently rank allow match densely corrupt input reveal consider constant corruption obey probability arbitrarily implication follow randomized succeed pruning recover account outli tolerance equivalently regime almost fraction input corrupt highlight matter perfect many none recovery regardless performance report pca semidefinite enable correction condition error partially come coincide tolerance algorithm outlier reliably provide recall threshold p recovery nearly soon theoretic recovery significantly outperform sdp heuristic match well cluster apply matching sdp exact minimize sum nuclear norm enable dense denote ground sign set sign pattern highly non negative sign sign propose assumption cluster inter two experience edge comparison deterministic edge error recovery therein encourage sec result performance consider recovery well evaluate describe simplicity full object fix universe remain assess configuration carlo trial reflect blue perfect red denote failure trial htp cc c map wrong incorrect phase object majority theoretical recovery figure noise ability improve illustrate transition diagram unable allow dense correction fig benchmark building fig six benchmark house building evaluate show building contain different generating algorithm present setup previous house matching assess sparse map match neighbor raw building build map sift view sample remain distinct per house count percentage correct building evaluate deviation manual compute image necessarily manual pixel evaluation htp fig ccccc house house moderate initial truth contrast ground house condition rank quantify specifically q eigenvalue eigenvalue put yield probability sec prove analyze kkt optimality treat sub I space via submatrix corrupted version convenient introduce notation complement support complement respectively complement projection resp way index element sufficiently claim bernoulli copy appendix define reveal derive result j nc q additionally random n represent unnormalized laplacian small eigenvalue see appendix contain resp concentrate state constant pass immediately bernstein variable duality summarize lemma satisfy additionally theorem generate high dual optimality decompose incorrect encode construct produce p sufficiently set represent encode symmetric q l j l toy contain incorrectly illustrate fig cc b toy example shape incorrectly map contain check procedure furthermore q assumption ensure dual satisfy establish follow lemma universal mn yield entry lie il il justify thereby algorithm correctly recover ground tn mn constant similarly imply eq allow constant q sec moment method control nice specifically follow sum cycle treat ki occur exactly suffice examine repeat least twice relevant edge span edge add adopt divide non vanish determine cycle vertex cycle notational simplicity exceed obtain since exist complete adjacency except exist bernstein vertex exceed probability least eigenvalue kkt suppose complement condition immediately fact assumption allow equality possibility suffice establish since semidefinite take imply put compare ij besides ij contradict must establish claim ii claim claim feasible optimizer sec proof would bind operator random eq observe q suffice examine due entry encode incorrect correspondence affect magnitude amount within lemma affect entry bound row row affect distinct block ji bernstein put universal ij p n universal affected satisfying express p magnitude hoeffding proof lie convert matrix one disjoint diagonal quantify block decompose ij concentration state entry lie support norm construction constant since upper universal h augment denote translate identity sum eq expand n derive positive negative j aggregate shape collection globally consistent close certain provably advance recovery none theoretical corrupt mostly object demand partially view propose jointly pairwise match densely corrupt semidefinite truth program spectral match numerically solve alternate method round near ability guarantee work even dominant behave outlier furthermore succeed minimal complexity perfect matching achieve include example confirm usefulness shape mapping cycle partial relaxation matrix graph find relation across fundamental scientific field list rna compare rich shape joint multiple object object matching pick perform remain pairwise algorithm satisfactory practice gives rise question aggregate one compute order efficient manner object observation pairwise preserve relational compatibility consistency composition map connect recently detect outlier work experimentally inconsistent cycle corruption despite empirical work underlie ground truth reliably recover provide match several order accommodate practical challenge state provide theoretical rise applicability presence highly noisy source input corrupt match result dense correction information consistency pairwise map appropriately exploit ideal map outlier challenge remain provably dense good approach ground map scene different camera scenario pairwise map expensive unnecessary characteristic infer unobserved match incomplete tradeoff correction remain require match densely corrupted pairwise map paper aim concern object error fold evidence propose call constraint attempt compatibility semidefinite constraint rely total match spectral methodology essentially scalable optimization guarantee surprisingly admit recovery input finding near optimal error precisely fraction behave outlier besides incur exhibit strong nearly equivalent full scenario succeed reliably unobserve partial input soon map connect theoretically evaluate dataset synthetic example finding art matching matching shape graph matching focus biased isolated technique easily exploit noisy fundamental nevertheless none demonstrate provable accommodate recent denoise model another recover relaxation observe relevant specialized joint also enable broad inspire rank completion component analysis relaxation recover herein occur analysis tight highly need incorporate structure encode regard estimate graph nevertheless intrinsic property herein explore form doubly belong highly symmetric translate language inter encourage detailed theoretical comparison organize problem setup include recovery method follow alternate method admm round strategy also performance proof theorem introduce demonstrate summary setup matching partially algebraic pairwise formally define notion throughout match encode discrete paired particular map w partial input output matching describe input partial agree partially totally truth detect incorrect pairwise map aim propose tractable return
countable express exactly determine learnable minor index learnable complete determine vc drive force must set nothing determine deal issue definition computable say within index pac learnable degree part present show follow eventually dimension computable initialize require least segment look also add stage stage index path empty infinitely infinitely many set vc vc set disjoint finitely computable effective concept arbitrarily thm conjecture definition thm thm thm thm learnable class indice computable make precise cover property vc method characterize object carry machine learn least gold gold number computable function initial string instance strength determine recursion focus interest index learnable model correct much exposition subject two kind arise target learn distinguished arise neither aspect easily gold identify index computable segment neither randomness present model pac learn recursion calculate learnable subset call call every behave ask running call existence know boolean pac learnable truth assignment satisfy space learnable learnable arise exposition integer hull contain pair intersection half consequently show reasonable theoretic arise finite vc david denote distinct least every measurable behave behave pac learnable vc meaningful determine learnable arbitrary class narrow meaningful constrain enough usual framework result know give class computable subtree path subtree co subtree define equivalence follow say formula calculus regard boolean variable let construct subtree include check subtree intuitively fall enough segment detect unless fall uniform representation take real string interval computable set path include binary length effectively subtree define hyperplane computable coordinate represent subspace subtree natural space exclude exclude space requirement computable restriction absolutely hyperplane give hyperplane computable coefficient hyperplane hyperplane lebesgue suffice show cumulative hyperplane furthermore multiply many machine situation could certainly terminology concept term computable enumeration tree interpret tree index concept adequate need would like class computable reasonable class computable class weakly effective computable computable nd place see mention soon strong definition want computer something concept coefficient let effective concept computable set complement
label subset training unsupervise frequent tag annotation vocabulary average annotation image whole annotation compare directly approach use specifically rescale side keep aspect ratio sift feature densely sample extract patch fix sift unlabele visual vocabulary represent suggest spatial annotation vocabulary frequent tag section together treat descriptor adopt used layer architecture layer activation unit softmax conditional discuss since image sigmoid output probability pc unsupervised layer label base top hide annotation word feed svm belong average ap ap curve average class compute metric report connection follow find normalize input histogram rescale unit hyper unsupervise weight parameter etc overfitte adopt dropout maintain decay throughout gradient average weight put emphasis parameter annotation approximately annotation word section multimodal hide epoch epoch layer epoch present baseline map performance among epoch unlabele baseline epoch epoch epoch datum epoch epoch deeply hide deep beneficial establish explore property section annotation imbalance visual annotation influence weight annotation weight value hide epoch annotation equal annotation configuration annotation bad annotation increase get among annotation perform good layer epoch annotation value illustrate multimodal qualitative retrieval scenario image retrieve learn task adopt similarity correspond rest multimodal query learn image ability annotation ground truth annotation probable annotation modality extension learn visual moreover propose deep version model bag image extract meaningful unlike model model autoregressive advantage iterative confirm competitive multimodal modeling achieve multimodal benchmark yu zhang department university china zhang universit k model choice task deep boltzmann topic autoregressive demonstrate state extend multimodal datum simultaneous annotation first increase discriminative learn employ learn joint word annotation compare topic model deep version reach state multimodal modeling source increasingly model allocation lda generative great model share model extract infer topic word extract visual convert word bag visual py model use predictive leaf word use multimodal variant lda propose correspondence lda relationship annotation modality assume multimodal lda learning regression module relate multimodal document multimodal corpus annotation word embed annotation power improve heart topic generate produce extract observation solution disadvantage sophisticated inference trivial expensive must approximate variational mcmc yet simplify visual image approach visual distribute representation artificial bag datum text annotation jointly boltzmann multimodal achieve extension time generative approach document autoregressive estimator directly joint chain modeling potentially inference perform instead document simple feed value network text rs text retrieval deal annotation illustrate incorporate visual highlight discriminative objective result confirm lda extension deep discriminative word illustrate datum imbalance histogram fed compute discriminative global mention multimodal lda lda modality describe image globally label inside city etc annotation content margin formulation line extend extend context multimodal modeling topic autoregressive network increasingly model review autoencoder multimodal though outperform favorable reduce multimodal especially paper deep version outperform predefine vocabulary data image convert visual sift descriptor densely detail image thus bag word sift descriptor descriptor contain distribution conditional model learn feedforward wise activation bias respectively compute equation visual word address issue tree conditional logarithmic leaf tree model reach binary regressor multiply right choice specifically root leaf internal always binary subtree contain bias logistic sigmoid guarantee could attempt organization word tree assignment leave v I document document descent word bag average exploit conditional previous hide since regression total practice unit eq representation could fed classifier computer highlight jointly concentrate single discuss deep incorporate discriminative feature visual describe text annotation feature unsupervise lda perform directly use pyramid reason statistical might computer address variant lda supervise unsupervised propose attempt architecture regular output softmax layer label put regular neural crucial difference hide visual conditional discriminative term understand encourage explain visual word hybrid generative term hyper perform descent multimodal visual word annotation performance note notation word annotation section concentrate unsupervise supervised mention order many view connection employ across interpret factorial ordering notation joint order treat explicitly random stochastically across autoregressive conditional rewrite summation conditional split first index order rewrite note equation simplify perform training expectation expectation require representation annotation random annotation perform stochastic specify separate conditional conditional notice annotation vary value vary procedure prescribe exhaustive sum predict stochastic predict implicitly sum gradient share permutation show split two network approach mention conditional summation layer histogram representation word original training instead hide effect complexity layer feedforward neural bias layer equation obtain representation implementation efficient binary tree regression however go back softmax conditional preferable conditional softmax softmax softmax amenable gpu end experiment extension softmax deep specifically negative sampling supervise regularizer importance unsupervise annotation framework treat setup annotation annotation problem example annotation ignore huge visual gradient come word conditional annotation vocabulary annotation annotation histogram element hybrid rewrite replace weight annotation weight annotation pay annotation cause imbalance hyper select annotation heavily improve besides annotation embed global play multimodal datum thing complement specifically image possibility matrix specific understand bias condition word multimodal layer world test retrieval achieve code publish ability multimodal measure simultaneous image annotation world datum annotation annotation benchmark extensive lda multimodal lda performance also support compare pyramid construct set tool image city country evenly maintain image evenly test occur densely sift feature size dense sift sift vocabulary grid spatial position produce pair use annotation annotation
principle relevance discussion statistical employ determine order refer determine reject hypothesis monotonic consistency propagation infimum variable line follow similar enforcing complement sx dd dd sx f x introduce parametric version kolmogorov consider parameter identify refer tailed ks tail ks test etc assignment satisfy ks hypothesis hypothesis constraint monotonic consistency parametric line h sn sn ij jj sn sn x propagation enforce complement application statistical employ classical hypothesis non parametric application mean sample sample accomplish one h reduce significance reject range fact work specific singleton illustrate illustrative domain feature singleton inspection desirable inspection plan unit plan horizon inspection inspection j model inspection inspection inspection interval inspection inspection enforce u inspection consecutive day inspection carry month inspection plan assessment day horizon plan assessment interval black dash h use encounter statistic fit look scheduling designing management instance desirable statistical schedule order constraint stochastic pass priori policy stochastic constraint property instead operate assumption specify parametric constraint constraint statistical constraint instead set work spread constraint statistical exhibit statistical e median bridge link program nature inference identify assignment specific discusse enforce consistency span encounter inspection scheduling inspection anonymous valuable suggestion university scientific project university publication part research foundation introduce modelling constraint programming discuss encounter statistic novel inspection scheduling desirable informally assignment prescribe first embed kolmogorov filtering enforcing discuss discuss application span encounter inspection scheduling aim inspection desirable modelling world consist outcome triple denote outcome sigma algebra return function probability mapping set element event often transpose distribute may time replica generate variate outcome variable define adopt follow statistical cdf statistical also cover outcome operate observe determine adopt carry statistical select g generate data hypothesis datum formulate suitable distribution determine obtain extreme outcome highly hypothesis great evidence collect insufficient reject follow survey test test kolmogorov student hypothesis compare hypothesis student inverse freedom respective tailed test great test assume statistic pool variate size student reject note range variance test parametric compare reference cdf hypothesis draw define cdf supremum set target converge kolmogorov nan reject cdf kolmogorov numerically approximate tailed stochastic reference I case vice versa kolmogorov employ band band entirely contain cdf tail stochastically dominate triple variable map combination domain use kind logic linear constraint constraint constraint among dedicated able remove infeasible enforce g arc consistency generalise arc exist compatible domain filter repeatedly solver heuristic search engine solver explore partial assignment order
set sum fraction correlation nk k let structure distribution acknowledgment office correlation uniform assignment half proof subset node eq choice cardinality union value prove fact david laboratory information electrical science management institute technology mit paper result unconditional computational bind class et al construction equivalent difficult noise computational learning bind exhaustive significantly restrict class aside assumption tree paper property focus ise show interaction strength exceed result impact variety application high dimensional biology finance determine access identically distribute undirected graphical vector question mostly undirected model search neighborhood decade ng give graphical kullback present learn algorithm perform neighborhood recover graph scale guarantee reconstruct sample complexity well exponent run great deal graphical ask answer show unconditional et al understand apparent plant clique show exponential computation algorithmic include chain fairly follow theorem algorithm tractable writing form exponent primary importance think require restrict class interaction seminal liu model restriction generalization assumption distinguish knowledge require informally property decay exponentially fast decay high temperature ise multinomial exponential correlation neighboring variable ise set candidate neighbor roughly correlation set exhaustive search neighborhood mention cost algorithm ray similar number reconstruct ise begin ise connect graph os benefit non negligible dependency wu et remove generic base regularize logistic algorithm logistic provably certain ise simple optimization incoherence restrict isometry difficult ask follow second general even ise whereby prefer opposite state correlation algorithm base prune pairwise correlation decay anti improve first contain kk lemma end basically reconstruction graph degree give rule require know let star nod star center none edge loss generality call edge upper probability star include star center non arbitrary cardinality match share disjoint event note variable mutually discussion error successful function lower begin observe include define take map reasoning section consider anti ise real working configuration amount example recover think soft large modify core eq estimate structure determine subsection e probability possible define restrict analogously z define set particular map neighbor configuration z ab prove iii ii z hz algorithm run core infinite obtain effectively conditioning conditioning subset law g u u corollary conditional estimate conversely ingredient show restrict attention zero start compute equivalently monotonicity together bayes denote last quantity number obtain stochastically dominate inequality reduce remove node condition strong e u b integer ising model use sample argue tolerance consider h statement eq next choose corollary inequality last show complete statistical taking
curve ranking hash hash noted rank well hash sub lsh important next subsection actual four hashing scheme parameterize retrieval parameterized lsh generate meta hash meta hash form hash hash consideration scheme randomize lsh preprocesse design intersection affect bias asymmetric correction asymmetric hash hash undesirable lead provably inner literature experimental evaluation advantageous scheme require modification asymmetric weighted weighted intersection general value vector asymmetric weighted similarity use new asymmetric promise hash corollary definition department ny department computer science nj usa widely index suboptimal desire overlap inner set propose hash scheme utilize asymmetric traditional monotonic inner product provably traditional hashing product evaluation publicly available claim easy matching operation web popular technique big estimate set later sensitive hash spam detection collaborative linkage representation common largely bag power number combination rarely often absence lead search representation eq component absence attribute binary application scenario desirable measure instance description five york base common typical name new york suppose query five I record match record clearly size short record matching scenario undesirable typically big often unnecessary record lead ordering order order intersection near neighbor interest collection universe interested product problem search eq refer search practical significance heuristic problem notable recent among use approach index record web large huge vocabulary quickly extra size query instance computing set cover hard greedy heuristic heuristic usually practical query locality sensitive lsh successful efficiently neighbor suffer curse dimensionality indexing scheme provably sub search impractical due hash indexing stream ideal modern inner inner product hash heuristic general inner locality hashing scheme product hash elementary argument inner special provable efficient asymmetric suboptimal hence asymmetric locality product likely suboptimal like common indexing product existence provable lsh investigation reveal result unlikely hash hashing show usefulness transformation construct provable hash remove undesirable bias towards eventually hash scheme binary inner call hash comparison binary web asymmetric hash provable improvement hash product construction asymmetric hashing neighbor four real world modification adopt practice efficient neighbor partitioning turn massive due curse theoretically exact near neighbor propose adopt near near near report neighbor near term deal similarity near similarity optimal guarantee underlie lsh lsh property high hash function mapping hash family follow obtain resort sensitive hash nn lsh accomplished neighbor query lsh monotonicity condition satisfy lsh determine neighbor note ready lsh family lsh hashing lsh known view apply permutation formally hash hashing even really retrieval nevertheless popular scheme intersection binary product argue undesirable suboptimal asymmetric undesirable dependency intersection novel lsh family lsh formally choose generate normal hash dx cumulative cdf standard normal distance monotonically distance lsh lsh parameter sign popular lsh concept give vector utilize component I normal seminal cosine reduce random lsh interestingly show actually non binary significantly near motivate asymmetric overview asymmetric lsh product lsh elementary show locality hash lsh unnormalized product inner vector hash valid lsh fix allow hash extended framework asymmetric locality hashing hash locality hash family sensitive nn query collection asymmetric lsh lsh preprocesse structure neighbor start scale enough lsh concatenation provably bound query depend realize idea convert cosine efficiently well start scale sign cosine query concatenation form follow follow end provably efficient l lsh neighbor thus outperform lsh sign structure problem asymmetric transformation sign projection also inner common typically prefer hash asymmetric hashing mh suitable indexing intersection general lsh unnormalized product proof lsh inner lsh inner lsh construction lsh hamming ordering distance monotonic lsh monotonic binary extra trick co present hash sample independently integer give independently happen hash lsh inner sparsity ensure web run hundred thousand therefore word hash scheme domain observe denominator way lsh sample thus almost argue thing later provable sparsity likely handle bound number lsh inner product f hash remove denominator worst meaningful providing provide asymmetric name asymmetric hashing mh monotonic inner hashing scheme theoretically exist scheme inner define query transformation concatenation zero power asymmetric inner unchanged query eq nature neighbor instead denominator large set sparse likely thus asymmetric usually lsh scheme asymmetric lsh lsh asymmetric lsh family neighbor product want quantity difficult denominator get formally eq new become monotonic formal complexity asymmetric expression preprocesse cs hashing intersection exist intersection space give negative transformation add chance match plain transformation create hash lot time sampling generation transformation eq similarity therefore nonzero weight problem know hashing sign theoretical asymmetric hashing mh lsh unlikely ignore asymmetric lsh maximum asymmetric advantageous binary query cs immediately follow query explain retrieval product plain approximation curve solid sign dash term irrespective general inner product compare product sign projection value compare suffice asymmetric sign binary inner summarize convenience although perform identically clearly irrespective asymmetric outperform sign theoretical powerful hashing base comparison hash binary rank
differently fast train part c help filter third fused rule fuse drive contribute two share part contribution feature complexity make part jointly slightly sub use evaluate convert relationship cosine formula make derivation unbounded make cosine function invariant cosine widely many binomial choose binomial recognition fisher training binomial criterion formulate denote subject numerator denominator total eqn learn separate far compare eqn binomial classified focus similarity equally make likely fix cosine cost determine eqn cnn connection train totally mini batch batch batch fast training eqn learn sgd backward top contrary specific branch eqn describe assign label pair tune positive build among protocol shoot setting intra dataset train intra conduct illustrate basic experiment illustrate conduct training per come camera camera camera disjoint subject split number epoch test report camera camera camera subject camera follow protocol subject probe subject camera remain camera repeat time split split report camera camera merge generic pair assign mask redundant one probe evaluate use view important factor asymmetric negative besides widely neural double although trick performance compare image similarity final fuse c rank without augmentation rank dataset crucial network know geometry person person pose virtual performance direction identification year pair batch positive batch art remarkably elegant pose information segmentation contribute bp simultaneously similar propose outperform art method superiority reason pair quality recognition dataset coincide capture capture capture totally device people image capture image person illumination subject small resolution image camera randomly include recognition c tr improve c tr improve I train comparison outperform par rank improve significantly quality similar combine similarity score intra result remarkably performance cause big dataset raise dataset cross number drop train hard generalize property texture dataset diverse besides experimental also adapt another target improve research learning extensive intra dataset cross conduct person person cross identification outperform significantly improve moreover research train person engine good sub share therefore cnn gradient eqn derive see cosine similarity eqn eqn eqn eqn respectively substitute eqn eqn eqn define eqn although asymmetric experiment reference denote sample view derivation gradient support national science foundation national science technology support chinese sciences project technology support project degree university china receive degree research face recognition recognition learn article international develop face b china chinese sciences interests computer processing international receive national university technology china ph degree university security chinese sciences work microsoft prior associate research interest recognition machine recognition surveillance publish paper international books li metric field person compare propose way pixel propose jointly learn feature texture framework two cosine function big variation person image use similarity label prove compared set dataset person identification intra illustrate person identification metric category identification person subject practical two view closely camera analysis retrieval algorithm recent year person essence recognition good evaluate similarity compare person identification challenge due person image identification usually person surveillance work mode low unstable pose cause image surveillance two large variation inter class summary challenge come aspect camera pose variation unstable illumination person collect first video person identification training source test domain totally capture different different identification good generalization dataset important unstable important identification exist method sophisticated histogram discriminative discriminate exist usually come texture strategy contrary propose module together feature unify deep originally signature verification person neural assess connection figure carefully image denote connection share identification follow advantage metric pixel layer optimize channel learn color texture concatenation structure switch specific dataset evaluation par cross significantly exist setting conduct strict experiment person identification far deep identification four review feature representation person identification focus person color texture sift fusion contribution final advance aspect color histogram structure person improve color texture feature extracted predefine localize part method localization obtain significantly configuration explicitly spatial constraint large history face precise pose normalization naive usually achieve rank pls among metric stream discriminative task variation metric divide sample accord pose metric explicitly obtain high loop style identification improve drastically intervention reproduce researcher early neural propose similarity signature verification compose network verification group property neural objective end network metric similarity
hessian therefore variable pg surprising correlation extremely depend two informally speak distribution inference c z w quadratic form analyze correlation posterior joint pdf px pz pz pz z deterministic auxiliary variable pe sake clarity completeness px pe e x pe eq square correlation hessian hessian log pdfs child determine parent child parent hessian l pdfs parameterization distribute hessian equation variable child general fast log case dependency
feedforward hide discuss behaved depict applicable multiclass multiclass sigmoid binary label scale minimum pt sep draw circle dash line min node mm name hide hide name name observe name name observe name black h line mm red red width red mm red line x line width mm width red width width blue mm blue line mm e hope moment start order learn label score mild regularity glm activation mild regularity assume row elliptical project span restrictive challenging distribution hide provide problem vector network provide note represent nonlinear neural feature build stein stein second law expectation follow stein last chain stein random differentiable gx px px assume expectation exist integration part result provide nice form auto shown regularize pre use correlation score encoder explanation elliptical project even gaussian improvement present approximation recover retrieve identifiable pose row problem class degeneracy least derivative activation deep network sigmoid assumption initial usually million satisfied weight sparse word neuron net argue sparse connectivity scale nevertheless direction back dense scale back propagation need weight feedforward hide hold uniquely recover version efficient algorithm traditionally formulate equality scale projection deterministic matrix bs ij bs additional detail node depth multiclass softmax sigmoid function network layer stein nonlinear network uniquely recover stein use rule convolutional assume consist parameter neuron therefore prominent progress subset nonlinear hide go challenge learn full weight middle layer weight manner future hope investigate new introduce paradigm literature unsupervise discriminative lot investigation continuous although discrete difference interesting small degeneracy class neuron layer small hand degeneracy square matrix hence weight tensor highly range topic challenge microsoft fellowship nsf nsf award award award support award notation bold time california approach feedforward network adopt operate moment label show factorization provably deep practice output gradient feedforward stein paradigm challenge variety speech understand deep net non problem involve million view guarantee contrary guarantee back paradigm unsupervise tensor survey tensor factor employ network dag paper employ net sparse theoretical guarantee prove correctness stein statistics stein show effectively base analyzing learnt exactly result feedforward neural label stein result state expect stein essentially integration employ stein row matrix weight degeneracy matrix neuron dimensionality note rank significant requirement argue performance practice recover efficient optimization early topic model dag establish also stein training connection encoder approximately provide propagation learn result correctly span weight vector improve score improve kernel exploit
quite namely intra layer therefore lead learn connect boltzmann six four hide standard c mean top bottom zero mean bias similarly begin break machine point computed instance limitation mean although acceptable small quality quantum probability deviation unit bm qualitatively see wide illustrate fail notable contrast perfect state expectation seem indicate success systematically sample evidence instance state confidence somewhat surprisingly percentile state percentile percentile data fidelity qualitatively although qualitatively similar rbm value fidelity rbm arise case take experiment boltzmann inefficient important provide insight database handwritten digit digit directly require compute configuration coarse grained image resolution digit distinguish order confusion digit appear pixel divide pixel set corresponding pixel pixel threshold subsampling procedure optima cd ascent experiment relative difference optima vary percent difference vary half percent observe optima synthetic body evidence grow approximately linearly unit constitute exclude optima rbms train mnist experiment towards find qualitatively surprisingly job predict estimate roughly mass substantial exactly vast case unlikely preferable b rbms mnist number mean stable median bad nearly variation unclear small example law numerical distinguish two possibility scale qualitatively nothing qualitatively mnist correlation field approximations fidelity issue choose training find uncorrelated minimal kl divergence distribution main benefit mean rotation mean call function equation implicitly solve iterate jacobian analogous gibbs configuration efficient forward exact methodology case visible mean take expectation derivative easy product approximation kl note low field approximation graph partition accurate small boltzmann approximation reduce error albeit suggest probability state approach limit strength correlation vanish continuous function therefore h v b h follow field unique find optima success apply behind sample ideally proceed hide unit newly place sample gibbs cd work unit unit sample repeat probability average compute approximation gibbs step rather true gradient try approximate ml cd optimize become log absence asymptotically approximate although approximate gradient gradient yield objective inexact efficient drawback main drawback cd visible class take hour day machine wise training employ potentially sub parallelism accelerate training rbms extent sequential update quantum restriction information store bit classical bit linear superposition dirac normalization condition measurement entry amplitude live superposition representation note write product four ability represent superposition essential massive parallelism quantum proceed unitary evolution unitary quantum reversible quantum knowledge quantum gate unitary acting gate gate need gate complete unitary computation correspond axis gate unlike rotation approximate within small em deep intelligence quantum computer show intractable conventional computer quantum boltzmann machine rich comprehensive deep lead efficient boltzmann multi connected counterpart introduction quantum conventional art classical recent machine intelligence task model ai task several layer raw train detect car accept raw pixel subsequent shape next shape aggregate shape learn nested layer hierarchy concept abstraction encode highly complex training fall machine network generative boltzmann ise encode feature concept ise interaction represent dependency feature output call visible node latent feature boltzmann bipartite boltzmann layer rbms discussion visible unit binary boltzmann configuration visible hide unit gibbs normalize energy configuration visible hide unit visible take hide observe modify likelihood boltzmann training process weight bias ml size regularization denote bm derivative take form computing resort divergence know lead suboptimal train full boltzmann provide framework deep illustrate elementary accelerate process lead well quantum quantum analog gibbs machine formal description appendix exist clear evidence consequence overlap complexity along probability configuration fact configuration use mf configuration approximation good refine exactly sufficiently mf minimize leibl tractable equality achieve mf apply another let configuration eq multiplied desire operational add quantum state right success amplitude boost success number quantum evaluation number logarithmic linearly number edge visible constrain asymptotically contribute operation require gradient scale layer number bm quantum show often make parameter exist algorithm store otherwise logical logical computed substantially reduce exact always hold bm state relative however probability small reveal fidelity gibbs continuity fidelity state formalize energy fail theoretic unknown example unlikely efficacy mf weight cd violate operation execute unit algorithm form case via allow datum superposition superposition amplitude estimation quadratic gradient consequently improvement allow process quantum training think quantum subroutine quantum boltzmann quantum simulator store computer superposition train entire set oppose sequentially accurately quantum oracle oracle query oracle superposition convert repeat expectation measure amplitude directly probability string scale analogy arithmetic success constant boost probability reduce preferable almost cd preferable circumstance parallelism energy energy layer depth see depth execute via calculation depth depth depth price circuit divide mini cd depth cd feed question regard behavior differ substantially question grow restriction computationally unit visible h h dash line line confidence impractical four add set flip contain digit bias show visible substantially increase space dimension differ illustrate primarily mf similar appendix typically close gibbs appendix reduce roughly rbms show rbms second issue determine boltzmann rbms tend rapidly scaling take set suggest assess benefit average find quantum optima find deep improvement ml constrain nature make local cd improvement optimization model bm outperform term quality machine bm quantum train enable rich fundamental work machine notably divergence approximation quantum significantly provide elegant approximate gibbs state mf desire gibbs bm operation depend reduction full quantum processor address result encourage advance quantum assess ability divergence set currently deep boltzmann regardless deep perspective begin computer unbiased allow probability sample quantum quantum approximate rejection quantum numerical begin state quantum likely inefficient depend state uniform joint boost success acceptable quantum number achieve fix initial refined gibbs coherent discuss generalization refine unit boltzmann unit form string distribution approximation distribution variational approximation field gibbs quantum approximation define let satisfy configuration analogous appropriate visible boltzmann visible jensen show lemma give success analog boltzmann success state configuration mean uniquely field partition coherent rotation eq step refine approximation quantum circuit measurement quantum reverse quantum save normalize distribution measure projective unit remainder proportional state normalize note valid value replace exact protocol place place success visible bias boltzmann vector mean field h add ib h h hz generate quantum machine visible visible unit compute x ib ex x ex amplitude provide reduction need occur amplitude quantum iteration proof compute derivative respect algorithm adapt compute bias superposition claim quantum hence success also visible require gate law probability ensure error error repeat state amplitude compute triangle inequality pick require query assume mean circuit visible bias edge training vector specification edge calculate within p alg superposition use amplitude learn amplitude learn amplitude j pt qualitative sample provide amplitude amplitude create evidence forward amplitude randomize case success use amplitude amplitude success version original circuit infer probability step explain corollary visible hide boltzmann connect within scale scale corollary iteration search boost success amplitude success find let identical apply amplitude measure success probability equation unless choose guarantee inverse assumption estimate propagate large error general propagate derivative exist monotonically increase taylor hence overall require repetition circuit produce amplitude overhead cost claim gradient bias quantum call boltzmann machine scale component compare cost incur superposition compute expectation operator compute run time rather vector imagine set inherent value mean although assess cost implement computable complexity implement learn oracle quantum process require use store query database issue may preferable gradient begin randomly gradient comparable error since two cost case majority uncertainty come obstacle assign magnitude configuration example boltzmann excess gibbs root mf reflect adjust particular regardless perspective transition confidence mf update wherein essentially unchanged use difference specify method field uniform overhead substantially mf break quantify divergence algorithm body investigation rbms evidence body comparable handwritten mnist ml optima optima technique verify result lie optima optima many perturbation compare point say fix decrease cite value repeat process perturbation second find use optima analogous training ml step subtle consider comparison determination divergence gradient ascent ascent calculation optima fix converge running varie least epoch learn calculation introduce absence bfgs ascent derivative choose occur c optimizer optimizer ascent objective bfgs objective gradient ml estimate small device rbm proceed ml add zero training point epoch stop meet noise ascent highly suffice yield optima find error adjust improve reduce rate suffice reduce error multiple reduce error need negligible c average rbms qualitatively quadratic relationship rbms weight vary weight bottom ability depend choose derivative differ substantially analyze result rbms unit know strong strong correlation introduce weight normally boltzmann mean weight find train recognition divergence set zero unit suffice state error case slowly function number rbm deviation deviation necessarily problematic deviation practical extract scaling choose cutoff residual bias unit variance datum random rbms datum scale zero chance bit optima average table give mean divergence field rbms less kl divergence slack see
take easy h restrict theorem third course triangle change cloud separately svm learn refer demonstrate improve oppose use traditional pre scaling variable feature range feature empirical empirically deviation variance distribution justification pre method precision less error important although comprehensive treating adaptation empirical observation learn diverse computer etc pruning decision rule robust reliable give specifically consider scale available coordinate value center integral equal discretization bin distribute value belong realize although clearly give decision discretize dataset decision dataset balance specificity discretize handling without direction key space consist segment disk four axis line shape line part triple plus sign perform eigenvalue correspond eigenvector yield per record six reason six every homology triple length train example category category evaluate performance computed maximum recorded sensitivity specificity type report table low rate six exception triple discretization appear six low triple triple shape location within dataset homology consist triple largely death respective homology high around triple former cloud contain disk indeed indistinguishable another decrease attribute triple point densely sample segment synthetic point point line segment either line densely segment line second case persistent total segment reason behind feature point densely segment persistence homology segment datum previous linear category low rate alone feature vs example bin discretization lead feature alone employ slightly feature collect ground see purpose randomly group scale persistent persistent yielded type ground rate feature feature lead greatly compare feature alone result promise multi feature consistently real synthetic turning persistence diagram treat diagram image algebraic geometry advantageous advantage bin discretization discretization lead real rule difference test life often slightly different allows retain patch however strong typically happen irrelevant match suffer identify plan future rely feature define construct beta preliminary show rate utilize continue investigate combined summarize table experiment bin indicate bin c max h max bins h bin bin max bins max xt proposition section extract feature within dataset topological capture diverse subsequent machine operate dataset construct correspondingly extract dataset synthetic context typical classification problem thus relevance technique assess result classification measure sensitivity specificity error feature extraction processing feasible computation mechanism remove add challenging difficult approach extraction rule mutual manifold benefit stem shape topological machine performance technique cloud return direction output onto subspace span intrinsic dimension look course line notion intrinsic want thin look look simply true sample intersection return reasonable notion prove build topological example chapter assess among nice topological differ move deal information local concept one depend entirely often impossible scale address radius version differ exposition good knowledge attempt dimension dataset quite different understand drop information effort learn learn use construction road map reconstruction outline follow briefly review utility result experiment cloud compute cloud ball around process repeat multiple persistence extend extra infinite diagonal dot diagram diagonal diagram define infimum diagram exist infinite dot tend infinity diagram perfect efficient state persistence diagram diagram red circle diagram red red dot close bottleneck dot diagram function function space non integer technical theorem stability homology persistent version fix non integer sphere radius simple take point plane zero complicated circle homology depend turn change unstable homology radius choice point center gradually line rank fortunately persistent
determine stop point f automatically statistically bold entry average unweighted column label represent fully supervise passive entire drop drop set implementation method fact worth average annotation average early high cause mention average cause subsequently many highlight regard various stop parsimonious annotation stop largely except sp unstable sense major failure stop g spam way early amount e g ls protein always clear stop tradeoff extra measure versus annotation know last clear annotation high sp sc depend annotation tradeoff promise develop preference much conservative user pick criterion suitable sp gap seem conservative sp stop likely method provide behavior sp control point visualize perform representative axis measure annotation axis stop sp stop exist without dna range demand dataset table cutoff requirement sp enable adjust stop fashion behavior area precise expectation intensity intensity annotation control intensity level determined fold average dataset c annotation window annotation automatically stop display stop measure investigate sensitivity number annotation fold automatically display ls fold avg stop learner annotation bold statistically significantly bold sc method margin learner entire training stop random stop fairly steady plus work fold simple work think determine stop develop create density meaning example make sure another algorithm stop stop unlabeled pool maximally stop efficiency get adequate representation latter accomplish perhaps stop add stop sp widely table model conclusion experiment statistically sp favor representative fold stop action fold tc corpus range crucial annotation identify improvement stop sp address sp widely stable sp exist informative criterion also informative demonstrate rigorous annotation tradeoff area user enable user stop user pick tradeoff substantially behavior sp center university md usa computer information sciences university usa reveal annotation stop al address furthermore stop handle range annotation tradeoff despite exist dominate conservative little provide stop al provide user stop nlp considerable annotation effort al enable must mechanism annotation motivate stop annotation axis see figure generalization perform stop make wind human annotation effort stop conservative tend right conservative amount far leave try reduce unnecessary annotation automatically stop improve restrict use applicability exist lead tend find far dataset break stop stop paper present new address area provide user essential idea applicable save annotation user stop al discusses explain sp detail evaluate sp estimate probabilistic learner confidence generally table figure fall stop margin stop denote table figure stop equivalent margin show explicitly al evaluation sp method applicable base tendency confirm say stop confidence consistently drop pointed criterion however two stop report max stop unlabele exceed generalization min qualitatively translate measure allow point stop idea experimental separate development cutoff agreement chance percent adjust measurement agreement human receive context drawback percent take agreement account agreement metric agreement metric differ statistic agreement chance category formally compute estimate instance label find require move separate development cutoff work experiment current paper agreement cutoff cutoff cutoff intensity sp intensity cutoff see give user intensity cutoff sp behave another maintain sp conclude stop check stop exceed cutoff ideal happen propose average window call window average work table tuning default fold vary requirement
video outlier return open circle table blue line viewpoint frequently rank removal top lasso order competitive video importantly indicate dataset consideration reference reference come publicly live totally varied comparison internet pair show zero leave corner table outcome table lasso image database agreement pairwise voting vote example l method integer score c far suggest contrast inspection confirm show show ranking return table rank rd four appear unbiased lasso successfully bias rank winner col usa country david usa usa col supplementary material proof mainly change column normalization assume gaussian present self leave turn restricted side whole less give achieve c sufficient negative outli side mean piecewise sum min min consistency fx right bind cl min min root comparison spread crowdsource crowdsource outli detection become huber cyclic linearize bregman achieve less detection os setting detect support simulate promise robust scale crowdsource vision machine crowdsource pair robust outli linearize random statistical aggregation rating back area various voting theory economic machine pairwise spread crowdsource g enable collection large rating scenario must address difficulty interest data iii online streaming iv among able characterize intrinsic pair incomplete sample vote rapidly assessment quality experience setting traditional environment equip development mathematic enable instead quadratic pair pair connectivity infer pair loop complex measure ranking setting propose classic method together track pair rating hence efficient batch deal despite lack supervision crowdsource single long decision significantly decision global crowdsource detection circular triangle triplet divide apply apply pair pair incomplete miss detect pair robust sparse uniform outlier angular embed approximately recover score history fill present lasso huber robust outli sparse component interest meet challenge crowdsource develop yet easy outli linearize contribution highlight huber outli approximation projection scalable linearize biased lasso suitable statistical consistency detection establish huber linearize bregman iteration methodology simulate world paper conference problem tend cost prohibitive crowdsourcing linearize iteration outli inclusion fast organize review work systematically robust huber linearize bregman consistency case establish conclusion pairwise outcome element aggregate ranking study rank centrality mc compression image assessment fitting consider set pairwise algorithm optimal aggregate global provide ranking pair angular theoretic pair comparison flow flow triangular flow local harmonic pair help various outlier score graph problem exploit os r enyi random refer computer literature occur large instability outli many applicable pair appear literature outli study well combine square discover regression huber outli modern selection linearize firstly variational imaging sense know estimator always bias contrast variation denoise view discretization combine gradient soft thresholding soft g reference therein biased lasso tune choose well achieve boost save greatly thus suitable computation systematically rank huber lasso linearize first successfully widely lasso cyclic statistic linearize iteration set paired datum miss degree generality assume vary quality assessment scale machine surface follow pair comparison map flow gradient difference gram view flow scaling contaminate go gauss tell eq score however put mathematical outli sense element achieve different type huber robust differentiable derivative pair solution z variance bind loss hence note get leave random graph comparison os enyi surely optimal huber loss q regard regard bad one contaminate reduce rank huber robust comparison ij I outlier lasso ij huber partially huber package solve fortunately two group column complement involve projection onto complement orthonormal orthonormal provide precise split solution obtain inverse full svd hence via correct pairwise say pair admit particular harmonic triangular ranking call cyclic ranking e unitary orthogonal cyclic detection via decomposition outlier complete huber propose jointly follow pair play role suggest robust normally hence efficiently huber estimation prove application sparse validation highly associate outlier leave outli validation projection subset projection random thank os enyi graph position consistently identify projection validation cross fail become dense magnitude nonzero tendency outlier regularization path example like drop percentage appear regard drop moreover magnitude outlier outli development projection optimal scale score suffer prohibitive item remove bias contaminate estimation introduce ordinary unbiased discretization scalable imply remove linearize euler discretization rule meet size estimator sign reach see sparsity large empty early overfitte early update exact stopping meet efficient establish outli share property e outli consistency condition speak incoherence magnitude however limit magnitude exploit choose pair restrict os graph tend unique hold contrary fails assume outli sign necessary sufficient outlier uniform lb path strong enough k sign play play six methodology latter exploit world collect crowdsource pt sn op auc sn data p first total truth add paired subset simulate pair comparison possibly two dataset total number pair sn number outli sn exhibit sn op compute vary give tendency outlier outlier roc plot different level create sn op see sn op auc deviation detection return accurate indicate half edge rapid decrease random guess edge perturb impossible distinguish op drop sn tell could case grow compute scale experiment image pair contribute outlier effectively however could simple scalable datum reconstruction first use ground truth obtain
align imputation minimize reconstruction reference corruption cell partition residual depth oppose split scope study comparison performance term digit report project test eigen clearly separation corrupt imputation improvement accuracy set avg std avg improvements nn map breast cancer spam digit emphasize first localize complete phase free separate test affect attribute comprehensive cover solution fair comparative nn separation utilize instead imputation nn neighbor corrupt instance attribute node utilize imputation detailed corruption separation step split segment anomaly segment anomalous segment segment attribute randomly split scaled instance corrupt attribute attribute instance corrupt explicitly compare imputation euclidean since low nn euclidean common method split remain split train svm compute corrupt deviation average improvement std std term split clean test corrupt set clean randomly choose split corruption observe successful corruption imputation even gain imputation correspond improvement relatively cf experiment algorithm corruption separation nn propose nn method outperform nn imputation corruption even nn superiority issue nn corruption cover portion g would several false corruption cover portion g choose anomaly detection use insufficient detecting imputation insufficient load corruption propose search fast experimentally corruption separation capability lastly imputation asymptotically maximization compare dimensionality yield estimator neighborhood easily achieve cross map imputation perform imputation opt clarity imputation sensitive attribute near imputation e experiment near lower corrupted pick imputation imputation potentially experiment superiority become superior imputation g set demonstrate devise experiment unitary covariance generate sample attribute therefore size nn part original mse term accuracy summarize imputation coincide imputation imputation consistently imputation mse sense improvement original accuracy around comprehensive localize novel I jointly identify local corruption binary characterization anomalous observation combination assume model drive conduct alarm alarm detect independently generate set experimentally purpose corruption capability algorithm training phase condition edu tr department comprehensive treatment severe noise framework novel efficiently give instance corrupt propose novel rank deviation among superior separate split partitioning iteratively detect anomaly clean anomalous vs pattern tree characterize affected attribute structure binary rate test experimentally remarkable classification purpose corruption capability typical localize corruption anomaly variety process even severe loss transmission channel localize result face effect novel emphasize neither existence corruption operate drive manner deviation nominal corruption external factor outside nominal consider anomaly specific interval attribute localize corrupt corruption property variety characterize formulate detection localization anomaly detection introduce false alarm nominal clean anomalous anomaly identify generate organize tree correspond anomalous pattern corruption nominal multivariate distribution success coincide alarm test direct acyclic multivariate derive alarm detect constant require corruption localize replace attribute attribute map exploit local dependency encode map load extra computational utilize also rank generalization label anomalous distance compare standard conduct severe indicate achieve imputation typical also empirically strong corruption separation capability corrupt statistically unobserve counterpart know corrupt attribute readily treat framework datum study impose solution imputation tool replace attribute draw density certain expansion contrary either introduce unlike approach incomplete attribute I precisely localize provide target object priori exhaustive result manual inspection miss algorithm jointly detect well imputation framework generic imputation imputation imputation completion neural study image statistical approach valid attempt enhance globally localize operation exist corruption common phenomenon application regard corrupted previously study study visual imputation descriptor part descriptor weighted handle partial solution extract error template map significantly fail remain applicable source another study imputation corruption improve processing stage classification handle localize affected attribute alarm detect acyclic estimator computationally utilize corruption load imputation anomaly notion description corruption imputation conclude corrupt clean observation f nominal severe multiple suppose corruption localize attribute corrupt distribute distribution statistically counterpart corruption model datum attribute vision provide uniformity draw scenario realistic consider include generally model mixture density derive nominal corruption distribution instance replace nominal localize appropriate corruption create corrupted corruption strength corruption corrupt modeling pose incomplete corrupt irrelevant exploit deviation nominal detect miss end formulate anomaly detection draw example anomalous corruption alarm well framework without single generalization onto give description algorithm tree separation imputation false alarm localize corruption affect affect vast algorithm corrupt attribute anomaly instance reference anomaly detection approach novel distance corruption localization corrupt nan nominal corrupted mix anomaly nominal unknown set hypothesis realize anomaly maximize alarm purpose score distance near th neighbor score function anomalous mix estimator remarkably volume test false alarm improve training point detect corruption test corruption imputation contrary truly train high corruption property anomaly achieve alarm first issue list propose metric sensitive give corruption attribute instance turn distance include create ambiguity term localization exhaustive possible attribute overcome result attribute responsible corruption permutation attribute make precision anomalous check noise attribute less exploit investigate euclidean rank euclidean sense satisfied derive consistency estimating level density characterize present corruption corruption might check anomaly anomalous alarm propose attribute characterization separate corruption scenario split simplicity attribute r rv v r strategy corruption separation separate tree create course expansion correspond e check reference rank pre encounter expansion binary unbalanced instance emphasize binary creating need continue decide corrupted node circle square anomaly wide spread attribute anomalous regard corruption characterize corrupted corruption unless create corruption since represent realization underlie attribute map maximize posterior hold w arbitrarily small drop maximizer approximate map nonparametric neighbor knn estimation neighborhood near neighbor lebesgue variation unnecessary approximate true underlie corrupted tree corruption namely I test reference associate ii detect attribute limited derivation detect calculation addition support corruption issue use rank neighborhood possible desire recall exploit imputation meanwhile decrease localization trade imputation localization investigate detail imputation bring result cf node generate step alarm corruption detection false alarm dependency anomalous partition imputation certainly false alarm correspond occurrence alarm anomaly detection tree separation operate anomalous pattern anomaly present rooted pattern reject reason false alarm rate must anomaly false rate detecting anomaly output correlate section label anomalous normal partition direct achieve certain dependency structure derive alarm detect globally rate encounter leaf anomalous describe corruption localization well imputation capability propose corruption algorithm detect task alarm rate detection nominal data nominal distribution depth root anomaly child root tree observe binary vector ease exposition map corruption label equivalently complement nominal probability mass tree binary acyclic node knowledge label leaf binary assume follow u obtain cf u define note alarm analysis localize definition root anomalous anomaly due corruption phenomenon therefore simple fig straightforwardly tree anomalous label root collection associate tree node equation factor last expand derivation calculation require e let child independent dependency generating would like attain hand provide dependent introduce derivation generate probability function parametrization alarm practical opt simplify corruption root solely depth symmetric child simplify note termination obtain short hand algorithm unlike multipli search stop corruption local anomaly initialization recursion focused localize exception straightforwardly incorporate regard change corruption pattern corruption search eq recursion stay valid recursion calculate node recall false alarm corruption subtract false alarm corruption simplification conclude anomaly map false alarm corruption secondly even hide label identically parent child obviously plot alarm detect anomaly experimentally discuss efficacy represent moreover depth uniform partition depth model acyclic anomalous labeling rank distance fitness dependency complexity main building anomaly compute train matrix distance operating distance score computation sort anomalous image distance let si si si si position compute train sort sort node expansion define sort load multiply constant illustrate separate detect corruption efficacy step corruption imputation improvement digit task gray intensity corruption corrupted specify region randomly provide clean test emphasize
dual give section establish global linear property regard convexity lipschitz continuity compact compact boundary convergence iterate supplementary condition subgradient translate si x si point condition vice versa matrix satisfy exploit similar lipschitz however show compact satisfie iterate belong compact begin iterate hence inductive argument appendix positive assume bind positive iterate k iterate converge conservative much conjugate primal hx use gradient establish use lipschitz continuity two call divide approximate inverse multiple solve solution dc algorithm big divide light base speedup therefore utilize spirit dc alm algorithmic alm attain solution comparison attain nesterov regime alm algorithm use speedup strongly function c alm substantially show fair superior establish regard validation edge criterion regularization probability graph sparse model consistency mle leverage already consistency refer regard dimension recover underlie proceed framework rich formulation generalization bivariate structure numerous marginal quantity inverse define operate specific knowledge matrix compare covariance layer regularization process achieve methodology regard underlie edge covariance correlation equal either reference estimate partial graph interest provide graph incorporate term generalization bivariate translate constraint somewhat generalized allow mle break away framework allow regularization base domain knowledge primal formulate modify covariance interval third note provide problem convex set constraint ij choose linear constraint constraint might always translate g formulation penalty consider regularization easily formulation equality constraint covariance decompose alternate b condition z second optimality substitute optimality proof previous new belong valid also modify operator x useful prove generalize iterate f converge appendix synthetic convergence demonstrate convergence iterate duality slow number solve synthetic library compute cholesky processor available resource solve conduct synthetic procedure matrix entry uniform obtain desire percentage level use small well condition size illustrate heuristic point use start well large decrease iterate dual feasibility consistent display require start majority case method improve therefore iterate optimal machine intel cpu gb ram table c sp sp sp e sp h e sp sp sp e e duality termination solved require well duality gap require prominent ill benefit work highly condition compare three consist temperature algorithm varied get yield ill ability duality highly unstable reduce tolerance solver report gap report significantly gap subgradient significantly slow converge c c e e na e na na e na na na duality achieve tolerance solver gap subgradient high ill report experiment gap achieve duality gap unable subgradient suffer issue perform high fast almost around speedup problem solve time determine penalty require field addition uncertainty quantification optimization times solve financial regularize critical ingredient portfolio require repeatedly shall efficacy financial context portfolio portfolio weight focus portfolio return asset period price divide price covariance asset portfolio period minimum portfolio selection tw return define associate portfolio budget close assume stationarity return account stationarity minimum portfolio portfolio divide horizon consist time start portfolio solve horizon constant hold period entire period update portfolio hold j asset return hold period application study stock stock average trading day recently illustrate interval begin begin return trading horizon consist hold period penalty versus condition covariance method metric excess expect portfolio switch measure weight choose metric appendix c fig normalize trade five table return realize risk ratio standard period fig cc c htp growth substantially across advantage rise portfolio balance transaction cost growth propose inverse inverse literature fast order magnitude regime recently moreover condition converge extremely slow poorly highly attractive modern inverse multiple maintain definite tolerance attain gap duality gap terminate inverse criterion tolerance progress iterate primal k tolerance impose step iteration bb maximize equivalent solve k numerical acceleration might feasible satisfy descent satisfied three case reverse inverse entry contain argument satisfy iterate use ingredient c fix define leave hand side c expression jacobian h inequality complete repeatedly inductive iteration rewrite term c subsequent iterate lemma step thereby modify f accordingly easily modify prove use similar constant portfolio background definition take pt trading period eq realize risk deviation strategy entire trading period realize sr realize excess return strategy risk hold start th hold portfolio period I e normalize growth portfolio grow recursion transaction stock short short portfolio trade thank code section estimate graphical despite advance fast ill modern solve covariance approach several novel fast algorithms order global linear rigorously operate covariance
attribute include output without validation significance sign create centroid assign output generate centroid centroid select attempt mod portion nominal datum outline add set output set number centroid time centroid centroid output output na feature c output nominal I synthetic always statistically exploit contain output make specific chance outperform I nominal synthetic well significant uci set mod decision mod create uci set allow nominal derive nominal original act class act feature act scale output output output repository experiment choose nominal mode accuracy I bold statistically uci nominal feature output l c ccccc c total ni ni h ni h ni heart post op vote contain I output original significant na I time significant difference na I outperform I majority na I model perform well never potential model mod decision uci motivation mod stem business com business reproduce version three nominal value output datum instance contact business contact method outperform I see uci mod synthetic real mod representative little uci supplement sources mod na I I mod solve multi I mod consistently outperform model significance uci repository business future use mod problem mod new identifying collect mod mod problem know degree piece mod classification primary issue function readily apply near refine dependence mod traditional seek set possible output classification multiple assign structured prediction predict structured mod address output incorrect mod propose company generate sale customer customer switch company retain customer could sale customer offer customer express course incur certain customer help write customer mail company generic mail e mail person see problem important approximate traditional define output output still acceptable mapping relation give choose solution many induce per output give without induce tree produce output induce neither approach output neighbor perceptron mlp approach algorithm auto network unable handle input contrast hierarchical neighbor hierarchical I approach traditional learning make focus nominal feature mod problem model modify use output near examine although label correct consider problem multiple dependency model overview classification first problem single already define mlp output may output mlp adjust towards whenever encounter whenever encounter adjust weight adjust ever correct try solid curve training dot branch arbitrarily however cc learn potential possible neighborhood towards extra provide mlp mlp near neighbor knn final prediction prediction feature output either output modification knn dependency mlp classifier part knn majority dependence claim dependence output loose dependence consider scalar conditionally vector output output dependent py py contradict must vector mod different type synthetic uci repository world compare single I separate combined output instance return otherwise correct
entropy shoot restaurant event word dnn representative heuristic shot shot dnn embedding confirm shoot measure auc area curve train scalable lead svm well linearly input shoot base use amount query log learn embedding without access datum experimentally propose semantic classifier category learn engine log learn semantic supervision shot semantic demonstrate effectiveness shot collect system aim request predefine semantic category instance system might query method svms model produce state require amount label mostly manual costly applicability problem semantic examine see domain add easy shot semantic class typically least category domain input classifier must without knowledge shoot none see well automatically neural use amount advance network result search click reflect query property call shot embed shoot weak supervision shoot discriminative produce semantic notably reach feature next quick overview shot section present introduce shot embed semantic classify speech upon maximize formally variation may express information priori system hand interpret weather function aim capture binary likelihood gram express user traditional text categorization devise maximize shoot concerned novel training label class predict knowledge input euclidean vision might semantic predict semantic class value train semantic match nn semantic none shot see semantic classification find semantic label would semantic interestingly semantic discover without label name sentence choose essence fact classify easy belong wish replicate net axis phrase relate movie detail shoot observation function property procedure space shot find match formally shoot z distance euclidean reveal mean space map sentence relate classify close properly semantic capture sentence class framework follow language interested method lda success semantic useful semantic click query behind query hide layer click include user query send engine engine extract thousand query million daily meaning meaning website query website movie scheme sentence task space embedding like sentence deep softmax hide word format index website q hide weight bias give semantic property semantic category novel encourage learn semantic without label precisely cluster measure class cluster hence minimize measure semantic good click task category close relate activity sentence pressure know visualization dnn dnn right sentence color location class name improve detail supervise label even bit supervision simple task train sum measure require category low require label data semantic space category space low entropy example around category low overlap shoot discriminative combine embed entropy hyper strength validation mostly determination phrase entity location phone large space boost good candidate first method result performance boost base baseline show big click feature entropy learn conditional label available minimize avoid label minimize zero shot produce generative evaluate shoot zero shot month query click embedding restrict word bag use contain filter
feature relationship normalization assume know output joint inference discard unfortunately necessarily give instead introduce strong bias optimistic cause explicitly account measure work marginal predictor significantly improve method minimize risk n user quantify usually e surrogate upper prediction define upper empirical second jointly form supplement augment optimize model sum belief respectively optimize start q operator may eq temperature structure semi eq obtain restriction comparison sequel provide insight select evaluation objective sgd concave gradient eq augment approximated belief propagation distribution expectation belief sgd lemma supplement unify control distribution sub sub gradient substitute sub gradient bp bp tb output weight vector calculate I define calculate w concave convex transform optimization problem naturally difference iteration new minimizing linearize eq expectation iteration loop learn ph u ph x simulate world margin uncertainty largely outperform especially small datum mrf discrete q graph illustrate parameter indicator vector output sample test hide mrf mrf train hamming loss high noting obtain mainly due sgd converging sgd ccc sgd sub learn converge within gradient much converge slowly effect mainly hard make nonsmooth sub slow sub descent illustrate converge sgd transform objective easy objective fair range training data mrf chain show outperform largely outperform sample outperform experiment relatively toy generally difficult enough model eventually seem since likely relatively test supplement uncertainty hide mrf figure fix accordance default default package encourage tune result table competitive significantly uncertainty explicitly outperform current chain mm e avg label label grid pairwise entry outer miss range evaluate list method increase see significantly consistently explain improve robustness accounting categorization microsoft pixel pixel correspond boundary etc model outline center treat patch texture descriptor compute sift descriptor patch word k color take patch testing find category explain superiority moderate building car variable demonstrate state art especially uncertainty optimize function also include nsf grant united air contract fa program uci edu uci edu uci edu give constraint slack unconstrained derive cut formulation proof lemma omit lemma objective unified framework paper bind q definition second denote sub temperature result complete paper demonstrate outperform example across likelihood datum high necessarily explicitly loss relatively instance pt marginal properly propose art uncertainty result smoother significantly outperform field real world furthermore unify case practice structured svms tool structure computer handle image segmentation predefine semantic expensive collect perhaps region partially relatively label couple syntactic annotation resource year variable perhaps notable hide field svms svms practical training violate hand procedure assign variable prediction even well perform max accuracy valid location improve category
represented call architecture topology statistical cluster standard activation radial architecture variable probabilistic behaviour layer historical unit dot perform operation operation provide summation sigmoid activation output represent modify window order verify obtain expect express window width width depend mean square represent preserve capability function radial basis rbf rbf weight radial intend second identical sum receive hide layer summation summation unit weight weight summation summation summation remarkably output selector effectively act one layer selector input score attribute text probable author probability selector adjust product output layer us part processing layer summation nonlinear fact layer task neural text attribute input exact number pattern match text summation equal people interested recognition database nn purpose since aim expand database identification dynamically database bias imply properly ensure adaptive reinforcement desirable human training give figure flexible model add time external tuning count group count multi agent identify concern text attribute concern probable information spread certain could extraction study exclude automatically enhance classifier filter probabilistic text correctly attribute attribute person performance probabilistic selector selection positive black mark mark validation purpose accord identify excess text person lack attribute text author assignment model dataset allow structure blockmodel report model replicate unable child attribute class class multiply identify lead would membership specie concern non parametric relational entity set moment sequential relevant use mutual class storage classification purpose recurrent show neural study result line learn analysis text consist successfully probable text sample complement integrate comprehensive verification kind try write classifier mean reinforcement follow correction agent supervision cope continuously feed adaptation reasoning support project project mathematic drive radial year title drive use radial basis reinforcement volume policy huge essence digital retrieval gain trust drive learn develop preprocessing period analysis recurrence frequency text radial author lie semantic apply lexical domain without modification external tune self adjust text author security availability digital possibility essence method digital text last decade classification intelligence language process intelligence orient programming intelligence ci ec optimisation problem management agent intelligence advance create analysis ci ci network use electrical control start strategy kind hybrid nn say work form cluster recognition efficiently fail efficiency result complicated general agent machine prove promise research purpose build classification rule automatic text one promise approach lie semantic category generate successively typical topic recognition appropriate classify involve business text belong party text technical devise solution extract text characteristic express obtain abstraction create analysis rely input evaluation concern historical period etc text use preprocesse grouping relate tool recurrence text lie lexical modification order reference suffice purpose reason intervention thank comprise e extract meaningful text mean radial nn feedforward detail implement preprocesse modification agent report perform related background draw figure agent characteristic text database database extract properly perform identification additional agent dynamically new firstly analyse text group mutually build ad hoc relation text word group predefine database contain dictionary return group identify new dictionary occurrence increase group start load load group database load break
objective lasso c call value statistically gaussian k k redundant tend tend exist feature need space large base reduce feature extension call propose moreover justify lar establish screening efficiently solve lar nn lar via lar lar standardized unit formulation modification optimization negative lar lar point lar center lasso redundant inactive lar lar lar base lar iterate change high lar ordinary least infeasible add regularizer lar complexity feature cost lar grow deal computational issue om reduce map advantage parallelism approximation universal detect two delta kernel delta normalize kx width normalize width classification delta kernel nystr om om eq nb bb bb nb bb bb nb bb bb bb regression similarly approximate simply entire lar om approximation helpful nystr approximation high e also lar distribute compute compute store output output k repeat stream fast establish screen method screening framework screen idea covariate choose value regard express exist objective large solution contradiction suppose obtain contradiction correspond large proposition connection feature screen select redundant iterative redundant iterative technique lar review exist mr feature relevance mr usually mutual relevance mr yet efficient mr input relevance redundant select overall regression interpretability redundancy relevance experimentally mr feature correlation base regard calculate backward selection compare strategy backward tend produce approximated window small optimal advantage show low case need compute computational nystr om propose experimentally om high setting mutual also elimination selection accurately implement optimal feature relaxed solve bfgs point necessary expensive high problem regularize base scale sample recently redundant find term exist feature selection method tend expensive moreover sparse spam feature globally spam spam closely relate multiple mkl potential spam deal additive spam need optimize tends computationally path lar computational lar distribute om dimensionality lar fold world dataset scalability scale biology lar baseline mi pc since efficient qp solver available matlab code size selection slow generate accord x variate covariance matrix regularization illustrate b om ghz core lar increase grow lar memory lar large case lar distribute computation hour baseline restrict comparative study baseline classification propose set type ar small p regression experiment rest run training report since dataset logistic gaussian kernel width chosen cross absolute check whether lar select kl red correlate many redundant ar r c lar spam p p show observe lasso high average red lar overall non next evaluate subject focus gene cause value use rest time select employ evaluate selection regression gaussian width fold finish lar lasso mean feature observe good lasso measure instead perform b auc use selection quite big used demonstrate scalability result goal predict activity inactive label determine feature sample sample classification randomly sample report area roc feature coefficient lar lar simple mr accurately lar solid lar linear call lar propose efficiently exploit variant lar normalize furthermore propose computation lar large feature experimental demonstrate lar promise theorem corollary yahoo com way specifically propose lar lar normalize lar incorporate reduce framework high lar evaluate high selection
evaluate standard mean remark detailed column respectively x n partition sum distance mean correspond formulate dissimilarity measure dissimilarity square I generally operational definition state dissimilarity object mean naturally rewrite square cardinality sometimes convenient calculate formulation form involve parameter euclidean interpret objective contribute become serious drawback solution mean considerable portion situation phenomenon experimental cluster cluster neither offer intuitive select framework drawback question accommodate develop drawback classical k cluster clustering mean characterize function centroid sample long vary play role formulation implicit centroid centroid confusion cluster universal exist centroid simple example p p verify gamma difficulty root difficulty try way optimal consideration motivated condition partition define partition noise feature make attain value respect make cluster noise feature reasonable state notation sample indicator belong indicator distribution uncorrelated obeys define minus state generate furthermore partition comment existence notice sample direct new result traditional theorem ii feature gap fact define feature ii suggest principle optimal relevant significance equation reveal vary partition rather reasonable intuition introduce special deal penalty mean example penalty replace difficult analyze overcome propose jointly word cluster surprisingly analyze theoretically solve cluster framework come existence partition variable variable responsible valid apply alternative solve iteratively respect fix procedure initialize q f formulation come like specify k sequence order identical directly set component element select operation approach cluster w old w nd ii kn cost cost condition relevant problem portion mean support assess theoretical propose mean main mild algorithm generate namely seek select follow partition consistency ff relevant partition relax whenever consider possibility condition certain prove follow show equal positive probability thus follow remark satisfie proportional degenerate grow growth sample notice condition gaussian generalize subgaussian k justified reveal mean high problem performance mean mean concrete involve gap strategy four comprehensive comparison first classification use adopt I criterion zero weight yield correctly eliminate fourth nonzero relevant correctly select algorithm correctly exclude criterion conduct experiment statistic mean mean k mean related cluster method experiment respective uncorrelated design assess feature relevant feature figure standard see maximal small average estimated weight close estimate noise show gap htp mean mean assume element simulation experimental means mean mean standard always k explain mean keep completely fail keep coherent contrary detect mean price little well k short pca previous datum different consist sample feature relevant feature relevant show ccccc mean mean comment k reason principal include mean treat equally dramatically penalize consider feature cluster datum result relatively simulation suggest generally cluster eliminate independent life validate broad feature experiment among centroid relevant conduct mean perform experiment generate obvious strong capability k mean mean subsection capability respective apply develop voxel expression voxel gene record operation correspond brain voxel increase brain grow resolution slice voxel annotate brain manually h pt region mean k respectively set mean improvement interpretability keep minimal nonzero eliminate discriminative firstly investigate observe noisy k mean instance channel beta relate beta interact alpha go annotation include bind electrical signal transmission cell much reasonable feature function whole brain usage distinguish thus support correct also still identify gene database institute rigorous concept feature cluster problem cluster eliminate noise yield simultaneously realization suggest closed solution problem analyze assess experiment study main follow concept optimal partition rigorously concept grow number size definition cluster problem cluster could time acceptable real efficiency mean possess selection property theoretical success k ability yield mean interpretable many along establish theory may compare lemma bind apply therefore variable sub exponential parameter exponential next suppose know exponential decrease definition thus feature cluster proportion sample expectation cluster j n n x naturally easily valid relatively suppose ij obey standard accord q I moreover base decrease big complete standard reason term zero partition discuss first essential last interested reader small probability noise big partition estimate complete national research china program cb china zhang university old university helpful thm corollary lemma remark penalty dimensional
discard tc sec state upper show tight provably thm hold discrete correlation choice uncorrelated truly dimensional characterized separately consider construct decomposition efficiency ability bind multivariate information really entropie thm directly translate bound dimensional system decompose sec optimize tight finally estimate datum describe thm suggest way build optimally informative layer maximally explain correlation layer maximize hierarchy representation layer easily compare optimization obtain bound convenient probabilistic surprising optimization imply self consistent equation iteratively variable space tc lx iy look optimum expression omit information ai iy iy positive existence common dependent dag overlap information keep lagrangian normalization guarantee appear formal marginal remarkable problem say write term involve optimization practice limit iterate self fix imagine start update easily calculate summing take iterate converge point surprising rearrange value function call bound intractable share tractable rule principle order final sum intuitive unique present hide obeys previous equivalent add index solution incremental obey long update guarantee increase stop iterate quickly update discuss look see really define input depend independent say inequality demand tree structure connect informative latent introduce heuristic solution tree I py j py j py correct prediction set percentage approximate fraction empirically unique question diverse redundancy quantity somewhat typically except represent quantity sample require imagine give unknown estimate typically small seem estimate marginal simple chernoff bind exponentially specify see optimize calculate marginal require update label amount function marginal easily provide quantity raw along datum accord learn recover fig b within percent learn representation increase setup group cluster take quickly sec world look b variable word correctly increase synthetic scaling include real consider series take entire treat month iid representation learn edge thresholded edges proportional explain stock color accord classification standard clearly capture significant wide construct restrict boltzmann tc lx mean kind wise total capture market decade connection new latent form thresholded hierarchical overlap connect group contain department like strongly home improvement home stock two contain contain demonstrate maximally informative contribute construct representation complexity outperform state synthetic demonstrate promise diverse human biology language foundation previous enable overlap contain specify cardinality representation optimize trade include characterize rbms explore connection bottleneck combination domain agnostic foundation rigorous information correlate datum research nf supplementary maximally informative hierarchical decomposition hold kl expand entropy definition replace lx hx tc I write variable give next thm want drop lagrangian constraint reduce constrain p optimize functional derivative identity unfortunately functional indicate delta px py take identity py px p px x px perform sum py py py lead py py py recall formal marginal py py py py like appropriately constant calculate term iterative bottleneck iterate value functional px long optimize argument equation skip appear mind lagrange multipli ensure recover equation optimum objective concave concave include objective unchanged objective must optimum iterative procedure start initial compare initialization always boltzmann hide thresholded edge keep high node online setup pt definition pt ex ex ex ex plus plus minus ex ex plus ex em input variable bound informative lead optimization maximally informative establish new principle demonstrate representation deep becoming solve great challenge image recognition language method constitute usefulness instead making directly consider bound characterize efficiently representation optimize successively tight modular separately lead compete maximize define regardless detail generate usually mutual correlation distortion intractable lead elegant self theorem foundation theoretically representation framework recent correlation explanation introduce excellent diverse source optimize sec maximally representation sec idea world financial use capital domain whose
dot product cosine none inclusion normally measure triangle move continuous space embedding diagonal uncertainty function asymmetric inclusion distribution perhaps literature gaussian distributional potential space map region inclusion provide geometry discuss work present qualitative quantitative common concern people seven similarity task lexical also demonstrate training support new incorporate distributional vector distributional semantic language broadly probabilistic row give relevant bayes observe space distribution train use arbitrary asymmetric factorization learn combination metric effectively per fitting embedding apply fisher preliminary graphical pair quantitative linguistic semantic traditional count region strength vector popularity word indicate position onto generalize eigenvector variance word nearby map dictionary linguistic relationship precision vs distinction orient unsupervised token type nearby token word type vector vector context input parametrize pair high negative pair accomplish define negative supervision provide energy base representation token context often word skip gram word energy context treat context score word sample train context achieve desire effect word context type pointwise limited dot product depend treat rely surface absolute energy train rank rank positive negative terminology function represent pre train context word set represent context empirical estimator covariance mean practice necessary add invert note learn possess unsupervise inclusion gaussian context rank present independent valid dot incorporate covariance would model logical similarity behave product seem natural indeed appear probability history gaussian inner broad gaussian aim always quantity firstly ratio likelihood commonly work difference interpretable hard logarithm determinant covariances covariance trivial model store efficiently matrix intuitive geometric similarity measure distance measure mahalanobis volume span principle component interpret prevent mean encourage concentrated encode context directional supervision knowledge sensible lead vocabulary type skip baseline two output subsampling paper training constraint improve vs diagonal gaussian embedding large comment original use constraint performance examine query sort measure gaussian denote broad frequency word get word frequency name appear context fairly word word nearby mix mind top choose sorted measure great qualitatively source less mention begin section precision pick ap empirical kl learn e learn variance diagonal spherical symmetric cosine mean base learn embed variance pre cosine asymmetric measurement embedding bad cosine word variance count reasonable kl regularize matrix empirical cosine variance choice either leave make locate gaussian discriminative power examine variance note word model possible force commonly context well distributional move beyond purely unsupervised learn unsupervised manner text form lexical reflect phrase look appeal embed dimension capture hierarchical tree area simple variance parent child create tree come come directional else large capture leaf negative receive node evaluate embedding seven similarity art scope however match report dataset skip gram implementation much embed algorithm achieve distributional quality experiment variance spherical gram dimension diagonal parameter spherical overall slight edge embedding embedding plot spherical diagonal significant shift diagonal vector see spherical generally set cosine outperform cosine mean spherical covariance distribution never include dataset sg sg mc word type directly represent directly notion enable rich geometry embed demonstrate linguistic qualitative spherical combination rank matrix going enable keep semantic align capacity move stochastic warm g descent gaussian concentrate high dimension multimodal another future idea
plug obtain q strongly continuity sum convergent subsequence z j j proper furthermore relation convergent subsequence j minimizer take convergent subsequence eq convergent subsequence use function convergent global sequence choose point addition algebraic argument subdifferential relation moreover subdifferential inclusion relation relation whenever hand subsequence readily hand x since non together third relation constant hold trivially property kl exist whenever furthermore x proceed far since non loss concavity see claim notice increase apply last first comment proof indeed dr see third stay dr state semi algebraic satisfie derive dr examine exponent rate z last consequently conclusion case follow contradict happen positivity immediately I tt combine precede result existence give guarantee sequence boundedness splitting choose satisfy addition sequence series lipschitz relation readily boundedness boundedness boundedness bound consequently boundedness complete dr splitting feasibility nonempty follow optimization close continuity modulus inner expression follow infimum splitting feasibility termination algorithm computation classical dr splitting dr splitting compare point study author parameter dr close nonetheless dr feasibility nonempty x finish need justify q partial tv particular furthermore passing contradict see result result super classical dr splitting method function minimize subject look necessarily super super limit hard regard many consequently nonconvex feasibility find nonempty bound eq projection onto state dr general nonconvex feasibility sequence sequence exist algebraic set corollary definition subdifferential conclusion together consequently finally definition conclusion ii play quantify dr classical dr minimize dr splitting dr vs dr dr splitting functions convergent cycle pair corollary convergent consequently cauchy sequence immediately look initialize z numerical method nonconvex feasibility code matlab linear system dr splitting x benchmark projection proximal solve specifically close subproblem origin terminate projection dr splitting adapt linear first random describe report well termination failure fail value termination failure different threshold easy minimizer fail hard splitting fail e finally dr splitting splitting minimize criterion splitting terminate exceed solve average termination failure slow splitting alternate quality dr method projection r dr fail e examine nonconvex nonconvex feasibility introduce local convergence dr method explicit threshold sufficient boundedness dr numerical experiment indicate dr method usually outperform anonymous comment improve cm research grant dr splitting nonconvex feasibility study class optimization less nonconvex set direct proper smooth rate give boundedness splitting find intersection general minimize size computable split cluster point whole convergent semi algebraic function dr splitting method usually alternate projection finding take problem mathematic engineering aim find closed set call feasibility problem cast refer reader recent detail dr split powerful solve compete find proper close latter minimize indicator projection feasibility operation main splitting efficiently dr aim find two close heat splitting close set scheme examine popular proximal reveal explain therein dr apply sum proper example reader recent exposition behavior dr split moderately theoretical justification complete nonetheless dr splitting method motivate analyze projection alternate method despite difficulty important understanding behavior nonconvex exhibit affine regular convexity feature improve local dr splitting super regular specific feasibility problem seek convergence establish intersection basic perspective recall find closed interpret optimization length distance solve easy feasibility common study prox regular proper computable splitting give addition boundedness generate see introduce dr nonconvex split whose objective computable threshold splitting method furthermore convergent finally alternate projection dr usually take paper preliminary material apply proper closed analyze feasibility simulation conclude remark value function define never subdifferential immediately robustness fx subdifferential reduce continuously differentiable subdifferential classical subdifferential v groups resp subdifferential resp finally say modulus indicator limit close onto semi union fx cover nonsmooth property particular semi
semantic interpretation exhibit experiment lda topic distribution five hyperparameter number topic dirichlet parameter baseline unconstrained optimization baseline intuitively poor sharp deep neural mnist handwritten momentum layer dropout input optimize validation weight million evaluate directly momentum descent difficult tune various momentum rate objective measure reporting poorly introduce sharp rate momentum weight objective evaluate full evaluate validation chain minute core surface discover simple varied burn fix diagram integration show baseline configuration step yield proposal accept run choose perform chain choose correspondingly significantly constrained constraint observation formulate allow user tradeoff risk specify propose acquisition bayesian include meta constrain application product design meta mobile device speed usage objective evaluate possibly acknowledgement would helpful discussion experiment award center edu show optimization function motivate optimize dirichlet topic hamiltonian monte carlo pass black objective appropriate company design measure good want ensure therefore propose perform customer people people company general might speech recognition phone user speech acceptable material bridge subject margin use arise volume simple chemical synthesis combination cause discover discover boundary laboratory rather would specify valid naturally proceed develop global start likelihood conditioning treat belief next acquisition acquisition ideally relatively proxy evaluation acquisition objective spend well via well spend new model acquisition complete loop task high result meta acquisition bayesian address exploitation vs idea interested region high improvement ei acquisition strong b improvement ei objective predictive density improvement target ei encourage exploitation input exploitation predictive minimum expect constraint ei close differentiable compute maximize optimizer acquisition constrain ei formulation constrain acquisition optimization address problem previous constraint noisy probabilistic specify constraint objective predict require trial evaluation discover resource spend simultaneously evaluate would spend acquisition incorporate support expressive parameter space example total memory usage restriction encode constraint bayesian acquisition conditional ei ei improvement assumption density formulation encourage place propose pareto active pareto classified specify determine number objective constraint infinite feasible region infinity however limited aim classify find discusse failure terminate thesis introduce weighted acquisition constraint first therefore input noisy know accounting uncertainty problem return always namely contain uncertain whose estimate natural constraint condition concept represent function represent boolean indicate satisfied constraint may also constraint ultimately solution satisfied constrain remainder paper propose acquisition determine beneficial gps gp multivariate arbitrary gps see gps gps model need gps represent real satisfied represent transform lead likelihood posterior compute gaussian predictive marginal permit discuss type program nonnegative logarithmic g g imply corrupt normal choice convenience form constraint model instead link subject sigmoid mapping cdf binomial form sampling need follow mat ern model differentiable one characteristic length fully integrate markov slice form elliptical slice whiten procedure avoid couple hyperparameter give acquisition efficient acquisition depend conjunction violate acquisition function constrain line constraint full acquisition integrate q gp hyperparameter gp hyperparameter previous acquisition violate ei exist therefore ei acquisition intuitively probabilistic constraint ignore objective satisfy satisfied feasibility purely satisfy high either continue probe region drop lower indicate black circle minimum find next property constraint problem choose evaluate discuss identify box possible acquisition individually evaluate ei cause prevent region identify belief follow exceed likewise improvement belief objective become occur
parent free overfitte way help overfitte degree grid two aim exponential complexity challenge previous empirical suggest bound improve hold great burden model real search view adapt learn hardness score parent liu work mixed programming formulation solve formulation effective attempt great encode easy exponential might see mostly hope highlight avoid context generation clean solver additional previous formulation describe formulation formulation encoding elimination order node obtain solution program ij ji number order variable take elimination eliminate specification equally minimize clique might indicate converse constraint whether node order result elimination produces force guarantee elimination order also difference partial order bottleneck reasoning ij ji order consistent turn consider perfect value eliminate node manually cardinality denote dag consistent subgraph scope dag program partially topological variable constraint th parent exactly force acyclic respect topological tie break arbitrarily node order node constraint ensure arc appear graph constraint responsible inside constraint direct put reach follow formulation dag specify dag n iw formulation directly optimizer corollary resource employ branch cut formulation stop still solution quality monotonically decrease validate feasibility previously call reasonably uci repository detail run gb memory parent per experiment one implementation language use order magnitude fast able result much domain estimation error tw breast largely poorly cope contain become easy consequence problem load unbounded situation demonstrate empirically reasonable time number variable reach unable next limitation handle domain bound dag graph design large handle naive design bayesian rejection structure discard hard poorly ii structure discard score constrain fact report elimination order compute topological search parent set straightforward efficient property ensure superior option base node edge denote graph subgraph idea sample search code moreover mapping code code draw sample trivial element computable structure idea extend divide give function dag combine sampling theorem structure obtain bound initialize empty reach dag maximize time precisely hope say base unconstrained need unbounded implementation search give might boost explain main practical drawback version process propose per iteration compare define define path topological force interested way edge order represent node tree partial order dag ignore arc dag exceed affect correctness specify parent bind initialize arbitrarily clique call root clique arc create mark link sample arc create cycle unless run sample create cycle represent known decomposition iteration step place order time space version small would drastically decrease space version run decode theorem code order dag greedy choose exceed take cycle form although much region space compute hash closely characteristic unbounded avoid set breast letter hill empirically analyze compare uci repository discretized variable sample set column original data audio community discard audio different equivalent maximum hill nevertheless one parent pre run pre score consider memory limit gb three hour solution minute version sampling method sample seed version ten seed relative version find minute hour version score ten score ratio value well whereas converse raw table intractable top largely superior even tree probably allow much tree satisfactory set hill within minute hour datum ten outperform worth note version matlab formulation amount minute might produce efficient try suffer show even create new mix integer programming formulation especially problem network result indicate state art might fail large domain purpose propose double provide bayesian limit empirically collection linear bind certainly every permutation tree work closely appear exact alternative cutting generation improve independently partly support office grant grant ccc cc min median max letter audio hill ccc version max min breast letter audio hill default ji present structure method programming formulation consist graph subsequently subgraph structure outperform state fairly accurate describe distribution reconstruct variable practitioner refer known bayesian drawing inference probable explanation maximize inference inference hard provably exponential polynomial time algorithm guarantee quality raise consequence assumption provably inference reliable learning method resort perform belief inefficient great learning inference np extend maintain relative bayesian bound show hard develop dynamic learn bad combine heuristic learn address recently seem
transform remark contribution relate length precision common absence recall inverse prove convergence distribution highlight whereas sampler algorithm comprise gibbs posterior sequentially predictive distribution posterior purpose distributional property marginal moment exchangeable appendix focus pick measure diversity sample equal pick summing diversity shannon diversity gamma know induce diversity site induce illustrate compute closed numerically formal computation shannon index display asymptotic follow continuity path vanish variation specie prior vanish case use precision value moment diversity easily description seem hard achieve probability diversity assess goodness interest synthetic give site describe se rational quadratic burn iteration show diversity covariate triangle diversity space quantile credible predictive diversity line graph estimate towards true diversity grow difference attribute smoothness study numerically estimator thorough purpose replication draw time estimate gibbs sampler iteration chain inspection ht triangle covariate black gray quantile credible colour resp axis represent covariate ht square exponential set ccc consist operational unit conduct site factor diversity round mention covariate example full essential experimental sparse model deal observation linearly different covariate give c cm compute fully describe conditional metropolis proportional compare truncate left target describe equal obstacle example output test observe distinct appealing process joint output entry distributional trick follow site covariate resort compute extreme intuitively ht obtained transform square marginally continuous sup moment factorial reduce stationarity process constrain come stationary handle diverse result size bias dependent bias particular namely index probability indice distinct measurable biased measurable product averaging transform encode pick state mention look insight random importance distinct site resp follow eq run distinct element covariate one replace element permutation final step stick beta random knowledge reduce ii generalize covariate whereas marginally covariate dependent define completely approach marginally stick define turn function process type j sup topology moment derive simplify require computation generality proof biased permutation hand side double simplification distinct independence decompose group treat fashion hx virtue summing kind pick covariate universit e paris paris france model probabilistic modelling specie site site classify represent give species site environmental covariate improves use dirichlet thus stick break transform stem markov chain algorithm sampler experiment conduct study specie operational site composition specie site index covariate probability associate specie able impact covariate population measure index focus affected membership partially obtain observe covariate analysis compositional proportion chemical composition specify field biology physics economic health quantification mathematical mathematic despite parametric pre specie suggest drawback although often term specie applicable sampling nonparametric extension probability factor extensively three construction class chinese restaurant orient line collection second completely analytical allow elaborate study distributional strategy seminal success stick construction stem great hereafter weight dirichlet model beta transform model diversity could predictive yield discrete nonparametric specie observe draw conditional rare specie conditional organized discuss measure characterize model bayesian propose datum study study defer study orient sampling abundance diversity notion question measure diversity numerous way diversity group specie specie sample also specie later nonparametric distribution shannon index discrete observe diversity index value diversity indice diversity specie fix denote plug shannon vary covariate parametric undesirable size individual belong specie parametric estimate avoid dirichlet prior eq direction impose species aim nonparametric unlike sample directly resort last covariate diversity problem consist many entry index improve model specie describe covariate specie count site remark site specie index specie denote also denote specie site abundance abundance site denote number abundance satisfie equivalent covariate infer interested path covariate fy x n p jx propose probability moreover vector nonparametric introduction nonparametric stick break idea nonparametric marginally jx comes break sample specie observe site abundance specie site collapse variation site explain site exchangeable investigate extension factorize factorize posterior I v jx jx species factorize specie prevent introduction site prior expression permit across specie problem drastically burden require marginally exhibit describe introduce gaussian
derive denote intrinsic manifold depend yet rarely study validation non several yield improve justification geometrically inspire approach come manifold laplace datum geometry hence compare induced geometry choose build riemannian laplacian method guarantee graph weight bandwidth construct equation discrete heat follow riemannian endowed riemannian riemannian definite riemannian category whether refer mathematical geometry mainly integration ii original zero dimension coordinate manifold linear distortion map riemannian metric right application positive riemannian dual show encode conversely implement laplacian binary h il h h dual laplace encodes intrinsic propose capture data geometry must measure trivially laplacian choose maximize geometry dimensionality riemannian stand metric ambient space propose tune enforce identity imply mathematically find self limited bandwidth laplacian ideal object equivalence involve prescribe represent h would tangent subspace inefficient tangent reduce evaluate sample point express r result subspace serve chart pass consistency encode notion heat kernel conduct heat produce requirement must map row heat neighborhood implement design compute around principal riemannian one approach step improve minimal whether invert trivially riemannian metric unit metric perform fast make numerically robust show represent algorithm column accord result straightforward brevity generalize word project proper submatrix I algorithms enforce submatrix close unit review enforce exactly metric evaluate propose distortion distance distortion move spectral riemannian metric q volume element represent laplacian geometry geometry derive practically dimension heat compute laplacian nr step high data speed computation large complementary possibility distortion variance distortion mention working dimension focus mainly propose already mention guarantee relevant manifold parameter usually interesting laplacian self evaluate statistically principle translate vice behaviour rise subspace align manifold noise direction distortion variable curvature curvature point note high large dimension reflect exhibit show even dimension intrinsic dimension first datum plane find range space shall large use range grey seen lie detailed limitation range partially range find find upper illustrate phenomenon use supervise depend parameter minimizes illustrate range small increase upper intrinsic method cost low curve weak nine intrinsic projection one depend choose smoothing investigate choose noise noise form noisy embed obtain align embedding laplacian dimension value noiseless embed replication replicate truth along find low slight systematic tendency support manifold present theory supervise task choose attempt split consist set group simulate anneal highly non smooth proxy scale construct laplacian heat heat kernel reconstruction error r lx think confidence eps change denote geometric use dual te six te te cv depend split split c c cv digit percent six use bandwidth away te lead regularizer five despite variability outperform case dimensional rotation scale take cv estimate cv effect distortion case examine finding panel order magnitude even hypothesis well find laplacian encode geometry find add introduce become become supplement experience associate probably consider short lead tangent plane provide principled select apply parameter laplacian near graph interestingly parameter possible finite drive method impose geometric intrinsic mode many expect superior yet competitive cv besides experimental reason supervised label smooth severe require particular
symbol decision depend symbol state activation rnn compute conditional vocabulary indicator variable sentence although originally straightforward corpus paper train english encountered fix long track consequently decoder recover encode vector propose sentence translate rnn encoder decoder wish segmentation phrase confidence find good segmentation integer programming problem source compose phrase subsequence I encoder translate consider candidate translation reverse decoder target language confidence q phrase eq likelihood indicator include q ij number source phrase contain phrase contain totally make segment counting relation hold evaluate well definition ns describe segmentation approach clause unless source language roughly order english concatenation translate clause necessarily despite gain translation translation question issue heart purely translation drop present robust intuition multiple short clause unknown neural propose approach computationally phrase sentence phrase parallel phrase english translation word select news un two word website news news phrase first development neural english sentence vocabulary word english consider token neural incorporate specific without segmentation translation decoder train english sentence translate english segmentation score eqs conventional translation expect validate segmentation segment mean confidence ht random segmentation refer mean length segmentation score segment clearly agree sentence segmentation ht average score translation definition le ann es le le le de de les du de la les es le et la du la des es health les ann es de service le du le pick remove pick take de cr un ai cr pi et de il cr il de move focus make really bank build great bank united say segmentation move make bank building great bank united say des une une pour la le pour le la l les une une du without les en une le like extend line segmentation extend reference il la cr une force les il la cr image un de le tend les cr est specify respect confirm medium difficult segmentation specify star confirm difficult deal pr te de respect le pr la la star le pr de le un select overall quantitative observe decrease form clause additionally independently segment sometimes htp source request begin segmentation request il le pour il pr send il le de sa il il sa automatic segmentation solution curse sentence score base translation translation sentence translate sentence quality especially translate mark research translation translate acknowledgment would acknowledge cifar research universit de van universit universit cifar translation exist phrase translation system paper address automatically input sentence phrase translate neural network segment translate machine translate clause form
optimality pareto front develop datum mining sort pareto front query retrieval triangle first pareto hull set method differ also front concept pareto good knowledge widely similar pareto front apply rank utilize pareto multiple criterion another anomaly pareto depth dissimilarity pareto database criterion correspond dissimilarity single entry rank score item score list community also combine query semantic multiple set contrast useful outperform pareto front fusion tail document utilize front method outperform avg multiple retrieval multi usually although disagreement view view criterion however give may severe disagreement area multiple kernel typically pareto many include economic science sciences overview pareto front set objective problem evaluate possible goal find criterion combine criterion usually choice yield minimizer without employ search identify feasible find pareto every objective pareto dominate another item feasible pareto front denote pareto front pareto pareto front generally pareto front front retrieve though equally pareto front pareto point pareto rank figure linear hull pareto front observation deep pareto pareto query retrieval introduce notice non shape pareto semantic related query introduce front retrieval sample retrieval query query combine partially pareto retrieve successive pareto tuple dissimilarity dissimilarity vector jx convenience pareto definition pareto system key pareto front e sufficient retrieve query return middle pareto previous study distribution pareto optimal point pareto geometry cloud non due pareto call al cloud pareto denote pareto denote pareto simplicity count follow front exist belong pareto front nx kx kx nh pareto traditional convex randomness even pareto scale account minor pareto geometry pareto pareto say pareto draw sure encode pareto set characterization density yield cumulative density uniform proof instead quite involved completeness preserve random among hypercube constant almost surely complete recall proposition context log concavity theorem method largely demonstrate query retrieval pareto integral characterized solution substantially overview limit non extract indexing retrieval processing computer use sift technique pyramid algorithm let md iy assign score assign assign rank distance manifold sort distance add connected rank matrix force nearby term force query rank rank iterative ranking function repeat inversion graph rank anchor construct matrix datum denote column affinity final ranking invert matrix inverting database computational computing require storage matrix computation retrieval xu al provide update prior query algorithm rank final individually main pareto front propose give give rank dissimilarity vector construct pareto associate sample pareto retrieve front return point middle front relevance feedback enhance retrieval could pareto experimental pareto state algorithm develop query correspond semantic label many belong query normalize discount cumulative gain ndcg community ndcg relevance measure single relevance retrieve otherwise retrieval binary score relevance relevance score performance assessment multiclass retrieval relevance query speak multiple relevance cover retrieve retrieve object uniquely query uniquely importance query instance retrieve image effectively query rank query retrieval retrieve object let logical logical conjunction label respectively entry query unique relevance retrieve query query unique relevance query retrieve set relevance discount cumulative gain normalize ndcg normalize possible retrieve contain label query difference assign depend relevance multiple set uniquely ndcg evaluate video widely widely retrieval community provide manually annotate level correspond key characterize global texture frame image key label entry concept image label label belong exactly belong class label members zhang database evaluate randomly pair run algorithm compute pair ndcg use anchor graph run time ndcg experiment ensures avoid particular state retrieval figure compare avg max avg query classifier joint avg figure outperform note generating consider pair label image multi unnecessary separately take retrieve pareto pareto front adjacent front user front bar visualize pareto front relevance five pareto tail middle tail relevance front fix average pair figure deep pareto fundamentally multiple query retrieval middle front suggest version return point middle returning say hold front lead improvement certain choice advance available information decide simplicity modification two query retrieval pareto front user visually explore front pareto front pareto middle pareto point identify pareto method shoot retrieve front include code retrieval correspond query retrieve retrieve retrieval algorithm linear theoretical convexity pareto prove pareto use combination front iii retrieval query include semantic image semantic query combine pareto front state improvement concavity characterize concavity pareto retrieval manifold two problem retrieval image literature correspond image semantic concept possibly shoot angle idea utilize object query call retrieval technique multiple query involve average problem goal find image contain query semantic desirable feature necessarily make fundamentally multiple query image form average query relevant align query retrieval context word approach seem query time tend closely query rarely query
enough use also index long convex lemma conceptually latter exploit consider hessian hessian st update version express substitute result recursively substitute result substitute expression add repeat transpose product except transpose need determine third multiplication summarize recursive plus one gradient variation pair recursion constant vector element determination require operation cost compute link maintain rank common adopt variation q iteration attempt observe see diagonal cost product adopt numerical store scalar multiply constant scalar return product variation implement give loop step constant element loop compute loop second outcome perform implement loop yield multiplication likewise inner product multiplication multiplication multiplication cost summarize computation implementation step initialize estimate identity determine variation curvature matrix step property section svms develop engine tb l initialize cf variation cf cf subsequent convenient instantaneous association fact goal iterate prove result follow eigenvalue impose intend gradient variance unbounde rare progress towards argument stepsize elimination variation require rapidly decrease step strong linearity expression write convergence proof need variation lower instantaneous inner product stochastic furthermore ratio variation variation strong inner hessian inner stochastic variation include stay definite descent stochastic alone guarantee arbitrarily bound matrix oppose formula update relate per approximation use find give assumption trace uniformly time appendix write account fact constant recursion give determinant sum approximation constant eigenvalue imply respective inverse approximation approximation large exceed emphasize realization irrespective upper conclusion direction conditional method bfgs initialize aside sake relationship infimum norm minor nuisance take present matrix give infimum optimality surely e realization establish subsequence role proof roughly eigenvalue limit effect variation regular bfgs variation eigenvalue result estimate strong surely strong hold sgd theorem characterization bfgs define iteration descent give satisfy difference expect constant appendix sgd dominate gradient prove bad sgd theorem parallel adaptive strategy latter description refer former description set known find hyperplane separate set set feature class hyperplane loss measure support support term problem train objective sgd algorithms accelerate memory algorithm randomness curvature nonetheless alternative sag sag gradient direction performance sag five objective vector sgd sag sag sgd square feature class half belong component vector interval likewise choose interval class order advantage vector sag process represent five stepsize improvement stepsize individually since minor sag various individually result average sgd sag objective sag l minimum sag convergence sag attain processing respectively average hold despite fact stochastic gradient feature vector discuss value sag feature magnitude achieve matter figure figure performance performance achieve sgd sag however average sag sgd order achieve far even large become vector respective computational analyze l runtime sag sag sgd fast among acquire computational cost account respective sag order increase process contrary advantage repeat record processing target objective parameter use processing time objective histogram sag minimum maximum run sag stand mark sgd sag analogous histogram performance sgd respective convergence sag still magnitude sag sgd advantage execution large engine apply click search engine query specific appear user descriptor title keyword position page ad display specific include gender ad success ad display query user vector logistic regressor ad vector skewed benefit component total observed age structure gender position engine set select feature whereas parameter convergence rough parameter relatively gradient sparsity nonzero orthogonal average training classifier iterate sgd horizontal vector evaluation versus read index divide axis correction way illustration achieve stand illustration make feature predictive frequency click ad complementary click ad separate define ad ad classifier predict ad ad fall likewise ad prediction consider interval histogram ad fall histogram predict ad conversely ad frequency count predict ad click ads eps eps click rate ad click ad ideal predict probability ad click ad bin sgd acceptable ad ad histogram predict ad point predict interval click rate ad test inaccurate click classifier test sgd classifier predict click ad perform complementary click label imply complementary click rate ad classifier sgd point classifier compute predict interval inaccurate element ad minimizer log likelihood cost ad train label ad replicate label likelihood give ad implicit sgd times select ad prediction ad eps ad click ad predict ad click ad count would bin classifier make ad ad sgd histogram click ad complementary click ad show histogram click rate ad increase classifier click ad sgd less ad click reduce click ad vector predict ad frequency ad display succeed find limited version stochastic sure bounding trace matrix behave far determine stochastic limited ability smooth support develop result term vector process well execution logistic regressor engine problem present numerical test train less similar classification begin observe product equality hessian inverse multiply yield fundamental term except last multiplication implement three product third last product analogous repetition definition element likewise last common recursively equivalently write nest simplification obtain substitute repeating process repeat final instantaneous instantaneous hessian segment instantaneous definition instantaneous definition variation claim true time inner instantaneous begin bind update notation define trace trace trace substitute simplify already one derive appear bound recursive expression give go provide conclude substitute common lead bind make recall determinant define simplify notation determinant determinant product simplify know last simplification observer symmetric therefore substitute simplification factor multiply divide nonzero norm third normalize occur associated eigenvalue large imply coincide particular trace eq also bind right conclude index derivation need determinant curvature scale identity write follow substitute upper determinant approximation bind make recall initialization reduce analysis inequality consider step sum recall conclude less low inequality provide determinant determinant product eigenvalue equivalently conclude product inequality reference hessian write consecutive substitution take hessian bind right hand product state third fact second term low side low bind use statement reference construct sequence satisfy converge surely almost explicit q vanish subsequence embed limit infimum nan transform bound optimality eigenvalue expansion argument schwarz simplification since limit infimum state line theorem sequence objective proceed provide sufficient give rearrange induction hypothesis recursive relationship substitution maximum bind substitute simplify term formula upon conclusion assume prove convergence specify gap term derivation completeness hessian state take around fix hand whose find imply true yield gradient substitute result value side double conclude substitute rewrite hypothesis satisfie define identify substitute function canonical class machine hyperplane separate hyperplane argument gradient train large rely base purpose category newton ever large time regard sgd slow use gradient direction replacement alternative effort fast gradient descent convergence descent still practice randomness challenging curvature profile ill condition slow deterministic deterministic newton stochastic fact limit specific converge stochastic newton use remain quasi speed time without stochastic quasi newton online bfgs bfgs memory middle ground broad applicability irrespective structure extend gradient direction estimate generalization bfgs gradient deterministic gradient differ improve bfgs reduce try adapt curvature quasi newton problem hessian possible singular estimate eigenvalue progress minor possibility analyse fact introduction retain ensure valuable iteration limit memory theoretical main show argument contrast properly guarantee brief discussion bfgs bfgs curvature differ gradient reduce memory deterministic property assumption sample function determinant lemma bound hessian approximation condition sufficient realization important result ensure suffer almost fair emphasize convergence sgd regular introduce dominate bad curvature condition describe newton behavior comparative well number vector term sgd feature make claim regressor click search engine section
valid maximum follow easily adapt extreme hence systematic sign statistic systematic compute observation function location modify wu statistic iid censoring present sign likelihood version wu model function derivative nominal regression dispersion sub test value covariate size six sign test sample adjust test size rejection rate adjust rejection rate nominal present moderate value asymptotic p plot relative sign ratio well plot anti behave test turn note nan guarantee simulate nan test give correct rejection power htp c maximum extreme location dispersion sub covariate possible sign likelihood approximation size summarize sign size converge nominal grow test htp c dispersion specification location value draw distribution present scenario sign similar behavior evident wu notably htp c c ht ht involve table maximum wind measure minimum temperature day wind speed reach ht temperature wind maximum wind value dispersion likelihood nan sign test wu display accordance evidence favor r intercept minimum involve jump eq put present sign much small adjust conclusion adjusted sign adjust test reject nominal nan rely adjust ht r discuss jump put p versus modify sign case possibly unlike adjusted parameterization simulation result reveal ratio test nominal evident shrink wu test behave well clearly wu test behave wu good test conclusion sign adjusted test simulation practitioner adjust component analogously nr r write parameterization parameter observe parameterization instance location h r x v z adjust sign statistic replace formula pc mm derive adjusted sign likelihood approximation sign ratio statistic monte carlo compare sign ratio tend large shrink distortion real discuss word area reliability frequency environmental financial highlight necessity improve reference statistic formalize unify recent book reference extreme reliability survival moderate specifically dispersion regressor literature nevertheless practical contribution small make correction derive adjustment likelihood statistic testing side hypothesis sample adjust location dispersion parameter simulation type likelihood greater moderate sized test present similar conservative adjust case perform test focus sided sign ratio widely side scalar nuisance parameter distribution sample law usually lead sign test considerable sign statistic derive compare finite sign sign test suggest adjust modify datum paper organized section present regression sign derive different author sign extreme performance real end conclusion continuous random dispersion euler constant type approximation inaccurate suitable error significance test normal accurate function space derivative statistic sufficient function notation sign statistic exclude row correspond involve turn require determination impossible application concern model e derivative sign extreme introduce approximation require diagonal note always possible orthogonal parameterization sign ratio replace sign take log interest adjust sign covariance correspond accord sign ratio statistic base estimate adjust sign wu function derivative f cumulative first adjust sign present extreme
matrix hadamard define recursively fast hadamard transform time detail construct recover cosine describe play rescale parameter adapt relevance control rbf adapt type fast method equation behave diagonal specific treat free operation diagonal need learn composite parametrize backpropagation optimize advantage th simplicity backpropagation datum partial gradient respect note simply hadamard since consequently permutation matrix multiplication back propagation allow jointly deep layer greatly overall prevent extract feature affect composite layer considerable store particular decay composite think several layer diagonal control layer include understand powerful adapting within essential layer layer suffice within apply dropout mnist optical character recognition layer replacement second jointly train table implementation convolutional pool use train consist relu linearity arc follow kernel cosine rbf kernel r indicate parameter adapt vary mlp mirror layer network investigation layer convolutional transform jointly network layer adapt performance far reveal overfitte increase final densely connect softmax dropout softmax improve part mnist train deep convolutional layer less validation factor capacity control reference many achieve r imagenet joint ad ad reference jointly deep final imagenet class advantage considerable redundancy could network work convolutional author use regularizer encourage connect memory sparse store nonzero storage take representation dense compare author quantization layer quantization weight clustering represent column close ignore decode representation recover weight book decode actual singular l half svd ad achieve usage please deep convnet reduce memory communication constraint svd decomposition train directly imagenet experiment reference usage drop fully connect svd half svd decomposition drop half drop decrease svd half new architecture call convolutional reduction imagenet usage exchange convolutional derive replace fully network conventional layer low computational cost previous memory potential preserve effect multiple stack potentially improve even far moreover logistic layer far either capacity control deep thorough introduce acknowledgment google national probe http www probe definition definition remark equally fully deep convolutional contain reduce preserve predictive constrain environment embed device replace fully connect layer deep end conjunction convolutional architecture substantially reduce standard convolutional layer fully connect vast majority parameter evaluate imbalance address test separable speed gain additionally address approximation layer total storage require many convolutional network redundancy parameterization exploit represent call parameter jointly decompose memory apply processing contrast convolutional substantially network kernel particular method imagenet million since full memory innovation adaptive variant learn call convolutional able standard imagenet possible line combine deep neural advance previous doubly effective scale extremely imagenet operate jointly filter convolution feature learn kernel way replace neural literature architecture benchmark connect competition global achieve drawback difficult practice train feature imagenet tune motivate add linear adapt tune convolutional expensive evaluation recently usage apply optimization introduce connection remove drop memory usage since memory gain maintain structure overhead consumption memory also train main bottleneck scale storage usually store computing take important insight kernel approximate basis harmonic fourier density turn drop distribution
zero keep unchanged slowly incomplete deal online adaptive reconstruction involve one p tt collect index also likewise introduce signal since dimensional natural leveraging attempt datum minimize unfortunately albeit rank np hard optimize motivate solve th value convex possibly rank control scalable streaming effectively c become large costly computation see nuclear arrive c efficient online complexity storage henceforth dimensionality bound quantity accordingly argue later bilinear suggest span column c alternative nuclear eq bilinear leverage p provide towards adopt frobenius optimality recover globally global solver satisfy massive ability modern computer store analyze incomplete updating obtain recursively track subsequently historical placing measurement end adaptive weight distant tracking environment weight idea perform online leverage nuclear conference network traffic anomaly popularity factorization separation music name tracking towards recursive solver alternate adopt coincide scale justification suitable instant q elsewhere minimize respect fix row obtain via denote subproblem solve recursive define ty l term plus correction resort inversion inversion track incomplete p l p careful reveal computational burden stem iteration reduce symmetry operation multiplication overall typically cf infinite case far reduce tune resort heuristic apply accordingly one window effectively develop well landscape end retain identical unconstrained quadratic obtain cf show inexact show cost number desire expectation take drop matter update l nothing tune backtrack rule adopt whereby sequence geometrically quadratic filter sgd newton order l small nonnegative building popular speed penalty computational per iteration critical nesterov variant accelerate l k accelerate therein acceleration backtrack stepsize sgd missing clearly accelerate sgd backtracking case complexity mainly accelerate incur online memory adopt lie regard acquire online cf fact accordance boundedness natural impose acquisition extensive computer upon eq cost unbounded evolve identical estimator scaling affect note evolve increasingly subspace prior subspace solve g resemble quadratic equation l g whether importantly news nan fall batch worth note play satisfy semi desirable hessian l pc inspire online theory martingale detail proceed main establish convergent rely upper update namely g l l certain regularity establish convergence cost c c nesterov acceleration establish adopt basic surrogate coincide sgd convergence g q l expansion tf sense l iii locally tight q l cf condition eq outline carry reflect iii argument regularity sgd convergent subspace satisfy coincide problem accelerate sgd step proof outline claim establish far clear coincide recently accelerate sgd put subspace technique convergent instrumental online offer batch estimator exact subspace correspond via window require p establish next numerical attain p iterate satisfie normalize regard even parameter g choose grow upon hand vanish condition modern become increasingly structure index far time matrix mean data cube indicate interaction sample ii eeg represent frequency datum incomplete loose nuclear missing encounter many application capture call high order miss stream multi algorithm capable latent structure parsimonious decomposition sequel focus tensor exposition tt via iteration q recognize quadratic surrogate correction gradient stepsize tm diag b diag diag diag diag put observe iteration close reveal update updating incur overall tensor setting accomplish tensor limiting say remain closed require tensor slice low approximation tensor subspace learn initial offer slice formalize establish slice set I live ct tc tc c asymptotically coincide effectiveness algorithm assess computer simulation synthetic real carry sequel generate entry noise simulate per take entry examine strength validation optimal size apparent optimal nuclear norm attain per suggest matrix discuss essence subspace tracking capability algorithm figure upon scheme accuracy constant true rank exhibit behavior expect choice relative nonetheless numerically ls become ridge term stable term computational iteration table compare algorithms c alg relative origin large scale internet ip management study flow dimensionality flow periodic across massive traffic traffic traffic small flow measure flow traffic collect operation internet network internet measure flow spike anomaly end detailed subset algorithm miss flow track representative b versus true blue flow traffic th truth tensor slice diag column likewise coefficient n taking w accordingly acquire slice form x examine algorithm test imputation streaming slice adopt various accordance matrix depict evolution see apparent collecting amount slice observation highlight accurately reconstruct fraction also adopt tensor slice size main memory report amount lead short time need less computation matlab optimize reduction another tensor beneficial subset employ tracking scheme develop sake imputation entail variable entail online subspace remark r c real traffic monitoring serve image diagnosis heart disease clinical mainly hold time may inaccurate acquire consist resolution mind recover ground amount completion low intrinsic dataset test contain entire scan divide patch pixel slice randomly miss infeasible limitation operation store candidate imputation acquire illustrate reconstruct tensor subspace note assume dft fidelity diag stand linear fourier cc image acquire miss ip flow traffic anomaly due failure denote link completely represent traffic flow connect fraction traffic carry superposition flow rate namely experience anomaly measure link measurement count loss small index measure flow anomalous anomaly sparse partial count instant vector anomaly time fully count time matrix naturally th rank nominal anomaly approximation online fashion anomaly denote dimensional subspace learn diag slice matrix absence anomaly nonzero cf sparse absolute anomaly traffic tucker anomaly traffic section fix slice contain nonzero physical I adopt figure run average detection false alarm depict available become traffic accurately three anomaly depict anomaly correctly pick note p accommodate slowly topology desirable monitoring health dynamic network advance subspace tracking put stream incomplete subspace base nuclear norm leverage characterization nuclear complementary strength develop converge provably performance regularize estimator online miss beyond scope paper worth future accelerate sgd alternative real incorporation optimality change satisfy begin nuclear optimality form q semi notational accordance must q complementary primal feasibility iv feasibility dual readily verify last putting imply due say latent encounter big incomplete dataset pose major challenge benefit scalable imputation miss latent rank datum propose exponentially nuclear amenable complementary strength establish simplify technical asymptotic offer develop internet confirm efficacy superior relative alternative subspace track stream matrix site web internet device volume consensus economic life volume importance volume fact motivate subsample privacy streaming comprise noisy correspond traffic collect physical link movie netflix acquire sequentially deal online estimation equivalently completion solve indexing column modern dataset give rise array index miss analytic traffic medical aim capturing call order presence principle tensor resort develop first preserve array paper contribute towards analyze stream incomplete namely capable
implication weak apart instance fall formulate metric previously propose represent hierarchy contain represent subtree share leaf maximally similar leaf define margin discriminant project thus near boundary formulation widely root share implication learn ultimately hierarchy seek hierarchy poorly near root recover partially place tree correctly relationship require split allow splitting generate splitting constraint hierarchy fine grain flat supervise flat via margin en metric return explicit function apply outside cluster semi time semi pairwise link cl improve semantic clustering indicate dissimilarity cl train constraint link proceed subset uniformly split operate supervise unsupervised check availability require link carry semi check unsupervised learn form propagate tree iterate point child contain constraint constrained separate hierarchy reach accordance processing continue minimum back unsupervised metric learn straightforward feed track return combine splitting hierarchy node pairwise split formulation function hierarchical incorporate link unsupervised seek include additional margin joint label cl slack pairwise slack unconstrained pairwise constraint constraint high high scoring satisfy problem link hierarchy relevant decrease low hierarchy case trivial class wherein cl cl term significantly modify reasonable divide instead attempt simultaneously constraint impossible constraint subset constraint separate cl integrate subset variant optimize cl seek satisfied via formulation constrain constraint two replace constraint attempt maximize cl minimize ml eigenvector q return method near neighbor full forest likely empirically strongly overall margin unconstraine ignore leave divide data operation fully training give parallel ignore parallel neighbor test distance near significantly reduce single ignore candidate different tree much approximate neighbor moderately dataset well case quantify validate efficiency approximate carry comparison classification semi supervise cluster mid know balance x segmentation x handwritten cifar spread cifar cifar instance reduce metric neighbor neighbor grind near obtain precision compute report htbp retrieval cifar htb htb technique gb mahalanobis purely original validate tune use rule dataset gb exception stop criterion minimum dataset evaluate weakly training balance segmentation find test competitive perform experiment compare method near amount label datum metric drop dramatically though become metric evaluate effectiveness nearest semantic precision label weakly unable retrieve class drop particularly relaxed effectively discrimination many broad make ccc analyze metric order semi number train metric retrieve near neighbor return convert yielded good neighbor vary number nearest range spectral measure record name indicate demonstrate consistent significant datum base yield result competitive segmentation dataset notable difference metric much oppose much strong semantic semi distance metric forest construct relaxed competitive scale present approximate near retrieval greatly retrieval little hope well membership extend incorporate relative triplet constraint semi even long many application long dominate mahalanobis advance mahalanobis metric forest interpret introduce randomness hierarchy combine powerful robust nonlinear semi information unconstraine relaxed allow subset hierarchy algorithm benchmark problem cluster core availability measure ad hoc whether propose address traditionally dominate metric method primarily generally easy allow fast globally optimal meaning operate notably need video document representation kind unable true semantic linearity version mahalanobis handle limited reason alternate inherently handle necessarily learn modality early nonlinear metric resolve advantage datum metric technique expensive explore advantage structure train nonlinear transformation shift however overfitte area formulate pairwise metric yield implicit could scalability lack representation neighbor order overcome limitation metric advantage exist
hz h note coordinate three dimension reconstruct concentration eq choose save equivalent spatial bandwidth bandwidth substantially begin peak boundary apply small preferred right flat also poisson data time location bandwidth zero fourth multiply constant serve facilitate thus determine cross validation adapt study parametric model several smoothed facilitate population analysis default software determine choice smooth deconvolution procedure take approach design penalty balance fit error inherently problematic easy smooth estimate meet capture popular dimension functional dimension reduction essential impulse smooth random non increase eigenvalue b mt rewrite eigenfunction variability summarize eigenfunction reconstruct voxel recover automate deconvolution inherently problematic computationally viewpoint curve functional curve advantageous linear choice adopt parsimonious principal component analysis popular functional term many rate induce spatially multiplicative curve adopt modified impulse reconstruct impulse response impulse directly perform impulse however computationally expensive automate deconvolution inherently detail denote impulse response course measure voxel function neither thus assumption eigenfunction covariance non eigenvalue functional component deconvolution pose due positivity deconvolution perform eigenfunction considerable perform hundred thousand spatial voxel advantage note deconvolution allow deconvolution usual standard measure next balance complexity strategy many include deconvolution implement see eigenfunction need implement deconvolution reconstruct result bias smooth unbiased small cubic interpolation fast seem estimation multiplicative voxel mean necessarily expectation mean derive estimation estimate classic eigenfunction principal equation principal score number eigenfunction voxel eq l kt ad hoc fit summary simulation select reconstruct integration voxel detail course score select next truncate datum contaminate independent error toy equally space first observe curve matlab function eigenfunction deconvolution well utilize deconvolution work region interest spline deconvolution deconvolution sp suitably deconvolution within however deconvolution fold increase integrate square approach cc sp spline except input replace gamma pointwise context considerably simulation pdf eq gamma mark structure five region outperform outperform show preprocesse satisfied always close gain deconvolution misspecification situation region structure neighboring deconvolution account contrary robust relatively parametric deconvolution strategy competitive mse value ccccc region sp region shape parametric restriction curve perform focus see improvement study normal subject available main control build understand normal brain twice analyse applicable across truth relatively subject subject population quantification quantification relate directly well fit investigation subject acquire subject min mode camera spatial resolution datum reconstruct filter cutoff voxel mm acquisition event frame movement correction c promise voxel curve intractable consider scan shape three eigenfunction eigenfunction variation function among figure component need response function cluster concentration specific difference bias bias parameter dependent analyze population study patient tend bias model quantitative rely would greater evaluate cccc experiment v pool h test analysis roughly corresponding brain considerably flexible model turn considerable correspondence pool show level variation different density value less individual fairly subject level parametric deconvolution take single scan carry intel core gb functional approach mass deconvolution via principal component expansion realistic plausible modelling assumption good possible inherent course deconvolution deconvolution examine change methodology explore segmentation show segmentation marker spatial coherence clear identifiable situation identifiable suggest effect could replace decomposition show give interpretation reason concentrate note smoothed estimate yield possibility function principal score eigenfunction however asymptotically go albeit necessarily eigenfunction smoothed considerable raw curve deconvolution naturally appeal several candidate deconvolution modality fmri response function fmri could although would deconvolution nan operator non response approach applicable replicate curve allow treat functional acknowledgement author would ci part wang nsf dms dms support ep author thank carry simplicity location voxel voxel density integration practice laboratory university california laboratory email uk emission chemical change human currently describe strong deconvolution model functional principal relative voxel methodology goal quantification concentration brain emission process human amongst modality associate work design process presence throughout decay compound quantitative technique say fmri establish target lead diagnosis high something characteristic cancer indeed diagnosis body diagnostic usage understand brain camera involve complex many available system system control brain reaction disease response later take part role system great estimate throughout subject disease diagnosis treatment fmri consist lead million volume reconstruction process reconstruct reconstruct facilitate practitioner scientific clinical suitable modification introduce could incorporate come ordinary ode system abstract voxel transfer assume ode biological adequate voxel mixture cell lead addition stable voxel square linear somewhat order account fitting explore nonparametric deconvolution flow online purpose continuously relative measure sensitive produce deconvolution deconvolution inherent deconvolution methodology observation possibly
model highly whiten common preprocessing remove unit allow training gaussian component spherical model fairly used statistic whiten preprocesse knowledge improve success analysis component visible right position natural anchor therefore speed initialize scale first scaling scaling determine ideally show hidden bias much weight initialize experience work well initialization hyper impact speed successful training acceptable number big learning converge local place weight become visible restrict prevent divergence big twice datum hold even big even therefore practice matrix rather also momentum counter grow certain recommend regularization momentum add percentage old batch vary lot momentum use prevent weight converge momentum stage rbms describe approximation estimating ideally sample low stay previously therefore step increase rate suggest persistent persistent persistent analyze training convergence restriction pt well performance cd pt difficulty image several author various modification propose address analyze generative argue failure predict pixel model rbm unit dedicate regard covariance diag compare restrict although covariance argue limitation develop spike rbm split binary spike conditional visible diagonal gaussian shift along failure procedure show learn filter difficulty propose modification learning difficulty problem expert constrain gaussian representation insight capability blind source separation capable meaningful toy natural image comparable ica ica orthogonal representation success highly depend setup propose directly experience imply vice versa knowledge center integrate distribution start train usually successful network illustrate write n expert expert gaussian shift j unit lead correspond visible unit bayes formalize denote respectively jk sum vector exactly follow isotropic gaussian convention conditional visible unit component probability view pick hide location visible take exactly place order component determine indicate constrain independent super source dependency able visualize toy example source yield whiten calculate assess run experiment ica ht one code efficiently pixel scheme biological plausible natural empirically code grey scale training image mention patch trivial vector filter fairly filter
extensive subset obtain partition median part actual suffer report linear logistic normal correlation choose intractable implement np selection vary experiment pt htbp htbp htbp htbp clear message excellent case bootstrap subset mse median dramatically performance bootstrap competitive range small clearly average lasso median winner consumption measurement predictor predictor code categorical date subset inference square error exclude produce meaningful time stage later stage subtle performance fast exclude due see algorithm produce physics interest distinguish particle size rest parallel classification accuracy correctly predict test plot exclude quickly benchmark list model achieve propose flexible message message burden aggregation message performance simulation prediction theory describe concerned topological exceed k let select select estimator average incorrect take side b justification theoretical nonetheless insight limited attempt justify theorem alone routine yu part address correlate convenient assume feature sample subset article cause choice strong article conclusion simultaneously next might satisfy elliptical high dimensional set elliptical special spirit invertible assume elliptical alternatively hold proof magnitude iii set inequality w w chebyshev inequality therefore take immediately part establish chebyshev taking term quantify take need minimum index evaluate solely adequate cauchy schwarz combine q part consequently single machine size least procedure begin c x assume big continue strategy part need satisfied size alternatively hold subset simulate htbp htbp definition department pa david department nc commonly algorithm store communication challenge arise guarantee excellent practical general median selection subset aggregation estimator attempt solve parallel feature inclusion parallel scale efficiently theoretical consistency relative usual velocity challenge promise procedure parallelization partition full different process subset subset type gain calculation optimization algorithm operation involve likelihood lead computational amount gain across computer communication drive efficiency importance communication limit communication combine step simple issue communication free improve statistical slow entire simultaneously article focus particularly approach subset suggest use zhang subset well utilize median show sharp certain broadly useful inference regression feature current combine design detailed missing contexts imputation computational another bootstrap fix feature fix justification excellent computationally highly organize message detail scenario evaluate message extensive discussion family proof vector matrix error assume fundamental efficient subset subset carry aggregated produce two rich literature consider generalized criterion feature dimensional attempt solve regularity selection solve problem solve yu consistency lasso could ordinary square ol introduction median possess advantage feature aggregation motivated averaging hence feature interest variable selection recommend simplify inclusion include subset median indicator otherwise inclusion inclusion indicator true inclusion polynomial vector gain time heavy tailed influence selection influence outlier datum put aside average estimate subset spirit
one fall shape usual center raw datum process cf middle factored represent coordinate cf structure convenient interpretation xx xy yx yy first method subsection graphical lasso term provide rather contain along boundary considerable edge partial pt ccc use non positivity part precision point sign constraint severe perspective specifically indicate long high post thresholde interpretable fall behind finding prove future acknowledgment characterization positive extract symmetric spectrum resp irreducible note symmetric symmetric must irreducible symmetric cf remark apply eigenvector follow fulfil upper ball apply theorem state follow fourth moment spectrum matrix stay fourth jk claim constrained fall spectral function verify r x bound finally minimizer non negative negativity kkt consequently choose claim would q equivalently write order matrix entry hence apply equality yield sequel associate row jk right hand jk jj j thresholding I say triple note converse independence sequel global markov independence graph c u partition formula submatrix negative verify comment virtue non negativity successively ab ab cf ab theorem theorem remark sketch finite precision negative vector estimation precision constrain determinant treat size greatly simplify provide log determinant correlation b random role discriminant confidence interval inverse independence relation miss modelling aim parsimonious conditional independence graph receive considerable finance comparable size development inferential procedure try equivalently case independence purpose procedure base independence regression suggest reference amount penalize likelihood relate regularization scheme enforce propose enforce dimensional address cite consider semidefinite precision definite symmetric e partial correlation precision gaussian positivity precision attractive sub form inference specifically knowledge author precision element dominant equal positive identity likelihood discuss restrict impose curse partial correlation negative gene priori unclear misspecification side constraint establish existence uniqueness maximum likelihood case unconstrained thresholding constrain yet sparsity structure thresholde negative high absence tendency produce approach exploratory analysis sparse discuss application sign log determinant estimation subsequently develop descent convex extensive include summary proof letter letter letter use submatrix index invertible frequently arbitrary submatrix column permutation entry likewise result set denote trace block compose positive whereas symbol I realization precision bregman divergence induce cf henceforth constrain know sign diagonal omit minimizer unclear minimization make difference unless minimizer word unless exist check constrained determinant divergence though employ condition fulfil subsection view extend determinant positive semidefinite cone constitute constraint negative q lagrangian multiplier tucker q note duality derivative equality variable seek diagonal follow definite necessity possible find unbounded multivariate consequence mis specification fact see rather target constraint know mis investigate select symmetric cf diagonal conditioning covariance give hand partition must sub role complement equivalently observation remain unchanged combine variable negative partition regression non mark infer mark mark student achieve mathematics mechanic algebra minimization matrix precision appear adequate one pair exactly suggest increase drop performance divergence minimization behave mis specification substantial bias discuss issue level coincide apart leibler precision accord preserve ideally one maintain partial partial loss example item one section ask least preserve negative question choice q accordingly diagonal verify consider dd observe see partition inverse formula non entry feasible feasibility hadamard preserve sign accord ar process order entry even even odd diagonal equal odd kkt optimality condition p lm satisfie violate recursion stationarity inverse preserve tight appendix positive e guarantee recover set fulfil ar opposite mind simplicity effect result replacement covariance cccc connect estimation seminal interpretable sparsity among penalty approach prominent appear complement minimization penalty desire modification end sign version post processing combine precision cardinality correlation aim finite I scheme minimizer hard let definite perform fit constraint diagonal entry definite improve regard small cardinality sequel successful entail estimate obey wish element still enough depend classic consistency use realization random fourth moment minimizer determinant order less justify cf first observe empirically sample whose constrain divergence unique exist constant consequently single high decay tail gaussian tail identification sense scope though convergence available g proof regularization substantial see explain prefer penalization penalty bias affect yet diagonal thresholding achieve level aim percent keep percent entry absolute result hence common solution solve instance constrain determinant handle slow ten thousand thousand million devise solver graphical analogous recursively regression recursively least regression solver apart ease implementation foundation block coordinate exist establish sharp runtime increase call scheme jj jj jj block optimize keep repeat block criterion satisfied solver exist sequel routine indeed determinant q moment show decompose term drop well problem give definite negative third constraint long linear concave function minimizer kkt lagrangian multiplier substitute kkt resolve automatically jj iterate strictly provide respectively solve write hard handle turn one solver problem experimentally fast exceed thousand existence satisfie suggest support cycle operation principal solve terminate view quantify criterion dataset systematically positivity constraint element specifically cccc accord obtain know keep aside hyperparameter parameter use example encode set adjacency matrix grid adjacency grid grid percent entry previous long away k setup exception setup replication average replication large error spectral leibler kl cc approach include various attractive constrain determinant divergence minimization validation follow quantile small diagonal compute validation fit quantile pick minimized minimizer determinant minimized variant glasso denote thresholding estimator glasso describe via glasso yield manner glasso likewise denote graph conditional graph provide see sufficient try structure node whenever associated regression thresholding apply coefficient regularization regression grid accord compute covariance jk jk obtain jk exceed regard proceed vertex chain star pair series marginal perform significance level conditional independence run fact independence perform setting rather apply dimensional employ drop performance observe grid star optimal sparse figure stage glasso exclude c grid glasso five different trajectory different range reason run report tuning exclude step graph glasso publicly code penalize problem implement approach algorithmic project report reveal sound empirically one degree severe average run glasso hyperparameter measure kkt optimality time figure
iteratively image confidence reduce several mutually spatio temporal confidence recognition use image recognize place problem system give detect place adjust bin stand drawback input continuous call cause place advantage learn unsupervised hypothesis every description quantization translate technique allow parallel place model discretized signature integration auto account dependence step relation unique word main lie visual world represent figure recognition represent robot time place model posteriori recursive encodes account dependency call restrict unchanged node posteriori eq tt simplify place characterization algorithm discrete variable dictionary place give discrete tt nt word estimation word sequence posteriori due unobserve unified gram descriptor divide filter bank filter scale project explain descriptor spatial quantization perform line image neuron word parametrize map categorization task stochastic average time high close vector quantization visual image learn computed step result strategy visual first sub sample image subsample strategy replace word compression rate strategy call strategy simple online carry place database see place recognition sequence acquire human illumination laboratory illumination carry protocol hundred enough new acquire follow five class illumination similarly test perform illumination htbp office c training share uniformly among transition use influence varied laplace set small value laplace give note signature laplace interpolation result could generally several percent see difference bar bar efficient usually except strategy gram speak effect increase bit clear gram performance note drop performance high seem confirm intuition behind laplace right new temporal recognition model dependence gain seem quite high study simple combine sophisticated useful look pt pt language paris sup paris fr inspire field paper filter standard discussion highlight improvement relatively field aim robot high human compatible ease daily environment notably environment compose place correspond house place call
test step variable decomposition treat problem ensure subproblem involve sort solve subproblem supplementary grouping together induced point subdifferential behave handle grouping singleton group sort absolute exclude r join decomposition subproblem proposition reasonable smoothed somewhat addition give control slow correspond prior ideally expect particular encourage degree model hard approach use monte apply applicable address reweighted minimize regularizer encourage term continuous counterpart difficulty objective alternative convex aspect apply double loop em gives monotonically improve true perform regularize method admm glasso inner loop admm give identical array typically micro method dataset http www contain gene tune produce near visualization major connect scale subgraph center relaxation tight gene cluster reweighte produce great computing proximal point algorithm compute proximal input ba plot convergence rate show test clear subgradient converge slowly practical applicability admm slowly problem achieve practice decomposition method converge quickly requirement iteration method subgradient subgradient decomposition dominate sort dominate least solve square run rough ba graph vertex per error reweighte second submodular standard conclusion fall grow structured previous prior make tractable use reconstruction department communication digital prop university mm key determination reconstruct scale formulate sparsity induce function graphical efficiently improvement encourage scale reconstruction graphical undirected graphical independence fit fit context model problem induce likelihood body paper various objective development knowledge encode parameter link sparsity pixel explore recover structure graphical recover free network enforce formulation enforce prior envelope relaxed non pose challenge option optimisation operator experiment produce scale real bioinformatics relaxation superior undirected place graph mean bag natural family assign probability exponential graph depend statistic consider degree parametrization weighting encode distribution correctly choice rest see take form infinite put weight posteriori set property lead non function interpret increase add consider decrease note cardinality modular modular sum submodular concavity restriction ingredient allow enforce tailed place weight node aware novel rise convex envelope precisely cardinality problem weight connect notation sort natural ordering envelope q piece intuitive behave like additional edge problem case q matrix graphical rescale distribution cone boundary psd cone handle interior shown gradient definite optimize differentiable suggests submodular subgradient proximal subgradient simple optimize smooth convex function subgradient due piecewise primal method return intermediate limit superior convergence sparse proximal rely iteration proximal close relaxation solve minimization submodular proximal operator cut function algorithm slow vertex clearly propose optimisation method gradient proximal alternate direction apply multiplier admm optimize number advantage proximal presentation update proximal turn updating criterion practice admm fast guarantee restriction place step size degree
data line save save save file txt specify confidence use confidence use error bar macro mean test confidence know mean confidence well detail allowed level confidence across value calculate trajectory computation enable help performance performance identify name identify line generate external first external model model section performance calculate case classification contain file file test file classified name predict predict file file performance test accordance percentage wrong precision equally auc sensitivity specificity specificity tp class rate class roc auc area curve avg test cross avg test variance test file file micro macro comparison file square average comparison file comparison matrix lf lf lf lf lf lf lf lf lf lf lf lf lf lf lf indicate lf indicate well indistinguishable mean statistically different statistically accuracy big average know tail know statistically tail account read contain cross performance learn store roc curve name curve curve micro average average curve interval average confidence name recall curves micro macro average vertical confidence interval lift chart name curve lift class case cross micro macro vertical averaging interval confidence file file supervise test file contain name course correspondence name name true predict txt txt txt predict txt class file performance contain rand time time manually see performance file file learn file section provide example execute follow source code create test call library copy section set classification log set parent evaluate performance store file column name name execute validation specify validation specify validation txt specify file extension specify class name specify section eventually enable specify name classification performance fold load pre partition describe line execute nb txt rely two cv enable validation nb result txt specifies file file learn set specify partitioning partitioning replicate execute test txt file model compare nb file section synthetic trajectory contain assume make cluster use learn maximize maximum parent store file name execute cluster difference name specify name replicate em especially soft assignment scoring inference static method performance supervise front follow part analyze figure diagram represent interface define trajectory trajectory stand alone implementation class potentially specify implementation trajectory trajectory vc vb vb vb vc trajectory instant theoretically happen code column string string add time string vb stre vb new string vb vc string vc class management set accordance partition file classify class line parent set consider node probability line previous line file format define previously bn class n n class parent specify complete parent parent iterate parent network simplify interface define abstract implementation method implementation implement parameter specifie parameter implement search optimization search algorithm return interface implement see diagram search structural scoring define method score rely conditional likelihood maximization select scoring interface interface simple definition evaluate interface define local good neighbor I local search use generate individual interface code interface interface generate marginal score maximization individual maximization model new classifier boolean false algorithm I string string put tx prior learn alg alg collection double alg structural use marginal scoring learn string object put put hill false start ia ib b ic boolean false ia ib ic true definition structural string alg string structural alg depict simplify algorithm extend interface abstract implement implement assignment purpose algorithm rely interface define criterion em provide stop example assignment put tx px put false definition object string learn algorithm order calculate code interface define define classification class implement trajectory classify double double usually classification probable criterion implement interface unbalanced order depict calculate performance execute implement implement implement execute performance class implement performance class implement implement test approach double test use validation validate performance detail evaluate performance section hierarchy class provided generate simplified diagram performance depict figure argument performance interface generic aggregate provide definition provide section run performance interface cluster aggregate micro average micro performance run performance implement micro averaging implement micro macro average performance performance unsupervise provide class possibility calculate micro macro averaging performance provide depict diagram performance interface define run aggregate interface single micro macro average aggregate classification provide run depict simplified diagram interface define implement generate model test lambda double generate interface provide test test implement provide use provide execute return result interface performance generalization double double execution double nb performance string addition provide test performance front end front class due start implement develop array information necessary I private help specify validation cv fold validation line column I help enable analyze call public string pass link line link list parameter add code use program rely loading algorithm loading provide generation execution paper temporal stand alone performance replicate test description give library many development class one possible future possibility parent static currently direction focus possibility partially observable sl section corollary definition time classification streaming duration open stand library implement continuous classifier marginal score introduce include understand library help extension contribution develop paper valuable suggestion matlab make correctness relevance source stream engineering reference image video science system social stream trading offer analyze streaming model evolution analyze stream monitoring time diagnosis study fire streaming among hmms receive dependency suffer limitation discretize datum poorly represent rapidly point dependencie necessary multiple increase time ct cascade conditional ct cascade devote require parametric dependency significantly problem selection ct cascade overcome affect future event homogeneous markov stream continuous classification stream stream trajectory recognition purpose stroke stream period temporal address continuous include latent continuous continuous classifier discretization describe source library stand line interface purpose prototype problem score score marginal conditional expectation validation extend extended guide stand alone usage provide explain library section read cluster usage implementation possible step system time several slice slice increment always propagate consist evolve space become intractable continuous overcome temporal network exploit continuous probabilistic whose evolve continuously finite domain bayesian component direct cyclic node parent intensity intensity parent probability occur drug introduce indicate person depend yes one empty variable fully specify text center node style circle empty hour person empty empty stop empty minute hour state empty state transition state e continuous network variable inference variable continuous time network continuous marginal hold associate node fully depend part style align center style em ei si full edge pt pt right pt right pt leave naive bayes effort continuous naive bayes classifier bayesian address learn continuous max parent formally naive max max node attribute take data structure marginal learning account count q x x necessity static count consist run local optimal maximize attribute search possible conditional likelihood especially value scoring log obtain learn describe stream contiguous interval continuous q associate interval parent interval interval interval consist attribute fully evidence maximum classification stream py n nx rely formula calculated expectation sum contribution statistic occurrence trajectory contribution occurrence count without set contribution count statistic time attribute assignment step hard trajectory statistic calculate account probable maximization stand alone library read file gamma step file website library test site stand alone show section file possible paper collection free source code accordance file datum parameter address case file section section hereafter term path library file library suppose test make virtual virtual increase avoid increase dimension gb argument specify table summarize separately yes yes yes yes yes yes yes yes yes yes yes yes confidence argument pt p cm cm cm p cm vs x x confidence provide help ignore help test follow naive bayesian parent learn scoring stand model hyperparameter count relate transition default spend datum big error big learn structural add dimension ignore apply example penalty enable count max default maximize default enable penalty model file soon format see file model datum possible vs must validation test validation clustering activate validation random validation cv cross fold default hold parameter fold cross validation cv cv section cluster require complete set measure rand coefficient expectation soft cluster I default trajectory trajectory change default default threshold soft soft threshold hard threshold
capacity image recognition popularity current object benchmark support tailor image experiment continue grow rely ensure inform second thm pt em author author token university vector task part explore interaction svm connect extraction thing induce capacity svm pixel preserve locality demonstrate surprising expression recognition detector improve decade largely svms train histogram gradient feature upon high page visual classification layer complexity view weight margin svm add capacity pixel possible perform classifier filter take remove sensitivity figure type successful change try remove leave paper choose square choice great flexibility show becomes refer convolutional visual write input edge hadamard operator pointwise sparse operation bank orient edge response descriptor q normalization address piece show previously descriptor form selection convolutional bank projection bank eq q feature affine quadratic interaction ii prior form affine projection kronecker expansion weight unary induce prior quantify weighting therefore prior svm pixel interaction discriminate distribution however deal distribution covariance often stationarity translate pixel fall quickly improve conditioning interaction order set unary redundant redundancy stem pixel account compact impulse encode spectrum natural train pixel information one preserve illustration result pixel fail discriminate information separate distribution overlap local discriminate spectra image contain structure contour experiment show inherently interaction pixel exploit encoding capacity separate noise sample spectrum train distinguish discriminate pixel perfectly separate two svms pixel encode affine good however prior simply reflect belief absence actual inform decision assumption may detection comprise possible pose gender identity background change appearance detector intra class detector unnecessary sort require perturbation classifier learn specialized aim answer geometric reproduce effect network heavily learn number primitive capacity let learn could amount broad expression sequence neutral formation task discard different canonical pose error broad visual recognition recognition pose face heavily contain condition label ground flexibility amount geometric control identity example coverage accord aggregation quadratic pixel error collect region pixel face degree geometric introduce pixel pixel far normalization extraction component traditionally receive much attention literature play invariance contrast normalization instead normalize expression experiment storage requirement local quadratic local quadratic take implement parallel dual solver server use core ram depend grid search come converge towards variation span green amount
difference correlation sense need sample study adaptive set access correspond need factor instance correlation decision policy figure constant eq complete proof optimal policy error let analyze sr successive description sr round round next round sr arm leave output arm verify sr exceed budget sample use sr trivial ti ks sr universal constant assume event event occur rest organize two sr subset suboptimal last r k aa optimal k know sr n k kk kk k n se elimination description se dynamic sr se exist arm rather round sr se arms suboptimal se find set sample ta u ta probability least return eq event inequality enough event se terminate claim fact separately follow optimal beginning eliminate round suboptimal eliminate b sample r moreover arm eliminate thus arm complete work towards hardness correlate sampling sr se result known argument literature term suboptimal problem equivalent arm correlation perfectly correlate se challenge design general observed identification successive allow optimal unfortunately applied correlated identification accept early arm mutually arm identification remain arm impossible summarize sample chi random variable two absolutely value chapter recall distributions eq application r positive strictly constant correlation describe standard define two center conditioning distribute gaussian h w divergence h h department research financial university edu sampling strategy rely wide integer use correspond probability model make arm select observe value uniform agent subset arm solution subset furthermore suboptimal devise reliably realize outcome adaptive budget give ask subset soon devise return want explicit limitation stop regardless evaluate terminate expectation algorithms sr correlation call difference argue easy close feature classical attain difference base arm arm ir satisfy arm need precisely adaptive level idea suboptimal need budget adaptive way estimate fix set correlation exclude suboptimal arm component focuses correlate accurately subset mutual importantly contrary adapt heterogeneity describe correlation go see use allow largely wide correlation sx
thank constructive comment manuscript support centre national commonly pair correlate parametric test useful uncertainty certainly one clinical method use coefficient resample create multiple original method assume representative concern size however principal difference though uncertainty distance clinical clinical intrinsic distribution point course count attempt uncertainty carlo analyse importantly latter two exploit uncertainty consist individual coefficient sample rank eq coefficient test score score z calculate infinity go bootstrap resample create consist statistical new randomly g may assign set use probability two quantity simple correlation coefficient calculate width e deviation score also resample method obviously account uncertainty likely pair perturbation create new perturb randomly gaussian center uncertainty composite language real ray indice optical band magnitude apparent significance basic distribution plot may clearly simplify nonetheless uncertainty magnitude significance correlation take apply perturbation composite one find return wide value green fact well significance histogram plot composite plot method bootstrap test method general serious heavily datum point distribution perturbation composite size uncertainty regard distribution clearly uncertainty account uncertainty tend return composite resample understand difference composite distribution estimate sample uncertainty lack estimate correlation coefficient uncertainty population whole
scenario simultaneously take advantage inter competition study segmentation pixel object convolutional design operation spatial output weakly supervise predict output propagation cast suggest replace fully layer train pre train level experiment without last layer weight initialization fine act baseline quickly converge semantic background class classification none weight prediction produce correspond identify max coarse maximally score negative compete object set heat ignoring non maximally score key background simultaneous inter help refine intra pixel time image segmentation challenge augment held intersection percentage segmentation mask mask union classifier fine tune supervision common degenerate solution predict train momentum quick network converge iteration achieve relative baseline mean mean classifier b b output show quantitative union baseline preliminary encourage propose novel joint inspire network kind merely field super refine grouping could likewise segment instead convnet map encourage segmentation mining berkeley edu pt reduce annotation segmentation degree formulation fully seek learn weak train jointly pixel label convolutional input need offer exploit supervision evaluate preliminary challenge convolutional performance supervision progress segmentation annotate consuming collect supervision improve infer train supervision proposal problem max margin representation sensitivity
furthermore fully induction suppose true integer lemma connect either maximal adding maximal add new maximal clique connect gm gm maximal clique define find find span maximal spanning include leave end maximal weight span clique right everything interested opposed complexity reasonable come elimination consider gm procedure eq consider potential case leave already step proportional message correct exact know problem bp gm subsequent recover use order algorithm exact property group gm gm possible guarantee subsequent operation bp tractable operation per bp need graph message message graph last inequality prove unfortunately thing gradient like ensure stay version size q three case give entire hide variable interested maximum ml achieve binomial parent look set order ml amount index write use triangle size write write therefore quantity calculate learn q maximization possible direct likelihood read thus I liu rely undirecte exponential family denote observe write use descent formula gradient expectation question link penalize introduce highly trivial graphical day last distinguish stand variable parameter iterate parameter proposition france observation realization want end error equivalent call compute calculate much knowledge inequality make gm constitute tool allow solve alphabet joint eq q represent factorize affect represent conditioning factorize obtain graphical sense graphical correspond model factorize contain parent edge chapter realization type undirected gm represent edge dependence gm undirected conditional lead word remove subgraph link edge maximal clique mrf q clique equivalent n I equivalent prove two claim therefore show stand side take bayes rewrite hand depend value equal know write mrf way maximal q nothing underlie example factor pixel probability q pixel natural unlikely piecewise crowd use human difficult verify evaluate crowd assign worker collect answer worker probability reliable infer task worker ia conditional distribution know answer distribution consist hard core instance problem hardness maximize become quickly intractable exploit gm reduce connectivity gm number formalize elimination make graph gm fig colored subgraph decomposition gm calculate marginal require
function widely mathematic statistic provide brief definition description divide order end restriction continuous spline spline basis spline one interval nonzero spline knot approximated combination work split correspond spline product spline column vector number tensor spline maintain nice modeling every j k spline product spline smoothness approximated lemma power restriction later function j every element assume sufficiently satisfy fy kx pair observation interval apply affine otherwise co conditional density neither index conditional oracle knowledge prior covariate spline j kk component affect inclusion assign indicator construct put model let mass every prior value support active index term basis spline stand independently coefficient r j j x rx j identical induce coefficient positive prior include truncate case assume decay hold e index next remain necessary default independence obviously isotropic well zero truncate poisson default simplex contraction long polynomially fast result allow covariate sample stand know hellinger stand density true density say rate fix n distribution q establish contraction lie class respect minimax conditional situation take rate compare grow polynomially coincide additional logarithmic grow obtain rate hence variable procedure smoothness trivially fix involve contraction necessarily recover predictor scope contraction determine assign mass complexity ambient dimension contraction entire consequence function condition relax qualitatively different following propose smoothness likelihood kx fy view form denominator involve integral power whose differ co collection conjugacy dirichlet write certain take consideration let enter moment kernel jointly covariate spline since spline basis interval calculation term take histogram property smoothness save computational exchange smoothness choose haar prior range bernoulli marginal truncate poisson random integer part restrict calculation subsection generate randomly sum least square develop select variable replication compare directly sum l directly carry sensitivity choose covariate probability fall range max dominate lebesgue density hellinger respect contraction standard rewrite exist conditional call kf f fy variation j fix choose number cover k r j r fact concentration around restrict every simply h f combine regard f view distribution within vary n suffice determine sufficiently theorem outline main difference approximation result calculation concentration change calculation concentration j sequence verify stand number k product tail cut clearly hence requirement meet isotropic calculation calculation entropy eq k define j great suffice since look identical isotropic proof mathematically express hellinger joint density dominate lebesgue leibler rest argument way isotropic tensor spline lemma multivariate c fy kx particular b kx j kx analog theorem tensor spline expansion multiple norm form consist dual univariate spline uniformly bound relation define relation boundedness exactly lemma interestingly usually require related discussion model let function shape change possibly observation predictor construct product incorporate issue posterior adaptively level also degree smoothness predictor technique calculate moment chain monte carlo large sometimes receive attention scientific genome association focus literature approach obtain distribution exist assigning prior mixture generalize break transform gaussian multivariate generalization beta generalized possess conjugacy modern datum many literature subset penalization least screen sis establish include linear gain popularity variable efficient compare uncertainty prediction accomplish assign model combine extend allow sub contraction often require allow result largely bayesian recovery trivial covariate restrict isometry problem framework analogous
explore cnn particular convolutional much convolutional cnn order method layer scalar quantization quantization apply scalar well method structure quantization give additional gain explore redundancy parameter knowledge systematically quantization parameter make contribution systematically explore quantization cnn storage comprehensive quantization quantization product significantly perform ability compress deep convolutional object image great area art already human great interest adopt scene cnn hundred million huge storage publish cnn explore property cpu speed execution boost operation efficiently work explore matrix little little devote cnn vector quantization cnn parameter show accurately parameter neural parameterize redundancy compression realization prediction surprisingly decrease performance confirm finding dense connected factorization widely cnn parameter consider dense connect orthogonal reconstruct eq correspond singular svd control optimal frobenius approximated matrix matrix compression turn neuron turn geometric view hyperplane round neuron scalar scalar scalar codebook form cluster look reconstruct need store index codebook center bit encode use center need bit per cluster index assume assume codebook despite surprisingly parameter structure quantization explore structure many quantization assumption subspace redundant perform quantization wise several submatrix denote th codebook store thus reconstruct x particular quantization codebook negligible quantization quantization structure quantization basic center example every represent compute vector reconstruct center need store potentially compression different quantization km capture redundancy explore structure try explore global vector hashing among store rotation paper output filter grouping filter grouping dimension filter investigation find explore group default image object contain convolutional dense patch feed layer respective filter convolutional pooling relu network epoch start every epoch momentum accuracy accuracy goal achieve compression perform compression several cluster bit segment column change compression rate mention axis case segment align compression accuracy center size able obtain small method usually low quantization codebook account compression rate center always codebook example center e codebook rate result slightly improvement make codebook size next center bit balance quantization herein error tune segment compression rate km compression svd vary achieve center iteration lower result layer mainly two factorize still store somewhat surprisingly despite achieve km keep quantization far improve km high compression codebook size big compression simple work reasonably compression give km goal km km compression suggest considerable redundancy neuron poorly classification fixing layer report layer layer lead especially rate result present application compress retrieval verify generalization compress server number server limit able process image server compress compressed perform retrieval database process retrieval precision compress cnn activation layer retrieval cosine trend consistently work one surprising center high original special application robust reconstruct mean approximation report ccccc compression svd km center km center km center center storage apply quantization save unlike approach factorization method find simply quantization mean able obtain structured quantization method able address apply cnn embed device
remark paper new failure ps call failure power failure rate poisson distribution failure binomial power eps class ep special ability cover five hazard I unimodal shape several em algorithm discuss class distribution fit distribution real linear failure moment many introduce discrete continuous random say component exponential ep discrete distribution see family eps series complementary series distribution series combine propose special distribution introduce new compound series produce I new due representation system appear apply cancer activation failure model class iv class parametric monotone unimodal decrease failure common outline special expansion section estimation em algorithm conclude failure function cdf parameter discrete power q cdf survival hazard rate consider density hazard give cdf cdf number density know nx x x hazard eq use generate e dx tx dx te e dx dx dx te bn bn normal distribution therefore xt nc nc bn calculation xt incomplete case distribution distribution become hazard either increase function contain special exponential geometric exponential poisson distribution density power eps special geometric power series hazard hazard decrease put eeg monotonically density decrease unimodal hazard increase fulfil interior boundary remain valid replace use hazard element element binomial mle lie interval root k iv nx rhs root proof denote rhs expression true appendix call step tool handle datum joint cycle conditional likelihood n obtain therefore estimation root equation unique appendix unique distribution estimator observe bx bx b bx nc taking give bx bx bx c nc ic ic ax move v observe information square simulation give theorem calculate standard restriction absolute firstly em distribution value assess determined correspond information shown suggest consistently iv standard error sample standard observe close simulate ccc ccc bias sim std sim ccc ccc ccc std std sim ccc ccc ccc c sim sim c ccc ccc ccc ccc sim std sim em demonstrate datum study represent health state fit exponential
vector fact easy see existence recall interested determine guarantee nevertheless suppose iff corollary subspace precisely come immediately corollary observe iff row infinitely subspace e contain linearly row state subspace imply even subspace us setup one subspace many subspace h thus explicit instance easy despite column subspace want know dimensional subspace lie additional suppose surely h want guarantee column indeed lie dimensional want true condition condition guarantee question document analysis column lie keep relation surely iff comparable converse previous different subspace converse suffice correspond dimensional subspace tell column subspace conversely tell observe unless element show definition arbitrary conclude conversely size iff size tell belong subspace remain determine contain basis characterization come lemma contain subspace assume infinitely many subspace imply might even converse come direct converse satisfie exist key behind idea subspace must onto lie vector determine subspace dimensional contain lie idea give constrain since must focus block block eq even position elsewhere exactly always describe arbitrary entry function entry make linearly linearly orthogonal onto essentially projection simultaneously would project plane constrain determine contain one accord condition obtain would ia statement force minimal assumption confirm entry whose answer question concept subspace understand hyperplane hyperplane hyperplane align canonical zero entry affect able subspace plane could nothing plane little show low subspace infinitely subspace always trivial generalize straightforward iterate happen drop would matter subspace generic subspace section dimension would identify almost extremely unlikely exactly converse state almost surely projection dimensional namely aligned sense suppose z course measure zero set subspace lie align general course distinct hyperplane intersection e occur projection summarize incomplete lie fit partially incomplete vector sense tool derive consequence already brief interesting relate research incomplete behave complete one form uniquely determined precise say document incomplete validate really want say identify solely way would exactly subspace lie identify incomplete insight sufficient deterministic conjecture em really contain set set vector characterize concrete manner characteristic mention tool area corollary example corollary infinitely subspace describe relation hope combinatorial reconstruct like reconstruct arbitrary projection describe variable involve equation support support equation believe dimension algebraic ambient element index distinct subspace general belong row subset size subspace canonical restriction arbitrary lie onto lie union subspace arbitrary arbitrary arbitrary incomplete incomplete version incomplete incomplete entry draw draw circle font cm font electrical usa guarantee partially union really lie subspace deterministic sufficient certain partially unique characterize incomplete try try fit look really really subspace subspace collection indeed validate generic iff lie become arbitrarily subspace even counterpart really follow vector lie suppose vector span counterpart lie course without know arbitrary precisely e rest union appear task lie subspace subspace setup far counterpart incomplete subspace subspace counterpart subspace vector want make complete counterpart complete imagine also counterpart lie subspace way directly imply counterpart imply nice extra incomplete could set extra generic behave allow counterpart lie answer yes characterize incomplete one strict subset vector observe row characterization determine incomplete really lie verify row first counterpart counterpart indeed subspace essential subspace incomplete satisfied fail would complete evident fail satisfy dimensional subspace belong could happen word precisely need discover incomplete substantially vector automatically lie incomplete case generic position generic determine generic exactly deterministic condition guarantee indeed lie sufficient sense satisfie indeed lie conversely exist condition unique lie element lie uniqueness give essentially result document formally characterize incomplete allow final goal indeed motivation particularly consequence detailed exposition document result intuitive result generalization brief future symbol statement etc document alternatively index al end arrival challenge information quickly resource fortunately simplify thing achieve big likely incomplete fortunately subspace handle subspace give infer miss handling attract attention recent year detect incomplete fit even miss converse useful bold something sure reach conclusion conclusion correct chance outlier find something something many subspace fit datum really precisely task certain incomplete really lie subspace whenever subtle match missing determine fit using already know really lie exploit subspace fit subspace drop incomplete want really lie certain really lie subspace whenever subspace fit characterize problem answer motivation essentially completeness mention motivate give lie dimensional subset trivially subspace fit converse explanation immediately converse rank lie subspace dimensional generic condition completion algorithm detect fit provide check fit fit generic say already belong subspace require alone extend reason stop rank matrix provide check perform fit em answer important open complexity one see identify false circumstance datum subspace want know minimum iteratively minimal enough give subspace denote subscript index unless state correspondence keep get subscript unless run setup easy belong make collection set subset resp unless intuitively contain set convenience rather typically room confusion think row entry incomplete denote specify index room confusion equivalently I entry depend specify writing shorthand shorthand similarly example thing simplify greatly degenerate degenerate subspace iff form rank every subspace unless state example iff equivalently iff fit intuitively generic formally every fit iff notice would vector easy fit iff shorthand define say iff define iff every informally say dimensional generic formally every particular entry position span setup formally essential satisfy section unify mostly lie give assumption determine say subspace simplify generalize dimension could arbitrarily simplify easily generalize emphasize fortunately measure hold without lie precisely subspace belong hard achieve convenience simplify argument analysis working assumption notice nothing uniformly result simplicity notation fully unobserved always onto e easily generalize alternatively define say nothing row exist assumption motivation subspace fit certain determine really lie subspace non really lie fit really lie precise present lie ready advantage intuitive emphasize simple usage goal really lie subspace set incomplete characterization formalize kind dimensional subspace subspace subspace intuitive interpretation set precisely property converse satisfy guarantee generic intuitive em row one observation least example converse statement column belong none subspace determine dimensional essential uniqueness subspace iff exist notice requirement strong requirement satisfie know little case subspace guarantee subspace difference subspace essential far prove goal intersection subspace contain contain dimensional subspace clearly see satisfy iff study every hyperplane characterize orthogonal non easy orthogonal scalar say entry position zero elsewhere hyperplane characterize suppose zero orthogonal next precisely elsewhere
constant weight neighborhood necessity describe note property take strict strict necessity hold derivative constant h ie case xx consequence characterization strict prove fashion strictly fix slope interval convex suppose trivially reasoning bc leave derivative right thm thm corollary thm thm thm lemma proposition state university year function sphere multiclass clustering moreover modification un cluster variant geometrically interpret recover sufficient recovery hide convexity sphere convex optimization simplex spectral segmentation partition class base similarity important vast array recognition bioinformatics compression include well practical aspect refer eigenvector widely cluster simplicity simple spectral partition straightforward thresholding eigenvector laplacian matrix however hierarchical split use approach procedure spectral process construct look embed sometimes rescale clustered conventional spectral eigenvector matrix interpretation meaning explain relaxation cut map machine mathematic geometry space datum approach result embed justification exist analysis minimum actual output local propose second cluster point ideal perfectly separate spectral embedding use asymmetric weighted recover optimization still without modification broad overview identify sphere derive characterization describe admissible turn function hide convexity sufficient specifically sphere correspond function ica guarantee algorithm connection result think use structured weight briefly cluster spectral term theoretical discuss technical sphere correspond description necessary condition space thought unit weight description recover local thought simplex everywhere else continuously ref derivative origin strictly twice continuously differentiable local vertex maximum necessity let twice differentiable construct integer positive vertex canonical direction local geometric direct spectral work ideal vertex normalize something let embed exist f simplex recovery cluster spectral happen local maxima simplex focus cluster text arise spectral cluster denote vertex two vertex cluster set cluster consist whenever convenience vertex index index index take diagonal component cluster practice rarely consist truly typically entry row nearly diagonal simple suffice perturbation introduction simplify proceeding notation column vector euclidean vector product space angle angle domain denote projection vertex unnormalize laplacian help light importance laplacian definite equivalently consist connected index orthonormal possible choice class v invariant basis extend orthogonal simplex separating contain laplacian contain basis nj nz jx believe tp nx x j jx coincide belong ni z ti demonstrate unnormalized graph map perturbation similarity interpretation consist eigenvector low perturbation correspond perturbation row interpretation normalize version spectral cluster isolated define place generate eigenvector propose proposition applicable perturb similarity way cut vertex vertex empty minimize partition cluster cluster make plausible cut simply isolate size variant min min minimize cost alternatively way cut minimize cut see section whereas arise reference interpretation perturbation insight cluster cluster normalization arise walk interpret state eigenvector formally equivalent analysis nearly reference cluster set truly belong distinct proposition map aggregate simplex connection continue simplex vertex mutually orthogonal suffice simplex sign embed point line approach embed sphere term contrast equivalently random vertex simplex recovery turn form property recover simplex satisfy equation complete enumeration maxima maxima strict embed use clustering contain isolated construct contain scale orthogonal basis nz x spectral clustering let g complete enumeration local maxima simplify exposition orthonormal basis simplicity condition series structure induce substitution map simplex let extreme point polytope extreme strictly point translate everything make let strictly maxima simplex immediately u ti lemma u f hence correspondence relative particular relative strictly convex strict maxima symmetry enough understand behavior around take strict convexity piece upper pick slope piece slope piece piece strict convexity precise mh piece slope strict piece n strictly convex let w strict local show strict strict immediately main theorem give strict maxima besides lemma fail f sphere distinguish power cluster distinguish power spectral comes intuitively choose increase distinguish follow q asymptotically one become magnitude rapidly expect g origin gradually growth contrast distinguish power convex perturbation need new class spectral contain define either component eigenvalue hand choose u discussion maxima symmetric however origin equivalence maxima cluster vertex place local maxima belonging class maxima fu fu fu pt ascent ascent expect ascent perform unit expand f u order new find constraint imply orthogonal simplex center find maxima point second control cluster center candidate x u c pt loop run mn loop projection speed pairwise inner slow exceed discuss point factor objective make couple nice find always optimization landscape base fully deterministic algorithm however choose respect toy example point circle uniformly radius circle radius circle circle scalar multiplicative display similarity construct ij p j set convention inter similarity color color embed sphere maxima ray go occur opposite direction symmetry depict black white high similarity class display similarity class circle high similarity intra similarity two embed encode desire simplex maxima color embed local cosine please spectral cluster image spectral cluster image divide represent distinct various region result promising exploitation pixel label similarity pixel
around proposal distribution calculate ratio accept candidate code rough identify central pick sub randomly pick converge randomly pick sub sub region generate candidate rr rand rr rand strictly estimate note proportional evidence point locate simple toy two pair distribution bivariate two mode integrate entire simplicity keep toy concentrate validity assume given calculate contour fraction volume posterior locate provide reference credible region correspond cumulative toy calculation take diagonal matrix proposal probabilistic region identity toy model theoretically credible sample sampling green colour sorting mode two already sub density globally markovian detailed balance requirement fulfil concept enable generate mcmc method mode however require rough information rapid identify different form reversible jump thus odd efficiency modal global chain accept multiple obtain rough directly k mean apply leave density autocorrelation mixed method toy well separate efficacy sample pick belong credible region excellent toy toy mcmc practically investigate individually calculation toy toy mode ratio q write peak value posterior r generality high value aim valid long c big mode third furthermore keep algorithm sampling modal become markovian long explore global propose novel variant allow candidate markovian apply demonstrate increasingly range datum problem popularity ready availability advance analysis methodology set complexity dimensionality main posterior desire parameter dense grid medium high burden problem high implement method efficiently tailor explore distribution dimensional generally long sufficiently complicated inefficient sampler posterior unable isolate mode work novel deal mcmc explore detailed posterior surface retain enable communication different sampler jump require structure introduce discuss main property principle state space point already randomly specify solely previous candidate point accept therefore algorithm symmetric necessary relax number sampling sample chain trivially guarantee understand requirement I particle jump state strong requirement markovian chain proposal properly mcmc sampler become mode whole statistical bias modal thus motivate really efficiently novel modal posterior local maxima peak well global note require mode sample rough information guide absence
computationally demand sg new solving generate include perform form sample set follow relate via thing stand random standard quantity class combination stable noise symmetric sg transform symmetric sg purpose sg model sg model dispersion skewness random lemma position dispersion version processing literature design tail penalize dispersion select maximize acyclic graph criterion stable graphical contribution ie random variable parent separately network coefficient sample get therefore asymptotically generate sg stable graphical network transformation skew stable sg sg identical skeleton represent sg consider x z sg accord network structure sg term variable structure detail sg lemma initialize initialize order search db first direct acyclic graph every combination j dispersion dispersion minimum estimate dispersion constant p tb node parent tolerance co initialize ols co initialize buffer matrix regression regression vector change tolerance square briefly repeatedly solve weight achieve successive attractive rigorous guarantee several software package available least square though section manuscript stable implement numerical ignore constant term candidate estimate structure lemma shape sg initialize parent add pa pa fs parent add repeat optimum shape sg sg initialize order optimum delta score find accept swap update delta score search structure popular hill algorithms acyclic start order parent part subroutine search hill search add family least least ol explore order elementary swap score local optimum stable search numerical assess five topology simulate stable datum project microarray magnitude positive independently well standard consistently observe topology appendix assess infer performance ols show box basic estimate low estimate tendency dispersion stable describe microarray profile multivariate sg aim compare ol result tail sg quantify expression task popular method detect differentially usually profile gene assumption quantification observe quantify differential expression sg quantify within exist tailed gene ccccc gene usa china usa usa pre process eight group provide table intensity microarray quantile normalize original microarray intensity measure learn probe intensity intensity probe obtain transform ie standard technique gene expression affect center intensity assign probe decrease order stable transform select center log intensity rank stable de quantification estimate bootstrap replicate diagnostic assess heavy nature intensity plot ol network quantify gene center rank ol goodness fractional co efficient edge parent correspond negative test heavy b average fold ol empty also assess treat quantification contain sample curve ol finally discuss quantify differential sg model cross de sg change negative likelihood test training lemma guarantee equation report dispersion noise probe heat c population change coefficient sg log expression validate heavy effect learn ol recommend diagnostic assess applicability ol graphical quantify sg wide biology next sequence rna seq sg dna seq dna measurement seq seq finally mention model beyond biology particular processing instance relate region image fmri series brain network stable sg image skewed heavy financial promise lemma molecular molecular density unbounded variance generalization skew heavy phenomenon stable graphical sg represent encode major extensive lack density base learn demand theoretically tractable structure dataset five topology ol also apply stable microarray gene expression belong global group stable improve ol quantify expression gene expression model phenomena justification generalization previously bivariate sg multivariate stable density direct acyclic graph dag topology density establish criterion topology empirically sg improve parameter linear tailed noise motivation computational biology profile expression involve microarray intensity show describe intensity assume stable evidence model quantify differential microarray datum belong iii project rest stable section introduce sg challenging criterion dispersion sg symmetric density furthermore establish sg symmetric sg identical topology theoretical efficient combine implement microarray develop sg introduction model variable direct acyclic set parameter parent symbol acyclic factorization parent appropriately family normal use density primary motivation stable limit limit sum stable density implicitly specify characteristic characteristic exponent skew close analytical stable gaussian large univariate
effect due popularity parameter effect income point correspond quantify tendency edge normalize occur configuration edge pair edge make convenience observe configuration pair treat satisfied network hypergraph parameter hypergraph determine correspond e e e describe move dependent hypergraph balanced definition balanced reduce walk bipartite vertex precisely consider j nh q balanced balanced become balanced operation balanced balanced balanced observation balance observable observable balanced edge ss moves model generate move observable write graph skeleton e applicable close several need applicable balanced edge dyadic configuration return edge remove output generate coin remove pair correspond walk select move walk remove move walk output random pair choose great edge head simple subset random composition great one composition length edge head tail trivial move perform graph simple return illustrate type move move complete dyadic trivial move would stay place affect time many move return metropolis mix returning depend one research try reduce move combinatorial move applicable edge f sg return move edge walk lift analyze fashion statement state lift contain walk return step odd entirely contain similar observable edge random move combination primitive closed walk step walk connect tail choose ordering output dependent primitive walk walk share make part great edge q section take observed implement scale step choose chi statistic goodness fit make package graph produce intersection linear bad algorithm run goodness fit dataset reported burn choice chi mle distribution check explore sample undirecte depict enumeration consider direct edge undirected b b discovered start point discover discover step uniform th tv step histogram graph step sufficient purpose chi square convergence node provide distribution chi reach approximately burn collect survey vertex vertex send mobile depict individual social actor effect nan chain running return model edge indeed would show histogram chi statistic p number convergence paper specific work highly sophisticated rely source extensively fit motivated heuristic procedure goodness fit direct test depict also vertex web direct specie storage occur value significance set would reject histogram mid two year relation model study fourth social edge direct represent b perhaps seem remarkably chi walk explore broadly discover network analyze direct heuristic goodness fit direct network metropolis hasting distribution present infeasible contingency always move basis notable e dynamically hope ever utilize main motivation move combination contingency orient provide dynamic exploration rely move could move balanced edge hypergraph model algebra algebraic applicable move hypergraph situation hypergraph entire dependent derive relation hypergraph implement dynamically hope hypergraph dynamic goodness exponential acknowledgement grateful project support fellowship nsf award dms acknowledge grant air force office research advanced project nsf theorem conjecture theorem rgb rgb institute university technology significant testing theoretically justify goodness fit basis connect model entire arise algebra structure arise individual web representation amenable categorical particular bring literature access remain goodness testing count comparison observe author systematic compare structural statistic fit recently review various modeling fitting remain linear exponential goodness difficult asymptotic approximation test network realization combinatorial enumeration determine distribution enumeration goodness privacy basis sampling move random visit hasting carry argue goodness furthermore log equip unique markov basis literature two remain make challenge broadly address remainder manuscript challenge markov connect highly object unfortunately fast algorithm move arbitrary fast even hour less structural markov family end compound guarantee move connectivity move minimal basis inclusion second model goodness efficiently namely markov basis common take move connect entire handle suggest move ahead time strategy beta basic generalization os degree cast contingency table commonly marginal focus linear move sequential adjustment appear importance sis contrast exploiting allow extend sufficient necessarily marginal test statistic marginal presence methodology obtain part basis issue algebraic combinatorial network ensure connectivity sampling basis ingredient move dynamic fashion table walk way hasting implement whose desire illustrate methodology dynamically generate markov basis specifically receive link edge derive structural result remarkably ti currently fast software capable basis fit feasible traditional metropolis hasting implementation familiar network chi burn vertical denote chi dataset indicate fit histogram web value move chi square chi dependent organize dynamically mathematical develop network find section study family time quick network walk explore little move due walk minimal suggest basis hypergraph assume crucial hypergraph encode view edge suppose edge recorded hypergraph hypergraph vertex computation independence cf cell table column therefore equal multi hypergraph edge vertex vertice another edge color blue collection vertex appear red show move recall observe hypergraph degree vector equal connect hypergraph set constitute move study convenience notation cover abuse edge move connect edge need record simply add remove mention connect realization constraint presence bound zero move produce marginal arise world structural relation put restriction network begin allow edge per model clearly introduce another running walk observable occur basis realization reasonable expect pass cell usual entry statistic integer cell structural zero observable contingency hypergraph edge b appear arise connect observable literature distance contingency basis connect constraint move structural zero suffice connect observable case constraint rely move arise call element reader move never equal move minimal difficult another move
axis cs none fill sep axis cs ia accuracy ex art achieve step choose quantization insight sequential copy one symbol stream check know sum stream compare generate stream summation stream model stream stream generator symbol stream generate anti stream unique since technique compare hidden realization uniqueness anti stream problematic mis synchronization applicability show dynamic valid section distinct model identical estimating stream specify necessarily frequency behavior carry selective stream stream ignore read match selective distance source stream bring stream close font font axis style thick corner axis scale width height gray xlabel read ylabel align center yshift ylabel title eeg ylabel yshift xlabel style yshift title yshift font corner scale axis top color gray color xlabel symbol read xshift ylabel align yshift ylabel error name east xshift west xshift ylabel title sound xlabel yshift font thick corner axis top width height axis axis top color gray bottom xlabel read style xshift align yshift ylabel self name east xshift xshift ylabel title iii xlabel yshift yshift north west yshift xshift xx north west yshift north west south east xshift anchor west legend text font thick corner grid style dash gray width scale color bottom xlabel symbol xlabel style yshift style xshift ylabel ms ylabel scaled style format sep illustrate convergence eeg ii sound circuit show fig stream alphabet obtain distance length short stream font font axis top width height reverse title ia title yshift align font min meta rgb rgb rgb rgb rgb rgb rgb rgb rgb label yshift style gray scale yshift xshift width graphic figure thick rectangle green font text font align green font center shift font legend align style thin align font ii distance grid thick axis style top color gray scatter map xlabel ylabel yshift font ylabel style yshift xlabel style font ylabel style font scale format cs cs cycle legend legend open legend sep sep fill text rgb shift style font path south concept color text minimum text yshift xshift node south sep pt snp align center non child grow yshift south snp child concept text width grow south inner sep color minimum width xshift yshift south close child xshift grow yshift south align center xshift concept text white width yshift xshift concept anchor pt font yshift xshift current south eeg yshift every font thick width reverse xshift yshift align font min max rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb style style style gray scale yshift xshift xshift xshift graphics xshift yshift font legend cell axis black thin height yshift xshift align center font title iii thick axis color color white scatter color fill xlabel ylabel yshift font ylabel yshift xlabel font ylabel font scale font format view file heart yshift font cell align axis style thin title style align center font title ii grid thick axis bottom color box scatter use map draw black xlabel ylabel ylabel style yshift xlabel font ylabel scale scaled style format fix legend cell align legend entry style sep text font font font yshift xshift south root color black text yshift south text minimum yshift concept concept color minimum grow node south n child black grow child concept color width grow yshift xshift text width yshift grow node leave text text width xshift node south concept concept text grow xshift node text xshift yshift south width grow yshift xshift path south concept child color grow yshift path south concept child color text text yshift color width yshift south black grow xshift yshift path concept color south concept data anchor west fill inner fill width height pt anchor font xshift heart digital north label south yshift every style font font axis width height axis reverse false ia cluster title style xshift yshift align point meta rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb style yshift axis yshift xshift width font xshift figure axis cs thick rectangle gray font align rr title yshift align font north xshift anchor north west grid style black thin legend yshift pos outer north background gray box scatter map fill red xlabel ylabel style xshift yshift ylabel xlabel style h scale false scale false style font format legend none axis scatter mark mark white map white table p scatter scatter mark option red pt table anchor inner sep font label yshift xshift south west rr anchor north south shift font center cluster cycle child color text black grow yshift xshift yshift text font snp align child south sep text width font grow yshift xshift node align center yshift text path south distance identify projection identification star application dominate understand discriminative characterize phenomenon find characterize size subsequent learning algorithm additionally notion optimality quantify variation classic neighbor nn depend space heuristic information constraint reduction geometric space manifold infer initial heuristic make task independent universal observe quantify summation stream leave behind flat white require despite stream underlie establish causal converge distance process characteristic hide priori sample always self determine step give symbolic stream copy operation row selective check version pre specify generate stream inversion stream summation show length sufficient effectiveness quantization able produce well maintain discriminate stream stream estimate inter stream estimate circuit threshold pass sufficient stream identical prominent find causal stream induce metric zero distance lead deep insight generator similar phenomenon together operation scale linearly time stream similarity pass stream selective output test short stream ratio refer error self show independent converge four rate scale limit similarity occurrence time problem implicitly never find actually framework system ergodicity stationarity finite priori perform approximately stationarity example algebraic space particular existence unique ergodic see probabilistic occur transition guarantee similarly stream alphabet symbol rare difficult section quantization error alphabet demand imply observation limit quantization process quantization similar still identical stream test quantization synchronization stream begin stream symbolic circuit compute pairwise similarity quantization compute embedding summarize table brain electrical eeg consist eeg patient free interval potential alphabet pairwise yield clear close ec show difference rest I manifold contiguous segment brain provide picture classify heat sound digital analyze series ignore label verify supervision data rest mainly see b iv classification table reveal optical experiment supervise proceed start light curve quantization allow light marginally see projection embed nearly ask period raw yield beyond fourth visually eeg potential database object actual merely randomly subject unknown element library extent outperform knn svm database include possibly knn embed yield figure establish correctness theory process stream establishe stream correct discuss scheme quantization section specific notion often dependencie mutual stream dynamical reveal establish correctness ergodic distinct overview sake completeness denote alphabet symbol unbounde denote string empty word identity string denote stationary stochastic process moment long stationary moment formalize develop theory construction initial matter algebra string small induce stationarity ergodicity countable notational brevity string use extension equivalence string easy equivalence induce probabilistic state introduce notion probabilistic final initial mark mark probabilistic initial marked alphabet state transition symbol generation recursively extend impose state string nan state symbol similar probabilistic unlike latter lack assume remove ergodicity formalize marked induce measurable measurable immediate imply mark correspond mark generator relation extend recursively word choose imply hence denote conclude complete mark index equivalence immediately probability generator mark introduce remove dependence need transformation transformation transformation generation state symbol state associate probability state exist lead string unique canonical induce canonical construct transition induce canonical representation mark canonical sense exist set construction stay follow ergodicity state mark initial representation state see degenerate letting follow strong lemma mark induce remove element construction starting next introduce synchronization synchronization rgb rgb rgb rgb rgb scale draw black width edge bend loop leave xshift yshift xshift yshift edge bend south xshift east auto fill black text draw bend b leave xshift yshift loop xshift yshift bend leave bb b south synchronization determination symbol machine symbol current top bottom exist string notion symbolic observable symbol hide state symbolic string symbol symbolic string count particular overlap string count occurrence symbolic string generate symbolic refer symbolic derivative induce q use convergence read q imply notion canonical derivative correctness stream operation representation notion translate carry stream notion symbolic match closely generative formulation underlie sense establish metric strongly connect symbolic derivative probabilistic metric one metric almost identically immediate low immediate string replace sum noting complete algebraic material include sake completeness symbolic probabilistic state description equivalence symbol alphabet induce index explicit sequel x integer right p explicit encoding say perfect possibly realization neither encode perspective equivalence equivalence say sequel equivalent I general composition composition rgb rgb rgb rgb rgb rgb pt font fill draw text minimum text bend yshift edge bend yshift edge bend pos yshift xshift draw bend leave pos yshift edge bend yshift xshift c draw bend node yshift xshift auto font fill right bend draw edge bend leave h xshift east font fill minimum right loop leave xshift yshift b loop right xshift yshift draw bend xshift auto distance cm scale font black text bend yshift bend leave right pos yshift bend node xshift bend right pos xshift yshift draw bend right xshift yshift edge bend yshift f bend pos yshift bend pos xshift yshift bend xshift yshift bend pos xshift yshift bend leave xshift yshift edge bend leave xshift gx yshift south node fill text right pos node xshift yshift edge bend pos yshift draw bend right node c draw bend pos xshift yshift bend xshift yshift bend pos yshift draw bend left pos yshift xshift bend pos yshift edge draw bend xshift bend leave pos xshift yshift bend xshift yshift bend leave pos xshift yshift hx xshift gx east font black draw bend pos xshift bend node xshift xshift b edge draw pos yshift bend xshift yshift bend yshift edge draw bend pos bend xshift yshift bend xshift yshift draw bend xshift yshift bend xshift yshift f bend leave xshift xshift auto node cm width scale right bend leave pos xshift yshift bend xshift yshift bend right xshift yshift bend pos xshift draw bend right xshift yshift bend right pos node yshift bend pos xshift bend right node xshift yshift draw bend right node xshift yshift bend pos xshift yshift b bend leave xshift yshift draw bend pos xshift yshift font font anchor north south font yshift xshift north anchor font yshift font south font south thick dash thick hx anchor l yshift xshift summing realization anchor north yshift south node font text minimum text width edge bend draw edge draw bend east minimum draw left loop edge draw bend east cm font fill draw text black edge edge font anchor north yshift xshift south alphabet composition transformation structure underlie crucial account proposition next restrict subspace algebraic group group induce map small domain string restrict definition follow immediate precede discussion addition operation x closure existence identity inverse element x string string x px relation px px px complete proof zero element rgb rgb rgb rgb rgb auto node cm thick scale fill minimum width loop loop anchor north yshift yshift addition operation q composition probabilistic general g sum section stream main stream realization formalize pseudo copy pseudo invertible always matrix underlie realization imply generator since generator string give symbol stream generator stream symbol move read one go symbol input stream symbolic compute exact input stream infinite realization transition invertible pass minimal establish note execute state synchronization length bring structure see ref denote transition denote claim imply follow without generality state arrange bound minimal realization realization complete establishe realization generator stream realization pseudo pseudo copy distance distance copy determine copy stream symbol stream generator stream read symbol output move generator upper causal I loss canonical stream canonical mark symbol symbol symbol necessary hide output observe symbol point realization causal red thick xshift east yshift bend left bend edge bend bend bend left loop loop node draw node edge bend leave west east rgb fill blue text minimum text thick xshift east yshift yshift bend edge bend edge bend node edge bend bend loop loop draw loop draw bend leave draw west yshift xshift east node rgb fill green green minimum width yshift yshift draw bend draw leave bend bend leave draw bend draw loop loop draw g bend edge g gray text dash gray black thick black control node anchor north l south anchor xshift anchor l east construct jump cause generality alphabet possibly minimal string observe stream summation produce arc jump equivalence state whenever jump back probability back imply assume eq weight norm contradiction state none row row compute imply claim norm unity term establish bind bound establishe summation perfectly include arbitrary input deviation sum small sum conversely large proposition w g w w q arbitrary stream p h inequality corollary stream stream realization alphabet read current symbol distinct write output move position go alphabet realization let associate current stream symbol alphabet output symbol symbol imply alphabet symbol denote generation stream transition symbol arcs stream distinct stream synchronization initialization occur back distinct symbol jump symbol new stream effect next symbol jump jump occur transition conclude x note establishe conclude generator stream inversion stream stream copy symbol distinct read position follow stream operation stream copy summation proceed symbol symbol symbol involve stream copy inversion stream copy imply scale shift font anchor south cell align legend gray black corner top scale style gray height style fill xlabel size ylabel style align yshift scale format axis cs axis fine quantization pass self stream sufficiently specified rate quantifie stream obtain realization flat irrespective generate selective implie indeed generate scale next inversion produce summation observe stream state copy text copy output stream get stream I harmonic probability stream ss proof occurrence input stream upper maximize lower upon product estimator string white define partial white carry string causality stream denote hide generator string generate xx x imply complete establish causality string stream converge absolute rgb rgb rgb rgb rgb rgb rgb auto font fill text width scale text bend node b bend auto fill bend draw bend bend bend loop edge yshift bend xshift yshift bend xshift scale shift font anchor south align legend style gray font corner axis dash height axis background color bottom white xlabel xlabel ylabel ylabel yshift scale style format black axis cs complexity slow discuss scale stream due selective step occurrence stream hide follow symbolic specify possibly continuous stream symbolic accomplish symbol alphabet alphabet range quantization value belong incur small alphabet length alphabet consequence fact size stream stream inversion stream fall rapidly alphabet make since fine fig illustration eeg quantization symbol accord hidden occurrence fluctuation probability self observe stream error stream observe stream average compute tt tc choose state property maximum scheme symbol data entropy scheme alphabet slice slice approximately c slice contain entropy property mean alphabet fine discrimination self fig ratio useful quantization minimize imply font ts axis style thin false gray xshift xlabel axis cs ts xshift ts east thick style thin false width height pos gray background xshift style yshift outer axis style outer style xlabel axis cs ts xshift ts east black height pos gray fill style xshift yshift extra xlabel axis cs axis cs ts south west yshift cell align legend font corner style dash gray axis height axis black xlabel alphabet style ylabel ylabel align yshift xlabel style yshift format title title yshift east anchor legend align legend style gray font axis thick corner scale axis height grid style fill alphabet ylabel ratio self discrimination ylabel style align xlabel yshift scale format format title style yshift cs coarse coarse alphabet produce error identical provide stream self test two stream contain notion theoretic investigate extensively stream random mutual quantifie amount information variable auto node font fill draw black black text node red east font stream align font stream xshift yshift east align north font yshift xshift south streams anchor north font yshift south dash east west dash east pt auto font draw minimum width align center edge path east font stream yshift east east fill font xshift yshift align anchor north font yshift xshift stream north font yshift thick east dash thick east initial stream show stream near zero mutual similar generator significant formally py single vector mass mutual px py notion variable amount information variable mutual precisely know stream generate px px py sharing information conversely high sort synchronization orthogonal stream nearly anti generate stream copy stream inverse require fall see simple stream twice randomly generate symbol accordance probability imply stream stream nearly stream stream correctly stream generator significantly c mutual inf ex color style font height axis scale true font format name plot horizontal sep vertical title ylabel ylabel style yshift xlabel table figure dr stochastic value font label align center difference mean vs yshift axis south east xlabel ratio axis background thick gray white group name vertical ylabel xlabel min meta rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb label style yshift style gray yshift width font graphic figure graphic figure dr graphics figure dr north align yshift style yshift width height legend pos south east xlabel axis style thick color bottom color plot horizontal sep edge leave ylabel ylabel xlabel rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb style label style yshift gray width font graphic graphic figure graphics dr std label align vs difference vs variance normalize yshift south title yshift height scale axis pos south east xlabel axis background color style plot sep bottom ylabel ylabel yshift xlabel point min rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb yshift thick axis yshift xshift font graphics dr graphics dr graphics figure stream generate stream zero nearly finitely stream illustrate stream provide stream conceptually distance information stream easy tool dynamical meaningful measure consider deterministic widely primarily model accept trace see stochastic development assume ergodicity stationarity consideration fall apart x x consider parameterize reaction simulate system every dynamic change fig reaction reaction reaction combinatorial current population reaction probability terminate partially length imply remove restriction strictly value count might behaved parameterization third reaction index yshift east title yshift scale legend pos south east xlabel axis background thick color gray white ylabel ylabel yshift xlabel rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb style style width font gray title xlabel reverse pos graphic figure rectangle axis cs axis cs cs axis axis cs axis axis cs cs axis cs rectangle axis cs cs yshift center yshift center xshift yshift xshift center gray xshift yshift xshift simulation carry system illustrate perfect reaction death expect degradation probable expect large however truly oppose simply monotonic parameterize dynamical series generate series update specie map sequential datum stream positive symbol symbolic stream pairwise compute sum copy flat ai entry difference measure series iii notably ai trivial dependent stream normalization normalization bi normalize little ci fig change via rich would surprising complexity minimum simulation cluster identify perfectly monotonic meaningful categorization tool discover obviously function precede tool introduce similarity sequential model training expert tune consequence may specific tune strength lie ability heuristic big challenge cc underlie physics notion set identical look heuristic design expert classify star determination state might require fourier activity quantify source stream priori space symbolic always stream anti degree anti mutually application detection anomalous object principle exceed specialize hard quantitative stream anti stream tie stream stream reveal
accumulate calculate way either dynamic increase al regret continue play reward world understand biological principle underlie physics work partially advanced institute integrate national life institute technology couple sophisticated capability find stochastic fluctuation body validate statistical exhibit principle modern digital phenomena device band material noise law logical response complicated circuit require relatively logic effort division costly consumption resource look biological system couple physics physics analog paradigm constraint body decision making suppose example coin certain player give reward probability respectively play trial reward repeat armed bandit highest accurately past phenomena application diverse field communication cognitive monte game application originally describe essence thompson index class distribution class computing become et machine system dynamic inspire dynamic resource collect environmental expand shrink terminal inspire algorithm decision physics physical volume body entail among volume increment volume part show derive volume law well greedy modify good device physical physical law optical energy transfer dot dynamic kt body bar slot represent make player coin accumulate initial play playing weighting parameter evolve particularly machine play high left simplicity overlap area fig area incorrect expression must satisfy large easily confirm obtain therefore conclude rule popular update play probability player though machine bottom row table play machine note update play give follow reward transform expect reward coefficient term eq satisfied weighting estimate simultaneously dynamic machine feature rule derive reward advance simulation verify exhibit derive accurately dynamic essence generalize separate follow reward fact
training experiment change run concept validation model rigorous ease initialize encoder maxout network dropout regularization hmm network purely wise per frame hmm decoder development propose rnn remove transformation softmax acoustic use rnn rnn encoder initialize recurrent ease match input frame c basic independent adapt gmm search maxout acoustic end rnn search rnn rule mechanism save batch divide sort form batch minibatch network frame zero rnn end network mlp receive input frame hide layer adapt rescale gradient pass update procedure use stage effect begin threshold average correctly report preliminary require produce sequence process specialized extension machine aspect rnn rnn variant encoder decoder explicit alignment state alignment unlike generation generate handwritten character next input look sequence help speech subsequent similar frame relation search match probable however without time decode search width even narrow significance vocabulary speech decode stream put another directly search probable hmm acknowledgment like hill acknowledge cifar development university universit de universit de hide traditionally speech bi directional recurrent encoder recurrent stream alignment attention symbol subset demonstrate achieve comparable hmm challenging sequence acoustic short sequence symbol two length know building classifier predict signal recognize investigate work recurrent train alignment decoding model keep position attention decoding model input select frame summarize symbol achieve error rate comparable dnn error rnns however greedy search tune month work achieve couple hmm hybrid system act acoustic predict input frame target frame way minimize frame classifier whole acoustic hmm tune jointly sentence minimize decode hybrid progress adopt fully feedforward recurrent hybrid hmm require control contribution decode hybrid mixture generate force alignment follow train acoustic network estimate furthermore improvement necessarily translate minimize optimize path alignment base symbol sequence merge consecutive identical output symbol backward sequence prediction extension rnn acoustic compose part sequence symbol symbol probability sequence rnn token originally perceptron mlp combine performance dataset rnn generate attention text step correspond currently word compute match location sum interpret align translation assign high location allow long recurrent translation recognition soft narrow encourage mode attention move nearby near input give current relative position successive speed decode generate raw also annotation generate sequence sequence element illustration present rl output condition annotation conditional decoder annotation context vector recurrent layer gate keep track dependency prediction realize softmax context annotation mean select annotation selection score decoder annotation base mlp output mlp sigmoid output attention search input frame match decoder repeat annotation location prevent location important show encode preference input frame location automatically learn prefer future concentrated like constrain advance network monotonic add cost input already emission normalize monotonically increase alignment add sum penalty show dot along intuitively encourage select nearby input respect previously effect alignment consecutive without hyper coefficient work optimize encoder implement
handle class novel statistical relational output efficiency max procedure propose expect constructive applied overview structure output little hybrid generalization hybrid markov logic operator namely constraint hoc translation form amenable rely solver conversely design integrate specific solver expect efficiently logical g deal assumption make expressive arithmetic formalism restriction whether logical boolean logical assignment theory predicate otherwise fundamental domain reason operation include e g arithmetic bit efficient find underlie theory solver solver develop max requires maximize formulae formulae formulae expressive package specialized solver intel circuit biology counterpart goal solver problem structure generalize max output compatibility joint learn compatible maximization output train negative quantifie correct margin rescale training error complexity eq wrong output play quadratic cut plane infeasible issue constraint violate guarantee find qp iteration original rhs appropriate loss solver implement cp formulae include definition formula refine formulae appear formula value separation max solver cost function boolean numerical formulae clause issue since clause towards satisfy practical impact involve constraint environment robot planning modeling gene two flexibility expressive power formal publication restriction activity sensor instant activity task tv sensor home include video audio etc cast discrete mean world activity inter case factorial use training intractable cast scheduling logic event predicate straightforwardly translate express activity specify likely duration activity occur hour involve interact constraint scheduling activity soft interesting application consider customer planning build house company price design minimum requirement boolean rate distance public service quality distances max develop prototype interactive various customer encourage constructive could learn viewpoint interpret generate first logic language solver constructive implement arithmetic paper novel class rely language allow describe boolean output solver optimization perform max constructive method reason relational characterize soft first logic encode weight logical formulae maximum posteriori assignment predicate formulae play role maximum problem solver issue logic suit hybrid require predicate make I
scheme binary hash hamming implement tree structure neighbor hamming point near computation extremely order meet locality sensitive requirement function satisfy normalize un kernel normalization valid family hamming distance histogram hash rounding sample kernel representation commonly kernel gaussian vector rkh key computation accomplish solely consider realization covariance vector practice covariance must show center change hash evaluation limit question rkhs space base non eigenvalue imply could time performance solid resolve issue section simple powerful allow aforementioned issue dimensional may precisely lsh gaussian vector lsh project deep hash infinite truncation view inner ti jt I direction approximation come know truncation view product two project word compute lsh use point estimate estimate covariance center operation lsh project equivalent lsh central approximately draw project whereas lsh know obtain good algorithm explain avoid technical issue could retrieval performance arguably popular nystr om rank original computing similarity nystr om point briefly difference nystr om method kernel eq whose eigenvalue column representation diagonal column center format hash even though nystr approximate whole center especially empirically issue section present theoretical performance lsh perhaps importantly improve performance present lsh make truly refine approximate via central possible incorporate practice provide gaussian first formulate similarity refer want literature semi implicitly associate feature infinite inner eigenvalue explicit span empirically become normalize estimator fit lsh normalize eq cause instability issue optimal proof principal choice query dominate bind utilize bind ingredient km go get eq near neighbor retrieve observe always retrieval empirically namely lsh thousand expect result good choice lsh subspace may achieve project obtain component replace good lsh hash sensitive decay property rkh affect retrieval transformation explore well exponential eq ranking stay matter change decay entirely slowly reduce eigen carefully validate comparison nystr method report commonly use comparison million randomly represent million sift descriptor comprise size descriptor whose kernel bit largely make choice number kernel namely histogram exhaustive near evaluate proportion rank position indicate retrieve verify original represent measure direct neighbor performance hashing scheme semi improve variant comparison restriction ccc b rank nystr decay property good ccc color transformation sift show fix small confirm lsh performance however entire tradeoff discuss initially improve use decrease drop show sensitivity choice addition kernel examine nonetheless still recall sift affect trade different sift used make recommendation nystr om moreover obvious tradeoff rank observation indeed nystr om regard nystr full choose transformation increasingly monotonic change scale affect decay continue decrease decay speed decay drop usefulness largely function kernel sift original room summary absolute combine retrieval regard power technique interpretation locality sensitive hashing conceptual provide apply appropriately project first suggest boost large show choice monotone performance present piece least bind product complement principal map define I sample draw decrease space counterpart probability state lsh admit hash randomize use query time dominate computation ready lsh relate lsh k reduce decompose right apply locality hashing vision near neighbor reproduce hilbert perspective step project practical benefit problematic conceptual motivation formal performance bound reveal boost several benchmark standard similarity database play application object scale fast
mcmc nonlinear model proposal analyze tackle inverse posterior wherein log approximate similarly gaussians facilitate construction mean directly approximate combine inform forward covariance approximation global restrict treat complementary move infinite informed subspace use dimension independent work interest truncate collapse mean avoid error require smoothness decay entirely influence forward observational constructs model sampling capture description notion fundamentally reduction technique organized posterior matrix examine interpretation covariance characterize approximation conclude remark paper along technical consider loss generality q infer forward observation statistical likelihood fisher coincide mode note posterior matrix depend matrix prior similarly kk covariance prior low reason describe along prior might decay spectrum lie approximate set semi rank positive covariance advantage structure optimality definite matrix canonical invariant metric cone give f invariance moreover treat loss function kullback leibler distance aforementione addition differently class function element value tend loss distance find appendix optimality approximation familiar hellinger kullback minimize iff minimize hellinger distribution depend optimality hold state posterior covariance eigenvector correspond unique eigenvalue computing criterion minimum generalize subspace span generalized direction parameter hermitian eigenvalue result eigenvector accord analogous transformation notice involve pde readily direct eigenvalue reference implementation available rich literature g parallelism method factorization hermitian product induce maintain efficient implementation solver accurately refer straightforward efficiently discuss introduce variance covariance inform strategy use norm function approximation develop hessian alone draw kalman approximate approximate optimal approximation explicitly posterior following characterize direction minimizer minimizer precisely inverse generalized appendix define define minimize span j along value informative first span eigenvectors informative hence direction along furthermore direction maximize direction minimize maximize corollary simple theorem linear dimensionality problem particularly effective posterior parameter constrain locally inform subject inverse notice posterior project exact posterior square far frobenius matrix definite cone matrix matrix minimize frobenius direction large eigenvalue different particular solution eigenvalue maximize metric identifie maximize distance direction maximize prefer former direction frobenius entirely naturally perspective eigenvalue finally statement approximation use approximation numerically approximation natural limit closely effort dimensionality reduction effort discard remain discuss update datum negative likelihood decomposition eigenvector good rank belong original different base space prior take mean pair u actual consist forward suboptimal general project update inversion summarize hessian technique method take importance quantity account key approximation theorem illustrate condition interaction problem forecast gaussian observational bayesian computationally way solve system bfgs typically initial matrix scale bfgs positive convergent storage bfgs bfgs bfgs posterior use bfgs approach bfgs result rank distribution fast accelerate repeat inversion set propose efficient realization already accomplish art iterative solver function seek approximation consider class relatively single linear precision datum approach justify posterior mean instance thus offline strategy costly offline fast evaluation approximation optimal cf additional posterior inverse formula efficient bayesian mode equal risk error establish basic notation let loss incur estimator bayes distribution function mahalanobis account geometry approximation direction approximation posterior approximation repeatedly multiple realization eq consist statistic replace update shall henceforth either bayes matrix bayes approximation particularly analytical exploit proof develop independently eigenvector normalization bayes risk rank define realization posterior bayes risk rank approximation decreasing easily bound analogously interpret truncated bayesian readily triplet rank compute bayes risk optimal approximate realization dominate prior need approximation theorem yield precisely stochastic describe gaussian linearization forward operator bayesian result support algorithmic precise statement worth risk dimension define approximation negative low associate bayes include power great approximation accurate show estimator counterpart fewer low sense subsection mean ill inverse stop statistical say iterative equipped stop use result justify observe iterate denote highlight aa observation heart iteration algebraic I nearly generalize usually satisfactory informative assume beyond convenient stop similarity minimizer may perform goal quite concerned ill whereas statistically approximation mean framework mean adjoint approximation I input forward adjoint numerical precede continue inverse investigate negative semidefinite theorem hessian bfgs reduction scheme difference explore shall refer hessian think denoise ratio variance canonical great alone inform direction reduction hand variance direction alone determine reduction effective spectra important direction depend one distribution generalize full prescribe spectra case orthogonal lead decomposition schmidt standard discuss experiment run bfgs optimizer x bfgs covariance bfgs optimizer bfgs bfgs bfgs paper store bfgs optimizer converge optimum taking result numerical prescribed move increasingly informative distance realization row realization fix obtain bfgs flat direction variance bottom row move third increase quickly combine great successfully whereas remarkably configuration spectrum generalize situation spectrum middle flat direction parameter great almost towards decay restrict dominate hessian however generalize reduction prior spectrum decay slowly reduction either quickly decay bfgs theoretical htp value eigenvalue leave f versus realization bfgs difference f green update middle classical ray intensity detector object present synthetic ray enter instance www enter ray system move reconstruction image detector location cross basis carry model ray intensity detector object discretize grid grid cell integral cell vector length line though logarithm inversion vector iid setup rectangular discretize circular ray measure detector opposite evenly angle ct exponential fashion computing intersection circular gaussian discretized length computation well nontrivial define discretized pde white process identity length control prior compute first negative update first formally approximation seed due match posterior course show htp htp assess low shall yy datum reference top error rank low include confirm compare figure right panel snapshot correspond mean relative time require approximation regard realization divide posterior computing iterative scale inverse solver computational cost roughly report realization obtain roughly relative time approximation could take cost forward sparse different heat exclude iterative solver converge popular mean efficiently angle application cpu cpu compute iterative apply much leave unnormalized approximation realization distribution dependent bayes theorem decay two consistently nonzero offset datum draw figure approximation configuration wherein detector spread around entire object informative red angle configuration slowly angle configuration loss posterior update great rank limit approximation informative relative limited angle relative drop realistic ray extremely flexible device configuration normalize number cpu divide black green red panel htp eigenvalue angle ray center lead blue angle htp angle ray uniformly linear inverse initial heat equation heat heat initial linear heat pointing unit spaced dot figure infer space pointwise observation function evolve show discretized example non field truth gaussian dependent panel numerical trend figure encounter confirm good low rank perfect visually curve somewhat occur approximation eigenvalue relative snapshot example visually indistinguishable therefore accurate solver section cpu apply adjoint model heat pde versus apply matrix also apply negligible example cpu figure illustrate important characterize heat notice eigenvector sensor show direction direction update capture direction great mode concentrate around sensor great htp htp condition inversion heat htp cf fourth four direction space typical large relative prior approximation structure covariance negative semidefinite form broad loss function symmetric definite argue metric identify direction space posterior variance update optimality optimality hellinger kullback divergence develop approximation realization approximation minimize error computationally efficient low numerically variety ray observation condition localize observation prior already natural endowed understanding function current operator allow generalize possible research approximation technique assimilation linearize assimilation present evolution introduce tailor work support department energy
chain account introduce nonnegative symmetric preserve condition approximation chain satisfie initially enter part circle j c basis conclude statement statement fact work construct inverse multiplication represent induction let theorem proposition example remark fundamental graph theory access entry take work depth machine field randomness addition random learn sequential markov monte mcmc algorithmic unless significant require reliable design scalable sampling challenge characterize characterization formally performance sampling subsequent sample parallel generate subsequent sample total random obtain gibbs method randomized process gauss apparent important multivariate model vector analogy gauss non underlie precision perform significantly worst partly mathematical et al generalize dominant scale converge correct partition processor sequential gibbs sample old processor lead development nearly solver dominant parallel bad logarithmic parallel scalable extend solve graphical focus parallel field natural arise likely gain diagonal precision matrix worst generate sample random field newton expensive sample covariance framework find square root return efficient parallel implementation depth nearly step framework concentrate nearly depth construct analog differ generate generate random univariate sample approach main factorization preprocesse randomness generate motivate develop spectral graph theory give algorithm construct factor om tight matrix essentially ratio entry always precision need remove importance run depth parallel processor prior could symmetric dominant factorization directly field precision toward linear matrix wide matrix structural matrix relate diagonal matrix representation believe structural barrier random field parallel graphical connection spectral graph field large notation paper spectral radius large definite definite definite matrix definite equivalence graph weakly matrix rigorously carry matrix entry ensure ic guarantee multiplicative rescale factorize combinatorial factorization factor satisfy satisfie moreover depth mn adapt main construct break product operator individually reduce compute count couple multiplication algorithm major issue use error handle error e compose alternate decomposition factorization base half term naturally incorporate factorization lead issue half power idea expansion polynomial power eigenvalue resolve front introduction one dense key correspond walk average well existence technical theorem without specify construction approximation total com nearly linear subsequent improvement near propagation chain follow scalable construct inverse length expansion degree finally procedure multiplication operation complexity extend nearly dependency refinement order obtain polynomial length expansion expansion moreover approximation polynomial preserve start degree provide chain substitute expansion equation preserve
vector regardless appear compositional successfully series compositional distributional one compositional model description experimental present discuss prior learn capable model complex relationship input nlp capture sentence representation word generic form composition apply concatenation vector word activation compositional representation fashion sentence merge serve semantic intermediate instead feed reconstruct encode hide optimization reconstruction unsupervise fashion deep input word representation token merge evaluate represent word compositional network probable general methodology induction discover latent occurrence calculate vector create vector agglomerative sensible word meaning mean black corner mm width height black corner cm draw cm height fill minimum size sep em text text em I mirror mirror amplitude mirror black black round black right similarity et al phrase task similarity compute match human sentence subject object sentence dataset every noun l comprise construct ambiguity play specific aspect dataset constitute evaluation term neural composition implement additive model sentence take wise evaluation conduct way composite sentence apply specifically sentence similar use procedure classifier report fold validation list ht word ht word multiplicative c multiplicative multiplicative suggest bring learn compositional l carry subsequent composition encouraging choose imply generic act processing outcome word never decrease clear positive algebraic despite benefit processing work suggest compositional constitute certain although return approach report study try interpret sentence phrase length deep model deal text deal compositional meaning word benefit explicit architecture compositional ambiguity align concept result improvement compositional google neural compositional hope semantic produce fed compositional net evaluation deep
profile distribution elliptical structural illustrative employ model propose distribution profile look fractional variation image model estimator compare estimator intensive size three fold correct cox wishart correct estimator quantify list assessment monte experiment analyze structured complex wishart discuss cox correct mathematically subsequently compare exist literature summarize coherent entry transmission follow detailed complex multivariate essentially commonly look hermitian scale wishart complex complex indicate scale wishart distribution maximize associate wishart stack operator show order ordinary function close nevertheless account adequate application filter guarantee technique ml suffer convenient estimator look propose datum discard derivation address datum describe obtain improve ml cox base profile method refer density play ml adopt accordingly derivative direct information additionally mathematical cox bias possess correct ml q possess asymptotic modify technique two vector nuisance address often mathematically approximate likelihood problematic associate bias therefore profile bias issue modification adopt modification tractable wishart approximation likelihood nuisance aa cox profile improvement ml scale wishart cox correct profile mainly goal derive correct focus cox ml eq need simplify replace ml follow discuss technique scale quantity obtain replace yield column zero obtain moreover estimator consequence quantity mathematical derivation associate satisfy word linear resort newton look study ml trace cox three assess measure error cv iii subsequently separate indicated simulation suggest monte employ follow size look iii respective element adopt square window respectively affect estimate employ mean table notice estimator offer derive size mse cv also show present term mse cv evidence bias estimator yield procedure estimator correct variance value lead present size look decision consistently exhibit reduce bias increase look affected well looks propose identify estimator among technique c em em lr c lr em lr lr lr lr em em lr lr lr lr lr em htbp separate operate band nominal look present area visual inspection ii htb replacement display obtain small mse thus yield tend less efficient indeed report complex verify r lr lr region c lr lr em lr lr lr lr em lr lr show curve curve usual ml homogeneous htb notice table discuss possess improve scaled emphasis look advanced cox mean assess adopt figure coefficient mean result correct superior appendix wishart ii order bias accord cox third kronecker product expression yield q formula indeed expression subsequent discussion detail likelihood particular address subsection result scale complex wishart equation profile log yield approximation log q simplify hold sc statistic department adjoint interest matrix sm receive electrical engineering sc mathematics de de
threshold ordinal soon pass threshold implicitly main rbm ordinal specifically form rbm offer impose random user profile easy posterior ordinal observation behave rbms never level pose gaussian another consequence rbms e see persistent ordinal rbms model correlate ordinal typically inherently influence user specific user item specific rbm architecture long map visible profile world art filter public rbm ordinal ordinal presents ordinal rbms discuss relate follow ordinal ease homogeneous draw ordinal solely hide k machine rbms binary observation example fig bipartite potential utility binary form rbm translate deviation free introduce gaussian rbms whenever category automatically define low generative estimate cumulative restrict spaced free nice thing like rbms factor posterior conditionally since ordinal observation n exploit bipartite rbm run gibbs parallel finally posterior ph kn nh sn lead update recursion utility ph ordinal unseen integration intractable datum rewrite make either ik replace value define ik associated parameter standard sufficient ess truncate put rbm phase ess ess r learn persistent effective strategy update chain integrate factor advantage bipartite stability average phase maintain per store phase short need per chain discard chain chain maintain chain maintain otherwise instance collect gradient ascent threshold utility domain r read lower le use derivative ordinal scale c c il homogeneous threshold assumption example collaborative filter user role influence popularity item switch user item I instance column hide row binary hide incomplete denote incidence set factor di visible ordinal variate model product potential specifically g define except threshold datum e condition posterior factor likewise give di di di di column still rbm item although explore chain resort structure modularity alternate process estimate item posterior pg di di item mean posterior th ph di likelihood th improve likelihood e estimation posterior reduce trajectory posterior simple utility decay previous empirically learn progress treat example efficient rejection bias zero threshold space evenly ph ph shorthand well view discover profile people life social political country wide centre people processing day economic country improve remain heterogeneous question hide unit fit obtain ph ph ph representation project vector plane locality preserve reveal us china see world three public million nearly million rating nearly netflix rating user nearly rating scale star uniformity remove rating use criterion netflix ensure rating comparison mf mf cumulative ordinal assumption without item item neighbourhood rbm multinomial assumption protocol stop report rmse mae mf rbms map mae depict base clearly treatment performance dataset matrix rmse competitive mae ccc monitoring rmse c rbm rbm gaussian ordinal multinomial rbm rbm rbm partly research rbms variety rbms continuous ordinal work extend rbms input ordinal rbm treatment multinomial less offer generative mechanism ordinal ordinal datum well science especially quantitative refer
essentially perturbation sake without give formulation three assume svm issue task relate directly adapt extract compare adaptation svm sample example adaptation optimization integrate accounting allow space go less constrain impose interpret layer straightforward target domain compare available target domain minimize train individual experimental sect confirm employ quasi newton derivative base efficiently svm general structured ground sample run output accordingly correspondingly integrate particular must give apply part focus learn set decide latent latent formulation concave write variable object alignment base appearance part return play truth output concave apply write moment adapt testing time learn adaptation classifier adapt domain adapt source window divide resolution regard low high tend discriminative virtual world www evaluate pose consider evaluate virtual world domain world pool domain b multi detector supervise adaptation roughly available training respectively evaluation average rate vs curve setting time repetition ensure performance baseline challenge part challenge hold conjunction neither report describe da use image note criterion vs avoid classifier virtual mention pixel hierarchical extra pyramid pyramid two pyramid pixel pyramid pixel feature pyramid illustrate training resolution height bound virtual real real challenging apply pyramid finally combine c moderate easy da lp see accuracy single datum achieve train importance leverage hierarchy set additionally assess full multi resolution show demonstrate adaptation explain training learn multiple domain multi provide good quantitative capable detect outperform remarkable mention adapt virtual take minute approximately ghz full need number virtual building virtual build available knowledge adapt detector challenge complete challenge new worse quite clearly rest adapt experiment time actually final measure image pca histogram bin baseline summarize svm target domain example source rest domain optimization perform follow domain adaptation fig l avg tree avg dc c w adaptation cd avg adaptation locate amazon amazon accuracy target split worth totally one setting clear importance style method domain domain domain per domain example adaptation style w perform analogously ad ac potential agreement hypothesis underlie hierarchical domain focus see three hierarchy improve adaptation tree show adaptation good two layer well previous study similarity quantification shift show similarity measurement yield good capture underlie w adaptation tree agreement label priori sect mix remove recent discovery latent call discover b b qualitative category domain discover pr hierarchy indicate correspond classifier configuration operate domain label discovery point domain category unlabele target mix pr predict category category domain discover source process domain want compare discover sub vs distribute predefine sub domain discover fair sect c domain discovery give layer pr pr layer pr pr among possible pr accurate predict pr see pr difference pr pr pr equally sect target must perform domain sect pr discard category require label datum reference sect three latter good configuration result comparable sect reader convenience discover domain outperform pr discovery accuracy see due train domain hierarchy error purpose object example present domain adaptation sub domain adaptation key target increase structural classifier adaptive exist sample adaptation apply detection recognition application involve imply category show effective accuracy ignore focus recognition target incorporate aware sa attribute support xu grant grant rgb computer vision es topic loss produce recognize vision classification object hierarchical idea exploit difference svm adaptive domain together source perform adaptation proposal detection category apply classifier show adaptation ignore structure incorporate discovery object recognition classification increasingly problem make testing challenge variety method increasingly machine computer image usually adaptation target domain domain adaptation allow adaptation consider intermediate domain adaptation adaptation single domain multiple propose label adaptation cover much case leverage multiple target domain criterion build domain implication adaptation main illustrate hyperplane traditionally option pool target single one strategy single domain adaptation perform propose target difference multiple hierarchical tree adapt source jointly sub approach b difference leaf hierarchy hierarchy imply adaptation strategy domain worth reduce domain divide target would ignore reduce time fact require svm svm term method vision field category wide category case increase detection ignore target focus recognition target domain discover organize discover domain experimental result detection recognition respectively despite propose decade comprehensive vision broad base attempt matrix classifier transform jointly learn discriminative classifier concentrate adapt parameter often base projective transfer svm variant require
dd dd conclusion follow many distortion weighting weighting factor infinite q issue address rescale rescale weighting tend distortion prop implicitly one time causal finally concentrate measure retain associate eq contrast causal state retain function q maximize whereas replace reverse causal state since prop distortion measure I information satisfy prop distortion measure weight average distortion reverse causal state notation mathematically sum integral vanish distortion divide treatment instead consider retain trivially proof prop time reverse state infinity ref reverse causal state limit counterpart recover lemma prop service clarity shorthand leave expand measure theoretic elsewhere solution objective minimize coding predictive predictive multiplier lagrange lagrange multiplier code predictive multiplier main r first eq combine straightforward calculate thus finally calculate eqs divide replace since start abuse notation somewhat justify bayes r rewrite allow slight enforce normalization constraint mean partition meanwhile formal codebook map codebook codebook iterate codebook realization practically careful variety converge cm every edge black state line width font loop loop style loop loop definition color rate distortion process suffer curse retain resource grow exponentially process dependency algorithm fail dramatically underlying correlation memory curse distortion objective term mechanic demonstrate causal distortion substantially rate analysis keyword optimal filter mechanic predictive bottleneck device predict either guide strategy adapt environmental ultimately due resource limitation coarse partition store bit store grow perhaps infinite storing exceeds predict resource shannon introduce analyze trade encode rate without prediction distortion theory principle calculate achievable distortion give test whether identify useful feature understand natural current calculate distortion arbitrarily long avoid avoid class future one retain length yield distortion long address generally predictive algorithm distortion maximally identify alternative demonstrate maximally dramatically improve distortion associate discover hierarchy mechanic distortion introduce distortion forward reverse state illustrate summarize application entropy fall capacity shannon coding say exist encode regime introduce distortion system capacity free shannon certainly never experimental guarantee quantum relation capture organization many adequate capacity said positively reduce memory act focused determine reproduce extent future adaptive simple require coarse collective root identify environmental human identify variant theory state review finally ref mechanic ref main generate behavior x contiguous exclusive x past tt well indirect internal mechanism forward causal state sufficient group equivalence relation shorthand denote forward generative build causal give start since output transition effect difference index alphabet hmms mathematical ref note process finite causal constraint next know next symbol similar minimal group together shannon reverse causal statistical predict process htp aspect future function forward causal reverse state form relationship symbol generate identity entropy residual employ amount information vice versa store shannon various information quantity function importantly algebra information random atom pose capture correlation capture hide coarse grain block process dominate vanish excess causal complexity hmms finite process past parametrize equivalence length quality convergence mutual chain rule information causal review distortion theory refer ref detailed serve theory combine noisy channel shannon channel encoder biological signal natural information processing post htp past imply chain source symbol encoder require source evaluate distortion quantify long period lead distortion codebook code distortion measure code rate distortion desire require achieve minimal distortion possible process successive semi series distortion forward source variable information source output capacity codebook decoder reconstruct input capacity enough irreducible rate regime positive regime ask equation view realization codebook solve numerically minimize one trace precede measurement sec denote symbol arise distortion use less familiar square like recently though definition retain aspect variable assume markov distortion q distribution straightforward chain since finding maximize ib since decide information realization relevant markov chain measure rate distortion determine limit calculate maximize constitute refer distortion measure confusion ib explicitly analysis vary coarse identify implement zero causal maximal bottom illustrate predictive soft predict fidelity indicate relevant thm b length retain information length limit capture ec nm though store storing limitation current thereby restrict distortion topological practically limited notably practice account measure sequence probability gaussian function call calibration state ref uncertainty mix associate h hmm take use ref information whose correspondingly slowly capture figure contain pair turn translate ref consequence simple notable compute sec challenge even become extreme transition temporal correlation correctly calculate curse dimensionality critical concern generate hmms away prototype biology finance quantitative highly markovian rather gap process countable infinity state ref heavily short likely fail curse require structural reverse proposition result lead distortion information equivalent theorem intuitive appendix sec illustrate partly partly analytical insight inspire argue temperature lead rate distortion set state retain somewhat markov fig imply distortion function say equivalent certain type measure htp bottleneck notation fig distortion measure distortion though instance distortion measure actually temperature great ask arbitrary causal long statistic temperature limit forward ref state without distortion reasonable distortion measure difference apply estimate incorporate entirely apply replace future sequence sec certain store unnecessary see process almost finite well concerned reasonably function causal state unnecessary htp hmm presentation process htp intuition side process hmm approximated feature approximate replace infinite discover single feature nonetheless imply infinite classic aware latter ref instead time past b ref process transition second phase transition critical causal state discover codebook change break key phase transition detail qualitative straight line obtain code fig bottom show feature indicate transition incorrectly suggest phase ref curvature function scale information scale reason exhibit straight recurrent forward state odd last causal fig ref whether odd one successive know uniquely causal versa causal reverse state conclusion directly machine result argument periodic ref noisy periodic process periodic relationship prediction key feature fig order periodic cyclic process causal nothing iterate attempt maximize however recall c hmm maximize forward causal state optimal sum code first critical temperature htp state even memory increase reverse feature block transition new process causal form g generate entropy many irreducible word restriction easily block forward reverse state odd causal give one must restricted state map add may sharp transition ref suggest contrary variable diagonal gaussian would curve see curve expect difference matrix discriminate phase transition sharp transition curve noiseless limit often bottom reverse htp reverse reverse bottom forward legend solid line circle color various htp describe change prototype rip therefore know joint forward causal reverse state statistical statistical rip excess bit invariant state investigate reverse causal function put disadvantage function greatly rip reveal correct either forward reverse time feature curve reveal regime reverse feature show phase transition reverse add codebook state reverse causal state rip fig ref causal fig phase end curve remain forward codebook reverse look sharp reverse time codebook reverse state codebook phase emphasis select prototype apply gain insight symbolic analytically symbolic symbolic markov know describe htp describe elsewhere yield show coarse new function typically low top bottom reveal representation htp black guide state state identify mixture identify original identify underlie feature inverse temperature fig state compress phase transition intuitive move add forward time causal finally implication hierarchy hierarchy equivalent predict probability fig say calculate estimate calculate reverse distortion viewpoint step recall refer machine answer polynomial np hard problem relate ise infer np complexity hardness infer exponential np seem though complicated common length retain basic implicitly sequence suboptimal sec without leave histogram rather build also deviation predictive causal identify forward suggest study series long correlation calculate derive infer exponentially long temporal sequence need curse head unnecessary adequate effective underlie process either use order produce inaccurate markov dominate well interpretation htp estimating curse word severe instead derive predictive accurately alternate directly mind predictive distortion quantifying curse sec new highlight side strategy analyze event predictive extraction ref dimensionality rule yield causal benchmark performance predictive infinite markov predictive state deep useful grained aspect appear machine switching determine function perhaps importantly machine state
low matrix technique hierarchical decomposition exist calculation force associate analytical consideration rank hierarchical construct dense scheme direct determinant symmetric etc exist low modification modification cholesky I entire direct reader paper linearly yield investigation worth point cholesky factorization semi scale symmetric factorization update identity extend product low detail compatibility matrix section factorization summarize discuss extension hierarchical incorporate perturbation hierarchical briefly know identity allow rapid inversion determinant update matrix low calculated formula simplify perturbation identity perturbation identity furthermore rank advantage perturbation cost worth formula kalman filter square direct solver calculate determinant expansion operation decomposition cost reduce recently determinant calculate determinant formula relate determinant identity often serve normalize evidence determinant right determinant precise determinant spirit determinant enable factorization perturbation factor rank relate positivity positivity large semi definite zero note matrix invertible exist definite fact directly I tu criterion meet prove criterion furthermore choice therefore calculation require arise give factorization satisfie simplify ready immediately equation substituting address restriction symmetric factorization symmetric symmetric ingredient square factorization matrix satisfy root combine recursive divide next yield root useful corollary perturbation extend update symmetric inverse fast also ti tn numerical demonstrating contain formula factor perturb matrix equation storage cost vector significant interest scale associate dominant product step tm pl make perform decomposition indicate dominant htbp c decompose decomposition proceed internal formula diagonal structured division inherent tree compression technique article hierarchical deal matrix shall restrict matrix matrix purpose diagonal block level numerically frequently level depict actual form rectangle rectangle rectangle rectangle rectangle rectangle rectangle low eq block diagonal feature factorization block update level rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle zero symmetric form ii ii tn worth recall sub block right I size step matrix need write step right side rank identity cost repeat yield rank exist represent separable computational obtaining factorization speedup cholesky rank entry give matrix scale ta context give xlabel ylabel seconds legend anchor south west coordinate coordinate coordinate corollary need versus consider q cholesky factorization ti compare versus perturbation plot take obtain factorization versus htbp xlabel ylabel seconds style anchor south east coordinate coordinate scale xlabel system ylabel style anchor coordinate scale xlabel ylabel time second south coordinate rank middle xlabel ylabel legend style north coordinate coordinate rr numerical benchmark encounter fluctuation act point dimension obtain occur perform radial force manifold embed test priori vector order diagonal slight modification variant arise frequently nonparametric find q inherent measurement cube arise gaussian process model eq definite table summarize tensor frequently brownian radius particle translation coupling refer show definite root tensor approximate chebyshev spectral square tensor scale locate volume gray white cd gray cd time error tensor singular rank diagonal grow symmetric factorization cost symmetric scale manifold benchmark
field office surveillance record expect correlation characterize datum intrinsic structure kind video geometric sample multiple reveal original extract view video individually method include complementary view integrate video intrinsic human video intrinsic htp multi framework view firstly unify video metric simultaneously capture real learn project greatly video preserve intrinsic view specifically maximal disagreement minimization summarize extract receive considerable past decade devote survey merge metric among disagreement maximal involve trade aim find margin superior traditional criterion approach optimize measure kernel metric finding metric base kernel additional pure utilize disagreement criterion study past decade comprehensive dedicated focus tracking move overlap view fu systematically video surveillance video use hypergraph view view suppose level view k unified matrix structural disagreement loss control objective former part problem preserve datum original disagreement criterion instance supervise achievable introduce graph ds tx sx ig cluster metric therefore formulate disagreement one cca matrix introduce variable simplification cca lead classification yet disagreement base learn introduce tr metric implicitly mean accord metric cut turn solve descent via eigen decomposition solve quadratic convergence assume real semantic space latent video dimensional feature usage impose disagreement minimization criterion decompose record k dx kx cluster I summary conduct road office hold scene frame show important object highlight video view construct view cluster select representative ed space dm diffusion important event office define length run ed dm ed office summarize video run frame video five participant give normalize comparable part multi view baseline tp office run cm ed dm reconstruct multi separability view learn propose combination metric define trade similarity original one science ns traditional design summary
ann feedforward pass ann activation lf b l l l ji I tensor extraction result rigorous back propagation norm natural machine character mnist digits activation th instead deal derivative herein write u towards begin function pass apply instead next constant embed play edge embedding similar connect belong however soon herein key herein projection employ herein advantage benefit instance overall different sum thing distinguish herein discriminant minimize less write ann layer shall henceforth unless implicitly incorporate addition regularizer substitute formulation space ann widely output negative ann separated belong different wish negative result sign pre term wish make extract weight decay prevent become help prevent overfitte finally control significance objective achieve naturally arise computational feasibility appear equation shall see sum involve carry minimize term undesirable grow go heuristic sum point ann sum distance near class thereby rise machine extract contain n pd ann vector activation clear expect lead gain heuristic situation wherein kx kx heuristic feasibility heuristic constitute equation think well develop especially fact clear back derive write shall illustrate since computation begin third compute ij plug introduce j w q notice permit write q round give shall propose minimize respectively use descent style compute l ij devote focus precede form stacking column denote transpose ann bias go equation chain dot column go equation particular consider herein ann bias ask compare reader simply replace latter role ann aim try supervise write clearly ann bias side equation perfectly fit put q require partial arbitrary weight essence task w ij c c play output proceed ann bias evaluate give bias consider play mind aid rule try compare ij value w w write b equality highlight stand truly ij l b ij know true essence compute l involved minimize indeed follow thus u u ann u require save stock far natural second quantity w b turn w go thing require advantage write u consider perfectly w ij analogy clear analog combine equation essence u backpropagation quantity call backpropagation pass train current vector ann flip argument backpropagation role current activation ann play derive b w ann ann activation end feedforward ann activation network l ij la repeat rather rather thereby obtain play compute final w correspond ij w ij proceed keep mind write write less reader keep plan backpropagation computation notation first feed relevant transfer sigmoid depend show derivative arbitrary weight
conjugate conditional xy ts yy enable c df begin define partition via df proceed observe default assume center scale update surrogate gibbs full c posterior approximate yy expression hold model df accuracy comparison draw df converge excellent report coverage length one treatment group eq sampler new I kt gibbs conditional conditional specific statistic df df specific mean I nc define set add simulate model estimate c df similar df inference extremely sensitive estimate perform parameter estimate uncertainty table occur make joint way df approximation conditional coverage kt coverage credible average arrival standard df full depend yield mcmc correctness approximate sample datum posterior closed increase often df model regression close form variational introduce additional pass augmentation via probit conditionally auto overcome storage arise section provide excellent example df apply conditionally finally consider poisson augmentation admit quantity online df apply namely ar default u hierarchical good backward kalman computation strategy intractable time grow univariate key importance sampling less grow propose move exploit latent otherwise poor resample window shift estimate df evolve dynamically define conditional distribution quantity enable approximate draw df metropolis hasting df metropolis within sequentially b j hasting bc generate experiment report prevent df propagation pl density pl df df concentrate pl albeit variance pl coverage df near credible pl see df substantially require pl use df produce contrast pl spread window expense hand avg mse df pl tn h c df pl metropolis pl memory ram propagate pl report big df probit augmentation sampler follow sample variate full truncate conditional present bottleneck size latent df surrogate budget index final observation partition dynamic I c df observe conditional xx tn I z collection draw c xx xx z b latent index example first generate table inferential case although coverage I predictor df addition good estimate avg coverage df arrival scale linearly observation storage score full conditional storage h additive mixed effect predictor q I implement df close surrogate propagate make increasingly restrictive restrictive seek posterior ignore among joint posterior completely specify df specify replace fix variational df draw prior u update rw u cccc report experiment case choice map default low df competitive increase predictor predictor suffer dense df excellent estimation high report coverage predictive df vb suffer restrictive df error replication case vb df error case c interval replication surrogate quantity sample namely report simulate c c df mb mb mb quantity finally update quantity surrogate df df online linear variable predict action base relate status angle dataset relationship vb entire mode sequentially obtain df effective comparison show df consistent summarize table partition observation horizon test method competitive predictive mse interval coverage df variational bayes vb suffer predictor select df partition horizon df suffer h c coverage df vb df identify study class mcmc df present limit stationary deal class sample proof time take borel tt omit namely tv characterize kernel finite horizon kernel calculate c df initially assumption kernel stream partition p p df require approximate admit quantity conjugate kernel proceed gibbs fashion conditional available transition hasting df stationary conditional scenario model conditionals dynamic present admit although hasting c df accommodate increase admit asymptotic produce establish latter case produce excellent draw n n essence refer lemma ergodicity ergodicity transition regular approximate chain formalize outline state neighborhood domain almost say grow rate regularity stationary df distribution mean get primary easy sequence conditional filter novel streaming df produce draw conditional posterior rely surrogate sufficient statistic quantity gain massive bayesian popular often propagation ep particle sequential approach face substantial difficulty exception df demonstrate illustrative c df characterization df runtime sampling improvement support table relate use relationship tv use r tv tv compare note follow gibbs stationary identical manner take account obtain q complete generate one obtain continuity assumption yield law consider evident law algebraic q similarly eq tt motivating coefficient representative group mean j response lee generalize pareto bayesian via fisher scoring filter nonlinear gaussian tracking stream variational process inference filter f g particle smoothing sequential compressed f wang theory poisson chen mcmc cut metropolis hasting propagation mix machine p expectation linkage cpu gpu e regression virtual j variational framework fitting generalize west forecast dynamic wang dirichlet gradient langevin b monte carlo pt pt lemma definition remark em em depth david filtering inference df mcmc online sampling surrogate conditional need store offer desirable memory requirement improve state art demonstrate dimensional df target sampling proceed data keyword approximate big monte measure statistical increasingly curse dimensionality truly algorithm infeasible probabilistic characterization lack scalable inference guarantee sample gold standard emphasis monte performance size mcmc number grow evaluation mcmc accommodate adapt transition draw sample joint mcmc require storage sufficient face storage scale architecture likelihood another monte chen sequential hypothesis approximate metropolis hasting conjugate model efficient updating obtain density broad approximate approximate posterior tracking error propagation ep assume convergence posterior predictive uncertainty variational et streaming combine stochastic inference variational method uncertainty accurately except carlo online rely sample smc involve large prevent degeneracy expensive degeneracy particle pl degeneracy satisfactory estimate add complexity propose mcmc streaming df proceed sufficient surrogate point sample df enable df definition identify update demonstrate online apply df regression way sophisticated section extension probit df increase also variational storage mix art df dimensional inferential extensive finite mcmc figure appear observe nan j j j value sample sufficient storage overhead multiple structure create computer ensemble updating cause address propose statistics j c c df proceed draw distribution potentially expensive conditional fashion
one receiver operation roc receiver error decide generalization base cross problem method overfitte hardness amount problem address case reservoir address aspect issue experimentally attempt develop reservoir paradigm reservoir successful machine aim understanding work reservoir study completely satisfactory result vary power delay perfect past computation finally reservoir carry implicit represent dynamic functional delay resource error long dependency narrow line fourth resource resource delay achieve reservoir computing trade relation nsf grant surface fit fit fit statistics h correspond matrix minimize represent fit fit goodness given map reservoir size task training error delay line measure c training measure measure testing training measure test training measure test I w reservoir node compare paradigm computation integrate method power ordinary dl memory arbitrary neural step reservoir problem series square error broader systematic autoregressive provide solid evidence computation device neural reservoir reservoir connectivity represent rescale radius sufficient input connect reservoir weight later propose assign et use reservoir scale weight ensure also contrary reservoir optimality demonstrate experimentally affect study reservoir performance depend heterogeneity performance communication reservoir scale world topology significant performance song coefficient odd circuit study study simple reservoir node cycle homogeneous reservoir arbitrarily find address reservoir demonstrate deviation normally distribute reservoir exponent optimality dynamic solve understand reservoir computation power computational delay line signal respectively delay line dl allow delay version reservoir dl connect read dl architecture delay delay fed teacher use teacher follow row dl row teacher dl augment constant initially dl state set autoregressive neural delay hide bias function linear transfer train perform teacher output previous value architecture effect fix vary reservoir reservoir connection reservoir reservoir distribution input reservoir evolution reservoir discrete time nonlinear reservoir row reservoir weight target linear dependent experimentally evaluate model act storage computing dynamical performance power attempt create dl task map lag dependency compare performance requirement performance easy white require computation baseline move discrete time present nonlinearity lag adaptation stage scaling systematically show averaging result run combination nonlinearity sensitive change heterogeneous weight assignment deviation experimentally power result power power sensitive map behavior qualitatively division understand attempt size device memory theoretically computational input universal reservoir clear computational dl perform functional comparison dl normal create weight reservoir choose optimize performance non map pattern simulation test step average follow line change delay line network series hide reservoir five error five reservoir reservoir use show measure normalize task decrease soon system sharp drop curse teacher expand delay overfitte high line highest simple logarithmic axis behave differently test around begin characterize increase test low time test reservoir dl reservoir task sign layer merely system find report dl comparison figure task map achieve delay map line narrow delay average task pattern delay task h delay much resource achieve figure comparison map dependency network result similar
theory generalization extend probabilistic discuss wide class trivial design operating evaluation recursively construct variant probabilistic estimate initial triangular matrix compactly point construct naturally treat posterior arbitrary distinguish property taylor coincide term numerical xt p method order strong ode currently mark parameterized introduction standard mean px sx k close map recursive analogous recursively form incorporate xt hc hc final describe ode estimate share structure sum evaluation careful gp ode perhaps naturally choose ad hoc guarantee choice work square kernel give euler model integrate wiener improper limit gauss markov posterior proper process give interpretation gauss although ode distribution gaussian uniquely suited banach add reproduce issue put posterior find second force albeit lie associate enter gp affect importantly interval solve begin error order term fan chain solver rather try option question pose hard answer means return method degree answer bar proceed main gaussian integral wiener process shift conceptual prior gray blue empty circle mean third green respectively line method final proper intermediate ode integrated wiener evaluation give rise euler corresponding hold observe value evaluate directly algebraic straightforward integrate wiener move fortunately limit lead twice integrate wiener choose evaluation node rise family limit integrate wiener gauss turn improper hold constraint extend show find second limit improper policy placing wiener evaluate specific rise limit entirely analogously match u table posterior mean q final node regressor highly integrate wiener wiener surely seem performance gauss trading good high integrate process model integration leave open conceptual probabilistic accept solver continue first evaluation oppose euler form next interval three option suggest novel classic joint column hoc run I evaluation first remain gradient posterior go point produce confident global perhaps probabilistic framework else strength continuously establish iterative lot weak figure result approach I lead distribution function naturally global simple case course least operation planning follow publication form bound favorable ad hoc fig se calibration close true se bar cover confident advantage calibration due choice framework show se line deviation show interpretation limit gaussian wiener class return particularly ode solver add point open solver rd ode solver proceed stage acknowledgment grateful omit paper additionally website publication symbolic toolbox multivariate analogously method work value derivation supplement carry case consider model separate kronecker define dimension dimensional wiener lead derivation wiener derivative eq last iterate derivative kernel necessary form form derivation symbolic integrated eqs observing euler generality convention covariance formula throughout appropriate simplify formula omit state formula sec perform symbolic toolbox write covariance covariance distinguish theorem covariance value integrate integrated wiener process infinite list function rbf kernel euler choice yield euler predictive weight euler even third order insight everywhere contrast wiener process interpretable gps markov model cost cubic reduce rgb rgb department systems ordinary art return return define ode work construct output good remain ode light solver provide rich output dynamical
measurement illustrate fig death illustrative grey analytically birth death absolute magnitude difference mean eqn consider absolute birth copy distribution proceed quickly note birth death process ref relation g z z expression compute likelihood measurement copy probability limit increase later posterior facilitate comparison abc posterior analytic expression posterior give support fig parameter birth analytic assume posterior structure right posterior thick vertical line show parameter alternative likelihood employ square eqn refinement abc abc first within guarantee reject vary reduce mean abc match low resemble analytic kernel enforce acceptance substantial away prior include parameter likely enforce negativity restriction step abc latter yielded investigate yield density support protocol allow lowest straightforwardly computable responsible approach abc speedup phase inference fast test approach parametric analytic result likelihood population assume start division uniform represent rna may rna loss copy straightforwardly ode use stochastic give copy ensemble trajectory compute average copy series prior discrete prior wider ensure range enforce range exhibit uncertainty rigorously mean skewed suggest possibility original posterior increasingly converge acceptance trial reject first implementation number inference experimental model rna number induction subset measurement line inference centre parameter ref assume infer posterior support quantify threshold efficiently infer stochastic biological analytic form likelihood abc abc mcmc avoid region straightforwardly trajectory equally speedup volume trajectory approach include sequential conservative approach alone exceed distance half contribution trajectory mean protocol mean refined threshold exhibit rejection achieve conclusion include variability allow powerful question propose arise deterministic exceed exceed negativity deterministic magnitude mean reasonably assume expand measurement variance measurement validity assumption biology desirable observation mean variance quantity circumstance analytic form usage implementation abc powerful tool calculation computationally rely repeat stochastic simulation address simulate demand behaviour lead speed synthetic variability much recent evidence biological cell influence remarkable example include within cell cell drug cancer information mechanism magnitude measurement stochastic description fall inference typical simple biological system simulate behaviour datum biological variant often require therefore simulation noisy biological measurement individual mean variance statistic parametric abc perform dramatically inferential decrease stochastic approach gene experimental quantity biological interest appropriate assume begin consider measure variance quantity imagine cell measurement rna take individual measurement constitute measurement measurement time develop series system quantity uncorrelated meet analytic sample assume record statistic uncorrelated distribute deterministic assumed estimate overall associated individual ref specifically log measurement overall associated eqn assume underlie associated sample propose protocol approximate bayesian abc abc computation likelihood true posterior absence explicit measure measure explicitly write trial complicated dataset distance summary summary threshold record posterior decrease computed posterior leave metric observe simulate simulated rise therefore difference facilitate comparison multiplicative allow comparison different magnitude associate mean versa square include model weighting note always higher change exploration perform inference begin wish abc rejection trial simulation fall trial yield ensure yielded accept possible perturbation represent propose perturbation kernel optimisation rejection times compute simulation times condition step pick accord anneal search initial ref parametric heuristic adequate search reasonable distribution employ algorithmic stochastic contribute final posterior desirable whether reject biological analytically easy perform simulation advantage perform
manual tuning network across patch also determine subsequent add horizontal work significant reduction b notation cnn filter channel dimension output channel convolution assume one channel accelerate multi convolution rank cross filter channel vertical filter dimension filter separability filter rank increase component solve eigenvalue filter alternatively could restrict connection field separate upon apply convolutional neural modification feature define convolutional cnns convolutional layer convert bias equation clean important remove slow three propose model alignment cnn classification experiment environment handle gradient update baseline model perceptron cnns minimize double perceptron filter layer prevent adaptation augmentation concentrate learning capacity initially eight one cnn achieve result table dimension baseline bias add operator layer cnns construct describe perform convolution channel layer horizontal whose convolution channel view post process fine tuning need structural filter except bias replace filter result drop dataset commonly cnn predict reduction distinguish discriminate feature find removing imply toward essential set vertical horizontal set achieve standard layer consist stage decrease model depend difficulty reduction configuration baseline decrease adapt vanishing cnn layer provide flexibility help path accumulation cause decay gradient trend channel update gradient weight cnns convolution vanish gradient handle initialization initialization balanced pass yield successful heuristic initialization table cifar solid mean indicate deviation variation model small illustration proper weight baseline figure baseline decrease baseline early region accuracy seed result variation backpropagation filter cifar though layer convert reconstruct cross two filter necessary surprisingly color sparse penalty effectiveness explain comparable baseline cnns trade convolutional stage channel spatial filter dimension ratio convolutional left layer layer portion smoothly cnns decrease layer channel usually begin perform second though convolutional achieve term r baseline c technique layer layer parameter optimize consumption concern affect parallelism scale intermediate backward pass whose consider break convolution produce intermediate need na I loop optimize memory usage highest adapt scientific I exploit memory real cnns many baseline multiplication memory convolution layer parallelism resource feedforward oppose convolution baseline convolution convolution break number cifar cifar mnist dataset cifar consist testing contrast normalization whiten whole reach outperform baseline mnist consist digit training apply per whitening cross correlation cifar highly simplicity almost accuracie cifar baseline structure cnns computations cnns baseline feedforward backpropagation pass present cpu gpu check parallelism measure intel cpu gpu model baseline piece filter ht feedforward pass convolution filter convolutional layer pass acceleration tend overhead become negligible speed reduce computation acceleration efficient feedforward acceleration convolution process consider effort access imply cnn channel increase use reduce training cpu gpu backpropagation consist update access convolution feedforward however accumulation convolutional operation frequent access acceleration negligible gpu technique convolutional feedforward acceleration convert layer channel vertical horizontal filter successfully ten accuracie cifar cifar addition effort manual difficulty learn reduction accelerate model remain acknowledgment office grant pr edu neural redundancy weight filter convolutional extensively heuristic rank train consecutive dimensional filter across obtain comparable conventional convolutional substitute filter feedforward parameter effort manual recent success convolutional network cnns enable researcher network cnn audio understand security system mobile accuracy execute cnn
complement construction capacity pose question term measure cloud learning involve functional minimizer functional involve minimizer correspond limit functional set imply notion convergence extensively calculus phase material science convergence establish cloud symmetric decaying rescale particular connect assign weight define graph capacity variational description total point cloud typically total vertex obtain lipschitz boundary support consider weight give z connect order limit rescale scale variational sense contribution identify empirical point measure arbitrary denote borel q borel second provide convergence pl pf consider grid cell grid grid consider isotropic radial pt kernel broad coordinate vector replace expression side otherwise rest denote main connected let let tv variation subsection definition sum become write topology good topology theorem uniformly precisely relatively subset hold allow tv conclusion functional scale converge theorems point total appropriately scale converge usual appropriately scale notion pointwise convergence smooth convergence obtain converge pointwise scale pointwise pointwise directly theorem imply subsequence present point distribute theory probability remark surprising describe task minimizer minimizer example may setting dd n convergence minimizer approximate functional minimizer limit extensive exposition book classical functional convergence free energy phase transition energy term kernel consider system show functional study one work conceptually work elegant complicated discrete set functional include interpret limit functional rescaling energy consider get depend size van discrete lattice functional year develop task important desirable limit procedure limit euclidean pointwise work von von eigenvalue work allow converge pointwise admissible regularity requirement show convergence cut knn graph consistency algorithm algorithm involve cut total functional illustration context consistency primary investigate satisfy problem find minimize domain minimizer complement describe minimize functional consider extension constraint inequality satisfy subsequence guarantee minimizer depict resemble present minimizer take minimizer htb variation tv function support function function define graph variation function approximate map preserve precisely subsection rely little far need variation functional weight analogous functional preliminary convergence space fact variation list subsection subsection notion space introduce prove result convergence functional tv functional prove extension necessarily independently distribute point case correspond context write abuse replace expression finally weight throughout restrict bound positive open equal norm also surface integral belong finite measurable uniquely condition variation associate call check depend derivative distributional supremum continuous functional respect finish approximation open every open algebra borel give set marginal refer wasserstein distance minimizer distance reference weakly borel denote borel integral tx absolutely induce via lebesgue measure equivalent sequence map inverse give third refer composition note point want point mass let lipschitz boundary rich sequence seek turn infinity maximal map infinity match rich matching form regular cube volume point point exist meaning range distance os dyadic dyadic composition consecutive nature length locally hand globally behavior scale source scale conceptually proof result domain borel lemma imply follow subset boundary measure let sequence let map discuss notion functional converge metric functional inequality hold q collection point check f identify measure pd pd pd dirac delta absolutely measure closure pd introduce pd remark lebesgue induce plan deduce f f n pd pd pd pd attention absolutely lebesgue eq metric subset topology convergence lemma prove map measure class bound know note vx obtain prove detail mention end triangle finally satisfy deduce statement every moreover measure absolutely lebesgue measure equivalent previous statement claim absolutely map nd f metric slight abuse notation compact absolutely continuous lebesgue u last one light remark finish also provide follow nevertheless decide present canonical distance topology endow complete endowed characterization moment completion geometric lebesgue converge remark distance remark functional weight variation open bound constant tv follow idea specifically first inequality functional argument presence boundary consider function regardless regularity definition family notation number converge zero limit simply limit let arbitrary purpose taylor inequality deduce diameter set finally straightforward claim diameter zero go claim q eq deduce right zero imply enough kernel suppose u generality bound lemma plan function limit gain function support bx u u x x dx infer estimate second jensen chain inequality assume conclude u open set compactly idea lipschitz bounding case conclude desire indeed define suffice right proposition u compact support assume k last change note transform equality thank symmetric straightforward constant implie therefore conclude deduce support kernel variation function moreover continuous simple enough imply q dominate note establish regular open relatively compact assumption function bt change assume constant enough x dx px establish geometric dx dx dx bi establishe outside radius straightforward yield eq role small eq follow bi lipschitz fact second inequality proof sequence remark proved bound assume function boundary remark bi lipschitz domain bi variable assumption compact set ball compactly lemma set compactly hold union set compactly cover finitely ball boundedness approximate l convergence bound lipschitz positive use let hold complement two matter estimate take slightly radius match almost u u un deduce tv remark proposition prove continuous nx h proof piecewise compact denote function use step compactly satisfy constant nu nu lipschitz analogously function proceed analogous inequality nu assume lipschitz tt n bound true nx ny ny ny nx ny ny eq change step conclude map nz enough note deduce l corollary n n inequality lipschitz restrict characteristic follow take advantage formula energy measurable exist nu tv verify functional satisfy tv n tv approximate characteristic key substitute follow follow approximated characteristic measurable class specify volume argument remark consider distance theorem assumption able distance would translate
ideally alone devote density drop quickly mathematically say function rigorous density multivariate preliminary construction system haar dimensional construct alternative necessary high basis exponentially partition illustration haar wavelet scale wavelet eq denote vertical translate l l j together dimensional haar wavelet let haar construct expand respect haar wavelet size impose function wavelet control decay widely use characterize dimensional next want localize cube seven function still level haar coefficient display decay trend generally haar condition perform haar plot haar coefficient coefficient absolute clearly haar coefficient estimation correlation mode haar haar summarize law try estimate spatially concentrate likelihood partitioning detect unknown structure rate partition condition normalize support rectangle negative haar basis upper therefore density support reach achievable rate small theorem depend oppose agree acknowledgement author thank discussion department policy stanford problem introduce estimator constant partition learn analyze reach conclusion curse adaptation calculate circumstance fundamental inference natural straightforward develop however currently increase size great may geometric dimension like suffer difficulty paper employ still paper thorough analysis convergence method datum establish density design low approximate sensitive window good need depend datum especially multidimensional difficulty cause classic show rate smooth method parametric large density still slow optimal curse seek convergence large indicate order depend result establish perhaps basic appropriately origin width histogram bin allow bin histogram generalization geometry density estimate density learn recursively partition partitioning allow opt prove support variation posterior yield opt tree p successfully computational recursive extremely resolve base partitioning develop major distribution analytically asymptotically correspond kullback leibler employ sequential importance dimensional eq density kernel kde apply compare hellinger estimated kde comparison unknown density histogram opt essence flexible enough geometric overcome thus density partition essence interest lie area perform dimension class partition achieve quantify representation quality formulate error density partition demonstrate optimal meet challenge far reveal nature explicit respectively insight high rest paper organize main section devote spatial respectively density measurable lebesgue scaling unit cube f space restrict one partition grow precision idea integrate search partition unit cube step recursion partitioning region choose along range partition display possible binary partition support size constitute approximate space background induce hellinger hellinger distance say well approximate satisfying let convergence section binary become determine log partition incorporate select promise accord score likelihood ready study converge lie cover precise fact decay empirical one part ratio difficulty truncate truncate truncation maintain key likelihood guarantee behavior truncate exist fail truncation omit truncation log expect low truncate log likelihood ratio large n fy fy hellinger induce ratio inside hellinger let see apply deviation inequality change rely class far satisfied likelihood explain probability ball radius beginning zero converge function denote easy check converge determined return expression lemma iteratively achieve f f k I last need dy dy mn du du du n apply establish argument automatically similar du kn n n I replace satisfied kn n condition assume n kn kn exactly order small optimal order select greatly contribute simplify improve reduce overfitte formulate dimensional essence true effective section calculation compare kernel regard relie make exist question class haar density approximation true select haar certain criterion tensor haar volume support wavelet involve derivative improvement variation condition replace mixed h controlling develop approximated density accurate description notation haar simple wavelet haar scaling haar haar haar scale turn orthonormal product haar respectively haar calculation heavily respect haar hellinger calculation hellinger haar basis haar support special denote closely approximation
different cross provide function average distribution leverage common elaborate end predictor label label framework svms however label information base conditional give learn challenge convex hard addition incorporate generative lead improved discriminative extract change set suffice consider regression set derivative label label vary locally valuable contribution derivative use establish high derive call score incorporate subsequent correspond many discriminative model feedforward neural setting result present supervise unlabeled distribution mechanism assume model code unlabeled rich coding learn assume score new relatively straightforward transfer estimate unlabeled estimation transfer propose feature high semi unlabele form high order apply several thank support support award microsoft fellowship award nsf award award proposition derivation claim example time bold department california ca usa department science california usa forms speech computer novel value unlabeled sample efficient extracting label theoretical characterize nature labeled employ tensor extract employ rich overcomplete thus discriminative feature supervise score representation achieve machine domain computer natural traditionally engineering tailor towards task consume instead automatically feature framework component ica exploit vast incorporate prior typically model incorporate explanatory associated generative boost discriminative task approach focus unsupervised employ expect scenario unlabeled one framework transfer learn adaptation dataset framework extensive learn challenging computer vision huge unlabele label one syntactic semantic access amount unlabele human unsupervise purpose without goal specific task human extract general purpose capability design unlabele give question concrete answer value general pre label present leverage discriminative information spectral discriminative extract pdf capture local score high tensor rich value allow characterize nature extract input label input moment label discriminative task employ decomposition find tensor analyze suffer spurious optima convex maximization overcomplete representation argue overcomplete get advance discriminative framework scalar tensor continuous handle structured problem unify end extract pre present gray shape corner sep purpose moment gx find width width line pt line width draw dash corner draw corner green width c extract discriminative characterize score label vanish carry however label degenerate certain derivative vanish even derivative average carry function moment useful discriminative discriminative model challenge establish discriminative work spectral recover challenge discriminative generative discriminative feed pre classifier fisher feed behind information learn classifier prescribe label sample conjunction sample run converge good solution
experiment ideal change condition potentially create different indicate task realization parameter normal reflect sophisticated manner individually element dedicate psd applicable psd newly formalize frobenius norm use use kind problem formulation weight reflect intuitively value deviation weight introduce balance variation small subtle deviation psd width cm experiment eps color distance matrix roc method bar eps specific learning modification replace regularization computational estimate solving guarantee exponentially overview hundred previous directly hadamard duality relate primal problem conduct alternate direction admm method density matrix try discriminate state note element normal fluctuation limited state obtain depict nm outcome contribute convert add diagonal density fig experiment numerically pair reduce density matrix state color trace state dash line fig roc curves width bar real eps select normal experimentally performance dataset consist five matrix tune value raw matrix naive estimate introduce case ed fair remove statistical fluctuation invariant distance color fig understand discovery fdr give true without detect colored receiver characteristic roc vertical stand leave corner fdr naive indicate reliable quantitative use percentage roc curve fdr frequently study value value dataset demonstrate method fig raw naive th th raw contain bias calibration accuracy place mode depict typically physical bias reflect distance fig well naive method particularly fdr curve higher moreover experimental state matrix state experimentally respectively demonstrate performance experimentally generate computer simulation demonstrate accurately matrix fluctuation naive matrix statistical auc experimentally computer density show ed datum mine key broad area quantum state essential interested quantum manuscript try detect state anomaly anomaly reconstruct matrix cause maximally process error cause anomaly anomaly matrix able quantum physical system circuit anomaly detail etc applicable wide focused anomaly detection unitary acknowledgment quantum project foundation technology reconstruct technique accurate datum propose accurately deviation reconstruct contain intrinsic fluctuation check trace average method detection grow rapidly computation
generalize modify li two population generalize p devote side log normal test normal examine discussion give deal nuisance impossible nontrivial test density nuisance xx stochastically increase xx stochastically book ij chi generalize approximated simulation set generate calculate u mt lp hypothesis side consider experiment normal summarize generalize al study power close amount random model fit log value reject em example journal department department decade accept reject test normal calculation find analytically statistic illustrated normal inherently life application analyze biological medical distribution term log normal researcher also article
structure clean weight direct simple bit flip able flexibility match graph difficulty demonstrate flip process version match copy measure match correctly vertex vary seed seed seed performance flip decrease increase study simple system make map neuron chemical chemical electrical potential chemical electrical individual across hence undirected self undirecte neuron remove leave vertex match utilize dissimilarity performance unweighted graph incorporate seed seed run direct seed match remain chance run weight graph different year across time realization seek period common unfortunately graph order connect union two utilize seed run mc statistically tailor real http www www finance enyi robust difficulty space flexible real simultaneously simulated well cutting procedure future potential appropriate future plan principle heuristic seed working approach greatly limit big scalable essential application lastly justify dimension automate approach partially national security engineering fellowship university technology project air force research laboratory contract theorem open joint fidelity department mathematics md university nc novel incorporate paradigm optimization fidelity euclidean match simulated matching graph characteristic many give seek correspondence matching preserve matching document processing name efficient even easy applicability numerous algorithm excellent survey exist matching often graph know cut algorithm graph contain match thousand vertex excellent achieve modify weighted generalization currently handle number arise aforementioned robust effectively match herein joint fidelity algorithm flexible inherent performance simple potentially section graph problem datum example outperform handle match chemical electrical procedure outperform across ability incorporate classification realization demonstrate outperform match x permutation doubly graph seek find minimize minimize adjacency respectively seek permutation allow quadratic assignment hard graph know without seed seek restriction equivalently seek permutation weighted problem know matching begin doubly form utilize frank wolfe methodology efficiently relax finally relax onto matching procedure herein attribute excellent survey match fidelity task perform seek provide maximize fidelity general certain assumption necessarily let graph generality label assume pose classical enough handle match vertex match actor communication graph neuron true multiple match vice versa see task vertex newly reformulate graph match intuitive often amongst incorporate order simplify sequel also aim perform match multidimensional scale common space readily dissimilarity representation dissimilarity ideally choose dissimilarity dependent dissimilarity although address choose dissimilarity match achieve excellent sparse detail os r enyi dissimilarity neighborhood mark global shortest expect dissimilarity embed space preserve contain within essence embed match dissimilarity match though imputation graph dissimilarity dissimilarity increase complexity access full j equal additional unknown treat miss procedure vertex methodology present describe automate labeling embed embed point embed match capture poorly preserve dissimilarity capture fidelity closely fidelity separability separability error preserve across dissimilarity match embed target simultaneously control dissimilarity raw cost represent vertex simplify fidelity separability error embed preserve dissimilarity preserve seed dissimilarity essential algorithm multidimensional outline vertex embed dissimilarity involve procedure simply dissimilarity suppose ideally preserve argument ideally procedure preserve triangle match amongst embed embed procedure seek minimize stress configuration represent dissimilarity graph e sum neighborhood euclidean distance amongst vertex approximate vertex solve avoid impose generalized np example use many set vertex seek classic assignment solve via present lie appropriate original chose tackle dimensionality dissimilarity procedure automate spectral present principled choosing work initialize choose iii iv solve j output v matching amongst approximate reasonable significant boost
decompose e plus compressive scope overall approach global compressive acquire essentially matrix exactly location outlier far section numerical several recent work utilize technique salient scenario seminal parsimonious decomposition salient image salient identification examine image drive cosine transform demonstrate salient spirit note work propose salient compressive formalize establish compressive outline proof comprehensive experimental auxiliary bold letter matlab notation form extract index etc exception indexing note notation throughout exposition bold letter usage clear context sum singular column matrix denote transpose integer may formalize admit matrix rank outlier lie span span dimension let onto assume cardinality aside nonzero assume column aggregated nonzero criterion informative subspace distinction index column potentially case contain matrix task etc issue subspace rank incoherent basis rank seek column seek like spread subspace incoherence low formalize notion follow column incoherence column svd matrix say satisfy q basis limit element canonical undesirable implie describe single distinguishing assumption satisfy nonzero condition outline algorithm distributional distributional preserve length multiplicative factor least union argument row note randomly gaussian subgaussian whose gaussian bernoulli position proof accurate structural column measurement satisfy bind set identify salient interesting outli pursuit succeed subspace location satisfy analogous identify identify outlier interesting achievable appropriate parameter algorithm identify salient column comprise fraction compressive salient operate directly matrix specifically succeed probability n column outside comparison address matrix observation form measurement author assume nonzero column row normalize sufficient approach may performance effect simplify analysis use recovery component condition regularization simultaneously identify salient measurement great leave reader prescribed function way action intermediate argument analogous provide appendix satisfy structural fix distributional column incoherence property orthogonal onto succeed provide satisfy structural sufficiently close simultaneously satisfie third support set produce salient matrix satisfy distributional overall intermediate union bind conclusion hold hold comprehensive motivated map outli pursuit op employ optimization low nonzero column implement method use accelerate alternate admm inspire op execution op procedure implement pdf pdf pdf percentage increasing allow recovery outli experiment follow I matrix notice square norm sampling rate fixing choose row parameter column regularization algorithm whether step employ outli pursuit identify true rate assess recovery achievable parameter outcome regime examine fraction observation result interesting somewhat intuitive efficacy parameter keep move top increase matrix moving outlier recover trivial see outli rank background pdf match far compare outli identification simplify applicable high notice vertical complementary may column relatively favorable curve yield successful recovery small identify discussion section difference due large two inherently operate scenario permit combination entry albeit individually op entry execution op step process task arise vision surveillance identify map image salient object image transform color gray decompose overlap patch matrix correspond notice gray scale value input collect architecture experimental approach somewhat necessarily bit heuristic due may exactly reduction subspace span column learn retain small leading singular generalize salient norm nonzero heuristic select qualitatively positive three regime examine rate result entry benchmark visual method datum well op identify visually salient region image identify salient validate use plus also compare op detection perform still produce reasonably sample moreover acceptable map deviation ghz intel core processor run os execute procedure discuss overall fast consistency promise salient task pdf b increase rank outlier increase variance estimation outli performance formally pdf pdf pdf pdf pdf pdf investigate experimental methodology euclidean essentially level row column sampling fix correspond perform trial record albeit reasonable might perturb energy result difficult support scenario require well support choice normalize observe degradation notice level variance inference step demonstrate extension amenable scenario characterize underlie observe subset formally denote let location operate procedure operate sample set matrix subsample comprise insight composite subsampling express operation subsampling specifically solve identifying span low component analog orthogonal operation column available element row submatrix form index orthogonal span residual energy th j column recent examine subsampling pdf pdf pdf pdf bottom column sampling parameter right empirically subsample column subsampling row cardinality figure outli outli degradation increase approach compressive cs follow reconstruct compressive somewhat simple kind recovery albeit background sufficient identification insight exploit operate compress original ultimately successfully location identification conjecture procedure least additional structural column contain location limit approach compressive principal pursuit comparable direct prohibitive rate storage element implement establish row compression incoherence compressive approach compress suffice investigation effort complexity op comment briefly complexitie examine op solver utilize accelerate op scale step solver step along would operation projection summarize operate full additional operation step mn p mn n multiply factor similarly operation platform could effect form implicitly light camera overall may embedding visual application likely salient element proceed formalism dimensionality sense stable comprise part embed second lemma least choose value follow albeit different throughout portion turn embedding stable embed imply stable embed nonzero embed satisfie recall svd nonnegative matrix strictly incoherence state norm row true account lemma space dimensional span third claim salient salient equivalent operator utilize intermediate term interference adapt let subspace complement embedding complement result useful stable embed directly stable embedding embed ij coincide establish establish generate specify approach begin brief geometric discussion embed appeal stable embedding comprise take word establish entail appropriate approximately preserve length comprise subspace affine embedding linear well weak result though embedding subspace receive fortunately may former latter recall discussion establish establish strong follow column combination affine linear zero sufficient word factor square length union unique subspace union adapt denote subspace suppose comprised part two follow directly five arise compact subsampling specify arise value ease exposition analogous replace albeit portion main lemma outli pursuit whose satisfy structural span column operator onto complement span column shorthand lemma rank condition large since follow span bound incoherence orthogonal operator use idea incoherence follow recall comprise subspace span span finally hold estimate column column span part entail binomial tail pr number draw denote let pr pr pr note utilize small value within bind bounding since singular chernoff let hermitian incoherence calculation pr put large realization random variable identification accomplish representative cast context formalism without let stable denote unique canonical eq randomly generate embed probability straightforward imply least denote positive acquire draw replacement kn nk nm know success result exhibit predict tight population somewhat analogous
applicable arbitrary later derive entropy ball completely spline applicable trend verification rate convergence fast base step roughly maxima gaussians recall use apply next reveal potential advantage gain linearize linearize side special adjust possible singular increase incoherence generalize estimate average squared leverage linearize hold incoherence basic replace grow reciprocal minimum nonzero something reciprocal large strong fourier scan require singular assumption incoherence regular roughly network likely vertex exception choice fractional provide deriving rate style seminal denote recall estimate main motivation radius cover closely fractional error small guarantee become reproduce rate trend entropy well k univariate hence tune trend filtering early standard boundedness way univariate trend minimax match establishe entropy embed contain derivative variation proof unlike previous directly boundedness signal instead evaluation hard manner currently limit analog difficulty merely concern entropy strategy number rich purpose bound decay nice display reader aforementioned strategy regard beyond scope demonstrate capability third covering recover optimal univariate fuse atom atom respect meet concern univariate wavelet usefulness univariate trend filter alternative wavelet evidence laplacian superior wavelet smoothing basis understand future rate reach acknowledgment national foundation international centre office nsf google support dms assumption projection consider first quantity note write establish rearrange univariate difference h k factorial basis evaluate column prove result row odd dimension nan nan become one operator order eigenvalue laplacian odd k dl suffice complete difference factorial form difference define factorial matrix order evaluate mind expand read solution want hx p use discrete matrix factorial application inequality place linearize claim argue arrive quadratic root bind mean complete decomposition establish vector decompose eq fact bind term th incoherence gaussians put term together associate vertex combinatorial pair normalize obey remainder proof include completeness accomplish apply follow inequality claim kx x rearrange spirit argue thus eq plug q desire hence straightforward second back closely care apply first mm x important form entropy equivalently scale entropy translate scale author desire stick input form orthogonal onto span hand analog polynomial inner factorial function ff therefore sup norm result constant r td latter break conclude prove main content rest reading cover j cover ball original noting secondly claim n denote column immediately apparent ball center ball cover ball cover ball ball com edu pa department pa mathematics department california pa adaptive graph generalize idea trend nonparametric analogous trend readily graph trend theory trend local rich statistic trend filter filter trend adapt across stand laplacian regularization enforce smoothness globally much hand yield either else throughout computational trend filter regularize penalty nonsmooth enough efficient large computation trend possibly difference node trend filtering fall call framework define alternatively synthesis first construct observe basis wavelet likewise kernel laplacian analysis mix motivation denoise census pa arrange connect spatially trend filter signal smoothness peak panel figure mixture gaussian underlie noisy right fit filter graph smooth quadratic define ccc observation graph trend filter df laplacian smoothing df df unnormalized build effective degree df measure complexity model top df adaptively fit peak graph df substantially peak center df middle begin high neighboring smoothing perform poorly df affect quantitative assessment difference trend spline df smoothing due consideration trend filter well yet sufficiently regime demonstrate local flexibility trend filter article section trend cover basic filtering estimator look simulated trend discussion row extract complementary vector nan rectangular begin trend univariate role suppose observe input location evenly spaced order employ operator filtering reduce dimensional fuse recursively operator take difference th fit nk th polynomial evaluate location formally verify examine analog graph edge observe order trend broad matrix difference lie entirely operator difference difference achieve oriented incidence row sign construction graph trend estimate node recursion operator univariate multiply transpose square exploit nk univariate remove row recover odd remove row intuition difference polynomial graph section sparsity difference specific piecewise since correspond value interpretation piecewise might ask component piecewise define question piecewise construct many difference across note exactly property sparsity orient incidence piecewise linear structure number require linearly neighboring notion linearity require would linearity euclidean piecewise polynomial piecewise difference mostly likewise cubic second extend lead odd exactly difference illustrate node compute plot graph penalty explicit detail htbp penalty n trend laplacian replace penalty define laplacian smoothing graph lie high leave part penalty laplacian difference choose graph small analogy comparison filter spline smoothness laplacian strongly throughout community generalize variant mostly discrete functional continuous counterpart quite say may meaningful embed aware trend filtering estimate k follow filter odd odd incidence augment odd likewise admm iteratively linear system condition gradient method involve laplacian augment linear solver solve solve subproblem trend problem tv parametric max underlying promising employ stack dual trend filter adopt interior update hessian issue grow problem poor condition experience experiment trend order flow parametric compare moderately versus preferred max thresholding prefer run naive admm soft cc equation mse achieve spatial examine behavior trend filter stanford project compose facebook real facebook user truth evaluate compare favorable entry draw nonzero draw run decay walk assign send number choose noise favorable design favorable design smoothing adaptive estimation wide level achieve square summary estimate laplacian sparsity nonetheless competitive smoothing walk wavelet trend motivate rely regularization semi supervise goal write observe node observe fall trend regularization observe row class th encourage class behave smoothly last criterion prior principle act fix still interpret imputation perform unobserved large htb ratio misclassification laplacian imputation popular uci c c car breast heart ad misclassification rate imputation uci repetition draw serve pair case highlight case specification place regularizer think heterogeneous smoothness laplacian design might broad run machine repository near distance serve choose tune wide experiment summarize rate imputation seed mrf smooth alternative sometimes laplacian uci select entirely base popularity belief favorable heterogeneity label nonetheless broad illustrate regularizers pure trend filter proper smoothly zero observe represent smooth
mu sigma mu factor claim analogous explanation alpha symbol x symbol symbol expression alpha beta alpha beta gamma alpha alpha gamma alpha beta alpha gamma beta alpha alpha alpha beta alpha gamma beta beta alpha eqn expand beta use gamma expand gamma alpha eqn q expand eqn answer eqn prove branch branch branch recover p finally gaussian branch take sample like suppose deviation bound additive algorithm generality normalize true moment lipschitz chebyshev estimate affect thing mean last give cubic show case odd root root perturbation coefficient large root I large sign algorithm reconstruct get remark theorem ignore ignore n pt usa university consider upper giving moment pearson denote necessary sufficient estimate error provably logarithmic yet simple dimensionality far bind separate reduce apply learn strong previous gaussian additive among well naturally arise population vary gaussian biology economic deal case mixture collection biology pearson gaussian pearson empirical distribution th define sufficiently true mixture dimensional parameter identify pearson locate root candidate match moment among pearson pearson prove extended reliably moreover sample complexity quantitative constant sufficient interpret year arbitrary novel surprisingly dimensionality allow logarithmic show necessary important six suffice identify gaussian moment differ first moment suffice provide lead polynomial extend dimensionality within gaussians specify mean pick dimensional coordinate parameter additive hope component simplicity combine say algorithm indistinguishable separate overall weighted average analogous statement precise characterization estimate simplicity component variation ff ff proper parameter estimate gaussian parameter tight characterization sample guarantee total interpret also technical quantitative dominant ignore measure variation distance benefit nevertheless facilitate previous component rather approximate easy approximate discuss gaussian tight bound regime corollary mixture gaussians bound dependence arise th reliably separate deviation mixture gaussian bound also desire smooth corollary theorem gaussian away output n well I essence gaussians reasonably separation relative accuracy second mean treat theorem bound away simplify syntactic level polynomial dependence separation essentially inefficient mixture bind hellinger mixture variance hellinger satisfies square hellinger rule statistical confidence impossible corollary algorithm gaussian learn mixture require tight mean approximate variance algorithm use learn gaussian away note low gaussian small away learn notably essentially incur dependence bind quite notably simple extend different copy result variation gaussians variation dependence exponent nonetheless exponent polynomial dependence interestingly improve isotropic let gaussians matrix far exist constant learn survey reader helpful discussion prior polynomial bound gaussians result least prohibitive moderate mixture gaussian component variation distance improper learning gaussians component however unlike component impossible general state nonetheless strictly good also assumption strong aware improvement parameter outline start moment system equation recover block unstable arise exactly rather exhibit set recover gaussians gaussian add formally add subtract unchanged mean leave independent accomplish call inspire well correspond four regime regime know applicable regime variance apply gaussians indistinguishable parameter regime appropriate algorithm excess fairly depend unfortunately root pearson compute root solution variance get valid mixture moment choose moment moment differently bind match moment another prove small perturbation root nearby perturbation argue nonzero need excess think interested correspond match six moment moment suffice show root compact give differently set region equal high degree coefficient get extend simple straightforward algorithm pair covariance shape iterate dimensional accuracy additional tell associated know newly position must ensure mix simple work use pick valid verification work project four anti form gaussian far true matrix identify matrix give overhead match close hellinger absolutely measure let subgaussian identical subgaussian eq like parameter denote hence series term inner bound constant returning imply approach q sample probability appeal relation hellinger way least probability sufficiently small variance constant match alternatively numerically certainly yield something plug mixture claim require learn gain hard distance group independent probability probability coordinate mean least guess incorrectly overall give sample probability coordinate specify would claim gives extend low gaussian main issue input get separate mixture nearly formalize follow gaussian subject moment expect well separate lemma useful use polynomial degree magnitude zero show randomly draw set consider draw arbitrarily bind singular without determinant product determinant formal know appear constant determinant close minimum result exist mixture match mixture far mixture free gaussians means relative match gaussian constant free parameter gaussians kp pz theorem different mixture match almost mixture minimal singular z f desire low gaussian differ apply desire give algorithm learn theorem result one denote constant similarly denote gaussian simplicity moment away eq make use gaussian increase amount make relate design gaussian plus definition correct also define excess excess section estimation bound zero mixture give satisfy statistic true value estimate I thm sample parameter proceed estimate however work cause estimate good nonzero job bind improve gaussians indistinguishable figure regime invoke simpler weak return additive x p max pearson substitute clear denominator pearson rescale excess root multiple root five moment suffice uniquely identify moment analogously expression remove explanation combine excess moment say fortunately exclude enforce sign root cubic otherwise solution suffice exact excess moment perturbation intuitive excess inspection bound convenient lemma full generality normalization proof recovery extend claim constant max max condition hold either recover approximation getting normalize denominator low denominator trivial rescale p perturbation perturbation p recovery moment examine therefore q gaussian mixture variance excess moment I imply x f estimate decide branch take ideal actual ideal factor perform algorithm gaussian appendix simple dimensional show reduction need factor accomplish f multiplicative reduction assume learn learn mixture gaussian mixture obtain restrict ii order terminate put determine number restrict coordinate ji first k terminate put exist k k total successful variance accurate show succeed case describe occur accurate step coordinate I case far p additive coordinate suppose described step accurate output must variance indistinguishable matter ij invoke powerful let normal eq degree see conclude claim direct consequence vector matrix constant every analogous lemma anti constant probability reduction gaussian fc gaussians rescale mixture grid check use previous lemma grid assumption let net coordinate contain definition sufficiently mm terminate following reject let failure terminate choose p problem accurate permutation enough claim every element accept need constant eq sufficiently union accepted eq hence accept sufficiently symmetric argument hold finish distinguish case center span correspond large probability element gets accept establish probability gets reject reject union reject accept argument neighbor finish distinguish c center distance pair span cluster hence pt pi one randomness possibility gaussians sample theorem learns gaussian gaussians empirical sample tt sg sx good variation prove gaussian intuition work direction subgaussian overall normalization identity eigenvalue therefore permutation approximation since approximation frobenius approximation branch find generality correctly correct classification lemma result measurement complexity dominate improve covariance eigenvalue g
properly model include member phenomenon occur vary underlie among specification effectively address phenomenon rather example specification cluster selection probability variable yet place evenly predictor step let contain reach cluster far choose evenly share piece weight mr mr mr mr mr mr follow stop iff strategy proceed induction easy procedure inductive next end q p us marginal q iii equality iii predictor latent jointly define sequence random final mathematically event eq expectation take final decision event claim see p u put piece map ready mapping find final prior regression bf update numerically prior particular form generality zero place intercept setup show bf versus eq coefficient determination undesirable feature propose prior introduce hyper put correspond bf versus notation
ga see power take high control node country flow country I densely mean translate triangle united china dominant united china steady suggest country volume indicator strength change consequence relation major indicator look middle panel pearson product volume year strong connection however positive south find united take country term economic exception find large major material produce country gate major rather highlight network economic combine conventional deep insight economic total either quantity relation make economic growth development approach big economic live challenge algorithm individual balance payment account activity shift lead forecast centrality identification indicator indicator novel suppose provide powerful monitoring early combine network long portfolio financial product standard ahead derivative product investigate international major local topological economic development individual gate keeping power volume stand establish economic amount gain understanding ever acknowledgement like european open financial cm centre engineering study sciences department college university usa global european policy public aware strong global economic architecture partly central conference macro fail maker limited help go far face conventional link trade size magnitude trade flow logarithm country china triangle trade connect rise availability large quality system component vast cover merge current economic behind generate interaction financial interaction whole science couple main major agent market two change stock portfolio network achieve qualitative gain international trade country logarithm country link size magnitude directional clearly triangle trade china remain already view intrinsic study picture merge indicator fusion trade financial indicator link several indicator result relational evolution describe demonstrate counter financial playing design financial come back relation use economic indicator change economic growth economic attract grow explain economic asset different macro face substantial large e underlie computational complexity total year track flow position technique big predictive power begin apply methodology particular availability fulfil consecutive relational financial indicator country constitute part country eight aggregated show track year position focus country position total china china usa indicators adjust change network represent node allow take call capture topological symmetric direct require reciprocal connection account feedback kind ready single balance account network financial describe indicator interested find coefficient matrix number indicator intercept indicator appeal accord criterion super exponentially number indicator big enter much availability amount time regressor describe likely contain optimisation perform regressor generate model additional regressor order significance regressor factor condition normalise go back final number regressor accept crucially maximally general maximal regressor side balance reject availability cumulative eight together meanwhile union hold total unfortunately partly incomplete expect enough achieve fit great majority indicator see moment strict criterion order accept single test maximal final fit maximal test perform year shift regressor nine fitting forecasting indicator error median fit indicator connect meaning separate cutting tell indicator couple network cover indicator cluster could valid outcome picture globally behind united china indicator fit true node lie tracking capability eliminate potentially predictive capability stand alone country applicability approach indicator dot point regressor colour code capability indicator highlight position indicator regressor axis actual physical portfolio portfolio use measure structural global financial market derivative market early financial relate aggregated term market amount range financial derivative underlie formation lead international participant leave large remove strong link find connectivity contribute indicate threshold market financial derivative evolution edge market expect couple cross correlation market mechanism translate class lead mainly united role international financial market create dynamical view would channel increase cascade get couple good collective may outcome cover network report country country edge aggregated country country dominant global edge million remove edge weak change world year stay remain connected apply threshold mean global describe connectivity growth depicted large temporal evolution take value continuously expand node track aggregated counter product product lag behind rate use market scale linearity scale account arbitrarily square value derivative positive relation indicate derivative market net market availability amount relation financial high maximal threshold safe reference rv market product multipli setting exceed must reduce transaction hold especially trade soon describe amount derivative grey dash line consistent signal red generate product rv variety description derivative may present achieve description around threshold fraction set signal note range applicability applicability description derivative product detail description link derivative
lack historical work address content base recommendation available formalize pool available budget assign rating order optimum verify netflix outperform baseline database mining recommendation increasingly try choose read modern service service user preference database common recommendation collaborative cf g transaction feedback exploit popularity trend much one arise employ deal lack transaction history focus availability content information user rating item movie book evaluate pool available assign item period receive rating adaptively associate select characterization rest survey work introduce notation optimal algorithm common cf item user vector user hold latent translate mathematical new reveal item latent cast budget constrain stand expect prediction devise validate result simulate netflix effectiveness baseline turn problem item indicate high portion item notation denote seek bias vector item user intuitively strong preference vice versa denote aspect train rating term instance verify assumption employ significance formally pool available budget constraint allow rating budget formally rating notation user set translate inherently generate user item divide item give pool user reveal rank latent ground optimal detailed exist well baseline evaluate approach approach unify whole paradigm aim conduct wish select minimize seek actual subset mse basically optimal rating equation least estimator give user provide model assumption rating assume rating user whereas new therefore estimator seek denote concatenation shall stand notation yield column since invertible practice usually add sequel ease emphasize regularize formally notation adapt abuse notation vector whose column vector notation divide two term assume follow key observation optimization simple assume use prove continue mean isotropic invertible transformation item invertible merely simplify statement without compute implement I square estimator term inherent avoid p represent turn available user use vector follow substituting result within r noise equality follow equation equality optimum refer user user initialize j alg j state definition definite u f definite say monotone last extend correspond user alg b mse additive sub optimal notice dominate follow define whose column correspond latent show follow three equal optimum third left prove minimize monotonicity rise insight mse user translate rational marginal subset e eq psd eigenvalue prove specifically column derivative resemble albeit plug monotone operator finally generate substitute alg inequality rely heavily sometimes assume respect model distribution analysis advance rating estimator generalize correspond scale sequel r b identically equation generalize least subset assign rating substitute square rewrite motivation establish adaptation refer root user notation define alg b item correspond user albeit omit advance recommender utilize formally item rate estimate baseline baseline follow large movie dataset netflix competition dataset contain million rating anonymous netflix customer movie process rating henceforth choose contain million rating user rate movie rating rating exist root rmse metric prediction result model descent sgd detail regard evaluation omit offline netflix movie henceforth movie recommender netflix rate dataset set movie rating movie netflix coincide available user conceptual portion actual rating actual remain note task carry metric rmse monotone mse minimize rmse experimental rmse separately
gap order design counter happen finally bound main gap gap decompose cumulative two gap analyze equation similarly trivially gap episode asymptotic bandit bandit low item item draw I distribution bandit bandit item small gap formalize notion consistent number choose analysis loss generality perform low bound inconsistent claim partition bandit parameterize regret bound follow bernoulli variable separately regret sublinear practical match gap free upper match bind adversarial semi gap upper armed bandit comparable armed major gap extremely efficient ccc minimum optimal node bandit experiment episode select observe update environment measure episode cumulative episode divide baseline maximum weight basis notion baseline common internet assumption span formulate six table contain cycle record exponential expect tend unlikely cause high report trend episode outperform episode report learn network span tree therefore observe episode cr dataset bipartite graph connect several region return episode select assign assignment study family call bipartite leave vertex bipartite maximize overall bipartite handle represent bipartite united states constitute bipartite top bipartite handle episode choose maximize overall success rate success assign list policy success learn report trend approach episode increase outperform policy movie title movie american child popular optimal movie return movie episode diverse recommend movie likely movie diverse interest formulate dataset people rate million movie attention rate movie cardinality movie vector indicate movie movie movie diversity ie dataset episode movie list movie cover movie appear diverse suitable diversity report trend episode increase greedy combinatorial semi combinatorial bandit ucb chen regret regret tight chen tight adversarial combinatorial semi main limitation efficient need exponentially need project hull efficient special bandit combinatorial past year submodular monotonic propose algorithm suitable first specific weight unknown learn interact repeatedly sublinear world practical practical efficiently introduce case combinatorial generalization combinatorial optimally one idea quite applied involve lem basis exist ia constructive exchange augmentation exchange k ia main finally step set contradict item event counter happen follow equation happen eq combine claim due sequence therefore conclude lemma proposition fact remark com notion independence closely modular bring propose combinatorial bandit problem maximize modular problem prove bound free bound sublinear interest prove world application resource designing protocol modern problem polynomial fortunately combinatorial closely efficiency find common forest modular modular represent sum item vector unknown interact world span delay stochastic finding unknown perhaps explore network return contribution bring concept bandit bandit new broad combinatorial optimization solve solve explore face computationally efficient episode sort number sublinear episode linear maximum network network maximize third movie recommendation efficiently framework real adopt subset denote remain set cardinality negative solution design principle greedy find weight basis optimistic give begin episode q episode estimate item episode order dependent item confidence upper exploration episode exploration avoid regret extremely episode sort motivated work challenge regret basic notation decompose rely heavily
gradient nonlinear behavior example discretize enough acquisition behavior map several acquisition exist experimental ucb ucb acquisition trade ucb emphasis easy adjust see across weighted area space may trade code bayesian available degree position control mx front back control robot fast operating system simulator dynamic physics flat segment robot simulator ode angular govern periodic amplitude cycle cycle period signal hz amplitude cycle signal filter sharp angular send every ms third angle controller parameter numerous different purely setup type controller controller classic self controller design keep balanced least place repeat cycle static parameter reference controller keep cyclic orientation subtract angle actual important performance tend fail chance behavioral descriptor contact factor controller simulate record whether contact contact behavioral descriptor store cell behavioral descriptor space behavioral descriptor actual discretized behavioral descriptor robot characterize angular position robot proportion interval roll frame additional robot end interval ms second movement return argument exceed return discount around orientation exceed robot possible parametrize move pre behavior thank result robot adaptation measure mapping fast compare reveal measurement robot flip backward distance great adjustment consider behavior additionally inaccurate outlier supplementary substantially promise maximum physical controller robot unlikely decide worth well controller discover controller perform stop eq location behavioral predict terminate select alternative way robot equation event occur robot text solution criterion strictly guarantee stop bad behavior map test trial adaptation drop choose area controller parameter controller increment behavioral behavioral million physical extended fig robot release ball classic assess place camera tracking track color eight robot position control maximize heavy near mx mx mx ax simulated robot way dynamic version simulator ode library result map controller target position controller parametrize eight angle motion range joint activate drive target choose make reproduce highlight trial recovery advance realistic experimental trial controller controller parametrize chance change value robot behavior position behavioral position position measure discretized compose cell experiment arm behavior map step performance work location bin accomplished performance movement specifically minimize angular joint mean angle map accomplish measure create distance well descriptor target bin joint angle use create behavior controller robot distance external camera bin physical position controller controller position outside camera marker rare corresponding experiment experiment frequent adaptation purely random angle continue time minimize cost run independent algorithm replicate detect low auto trial contain trial simulate release bin adaptation stop bin cm stop adaptation within controller controller dimension evaluation million execute condition physical robot measure manually measure dash manual minimum bar text report section come robot behavior influence single affect magnitude test affect behavior affect value extend paragraph test conduct behavior affected affect repeat testing replicate dimensional stop adaptation pass stop text criterion increase change matlab range explore around dotted red large change algorithm rarely iteration occur many cover entire search fast risk promise area space minimum physical robot choose search already largely step map area thus avoid choose exploration experimentally conduct intel ram take across robot happen robot map robot simultaneous localization fast slow million per second powerful computer frame process computer accuracy computer step much easily arithmetic robot second adapt conduct robot overall physical robot second second initialize robot second allow measurement second identify controller time arithmetic column conduct select second second investigate physical text datum scenario scenario robot generate map run consist directly parameter time experimental control need trial model inform enough effectively empirically dimension policy algorithm trial allow highly illustrate performance variant search previously directly search high show error produce high space search initialize map initialization allow evaluation typically previous art policy variant improve still publish automatic dimension trial run original evaluation gradient bayesian powerful provide prior optimization component trial significantly recovery state environmental robot recovery flat supplementary create also eight map increment supplementary roughly robot perturb multiplicative trial design classic slope trial find slow behavior learn trial error outperform flat every slope angle controller trial reference controller setup map increment replicate degree increment trial variation slope trial error find slope find fast slow positive ascent algorithm perform trial cause controller sensor controller keep science course perform behavior keep vertical reduce nevertheless trial outperform median performance discretize map map point behavioral randomly location store intuitively far keep intuition understand advantage map controller behavioral descriptor diversity behavior average behavior map million evaluation robot distribution difficult supplementary median percent cell percent appear fig random discover numerous also million evaluation average cell whereas reference controller robot diverse search measure performance million behavioral descriptor dimension behavioral behavioral behavioral descriptor proportion contact create describe test performance alternative behavioral descriptor descriptor evaluate affect behavioral descriptor test behavioral descriptor behavioral descriptor contact denote boolean contact contact contact orientation behavioral descriptor characterize change angular measured proportion roll angle dimension roll angle robot movement return exceed return otherwise motion around angle exceed orientation angle negative dimensional behavioral characterize proportion ms intervals robot along axis robot interval second simulate movement less mm dimensional behavioral descriptor second utilize robot second movement measure simulator behavioral descriptor capture move movement utilize second movement deviation descriptor capture robot location robot straight speed robot center final axes maximum robot position dimension second axis robot expect robot speed behavioral descriptor multiply reaction force behavioral descriptor ground reaction force movement ground reaction force behavioral apply ground reaction generate average second simulate movement angle descriptor capture ground angle contact ground angle normalize low roll angle descriptor roll low ground coordinate roll angle time low ground roll range ground angle global coordinate average second contact angle behavioral descriptor differ knowledge randomly behavioral descriptor intend little behavioral descriptor quickly pick descriptor consideration instead generate list fashion randomly different select random without descriptor descriptor descriptor available descriptor behavioral descriptor physical robot repeatedly break robot modify simulator remove map million map behavioral robot generation store cell behavioral descriptor behavioral descriptor behavioral descriptor actual discretize eight behavioral descriptor map behavioral descriptor ten replicate descriptor behavioral behavioral descriptor therefore replicate descriptor replicate descriptor randomly perturb noise deviation fast behavioral descriptor trial robot require trial experiment achieve choose behavioral descriptor similar behavioral descriptor lead median descriptor descriptor roll angle performance discover descriptor significance remain descriptor additionally discover alternative choose descriptor lead well evaluation experiment extended trial difference behavioral descriptor behavioral three choose low angle median descriptor performance orientation relative relative roll descriptor remain behavioral descriptor angle descriptor case statistical significance statistically descriptor random behavioral descriptor descriptor descriptor descriptor performance significant reduce descriptor factor description choose behavioral descriptor behavior discover descriptor reference experiment show selection behavioral behavioral descriptor randomly median prior reveal algorithm descriptor trial robot video behavior produce map classic robot deal finally video illustrate robot condition video type classic form reference extend figure table france l fr leave environment however adapt variety think box behavior specify failure diagnosis contingency plan introduce trial adapt require self diagnosis contingency novel create detailed behavior behavior guide discover experiment successful robot way break broken way enable suggest principle economic notably distant deep obstacle complex environment behavior behavior behavior effective cope broken occur case robot straight begin robot type behavior automatically generate behavior behavior well robot rapidly diagnosis design contingency self monitor sensor possible situation fail diagnosis plan differently trial allow discover behavior limit occur impractical curse dimensionality fast algorithm constrain tuning require minute limitation adapt rapid adaptation automatically compute thousand behavior insight whereas start survey understand behavior previous enable validate store knowledge robot try behavior performance behavior end predict behavior discover quickly way without understanding occur trial behavior create robot automatically robot describe dimension behavior behavioral measure speed contact demonstrate degree capture behavioral fill performance search perform behavioral extended fig simulate million behavior need perform per robot assign behavior try robot drop select promising measure robot behavior nearby assign extended continue satisfactory idea capture gaussian approximate acquire search select behavior maximize information select uncertain exploitation select whose behavior physical record behavioral update update affect test whose measure good performance behavior robot need camera estimate supplementary parametrize parameter cycle joint supplementary space contact supplementary fig failure run independently behavior default factor behavioral description adaptation time independently generate alternate behavioral see fig lead search perform creating robot experiment space six contact behavior robot promise update performance behavior map nearby behavior confidence similar robot well predict performance fig behavior high overall behavior affect behavior c c central box extreme create factor behavior per supplementary method condition test create body orientation behavior box trial robot drop ball joint degree offset broken offset joint condition create behavior robot dynamic classic reference median vs suggest trial produce robot aside effective reasonably speed time vs vs vs demonstrate robot initially fast reliably physical reveal art adapting environment trial less second physical four include randomly descriptor b fig performance map standard bayesian work initially area adaptation work robot error quickly identify high approach robot arm condition arm behavior behavioral specify angle approach less minute second fig fig map behavior portion contact ground behavioral space discretize dimension color highest discover behavior behavioral dimension legend leave behavior leave dynamic engine simulator http www ode matrix pre adaptation map adaptation conduct robot right matrix discover circle represent behavior physical robot red discover amongst behavior find error predefine cope condition new body behavior work quick behavior try modification additionally optimization procedure similar employ human strong evidence combine prior bayesian trial error period idea day quickly improve simulator predictive exist differently rapidly adapt circumstance thank discussion european research european union horizon innovation agreement correspondence request material email behavioral controller far repeat depict newly rarely map expensive perform per robot create multi computer supplementary running behavior simulation dark green uncertainty predict light green band actual robot dash balance behavior perform try behavior acquisition initially maximal performance physical behavior performance uncertainty nearby robot maximum behavior threshold performance dark occur physical test expectation occur variant behavior prior equivalent map map none none bayesian al none gradient al one art et et al colored discover evaluation robot panel pool removal show require slope angle physical trial outperform black six scenario pool condition median behavior trial reference slope line color dash classic reference try median colored area supplementary experiment speed discover descriptor simulate remove six scenario pool across behavior descriptor contact robot instantaneous velocity robot robot straight vi reaction force angle ground without replacement descriptor design descriptor bold line color median colored area extend colored circle supplementary experiment color represent performance high map black circle indicate indicate performance versus reveal robot legend color map black circle robot behavior test performance versus panel point last behavior robot previous whether leave typical performance produce map behavior behavior angle joint arm reach tend reach behavior nearby robot physical robot replication pool experiment success replication replicate robot reach cm bin center performance physical condition controller behavioral space discrete behavioral physical performance encounter controller currently store behavioral descriptor reality robot robot user deviation gaussian major behavior adaptation adapt new environment behavior introduce map create search high
entry edge node stand lie define graph nevertheless universe cast adjacency sample triplet feature good label interact get easy thus partition four depend whether node involve resp resp predict four represent unseen unseen unseen family four undirected two prediction validation evaluate network procedure evaluate global adopt practical context tree ensemble presentation assume derive class apply sample concatenation straightforwardly new unseen homogeneous graph handle sample without constraint symmetric separate try around constructed learn model exploit combine arithmetic make node learn node r cn train symmetry prediction model try scheme model take max prediction lead improvement building sample conditional estimate set threshold proportion edge versus specialized homogeneous asymmetric still node could base classifier ensemble context briefly several tree tree feature terminal instance output associate reach instance identify split sample output typically make competitive term extremely select good randomly split root candidate attribute feature output instead method local global learning sample feature come tree grow construction leaf result rectangular submatrix submatrix figure illustration ensemble straightforward variant build tree output model subset build sample r make require tree output submatrix base cope miss us case global region root leaf input partition matrix profile partitioning pair submatrix furthermore feature tree case local grain another give interpretable rank local multiple provide ranking one row output separate therefore complementary interpretability prohibitive network million fortunately go separately relative relative explicitly gram computational complexity tree tree practice relate sample computational complexity multiple output complexity however output carry six biological homogeneous bipartite four find assess relative approach method mn ern three bipartite main interaction protein highlight feature datum localization use gene edge growth fitness environment successive feature ern bipartite tf gene tf connect expression drug target connect drug protein vector presence chemical drug absence fold cross robustness run fold cv assess performing time fold illustrate return choice discretization precision roc curve fold cv extremely tree highlight high node expect node involve want assess realistic usual evaluate degree pair pair degree evaluate protocol baseline successively homogeneous bipartite local output last curve approach mn protocol similar curve appendix auc l ts ts ts ts ts pair informative interaction baseline network highly curve informative performance roc approach mn multiple indistinguishable result line indeed multiple grow single able tree additional precision obtain ern protocol appendix cv explain difficulty cv replace fold burden randomization extra tree ensemble bootstrappe rp ls ts ts ls ts ts ts ls ts ts tf tf baseline drug protein high node pair prediction ie significantly ern kind prediction generalize tf kind equally due kind ern intrinsic difficulty generalize family four generalization protein relative number well baseline degree prediction generalize local ern multiple term additional several method literature ensure fair avoid tuning paper summarize publication protocol measure result cv roc mn ern cv develop apply local predict mn exploit performance local multiple ensemble infer mn inferior mn ern focus know evaluate cv ensure belong close slightly good regularize classifier one protein prediction prediction well additional comparison method competitive notice test tree achieve randomization explore ensemble biological train network family train single carry compare state intrinsic importance score almost nature reasonable computational requirement turn method less model approach term advantage possibility introduction one loose however possibility method extend local unseen step kind train prediction svms benefit reduce potentially compute well improve exploit potential correlation interesting sl local approach focus biological network evaluate method try date incorporate tree base ensemble method however comparison like family protocol term prediction want merge family rank novel confident question largely biological record unlabele negative notable exception unlabele theoretically account example acknowledgement bioinformatics platform provide resource network class channel protein nuclear similarity protein similarity chemical structure number protein size edge curve l ts ts
good candidate create corresponding rank candidate scatter size increase repeat increase size small candidate dendrogram feature default choice result candidate size auto instead true dendrogram still htbp propose simulate contain x p organize cluster control separate remain clustering linkage partition cut dendrogram measure candidate framework either candidate specific pre candidate average standard different setting average produce accurate affect cc another cluster specify choose true variable candidate sr produce sr indicate selection selection accuracy affect ratio affect cc average deviation auto candidate select candidate natural sr auto candidate may sr sr drop sr affect sr sr investigate r naturally parallel computing rank currently boost simulate fix true variable set combination setting linkage setting number different second figure conclude long computational time cluster relatively explain complexity approach computationally demand give scale benchmark perform complete linkage classical feature gene auto select gene auto specify cluster detail use also choose candidate number three case specialized gene level process step pre exclude logarithmic transformation datum array nearest neighbor miss value feature misclassifie case misclassifie auto feature satisfactory hierarchical high conclude produce accurate htbp breast breast tumor identify class er observation belong four publicly result mix auto select case misclassifie marginal conclude microarray gene identify type patient distinguish significant patient classical hierarchical feature misclassifie misclassifie poorly feature produce misclassifie subset default expand default classical feature high marginal mixed cluster thus however analyze breast publicly discriminate distant within classical clustering misclassifie high relatively misclassifie case set perform well interpretable less framework select cluster hierarchical reference flat contain collect store medical tool cluster flat coverage feature explain underlying hierarchical observation adaptively limitation complex sparse framework propose produce compare sparse datum exist furthermore interpretability use selection demand hierarchical cluster hierarchical observation dendrogram broad microarray imaging mining etc cluster brief survey proposal row additive unless equal employ version automatically attribute variable extension complex multiple note truly variable propose new criterion j j nj feature weight directly difficulty dissimilarity proportional dissimilarity criterion obtain sparse dissimilarity remove component spc column alone spc dissimilarity dissimilarity feature obtain classical cluster dissimilarity show genomic interpretable dataset limitation criterion reduce hierarchical hierarchical spc dissimilarity e dissimilarity calculate mean fully illustration issue example contain x organize represents gradually increase choose feature include dendrogram cluster dendrogram hierarchical many naive compare satisfactory feasible rapidly performance difficulty subset several number size candidate cluster loading spc spc directly transform loading spc relate low refer candidate criterion dendrogram result cluster accept split node dendrogram dendrogram leave leave terminal leaf height dendrogram repeat cluster label compare cluster base area knowledge specify default replace conduct whole datum result dendrogram main selecting give candidate choice candidate present subset fix size candidate reference assign apply spc say assign step f subset contain conduct hierarchical use dendrogram select good dendrogram label leave choose record rank candidate cluster label calculate use candidate scatter value rank local discard one create scatter rank increase decrease j otherwise repeat small go small rank candidate loading potentially variable different use spc feature different rank set
work surprisingly regardless degree expansion reasoning provide extension clearly treat input dot inner permutation next behave variance albeit dependence input evidence simplify term product begin square extension stack matrix defer subsequent pair decompose goal compute hence orthonormal matrix follow u identical cholesky addition explicitly compute integral moment analogously construction mean combine write computing correspondingly plug simplify claim lagrange plug act hadamard agree permutation randomly randomly subset fix k calculation act put claim denote map stack iid copy q average arise omit theorem show give block weak believe could improve considerable analytic say guarantee performance work practice confirm nonetheless immediately benefit let x dd demonstrate almost net use random projection function note rather set relative decomposition arguably approximate inferior direct hadamard weak conjunction matrix motivation end isotropic albeit purpose kernel dd case rbf dataset via spherical contrary kernel fourier concentration rbf kernel actually rr speedup ram x exact rbf rbf rbf ct slices n year forest n cpu feature rbf perform variant kernel useful tb par computation offer eigen algebra library interested go take around speed large problem evident confirm dimensionality importance expansion cifar dataset pixel accuracy feature achieve expansion slow total slow demonstrate expansion raw class use test offer overcome obstacle competitive problem require real prediction run include formulation practical tool offer method research multiplication near see ex mm claim remark definition scale store compute decision typically expensive prediction difficulty propose approximation computation exhibit property unlike hadamard multiply store propose computation kernel dimension improvement translation dot polynomial experiment achieve full less memory memory make set prediction successful range extraction heart inner infinite idea show nonlinear separation body literature hilbert rkhs norm penalty furthermore one interpretation via gaussian detail employ trick day ten state expansion finitely coefficient must fairly whenever effectively space exploit frequently solve expansion instance show number many problem linearly consequence expense grow method exceed instance large solver albeit limit kernel issue compare access solve reasonable almost average nontrivial fact subgradient associate nonlinear expansion expansion full problem cost subsequently discrepancy expansion basis exponent arise reduced operation storage subsequent aim find expand suboptimal evidence well basis function extract good reduce expansion encode likely dependency covariate projection project albeit method storage able million gb memory store obtain minimal recent provide guarantee expansion offer one efficient expansion relatively dimension curse tune localize rbf kernel promise compatible choose iterate data suffer membership observation optimistic summary function potentially promise exceed algorithm discuss promise translation invariant kernel nonlinearity storage operation reduce expansion show conventional rbf simplify code expansion offer convergence decay satisfy key trace kernel normalize basis function establish prove limit function relate via exist e class computationally kernel action eigenfunction kernel basis reason whenever find efficiently decompose irreducible decompose irreducible eigenvalue invariance dramatically simplify construction unitary orthonormal concrete kernel translation fouri unitary kernel expand particularly fouri transform many may fourier expansion gaussian harmonic choice fourier transform gaussians equality fourier spectrum express multiple convolution polynomial one rotation spherical term provide radial contribution derivation kx degree linearly homogeneous polynomial dimension coefficient expansion expansion l always since unit sphere expansion homogeneous expand accord prove correctness sufficient matching follow establish equality line representation key depend case may isotropic product isotropic vector accomplish construction denote obtain expansion linearly homogeneous variable uniformly able efficient dot kernel symmetric invariant kernel satisfy symmetry efficient mean rapidly case expand symmetry undesirable instance fouri undesirable deviation phenomenon distant observe desirable expand function achievable expansion likewise expensive possibly efficient alternative generalize derivation nonlinear multiplication computation symmetric depend norm inner provide operation special term well clear express product basis need obtain expansion approximate follow integrate conventional conventional explicit evaluate approximately random cost subsequent operation divide proper weighting spread expansion tail easy rather express kernel evaluate directly expansion exist case draw sphere price function rather tool fix polynomial degree respectively cosine x follow homogeneous rotation follow integral odd integral vanishe integration sphere via rescale exponent curvature g detail offer simple immediately without form commonly x p considerable sufficient fast expansion introduce begin yield following discuss previously average method operation seems really multiply rbf summarize two reduce avoid store take gaussian cumulative replace generator progress need simplicity gaussian general subsequently without generality hadamard hadamard dt candidate cosine dct main diagonal scaling compute store hadamard multiply hadamard variant hadamard replicate random stack enough feature prove verify computational cost storage matrix entry operation hadamard storage implicitly transform operation carry block compute basis cpu budget establish sufficiently row long hadamard permutation see amount operation also iid act isometry ensure hadamard incoherent store sorting
power consumption explain researcher resource consideration fluctuation price day name automated area machine reinforcement different state electrical prediction instead section literature behind section conclusion vi work learn demand consumption notice want play device device moment paper relate publication learn train world individual consumption reduce energy consumption purpose exist appear numerous introduction detailed subject behind learn section reinforcement decision process consist perform type agent aim lack knowledge environment affect future develop would action good reward stage reinforcement learn particular make correction give action action state agent action h initialize pair state episode choose action next update episode hour episode current hour day go state control decision connect dynamically early hour several number give moment thing place total amount ready pay state initial intermediate pay car section time come expect ready assume energy moreover amount price varie learn typical total demand customer day characteristic queue type update arrive among place application hour maximally possible speed hour must go whose remove array sort tie occur type word array decrease every jt detail gradually decrease number system newly manually dependent newly day hour whose origin describe reward q latter reinforcement approximate state q describe essence initialize initial exploratory state cost pay pay price want type represent family pay extra family sigmoid refer empty pricing energy explain wind wind panel average wind generator day file wind generation france sake consistency number wind france generation france amount hour panel generation generation assume pricing france website price half hour decision depict measured day result purely strategy easily formulation percentage evidence function medium rt take h try several
priori technique isotropic class choose volume identity shape structure gaussian find limited sample produce quick little expense number low computationally take subset datum block incremental storage whole package handle large datum fail sample cpu computational expense limit heavily overfitte information contain bayesian outcome homogeneous numerical integral formula integral lead prior respect treat hyperparameter uncertainty account imbalance imbalance imbalance maximum entropy observation subtract new transforming bayesian hyperparameter hyperparameter evaluate integral unit towards term integral predictive dimensional integral integral large limit variance dimensional integral line evaluate integral lead negligible get lead simplify simple propose consist classification varied increase dimension low last handle dimension mass move away remain separate class high dimension allow class peak move increase constant increase mean variance indistinguishable dimension class classification number class imbalance observation h version simulate dimension receiver roc suggest lead dominate continue dimension expect neither class improve give subspace although use illustrate notice overfitte overfitting demonstrate well mm accuracy ex ex ex ex ex ex negative breast patient uk record exclude yield initial diagnosis group breast cancer year aim patient expression predict patient patient imbalance h priori outcome candidate report outcome order coefficient rank reduce set shown perform cause accuracy decrease test although decrease per h ex additionally posteriori information lie order less accurate gene handle rank information lose technique priori dash line dimension dimension three formula predictive produce superior efficiency succeed overfitte dimension extension take account instead apply training outcome limitation expand approach heterogeneous also choose optimal gmm uncertainty hyperparameter could incorporate risk heavily overfitte remain discriminant dimension parsimonious uncertainty overfitte remain imbalance dimension take resource result formulae institute molecular college mm uk ac uk centre mm department mm outcome high datum problematic imbalance overfitte become computationally apply overfitte level bayesian reduce integral obtain integral use simulated datum due computational efficient bayesian integration outcome classify use automatically observation test assign model discriminative variety context involve conditional base optimal validation efficient set number arise overfitte combined outcome prediction know datum class class observation belong x belong assumed independently typically probability belong imbalance accord belong
fields mrfs must resort iterative chain classic start repeatedly pick sophisticated sampling wang principle space focus univariate simplicity univariate define configuration vary define guarantee sequence run gibbs total extremely make gibbs ise strength strong main distribution gibbs sampling update mix time systematic scan way establish example verify case require computation form interaction ensures see fx prove summarize ise fast mix spectral absolute imagine necessarily like another derive projection measure result close norm singular frobenius v obtain close ise parameter issue general dense project needs control provide take setting otherwise enforce obeys enforcing find close derive triple dot close guarantee mixing notion closeness vary convenience solve something ease parameter ise theorem probabilistic interpretation divergence perform also project strategy iterate dd dd calculate project case sampling sample chain gradient estimate dependent descent markov total number divergence avoid tend assign assign nonzero easy however slow mix difficult toy graph inspire tree tractable motivation define remove continue distribution onto derivative divergence compute use maximize flexibility select simply tree cover make kl place reweighte variational mf factorize tractable much theoretically marginal run long take long gibb rapid asymptotically comparison topology grid graph node draw strength mix attractive respectively average random calculate divergence subgraph tractable grid horizontal vertical subgraph also random span tree cover original subgraph stochastic divergence pool repeatedly single affect pool mirror descent intensive iteration perform reasonably gibbs pass scan account effort total euclidean projection none original accurate scheduling interpret rapid mixing weight practice still rapid mix property give original distribution cc high intractable motivated approximation tractable family notion obey ensure gibb rapidly rapid matrix strength intractable family onto solve iterative divergence firstly consider novel piecewise subgraph secondly kl generate project gradient experimentally gibbs sample approximation enough sampling original accurate onto give extend mrfs secondly project onto mix also mix apparent term interaction strength know threshold roughly know quickly spectral equal conservative tight mixing occur minimization minimum fa ise absolute pass problem lagrangian return multiplier enforce matrix enforce duality saddle exactly minimizer establishe similarly dd divergence firstly dd fx px observe thm university new south inference ise model high gibbs time guarantee divergence gibbs sampling strength available high cope tractable intractable attempt fully factorize kl distribution tree overlap cluster way propagation reweighte
isotropic kolmogorov scale analogue hold ball graph one ball check note ratio integral manner quantity isometry metric embed reconstruct non neighbor short graph call von set straightforward convergent weighted geodesic metric multidimensional us isometry perfect neighborhood size graph degree code statistic centrality walk von neighbor density path von fail converge level perfectly estimator regime example figure gaussian right track near black fails cope isotropic line figure probability concentration parameter estimator maintain dimension validate nonparametric graph ground amazon sale recommendation naturally asymmetric notion product category popular association sale effect closeness fail display multidimensional metric belong separation across notably history overlap expect music computer science book little serve suggest overall amazon suggest walk centrality identity stationary invert extremely rapid metric near neighbor theorem technique little world information test predict amazon product figure question graph require connectivity suggest regime perform perfectly suggest degree bind much density suggest combine may lead highly effective theorem thm conjecture corollary conjecture mit mit connect iff graph vary arguably ask recover underlie density degree least graph resemble centrality empirically perform well well even analyze occurrence cluster unsupervised recover unweighted direct drawing arc distance typically symmetric construction typical neighbor arguably friend property local density recover integrate show walk direct relate term enable may isometry near neighbor graph e lie region ball walk graph toward high density recovery primarily focus aspect believe offer analyze amazon draw co amazon extend shape simultaneously product similarity metric embed address von integration ordinal graph restrict near provide dimension apply strongly von similarity manifold example discrete neighbor latent prove entire trajectory probability let radius depend analyze unweighted direct edge observe specify latent possibly specify problem walk weakly process via convergence rescale walk closely approximate walk move toward region follow state control technical express solely radius center rescale necessary regularity discrete drift regularity time rescale converge space brownian motion equivalent enforce diffusion strong claim hoeffding borel converge uniformly hold regardless graph drift drift diffusion coefficient draw kronecker delta converge verify claim limit taylor expansion lie inside definition integrating result
infer subspace challenge specify dimension challenge embed sphere relatively specify sphere posterior consistency utility manifold characterize riemannian machine manifold intersection possibly space space manifold mixture special arise subspace mixture linear quantitative application communication infer mixture subspace reduction projection onto subspace go contribution fisher interesting statistical great address reduce summary summary combination statistic range algorithmic nonlinear challenging probabilistic multiple population distribution mixture useful model datum assume dimensional ambient often population degree number population address arise mixture restricted manifold offer limited subspace subspace penalize zhang significant difficulty address subspace approach require come jump immediate subspace vary sphere distance subspace remove occurs move tool posterior structure paper subspace use frequentist analysis bayesian prove procedure utility discussion notation subspace denote manifold column group letter normal orthonormal basis equivalence x article involve multiple copy even draw independent manner subspace ambient concentrated subspace basis state normal residual orthonormal mean estimate affine subspace serve generality diagonal observe ambient either equivalent convenient parameterization likelihood component specify entry entry give mixture specify parameter conjugate location normal term dirichlet triple inherent difficulty sampling want fix subspace prior triple specify specify conjugate von fisher operator von spherical conditional specify seem since assume change needs add close avoid raise subspace integrate nuisance specification subspace recall set equivalence appropriately dd embed ambient embedding embed nice sphere measure subspace projection sphere radius proceed compute matrix low triangle lie key extra subspace high aa half origin summary manifold embed section integer geodesic angle subspace embed projection h sphere coordinate first column diagonal act sphere coordinate useful provide projective structure distance sphere projective subspace projection invariant projective square everywhere optimization distance distance van distance maximize distance distance numerical instability sub easy principal computable value exploit euclidean computation simple coordinate embedding pl pl embed spherical place projection place low half subspace half low projection point image address pre subspace minimize subject compute equal eigenvalue point euclidean full low sphere correspond trace sampling subject parameter parameter sampler sphere lower efficiently difficult address sample projection matrix gibbs close obvious sphere efficient base loss th close subspace subspace put subspace temperature gibb orient posterior traditionally misspecification overfitte arbitrarily use hasting effectively sphere fact relation subspace correspond procedure correspond trace procedure compute proceed draw mm mp p initialize multivariate normal sphere normalize projection embed step dimension procedure z mm z mp acceptance compute step procedure k center walk respectively th analogous j subspace instead matrix inverse u ti ki ik kn k kk tu f respectively probabilitie latent metropolis adjust first stage burn decrease variance adjust burn period sampler autocorrelation provide procedure extensive extend trivial space datum neighborhood weak continuous hellinger hellinger neighborhood kullback leibler kl denote neighborhood sphere uniform von distribution project section induce subspace denote induce e dx f x posterior consistency weakly weak radius leibler result weakly need basis assign mass kl choice infinite dirichlet process measure model theorem open subset measure first large pick approximate finite sum neighborhood continuity k assign theorem enough strongly radius hellinger model element depend let fx prior q posterior consistency hellinger posterior distance j j eigenvalue give k cover ball center origin therefore j ii mn whose fall divide interval metric entropy bound prior
query nature directly study loss closely examine metric investigate weight loss discount next probabilistic derive loss base theory weight reverse fields mrfs pseudo propose mrfs piecewise upper bind mrf finally aspect site web yahoo question answer input functional functional losse multiclass logistic base approximate design multiclass logistic functional discover control properly another rating small lead frequent occurrence tie suggest functional rating group object aggregation group functional indicate max geometric maintaining requirement piecewise discount influential rating combine effective section present picture specification rank piecewise weighting describe approach loss design yahoo yahoo section provide aspect conclude ideally impractical I sort simplicity drop functional rank relevance score e query estimate lc ndcg rr position depend represent object separately modality useful relevance quality since transformation space transformation compute association use sigmoid nonlinearity option q functional reduce estimate conjunction parameter hypothesis certain feature conjunction emphasize ensure benefit unlikely impose pearson uv respectively relevance score correlation min arithmetic issue often presence relevance tie may useful operate tie object like object define aggregation function sensible necessary whose list relevance level individual function reason impose tie perform average complexity object computational g whose next clarity drop reasonable rank metric ndcg see position object rank position loss focus situation emphasize good situation reduce choice multiclass pr pr well sophisticated three wise sum metric directly metric sigmoid function drawback flexible sigmoid approach approximate wise sum weighting depend perhaps prove appropriate weighting ndcg pseudo likelihood logistic combine es piece depend study literature often losse fact unweighted logistic actually ndcg study effective functional effectiveness experimental design specification likely emphasize rating discount section function start specify query cccc accord rank object essence specify permutation clarity reference well respectively start joint distribution accord rule eq shorthand representation informally interpret object probability axiom choice proportional translate worth finally pl context typically pl likelihood however good performance put emphasis ignore suggest still representation part interpret irrelevant thought worth instead reasonable specification interesting interpretation object weight e instance treat rating variable clarity drop explicit factor accounting object e encourage rank graphical mrfs case resort approximate several approximation retain make easy eq proof reader general metric propose loss form evaluate case involve answer site aim experimental present empirical risk study also detail subject paper omit clarity prior limit like offer traditional answer yahoo collect answer question question answer answer test multi opt multiclass eq separate eqs answer frequent training answer functional conjunction multiclass logistic metric note per essentially multiclass embed space appear multiclass employ yahoo relevance perfectly document subset another pre feature fall combine whole set improve space functional weight pairwise table report overfitte since number free space scale number loss comparison c c aggregation mean geometric definition unweighted unity include rank report functional type competitive various rank functional report approximate metric ndcg assume object query performance comparable pairwise see n quadratic hinge ndcg ndcg score list appropriate weighting decomposition report weight however role generally help probable reverse put remove bad see effect chance object pairwise clearly demonstrates greatly improve quality rank list approximation mrf likelihood aim design rank domain aspect choose functional losse logistic approximate loss specifically predict good multiclass rank losse state history decade rather complicated pre compute reveal plausible assessment multiple useful detect evaluation confirm provide employ together yahoo space functional achieve different contain rating tie rating group score benefit operate level group much attain task optimisation metric complex convex make optimisation share message many piecewise locally variable particular rank discount weight influential pairwise rating effective literature notably net bilinear power scale bilinear combination fall work contribute place piecewise log loss weight element bind ndcg score ndcg weight investigate wider weight consider field hard pseudo approximation study approximate metric contribute rank tie little tie group tie group functional
sm represent cone span vector span half complete characterization example vector unit disk half space extremely grow sm sm semi coordinate descent optimize unconstrained least square solve dedicated nonnegative solve propose row therein implement converge exactly give although initialization dimension matlab semi nmf original nmf initialize mean column centroid column binary add modify decrease hence rather would initialize drawback sufficient nature sometimes numerical denominator exactly require product significantly slow observation coordinate optimize nmf strategy initialization strategy initialization construction theorem initialize compute via truncate flip motivation behind effect correction good frobenius directly fact algorithm generate decrease ht nmf good multiply b ij compare initialization strategy turn appealing point equal bad initialization equal completely solve semi nmf matrix properly zero contradict z column remove factorization nm mb yy loss discard influence interior generality fact orthogonal complement q hence modify prove since must note row since flip keeping ii formula remain strict rv r remainder sm nm ni solve nmf polynomial suffice check check whether inequality feasible check model sm sm polynomial form sm decomposition nmf approximate nmf good approximation contain truncate svd polynomial follow svd optimal time therein belong case propose although interval hence feasible feasible terminate semi stop soon smallest far initialize large ten choose precision initialization refine nmf locally implement use perform multiply optimization problem rv matlab tell semi nonnegative fact semi replace note strategy possible sm interior contain positive space matrix frobenius guarantee irreducible nonnegative eigenvector irreducible connect every vertex let irreducible nmf truncate corollary suggest semi unconstraine truncate svd challenge mean nonnegative nmf nonnegative really nonnegative encounter irreducible even likely exist nmf bring several author nmf nonnegative view optimization technique update guarantee however nonnegative initialization make impose would enhance per entry semi semi factorization sm sufficient condition algorithm semi nonnegative intuitively column column belong positive semi whenever nonnegative exist note mx semi nonnegative likely section positive semi semi seem nmf np equivalent since arbitrary nmf ball stein net equivalent necessarily semidefinite positive semidefinite let follow symmetric checking whether nonnegative co np reduce since nmf surprising nmf solve general nmf positive element matrix clearly nmf np hard allow negative problem nmf well question communication always example half however infimum tend semi nmf problem pose cone approximating require uniform matlab notation take cluster cluster add update constant stationary taking indicator seem difficulty initialization stable see remark initialize initialize optimal row irreducible subsection generate synthetic dimension synthetic initializations rd resp rd perform iteration rd initialization km initialization real error input nmf rank unconstraine far away percent nmf approximation nonnegative matlab intel core ghz go ram user convenience generate theoretical finding compute nonnegative generate interval note irreducible sm mean interval value display define perform initialization perfectly lead rd km rd km solution close nonnegative average nonnegative nonnegative semi sm normal distribution previous average absolute add proportional display box previous single ht cc always dominate km appeal seem much practical km slightly rd initialization seem beneficial case test quality good perfectly uniform half space take condition meet well meet surprising approximation rd km bottom well close nonnegative km gap bottom figure even rd km also example become vector nonnegative perform get far away nonnegative expand increase matrix point always within iteration confirm cc iteration initialization recommendation give follow nonnegative recommend rd semi nmf particular factorization rank perfectly try nmf take semi nonnegative sm imply see run solution match face datum set arguably use lee image nonnegative optimality uci use see illustrate compare nmf initializations rd algorithm quality rd km ten initialization generate average obtain run wave wave wave rd rd best km rd rd km evolution data initialization c strategy bottom leave right top bottom interestingly matrix although iteration would figure identify point bad figure lead perform nonnegative allow experiment rd well although difference significant face nmf lead exact contribution fold show error small approximation counterpart initialization semi initialization practice exact nmf solve give semi nonnegative
already finite excess towards within region latter large illustrate computing period return assume dependence return accounting st yield return limitation discuss direction historical record availability historical much uncertain systematic precision water rating curve systematic ii aspect systematic impact multivariate systematic unclear therefore incorporate explicit dependence alternative version historical high quantile generally decrease historical include inference however necessarily small quantile estimate take shape uncertain shape take existence joint angular dependent addition dirichlet allow variate probability poorly limit probability threshold range confirm multivariate period shorter asymptotically author thank part fp www project corollary notation com paris france national sciences pour et cs quantile typical return period face challenge historical approach peak marginal site site semi observation augmentation involve historical ignore historical highlight availability historical investigate significant asymptotic bayesian reversible jump variate problem maxima peak commonly model major site illustration record year whereas period additional challenge one complete understand extreme theory univariate issue site frequency analysis jointly several site extreme value record use site induce site ignore dependence observation site alternative use elliptical spatial beyond spatial independence compatible extreme shape parameter turn quantile compatible theory method record complement systematic measurement censor miss censor extreme location censor inferential carry partially family admissible extreme parametric restrict parametric logistic practical censor version readily subject modeling choice allow neighbor resort characterize write weighted parametric keep flexible combine datum threshold neighbor partially censor combine historical explore build previous extend multivariate variate maxima approach correspond record episode multivariate implement combine historical investigate scientific firstly relative quantile estimate existence historical nature strength systematic bayesian parametric dependence dirichlet mixture allow inference vary complete jump inference censor adaptation inferential censor practical advantage mixture need wide theoretically realistic moderate remainder dependence summarize feature describe inferential fit result discuss propose summarize finding france close period historical marginal extreme water record work simultaneous record daily course censor small error proximity four visually confirm obtain miss record censor leave censor comprise possibly upper period location temporal observation j j stand record arbitrary missing model raw daily term level threshold cluster maxima treat threshold fit detail propose identify size estimate index approach heavily inter arrival available censor adopt duration representative choose region censor exceed none position typically case censor intersect end marginal position temporal start cluster maximum special care censor duration temporal index coordinate variate scheme situation location maxima censor marginal upper bind prevent belong extract horizontal black grey draw consider location censor variate inter day namely inter cluster inter completely remain account classify homogeneous block day contain maxima together coordinate censor censor location study available available modeling another extreme occur corner suggest model mixture period exact gray superposition increase rectangle coordinate threshold description overview reader refer review margin location consider pareto threshold latter let possibly unobserved maximum water model empirical intra variate day marginal variate handle fr define transformation transform one extreme extreme radial homogeneity large away unit fr switch system angular component scale context angular call angular angle correspond behavior entirely angular variate set different angular distribution pareto density represent point segment blue weakly concentrated say simplex occur mostly contrary variate due angular mixture angular short dirichlet center therefore average q center mass center technical justification paper overlap censor miss write term poisson intensity precisely threshold observation extreme fr measure relate angular widely context take kind q region poisson poisson absence censor jacobian accounting transformation integral expression context carlo build censor involve whose assess option miss full dirichlet dependence historical confirm ratio value statistic assess add value account historical consider systematic total iteration require remainder obtain score together distribution concentrate indicate identifiable panel distribution frequentist historical confirm taking tend uncertainty shape return credible interval inferential framework take increase return
useful discover cross powerful flexible pathway framework dataset ht gene far gene fig lemma thm corollary thm pathway challenge knowledge throughput heterogeneous informative specific integrate hypothesis source include novel integrate attribute utility study mechanism rna protein pathway current pathway study decade discover novel component pathway find pathway development drug target pathway one major biology utilize high throughput pathway nature research medium reconstruct pathway hypothesis mind try throughput contribute comprehensive refine hypothesis characterize three feature well set heterogeneous second exploit datum reconstruct part pathway informative protein domain far utilize approach planning study assess biological propose extension pathway hypothesis demonstrate pathway feature distinguish pathway refinement methodology network reconstruction comprehensive integration collection protein helpful describe global mechanism pathway refine integrate heterogeneous reconstruction edge path pathway diagram mm pathway diagram expression sometimes datum bind protein protein completely protein interaction predict pathway cause relationship perturb effect nest effect dna individual pathway far utilize reconstruct pathway differ pathway pathway identify edge differ formal determine hypothesis contrary spirit exist biological support suggest probabilistic model pathway refine pathway graph three demonstrate explore content individually e pathway infer use pathway encode pathway except part adjacency observe edge primary adjacency although differ reconstruction informative hypothesis pathway set bayes pathway proportional prior sampler explore pathway edge gene product sample accord without complementary bind datum dna gene domain differential specific location uninformative likelihood essential exploit fact translate localization product process notion informative location additional map conceptual first protein cell second protein cascade factor protein enter gene binary pathway type molecular pair description collection five data index conditional informative protein specification discard carry little conditionally expression pathway aim read sequencing pathway whenever co similarly profile uncorrelated differently put emphasis beta list protein conditional differential relate neighbor protein indicate presence protein logit relate logistic pathway informative pathway pathway hypothesis pathway reflect protein interaction gene expression target tf domain goodness fit auto knowledge pathway extract summary pathway tf target pathway protein cascade tf containing pathway protein two contain sub mark grey target member indicate explore content analyze pathway dataset throughput protein pathway cover edge sensitivity see b tf target clear biological pathway gene show pathway type prior wide cascade expression tf target however informative explain complete pathway pathway observation valid pathway take sensible concept exploratory ultimately hope evaluate combine type support hypothesis pathway fit occur protein frequency logistic sensible default pathway domain pathway structure pathway initial also c fig strategy power separate node left pathway neighbor left edge leave assume encode agnostic presence auto logistic likelihood simulation hypothesis lastly figure recall informative poorly gene contrast high precision model able pathway pathway pathway specify relevant location consist pathway sub pathway interaction pathway currently pathway include sub pathway usual tf tf tf tf match pathway fix pathway specify node likely domain node domain reflect interact leave precision recall
see estimate pairwise parameterization effort specify importantly link correlation dependence covariate model financial application allow model tail dependence achieve copula parameterization particularly financial tail vary covariate allow link covariate prominent covariate without dependence covariate margin suitable connect link covariate allow parameter approach make interesting interpretation forecast furthermore reduce see detail specify marginal copula omit subscript first turn variable indicator informally express covariate variance independence intercept slope decompose prior normal prior intercept include parameter indicator one intercept put g imply intercept covariate technique two situation imply model parameter trivially intercept prior intercept copula copula logit intercept numerically mean intercept parameter case normally matrix trivial positive definite matrix normal identically distribute reduce experience shrinkage meaningful great explore comparison consider special dependent new put restriction copula tail inequality visualization parameter dependent independence decompose prior indicator indicator generalize logit beta beta logit extend logit two generalize function reduce logit link intercept decompose application link prior mcmc metropolis hasting truncate draw reject mcmc likelihood copula j jj jj u u component jointly joint update conditional metropolis hasting update indicator parameter tailor acceptance rule g ig proposal mode finite newton small matrix negative hessian currently exclude enter versa allow indicator indicator scheme copula variable scheme update show hasting proposal distribution model low copula copula implementation dependent variable indicator wise function update copula mc alternative independently margin margin see widely asymptotic copula fr initial obtain request daily stock return copula split margin copula daily daily stock market period include large establish subset cover small possibility capital great index percent total hypothesis dependence parametric nonparametric dependence little covariate covariate margin table covariate univariate regression mixture lp description week month geometrically decay geometrically absolute return l high measure decay return geometrically decay figure series volatility return financial depict empirical respectively fr copula mean copula appropriate usual copula efficient extreme near fr application conditional tail autocorrelation lag marginal similar link posterior copula refer marginal selection important inclusion low dependence part p margin tend highly risk margin select copula copula selection marginal respectively intercept include p index subgraph variation tail even contour copula financial compare hoeffding bind figure normal financial copula capture dependence model performance historical predict ahead prediction calculation costly forecast assume add multiple processor application smooth gaussian table commonly predictive covariate covariate copula copula well margin link covariate sample posterior daily return index advantage copula improve margin augmentation work visit department compute information science university present mcmc detail implementation great hasting embed metropolis copula cdf marginal link parametrization numerically dictionary upper section let link conditional complicated need substitute density obtain omit copula eq u vs q exchangeable derivative u direct derivative elsewhere cdf q function function calculate integral analytically advantage connect derivative asymmetric symmetric density binomial knot mathematic finance economics china copulas attractive multivariate density marginal particular area possibility dependence approach tail yield interpretable practitioner dependence function copula posterior method covariate copula dependence mcmc copula research term univariate copula cdf e g copula dependence construction copula widely survey review include construction tail bivariate copulas tail px u low tail bivariate copula explore copula bivariate copula another asymptotically fx testing detect event study vast extensive copula model article investigate cause many situation relatively copula model tail copula modeling obtain tractable practitioner general copula copula construction form copula stand copula tail dependence outline copula introduce copula presents mcmc daily stock fr hoeffding copula strong positive dependence dependence see bivariate modeling copula commonly copula copula bivariate copula bb copula tail tail density copula tail use copula autoregressive daily
run slightly feature correlated pair nature far design correlate feature predictive task thereby lack instability ensure correlate assign exploit clinical structure inherent purpose diagnosis record medical international code code event build undirected node incidence share temporal term iw graph rewrite simulate create model feature select finally list feature hyperparameter stability resample lasso stability fig affect auc feature bootstrap clinical fig ccc feature stability medical inherent cox heart patient focus solely heart similar study competitive gender age history clinical level rare result past diagnosis demonstrate entirely datum medical database integrate clinical pathway serve screen select future include investigate structure enhance mm clinical yet little attention cox clinical structure inherent record intervention hierarchical medical knowledge demonstrate efficacy predict failure patient clinical increase discriminative power report competitive heart failure serious frequent heart prediction variable hypothesis literature reach medical record aspect care history diagnosis procedure marker record time comprehensive deriving prediction unfortunately automatic particularly clinical cause instability estimation little investigate heart cox high improve exploit clinical inherent diagnosis disease structure edge relation sharing strength among correlate feature thereby stability cox train validate datum validate measure index exploit clinical structure stability regularize cox utilize structure cox database one sided bank pool feature database filter bank event time cox risk due failure future instant unlike patient
version choose choose view variant update every number time cr r r r super martingale follow inequality last eq resource allocate despite future uncertainty optimization extend general arrival permutation random competitive bid ratio et draw propose subgradient optimization framework application internet online resource new constraint modify program considerable online ratio achieve offline competitive online maximization case infimum many propose trivial competitive lp derive competitive lp often competitive restrictive permutation permutation formulate define subset proper technical utility impose optimization vector procedure avoid take online optimization solution algorithm tx na member literature follow simplify online distribution permutation simplify identically distribute lp online lp go grow example permutation achieve light dedicated permutation online lp briefly combinatorial optimization relaxation program important lp nk ip nk linear primal variable dependency union al give author name choose linear number constraint ratio adversarial adversarial paper aware recent similar also prove online program include general show achieve ratio propose lp apply general permutation give adversarial interpret subgradient call n allocation combinatorial matching lp online integer solution allocation online associate online allocation terminology equivalent server amount order pay view server utility associate order server decision fulfil total resource server make lp utility one online briefly review case thorough reader online bipartite n online bipartite special allocation et call achieve competitive worst prove achievable competitive online ranking achieve competitive algorithm study competitive receive distribution competitive ratio problem assumption applicable propose online show competitive propose explicitly primal concave version utilize primal problem bad fall many al ratio infinity bipartite generalization bipartite matching graph adjacency matrix lp without rule review rest ne ne simplify bernstein hoeffding replacement convex proposition replacement one sampling replacement see sub martingale chapter denote complement transpose denote function e e start simple feasibility let consider solution uniformly nx sx algorithm derive concentration sum simplify integer kn ni martingale maximal inequality bernstein algorithm sigma show super martingale small inequality give conclude follow generality competitive offline concave pair primal pair f np h bernstein close define ia f ta sx describe apply represent subgradient tx f choose similar analysis concentration sx nf sx define q maximal inequality conclusion k p main present define tt super martingale tf
mutual conditional find candidate check independence graphical degree search become much focused complexity writing importance exponent independent work propose efficient liu make generalization possible distinguish recent efficient algorithm informally graphical asymptotically distance pairwise ise multinomial survey possible efficiently away variety paper give algorithm require require base use regularize regression show certain incoherence careful fail ise term mcmc temporal mixing e conclusion informally speak generate ise graph converse generate np precisely know without determine pose answer ise model despite strongly problem algorithmic connection ise efficiently discuss state conceptual identify ise structural almost liu arbitrary node maximum consider ise exist neighbor information state actually consequence information provide mutual algorithmic ise arbitrary constant interaction strength complexity algorithm doubly obtain suboptimal section statement order node form neighborhood pseudo neighborhood spurious worth add crucial add create pseudo neighborhood range neighbor high many condition neighbor potential argument whereby pseudo pseudo easily remove neighbor connection introduce influence state lemma average correctness result proposition prove section discuss study ise name boltzmann approach boltzmann machine attempt find ise use network protein network physics rigorous base truncation expansion relate entropy message pass learn broadly statistic dimensionality parameter ill sparse underlie discuss optimization regularize objective popular approach infer ranking optimize fail far ising tailor often theoretical progress variety gaussian work ise function effectively learn soft joint boolean dependency ise model apply fouri nevertheless level total log exponential temperature range correlation assumption depend consider graph degree bound associated shorthand eq partition parameterize edge external edge satisfy field satisfy bound appear complexity alternatively think implication conditional write bound spin away statement conditional appear observe configuration measure zero meaning expect corresponding graph rather robust bad entire eq runtime use certain eq quantity obtain perform neighbor imply result show neighborhood thus work u ix replace version influence event influence within learn data step neighborhood accomplish high conditional I set neighbor use potential construct simple relating reduce neighborhood try correctness rely subsection terminate construction remove let suppose numerical probability return probability return correct neighborhood runtime obviously exceed state value give prove give correctness bind neighborhood theoretic quantity include kullback leibler divergence definition add node add x add pseudo term quantity prove lemma quickly runtime claim maximization pn negativity entropy mutual u x strictly large consist add mutual shorthand conditional x q jensen concave root inequality variation lemma conditioning add ix state use correctness graph parameter u hold pseudo construct node contradict line algorithm pruning pseudo state discard conversely I neighbor discard finish proof proof lemma guarantee corollary correctness event u prove proposition proceed adjacent x neighbor begin multiply gives sum last quantity give eq inequality appear proof proof randomness decompose concerned anti concentration anti concentration result os show anti sum variable os let decomposition lemma conclusion approach mention strength let z decompose uniform z mu represent think obtain choose hence probability placing minimize subject arithmetic average z prove lemma proposition subtract statement recall odd definition z apply assumption estimate concavity additionally monotonicity partitioning estimate eq subtract third q true follow bound second eq negativity low multiply plugging complete node configuration x application remainder hold bind inequality inequality triangle depend bind ab ab ab latter plugging hold early evaluate eq learn ise ignore show light large correlation plausible one
tackle nonparametric approach process aim infer infinite bayesian paper propose exploratory neither sample output summarize take function article obtain automate dependent case may grain pattern sub keep reduce post technique consists merge successively contain curve cost sum divergence merge create dissimilarity merge post processing technique hierarchical decision tool plot dendrogram pareto chart criterion number cluster introduce curve relate alternative technique artificial power consumption finally summary describe goal collection value observation exploratory goal reduce pattern regular shape low linear combination base simplification cluster use discover functional e segment find minimize constrained segment space similar maximum estimation induce partition interval partition abstract functional exploration build cluster interval summarize represent canonical draw organize multinomial fit nonparametric overcome unbounded computing approximate sampling chain dirichlet require parameter parameter significant expect reliably estimate approach much parameter available dp approach clustering treatment mode matrix contrast probable optimization secondly order one retrieve monotonic sense aim continuous model clustering curve realization discrete computation approach first use build size rank exploit experimental fine grain retrieve pattern principle detailed functional grid phase mining process time consume result rapidly reliably conditional supervise joint interval partition grid datum combinatorial curve store curve density discretize cross univariate triplet curve joint per curve joint tend group cluster discretization optimize family cluster curve cell cluster define curve interval curve point cluster collection curve curve dimension curve curve interval point curve dimension select distribution parameter choose uniformly dimension cell cluster distribution cluster give resp resp log unsupervised datum accord optimal value eventually subset kind stand way partition nonempty subset code description relate number specification partition cluster multinomial cell follow specification multinomial third likelihood line cluster follow rank resp heuristic heuristic bottom heuristic grain curve per cluster cluster merge decrease merge post step embed meta heuristic mainly different initial improve extensively evaluate true detect grain pattern approximate instance train accord lead exploratory still fine interpretation aim simplify information retrieve agglomerative locally exploratory highlight set datum successively uniform white distribution p x collection per triple randomly curve accord apply functional method subset size per display average point subset discover interval start curve method recover pattern cluster curve distribution despite retrieve point may totally curves systematically place grow retrieve show well pattern property regular moreover deal distribution cluster one consist consumption home consist give power consumption day minute aim characteristic consumption grid interval record day group discretized power measure power segment highlight characteristic day home represent piecewise line power consumption segment power grey grey read highlight within segment prototype locate dark power rarely prototype segment multimodal segment highlight multimodal extend ht grid yield characteristic easily interpretable consumption year may agglomerative represent dendrogram chart percentage consist cell accord percentage keep cm opt dendrogram chart concave keep information ht highlight retrieve propose four processing power consumption time four prototype red solid curve compute discretization consumption conversely make certain cluster segment common consumption enable period pm pm able term consumption period day curve locally segment prototype average prototype estimation highlight segment distribution consumption power consumption locally segment display density consumption retrieve power dense around unique consumption consumption function also peak translate interval highlight power around retrieve compete approach track difference color difference use figure retrieve certain indeed way group highlight link weather france period rest year consumption appear early may day classify interestingly home consumption cluster period ht ht color consumption day show retrieve cluster year power consumption day cluster consumption characterize day home intermediate show immediate track retrieve scheme hand user prior powerful exploratory thorough understanding locally cluster complementary focus exploratory curve paper point categorical curve cluster point select
derive relation arrive excess square series prediction satisfactory adaptive vi goal output mapping hypothesis kernel hilbert rkhs definite may control smoothness solution rkh satisfie namely f k f kernel calculate j solving square usually necessity input algorithm alternative compute gram express step ei upon time current nice sample fit current incremental adaptation essence solve square adjustment involve step assume gaussian factor define depend look close size look inner product unseen fall study recognize build current previous rkh correction efficiently change motivate component formalize propose sequentially optimize unknown product draw define minimize practice alternative measure course batch present sequentially across previous size size sequentially iteration initial size unchanged addition new use new vary iteration adaptation rkhs learn cycle center remain term rkhs jointly optimize datum I depend train independent mean square condition I error iteration size minimize optimization input q residual mapping iteration condition noise space denote size iteration size much size density theoretically desire mapping impractical solve optimization importantly usually develop without discuss minimize mapping current optimize iteration prediction optimize minimize instantaneous stochastic readily denote kernel size adaptation residual case follow kernel sequentially kernel size center old center remain initial manually roughly advance follow observation sign sign sign size successive contain desire desire mapping easily input adaptive filtering provide tool rkh mapping derive energy express residual rkh contain correction term nonempty interior consequence contain rkhs far side q energy relation form relation normalize satisfie monotonically square decrease far reach excess e two assumption another uncorrelated become steady state easily map steady state close ii v related observe neighborhood size little confirm kernel little accuracy influence speed reach steady case accuracy steady section present result static estimation output generate size see kernel influence rather speed work achieve final size almost still final fig evolution curve adaptive size plot value size map plot purpose well desire function visible desire detailed summarize speed still select see kernel table little effect steady training confirm prediction except run average list steady attain steady state case steady state convergence speed steady kernel steady size q pick predict current step fig simulation run segment series train iteration mse compute iteration kernel small mse final expect yield satisfactory interestingly desirable deviation quantization apply network experimental set quantization curve demonstrate obtain show size converge dotted network quantization testing mse c define role adaptive filtering radial usually default kernel mapping influence crucial square efficient algorithm iteration computationally base energy relation mean prediction automatically proper converge achieve sequentially optimize function idea interesting line optimize cn edu cn edu filter filter develop gaussian size still square optimization develop sequentially mse theoretical convergence confirm static short learning nonlinear inherent space space kernel thank popular include vector principal component kernel etc significant counterpart learn extensively statistical literature nonlinearity system cost filtering filter reproduce hilbert rkhs filtering nonlinear algorithm affine recursive square create radial rbf adapt simplest fast effective main grow increase requirement especially growth important datum accept approximate constrain compact desirable selecting address implement filter two kernel adaptive universal approximate capability stability normalize kernel
model free online unlike agent strategie continuity truth employ empirical ne provably convergent ne also question concept indeed stochastic game learn observe prove game extension discount stochastic tuple agent product space state action go player choose discount influence reward obtain agent represent various stochastic game xx ix ia stationary strategy suggest transition probability agent randomize write expect goal strategy nash stationary nash equilibrium game nash discount well result discount strategy shall stationary nash programming write q picking act distribution behind formulate add ensure feasible nash equilibrium optimization objective minimize agent isolate meaningful agent natural ensure vector feasible problem valid equilibrium useful maximized combination implicitly imply formally give ensure policy ne point nash equilibrium correspond discount work manner remark difficulty solve quadratic summation multiply inside constraint easily see non player game page every game requirement optimization apparent two player descent convergence minima nash equilibrium strategy two player game case sum player contain arise newton solve hessian objective infeasible require hessian necessarily present break subsequently stochastic sg sp agent along state ensure bellman tuple let bellman let ix formulated derive nash strategy sg sg sp feasible sg sp sg sp equilibria intuitively objective summation zero imply sub ensure bellman error turn agent combine sg sp sg sp lemma sg nash sg sp nash tuple optimization problem suggest imply nash nash sg sp nash optimization sg sp point nash feasible satisfy sg case agent derive sp sp condition sg sp lagrange multiplier slack lagrangian kkt corresponding kkt necessary sufficient equilibrium underlie game strategy tuple linearly sg sp condition impose additional independence ensure establish sg kkt kkt sp sg kkt sg sp kkt sp feasible problem substitute eliminate reduce feasible sg equilibria sp sp sg sp sp impose linear requirement introduction actor operate value operate along slow sg sp dynamic update tuple follow operator stay I sign project outside small around continuity technical help provide recursion slow recursion proposition g objective q taylor amount show second infer xx let matrix characterize recursion present ode actor evolution ode precise point ode far see infeasible ode asymptotically unstable asymptotically limit nash surely ne discount rectangle mm join height minimum cm draw minimum fill environment fill red dot right dot fill south node n leave start south south south amenable rl neither transition operate free illustrated represent discrete state agent localize agent decentralize spirit rl presents operate agent temporal td recursion note recursion operate similar operate variant know slow motivated proposition start size play reward respective value quantity derive light converge nash strategy number multiplication behaviour appear state strategy avoid typical multiplication iteration complexity confirm iteration complexity finding game result scheme rate scheme show x see update operate model td stochastic approximation argument early literature update recursion change operate model free hence access update track handle proof analysis underlie transition discount multi agent rl detail later modification comprise size ensure quasi static update appear infer rewrite recursion eq cn space one trivially track ii use theorem analyse recursion system column reward globally asymptotically stable limit rearrange identity ode eigen particular system globally ode converge start see euler give quite prove dynamic programming latter variant converge rl govern globally equilibrium lipschitz h origin asymptotically stable show unique globally asymptotically ode term assumption ode trivially satisfy asymptotically stable point update recursion converge correspond strategy limit bellman ode recursion infeasible limit sg feasible point ode partition unstable unstable gb unstable since since value long equilibrium feasibility sp return ensure point induced spurious xx operator suppose I iv I iv consider possible contradict solution result well result recursion project onto ode ode make continuous surely size satisfy bn bn ode compact asymptotically stable converge almost update slow scale rewrite eq assumption recursion continuous since continuity fact continuous spaced trivially upper bound claim iterate govern unstable observe converge stable equilibrium offset policy compute use every numerically convergence govern stable outline crucial detailed establishe td update converge involve consequence fact free access analysis govern surely globally point sequence write argument q assumption assumption govern globally asymptotically establish estimation recursion technical result aggregate recall theorem almost surely let update write n v ny j g r martingale square integrable ensure ergodic verify martingale surely almost natural ignore main paper treatment recursion x I converge estimate allow proposition I order verify since verify straightforward space quantity verify update converge algorithm two variant involve intensive solve simple payoff individual agent game pick constitute ne pick either ne stage payoff agent accord payoff perform length stage aggregate evident ne strategy tuple ne iterations b b inner sep simple sum discount game name stick short locate rectangular like stay precise description component specify agent rectangular product action agent neighboring action one transition agent agent function next action reward agent state thus show sized game sequence q correspond slow constant step lead fast also domain equilibrium policy follow offset pick xlabel number ylabel height marker font file plot txt xlabel ylabel avg distance grid marker legend style legend column restrict file txt txt file txt evolution go nash equilibrium evolution algorithm homotopy exponential practically infeasible evident strategy converge grid within imply get iteration get grid drive corner grid short explore grid exclude take minute take nearly involve nash equilibria iteration q variant implement xlabel ylabel height marker plot x simple full game assume reward state transition intermediate case albeit know particular reward position runtime suggest sg equivalence nash necessarily avoid minima play equilibria certain differential ode stable coincide stationary equilibrium underlie rl experimental quick future successfully state huge strategy tuple reinforcement mdps action space aid bring system use extension constrain game stochastic game additional constraint function tuple might arise application detail sophisticated pt nash equilibria ne game ensure bellman characterization nash equilibria underlie game sufficient sg sub develop descent descent avoid minima converge self equilibria certain ode coincide game game establish consistently outperform discount stochastic nash reinforcement stochastic approximation game single shot prefer several action intermediate stage one stage chain random system next popular scenario decision process mdps give allow suitable reward incur influence another merge mdps markov behavior game select receive agent game agent treatment game pay criterion like game several model game stochastic also markov cf evolve act simultaneously transition agent get know aggregate include agent individual agent objective maximization discount reward couple vector
q lemma lemma rl well part follow fact theorem corollary institute pa usa learner prediction address analyze regret neither benefit develop interactive learning extend reinforcement commonly suggest broad exist learning increasingly notably game ai easy expert translate behavior perhaps develop space g list parse understanding ground policy execution performance supervise require benefit leverage provide make error expert equally drive expert expert choose go learn poor reason agreement poorly mistake cost learn user intend task contrast impractical additionally policy statistical rather leverage sensitive approximate policy develop theoretical iteration despite regret stability broad view horizon decision policy action use action state induce state system typically sample learning influence technique learn go expert form observe uniformly explore observe perform action train dataset iteration interaction begin uniformly explore expert continue new cost go visit train learner iterate policy expert action explore detailed henceforth assumption contain policy good future go lead low observe action favor initialize mix collect uniformly start execute execute current execute estimate go start expert cost view online demonstrate problem policy policy randomize deal sensitive sensitive online algorithm descent description default learn classifier surrogate sensitive stable regret highlights standard indicate predict sample go cost state optimization weight machine good cost ns nn squared loss go regressor common include sensitive handle bandit must go inefficient feature current exploration carefully set traditional care bandit algorithm finite many show procedure interactive strong analysis policy policy competitive expert rely connect adversarial online property sensitive choose policy incur adversary go exactly collect provide n policy class regret policy n beginning trajectory uniformly randomly policy trajectory visit go hold infinite iteration collect amount policy sequence j policy sensitive classification aggregate dataset policy interactive unable bind linearly classification play sample analysis depend reduction sensitive ranking relate task action random reduction particular regret aggregate example optimal regressor regression use pick regressor regret relate task performance regressor regressor training regret use particular present cost regret aggregate go obtain regret mention case cost optimistic future fail policy even policy albeit policy driving scenario reach goal shorter drive narrow road policy go time enough collect policy loss example iterate iteration initially guess use go specify detailed algorithm initialize collect point execute execute state aggregate I online loss go current correspond sensitive algorithm online variational step td find regret cost thus find good distribution state limit provide performance guarantee result present single policy execution allow generalization efficient impose strong requirement regret sensitive instead cost learner class cost still online reduction regret performance strongly limited quality iteration mechanism distribution policy guarantee theoretical observation first perhaps crucially suggest approximate policy theory counter understand share ensure like rely future go understand heuristic wolfe achieve performance variant effective batch suggest instance approximate stable cost go seem counter intuitive however divergence approximate ensure many broad form piece grow picture analysis batch analysis online seem concern performance robust become dynamic execution concern method rely cost impractical collect estimate state action expert visit trajectory per setting expert heuristic analyze combination first loss minimization provide expensive provide class compete mdp remain learner trade must explore detailed detailed begin lemma need bound expect query expert fraction let argument integral px qx fx qx additionally rl fx qx qx qx fx px qx make lemma encounter collect continue execute expert td rl useful present policy let execute time rl v u reduction sensitive classification achieve class pick number policy q q u inequality similar argument rl well e follow finite explore uniformly sensitive loss go action e linear regressor regressor regression j use regressor pick cost additionally
present compare intuitively output face produce need produce negative weight weight without reaction target prevent must actual concentration adaptation adaptation originally distinguished adapt residual input signal could happen soon contribution input shown distribute input contribution specie perceptron combination input adapt weight reach update reach point beneficial formal perceptron concentration negative entire present manual would genetic ga cross mutation use mutation fitness reflects give encode learn fitness task force utilize otherwise ga tendency opt utilize capability input choose target safe region far allow weight integration proceed define performance chance generalize sufficiently primarily drop distinguish draw among function function perceptron constant zero fully eliminate consumption branch perceptron discard adjust act consume part output function fairly would impossible calculate formal perceptron question count weight concentration trace output error chemical asymmetric asymmetric signal perceptron analog learn feedback provide et al simulate base capable compare system cross besides input utilize shift general bias would oppose employ system evaluate employ fairly chemical plausible suggest carry problematic feedback need introduce variant separate reaction chemical automatically transformation dna circuit dna range art dna circuit oppose adapt integrate delay could tackle series chemical event relevant drug adjust cancer cell material grant pre determine manner feature compute challenge implement extend chemical analog analog asymmetric perceptron perceptron simulate capable specie actual dna perceptron analog supervise build program chemical machine change limit applicability limitation environment adaptive chemical learn external imagine million molecular broken system design predefine specie could inspire chemical implementation adaptation chemical calculate e chemical achieve desire previous work first simulated artificial chemical learn teacher chemical perceptron system input step aim simplify implementation employ asymmetric thresholding flip side perturbation structural redundancy real application subtle among achieve perceptron model improvement necessary analog asymmetric perceptron original mass learn environment modular number demonstrate learn nonlinear analog combination chemical line chemical reaction formalism consist molecular paired rate symbol molecular importantly reaction treat concentration quantity require chemical constant order reaction reaction define speed signal contribution reaction rate total perceptron input contribute output process dot product chemical weight difference require perceptron output calculation reduce concentration formal specie input since bias represent weight three oppose formal asymmetric addition sign concentration weight otherwise asymmetric specie leave negative represent monotonically increase initial specie weight share effectively implement replace decay specie embed previously encode complementary opt reduce half complexity underlie concentration limited qualitative formally act weight impose negative pressure weight branch contribute concentration branch concentration reach
topic vary result figure bind vary true truth partly change dataset effectively narrow vary model investigate provide direction analyze generalized mixture weight sum product analysis component order derive structure singular example conduct mixture component gmm assume generated spherical assume gmm method likelihood similar provide namely maximal integer similar detailed omit excellent work analysis lda present bind order topic analyze q proof analyze value moment thresholding solve inequality bind singular especially bound ij ij set least probability follow least element study proof proof gamma tool degree freedom proof tail r shape follow hold gamma chi r r v hold n n c overall intermediate lda parameter eq q omit ambiguity also representation th document diagonal separately section analyze large corpus selection set topic advance topic first bind topic text demonstrate use easily generalize recently model variant prove extremely corpus word generate mixture latent multinomial become text lda variational topic play successfully apply show large incorrect computational lda grow equally unfortunately topic lda approximate via mcmc selection aic bic though achieve success asymptotic run dataset hdp alternative select inconsistent topic amount theoretical latent model spectral outer vector correct topic topic lda analyze topic provable guarantee utilize tensor moment upper term result computable contribution valuable determine sampling spectral information true inequality regard constant organize main synthetic validity generalize powerful building block recently moment explore lead topic directly derive properly third estimate decomposition line discover empirical document multinomial topic collection th document document topic hyperparameter topic generative lda denote generate natural meaning learn moment q outer moment show outer product topic since summation linearly large singular value direct way access estimate moment large overcome obstacle study estimate infer estimating size large approximate enough pick threshold simply count achieve examine investigate generality namely also simplicity definition singular variance lda chain step semi defer ii matrix frobenius therefore expectation proof task discuss high relaxation keep dominant statistic examine document one pay hundred k matrix diagonal row order chernoff bind appendix great minimum random proper conclusion assumption utilize
split ghz cores gb image cat specie table observe surprising example assignment contrast vs second suboptimal learn difference cat image assignment art approach head box segmentation well image segmentation make ht ground segmentation head extract encode total denote improve extract segmentation dataset specie consist class already well improve segmentation worse annotate information confirm effectiveness training imagenet train report table ht sift template propose art mostly implement extraction share comparison operational cm method summarize take employ stochastic method metric come aspect become significantly class image separate contrast class appropriate method address challenge arise stage arise many extend projection address challenge performance significantly art approach plan combine segmentation approach feature nsf com com fine grain categorization basic challenge occurrence distinguish intra paper propose metric address embed different apart address flexibility portion however end subproblem computational benchmark grain categorization aim distinguish class classify handle somewhat requirement many subtle deal intra variation pose example feature extraction extraction localization choice include recent development cnn e imagenet state art dataset difficulty training dataset benchmark thousand deep segmentation aforementione classification exist directly grain strategy strategy grain vs scheme effort variation metric approach occur work different far neighborhood effectively handle tradeoff inter intra metric find test limited dimension straightforward dimensionality analysis pca projection problem unable take dimensionality suboptimal challenge constraint usually require avoid overfitte total triplet example dimensionality datum lead computational challenge optimization ensure psd intermediate storage save gb store complete framework dimensional dimensional divide original optimization difficult classified currently metric adaptively improve metric optimize handle subproblem dual develop enjoy random learn dimensionality finally storage copy randomized matrix process metric overfitte extensive efficiency section describe conclude direction many develop find two survey paper base triplet adopt serve address although devoted examine project rank rectangle advance apply directly less address assume storage suffer high cost propose focus triplet near dy triplet assignment tm problem significantly study loss appear effective hinge loss benefit l challenge psd projection first psd project end follow j triplet constraint dot product summarize reliably determine overfitte triplet number summation term solve help image category visually lead mistake address process stage metric triplet triplet optimization improve learn solve optimize objective stage obvious strongly chapter solution optimize finish metric optimize stage original problem number summarie dimensional subproblem dual technique simplify investigate analyze introduce dr triplet project projection double projection preserve pairwise variable low b learn significantly develop estimate although expensive impossible save save method suboptimal save popular avoid metric efficiently challenging address recover step express summation multiplication triplet correspond matrix verify second exploit efficiently eigen accord independent appearance compute keeping fortunately
optimized mention capable achieve bit image possess complexity arithmetic computationally dct future consider approximate transform dct high adequate datum frequency quantization restrict significant propose dft pruning time ignore involve avoid pruning discard apply design wireless communication another pruning method dft dct pruning originally propose wang consider generalize extended pruning technique terminology vision dct base dct theory advanced propose architecture dct architecture operation avoid pruning refer operation discard bit architecture address prune discard distinction terminology grow furth dct present introduce dct realization version propose embed video modify associate dct approximation particular successfully mean noticed concentrate image low image quantization frequency effort frequency keep transformation derive associated transformation compute correspond compression computational overhead since quantization graph relate signal signal set zero transformation mathematically accord input sp font lb lb lb lb lb lb lb lb counting multiplication assess obtain complexity state dct dct chen dct full version version retain coefficient technique transformation mention section compete coefficient transformation compress process evaluate degradation zero nz quantization nz translate long zero beneficial subsequent encoding code stage contrast adopt average table value percent value set significantly maintain qualitative comparison compress describe capability separate modify approximation hardware evaluation transform transpose buffer test matlab realize gate array device validate hardware loop interface approximation prototype use complete agreement verification hardware code nm technology synthesis implementation respectively flip ff critical delay area time operating frequency synthesis place tool file design frequency propose dct show area consumption dct synthesis nm reduction reduction normalise metric clear use dct frame hz assume rgb resolution ff modify modify propose video embed reference transform reference chen dct consist software show frame code chen fig frame db minimal degradation db computational point dct arithmetic chen dct compute rd sequences qp compute bit curve
induce cluster modal ideal population straightforward formulate informally mode population modal shift example include shift modal alternative version section population lie behind modal population reflect region high separate cc exp exp exp node exp right node split split plot exp exp exp node node node scale final set methodology identify cluster density cluster level correspond hence single increase reach splitting branch panel component core probability part assign depend point line difference equivalent clustering singleton leave branch two branch panel core component node clear splitting cluster panel correspond precisely formulation define minimum attain solid circle panel notice level core constitutes approach color cc sigma sigma normal pi exp pi x grid smooth gray axis cs cs axis generalize high dimension follow normal distribution natural separate density branch differential topology topology study critical useful application range represent indicate flow effect summarize follow enough degenerate enough time nan degeneracy point index negative critical write sign precisely example possible figure minimum saddle index ccc x name width view title name grid domain title name view domain title unstable manifold explain minus smooth integral satisfying minus descent water unstable curve start analogously integral note form manifold point stable manifold unstable manifold contribution modal unstable gradient maxima modal flow cluster clear saddle point index associate unstable maxima unstable manifold rw r u cluster apply univariate example maxima manifolds minima unstable manifold gradient curve satisfy unstable manifold become stable manifold could xt give example ideal modal look bivariate normal terminology iv mode range true population modal cluster cm contain contour triangle point plot mark triangle point thick pass compute numerically make thorough density location mode take mean different connect curve finally numerically value shift saddle clear kind essential every twice redundancy add partition compute thus interpret penalization cc node scale thick idea even shown match depend permutation component empty obvious minimum match match estimate replace measure distance clustering differ include transfer extend population counterpart minimal mass need transform connection clustering hausdorff note empty hausdorff equivalently ab analogously take consist set distance identify clustering subset hausdorff distance whenever hausdorff regard distance hard standard demand meaning hausdorff value instance clustering hausdorff mainly due picture hausdorff obtain wise minima add copy minimum far involve match solely analogue obtain probability lead consider distance clustering cluster understand methodology population clearly consistent mode stochastic convergence represent clustering define sensible clustering replace include nonparametric mixture fit unsupervised plug dimension study easy nevertheless important development plot exp right figure estimation intermediate line density estimator density close sense clustering suggest cluster easy density modal clustering density univariate support modal induce sequence almost derivative modal clustering surely proof state clustering boundary minima density dimension scope manifold estimator bandwidth necessary time truth represent try close specify depend notion whereas population goal like modal modal identify make tool partition maxima modal need smooth certain degree specifically time differentiable function extend notion mean smooth critical treat resort mapping cover book present play gradient goal clustering ideal introduction aim methodology approach consistency mild study minimize distance base measure method aim perform modal necessarily rely adapt estimate connection part project grant use finitely many isolated critical modal minimum sure interval hand strictly big critical since previously critical similar argument also neighbourhood mode convergent subsequence subsequence neighbourhood critical neither critical neighbourhood contradiction converge n j write minima minima theorem lemma corollary remark despite recognize investigation theoretical reason problem specify target seek population base cluster algorithm try aim theoretical focus ideal population modal region new methodology evaluate population mild estimator modal branch research rigorous methodology recently express concern paper contribute regularization state cluster even seem method statistical cluster solely notion gradually merge agglomerative cluster graphical successive group know inter merging linkage complete linkage linkage notice usually pre specify seek center goal certain function represent extended determine distribution state need statistical nonparametric cluster parametric mixture generating use bayes rule region separate low concentration mass sense mode function mode modal understand definition goal modal introduce datum connect usefulness advantage drawback population recognize author like matter think many show univariate phenomenon modal visually identifiable none three usual recommendation several value tool orient graphic function detailed explanation plot
soft display soft green intensity vector provide plot soft assignment location plot across likely chance domain cluster distance determine soft choose base boundary tend color mild implement visualization technique distribute isotropic cluster coordinate identify display visualization pick sc consist chemical feature region area find use chemical measurement cluster normalize select show gap gap move cluster cluster cluster connectivity apply point area see cluster dominate type produce contain cluster south observe cluster connectivity map mode reflect capture structure produce two observe edge connectivity hide interaction width edge degree dominate connect spread south north north west cp I om pp database repository protein different protein class cp om pp feature find standardized rule filtering sc plot cluster visualization q visualization panel connectivity confusion versus panel confusion panel third overlap protein e overlap third successfully identify apply uci machine database repository later camera inspection image pixel picture wavelet extract datum figure separate seed little connectivity panel seed high connectivity mode cluster select new visualization high also establish high cluster visualization chi chen grant support nsf number nsf grant dms need assignment gaussian variable unconditional soft kx pz data introduce latent representation soft representation infinitely representation density lead mixture thus depend upper define mode define closeness specifically mode l py l ii profile eq control distance boundary illustration method replace kde summarize contrast coordinate free soft data bandwidth contrast shift evaluate level py soft assignment less cluster correspond weight define norm scalar function ordinary consist mode must correspondence mode bound note eigenvalue local density less must mode generalize local mode triangular mode estimate mode triangular inequality estimate vice versa sufficient require need kde theory eq constant location location mode approximate focus derive generalized third eigenvalue away invertible pointwise nonparametric theory page mode rate thm pt proof mode chen department mode define density mode soft variant assignment cluster visualization secondary cluster method mode relative population estimate estimate mean algorithm choose depend despite advantage room improvement mode cluster hard uncertainty visualize cluster mode tend call paper visualization shift segmentation cluster statistic idea select bandwidth rule merging mode soft assignment estimate measure selection mode occur frequently grow high multidimensional example assume degenerate unique mode e path path standard integral intersect contain lead saddle local minima kde kde bandwidth easily using become assign attempt assign soft belong soft type population intrinsic variability strongly mode boundary associate soft sample level uncertainty come soft vector capture uncertainty remark distribution latent discuss mixture soft straightforward obtain soft idea mode describe method appendix soft way start diffusion start mode mode reach easy interpret eq define actually run diffusion correspond th assignment discuss cluster despite literature mode mode adapt hausdorff hausdorff hausdorff generalize non condition l commonly kde grow estimate derivative kde local mode local hessian modal constant hausdorff assumption condition eigenvalue mode consistency number consistency explanation fact kde mode apply taylor local gradient estimator hausdorff distance decompose bias technique soft among generate hard mode assignment connectivity high assignment analogous classification class might row matrix connectivity cluster useful summary overlap else bandwidth gradient standard estimate common square generalize rule reference smoothed validation reference slight modification deviation reason cluster vector difficulty interesting consistency consistency supremum norm mode consistency produce cluster call kde converge slowly create kde turn consistent slow example mixture coordinate order smooth gray four hand filter gray gray want approach deal increase merging may cluster quick simple curve merging enforce threshold merge large cluster recommend intuition recall well diagnostic sc display gap cluster induce know gray reference rule signal structure addition denoise remove persistent persistence extremely computationally intensive
problem without gradient programming method programming experiment conclude suggestion subsection convex application non implement particle particle particle movement reach movement evolutionary particle particle respective particle velocity equation velocity varied iteration particle well update velocity iteration position update interest determine shortest short path ellipsoid pose region case distance suppose known reformulate ellipsoid outside ellipsoid ellipsoid determine particle close ht outside ellipsoid dotted evolutionary reach initialization update equation algorithm modify addition evaluation away addition advantage computation however particle move counter velocity modify velocity varied iteration ellipsoid surface sphere form velocity vector direct f minimize present position split previous objective dependent minimize direction velocity update relate include reduce purpose direction shortest place determined region particle place space particle point path particle reach often go position update particle lie within search intend position particle propose multiple gradient problem solve reach consensus discuss share alternate direction multiplier one constraint solver format present randomly update position global store maximum iteration specify experiment initialize good calculate range update velocity position section reliability test quadratic programming lp constraint lp reformulate identity ht iterations lp except reach monotonically ellipsoid length scale reach figure ellipsoid ht two lp iteration c ellipsoid position x neighboring space carry neighbor high fitness become ht algorithm deviation apply generally suitable classifier choose weight use classify test pose solve show class mean equation kind data feature hyperplane hyperplane region platform implement svm svm neural hyperplane learn show layer perceptron epoch training determination hyperplane class mahalanobis point ellipsoid class placing evaluate close optimize reach boundary determine hyperplane hyperplane observe hyperplane place hyperplane bias method svm equal network svm test uci repository dataset categorical case hyperplane characteristic network estimate classification hyperplane approach develop bias move recall varied nature attribute integer length linearly whereas linearly physical property colour intensity whereas chemical property content content chemical eight parameter value age value problem dataset database contain binary vector input perform formation sample step eigen decomposition eigen perform dataset independently subset validation project class eigen correspond covariance project test validation use remain fold ten subset subset equation hyperplane turn classify average cross
standard interact worker dynamically offer utility know outcome value time contract observe realize receive feedback assign compare utility goal subset infinite natural example integer offer minimal payment worker time let value outcome order increase tie arbitrarily convention nan outcome cost production obtain outcome effort let contract function outcome non contract sample payment ix e worker utility expect worker type population density many effort outcome effort type multiple effort worker contract break consistently worker effort contract minor minor throughout compare many benchmark reduce good contract contract maximize utility outcome regime quality fine nan outcome study special pricing obtain improve bound monotone appeal generality instance monotone appendix far monotone typically contract reason crowdsource restrict attention satisfy achieve guarantee relative machine useful benchmark benchmark relative guarantee relative x optimize special unclear extend contract crucial maximize utility economic often amenable rigorous suggesting may rational pointing heavily regard worker behavior serve enable guarantee collective worker behavior property use increase increment payment increase enforce form complete outcome pay formally increment splitting half dimension carry version task discussion contract mab additional basic mab repeatedly choose traditionally call specifically round select arm receive reward reveal mab arm arm round regret arm define reward dynamic contract design naturally model stochastic reward monotone mab assume upper precisely respect supplementary mab set reward payment determine outcome determine worker effectively supplementary structure sense numerically call area action know state discretization space several choose treat space use monotone bound describe discretization monotone contract nan outcome non convention contract bound contract monotone increment representation lie contract necessarily bound cube convention weakly discretization align dimensional increment henceforth cell least contract cell candidate contract denote increment maximal increment atomic contract advantage problem essential composite ideally maximal difference utility proxy useful follow prove type satisfy consistent break composite cell place worker behavior development worker behavior outline cell comprise increment round confidence index contract half corner cell introduce round algorithm consider round round utility composite payment similarly payment round anchor accordingly estimate virtual deviation suffice high average expectation algorithm active cell confidence contract become uncertainty due express virtual width cell summarize initially contract post contract allow amount composite minimal near allow contract corner issue uniform mesh preferable composite maximal corner contract contract minimize minimal anchor corner go omit issue virtual write pointwise dominate consider weakly resp contract resp contradiction type denote generate I xx contract give contract combine obtain note equality contradict outcome contradiction complete place directly break weakly worker effort respectively assumption either rule proof payment increment expectation worker finish contract significance corollary compare parameterize goal respect candidate regret rt contract utility optimal x cell feasible implicitly overlap sometimes contract outcome parameterize constant practically small outcome exponential notation candidate policy exponential type prior dynamic pricing bandit approach tend arm also fine equation apparent regret bound bandit state covering subset relative cover number cover cell virtual width contain cover collection minimal small cell cover size feasible observe exist minimal cell cell easily yy literature cover regret precise polynomial performance bandit appropriately notion dimension capture problem typically shape notion eq equation obtain consider dimension bandit approach contract design mab mab corollaries loose bound rt rt equation apply factor may well corollary setting discretization interval bound yx bandit metric discretization significant advantage argue without know coarse probably aggregate accord predefine formalize hierarchy subset contain hierarchy contract splitting half contain shape aware similar coincide regret focus spirit slightly structure cover ball ball albeit determine bad nice problem instance obtain match bound aware aware algorithm information special basic nan worker whether reject effort contract completely price distribution nan outcome rich discretization improvement principal somewhat rich worker salient contract outcome nan outcome bring outcome bring cost break way incur zero outcome high simple worker effort outcome high pricing x function note maximize know reduce essentially price high low lipschitz continuous fairly worker small width regret consider low give natural contract achieve range rt mab contract worker worker effort receive payoff therefore effort effort call inside non decrease discretization dynamic pricing discretization contract result contract denote p conclude width need count cell increment virtual width small denote benefit axis increment contract also contain characterize virtual cell expect payment value two definition represent simplification count cell virtual large care cell therefore relevant know virtual version alg alg arm arm randomly alg discretization xx arm argument mab regret execution event clean involve probabilistic argument tend simple execution cell contract essential ensure execution round execution least concentration chernoff need careful number activate th activate activate round fix cell claim choose round event cell tuple tuple consequently chernoff let integrate precisely let multiply side cell bind feasible union proof sketch one namely composite round anchor select anchor notion technique establish event play sufficiently often auxiliary enable choice precisely hold tc equation rest execution execution contract b claim anchor clean execution contract execution round atomic composite contract fix part clean execution fix contract composite atomic unique contract clean execution cell round else recent cell contract regret next upper bound hand equation clean execution lemma execution cell depend cell follow atomic contract candidate contract easy lemma ever activate order choose cell cell contribute focus cell never active activate cell call whether leaf since claim therefore cell leaf parent since case claim plug make simulation bandit thompson replace change observation consistent pick composite within increment minimal thus effort worker market extremely diverse third represent ground market market market across suffer near consistent intuition focus explore region slow eventually achieve worse initially eventually suggest know advance tune advance confidence confidence play simply reward plus try well across three market constant long utility round various first horizon outperform version market advantage specifically example adequate large fast adequate focus promising region achieve know advance optimize approximately calculate advance small vs homogeneous worker market show thompson logarithmic converge payoff confirm decrease round utility run round consistently outperform two bandit different discretization market discretization know discretization still outperform finish run set run thompson round opt offline plot large bar discuss see dynamic contract family perform price sequentially principal item agree item derive utility minus round know fix know principal pricing contract outcome special one effort generality effort level non crucial simplification contract design discretization easily mesh case implicit pricing modification pricing bind translate pricing pricing idea achieve match low sketch low lemma contain contract summarize nan worker expect contract loss price price interval virtual reasonable choice dynamic pricing regret sketch key call former happen otherwise mutually cell virtual red cell virtual dimension argument conjunction correspondingly bind task pricing horizon achieve give interval cost pricing interval partition sub dynamic pricing achieve continuous follow cell virtual diameter least diameter diameter j p feasible virtual feasible cell diameter overlap ok moreover less feasible theorem case instance rt nice two interval adjust two discretization specific principal derive utility fix p complete consist argument rt imply rt rt paper contract crowdsource outline area extension principal contract classic agent whose production describe specify contract outcome effort stochastically outcome maximize utility contract observe outcome effort level creating contract expect payment make maximization constrain problem principal assumption type know exist problem focus principle principal offer choose reveal type problem principal utility principal agent restriction must certain range capturing must agent paper neutral instead set hazard space know characterize decrease interact multiple agent principal adjust contract author agent agent offer variety set principal agent effort level uniform contract version principal single repeatedly agent effort outcome effort effort observe contract work agent learn type design accordingly study online principal focus uniform discretization approach study set outcome issue worker verify outcome standard know worker verification payment bundle outcome assume relax adopt david journal recently encourage high work crowdsource explore award improve user generate encourage internet content motivate attention sided encourage crowdsource market crowdsource market differ worker effort close crowdsource pricing task arrive price worker goal learn single generalization pricing outcome examine financial crowdsource market potential explanation phenomenon concept worker cost complete worker task complete worker worker appear payment mind research run screen worker quality effect existence overall demonstrate crowdsource market traditional people market rational worker economic demonstrate collective particular decision algorithm decision multi mab dynamic pricing mab operation economic branch computer theoretical science ai economic survey work beyond refer mab background bayesian mab lines mab mab reward neither know formulation understand handle arm information similarity arm arm relaxation discretization use particular general template lack priori require mab immediately search shape explicitly reconstruct part direction mab setting principal offer price transaction principal formulation version note initial improve specialized unconstrained crowdsource market define round agent treat bandit conceptual aside adaptive assumption provably case illustrative theoretical dynamic contract design far clear provable structure need order argue optimality contract currently fine cell sophisticated sophisticated bandit incorporate cell sample deep principal primarily natural mesh open concern x mesh mesh interest monotone prove significance unclear would characterize scenario contract bad contract result latter scenario need much extensive special difficult access appear direction corollary bound optimize apart design derive case mixture belong parameterized family unknown parameter direction budget extend corresponding dynamic pricing difficulty setting much contract adaptive discretization conjunction general bandit bandit choose mesh price budget discretization bandit section provide suboptimal restrict non payoff I level high type subscript describe worker choose effort outcome equally likely choose equally verify simplicity assume worker tie effort favor break tie effort level favor worker worker break tie consider separately contract make worker choose nan contract contract break tie effort contract cause choose prefer effort easy would low effort case expect worker worker choose effort worker choose contract maximize value maximize appear maximize appear set easy occur plug contact expect utility strictly preferable contract fact unique contract contract summary exist effort level effort obtain contract payment neutral I payment agent principal payoff principal agent payoff optimal contract give contract
bottom propose paper start objective illustrate step embed cloud follow propagate merge body extensive topology transition study application conclude main objective unsupervise coherent automatic move body body represent voxel obtain pay attention representation suppose collection link possibly meaningful segment move segmentation place unsupervise human intervention segmentation entire surface shape consistent element adapt topological priori move recover posteriori consistency segmentation limitation explain spectral apply action variety compute dimensional embedding preserve reconstruct follow square computed minimize error q embed cloud matrix rotation bottom eigenvector discard ht despite originally unsupervise display geometric unsupervised cloud voxel neighborhood preserve dimensionality cloud property roughly live approximately map spaced widely separate branch cloud constraint link form roughly force radial direction space much compare cloud arm unlike base geodesic neighborhood relatively purpose coherent obtain evolve geodesic strict sense laplacian embedding form link neighborhood preserve motion neighborhood evolve affect illustrate pose transformation intrinsic effect easier embed shape body pose propagation much exploit devise consistent shape weak cluster unknown inherently dimensional embed propose step instant cloud cloud cluster branch intrinsic low embed segmentation segmentation instant go description map embed space employ mean segment embed happen adopt sequence shape segmentation automatic moving embed try branch look roughly mean distance measure triplet leave generally tuple propose notion graph pair nonnegative tuple measure analogy edge point tuple next weighted vertex hyper construct finally part hyper embed affinity embed cloud trivially back use automatically step form branch instant branch termination certain empirically point project onto cloud red branch termination side square g consider termination well embed transition move whenever happen accordingly return need segmentation change contact different part centroid initial seed cluster embed cloud centroid square left position old colored circle xt jt obtain embed cloud embed centroid embed cloud time colored circle seed cluster arise besides help contain body contact cloud possess way arranged belong sensible branch detection side suitable tool allow adapt topology instant branch cloud seed cluster branch seed yield rough cloud branch seed inside k mean branch termination seed branch termination wise seed get close distinguished make sense opposite impossible distinguish require merge topological move body validate change management summarize segmentation integrate section instant ht current data figure embed yield cloud branch embed cloud detect branch plus embed cloud cluster group section seed use branch centroid branch splitting merge centroid embed shape centroid map centroid back algorithm synthetic qualitative quantitative test set sequence person ground body link e simulate form plus volume solid black drift inside shape motion usually span distinct body part phenomenon trajectory segmentation visually show relation sequence frames b subsequence frame whole centroid transition management arm contact number accordingly proceed smooth viewpoint several high camera long capture move around cm typical smooth centroid trajectory separate complicated perform body topology person remarkable possible three cluster embedding mean ground truth voxel close capture multi camera phenomenon gap figure middle body head middle adequate score high method show remarkable compare drop brief trajectory top show middle gap affect quality segmentation embed cloud stable three challenging sequence apparent strong corrupted topology bring size neighborhood former affect stability embed shape along low notice embed remarkable general embed varie neighborhood body different anomalous belong distant neighborhood top right value yield neighborhood method use computer graphic tool analyze surface virtual reflect classified encode operator mapping vertex neighbor point vertex edge th trivially represent also operator point affinity laplacian w locally laplacian eigenfunction geometry underlie eigenfunction domain set eigenfunction discretize follow eigenfunction coarse partition interest affinity link associate determine cloud choose dimension result make shape arbitrary able visually visualize eigenvector color cloud determine top four brief dark peak different eigenfunction locate eigenfunction peak result may possess branch eigenfunction able resolve head separate head appear eigenfunction argue segmentation branch turn amongst illustrate voxel fair run product voxel limit natural segmentation head show top bottom stable subject consistency consistently body move top bottom point laplace sample produce distribute along regular voxel neighborhood cause instability embed cloud particular main index segmentation sake simplicity first eigenfunction w segmentation figure plot segmentation apparent pose come contact cloud truly consequence measure distance along body path cloud illustrate split merging segmentation preserve embed change situation preserve change recover distinguished compare shape body synthetic sequence plot exhibit smoothly virtue plot space embedding exhibit body perform geodesic spectral transition clearly advantage cluster shape embed choose sake change dramatically see building motion framework segmentation body look clearly smoothness track feature track figure model estimate classify one segmentation coherent segmentation chain output reconstruct rough model free motion fit principal axis position part voxel set represent environment hand interact virtual object virtual environment ellipsoid implicit representation time interactive object voxel note look construction identify remarkable hand evolve become isolated rough show comprehensive present dynamic segmentation cluster temporal consistency exploiting characteristic estimate topology transition algorithm versus k synthetic generation extension separate contiguous pose quite straightforward manner propagation different object exploit method remain preserve near future universit france order motion pattern compact discriminate context learn recent time motion capture voxel gain segmentation move entire coherent robust construct bottom move body track motion locally embed useful volume shape dimensional easier unsupervise coherent segmentation shape embed coherence merge accommodate body real voxel datum consistently totally unsupervised robustness quality series discriminate infer learn fashion time set calibrate yield gain attractive inherent invariance motion flow kind vision computer graphic visualization sub mesh segmentation application graphic link representation mesh actor reality texture map reconstruct may topological scene actor extraction obtain smooth suffer surface curvature sensitive shape graphic applicability visually acquire remark interest recently segmentation visually reconstruct segmentation static mesh mesh propose probabilistic method motion segmentation mesh advantage embedding mathematically voxel think describe shape connect vertex represent adjacent point voxel address partitioning provide extremely powerful framework allow partitioning laplacian suited space span eigenvector laplacian problem allow circuit partitioning segmentation mining method laplacian characterize perturb ideal datum infinitely link locally map attempt graph circumstance partition kind graph strongly make mesh segmentation spectral cluster number firstly difficult completely secondly eigenvalue symmetric matrix crucial finally eigenvector interpretation mesh domain view eigenvector applicable measurement estimate pose discriminate motion mesh surface volume approach availability set volume surface geodesic volume invariant pose surface sensitive topological often occur visually less mesh local mesh volume consistently partition surface pose community curvature derive segmentation reliable visually reconstruct consistently segment point wise rely use explicitly
spline outperform foundation grant f theoretic smoothing spline reduce describe curve remove outli spline dynamical result problem risk describe convex solve via numerical control spline widely process particular give curve limit curve well minimize curve theoretic spline spline control spline linear theoretic spline rich curve give robot drive draw smooth find expect motion theoretic spline trajectory mobile contour distribution estimation name application theoretic spline see conventional theoretic spline drawback fit crucial drawback conventional control spline outlier overcome propose parameter utilize norm absolute shrinkage robustness adopt norm assume tail assume study matlab computation matlab program effectiveness method organize review control discuss drawback spline draw conclusion dynamical na suppose sample data control dynamical follow cost regularization specify smoothness define second weight loss theoretic spline formulate control e impulse response dynamical define control spline control compute offline formula draw curve drawback spline noise adopt design spline error spline coefficient lasso p pp additive assume heavy account unlike spline solution represent within extension nesterov achieve still use efficient software seek minimize basis variable optimization curve control trade fidelity zero chosen error cross section example spline rate
derive corollary fix section eq ix x n kk x probability continuously differentiable statistic quantile estimator assumption probability scalar pac arm eliminate total refine upper k derive regret convex measure average law coherent express integral guarantee risk depend choice either quantile section theorem almost arm assume small q derive give recommendation average bind rt h functional quite language maximize bandit describe focus refine countable entropy denote number consistent entropy plug matching entropy function support let near estimator theorem proof result recommendation rt information functionals r enyi entropy notion shannon cf divergence predefine arm tackle elimination arm elimination particular assume efficient arm stop refined use management plug equally want probability return advance proof lemma well eliminate drop episode arm arm eliminate denote number per l triangle follow bind finally triangle equation chebyshev v p v triangle eq verify two theorem yu arm unknown arise entropy language new combine arm tackle achieve provably illustrate method number management refine stochastic slot reward independently randomly arm high reward budget accurately range identify medical trial transmission mab optimisation find high arm functional arm finance risk capture functional metric arm functional expect estimate consistent bias functional entropy risk background call identification optimal propose efficient elimination optimisation arm trial algorithm generalise successive elimination sequential theoretical guarantee refine scenario functional applicability management risk management variance functional functional widely follow contribution introduce identification bandit generalise arm propose elimination regard generalise optimisation theoretical refine risk management discussion bandit variety adversarial survey bandit type average pure exploration relate pure markovian armed bandit risk utility risk functional real maker variable arm line notion finance optimization bandit consist receive unknown repeatedly arm result sequence functional goal identify high within exceed observe round total arm elimination sufficiently value I efficient one sided property guarantee far functional elimination obtain wrong order arm wrong addition eliminate eq arm eliminate round eliminate main state main eq probabilistic strictly success correctly sketch denote arm eliminate conclude decrease follow corollary follow pac probably correct pac former hold sketch sketch let conclude result comparable arm design expect compare follow guarantee exist value indeed l recommendation straightforward roughly speak outperform exponential exponential imply sufficiently one comparison elementary algebra
specie datum distinct similarity ignore closed lr procedure bold column hypothesis reject approximation literature information gain look hypothesis adjust elementary adjusted level overcome assess whole principle possibly real practitioner offer assessment whose gain remark type application provide know advance investigate procedure family contaminate gaussian p eigen popular cluster year remain likelihood tackle use alternative flexibility applicability along way develop reference lr considerable separated accordingly generalize procedure assess model advantage via application mixture eigen test gaussian see sect consider assume class original allow popularity largely theoretical significantly increase parsimonious model impose popularity contain member member axis member flexible parsimonious family see sect ml sect relax family base decrease eigenvalue ml constraints sect lr statistic compare family unfortunately solely restrict herein adopt hypothesis sect approximation lr component sect model line parametric lr sect drawback discuss pairwise benchmark overall member preferable mind lr recent sect aspect lr sect sect procedure demonstrate advantage variate th j parameter introduce consider eigen decomposition scale sort orthogonal column accord eigenvalue element geometric volume orientation impose constraint right side parsimonious herein triplet variable write different orientation restriction k pp kp kp kp provide representation model denote algorithm work I come indicate closed ip however characterize state fundamental family sect motivate orientation r scale component configuration family motivated end sect estimation procedure log group scatter derivative preserve programming programming primal active apply update common methodology sect transform adapt way test lr regularity commonly asymptotically distribute freedom specify ht start kp k kp kp kp kp k k kp k kp kp kp kp k kp kp regularity reference examine aspect come shape orientation eigen necessarily fix regard bivariate ex note shape eigen volume parameter shape orientation far generate regard choose orientation variate numerical guarantee overlap adopt measure overlap take value overlap overlap take computed model ht c c arrange move increase nan approximate uniform gray result assume size increase provide small encounter practice method bootstrap proceed replace compute bootstrap turn repeat successive assessment distribution distribution accurate concern solely rejection specify replication approximate replication approximately assessment hence sect bootstrap mm refer approach regardless prefer procedure respect entire generalize lr completely label hypothesis play true hypothesis nan false true false true mm indicate restrictive e parsimonious bf df say lr restrictive significant denote report reject significant strong evidence interpretable comparison hypothesis powerful among strongly control probability partial false hypothesis lr bootstrap lr environment cf strategy ik soft way one multinomial implie initialize force algorithm hence bootstrap sample correspond
normalize node node unseen unseen class markov come markov chain canonical form describe describe unseen leave zero shoot predict semantic involve perform shoot semantic connect connection probability specifically image see visually use classifier extend canonical meanwhile extend see node write extend transition state extend semantic node unseen g reflect unseen probability compute extend chain inversion formula eq start canonical block whole dataset image store probability equal stack final pre variable image therefore method linearly unseen maximum paths semantic whole semantic stable similarity zero shot see class compute c roc cat auc good per indicate bold zero shot provide class source class alternative first direct base see class category model bipartite image new vector test shoot prototype near shoot training rather apart publish first unseen name train skip gram text corpus word unseen english concrete website gray word unseen use training apply toolbox semantic choose unseen search near construct subgraph class choose see subgraph top cosine ensure unseen connect code shot area auc semantic auc six individual class ten result direct method nn almost class dataset class attribute propose comparable especially note visual attribute category manually exploit free word linguistic basis manual annotation encourage visual exist method c vector attribute ds see value see image influence parameter also totally divided fold fold run ghz average number shoot graph visual category shoot chain effective stable bipartite via transfer explore unseen class differ embed space image free space former focus work focus relationship unseen semantic word unseen view incorporate one probability unseen effectively shoot achieve linear image zero receive recently via social medium sharing image build visual large visual recognize make connection unseen class unseen semantic space embed early semantic attribute approach embed define binary attribute similarity attribute share class manually intra poor scalability attribute alternatively start popularity learn language scalability adopt remain similarity datum shoot unseen class training class need option first datum learn representation image belong unseen intrinsic limitation learn unseen domain adapt function unseen class option embed use low semantic feature embed purely unseen semantic superior direct vs shot object unseen unseen semantic graph see previous see unseen model fig unseen structure ignore view exploration modeling relationship flat hierarchical compare bipartite graph semantic stacking probability together zero shot linear image approach give section paper conclude zero semantic attribute employ embed knowledge exhaustive attribute work automatically discriminative visual embed attribute overcome explicit recently extract basis wikipedia attribute prototype target project use prototype shot learn word bi space manual annotation scalable scalability differ transfer experimentally exist label contain see exist attribute variant take mapping low embed learn mapping space image unseen class semantic prototype mention early strategy level test classifier similarity
sequence subject target indicate presence confidence information need human visual exhibit strategy location presence target ask human take present learn strategy optimally reinforcement setup operate regime inspire little challenging ground truth region available consist absence image human subject ask action available begin brief localization evaluate discuss experimental information extent scene play fall onto person background human weak localization overlap segment bounding box average action search signal segment pool analysis segmentation global vs image index represent index belong image image bag correspond region human image represent bag correspond contain region know none region detection target wish simultaneously separate hyperplane positive one bag inside positivity encourage separation bag response localization target distinguish contain negative poor localization positively label label slide detector exclude ground greatly localization overlap threshold eq formulation hyperplane eq k remove label treat instance g ik g n change iterative alternate assignment subject constraint bag iteratively maximal remain line positive bag enforce label maximal instance bag always label negative line detect bag long iteration equation exhaustive weakly aim minimize load restrict methodology experimental pd td f te pe response encourage formulate final incur execute penalty reward model start set approximated sequence reward strategy one please supplementary material bag consist segment bag consist segment evaluate svm c set segment extract image max convolutional bounding descriptor box image mask rescale pass record fully additionally build consist normalize aspect final neural bounding descriptor train function segment remove constraint train confidence segment restrict image label investigate setup bag segment pool image return truth box object action within bound play head mobile phone fig trivial evaluate metric response predict bound segment fall truth report classification outperform baseline metric large supervised precision metric restrictive suggest supervise approach recover invariant pattern within least absence full extent actor topological learn substantial metric additionally leverage movement annotation classifier good learn supervision mi fail human movement task train seq bound bb seq case learn image alone optimize bfgs optimizer regularizer initialization run metric measure segment total use intel cpu require approximately half segment leave base exhaustive bb performance affect extraction optimize operate reduce search form novel base static box annotation topological constraint accurate classifier movement additionally novel sequential achieve detection extensive progress speed supervision institute mathematics science mathematics department se recognition detection increasingly therefore expensive manually keep run address issue weakly supervise segmentation detector signal provide localization system use inspire operate detection confidence exhaustive fraction constructing detector decompose confidence detection search set search suit train require manually algorithm increasingly need search practical rich supervision development hardware less expensive annotate acquire training usefulness absence additional annotation insight operate paper detector confidence weak manual image annotation contribution learn static associated movement require annotation box segment movement integrate strong supervision target box develop sequential operate improvement accuracy slide window accelerate prominent branch neighboring region use detector base vector image prediction formulation supervise svms see review movement collect video include face action successfully boost action segmentation availability movement human use object detector method employ supervise human reduce box play role annotation ground truth limit two recover ground confidence require availability bounding box visual idea problem recognition face novel fully setup show want list tight bounding box actor additional spatial
initialize ap successive coincide red ap similar randomly lower constant lower bind linearly norm lie extension make pair small state duality turn duality gap translate discrete linearly require q minimize separately value desirable reader running times oppose one exploit combinatorial algorithm frank wolfe gradient combinatorial run integer value function frank wolfe subgradient behave berkeley submodular variety discrete machine processing vision minimize pose challenge extremely linearly upper rely geometry submodular spectral recent submodular set inequality high potential submodular minimization q exist inefficient scalable optimization frequently possess admit quickly example kind function count joint term subroutine minimize modular box cut certain sparsity induce covering express submodular recent demonstrate offer benefit admit minimize separately manner decomposition decomposable close bring together yield empirically implement case performance heavily well alternative leave geometry approximation prior relaxation problem importantly associate submodular draw submodular beneficial submodular consequence ellipsoid polynomial fast integer require address decomposable integer maximal cardinality simple approach nesterov sublinear integral algorithm study convex feasibility good problem survey subspace characterize angle rate alternate arbitrary convex understand give condition alternate case unclear challenge rate uniform submodular largely study relate thereby obtain generality relate subspace connection useful submodular identify modular modular polytope many extension problem relax extension round continuous indicator amenable smoothness alternatively formulate proximal recover indicator discrete bf bf imply project project project projection projection simple live set note submodular omit dependence simplify bind uniform submodular ap descent start via implement solve analyze eventually pair ap convergence ap motivate approach simplify setup subspace span case angle slow higher relevant cosine subspace arbitrary converge linearly rate ap rate generalize consider pair show rate ap nonempty face generalization angle two part relate ap faces general ap theorem angle corollary angle matrix tool specific state bad rate ap ap result weak assumption join triangle cm color color pattern pt color color pt color pattern color width singular product bf rbf rbf proceed characterization face base submodular disjoint immediate disjoint write follow directly multiplication less remark less equal matrix index row remain view matrix symmetric weighted let index vertex laplacian small eigenvalue appendix bound hence appendix ap converge probe submodular slow augment submodular f r zeros cosine around optimal behave subspace pick initialization angle low exists decompose ap generate objective find define multiplication map fx fx translate discrete set small value give number submodular highlight structure serve work aid grid hard additional suppose nonempty subset relevant applie run future dr subspace converge oppose cyclic stochastic award award fa amazon services intel microsoft yahoo office research grant nf nsf need map close nonempty subset suppose eq follow similarly lemma part rgb round join triangle cycle pattern color color color width pattern pt pattern pattern circle fill circle node fill pt segment convexity middle angle inequality face face far point amount toward interval every either contain face whose relative face contain unique face state ap possibility intersect ap terminate terminate otherwise face j inductive ap ap along see result face face amount affect angle
generalization cover return encode piece encode second part possible construction towards label diverse want encourage change correspond assign label diversity indicate envelope take value index pairwise message bp diversity cut plus minus berkeley origin negativity achieve achieve space svm extract often slack svm primal primal w equation add problem equivalent enforce negativity primal solution hence expect negativity plane label diversity presence scoring contain diversity contain ground define item belong adjacent label transition marginal gain become eventually maximize cut look front show different diversity set solution understand different fs fs prove bad case let fix obviously submodular cardinality constrain optimum sized correction least monotonicity submodular maximizer I helpful fa monotonicity rearrange
equivalent subset stage algorithm add variable consider stage lasso induction lasso know sign support recovery hold lasso lasso abuse show invertible al know indicator function trace solution parameterize probability apply al weighted apply always assumed lemma et new probability combine force lar al lar force allow algorithm analyze x x lar lar estimate lar assume variable hold lar suppose correspond lar active lar iteration slight guarantee variable point w zero x strict contain argue induced base x variable select inductive lar intermediate since run lar stage variable add stage since intermediate w stage always follow add principle end stage lar know recall give partition g define constraint union decomposition permutation product permutation b b permutation unique sorting specify construction q b permutation notice tuple permutation group b consider piecewise linearity supremum set piecewise lie local minima tuple permutation induce replace x absolutely shows immediately seek boundary lie concave switch interior boundary versa purpose derive contradiction concave lie hold unless contradiction lie wrong minima lie concave path path b nonempty union solution b suppose absolutely unique however produce produce homotopy maintain identical lar modification note notational convenience carry homotopy
portion portion memory single master machine similar repeat amazon ec master machine core gb ram run hardware acceleration netflix row column zero report large want remove low running unit storage implementation ni miss goal row variance one simultaneously consider standardize via model standardized realization random would realize unique include likewise overall refinement attempt center complete issue learn center latter important completion reverse center prediction miss full although centering similarly conclude lemma place replace lead consider converse appear use say hand analogous true thereby use lead rate complete q assume proximity rate interpret locally eq rate property towards limit simple problem suppose limit k fa kb line follow contradiction e limit point lead problem much load want center sparse class introduce old rearrange get similar modify multiplicative symmetry get similar equation amount iterate four equation zero algorithm remark proposition lot largely netflix competition solving margin factorization bring together completion software implement approach environment index element preserve replace onto complete nuclear singular rank develop iterative solve step replace entry correspond thresholded operate replace element set solution matrix svd netflix storage number pose netflix use svd singular reduce svd compute alternate exploit piece hence store piece alternate step multiplication well svd time use warm start get warm start path use solution warm additional different use alternate algorithm solve separate ridge regression response predictor ignore amount regression remarkable tie rank block note orthonormal rank give draw idea step use fully ridge reduce regression use alternate amount shrink component offer rank property step recently approach see simulation remainder detail large superior include netflix highlight publicly implementation describe center incomplete develop svd adapt likely expert proof convenience inequality value denote suffice consequence von establish problem optimum appear fact solution characterize factorization consider lemma conclusion theorem inspire alternate algorithm reduce present ridge initialize randomly ridge simply coordinate compute multiplication follow svd keep solution represent need subsequent ridge trivial essentially alternate necessary naturally block many leave likewise plus multiplication determine singular correct evidence shrinkage accuracy bias case index solve initialize randomly alternatively warm start follow repeat suppose wish side current use importantly eq sparse dimensional problem multiplication modification step computation version predict multiplication rescaling row solution change c significant final solution soft svd ridge might reveal soft discuss lack warm almost package stationary denote play rate produce algorithm correspond problem derive low formal update lead derive description first produce thought style step upper loss recall objective ab outer equality observation suggest fa b ab ab ta bb bx ab x z observe procedure proof easily establish iterate never function monotonically iterate lead q aa put use argument complete previous derive elementary update every limit sequence point problem reach point respectively consequently fix follow thus quantify close stationary algorithm make improve monotone sequence characterize zero decrease converge quantity rate establish arrive theorem role corollary employ closeness stationarity respectively successive iterate understand guarantee remain across avoid appear estimate property begin notion point say order point follow fix update point singular tie stationary let limit point sequence subsequence sequence converge limit subsequence converge partial uniqueness point converge must converge versa generally technical bound leave update unbounded objective implication imply modification modification implement step idea modification also take choice hold decrease overall compare follow factored decrease nuclear sequence condition factor converse true estimate upon iff point satisfy convex stationarity stationarity rank stationary point see verification converse point condition stationarity closely nuclear operation constraint matrix task attractive plus low generate let limit solve solution broken forming require form require factor multiplication mention solve factorization algorithm instance descent separate ridge initial ij x section may decrease criterion slow factor experiment show netflix show result increase last least green svd svd iteration likewise exist instead fraction operating algorithm involve alternate ridge regression third alternate orthogonal operating perform model true rank k movie pick operating criterion coincide degenerate fairly consistent reason use execute early even though warm svd netflix competition user make missing probe subset rating leave netflix imputation shrink panel calibration light far dot competition compare restrict shrinkage rmse score somewhat win improvement warm
class sl sl neighborhood data sl method train resource table several baseline sl ne retrieval popular linear classifier sl ne achieve much accuracy complete reasonable amount embed training effectiveness accuracy attain sl close sl sl project sl perform sl ne similarity space may sl sl similarity function sl sl function retrieval retrieval curves sl ne consistently sl achieve sl ne b sl ne sl scalability visual challenge contain object validation test set sl train evenly pca prefer give fair short flat error baseline include svm retrieval two metric sl ne sl ne rate similarity huge sl ne improve lot fisher feature representation learn sl ne weak reduce indicate mm sl sl de sl subspace serve ne sl take sl de hour exclude retrieve train distribute computer sl ne day novel investigate neighborhood margin relative similarity training irrelevant ensemble scalability dimensionality validate classification achieve importantly demonstrate scalability method neighborhood efficiency potential direction neighborhood great complementary metric token lin research wang classify category huge amount approach promise study decade impractical handle paper scale image descriptor end discriminative similarity induce exploit dimensional process parallel learn quick validate scale thousand art achieve efficiency scalability internet automatic categorization become conventional approach versus paradigm large mention size web vast densely database result include parse face classification near knn successfully imagenet challenge measure characterize distance yield learn theoretic margin supervision comparative triplet grow polynomially modal usually single insufficient correctly throughout metric apply part datum space assign metric discrete parameterize extra besides dimensionality descriptor another scalability algorithm mahalanobis distance form computation dimension calculation cubic semidefinite impose mahalanobi save non reduce free feature scalability finding manifold sample local neighborhood try discriminative save focus original high subspace superior scalability datum validate sec efficiency scale thousand sec contribution paper embedding similarity scale reduction manifold intrinsic represent weighted graph laplacian construct refer dimensionality regard case distance learn label distance exclusive weight embed sample transform purpose similarity number quantify similarity bilinear similarity parameterize low place find embed cast weight encode commonly whether come partitioning impose pairwise sample knn triplet assign neighbor cardinality weight triplet hinge similarity hinge require measured transform solve eigen decomposition approach eq instead use gradient iteratively whole sample contribute gradient triplet update performed iteratively terminate upon triplet though strategy triplet speed accord optimize return triplet relative similarity sl ne data set normalize norm sl ne introduce objective base triplet constraint key scalable advantage sl ne sl ne computational progress benefit build dimensional impose low unfortunately bilinear simple quadratic computation ensemble low subspace projection learn function similarity ensemble project however scalability practice build energy guarantee sense direction efficiently approximated entail computation metric perform extra iteration projection attractive dimensional data similarity function impose similarity try summation constrain subspace explore combine scalable employ combine metric metric parallel computational forest distance binary output parallel base partition subset scalability sample learn ensemble function apply sl ne induce gain sl advantage distribute manner sl ne similarity way potentially reduce consider overhead parallelization function helps quickly offer resource carry optimize coordinate lastly sl accelerate metric sl ne propose sl ne version sl locality constrain code spatial pyramid grid dimension unless specify project space normalize metric sl ne sl de step sl sl open neighborhood neighbor search hash neighborhood sl soft voting knn classifier similarity specifically index class voting sample select entire testing sl ne sl svm retrieval generate neighborhood code publicly rough five
boost ratio supervise high observe improvement augmentation yield improvement convolutional auto train unsupervised fine cifar dataset method subsequently initialize call unsupervised fine highlight unsupervised auto learn input bias unit commonly choose include sigmoid function meanwhile back representation bias learn via descent constraint encoder tend impose upon structure form regularization include noise input hide activation model auto show relu cause activation use thresholde auto encoder regularization connect address convolutional cnns responsible neighborhood visible dense extraction pooling stack network learn aspect convolutional net possible extract localize fashion include introduce pooling map influence limitation procedure inference deep feed convolutional showing cnns show framework break pre supervise fine describe incorporate salient yet network convolutional layer bias describe architecture previous involve module follow module encode module nonlinearity follow pooling encode module decode module pooling filter decoder fashion reconstruction network would express learn representation fix bias convolutional activation auto knowledge train initialization neural network training relu relu set bias initialize draw patch layer sample singular value cnn account additive overlap hamming window intensity build weight decoder module module layer softmax module layer unsupervise supervised descent decay select duration pre center natural cifar color draw object category object benchmark learn relatively per class unsupervised contain per example cifar structure similar show modification consist convolutional layer size layer size pool full softmax output net train neural library state unsupervised supervised tuning report overall qualitative result bias convolutional auto perform encoder auto analysis regularization completeness report full visualize filter directly filter present convolutional auto layer indeed interpretable orient color center quantitative performance pre use augmentation experiment train initialization one pre train cifar cnn relu interestingly subset compare unsupervised experience subset cifar cnn bias supervised regularization affect specifically horizontal training zero zero cnn regularization cnn figure individually subset cifar supervised fixing unsupervise vary setup augmentation dropout virtual large unsupervise notable unsupervised pre augmentation dropout individual iii experience large gain observe sample figure effect iii supervise decrease elaborate effect improvement rapidly surprisingly unsupervise learn supervise cnn unsupervise unsupervised cifar worse unsupervised sample pre convolutional assess experiment beneficial design benefit network comparison network layer unit pool layer convolutional layer tune split consist class set accuracy subsequently average final recognition cifar train split highlight unsupervised increase extensive augmentation rotation color additional value pixel like six image modify pixel additional data helpful pre maintain accuracy convolutional convolutional hierarchical match task bayesian bias cnn cnn auto
relative primal dual tolerance notation r r r indicate version tv use acceleration nesterov fista manually fully explicit variant primal reconstruction dual manually primal cp e implicit variant cf section cp perform tv furthermore pre si however q problem warm filter whose effectiveness validate ray ct dct ii adaptive motivation initial choose fix strongly influence speed algorithm matlab execute ghz gb matlab build matlab limit single algorithm evaluate object via radial angular use dimension compute apply minimizer ground iteration influence inexact due may speed expect iteration relative threshold adopt individually reach truth show data measurement event cpu since plot performance cpu present evaluation backward projection nearly cpu indicator performance backward projection compute step consume thus projection evaluation crucial follow discuss tv show inexact tv problem lead also well accurate indicate algorithm trade rate another concern accelerate modification em tv tv actually improve em tv number iteration parameter increase tv result denoise tv inexact variable metric view tv error second cp cp contrast em tv influence inexact decaying observe affect yield convergence e suit achieve versa plot figure acceptable trade asymptotic convergence latter aspect case natural acceptable trade acceleration cp cp line size evaluation iteration right pre figure si observation tv choice trade cp si si evaluation cp si achieve iteration efficient lead tv denoise effort thus practically choose optimally balance proximal evaluation error second within tv algorithm dependent augment yield achieve see row cpu algorithms perform cpu get relative make tv evaluation tv cpu decrease problem approximate increase tv denoise utilize rough cf improve tv tv evaluate need high result slow practically evaluation term cpu cp performance cpu time evaluation decrease step improve remove single evaluation step increase number total table display second evaluation error performance cpu value coincide em cpu cpu em tv cpu tv cp cpu si best cpu cpu stability regard choice value smoothed smoothed table describe parameter disadvantage solve solve increase I lead cp si rough em tv scheme able respectively high carefully additionally accuracy proximal tv si tv across addition proper different parameter time require evaluation tolerance tv tv cp cp si base regard cp cp cpu tv tv cp si tv tv cp cp si cp cpu cpu tv tv si conventional ray ray intensity ray clinical practice decade carry understand valuable object distinguish type agent detector ct energy thus eliminate obtain spectra layer integrate detector usefulness image ct advance technology detector individually provide availability development lead image name material practical ct specific refer overview projection mean energy reconstruct material acquire maximum distribution number material consider accept integral reconstruct projection element inter compute decompose high fact exploit fully spectral ct proximal material penalty admm preferable ray ct g example apply employ ct simulation measurement spectrum response counting prototype ct scan current detector height source detector view per energy spectral decompose effect decompose via fisher information treat filter admm middle latter reconstruction material middle fully material reconstruct row good iterative exploit possess reconstruction strategy manually indicate preliminary advantage exploit inter experimental ct simulate six bin em filter describe middle cross decompose covariance effect right recognize difference maximal intensity reconstructed show frank university monte manuscript work ct university award research north theorem develop quite flexible allowing denoise resolution optical flow form eq fidelity depend often operator nonsmooth functional successful technique nonsmooth generalization norm coefficient reference therein efficient imaging structure choice like augment overview result computational providing discuss notation average operator discuss pose present ct notation definition chapter whereby operation convex semi continuous real fx fx subdifferential set consist conversely differentiable element global minimizer conjugate positively conjugate usual conversely norm digital common basically kind comprehensive overview proximal proximal prox define envelope straightforward ensure minimizer characterized envelope exist unique envelope continuously minimizer iv prox prox univariate shrinkage threshold know huber label node scale bend fy f proximal subdifferential function df rule subdifferential calculus uniqueness proximal operator introduce positive involve efficient frequently appear mapping operator empty rewrite consider next substitute lead see back substitution db proximal orthogonal onto x start similar describe projection completely consider operator q threshold sort large index simplex previous group norm consideration sometimes shrinkage let prox prox elastic set operator whole proximal evaluate involve norm unitary characterize positively homogeneous vanishe origin permutation analogous symmetric singular interested define x ij x singular nuclear norm find singular determined q determine construct solution diag diag diag diag u diag continuous e onto convex similarly prox prox prox f prox x prox prox banach fix contraction contraction property start continuous converge unfortunately follow example show operator obtain well x eq back name average yet operator space every operator lipschitz average respect parameter operator average assertion ii rx ix iii rx tx ty tx ty ty ty therefore ty tx ty ty rx rx rx rx expression conversely complete average composition composition average also average concatenation operator boundedness sum harmonic asymptotic mapping asymptotically q exist subsequence hand r j bt strict tx l result replace weak short proof simplify convergence follow regular iterate hence whole sequence converge e combine theorem main result converge theorem characterize rise proximal back since converge prox fx generalizations cyclic parallel lipschitz lipschitz e subdifferential calculus know minimizer proximal proximal splitting prox special function non become also detail modification extension lipschitz continuous lipschitz exist average concatenation sense norm hilbert algorithm function convergence indeed fulfil algebraic process arise solve time slow idea rewrite prox r modification convergence see minimizer progress know variational characterization proximal ii add main inequality q give thus single regard yield generalization algorithm nesterov nonsmooth fast iterative metric strategy replace symmetric positive definite search strategy mention approach overview cyclic ask rewrite lagrangian infimum saddle lagrangian problem problem converse primal constraint saddle lagrangian optimization alternate minimization know precisely note differentiable lx version type allow overview primal r r r new admm furthermore correspond theorem proof lagrangian saddle produce saddle lagrangian arbitrary consider g r subdifferential r h r p gx gx r r gx gx r x gx r gx ax r p r well impossible keep explicit p cauchy schwarz p p r r r r x p x r relation get sum regard x n r n convergent subsequence cycle concatenation affine operator operator jj saddle x r p n n j imply equation linearize proximal interpret admm variant proximal positive introduction provide flexibility operator induce symmetric definite admm also classify inexact monotonically analyze additionally allow inexact apply inexact use straightforward see primal hybrid bregman method process worker admm linearize bregman yy bregman interpret first order special bregman distance bregman shannon discrete generalize kullback leibler proximal bregman proximal read fy r converge initialization minimizer attain convex convergence objective x eq use equation imply split p obviously split approach type inverse fidelity regularization generally banach cf unnecessary bias contrast loss variation iterative pose saddle close inverse pose raise issue practically expect range reformulate standard paradigm iterative increase decrease saddle version optimality would abstract subdifferential know cf reason approximate iterative iteration topology ingredient primal variable use understand iterative analyze augment saddle superior imaging eliminate fidelity bregman together detailed consecutive bregman prove topology variant approximate linearize analyze however restriction iterative ill pose completely open future well variational step splitting dramatically reduce exist saddle decrease estimate convergence speed discussion aspect far heavily force rough algorithm classical imaging vision analysis application natural science area present emission x ray ct detail reconstruct worth emphasize proximal splitting algorithm broad application last decade mainly cause ability distribute big intelligence internet finance another boost popularity splitting regularizer problem formation derive quadratic
number due snps selection application linear big feature general np therefore resort greedy use sure screen sis select independently correlation redundancy correlate penalize introduce regression penalty lasso regression penalize method interpretation penalize penalize maximize double exponential exact intractable result relevance crucial inspire success computer study strongly estimator prior approximated reason ise posterior probability mean rapidly intensive validation procedure especially well suited modern often approach traditionally focus weakly effect assume tend weak regime penalty strong influential suited usually validation prior weakly perform well exceeds greatly exceed prior strongly physics derive series expansion posterior associate present simulation correlation predict c predict quantitative trait datum aspect many g describe gaussian generality throughout paper regression regression strength choice penalty include ridge maximum least instability compute l end exactly describe conjugate give close indicator simplicity coefficient ahead reflect identify bayes theorem error expression identity row residual error coefficient condition specifically prior bayesian penalize regression partition x variance directly relevance could maximize posterior distribution setting estimate expectation evaluate analytically computation often long expansion ise external field define pearson variable assume enough necessary detailed derivation expansion long regime strongly restrict matrix standardize relate covariance rx natural penalty expect order expansion reference map ising perform calculate ise approximation lead consistent field computationally tool approximate feature calculation correlation penalize expression depend determine usually value ahead feature selection variable n computationally eqs pearson aspect require storing coupling potential adaptive requirement even first relevant specifically therefore rank pearson sure first order approximation strongly spin represent enter couple negative coupling include couple redundancy feature first analytic expression affect percentage body small enough posterior applicability expression gene intuitively correlation demonstrate impact inter correlation selection tractable correlate pearson regression coefficient study expression feature feature stay path generate feature correlation figure feature red high clear gap separate irrelevant correct feature visible inspection feature beyond performance demonstrate relevant black irrelevant black vertical line accurate correlation feature point root relevant suggest posterior figure small calculation rank relevant ranking highlight couple reverse engineering predict expression consist team correlation blind elastic net predict response actual value parameter validation elastic regression feature highlight present achieve benchmark compare validate path prediction phenotype expression sample probability line net rank gray light gray outer quantile gray middle dark gray line mean posterior identify vertical compute posterior lead team challenge choose rank rather zero plot compare et gray represent quantile gray represent quantile gray outer gray dark inspection figure al high posterior among show percentage demonstrate generally feature probability although gene expression identify validate posterior range chance highly variable et selection surprising gene root correlation critical compare large ignore correlation relevance highlight importance genomic penalize regime call ise ise expression control analytical technique develop study analysis study bayesian technique practical efficiently path probability relevance dataset genomic study work ideally suit genomic feature much regularize regression relate focus identify mean occur transition regularize influence highlight accounting correlation assess statistical significance even small correlation cause huge reduction probability expression represent moderately likely implication assess ignore implication result correlation probably significance high dimensional manner independence screen rapidly low intensive procedure infer penalty suit procedure briefly review aspect entire argument detail main text appendix distribute without loss generality intercept regression function include penalize square least instability automatic observe use conjugate distribution p posterior specify flat simplicity ahead reflect identify relevant expression variable square residual error regression selection posterior expression use expansion x term ask eq calculate therefore constant thing together posterior thing little truly bayesian reflect penalize choose inclusion penalty help indeed maximum therefore combination definite series log posterior expand need place standardized variable go column converge correlation plug result namely
svm softmax cnn svm output combine eq loss understand kernel weight enforce hide sensible eqn margin hinge svm front notational eqn margin square note decide course overall produce output act hyper overall reach vanish role importance addition use simple enforce second vanish epoch although summarize follow want filter classifier feature map depend satisfactory classifier restrict part consider layer highly proxy performance eqn wise greedy perform overfitte state benchmark particular advantage capability train gradient parameter gradient conventional come layer supervision next formulation eqn ease advantage discuss function lead vanish gradient deep neural loose formulation hope advantage deep highly convex objective energy dl observe large often strongly step strongly function attempt convergence eqn flat convergence locally regularization refer reason two fold encourage layer directly prediction keep vanish direct supervision ground stage filter necessarily illustration b give loose comprehensive study name one w w j layer see illustration every mean lemma around make necessarily however observe flat ultimately really care gp directly see eq base assume readily start rate small strongly eqn use top show compatibility equation derive improvement convergence layer help discriminative benchmark mnist cifar cifar follow protocol et mini batch comparison illustration effectiveness match architecture layer convolutional error epoch schedule difficult engineering widely setup adopt scheme equip show boost softmax svm cnn softmax performance presence burden require augmentation cifar augmentation without average exclusive method illustration ccc c validate mnist handwritten digits widely extensively adopt class figure cnn softmax softmax softmax margin svm margin svm cnn svm show whiten augmentation compete train training gain softmax sample comparison generalization htp stochastic network cnn cifar consist training normalization corner average phase center table error augmentation add robustness layer fast rate burden hyperparameter tune cnn observation feature example cifar show map cnn cifar classification maxout network network error network network cifar similar cifar cifar make classification task setting boost consistently show cifar demonstrate advantage htp pooling maxout network view house number digits digit testing follow select sample class follow normalization augmentation table augmentation net view understand nsf award nsf award lin wang david proof example em plus height width zhang microsoft tu net make hide boost cnn architectures effectiveness presence vanish objective layer different gradient analyze evident exist e g art mnist cifar cifar network dl form gain amount deep technique hierarchical demonstrate automatically thousand million pattern concern fundamental current dl framework layer difficulty due vanish thorough algorithmic attempt make availability amount manual dl capable activity open sharing greatly adopt dl technique dropout pre augmentation enhance dl variety fine tune learn demonstrate observation initialization difference presence vanishing gradient dl make study follow discriminative display less hide layer discriminative train hide feature serve map make quality feedback able weight filter favor supervision quality
tracking approach update approach publicly second frame gb table four approach success rate show c boost phone chart report observe boost approach detection result tracking curve grow horizontal show track material clutter challenge perform change chart frame camera motion frame illumination variation frame object book contribution performance approach part mt exactly experimental phone chart book table find verification situation produce tracking capability separate consist present task output metric joint simultaneously discriminative verification learn model scenario structure complicated scenario discriminative enhance create benchmark video challenge video experimental correspondence address li support national science foundation china national basic china aa china engineering pt plus pt wu college computer science china edu cn computer graphic tracking spatio statistical framework effectively balance aspect manner coherence across frame consistency frames discriminative issue spatio task structure drive characterize adjacent model verification base feature intra inter separability three module simultaneously learn demonstrate due motion object augment ar object compression structural object appearance various cause illumination spatio structural property appearance construction variety descriptor encode invariance appearance sift speed extraction extraction descriptor effectively adapt appearance variation cast tree boost tracking intrinsic geometric frame address issue al structure track geometric propose tracking simultaneously underlie frame interaction coherence track joint capable balance important part coherence across frame frames coherence encode optimize frame propose explore spatial consistency perform verification structure output cross frame descriptor adapt situation structure discriminative power inter separability summary track task structure task learn structure spatial temporal present learning release video challenge video cover several experimental video tracking manually annotate besides provide track mainly compose prediction object structure discriminative feature compatibility feature train f hc hc nonnegative determine tracking result without task multi task task case rotation temporal model coherence stable tracking result well tracking feature formulate metric map replace enhance discriminative show example space frame template map another large map task utilize consistently transformation f feature frames feature j j jj frame correspond template norm regularization incorporate framework scheme approach form learn transformation predict maximize added training use frame track online adopt let kt eq convenience let tc w solve th trace convert diagonal element descent update current show supplementary file k eq material gradient k material sample descent supplementary algorithm material calculate model transformation predict maximize collect solve nine book
produce period duration prediction daily monitoring http www analyse daily data year daily record account effect assess water management discuss ni anomaly temperature cover west normalise mean level pressure south calculate anomaly another strong place restrict site obvious discover focused effect study generalise variation recent model daily range summary explanatory variable range year period location daily daily zero small day show stationarity site area section hierarchical spatio model statistically complex relationship observation introduction denote observe process three layer observe censor alternative model hierarchy represent intercept regressor stage spatially correlate th generalise define vary autoregressive model single writing value process follow temporal subsection model spatial vector ts weather ts kt x n spatially correlate x kt nn diagonal corner ts ts covariate ern gamma rate correlation control smoothness isotropic detect one spatially model generalise cholesky corner correlation triangular q generalise skewness introduce identifiability effect overall product routine calculation give spatio tp denote give compute likelihood censor censor observe information time carry regard know eq augmentation embed integration also assign prior uninformative variance prior spatial decay parameter prior set range km assign uninformative generalise similar prior sensitivity use hyper hierarchical model censor censor metropolis full conditional show global covariate result std ar skewness spatial decay intercept brevity vary skewness ci vary effective dependency km analysis significant impact posterior distribution check sensitivity truncate original uniform result show sensitive hyper brevity omit prediction period return return period event past calculate produce period weather previously sample spatial return ci period curve return period site period dash ci directly observe daily quantile observe year quantile graph site include shape important return period model construct extreme period daily datum weather present conclusion demand extreme daily return period might error model spatio daily closely worth note daily another use duration threshold daily box plot show match efficiently short weather dot observe number prediction produce map threshold prediction define study weather spatial exactly value represent show close region develop generalise prediction spatio datum heavy tail substantial vary skewness theory maxima sized threshold fit daily across south west propose accommodate volatility skewness dependencie future plan process coefficient dynamic spatially often address covariate vary whose generalise similar adopt censor modelling zero approach alternative mix compose use censor couple generalised modelling limitation first fail massive spatio non model assume spatial stationary complicated interaction model h mm h mm mm proposition planning produce uncertainty spatio skewness environmental generalise construct duration sized block fit entire spatial spatially vary volatility skewness drive covariate spatial inference temporal modelling process model require duration period model require short term realistic associate variation cause environmental often data resolution cover valuable flexible statistical accurately shape skewness reliable serial dependency objective unified move towards objective generalise process significant model build propose event unify model efficient involve finite threshold large contain contribute spatio model methodology adapt type potential city development planning assessment extreme return measurement denote recurrence period calculation occur past take return extreme assume occur occur find occurrence event period uncertainty duration aggregate threshold measure measure extreme environmental physical process widely coarse current often extreme theory model tail model argument generalize univariate let normalize distribution converge justification maxima size maxima maxima china south individual weather spatial period calculate spatial sophisticated weather daily maximum pareto threshold get parameter control weather extreme isotropic induce walk describe temporal smoothing environmental level distribution shape scale amount discard usually diagnostic show also ideally like spatio temporal law tail well centre meet generalised connection model lebesgue generalised say normal random generalise gaussian conditional scale skewness tail influence contain appeal property moment generalise eq suggest case many laplace
dataset breast cancer division stanford school treatment take pre gene measurement per patient copy measurement six correspond tumor area select expression copy gain perfect employ issue pre variance gene copy measurement circular algorithm smooth variation fused fuse lasso slow consecutive triple copy average copy scale follow q coefficient examine plausible select coefficient smooth mention nature objective often term box focus optimize solution equivalent iterate many package package subgradient respect divide substitute xy b proof penaltie solver iterate total computation try fit follow mean solve solve far build package run independent normal net take second augment version take second speedup come introduce model collaborative response framework decide suit canonical seem increasingly generalization suffer issue real issue performance acknowledgment thank comment regard development scenario observe deal canonical reach optimum supervised problem many study researcher outcome expression copy patient type come may need make future copy future prediction canonical correlation collaborative characterize penalty possibility seem case simulation improve prediction advantage secondary look sparse compare biological penalize collaborative regression covariate partition response response store length collaborative minimize seem basically want base easy calculus satisfy order find closed assume none xy classical way series instead happen substituting get perform onto space also expansion help build actually projection column column projection onto set onto intersection space case eventually thus shrink part picture parameter act shrinkage one nice aspect add penalty penalize collaborative regression penalty penalty know introduce sparsity namely know fuse smooth helpful meaningful copy convex also linear penalty lasso ridge penalty often predictor lasso fuse combine discuss penalty term penalty potentially available contain identify useful costly amount noise alternatively different brain hard information current patient perform basically agree would framework fit fitting look solution reduce ultimately decide problem appeal analyze correlation generate datum follow iid discover issue cca seek third supervise correlation dataset suggestion make optimization trying iteratively maximize rest long solve iterative people take coefficient ridge replace identity matrix solve ridge replace order adjust sure lagrange convex regime add regime sort variable selection useful gene pathway particularly value fairly noisy statistical follow offer desire coefficient penalty add negativity iterative problem hard high dimensional ideally optimum coefficient start believe globally logic sufficiently search exponentially start output criterion generate extent get dataset x z run sphere line correspond start penalize solution especially thing start default start suggest supervision threshold pass cca also use cca supervision three objective cca cca bound form enforce variance constraint cca characterize enforce note
ref eqs relationship q derivation seem algorithm formulation hamiltonian hmc require nonlinear metric lagrangian mechanic ref solve equation avoid hmc equation agree ref splitting differential discretize linearly implicit scheme geometric effect decrease speed region unchanged region must often barrier approximate location saddle direction curvature assume negative neighborhood perhaps barrier ref use switch define help integration elegant practice splitting gaussian dimension gaussians dimension dimension distribute simple problem mode especially sensitive dynamic implement efficiently choose ref integration sample auto time mc hz n practical eliminate system hmc variable aim show feasibility acceptance split ref ref free either impractical stationarity benefit self understand balance recover easy get transfer operator p f sample transfer self adjoint weighted markov process replace get eq similarly name eq hmc satisfy illustrate langevin sampler way impossible generalized conditional verified mcmc balance balance make modify detailed balance langevin formally balance splitting add mean variance equivalent show exploration thank contribution investigation work project definition one generate normalizing often resort prescribe configuration step acceptance metropolis hasting acceptance rate configuration dimensional numerical essence monte present volume possibility derive couple barrier dynamic base prescribe protein unbiased structure task whose multiplicative energy require chain enforce method correlate produce offer possibility correlation integration requirement theoretical justification dynamic reversible preserve phase literature ref ref volume jacobian fully hmc mcmc rely ergodicity volume phase rather requirement special weak expand design move dynamical make hamiltonian ref begin probability marginalization volume hmc one efficient unless article formula special occur ref worth derivation calculus geometry mcmc style highlight mathematic substantial chain clarity enhance mistake avoid avoid multiplication requirement remain already couple also simple potential benefit generalization develop molecular mcmc md set characteristic rotation design tailor basic hmc also introduce generalized hmc three modify defer stationarity balance importance detailed idea configuration unnormalize aim estimate discretized equation produce systematic markov acceptance systematic surprisingly tend efficient challenge ref move stationarity ergodicity chain ergodicity inclusion stochastic follow inclusion example give generate auxiliary choose simplify scheme aim covariance basic identical propose extension give mala latter reference asymmetric example diffusion slow moreover possibility flow marginalization hybrid stem change auxiliary become momentum configuration hamiltonian duration hamiltonian arrange introduce additional hybrid hmc numerical scheme piece system general hmc density density hmc stationarity reversible mapping hmc jacobian second difficult alternative ref alternative hamiltonian hmc modify slight important issue size ref theoretical generalize mcmc phase space partial variable next allow duration momentum choose determined calculate move choose course outcome go trajectory slowly possibility hmc generate extreme random walk discrete ergodic mc given eqs langevin tensor scale result integrate hamiltonian denoting obtain sample far develop tensor diagonal explore molecular ref use mass hybrid monte statistics mass fisher always numerical illustrate bayesian statistic general previously state density mapping dimension practical density may configuration space generalize reversible w preserve mapping cc cc choose require specify may dirac delta represent law enforce
restriction nonzero entry satisfy vector zero identifiability establish linearly recall form large I column row follow nonzero without nonzero row denote nonzero linearly nonzero since column combination column linearly linearly row contradiction generality entry column accordingly thus back appear note write certain canonical case linearly entry linearly dependent imply subspace closely relate completion rank entry span column draw absolutely lebesgue matrix submatrix uniquely entry complete relaxation column hold submatrix submatrix result behind simply canonical easy follow projection certain entry g incoherence method complete low agree aware entry completion scenario show show completion remark observe necessary condition corollary incoherence uniqueness randomly comparable condition necessary identifiability come identifiability draw connection subspace theory insight yet insufficient subspace identification bipartite disjoint vertex nonzero title column row vertex vertex label vertex vertex font edge edge r r c recall vertex vertex theoretic verify vertex title title title row vertex vertex width pt width vertex label vertex vertex pt font edge r edge c interpretation extend context corollary state connectivity insufficient identifiability adjacent one identify projection subset random scheme meet trivial gives verify incoherence sampling remark conjecture draw circle font projection canonical necessary deterministic identifiability new completion arise variety signal situation grow subspace incomplete projection contribution sufficient implication organization state main illustrate implication result present viewpoint column nonzero indicate onto hence without generality
indicator principal incorporate intensity dimensionality well identification principal principal refer result ol projection subject basis two slice slice slice gray matter white cognitive ability maximize feature subject list mean lastly classify cdr great c feature age white gray coefficient pca coefficient coefficient recall correspond perfect cross test subject repeat time report pca result increase suggest single set radial inverse gaussian classifier classifier decrease solely pca component feature recall classifier set reduce indicating achieve voxel severe make reduced feature attractive classify truly cdr patient reduce feature improve technique acknowledge grant rr mh additionally would ng insight classify moderate cdr support machine act reduce subject segmentation feature contain classifier linear training test precision correlation coefficient cdr component describe classify accord clinical cdr train open clinical cdr work separate ad mmse report recent identify structural work utilize subject contain subject raw image raw voxel value feed scan voxel million raw voxel intensity severe bias implement classifier subject cdr score moderate describe thorough diagnosis please radial function svm cognitive via mmse clinical rating cdr subject cdr rating rating none great gender normalize brain process brief post process brain
py box dot box probability dot box probability one step possible probability apply hence py bn compute detailed refer appendix b instance bi th instance depict figure newly py bi use forward approach swap step bag inefficient lead compute py bi b b swap initialize b py b b bi bn bn bi bi bn bn forward swap th instance conceptual illustration forward substitution overview substitution b b p py bi r bi sort cardinality contradiction low triangular structure lower triangular element computing substitution th th h bag three set exclude sort sort construct proposition py bi bi py l bi py bi bi bi p combine substitution algorithm substitution e step bag efficient inference instead bi b initialize bi py bi py bi increase function follow determine use backtrack backtrack step backtrack step transform original multi regression transform space space yield bn regression require rbf instance tune note version make step py bi bi bi bi bi bi b py bi bi instance prediction instance prediction predict instance know inductive mode bag inductive maximize feature absence label ti ti programming bag bag compute prediction union follow compare approach annotation maximum sim svm nn expectation maximization sim tune search set norm divide feature training phase confidence predict confidence access predict ti classes rbf nn letter letter bag bag create instance letter bag create letter uci repository two bag class instance bag letter letter metric annotation instance divide predict predict label sim method logistic instance accuracy upper every dataset without inductive namely hoc parameter bag note free scheme baseline method parameter inductive hoc bag present post hoc instance accuracy table letter sim whose performance comparable high outperform low accuracy low union dataset outperform letter dataset per bag avoid effectively sim max softmax ignore mcmc challenge stop criterion bag ignore useful constraint union equal preserve dynamic classifier work well instance accuracy since accuracy table perform bag measurement instead directly level follow parameter tuning propose understand drop tune case bag metric bag phase divide fold inductive letter sim respectively dataset outperform inductive sim inductive report accuracy union sim test level p p mm l dataset sim level sim l examine ability sim exclude hoc tuning bag accuracy percentage bag runtime training percentage value accuracy depict bag practice label costly training sim sim use amount clear gap less efficiently information achieve small bag sim indicate softmax instance instead come result sim information explanation em another possible maintain union bag bag small number class occur outperform objective section indicate mm sim svm ap union rl ap rl ap letter ap co letter rl ap rl ap rl ap level examine propose bag level letter letter compare sim compare logistic optimize evaluation hamming note assign output classifier bag bag coverage classifier output bag confidence bag independent confidence sim bag maximal confidence instance w ti ti ti ti ti svm nn report measure bag similarity representative bag dataset instance sim instance information consequently approach bag helpful predict include sure overall scene annotation annotation bag level nn seem svm feature instance iteration sim tuning iteration number iteration higher lead runtime performance set increase runtime sim report letter improvement compare annotation letter letter significantly compare classifier instance feature unchanged compare high running feature time long run kernel union k sim level optimal reduce approach consume bag term bag letter bn bag l substitution l bn b bn letter letter pruning remove bag contain large class dataset sort value percentage bag validation every bag due ambiguity set pruning runtime decrease prune use letter letter sort bag bag contain bag ratio b bag constitute letter letter cm letter letter bn letter letter bag function bag letter significant letter letter dataset letter letter time remove letter letter bag letter remove word affect bag constitute runtime maintain bag runtime decrease keep runtime decrease linearly keep high proportion bag number computational per substitution consequently depend bag maximization iteration bag label costly runtime compute bag new compute q since label subset bag size runtime iteration bag change bag select runtime bag letter percentage bag percentage sample bag runtime almost dataset stochastic ascent sample bag bag sample runtime keep letter b b compute call dictionary long take converge one lot result use create em iteration become consequently convergence propose accuracy runtime letter dataset create similarity instance accuracy runtime exponential factor number reality additionally several speed pruning reduce runtime accuracy unchanged pruning technique runtime bag letter letter annotation problem multi ed logistic regression facilitate likelihood focus challenge exact instance label program bag experiment dataset approach outperform art especially dataset approach sim tuning bag get around training bag achieve sim classifier approach extend pruning technique datum mr liu application appear b py lp py bi iy bi rule become bi rewrite rhs follow bn n b n conditional rhs n bn bn bn rhs macro labeling labeling instance bag algorithm infer bag refer challenging ambiguity regard label discriminative expectation inference probabilistic bag label instance evaluate world song activity recognition outperform sometimes state bag label annotation expectation graphical programming multiple instance carry uncertainty conventional training bag bag segment name object house contain list providing I bag classifier level classifier refer reader construct without explicitly reason citation knn knn training bag cluster hausdorff encode bag hausdorff distance apply citation knn knn focus paper label annotation exist resort bag score bag maximize bag bag small directly annotation example sim max score bag graphical annotation application inference employ maximization discriminative achieve lower first instance annotation propose computationally calculation finally superiority various domain song image annotation bag level multi instance address process dirichlet allocation lda know processing corpus graphical lda proportion topic randomly select topic hide bag bag label consequently instance express propose learn mapping preserve maximum formula dp model bag
instance compressive proposition purpose equivalence notion algorithmic equivalence namely equivalence algorithm machine illustrate analyze regression fundamental property object field several theoretical concept accuracy algorithm concern raise define characteristic could train smoothness equivalent formulate algorithm adequate algorithm exchangeable even indeed restrict optimization ignore set never rigorously equivalence learning notion particular sufficient generalizing weak related equivalence hold equivalence set advance sense equivalent study regularize algorithm hilbert rkhs particular regularize rkhs regularization variable efficient call concept relation ridge contribution formalize concept equivalence notion algorithmic sufficient allow transfer stability equivalence transfer equivalence solution weakly section banach reproduce space rkh notation take learn output discuss equivalence problem optimization function limit solution small constraint function unique optimization equivalence focus occur optimization associate decide equivalence support equivalent stability view imply underlie share learn relate question rigorous concept equivalence notion equivalence optimization definition convex onto q convex unique equivalence naturally equivalence set algorithm associate optimization solution solution guarantee varie mean pay algorithm depend crucial controlling widely regularize learn indeed value regularization define different index weak extension definition equivalence basic section equivalence equivalent assertion become strong say equivalent weak frequently encounter machine naturally occur lagrangian duality method immediate supplementary natural whether know weak question study consequence algorithmic notion equivalence allow algorithm present weak weakly regularize proposition follow proposition mean however assumption weak equivalence indeed vary make increasingly consistency whether weakly follow get algorithm regularize equivalence sufficient transfer stability regularization widely uniformly nz property decrease decrease stability important unlike equivalence case lemma stable uniformly stable equivalence weak equivalence sufficient ensure transfer transfer express assumption knowledge easily express stability hamming usual hamming z n z z nz ig nz z one move permutation proposition see generalized notion regularity function I lipschitz stability satisfy assumption admissible locally proposition extend depth case equivalence algorithm hilbert space rkhs k kx combine square regularization term solve exponent classical ridge regression recover propose practical solve pose novel generalize ridge worth become ridge analytic spirit following cm basis ii transformation reconstruct weight orthonormal let material fast accurate exponent show weakly problem strictly lagrangian weak weakly unique unconstrained equivalent eq problem weakly easy map f depend weakly satisfied stability supplementary subsection conduct real world algorithm compressive strength instance world repository additionally attribute dataset input uniformly compute x part equivalent positive showing regard algorithmic property way namely notion stability transfer concept equivalence algorithmic robustness aim far quantify efficient two learn weakly strongly equivalent algorithm consequence say curve ensure minimum easy sequence likewise generality third fourth use try exponent reader theorem go repository root eq notice q also due combine determine formula derivative since semidefinite depend definition need write
approach curve flexibility produce procedure assumption future non model mm receiver operate measure indirect roc population lead flexible account use close show frequentist roc algorithm mixture model operate roc gain popularity detection world phenomenon justify necessity performance diagnostic tool diagnostic roc present practical roc desirable reason theoretical many author way curve issue model roc two category indirect appeal construct population divide mention curve certain monotonic increase overcome obstacle population use smoothing reduce distortion smoothing roc curve assume population derive construct parametric semi parametric population obvious adequate cancer distributional verify assumption roc assume population variance distribution determine advantage curve attractive presence population solution technique generalize smooth roc curve algorithm category ordinal suggest diagnostic patient relationship directly model diagnostic population status disease probability use like lack issue semi confidence band population motivation give roc flexibility natural mixture population enable shape gm replica curve bands roc curve mixture flexible carlo apply absence form band construct curve idea method replica simulate accordingly q major estimation consist prior bootstrap mixture modelling score respectively suppose follow carry via maximization parameter generate repeat monte simulation roc curve similarly compute estimate empirical averaging indicate therefore band curve law theorem several simulation flexibility fit e choose evaluate auc visualize method discrimination examine strong moderate poor practically patient different stage disease replicate curve discrimination commonly curve furthermore case closely compare figure band curve average without monotonic auc curve htp htp htp htp htp two calculate auc relatively simulation
method svm also space baseline exponential amplitude smoothness hyper noise svm use hyper smoothness turn validation search svm validation guide competitive metric independent website text task rest sample testing text information use explore different description frequency tf extract descriptor codebook text representation architecture sentence construct codebook cluster elaborate sentence planning explore future dimensionality domain keep deviation outperform however clearly network due svm already couple difficulty hyper mind svm hyper separate tb domain highlight equal mean method small task collect search category description colour attribute class retrieve category provide contain gain year due capability extract predictive interested power visual domain convolutional fashion feature provide pca dimensionality keep principal attribute one small average rank support rank also statistically four still enough evidence attribute dimensional base classifier principal domain term baseline appendix dataset advantage deep feature semantic performance illustrate sigmoid differently example gain importance one importance test four evidence reject comparable slope way accuracy term advance network representation plan extend multiclass speed quadrature free propagation repeat image domain neural highlight r v cat cat v cat cat cat cat cat v cat cat v repeat binary task highlight cat cat v cat cat cat cat cat cat cat c technology attract additional treat nuisance training slope sigmoid knowledge svm advance provably many integrate develop certain prior parameter training rely adopt see prediction attract considerable within reflect increasingly situation expert train request customer order expert use information typical service inspection classifier processing play role build mobile device operate constraint generate increasingly data care diagnosis available drug trial subject obtain impractical noise final require aspect influence become interpretable training directly read slope shape example fast sigmoid try fit slowly slope put interpretable higher relate originally think turn need classifier slack lead improve several categorization setup subsequently improve characterization constitutes commonly use predictive long analytically deal markov carlo posteriori bayes gps reason section propagation classification self review particular emphasis latent latent elegant aspect intuition make test input noise latent classification specifie property function evaluate crucially input contrast approach term assume input task flexible numerous upon knowledge nuisance view flexible dependent suppose say reflect contrast say sec posterior solid easy difficult view context however classification actually sensible investigate equivalent clear transition high slope slow uncertainty likelihood sample uncertainty n hyper amplitude smoothness parameter unseen label associate newly datum decision large interesting bayesian predictive marginal analytically tractable adapt quadrature please
conclusion birth ol ols ols pl black relationship birth weight nonparametric specify fit age record partition curve age available cell component estimate fit remove panel response birth quite particularly birth birth quite dramatically gap gram birth older optimal choose generalized sided response bootstrap response among pl covariate pl estimate pl exhibit birth gram difference statistic black marginally effect concept consistency though nuisance estimate configuration reasonably nuisance result fairly stable profile ratio method note greatly simplify predictor partially insight tradeoff estimation estimating nonparametric nonparametric little component burden loss efficiency negligible burden bias generalize broadly utilize theoretical derivative cm discrete covariate category lie simplicity dimension mainly one correlate corollary imply follow proof corollary proof corollary least form need first analyze numerator denominator conditionally sketch analyze eq use ng positive lie total large small clear argument lemma combine complete conditionally j together give complete give thus equal combine analyzing show eq q ix j ix j thus mm pt remark utilize theoretical statistical modeling estimate partially consistency phenomenon model via balance burden model bias approach square little behavior test satisfactory study birth weight analyze partially categorical normality estimate dimension parametric regression term surface practice effect requirement determine functional misspecification estimate contrast form influence bandwidth local kernel essential versa suffer curse exponentially design categorical appealing alternative specification covariate part linear nonparametric assumption estimate affect gain great popularity economic science lot devote model partially introduce estimator assumption extend specification first investigate nonparametric comprehensive partially expectation subtract way expectation estimate dimension obtain root nonparametric curse dimensionality call theory correspond true every estimate desire nonparametric efficiently convergent bandwidth procedure unstable correlate partially problematic fan fan lastly structure example interaction biased model complicated idea inspire property average square classic property moderate estimator almost component easy incorporate covariate explore regarding paper follow follow review consistency regression parameter consist univariate categorical analyze offer method far technical refer statistical nuisance nuisance phenomenon terminology nuisance consistency phenomenon mix longitudinal etc discussion theoretical microarray q assume grow nonparametric error sample fan show estimate consistently moderately pay nonparametric function variable assumption sort realize interval interval realize density narrow sub ji reformulate term ignore almost efficiently express profile profile similar least degree freedom use formula plug update estimator readily extend several express categorical categorical subset categorical subset partition sub model profile square use regularity condition e least define eq bivariate extreme model consistency outline component highly eq fan fan smoothness approximately denote model regularity corollary result fan replace unconditional corollary defer appendix extent resemble kernel view corollary limit classic partially naive cost efficiency factor increase remark average resemble testing quadratic suggest partially eq nonparametric greater critical reject critical hypothesis testing test slowly hypothesis bootstrapping suggest replacement bootstrap use statistic repeat replicate frequency replication bootstrap quantile confidence band examine effectiveness propose model mix categorical compute test hypothesis power examine alternative method simulation package fit nonparametric curve fit generalize validation bandwidth I example sample produce package first illustrate relationship logarithm slope size moderately propose increase decrease result study suffer capture theoretical follow examine alternative nan size simulate figure empirical theorem bootstrap section sample behavior power value leave value similar confirm least statistic linear consistency h cc estimate cc generalize additive mean z u u go simulation example property test indicate outperform package sensitive sample size nan nan distribution also nonparametric increase also reduce power eq q equal independently bernoulli independent continuous sample purpose bivariate pseudo nonparametric nonparametric see additive nonparametric estimation np tries estimate bivariate simultaneously parameter study np hand nonparametric interesting outperform nonparametric wrong panel curve relative parsimonious nonparametric extent regard table sd propose cn ci ci ci sd sd sd sd sd sd equivalence two component z test estimate use formula x r package bandwidth smoothed estimate plug formula associate bootstrap
user accurately predict score less predict suggest predict missing use recall per user average user metric give value recommender system dataset website user review user social utilize dataset rating friend rating undirected rating item review specify trust utilize dataset convert direct trust link link movie rating result density user connection identity item user gp I trace pmf pmf gp pmf pmf show outperform pmf five fold find select vice versa match researcher perform rmse gp factorization baseline term highlight shown suggest rank dataset table mirror dataset model approach similar trend highlight similarly slight improvement constrain variant recommender dataset nuclear gp introduce predictive variate nuclear recover gaussian norm constraint association task recommender perform characterize necessary condition norm constraint explore interested gaussian nonparametric interested nonparametric constrain inference biological implication disease expert acknowledgement grant thank u manuscript constraint duality subject general recently constrain term banach duality denote define match theorem conjugate q pz z dual ensure normalization dual optimization often alternative separate optimize unlike approach density satisfy empty subject constraint constrain write minimizer achievable associate density consequence state pz ensure denote feasible density give pz key fully specify parametric family give expectation constrain follow specify corollary convert optimization hilbert prior similarly gp bilinear admit spectral decomposition value spectrum common induce define functional regularization theorem optimize evaluate e covariance evaluate modeling approach constrain optimize include noise function variance q solution similarly column parametric represent optimize gradient respect gradient simplify collect similar hyperparameter computationally challenge optimize update require matrix proposition represent interaction entity information vector column available interaction entity axis modeling model noisy generate construction flexible side information row association disease set observe association covariance network recommender involve prior experimental highlight performance constrain variate process approach domain organize entity row entity sparse primary represent encounter addition matrix include entity entity graph describe column graph matrix observe low modeling recent variate compactly correlation row extend replace g variate new row gp describe scalar scalar collaborative learning despite gp structure rank prediction low product reduce improve low rank assumption computational concern size na I scale linearly factor method approach empirical inference enforce useful inference alone constraint margin enforce constrain relative capture rank characteristic combine bayesian nuclear distribution solve constrain problem gaussian finite iii art disease association disease side recommender network side begin discuss background variate nuclear constraint variate low art specific disease recommender domain statement building process section case bold upper identity pp p density let index index ny observe unobserved include bound node auto ex z free variate doubly index multivariate distribute notation decompose row covariance scalar value process nz nm distribute extend arrange covariance matrix gp combine noise see draw z n z interpret latent task posterior z datum index covariance close follow gp scalar appropriately variable scale storage require I probabilistic involve give often achieve via however may one impose intractable careful alone approach via relate sample probabilistic thus standard optimization enforce density enforcing pose give vector let set constrain bayesian distinguish unconstrained discussion inference provide constrain bayesian constrain relative minimization maximization study language discrimination inspire combine document apply prediction result intractable require variational make appear without require simplify oppose principal variant extract bayesian model linear covariance non factorization rank must prior correlation design variate far nuclear automatic implicit ex ex r edge r rank matrix process low factor detail baseline rank model nn computed optimization problem trace statistically interpret one probabilistic gaussian posterior quite practitioner approximation utilize priori recently intractable slow inaccurate large dataset approach focus prediction exploit model kernel base apply probabilistic relational alternative focus learn additive nonparametric instead represent term amenable fix nuclear optimization q ex ex represent form solver represent avoid storage full estimate reader refer paper detail increase factor far descent singular singular sparse power factorization nuclear norm differ number factor requirement nuclear complete dataset association domain dataset study consist column diffusion prior covariance validation selection result constrain trace hilbert trace contribution baseline probabilistic factorization factorization pmf pmf pmf covariance covariance implement representation outline cholesky decomposition representation hyperparameter spaced noise spaced variance estimate overfitte explore distribution g regression away gp column matrix multiply computationally observe implement allow expense norm regularize optimization implement limited bfgs design validation discussion refer either disease disease prediction recommender experiment design evaluate randomly partition fold set hold experiment experiment wise e test segment dna gene identify human interact disease include disease genetic standard expensive method predict list disease significant interest reliable gene share know feedback include like train similar differ hinge respectively svm regularization also pmf baseline implicit feedback dataset recommendation sampling follow randomly disease item set combine metric association laboratory consume costly rank metric remove gene compute label sort score test remove gene precision compute relevant retrieve retrieve fraction retrieve retrieve length metric set separately reflect dataset evaluate disease association database disease disease matrix extreme problem large set predict e even hundred database disease disease branch mesh extract disease database positive association result gene association density interaction disease disease genetic genome include protein interaction specie gene link
number coefficient iteration amplitude spike curve meet reveal reconstruction except return seem constant considerably laplace handle different level regard fix spike fig fast due origin heavily summary simulation competitive fast provide sense review literature generalize beta mixture signal bayesian prior maximization cs simulation outperform art solution beta prior sense hierarchical amenable expectation maximization em algebraic update yield experimentally validate recovery art method level sparsity observe sense reconstruct consider acquisition noise signal pose polynomial replace recovery method use algorithms support sequentially approach compressive know compressive sense proper utilize reconstruct modeling prior concentrate tail work area lead introduction prior utilize large require suffer computational low another induce widely exploit hierarchical facilitate student assign layer prior later work laplace student concentrate near tail coefficient model amplitude paper suggest prior capability amplitude level sparsity choice free origin heavy tail sort nearly hierarchical inherent advantage impose reconstruction convergence summary main propose compressive perform art superiority wide compressive analyze induce section detail propose framework signal distribution laplace student much model large shrink like toward statistical mean white normal joint assign formulation laplace tractable address problem joint pdf determine induce pdfs p suitable example stage student hyper precision represent method regard decompose q impossible adopt q conditional calculate p eq compute maximizer likelihood obtain suffer drawback inversion require suboptimal suggest model add fast calculate single hyper iteration regard hyper rewrite removed depend isolated looking maximize likelihood respect eq equation maximizer plus induce limit infinity origin maximizer basis local either root happen case distinguish discriminant cubic root discriminant real provide negative root negative root positive root scenario case point side axis root include root sum conclude two base subsection summarize reach performance degree non amplitude select exclude procedure great update sake quite algorithm remark mention hyper substitute analyze art
regret idea reference generalise notion name justify key regret condition generalise algorithm mirror crucially potential expert game expert reveal prediction expert player aim player keep close expert round loss player independent round denote make assigns assign predict expert make prediction reveal expert tp aa round expert round aggregate mixture set mixture eq expert prediction aa condition find result concern regret achievable round aggregate converse loss work proper loss main simplex loss two follow key result divergence formally sequence observation expert play sequence notion shannon entropy corollary uniform interpretation initial guess expert price exactly far mass expert aggregate recover usual aggregate generalised aggregate algorithm update performance understand prediction range study aggregation loss variant aggregate loss computational weakly aggregate regret observation aggregate also discussion relate aggregation vector observation loss loss loss product discuss offline make entropy call notion coherent mathematical throughout v conjugate readily fact entropy shannon countable concave proper thus original conjugate motivated entropy entropy call scoring rule say predict way construct express entropy proper associated shannon log reduce f see substitution inequality become definition divergence update condition eq expand result generalise aggregate strategy repeatedly next show process simply dual update constant arbitrary ignore relation sum aggregate substitute maximal substituting theorem update x series note something even strong state condition hence exist similarity generalise aggregate market difference bundle instantaneous price correspond framework bundle formulation say observe must market surprise outline main essence term divergence original achieve regret terminology bound bound seem bayes shannon entropy stand loss regret curvature discussion address large large correspond bind choice reference matter constant directly counter conjecture aggregate provide via convex kullback leibler divergence naturally replace loss bound generalise aggregate mirror aggregation bayesian view aggregate play prediction evaluate
conceptual operation capability mapping main tuple light gray tuple consist tuple return return tuple tuple look pattern observe location tuple way arguably convenient computationally eq denote white empty encode tuple tuple return illustration tuple consecutive shaped sequence formal requirement tuple network tuple issue architecture location thus surprising individual tuple eight ignore result tuple use network play tuple example alternatively construct tuple place representative tuple thank tuple tuple table computer world tuple external tuple input forest set evaluation interpret evaluation complementary color play purpose employ two play learn play black white vice alternative player output select position maximal whereas select minimal learn black lead position piece play black select move piece white piece player use output architecture rand rand rand rand tuple architecture tuple architecture computational n tuple evolution strategy weight individual evolution mutation double game play run compare tuple network game play despite et capable per thank finish run ghz repeat evolutionary measure last statistical follow significance correction compare present rand inversion regardless player architecture confirm statistically also detail moreover visual inspection plot reveal experiment rgb rectangle rgb rgb rgb circle rgb circle circle rectangle rgb rectangle rgb rand rand inversion performance score obtain heuristic shape white black line represent range black dot outer main possible tuple shape rand rand rand tuple make impossible pt rgb rectangle rectangle rgb rgb rgb cycle cycle rgb rgb rgb rgb rgb circle rgb cycle rgb cycle rgb circle rectangle rgb rectangle rgb rectangle rectangle rand x rand rgb straight n shape tuple rand performances fig explanation three pair reveal rand rand rand notice difference substantial vs lowest good see architecture robust variance rand cf rand attribute initialization process intuitively tuple short two time weight well difference architecture plot performance run eventually seem learn gap small among also six analyze tuple architecture give player obtain tuple result high player line player employ allow format date player name tuple tuple tuple n tuple tuple tuple tie ff position evaluation since use inversion performance good player come website performance tuple player game output able evolve name tuple consist straight tuple first care double game standard move situation game obtain good general variety evaluate utility player computational evolve tuple good player compare good method weight et find weight rand rand latter obtain result state mutation efficient elaborate progress claim suggest factor dimensionality considerably rand rand weights nonetheless three obtain high factor architecture finally tuple operator mutation although flexible add make hard analyze network systematically tuple shape tuple originally consist straight tuple weight usually weight advantage slow surprising since capture opponent piece three black obtain computational study nevertheless whether evolution interesting question systematic network advantageous reinforcement difference learn network support centre computation helpful remark early rand rand x rand x performance evolutionary run double heuristic player format contain tuple expansion weight use put pl network successfully game connect effectiveness network architecture tuple location provide n tuple sequence effective systematically place straight tuple obtain curve yield date evolution strategy tuple value function attract ai due mathematical early shannon lot conduct connect bottom game constitute valuable computational intelligence playing monte carlo technique employ quantify state look effectiveness tuple tuple long generate tuple exist evaluate way
multiplication costly subspace optimization use newton burden multiply span direction multiplication store previous enable efficiency substitute direction coordinate often burden change particular coordinate column still dense matrix need operation one pass multiplication multiplication wavelet transform coordinate descent without move along perform analytically multiplication via objective along another iterate elsewhere quadratic obtain always step multiplication comparable ill far accelerate every affine current direction several propagation progress step significantly nesterov figure adopt matrix fit ill beyond aim additional iteration explore present ill pose type converge fast go newton boost multidimensional looking approximate newton expansion accuracy inaccurate problem one k disadvantage line ill behave model fit far away direction therefore around iterate adjust dynamically actual fold find minimum time find unconstrained therefore ideal trust ellipsoid euclidean adjust ellipsoid proposal use use computing invert therefore quadratic taylor expansion iteration tn accomplish line guarantee decrease effectiveness tn internal attempt replace cg stay match tn step pose drive involve available ever become increasingly inefficient incremental contrast obtain mini large redundant sophisticated recently partially adapt unconstrained method mini batch mode mini bfgs still room fast plain trust region newton invertible newton fast hessian problem accelerate lagrangian constrain non trust constrain minimum time expensive ideal kind euclidean ball adjust trust region point expansion correspond go infinity pure step newton motivated propose current line method include step subspace take direction effective conjugate method newton quite burden newton newton square popular g moderately feed neural mention tn method internal cg go resolve replace step optimization way cg break outer stay tn tn approximately minimize quadratic eq gradient tn truncate cg come original cg tn subspace pass inner cg instead minimize extend monotone descent tn monotone reduce several step gradient order tn optimization start tn iteration stage cg cg tn approximately cg last cg iterate affine span tn last quadratic cg tn tn subspace tn sensitivity process objective trajectory cg function independently stop tn tn converge inspire recent mini quasi newton bfgs plan step previous mini batch gradient restrict hope mini responsible resolution accelerate computation first solution coarse grid formulation fine grid much iterate fine guess approach relevant linear e level gradient converge fast problem happen grid step cg fast iteration optimization previous gradient coordinate go descent provide coarse grid fine good cg acceleration acceleration lagrangian constrain look saddle dual current augment primal separable direction multiplier applicable include share regard nesterov possess optimal bad express formula lipschitz incorporate smooth nesterov composite objective direction gradient term objective spirit use direction outperform fista hope develop bad fista conjugate well nesterov however could extended function substitute method constraint indicator simple feasible follow composite pure tn tn tn cc tn tn tn depend cg tn tn converge cg truncate early
minimization one unique asymptotic depend second form kernel denote whereas empty set reason estimation pilot threshold considerably great density calculate intersect solve modify paper reconstruct level histogram cell dyadic nonnegative define denote lebesgue bandwidth empirical minimax course plug estimation h generate density model correspond density present asymmetric separate jump peak addition threshold although show therefore bandwidth selector facilitate follow horizontal small vice allow detect competitive density mean order error colour accord algorithm colour compare bandwidth behaviour sophisticated density specific competitive simple unimodal simple behave however bc previous chernoff mode mode ms optimum large vertical axis nx hybrid centre complement proposal convex hull hull method convexity restriction convexity hull classic smoothing assume complement context case union radius obtain represent axis unimodal case convexity hull competitive hypothesis restrictive case convex h smoothing density component seven quite promising model competitive unimodal conclusion extract one exhibit competitive quite present disadvantage depend hull competitive present bad hull competitive behaviour provide density none unimodal horizontal axis axis horizontal science innovation statistical system science estimate plug method nonparametric estimator thus view recently specific priori geometry excess selection avoid excess mass geometric restriction like plug pilot open exist review two behaviour extensive large respect different set domain concentrate great effective play crucial role scientific receive considerable component appear related mode cluster survey estimate level involve statistical two density include analysis outlier review outli belong effective scheme ba ba broad estimation motivate plug method analyze behaviour drive reconstruct set compare multidimensional counterpart excess method plug section mass section kernel gaussian bandwidth thus propose integration solve plug inefficient calculate empirical consistency methodology receive considerable literature ba many select next review estimation ba ideas population tolerance let assume work operation draw
follow chapter hand simply tb asymptotic dotted case involve continue hold check go theorem section standard non coverage require however validity unit matrix th row vector object modify replace significance significance concentrate paper determine explicit orthogonal first definition show pp mode concentrate prediction split series step look difficult general set point fortunately exclude absence euclidean intuitively hyperplane outside express expect typical data uci datum hold projection matrix onto onto even angle hyperplane element coordinate label exclude surely check let give simplify prediction cf except turn least corollary last contradiction complete opposite least many r n ni contradiction last component ridge lemma hand replace side replace allowed degenerate lebesgue suffice side replace prove assumption ensure condition satisfy q simplest give exact normality formula calculate last equality w x finally imply normality assumption early extremely perhaps extremely completely ignore reason type front condition indeed general intuitive residual iid vc disadvantage ignore eigenvalue true regression little interval illustrate informative different produce exceed significance level theoretical quickly small research extend ridge acknowledgement grateful kolmogorov project terminology support grant ep institute institute technology science uk proposition mm coverage iid considerably restrictive useful case little whereas asymptotically interval discuss primal early name prefer name also figure correct theoretically purpose assumption independently obviously original happen resemble nonparametric classical test turn efficient satisfied develop almost efficient violate different rather figure serious ridge interval apply regression result somewhat counterpart figure little gaussian assumption change difference point predictor empirical discuss notion validity two section ridge interested g theoretically theorem violate recent theoretical validity predictor ridge predictor conditional algorithm likely interest advantage regularity conditionally valid chapter noise case bayesian e perhaps apply objective toy simple tolerance stand form tolerance interval define show lebesgue difference interval however deviation z representation expression later agree involve g david normal david two ignore contain interval asymptotically train valid object validity attain predictor summary interested attribute particular random parameterized unit random element intercept recover add attribute object ridge object th prediction quantile enjoy equal significance object finally possible conditionally produce special g chapter reproduce length ridge regression residual row neither residual make score confidence give test object q score closure notation later guarantee instead observation define unconditional guarantee ensure prevent rely iid object validity generally guarantee conservative validity lower bind predictor validity difference
people forget else surface classified become error memory combine successful machine component read memory component introduce implementation answer index potentially component incoming internal old new input generalize intend produce example action word sentence audio internal feature test distinction test time potentially idea etc component make pre processing parse resolution text could encode internal dense simple store select slot part variant store store slot huge need store entity scale operate operate retrieve subset operate right variant implement choose replace memory experimentally yet component responsible reading memory perform e calculate example answering find relevant answer condition conditioning rnn poorly network component neural neural describe relatively implementation basic module base sequence text store available original return empty module memory old subsequent core lie module module support give support score support candidate input support bag modeling input separate module need response simple return previously sentence true rnn word see match task figure module retrieve fact leave would relevant give office place drop finally would score score dimension role map bag support model differently weight desire support label training give input know perform loss sgd specifically response support sgd employ use feed give simple memory subsection basic office office office sentence arrive stream rnn modify segmentation learn take far look proceed embed dictionary margin sequence first concept something training mechanism store equation hash hash hash hashing word embedding dictionary sentence word I consider share mean word fall word well word match memory speed take account memory slot answering fact capital france answer figure implement extra memory assume memory slot absolute relative success candidate triple older old win step compare winner human read lot ideally neighbor word assume similar idea incorporate separate word store bag co occur feature context learn new kind dropout embed instead embed efficiently word pair add learn another way stay extend match matching occurs actually build conditionally match matching matching context classical method document memory retrieval answer reference create graph fact knowledge kb question logical query approach recently base memory differ extraction principle kb follow extraction memory store choice embed potentially build kb away neural memory reference type model memory location store network module learn successively reasoning read salient fact operation design differentiable work incorporate change use allow network address relevant related read experiment memory whereas language reasoning task focus sort network know whereas toy relate method translation alignment representation predict overcome poor long perform recognition dynamically determine character one back single character retrieval look document look schema pdf box evident see consider highly time component recursively look path read create kb www stack fact create event I think highly relevant consist statement store subject relation triple triple corpus combine pseudo triple book perform framework return answer test annotate wrong human whereas ignore label support end also try bag feature unseen word modeling memory fact cluster embedding try hash string hash hand lot share match embed l candidate speedup hash k hash character object character around pick dropping object text label question task overall answer pick k identical testing rnns short memory rnns entity ask try actor drop ask answer true answer g I appendix object support statement sentence find drop support statement usually ask actor actor actor question question difficulty actor actor actor actor actor lstm pick drop room believe take office back office office rnn language modeling optimize fix margin epoch training answer actor task rnn without question bad question bad difficulty demonstrate encode memory high level distance rnns expect sophisticated limitation mistake memory wrong pick question otherwise pick person hard indicate successfully inference whereas without rnn fail multi whereby rnn demonstrate test ability previously word word train simulate except despite simple pattern base drop generalize mean stage unseen completely fail drop take drop grey pick office come find become go simulate naive train scale simulate simply output high score allow simple capable general question example answer future memory future develop evaluate multi inference try complex bridge require causal sophisticated architecture explore deal task sophisticated management sophisticated sentence important dataset supervision answer fact rich detail specific variant network apply vision acknowledgment discussion simulation behave game within simulation allow ground comment want understanding kind world secondly release evaluate currently within answer task location task simulation please pick please learner object object actor drop examine object object drop object actor something place drop something underlie act simple random valid restrict drop actor actor text e go office drop example correspond figure ask office easy question convert look look variety automate get replace take drop discard put replacement currently ambiguity article add lexical variation join compound join later language model model compound noun seem easy hope add complexity control hard substitute generate answer fact input
correction datum categorical subset test call hypothesis remove report false frequent insight discover combination factor question significant subgraph mining trivial mining vertex paper subgraph multiple frequent propose detect subgraph order magnitude far I consider dependence mining concept statement subgraph correction discuss algorithm summarize subgraph vertex restrict summarize comprise transfer subgraph versus graph set xx h p k effective subgraph give collection graph subgraph significant membership datum two membership occurrence absence subgraph frequency contingency cccc occurrence occurrence strength association quantify present hypothesis true rely margin count observe one tailed type error significant setup database many hypothesis subgraph wise error one multiple problem deal issue significance level common test subgraph despite know detect truly subgraph huge subgraph test correct level small subgraph ever reach one correction achievable subgraph x tail test decrease require minimum significance threshold never membership graph occur insight subgraph increase exclude candidate subgraph subgraph database natural subgraph decrease subgraph task subgraph al use enumeration apply mining next apply challenge subgraph use subgraph subgraph frequent subgraph subgraph ratio monotonically subgraph subgraph coincide frequent subgraph follow root subgraph root subgraph long frequency detect frequent subgraph search significance threshold find frequency root frequency expensive subgraph finish subgraph easily sort subgraph frequency level significant subgraph search search possible reach et al frequency repeatedly decrease one pass mining high frequency significance significant subgraph pt incremental decrease newly propose increase termination know admissible frequency large able terminate soon subgraph whole possible repeatedly search expect work number admissible subgraph quickly frequency finish full early termination publish search arbitrary modular fashion subgraph monitoring terminate use obtain without correction thereby search root root set set terminate termination admissible incremental efficiency current examine significant terminate subgraph combinatorial subgraph subgraph far computational controlling fdr recently become lead examine force bf na subgraph occur assess result subgraph bf comparison find subgraph efficiency bf strategy art employ fast always test use permutation ghz gb avg max min avg ptc mr eight dataset protein chemical summarize table benchmark previous undirecte dataset except ptc challenge chemical include originally design prediction graph label ce se divide subset mr fr fm use mr since property classify inactive protein database six top ec ec classify ec ec ec ec protein create classify non relatively compare national cancer contain chemical classified anti cancer balanced set retrieve effectiveness correction subgraph improvement detect subgraph permutation label varied subgraph bind factor number subgraph miss mark huge computation correction much large circle triangle stable subgraph factor increase exponentially might tend moreover confirm highly correlate factor dataset subgraph subgraph subgraph subgraph size maximum significant maximum subgraph reason significant subgraph detect subgraph furthermore long subgraph become correction confirm correction large test get close cross mark blue effective green triangle bf time summarize root mean deviation fast also subgraph permutation figure clearly show bf average subgraph candidate contribute find subgraph whole search order pass search bf average slow speed one often bf around repeat reach incremental search quickly incremental search reason frequency repeat subgraph effective mention subgraph case mean feasible within size subgraph computation confirm subgraph graph stem fact subgraph correct significance multiple control subgraph runtime statistical power subgraph exactly efficiently solve subgraph dramatically reduce statistical far reduce correction subgraph several frequent subgraph retrieve subgraph experimental find subgraphs art result biology believe foundation follow important integrate exploit dependence test string subgraph sometimes extremely fast dataset maximum subgraph fast bf pass incremental acknowledgment aid scientific start grant mining von grant ar department engineering science university di department engineering problem find transaction test pruning hypothesis significant statistical truly real exclude power
overall sample expansion sum total quadratic kind utilize correctness x correctness prove large mathematical correctness obvious get convergence currently system break fail instability achieve performance machine rate reduce calculation many massive traditional demand framework widely use machine apply iterative framework appear iteration name library improve storage improvement develop base percentage always communication solution calculate failure occur present able performance efficiency master node percentage master rather dramatically high tolerance influence asynchronous relationship statistic section speed mathematic balance efficiency performance conjugate bfgs exist apply example master example master first quadratic convergence quadratic
advance compress sensing achieve line algorithm reduce tool differ base dimensionality explore use representation parameter field instead discretized express coordinate continuous counterpart representation proceed equip prior inference particularly dominate propose quantify mle difficulty stem likelihood denominator bayes previously analytically primarily nonlinear implicit follow exponent quadratic note respect obtain taylor nd address adjoint base readily include section work order reduction employ jensen inequality construct likelihood readily lower bind express kullback kl divergence divergence maximize low likelihood aforementione imply inference functional form minimize kl divergence equivalently field approximation look variational statistical expression section linearization approximation ultimately posterior latent enable identify approximate detail section pointing achieve semi inverse approximate p unimodal nearby construct account employ specification span rescaling resolve require r imply equation equation covariance diag induce natural ordering permutation entry column order variance attain direction second hyperparameter adaptive addition coordinate prior induce smoothness spatial variability prior jump value neighbor adjust site belong correspond adjacent voxel jump pair strength induce weak penalty vice versa boolean matrix produce jump one diagonal hyperparameter jump combine natural hyperparameter product absence priori correspond limit jeffreys prior part overall discuss parameter absence low log alternate optimize keep optimization zero purpose aforementioned linearization previous readily deduce mean adopt aforementioned covariance orthogonal eigenvector orthogonal principal diag reduce despite uncorrelated along implicit aforementioned derivation unimodal subsequently relaxed employ gaussian enable modal posterior approximation combine component would possibility far along examine unimodal reasonable imaging discussion expression bind imply argument aforementione simplify make expectation q augment orthogonality constraint address iterative algorithm prove cost setting employ brief skew base aforementioned preserve orthogonality use detail inverting call computation update iteration far depend analytical employ expectation scheme appendix completeness location probability minimal neighboring follow determination derivative order derivative difficulty set purpose update current guess equivalently assume guess denote value increment approximation term keep depend concave quadratic setting exact improvement possible term exploration summarize basic variational converge equation update update demanding call evaluate derivative increase equation attain presentation fix coordinate question address accord coordinate convergence reduce batch achieve coordinate use termination prior belief employ distribution dimensional measure coordinate p explain metric add coordinate consecutive linear elastic material nonlinear employ scheme whereby one reduce capture subsequent basis precision follow accord absence purpose elastic material stress tensor modulus ratio consist modulus value breast spatial unit employ boundary employ encounter static pressure apply depict horizontal assume solution yield configuration contaminate adopt top row total depict corresponding row one infer practically identical say posterior employ figure quantile deviation c standard indistinguishable imply coordinate basis comparison infer depict depict information call determine function reduce coordinate basis notice drop small reduce coordinate early practically indistinguishable full forward call forward call reduce account evolution objective forward discuss monotone call update reduce small coordinate depict capture correctly contaminate nonlinear describe several employ characterize density modulus cauchy whereas term material modulus play integration contribution modulus strong constraint model remain see circular material inclusion small circular material domain discretize solve condition bottom vertical load vertical point employ mesh corresponding employ also contain discretization include result mesh high snr medium contaminate gaussian contaminate snr material depict coordinate estimate obtain depict posterior firstly quantile ground credible large snr operating dramatically relation h snr medium snr snr medium converge figure depict snr behavior variance example exhibit quick small number call function effort begin call measure call number forward call depict evolution three snr h snr figure vector decrease similarity basis dataset clear variability associate variance large small level high capability problem propose much fully validation information theoretic approximation limit solver consider call forward solver operate hierarchy refined call fine also fully enable medical diagnosis currently posterior posterior arise frequently noisy datum represent challenge aforementioned line offer appeal possibility subspace associate approximation combine accurate maximization scheme completeness proceeding jensen readily bs diag diag dimensional calibration approximate appropriately select two firstly enable forward discuss evidence validation demonstrate methodology problem material medical diagnosis uncertainty quantification dimensionality reduction pose several model calibration assign proper mechanic material identification guide inform assessment system reliability identify material exploration without rigorously consider without provide quantification solution aim compute interest formulation offer unified framework deal uncertainty incomplete assess inferential uncertainty engineer parametric field bayesian scale size inference standard chain monte carlo generally impractical simulator advance sequential sampler difficulty modal pose computational identification biological material diagnosis measure sample obvious ease patient ray variation general lead early accurate diagnosis insight modality progress propose new imaging technique aim develop rigorous image modality project pre post compression image position external load diagnostic alone appear coefficient raw noisy e bad derivative sometimes quantification employ alternative indirect admit problem discrepancy minimize raw arise paradigm development size resolution expensive tool content incomplete constitute advance progress improve efficiency tool applicable basis vb task machine community recently employ approximate solve appropriately minimize kullback leibler divergence enable closed identification signature correspond attract application develop representation dictionary achieve material exhibit variability e care scheme employ frequently equation solve accuracy capture physics mathematic impose burden general solution linearize need eq forward solution cost report numerical experiment particular converge solution inversion subsequent available low experimental location boolean pick since emphasize generally highly nonlinear inverse nonlinearity constitute basic addition scheme computation pde constrain work scalar employ direct jacobian need solve node repeat
particle depend child previous particle way total generation particle process expect produce point produce child vice versa similarly child child use criterion particle order make branching decision initial particle decide continue particle particle analogously expectation cascade estimate fairly smc statistic denominator tie n k likelihood summarize initialization bound make resample random particle resample resample particle cascade correspond particle assume proof backward computational implementation cascade particle gradually propagate truly introduce arrive resample eq particle count ordering good situation order order completely preserve na I incremental resample scheme dependence order reach quite first stage well cascade address order particle impose permit run count consume efficiently particle impose hard limit particle simultaneously particle generally process ideal vary hardware make scheduling across queue equivalently weight queue live control responsible particle likelihood particle terminate queue queue terminate particle particle model use exact posterior compute design stress cascade rather branch scheme particle expect particle filter compare bad particle instead particle forward particle statistical quite thus speed benchmark competitive iterate iterate smc particle algorithm provide repeatedly particle efficiency significantly non resample suggest previously total impose per true normalizing constant ultimately efficiency record marginal likelihood show amazon ec core intel v processor much fast estimate compete model quick filter incur cascade initial particle particle progress end simple cascade hardware forward simulation increase particle possible also interested large small hardware comparison hardware configuration particle cascade broad applicability particle appropriate graphical particularly smc recently appear reference suggest primary bottleneck barrier synchronization something entirely boundary exploit parameter nuisance grow smc leave parameter se cascade particularly relevant attractive smc acknowledgment ep grant frank material air laboratory agreement reproduce purpose notation conclusion contain imply u air laboratory theorem axiom claim theorem frame department science uk uk ac uk call cascade barrier particle throughput memory unbounde cascade straightforwardly barrier synchronization limit costly term memory asynchronous particle efficiency resample throughput synchronization follow act queue barrier choose particle previously proceed traditional pf particle completion termination simply carlo merging set particle iterate carlo merging produce smc run close nature suffer fundamentally inner suffer directly avoid nature cascade merging share bernoulli branch exploration method use filter propagate cascade resample total generation allowed gradually decrease branching generally observation order choose appropriate resample increase effort scale advantage remove collective particle cascade arbitrarily particle budget focus keep proposal allow particle normalize particle continue generation rely resample state markovian dynamical random observe particle proposal density intractable conditional complex black evaluation costly capable simulate weight posterior weighted particle importance estimate suffer degeneracy wherein mostly moderately traditionally resample progress weight discard many scheme resample overview approach think draw stage particle equal resample introduce resample prevent resample add unbiased scale number particle parallel separate must normalize require forward simulation compute weight collective lead large memory finite hardware limit capable run particle move must memory requirement particle great ram substantial disk cascade address
blockmodel node row belong let probability block adjacency covariate expectation covariate blockmodel nc next assume hold motivate graph sbm nc sbm say block graph blockmodel gender diagonal producing within definite stochastic tend edge within nc covariate eigenvector norm finally mis membership nc sbm zero kk th row block population equal derive mis eq contain appendix eigenvector contain eigenvector iii theorem bound mis cluster mis recall cluster rotation intuition mis rotation minimize mis cluster bound mis mis assumption iii contain choice suggest base bound notice iii contain simplify key result ensure sufficiently sensible sparse routine differ suggest theory alternatively grow graph value low attribute sbm allow number covariate blockmodel block independent kl divergence covariate opposite correctly least node block insufficient insufficient compare simplify give cluster probability achieve clustering require investigation cluster result regularize cluster simulation bernoulli edge block opposite probability covariate canonical spectral utilize regularize rsc covariate sc effect mis block conduct graph method perform simulation mis change specific covariate h tends regularize poorly weight structure final membership membership bernoulli covariate long differs align varie agree assignment robustness misspecification robust misspecification mis show achieve mis graph identifiability require graph graph voxel brain voxel voxel treat spatial covariate contain brain graph range density brain label region treat covariate demonstrate effectiveness spectral utilize covariate help discover block ability covariate spectral discover cluster covariate brain interpretability covariate mainly exploratory tool insight relationship examine relationship brain spatial utility covariate partition graph matching ignore covariate spatial distinguish brain covariate spectral give homogeneous partition favorable degree utility flexibility cluster stochastic blockmodel nc well regularize mis cluster relaxed overlap strictly graph accurate mis misspecification nc sbm useful study assumption demonstrate useful tool obtaining priori criterion brain spatially coherent relatively homogeneous community homogeneous balance decide analyst relatively homogeneous cluster easier focus partition align covariate could value still computational eigenvector potentially reduce step direction develop understanding graph informative useful ultimately thorough relationship structure essential deep social derivation change major concern determine clustering eigenvector approach let follow note j eigenvalue position result lead ji position position lead optimal transition transition transition argument symmetric covariate transition lead equivalence membership eigenvector b h exception symmetric eigenvalue spectral population apply individually kk next bound spectral variance entry otherwise eq norm assume expand restrictive restrictive supplement probability three term supplement bind establish q q hence supplement result term consequently five give q lemma onto span choose eigenvalue condition close centroid ki give use result simplify theorem decrease span assumption tuning investigate mis simplifying assumption recall tm k minimize eq mis depth suggest suggest analysis check assumption degree check finally find mis cluster minimum occur agree eigenvalue uses derive low specific covariate first divergence divergence bernoulli covariate block assignment rewrite easy probability b condition cluster satisfied high cluster clustering remark biological consist interact unit intuitively graph reveal insight graph network leverage help latent statistical provide joint blockmodel mis result without covariate large graph derive location region membership covariate easy node brain blockmodel areas vast amount contain valuable gene brain region relationship essential solve feasible technique diversity understanding block contribute common pathway common form network become social biological science network discover insight characteristic extensively aspect include unlike minimization modification spectral cluster regularize network certain bayesian flexibility block likelihood provide quantify ultimately computationally globally partition diverse modern often represent network brain brain potential utilize covariate graph covariate discovery covariate procedure enhance homogeneity within filter partition covariate natural interpret rely ad hoc heuristic estimation broadly heuristic focus categorical computationally expensive discover multi block binary update algorithm include space point cluster relative node node similarity approach covariate spectral laplacian square parameter adjust relative covariate section propose type graph without general covariate variant previously minimize weighted cut spectral relaxation decide chose laplacian covariate recognize tune blockmodel type paper derive performance method initially motivate intuitive covariate blockmodel combine stochastic model mis accurate covariate method laplacian cluster canonical correlation perform nc sbm however canonical correlation fast tune single covariate without intuitive determining tuning consider provide optimization set knowledge brain set use location spatially coherent alone easier interpret align connectivity vertex node edge adjacency restrict study undirected unweighted small direct weight graph ii treat constant improve graph parameter average cluster correspond regularize spectral prior run use node necessary introduction spectral
feature well comparable base object cnn part multimodal fix image feature deep stage future ms dataset deep cnn internal deep platform non tune via k across ms average sentence exclude stage core test benchmark level annotation tc take around people image sentence sentence annotation training testing provide five adopt separation dataset provide annotation crowd annotation previous testing task current release contain annotation sample validation testing currently set dataset ms scores b b translate reference similarly sentence generation remain evaluation drawback description generate sentence please calculation task material evaluate method toolbox version evaluation retrieval measurement retrieve give top top rank retrieve k important retrieve sentence tc metric please supplementary improve publication update serve architecture rnn representation input conduct evaluation metric mention accord give strong language successfully capture without content sentence rnn generate low indicate consistent perform rnn retrieval since report score future comparison retrieve sentence rank retrieve metric compare supplementary material tb ccccc base rnn cccc cccc c sentence dataset retrieval k art method devise image representation use feature avg denote confidence strategy help use feature cccc sentence text r rnn devise avg rnn b generate rnn htb text r random devise rnn ccccc ccccc b rnn dim yes yes treat version method appear result devise search keep layer five recent representation recently recurrent relatively compete advantage efficiency input store recurrent layer perform generate greedy rnn c rnn method server select time represent reference reference importance rnn rnn outperform strategy multimodal substantially improve supplementary main near rank rank share hypothesis search keep generation sign probable treat hypothesis hypothesis validation set generating hypothesis image retrieve near detail tb try two pixel window corner pool ten second feature calculate multimodal see scale neighbor feature feature rich visual row contain old retrieve consensus share rnn shared server rnn share rnn rnn suppose get neighbor consensus score treat image rnn calculate similarity near nearest cross validate server consensus point consensus get test variance image improve ten hypothesis hypothesis surprisingly room improvement multimodal recurrent three task image retrieval sentence model rnn interact multimodal rnn image flexible representation sophisticated acknowledgment ng yu technical thank anonymous center machines award cs l rnn rnn rnn rnn embed input multimodal capture level multimodal predict validate train rnn rnn denote whose word multimodal thus two rnn denote rnn whose two layer word embed rnn rnn connection multimodal embed multimodal layer rnn word multimodal perform embed layer three image rnn layer layer connection share table show rnn indicate visual practice hard train keep dimension sophisticated rnn adopt metric retrieve sentence image percentage retrieve retrieval neighbor query image space hard description subtle sentence pick image sentence sentence sentence figure word multimodal originally machine first gram multiply brevity eq length reference sentence generate strategy whole reference sentence close shorter xu wang com california recurrent network model novel generate recurrent sentence convolutional image validate four benchmark dataset tc ms outperform state sentence art page code hypothesis sentence rnn sentence description early education image retrieval thank development computer vision processing progress brief method treat retrieval semantic sentence lack ability novel describe image object multimodal recurrent network observe paper topic acknowledge address sentence part multimodal language part word recurrent vision convolutional cnn generate connect rnn validate benchmark tc significantly incorporate deep representation image sentence rapidly vision computer convolutional margin image task imagenet widely vision design layer perform substantially language recurrent task rnn machine extract sentence sentence previous method embed sentence image object feature generate feature global category language field model field kind generate sentence correct description generalize retrieve category learn sentence topic rich sentence serve retrieval task work embed layer multimodal pre initialization embed random see word three multimodal layer rnn recurrent space add element relu success training vision differ adopt relu backpropagation rnn appear temporal work heuristic truncate step early efficient truncated layer multimodal connect three layer recurrent representation activation framework activation multimodal space together activation multimodal
score average sentence positive purpose supervision supervision group review average predict score classify review sentence art training emphasize infer label sentence naive classifier review classifier transfer ignore neutral neutral sentence manually label sentence either half report sentence tool tool pre train web interface th supervision require expensive entity sentiment entity di context review movie force predict sentiment specific getting predict sentiment phrase actor sentiment role movie figure illustrate actor movie total movie score phrase sentiment desirable movie di sort actor provide work advance deep consider similarity embedding demonstrate deep label individual explore classifier embed modality well development application rgb institute advance cifar ac uk infer use learn grain area voting group illustrative vote city public policy individual present aggregate say concern privacy vote application privacy technology solve arise intelligence similarity measure instance objective function create capture success rating sentence group sentence review review detect comment toward service present overview create sentence document convolutional embedding sentence review embedding convolutional embedding illustrate regularize learn objective transfer entire review eliminate high cost label sentence success computer deep effort language processing considerable decade interest effectively convolutional neural network representation beyond block notable vector extend simultaneously move use convolutional representation document convolutional experiment method multi instance refer instance powerful extension learn variety prediction categorization translation object privacy bag predict prior instance make connect formulation instance assume bag group determine xu contribute equally bag bag generalization bag term bag label within survey multi vast disagreement terminology close formulation deep transfer set group instance instance assign measure label example essentially invert label advantage assign instance goal assign produce training set goal construct objective propagation manifold label similar semi term alone label group adopt loss ensure term label simple relationship say label element term act avoid trivial instance regardless group belong individual carry regularizer shrink compete two value fall second bound therefore order contribution equally assign label unseen label goal instance test instance group objective item similar transfer green embed sentence document sentence intermediate representation require refer sentiment sentence document sentiment attempt sentence contribute positively sentiment interesting explore causality automatic sentiment deep sentence outline previous sentiment sentiment obtain similarity recent distribute work show work extend text create similarity q expect nearby embed correspond similar sentence appropriate measure closeness embedding particularly match level supervision word sentence purpose figure use review group movie exactly keep incomplete child regard I
bayes study allocation family datum probabilistic free model hmm state introduce literature well derive collapse derivation let define maintain denote negative eqs z n k review collapse symmetric gibb take specifically start part concern object posterior formulate conjugacy easily r k l k I n denote solely combine q denote beta plug cluster assignment domain iteratively posterior stochastically statistic result derivation symmetric omit concern domain denote eq practice assignment z hyperparameter put omit scope employ posterior however never number detect valid manner difficulty slow nature sampler million mix network datum model slow need introduce sampler make implement reason motivate develop fast work sampler variational bayes quickly collapse vb vb vb hide posterior posterior variational assume minimize leibl divergence posterior variational posterior speak vb analogous iterative vb low w posterior vb monotonically vb vb break eqs vb solution doubly posterior number number virtue bayes unnecessary weight infer I b describe eqs mainly compare original formalize introduction truncate cluster simply assignment variable posterior conjugacy posterior v k vb equation derivative low eqs naive vb interest derivation vb variational posterior assignment variational j l sum vb posterior derivation present final q eq vb bind easy hyperparameter derivative hyperparameter k l l inference assume posterior collapse ordinary vb posterior posterior local first collapse gibbs estimation reduce evident good derive vb employ truncate truncate reader present presentation k require way membership k n k z z variational posterior concern original vb form vb whole conjugacy inference form variational naive vb reason form hidden variable inference resemble collapse gibbs vb inference one put difference sample assignment repeat rule cluster representation replace sampler exclude computed datum denote expectation likelihood prior rule evaluate expectation derive inside part readily obtain ready conjugacy denote statistic variational cluster equal rule intractable computation inference expectation side follow fa fa x st take posterior approximation employ consider lda obviously nd learn taylor I computation expectation variance expectation variance count eqs k l expectation variance compute expectation j l l plug eqs domain update completely expectation l l j l j l j equation solution iterate local inference code derivative concentration dp role never rule technique follow current inequality zero update observation hyperparameter eq evident update eqs update rule incorporate count vb parameter theoretically vb monotonically variational solution iteration automatically sound monitoring unfortunately guarantee far expectation see taylor lower bind sure procedure monotonically inference important problem convergence literature estimation many empirically monitor couple naive leave detection automatic prove stationary non viewpoint highly preferable devise practitioner easy inference algorithm em prefer guarantee allow user vb automatic termination guarantee convergence stochastic leverage fact lda contrary rewrite way l l x right membership degree eqs inference limitation miss within result inference linear inference correctly existence missing need entry another algorithm posterior trick collapse way update massive parallelization collapse direct computation computation usefulness experimental confirm fact inference model vb relational fast inference linear scale relational compare baseline deterministic comparison collapse sampler initialization fair comparison update gibbs report completely manner value hard assignment solution weight cluster solution gibbs priori assess examine explain evaluate average test matrix exclude relational roughly held likelihood entry evaluation likelihood test entry initialization hyperparameter compare solution convergence vb posterior relative quantity converge state employ utilize keep miss reference gibbs iterate sampling procedure discard burn repeat collapse gibbs would million obtain resource collapse practitioner size mail use study mail transaction company member send member transaction contain record fm service list tag relation dataset time tag relation tag inference naive vb tag co occurrence count tag name tag tag entry binary tag tag size deal require observation exist dataset evaluation c modeling performance solution test conduct statistical dataset vb c gibbs dataset vb reveal inference significantly confirm well dataset potential inference vb inference vb specifically vb well artificial small cross still estimation estimation dense general face analysis informative iteration expect bad optimum contrary sampler collapse million well perfectly collapse gibbs sophisticated index sort group horizontal color domain domain assignment cluster highest sort sort time convergence report trend average aside collapse magnitude nd unknown vb third count maintain able efficiently cache fourth landscape posterior smooth vb concern vb truncate fast cpu convergence threshold quantity fast illustrate plot collapse case practitioner computation gibbs tag c vb user tag far conduct dataset typical datum goal perform datum assume test time convergence level five relational large dataset tag indicate rating rate regardless rating rating movie subset netflix degree worst movie million entry c cpu linear xu xt netflix na netflix rate na present times average cpu times inference enable n general dataset take cpu cpu please row cpu grow expect nine cpu dataset number technique cpu threshold relative become fast beneficial collapse gibbs linear well million estimation also collapse inference process detect datum costly contrast require practical relational collapse variational infinite convergence inference replace vb base standard taylor derive formulation parameter never practically open start examine assess effective annealing average offer anneal mechanism equivalent converge lower stationary aspect shrinkage implementation inference application result offer precise inference naive vb enhance speed possible stochastically recently lda examine parallelization collapse sampler algorithm advance aside assignment toward focused dynamic subset filter easy observe relationship average guarantee practically useful collapse bayes inference number include hyperparameter study practically two study develop convergence inference enable automatic expert difficult costly manual monitoring inference describe large relational show performance infinite collapse relational analysis link social service customer record scientific statistical present among infinite simultaneous bi row dimension customer correspond item modeling tool datum without need careful bayesian sampler guarantee posterior infinitely stochastic often computation computation thank factorization approach easy convergence vb due factorize posterior collapse estimator integrate parameter inference collapse gibbs sampler fast sampler collapse especially hdp lda paper taylor simple sense report yield vb exact collapse paper collapse gibbs difficult preferable report collapse gibbs interestingly vb easy reason vb perform poorly partitioning problem promise vb focus topic bag style set formulate derive relational fast naive vb derive automatically optimize dp nonparametric study c c l vb comment seminal fmri gpu noise filter extension fully cover inference two practitioner aspect theoretically inference exception use valid lda however problematic practitioner familiar try manually sense favorable behavior naive bind pseudo leave log likelihood empirically serve develop anneal technique automatic detection anneal converge stationary equally preferred practitioner want art apply issue seminal introduce optimality valid first hdp hdp solution hmm attempt hmm implementation square object product observe item cluster user inference make impractical relational describe computational especially solution item denote practically guarantee experimentally offer comparable even naive variational multiple synthetic real relational magnitude fast vb stable dataset scalability propose relational convenient practitioner automatic collapse solution use precise behavior solution simple annealing convergence inference object gibbs solution introduce vb vb solution present convergence issue annealing discuss devote final relational dirichlet dp cluster unknown dirichlet mixture proportion break chinese restaurant crp crp employ collapse sampler collapse crp crp partition let partition crp k pz hyperparameter equation show rewrite allocate partition object number partition exclude object membership object new randomly partition crp result construction
ai package binary node node observation fine grid tuning parameter panel grid node calculate consistently proposal color pseudo proposal brain university project process occurrence computer select large term gaussian among term student edge node control fix criterion select six indicates explain occurrence word provide explanation relationship htp publicly available expression gene cancer poor patient among associated cancer reconstruct gene among likely role identify gene brain target identify identify fix present empirical covariance result network display identify gene decreasing estimate interestingly gene know role variety gene highly connect use recover framework framework three accommodate sparsity connectivity c define equivalent theorem sufficient solution remain eq follow desire proof present prove matrix dual dual ii equality compact swap min equality ccc subgradient matrix suppose solve support subgradient imply eq q obtain suppose ij cc jj feasible p last summing use arrive contradiction result ji consequently combine use consider objective lemma assumption v f solve graphical evaluate contradiction must correctly fail node htp color study extensive admm ghz core replication display iteration converge function iteration increase run never without htp f update ai gradient descent choose bfgs method unconstrained line evaluate computed solving note leave improvement future work htp lee consider author learn implicitly accommodate realistic network convex penalty framework three widely graphical ise model multiplier algorithm demonstrate illustrate proposal set gene expression multiplier graphical wide variety gene presence indicate manuscript type graph marginal independence connect conditionally marginal marginally without graphical model number specific variable conditionally independence marginally independent marginal density ensure sum determine interpretable encoding encourage take graph place double consequently approach equally edge believe certain world wide web world node power law example include social network li towards hundred gene result typically free paper densely substantial number densely connect contain node propose convex penalty estimate contain penalty ise author propose graphical however arise free less author section graphical see contain node much graphical apply graphical ise apply gene discussion general accommodate matrix assume convex order estimate non instance graphical sp penalty encourage estimate contain entirely zero model penalty encourage sparse entirely figure edge via form eq parameter control connection convex combine depend loss network dense value closely relate overlap context admm guarantee consensus solution htp p stop f soft operator ij z detail rewrite augment primal lagrangian usual lagrangian quadratic complete appendix ai depend special network assume take form serves extend lasso accommodate propose encourage contain connect update ai derive minimize respect treat domain solution denote eigen admm update compare admm ghz intel core take minute admm extensive run solution column condition depend upon result literature fuse partition block jj c j k check block bottleneck eigen block compute eigen compute eigen computational per exploiting take second apply connect large node optimization tuning denote p condition reduce tuning reduce solution diagonal decomposition scope tuning place penalty minimize q unique zero motivated fact degree involve estimate parameter degree freedom proposal proposal enyi proposal graphical start index node graph cardinality set highly estimate jj equal randomly column proportional world take natural scale free distinction n standardized proposal gaussian graphical model package glasso problem involve three describe previous section display average simulated square tune fine tuning parameter bic minimize involve tuning tuning display figure proportion simulation edge equal iid correctly proportion correctly highly ii nonetheless outperform select use graphical bic large colored correspond graphical performance proposal correlation screen thresholded absolute node thresholded purpose estimate scale tune parameter partial use package space combine author claim proposal iii average grid specify element thresholded node fine value note figure b c use fine curve error baseline include surprising graphical since approach implement iteration edge comparable network intend perform intend network intend graphical figure type criterion color
learn generalize another discriminative input risk nn note neural give source natural use probability correct label eq internal representation neural consider unlabeled representation sample hyperplane proxy see estimate logistic regressor denote target enable adaptation term solve hyper implement trade source divergence hyper parameter tune quantity network domain regressor parametrize way source layer accurately regressor unable detect sample option alternate simple stochastic sgd sampling update crucially opposite follow consist except bt sample layer adaptation neural regularizer domain network parameter experiment split source label use validation risk mention previously domain focus mainly hypothesis recently become increasingly study notably principle base denoise principle adaptation domain indeed directly optimize work hmm representation use inspire versus sentiment argue optimize divergence rely reason idea learn auxiliary also explore context belong identify result minimax learn minimax work assume learn adversarial generative datum work share pca toy color regressor experiment behavior distribution label way keeping contain unlabele dot capability experiment train use propagation execute risk minimization train regressor discriminate toy experiment boundary compare graph relate graph part relate column four detail label class sample contrary perfectly analyze affect hide present component pca map dimensional hide project two nn pca representation point point visible labeling easier correspond represent graph clearly conversely point locate opposite corner pca target resp difficult locate cluster pca suit classification regressor classify otherwise regressor discriminate prevent explain train regressor learn domain discriminate target although discriminant decision surface neuron th neuron three cluster allow straight classification observe regularizer prevent neuron indeed neuron corner corner c source name nn book book book book book mm svm standard equation hyper search consist target amazon process four review specific book rank star product rank star adaptation task example book book source domain unlabeled use unlabeled procedure logarithmic procedure adaptation nn svm use part show risk report one conclude help suitable domain brief unsupervise robust feature representation input find reconstruct original noisy counterpart show optimize objective experiment subsection pair source corruption execute three procedure concatenation encode nn representation sound test table foundation claim representation toy confirm real compare proxy run nn recall obtain construct equal first subset use large range low firstly representation experiment hyper parameter lead expect compare representation standard nn influence precede lead low lastly present notice give great data approach help seem representation algorithm inspire adaptation behind encourage predictive uninformative extensive toy sentiment show effectiveness strategy notably autoencoder turn representation believe incorporate extension deep adaptation task beyond basic denoise autoencoder thm introduce distribution domain suggest discriminate source propose objective implement task uninformative sentiment unlabeled domain performance either even input extract stack denoise autoencoder obstacle develop exploit one generalize focus context
trade overfitte perform goodness fit criteria information bayesian cv understand equip aic cv standard selection theory new recently impulse response realization learn counterpart selection paradigm robust aic type cv crucially depend covariance kernel process variety semidefinite kernel nevertheless straight framework fail lack system stability several deal stable introduce spline realization implementation concentrate totally kernel point algebraic connect completion particular band alternative stable kernel triangular diagonal interestingly spline factor admit covariance burden scheme spline organize spline briefly review matrix completion introduce end symmetric resp semidefinite resp matrix vector diagonal diagonal denote set submatrix index submatrix model fed impulse white noise collect dimensional express whose input represent impulse estimating system impulse dimensional model sample continuous covariance independent empirical paradigm marginalization joint density estimate impulse identification assign class covariance stable information impulse surely first stable see tune correlate tc kernel q therein also concern completion pattern graph entry specify every great band center entropy covariance admits complete fundamental main get ij top left optimization namely entropy partially band problem denote value mx band feasible admit property bandwidth namely positive maximum state matrix give lag also maximize matrix pattern problem admit close compute specify symmetric matrix bandwidth call follow extension specify bandwidth step suitable solution factor central extension partially central band admit triangular stable rely introduce form order spline stable spline highlight specify spline computed entropy nest prove dimension claim statement k k central extension band hypothesis claim maximum likelihood follow ij j spline covariance satisfy equivalent moment stable factor form theorem immediate consequence sum stable spline resp hand resp diagonal positive determinant I thesis recall lie likelihood nonconvex hyperparameter observe stable spline
eq quadratic root explicit formula constraint check omit detail x rescale generality strong upper conclude note lemma thus display easy generality prove acknowledgement would thank discussion grateful point barrier ex barrier seminal nesterov explicit construction universal barrier geometry concave elementary families interior main recall specific barrier prove sharp isotropic log concave bilinear likewise derivative self barrier convex furthermore addition barrier xt gx newton approximately minimize theoretical nesterov barrier barrier universal convex seminal always self also set simplex hypercube prove exist barrier self mid also construction barrier barrier barrier canonical barrier universal barrier give barrier tool dimensional generating body viewpoint prove mass proving derivative seem barrier inequality canonical barrier inequality play barrier inequality fact body concavity somewhat effect complexity interior consequence local sign dual universal barrier analytical barrier side analytical center universal know point geometry proper convex homogeneous furthermore recall characteristic immediate barrier homogeneous begin connection barrier satisfy universal barrier universal barrier hull different consider universal barrier body barrier barrier characteristic cone particular barrier barrier self barrier barrier barrier introduce unique amp exhibit convex riemannian barrier canonical barrier recall unique volume ellipsoid perhaps homogeneous coincide constant somewhat generally canonical equal affine conclude comment generality cone focused self tractable barrier barrier essential obtain universal barrier barrier immediately effort implement gradient barrier practical important efficiently computable barrier convex interior one barrier sequential known describe history possibly randomness player adversary compare cumulative cost cost costs play g challenging receive limited feedback bandit player observe incur survey view feedback seminal role good precisely run mirror originally choice barrier sampling key barrier hessian barrier proportional inverse scheme achieved support ellipsoid scheme good universal scheme make much ellipsoid sampling via discretization mirror barrier scheme strategy introduce exploration one property barrier prove implication noting obtain numerical whose concave move part self become word show eq proportional eq interior satisfy exponential associate weakly operator norm verify smoothness support technical rely log concavity measure gaussian give proof lemma conditionally stochastically dominate condition law equal constant learn exist confirm use find let x x x x thank proof let measure direction consider smooth support key thank differential lemma proof yield
deal different american category image style piece work th style art tend year style style style unknown lot classification quite image category style ordinary classification two comparative conduct review semantic level intermediate feature fine use different comparative evaluation evaluate three model bag generative use semantic feature use discriminative model discriminative model capture semantic employ machine intermediate generative specifie distribution new distribution intermediate step label discriminative thus avoid comprehensive semantic capture characteristic color texture color histogram edge feature formal intermediate level apply descriptor sift level descriptor localize generate intermediate bag create image codebook visual represent capture frequency level semantic content water denote existence semantic worth note color texture focus intermediate semantic utilize generative model like topic capture visualize document example characterized atom represent level descriptor describe constitute represent straight constitute topic tree region concentration color water subsection detail bag popular categorization document matter categorization typical several representation descriptor descriptor descriptor encode codebook machine overall presence pre classifier generative dirichlet allocation semantic categorization localization scene categorization fine categorization purpose dirichlet topic represent characterize graphical image topic total setting minimum tendency hausdorff asymmetric period construct represent direct indicate high potential define influence multiple space indicate influence influence generally influence influence truth influence retrieve top influence retrieve influence pair influence truth pair detect influence sake relatively influence truth mean feature descriptor descriptor recall graph percentile distance descriptor table percentile column similar generally manifold curve recall euclidean manifold three top recall l l l recall l l l l recall graph achieve visualization similarity purpose short scale dimensional graph htp htp figure visualization influence projection code plot reflect ground truth style cluster together mapping modern abstract differ leave seem dimension much distance broad influence similarity yet way style list influence ground truth close mapping truth less coherent lie consistent mapping illustrate suggest influence htbp surface automate discovery study pose question find knowledge qualitative quantitative measurement study present comparative semantic good task distance manifold comparative give different central link maximum formulate hausdorff central influence tool similarity annotate diverse publicly task lot search similarity many principle way influence difficult huge valuable examine expert period find influence connection influence contribution explore computer influence set present comparative comparative problem review model second feature compare level vs low investigate influence pose discovery purpose map work use concept basic art shape color line general sense seven principle art movement unity subject matter attribute work art art influence connection art continue art inspire body art influence similarity inference suggest ever inspire unless sake consensus cite comparison study study x similarity clear matter computer advance develop object categorization scene etc recognize scene category historical look expert landscape look color texture complex concept think logical tackle methodology learn concept automate determine measure task mention way describe description translate automated way van element art finding different automate influence quantified connect keep challenge people art suggest towards measure influence besides various orient volume database internet task organization retrieval properly become classify category classification speed significance broad view level indicate line composition square see position conclude make mean completely symbol symbol word express meaning work need image describe list many semantic opposed color texture influence subject matter semantic importance find similarity become prominent essential linearity paper computer automate influence set discovery methodology study different representation measure study collect contain time period collect ground truth influence contain claim use discover influence discovery would evaluate require compare different detecting influence contain truth negative different resort classify representation good classification determine influence perform comparative classify seven detail sec conclusion study confirm semantic useful task influence right similarity illustrate detect methodology ed look similar close look subject matter detect similarity measure discover however clear influence time influence influence result achieve structured survey describe use section describe classification include describe methodology evaluation automate automate fine utilize level texture study classification style define signature low among eight focused discover similarity influence publish experiment image classification pose annotation present et al texture exploit effective inconsistent texture visual image al set annotation keyword describe annotation class pose annotation pose human image query contain improve propose local problem et unseen annotated class retrieve visually similar
fix power remove investigate novel tensor introduction mention practical aspect model among iteration analysis overcomplete apply propose guarantee subset relate study gaussians list gaussian propose moment polynomial general challenge model improve divided spectral method among mention noise noise moment spectral recover general without degeneracy condition guarantee power third order tensor recover overcomplete vector factor asymptotic member outer mode refer row arrange vector slice fix slice rd multilinear particular mu multilinear mode multilinear combination tensor slice rd form tensor section exchangeable mixture state convenient vector q basis simplicity argument even style cm sep sep draw name view independent hide variable conditionally denote matter observation moment unsupervised problem distribution pass decomposition main perform multilinear thought rank alternate run initialization well return moment mixture tensor update multilinear set maximize iteration power output center cluster cluster center propose guarantee state explanation condition column noise universal guarantee assumption crucial argue conditioning strength sure ensure inside vector next propose let q iteration initialization initialization setting mention initialization rd randomness recovery therefore update require use polynomially main signal recover regime noise thus order initialization condition c relevant satisfied iteration different mean modify third observe q additional spherical state state relate hidden recovering follow first change asymmetric among appropriately version introduce new phase phase show component provide incorporate result initial constant correlation component simplify rd tensor dynamic tensor phase eq update vector w constant dynamic condition constant h recover sign resolve sign issue lemma follow satisfied generality initialization h exploit inequality desire initialization universal corollary make initial correlation result normalize intuitively roughly hand vector argument show ensure proving show residual randomness distribution enough tight enough argument hadamard product entry multiplication vector matrix operator space orthogonal denote analyze dynamic rd iterative write analyze evolution dynamic explain early step analyze since update provide tight bind evolution careful controlling amount exploiting enable matrix break power intermediate follow power rank break step introduce intermediate unnormalized thus analyze randomly entry follow middle update evolution fact entry projection randomness row orthogonal equal exploit direct evolution govern conditional lemma iterative column orthogonal constraint orthogonal orthogonal remain exploit previous characterize middle intermediate entry remove decompose q intermediate residual randomness variable reference throughout iterative middle denote copy similarly conditional iteration variable intuitively residual randomness characterize concentration component maintain analyze dynamic power update iteration black following assume iteration induction scope inductive ty induction end induction provide analysis show correlation latent challenge regime technical small residual sufficiently enable progress generalize tensor support part microsoft fellowship nsf nsf award award award nsf award formula n intermediate formula unnormalize remove column recover bt b r see randomness equation update part leave u b tv let partition first rest make remove induction vector induction induction hypothesis true hypothesis bound initial state prove hypothesis end figure scope flow start start show induction hold iteration induction tb concentration similarly matrix I know hand eq expansion step random dy ty b eq definition establish expansion dominate tw iteration hypothesis hold random bring randomness formulate lem subspace orthogonal sum random high prove sep black w w infinity bind induction norm know high hypothesis hypothesis exploit contribution bound combine bound guarantee formalize lem ir bounded order last inequality choosing depend immediately norm hypothesis tx unnormalize q notation hypothesis desire induction early random component subspace represent norm bind argue expansion triangle cauchy inequality exploit norm involve term bound x analysis combine norm argue conclude lem induction also hypothesis early bind unnormalized argue observe contribute term exploit induction involve bind gaussian inequality exploit second finally argue hypothesis bound total term lem basis x x p know know iteration induction bound even large know inductive step inductive parameter polynomial constant q generality assume induction hold inequality inductive step induction step enough constant addition constant point universal get step side say inductive section prove inductive bind value value random concentrate norm even project subspace high variance argue probability random least value prove lemma lem vector vector gaussian high p direction equality z rp inequality step argue high square entry restrict apply part product orthogonal lem p triangle third cauchy lem maximum th basis leave difficulty prove treat lem specify expand hadamard bind type form bound lemma bound induction hypothesis hypothesis correspond eq v bounding difficulty treat entry hypothesis subspace q dominant cauchy early desire rgb criterion derivation remark em time
face challenge prove correctness require minor section intuition problem cost distance cost start open center keep service detect conclude easy without new cluster expensive denote point convention optimal consist center cost cluster average vector contrary let cluster eq total trivially bound bounding phase complicated phase phase open subsequent triangle inequality r f I r n r sum center estimate succeed b assignment optimal cost stage vector start incur sum start additional contribute cost bound k terminate round conclude execution p r observation online stream small must two distinct dataset aspect ratio create center make arrive algorithm phase first conclude reach probability round discover set adjustment operate entirely ad hoc large center sum square close neighbor evaluate execute dataset website uci letter e e e uci collection engineering sake learn datum raw mean decrease slow large significant classification observation raw news letter particularly low improve enable advantage phenomenon mean result rather cluster value every setting range roughly interestingly choice roughly average reader online mean center algorithm sum cost inherently center decrease nevertheless function target different dataset normalization divide monotonicity plot center identical figure something cost fix significantly pick center improve rate random choose center surprising compare mean run invoke use mean mean term pass online online mean bad note dramatically everywhere cost helpful suggestion theorem observation one generate logarithmic much operate strictly study point cluster center cluster eq offline advance access obtain provably difficult see streaming allow keep logarithmic stream center algorithmic idea streaming assignment online allow priori arrive algorithm open consist conceptually choice unseen stream intersection algorithm output time stream hard algorithm trivially stream trivially sufficient mass stream overhead independent length stream even stream act assign trivially assign one exist suggest read conversely part line scenario yahoo news decide belong act advance practitioner simply refer recently well mean ratio search design base optimal effort adaptive reduce pass
level show figure htb htb quantitative bold remarkable theoretically central great derivative noise figure fail root lack derivative order contaminate use order quality demonstrate method parameter demonstrate technique demonstrate success show general desirable operator frequently inversion consider regularize previously iteration size use instability converge process transform equivalent initialize lc noting map ms use datum form justify prior pick estimation grid rectangular surface divide generating contaminate literature potential inclusion regularization inversion initialize weighting find technique condition meet iii practical knowledge constrain interval project newton freedom hand regularization whether e g overall rely interpolation typical demonstrate case reconstruct case solution less useful mdp accord confirm error average error invert noise contaminate except invert estimating regularization require ideally prior solution prior require ms provide alternative central curve efficiency require implement order find solve inversion iterative technique replace svd development nsf dms novel sampling regime present pair also correct determine value ts limit low contribution seek corner shift use iteration hold filter corner namely corner parameterize equation lemma goal principle inversion discrepancy augment weight fidelity regularizer size nan space intersect newton find optimal inverse noise map datum implement scale generalize result verify regularizer approximate unbiased predictive focus smooth property iterative data noise curve discrepancy experiment efficiency context show curve principle general principle desirable principle ill pose equation discretization space vector assume contaminate discretization decay responsible ill extensive literature literature ill pose w tw yield dependent tw covariance white map I tw symmetric permit white determination research generalize validation rp principle mdp comparison criterion reference vary measurement mdp assumption apply extension effective consideration nonlinear inclusion principle estimating mean np np interval around development advantage root converge algorithm scale matrix iterative extension non unknown may measurement principle mapping approximate distance inversion sharp preferable example inversion similarly minimum support ms support introduce iterate stationary iterate operator update residual iteration force geometrically technique mdp approximation ms reweighted introduce analyze ms reconstruct inversion update regularization find often mdp curve apply knowledge follow development strong literature improve algorithm use svd parameter technique present also impact ms shift intersect realistic approximate typically w singular introduce ease presentation vector state rank respectively generalize singular invertible indexing generalize note indexing result variable statistical strong non condition sufficiently large limit hold p tw functional centrality tw examine yield first g tw tw tw tw tw tc assumption denote consistent apply obtain transpose adapt extended filtering replace obtain pick filter already note statement uniquely order hand particular order spectra matrix calculate ordering specific order standard p difficult collect estimation assess parameter review aspect section literature completeness formulae technique inversion residual valid adopt validation measurement set regularize formulation yield minimization associate objective flat create compute numerically trade norm residual problem corner difficult find curvature plot unbiased predictive risk successful use simplify
instrumental state mm hyperparameter assume generate covariance attain analytic risk recall goal therefore j follow use derive section assume submatrix row index use prove second solution dual write objective last hypothesis transfer accuracy kernel regularize forward follow notation brevity truncate q risk label hypothese j last jensen inequality prove
p cm full vs rw propose similarity two compute fix feature subgraph arbitrarily accurately sample increase process normalize histogram similarity two recent induce social improve subgraph normalize subgraph induce subgraph size histogram node generate dimension dim reason increase representative actually graph costly see importance subgraphs subgraph histogram subgraph computationally practice walk similarity number rich regard domain cauchy kernel two adjacency define computed close recommendation value eigenvalue worth dominant take eigenvalue eigenvalue adjacency normalize inner evaluation consist run svm standard combine fold th validation performing value svm fold fold act repeat fold acting partition error show outperform compete state art capture capable accuracy three range variation significant ideally tune easy walk sometimes subgraph expect count subgraph significantly interestingly count subgraph size except vs histogram loose computational counting subgraph consistently demonstrate superiority argue incorporate big complex along adjacency sound similarity dominant poorly compare eigenvalue graph different different eigenvalue seem right describe graph characteristic infer analogy explain covariance comparable perform key require compute dataset record pair similarity network similarity take summarize ghz cpu machine gb poorly fast rw method quite surprising rw slow cubic time complexity linear computing representation histogram counting subgraph costly count subgraph histogram sample subgraph graph rw magnitude histogram subgraph every histogram sample induce next step match subgraph structure graph sample solve graph start become intractable expensive count capture quickly loose count big nice capturing space positive semidefinite characterize graph covariance matrix adjacency indicate matrix naturally overall procedure edge superiority state approach balance representation tractable meaningful believe provide mathematical allow graph representation algorithm like semidefinite entry normalize vector adjacency encode underlie contain sub triangle addition suitable represent graph naturally similarity graph similarity measure state make practice believe empirical study adjacency characterize become increasingly whole gain analyze neighbor recently attention social network informative collection close center scientific individual scientific link compare dependency likely lot exhibit densely compare people reflect belong physics thus characteristic utilize possible discriminate recommendation discover citation recommendation network right different similarity measure meaningful mathematical embed structure fundamental representation spectrum world scale spectral density consist sharp peak mathematical graph compare common eigenvalue characterize comparable eigenvalue compare occurrence subgraph representation social inner product lead clear histogram counting capture subgraphs size computation know histogram subgraph size subgraph subgraph computationally require give capture behavior computationally challenge requirement map count count take alternate adjacency vector one generate argue covariance covariance adjacency covariance count histogram semidefinite covariance kind similarity multiplication compute representation outperform example subgraph perform poorly study power adjacency social researcher experimental scientific explore contain scientific work undirecte unweighted connect represent vector default transpose component two adjacent associated permutation operator multiply row multiplication e adjacency graph adjacency matrix represent structure vertex adjacency eigenvalue wise eigenvector path consecutive term I path path path whenever path hold exist contain triangle highlight quantity fully characterization vector show generate truncate order matrix fast web domain include page truncate similarity sufficient describe associate represent common basic map characterize adjacency associated graph permutation structure order entity perspective perspective turn covariance value spurious start compute later graph structure give first generate normalize since normalize equal ease ie h input adjacency initialize x nm tc symmetric semidefinite symmetric semidefinite permutation yield q imply converse theorem true hope small etc adjacency undirecte number triangle number distinct variance term quantification lemma path count twice length edge type path figure repeat ii repetition repeat path possibility contribution path total length path count path path explain contribution double count twice path triangle loop graph triangle b loop length node triangle contribute path generate many path correspond path total twice therefore eq add algebra substitute expression clear small count triangle along observation behind count path length disjoint separately extend involve computing term along sensitive empirically publicly available twitter consist user twitter edge generate graph add edge twitter value se network social high closure induce ac ab triangle theorem infer encode discriminate structure tell dimensional comparable structure fix common mathematical semidefinite property well notion two respectively summarize compute similarity adjacency matrix adjacency matrix semidefinite valid similarity semidefinite similarity semidefinite algorithm operate social later determined spectrum right graph consider fact semidefinite matrix mathematical detail c physics node symmetric semidefinite look eigenvector converge matrix avoid choice general recursively inside multiplication summation complexity computing graph addition require compute matrix graph computation argue value graph reduce costly step matrix multiplication proposal graph describe task publicly meaningful label availability structure create social
vote going apply hyperplane dimensional item formally unnormalized shift label directly general follow suppose item hyperplane know error bound worker aggregate factor worker one need give list corollary condition mention vote important special cover several simplicity case constant assume constant voting worker model meanwhile plug error theorem theorem hold well critical reliability weight choose us room weight detail far crowdsource majority error voting uniform meanwhile tight obtain letting simplification desire exponent second factor tight real crowdsource enough reasonable crowd similarly omit possibility control small majority crowdsourcing case tend infinity label majority I vote corollary tell quality worker worker random label correctly probability worker property majority voting ensure enough reliable worker available aggregate majority voting require worker section discuss method crowdsource model maximization posterior well know optimal reality model true way class build parameter apply rule call refer introduce em estimate might estimate start thus relatively rule know label map bayes classifier worker well guess oracle map help em map rule rate oracle derive prediction map visualize model corollary show voting shift item balanced balanced rule voting mean section understand infer ground label via prominent estimating crowdsourcing estimate map predict item study design study weight majority voting version rule rate ignore class corollary imply relax item converge suffer local error obtain naive voting treat gold sampling predict step th item defer label hard apply concentration martingale exponent important away accuracy guess average population well make worker label error decrease decrease beyond intuitively score worker increase make error increase e accord weight step way assign alg plug weight practical noisy worker linearize stable meanwhile compare synthetic experimentally art datum majority em public code average experiment pc window intel core cpu memory crowdsource affect variation item error reflect compare compare b run worker label three ground truth uniformly accuracy distribution expect worker match worker control worker figure error trend true converge increase vary confirm result fast majority voting clarity temporal omit ccc ask rate scale label worker around consensus treat multi labeling experiment dataset varied plot performance label compare majority voting generally majority voting bound decomposable crowdsource good drive iterative weighted voting theoretical rate reflect trend real crowdsource superior voting perform low misspecification want similar description correspond error rate labeling obtain score function complicate manner formulate voting em algorithm difficult nature simple em crowdsource thank suggestion comment thank discussion thank comment suggestion since prove corollary require put section present simplicity simplify worker vote equivalent aggregated label discuss simplicity notation bound set label specific depend value subscript come item say drop subscript item eventually assignment probability major focus term relation want bound voting bound apply concentration inequality far rhs depend inequality variance apply note definition sum concentration rhs get result hoeffding bernstein since rhs inequality c increase function decreasing prove argument straightforwardly far practical high hoeffding bernstein hoeffding lemma go hoeffde bernstein give finish bernstein chernoff show condition nd step easily finish several assume prove course chernoff directly step inequality prove theorem result map rule posterior oracle oracle section special map step vote every item meanwhile label hard generalize omit clarity prediction majority worker majority voting worker majority proving need several final proposition use prove convenience enable majority vote agree label give apply apply argument bind agree majority voting mild th item give match vote close number worker close close voting get since lemma apply measure entry quantity mean trivially focus non trivial column go put j j upper case take proof bind achievable bound say score voting decrease proposition derive bound step get true label convenience denote algebra condition want step apply get next derive lower expand I drop irrelevant two argument step upper eq beginning depend imply crowdsource become effective tool human computation since worker crowdsource aggregate label exponential aggregation crowdsource analyze aggregation majority voting voting posteriori map show optimize rate set iterative majority voting optimize approximate oracle version provable real par method computational around crowdsource expectation weight voting hard computer visual video crowdsource people call appropriately aggregate crowd yield could one crowdsource apparent purely complete label truth evaluate worker may answer question assign beyond worker persistent finish drawback reliable answer crowdsource yes voting treat equal worker majority voting improve upon vote back worker confusion distribution give element represent misclassification worker principle true label confusion matrix maximization majority voting put confusion simplify confusion progress true endowed concept worker area label crowdsource graphical apply infer principle worker assign worker budget apply extend infer behavior crowdsource system investigate error various provide majority voting crowdsourcing apply minimax coin mistake final labeling focus global optimizer rule find optimizer provide finite aggregation crowdsource motivate main rate aggregation rule worker rate voting majority voting gain insight design optimal voting majority oracle maximum posteriori optimize majority em algorithm approximate understand crowdsource propose drive weighted majority voting guarantee implement art simulate cost focus error crowdsource obtained analyze worth focused crowdsource labeling multi meanwhile crowdsource set real crowdsource crowdsource assume worker task cat evaluate assume labeling item represent miss however convention true th th item kk I label item item th get call configuration flexible worker item item gets label cover general worker adopt literature chance refer require specific get match worker task select worker follow represent match ambiguity depend either general context constant bound denote bernoulli operator number meanwhile throughout locally cover case originally reliability model confusion represent probability note worker label true represent label item modeling worker flexible overfitte worker impose constraint worker confusion matrix model probability item simplify probability worker matrix parameter reliability worker another item actually toy confusion model vertical axis actual class horizontal axis predict color different labeling item correctly three refer coin worker label noise signal binary special crowdsource convention convenience confusion parameter binary introduce ambiguity item labeling item rule aggregate noisy refined label item worker great labeling worker treat voting extension majority vote differently majority write worker behind majority generalize way input worker item predict potential label potentially base maximize aggregated score label decompose worker shift gain label label item reasonable contribute predict thus noisy final illustration majority special voting express approximated also paper sample decomposable bound crowdsource weighted majority optimization bound guarantee simulate deferred section sample error high expectation decomposable aggregation main focus specialized straightforwardly observe base probability accord assignment process decomposable introduce section mind quantity measure play quantity associate
detail modeling task linear quite general result mostly realistic assumption may sometimes behave preference situation agent class error good output definition compression imply section learnable agent behave fraction pricing always apply broad price amount bundle example volume arbitrarily unit non item offer demand learnable necessarily efficiently situation prefer preference option problem permutation vector represent hypothesis complexity h see p learn reveal preference bit optimal linear bp concave formulation tucker kkt xx dual respectively bp simplify price derive utility per spend good price per unit per segment segment clearly completely segment allocate dx dx follow bundle drive price achieve unit good utility bundle get kkt class utility together feasibility h figure bp since enough bundle suffice bit numerator next calculate preference determine preferred bit upper bind extra initialize bp else round else round rational learnable reveal preference learn length segment make jk last segment learn budget sure segment segment price jx budget appropriately extra p bp return bp else denominator else j j bit good learn apply procedure learn swap good call segment sample complexity learnable preference surprisingly behave h ax bundle amount show price budget enough preference bp like ratio evaluate get optimal give budget show price learn reveal preference suppose learnable query acknowledgment grant fa grant microsoft fellowship award review establish recent part every compression compression agnostic outline compression consist subsequence h class admit compression learnable agnostic chapter learnable sample complexity satisfy learnable agnostic follow hypothesis low hypothesis price bundle bundle good utility pp see different utility two stand segment utility segment length priori utility reveal preference length candidate note segment accord order bundle segment fully bundle admissible bundle bundle demand mapping l follow immediately know segment learnable efficiently preference argue computational remark admissible bundle admissible bundle bundle admissible bundle optimal bundle h bundle sort segment section ensure segment thus admissible optimal bundle bundle allocation last good bundle good segment like decrease allocation finally good allocate second bundle segment class demand yield utility vector utility learnable preference sample class reveal preference statistical observe bundle relevant standard distribution h learner output value learnable multi implicit learnable sketch note point w value exactly set subspace dimension collection correspond vc every consistent suffice new system utility length linear segment create example xx xx learn predict employ linear xx define class learnable recall ax equality least utility going maintain algorithm xx utility define prove successful claim characterize example bx minimize minimize side bx minimize side define ax get algorithm successful utility function contain utility b since return requirement x sample ax probability algorithm output learn efficiently query set bundle output utility query output function since show suffice learn h ax query enough decompose constitute segment length segment slope segment easy give learn segment last thus check segment else else maintain inequality correctness separately learn total learn query x function function every kx nu proposition theorem claim recent line start learn reveal past price produce agent line sample number class draw connection advance tight solve numerous generalization ability preference utility common assumption economic choose bundle decrease classical reveal preference economic begin seminal traditionally explanatory generate finitely many seminal work algorithmic construction linear monotone note agree imply necessarily recent line explicit formal agent constraint produce hypothesis forecast future agent utility monotonicity concavity probably approximately utility importance focus class concave commonly sample complexity include linear separable significantly expand establishe connection reveal preference learn advance intrinsic compression yield believe variety game context establish connection e recent computationally tight guarantee utility price reveal preference setting improve concerning actually reveal structure much specifically independent linear class powerful generalization immediately necessary important reveal preference theoretic term agnostic set target fraction accommodate include readily preference power instance exploit optimal kkt order ability desire exploratory purpose able predict price point analyze utility reveal summarize reveal preference rp omit previously c rp value set set price price say intrinsic th amount bundle price compute decrease utility budget bundle utility preference bundle let bundle follow problem assume optimal optimal break utility define vector function class utility type utility function paper price utility learning assumption simplest reveal query h learning response oracle reveal preference utility due certain predictor learn term linear class scheme roughly produce thereby section class preference cast hypothesis class w yx x element tie handle tie remove distributional support machine xx www compute reason exponentially constraint violate scan svm h complexity I w learn h complexity return w w find well hyperplane maximal maximize quantity maximize unit hyperplane q since utility cast linear throughout section generating tie bundle respect utility generating reveal sample show class imply demand learnable reveal preference sample linear bundle always bundle also essentially price bundle decrease call
form available inverse resort sampling offer alternate methodology posterior accelerate gain address problem discretization scalable main contribution dimensional formulation aim minimize expected structure pde infer coefficient pde elaborate sensor pde expression assess computational objective evaluation optimal demonstrate scalability adjoint pde dimension seek average trace prior expression covariance independent problem form available covariance formally would problem cope data trace covariance possible expression computation sampling expensive applicability posteriori functional minimizer map posterior map approximated linearization approximate specify pde determine map describe objective trace operator operator address bayesian datum collect absence sensor place combinatorial problem assumption weight use inexact cg vector gradient objective efficiently adjoint derive lagrangian formalism elaborate infer coefficient pde interpret problem pressure log minimize comprehensive optimal design assess quality bayesian inverse design design show design design sensor adjoint solve numerically scalability flow setup th material solution dimensional hilbert borel borel operator adjoint trace say dd paper field measurable pointwise variance satisfie trace proportional average variance formulation infinite law model prior strictly self adjoint trace differential operator laplacian dimension trace endowed element dimensional denote likelihood probability experimental require forward typically pde follow q thus bayesian posterior describe parameter condition relationship bayes formula side measure observable finite pdf maximize extend ball finite dimensional minimizing follow argument point deterministic description depend experimental challenge describe experimental sensor measure assign negative location w determine location inverse candidate sensor regard dependent uncorrelated diagonal n likelihood statistic classical formulation probability candidate e sensor weight large experimental repeat place place sensor place optimization solve relax enforce binary weight penalty trace trace trace monte accurate trace covariance description estimator use randomize trace implicitly operator possibility estimator vector possibility gaussian random standard entry computation trace operator justify infinite analog operator condition adjoint trace eq moreover appendix carlo q take problem map additive depend denote covariance measure minimizing denote gaussian low observable map evaluate pde control sparsity various option possibility use strategy present optimal design nonlinear problem tractable approximation likelihood follow minimize infer vector follow experimental still experimental experimental general distribute inverse parameter describe average observable additive namely inverse nonlinear close posterior technique markov variance statistically observable computationally extremely observable map consider gaussian approximation design realization hessian newton explicitly implicitly function gaussian approximation experimental admissible design trace integration infinite integration replace q moderate later draw model enter hessian incorporate physical mode insensitive highly indirect dependence objective enter directly ik map hessian operator I trace orthogonal trace see tractable design approximation formulation problem corresponding penalty assumption differentiable elaborate experiment inference surely bound ensure space weak read subsection variable inverse whose formulate characterize hessian section optimization derive adjoint equation evaluate forward pde solve weight cost eq require minimization pde constrain minimization variational approach optimality condition functional multiplier emphasize formal lagrange multipli minimizer variable vanish weak weak form adjoint equation side equation solve q satisfied practice require coefficient field problem field note pde characterize pde note order order hessian application leave pde pde constraint pde describe hessian evaluating satisfy solve problem efficient follow adjoint variable pde derivation rather defer simply hadamard hadamard obtain right side hand side operator identification adjoint coincide describe respect exploit computation problem implementation easily perform computation ht trace compute ik ik ik discussion qualitative evidence term forward adjoint incremental agnostic forward pde solver employ solver pde pde solves dominate algebra negligible scalability like pde solve sensor pde scalability dimension identify hessian hessian solve covariance represent hessian inverse rank parameter dimension independence observable map denote grows initially contain insensitive mesh refinement solve objective inexact inner conjugate iteration cg mesh invariance cg operator compact cg turn forward pde pde newton cg newton approximately solve cg system computational measure pde evaluate mesh compute pde solve pde solve cost evaluate like pde observe side hessian application hessian pde argument quasi newton make newton problem dimension briefly comment enforce use problem solve cope convexity function guess subsequently penalty proceed precise function attain section section refer pressure u flow drive pressure bottom figure truth velocity th pressure state estimate compute prior square point allow covariance three correspond finish inverse available sample datum trace configuration design configuration solve deviation prior field study effectiveness design respect sensor conclusion design sensor randomly design indicate sensor entirely inverse noise square cc height axis e ylabel style north east red mark mark tr cloud txt color pt scale axis legend style font nod legend pos north east mark cloud txt mark marks table tr opt design red dot design blue dot figure respect effective try recover underlie truth address conduct effectiveness design sample get fm average see problem assess base design indicate compute expect point scale axis xlabel ylabel legend font pos east marks mark pt txt color mark size tr opt txt height xlabel node legend color mark pt tr color mark mark pt tr txt expect dot blue panel examine method location increase specifically solve adjoint pde building method discussion cost inner inexact newton cg cost report total cg outer numerical insensitive candidate interior newton problem dimension see insensitive sensor axis legend style north white mark mark size inner txt cs black mark x scalability txt width legend style north blue mark pt scalability txt color black mark size outer scalability txt width height axis xlabel legend right legend pos east mark scalability txt height axis xlabel legend style font pos north east color mark mark thick table ns scalability realistic comparative description physical field slice three four production corner domain production pressure corner impose boundary circle homogeneous condition remainder boundary field velocity pressure obtain solve ht b denote choose black center pressure obtain solve equation construction problem assume point corner production well boundary compute regularize tb b inverse discretize triangular finite freedom grid candidate location compute draw give estimator six method residual reach maximum bayesian optimal sensor truth triangular fine mesh degree freedom record pressure sensor datum subsequently use solve problem sensor assess effectiveness sensor use randomly design note width height scale axis ylabel font legend north mark mark mark txt color mark mark mark pt cloud opt txt scalable design nonlinear problem scalable measure pde additional forward represent hessian outer determine sensor optimally pde sense pde derive adjoint enable require expression hessian require medium compute experimental measure pde insensitive limitation define field case map accurate bayesian inverse expensive map challenge fact inverse inner limitation indirect sensor configuration problem appropriate pay otherwise problem tractable requires pde discuss characterize hessian operator solve suggest rank approximation discuss contain important grain solve additional iteration reason sample goal
chain point early problematic imply mdp iterate average mix convergence center exponential approximation efficient state expand iterate td available td asymptotically average td step dependency least temporal alternative classic concentration quantify bound without mix transition mdp mdp action function discount denote instantaneous reward action bellman td standard provably curse associate high dimensional linear approximate incorporate td incorporate markov moreover stationary markov feature column matrix eigenvalue satisfy assume nature make td assumption counterpart see upper convergence however upon problematic choice knowledge would imply state matrix latter draw white minimum em fill circle node thin auto align fill red right align right update gray blue center leave center block fill center align line join large combine average iterate stochastic convergence without constraint size exponent arbitrarily although constant remain minor choice would bound thus average dependency convergence td exhibit optimal iterate td algorithm follow fix iteration solution behind increment order larger iterate previous increment x f td finite sum discount make policy centering use td stationary mixing term affect finally fix sgd center difficulty overall start epoch epoch epoch update q mm td conjunction iterate exponential let epoch ex particular policy follow mixing hold mdps mix exponentially fast second mixing dominate first rhs reflect know dependency would choose split involve iteration first state depend fact td arise provide proof f rs x bound require complicate td project reader refer order present expectation q inequality bounding theorem mean martingale step establish lipschitz constant detailed show find bs integral td evaluation asymptotic convergence derive linear function bound high optimal choice td fast variant td incorporate center rate fill rectangle height em cm black ns f rule martingale policy plugging mix jensen eq equality deduce martingale algorithm rewrite sum sigma lemma ingredient invoke rs ns ng ii constant return instant equality two expectation q consequently bs schwarz conclude martingale function bound lipschitz eq obtain note regime choice bs c bs bs comparison integral bs q equality page detail last take give nm center solution epoch proceed rewrite sum epoch note I recursion obtain prove pt pt pt provide asymptotic know probability optimal knowledge underlie problem employ scheme convergence knowledge furthermore center establish process mdp rl solve mechanism discount hope function approximately constitute e actor temporal td evaluation sample simulate td representation entry state curse trick linearly parameter efficient td even rule td incorporate approximation
outlier occur overcome robustness enhance nearest knn subsampling cloud preserve topological outlier denote near average distance neighbor density cloud extent filter cloud persistence tb subsampling cloud take sequential pre sample previously select scalar persistence diagram confirm persistent subsampling another cloud colored environment selection persistence interval distance rough signal let sort decrease power n l snr cloud average use measure snr great snr summarize however cloud construct homology homology birth death persistence interval extraction persistence length scale invariant th deal indicator space topological optimization one interval space persistence diagram correspond length environment result surprising quantile density outlier classifier variety contain either investigate performance use negative scale environment environment whole justify degradation third retrieve point cloud metric topological environment mobile sensor whose enhanced piece information cope uncertainty cloud extraction persistence diagram system quantification uncertainty also work extract feature cloud exploit precise email paper inspire network motion mobile node inspire motion utilize extract weak build environment spatial feature manifold environment extract dominant topological persistence interval topological improve robustness outlier density base subsample employ diagram provide representation strategy agent enhance sensor network broad mapping attract lot decade mobile flexibility adapt environment model biological agent equip mobile sense formation behavioral distribute distribute requirement response make capability rough provide localization would fail computational extract require make localization topological persistent homology qualitative cloud take compact topological persistence diagram specific topological coverage stationary sensor neighborhood mostly static physical node network complexity look investigate employ homology topological mobile mobile hoc topology physical model topological map mobile sensor network sense movement expert localization retrieve status proximity point topological base subsampling use point topology extract persistence low dimensional structure cloud machine integrate persistence inference study classification persistence use construction organize follow present section inspire sense overview propose framework metric subsample feature finally discuss brief introduction present tb persistent homology way object connect space representation rank topological number cycle component sample represent cloud finite equip method topological cloud build ball vertex base pairwise complex persistent homology compute value class topological death persistence call persistence persistent homology represent stop stop mode short stop long stop characterize time model sense inspire sense capability combine wireless receiver body well boundary within radius equip unique agent occurrence furthermore able status rw state summarize exploration environment probabilistic motion describe sense power environment purpose environment free instead deal directly information base process build construct motion circular region extract cloud computationally expensive cloud possess topological persistent homology ordinary interval computation persistent homology use robust extraction cloud visualization purpose scale projection cloud tb assign network coordinate represent tuple construct undirected vertex exist encounter time limitation available base assign rough connect tb due agent stop detection redundant place beginning proximity occur inside agent exploration cloud color resemble corresponding cloud justify due accumulation uncertainty long period make environment could figure produce collect help estimation make worst short capture correct impose
symbol calculation ignore coordinate central word assumption expression ease exposition even independent moment relation calculation mean u independent rewrite eq calculate integral definition distribution integral taylor expansion final definition kind let term section term consider term appendix second section variance derive bind calculate abuse notation summation split different calculation expand term expansion use calculation obtain bind fu du fu du believe mistake notation paragraph power assumption
proof since get q showing hand schwarz inequality consequently tensor link q q triangular duality line combine immediately lead consequence let q adapt argument sharp deviation event define eq contraction contraction principle denote duality plug partial derivative sequence eq similar argument get mn md n term negligible finite nt matrix probability distinguish two use see therefore function dependence implicit packing construction inspired consider test first matrix replicate block matrix contain matrix construct pack satisfie e distinct kullback leibler either function get universal take value estimate entry work completion recover unknown real rank investigate take output generate recommender multi classification guarantee nuclear maximum advantage knowledge nuclear unknown minimax claim class arise application recover observation entry course proportion entry set framework amount least program trace value entry highly unknown rank voting preference survey yes agree opinion much completion observation constrain sampling consider random unfortunately recommender popular rate frequently important application yield fast rate obtain bit completion likelihood uniform unknown slow recently exponential family knowledge unknown nuclear penalization allow consider scheme unknown previous difficult matrix paper completion introduce establish bound minimax logarithmic factor bit extended alphabet coordinate recently introduce experiment entry order way tensor function simplex hellinger vector leibl parametrize coefficient reveal denote ease write instead likelihood regularization exist positive classical row constant q yield kullback leibler divergence universal constant logarithmic capture differ general set parameterize observation vector negative likelihood observation section function factorize binomial multinomial function ne k ny eq kullback divergence universal define give observation q without confirm negligible burden separable achieve coordinate describe support except equip value role triangular imply eq therefore equivalent implement iteration nonnegative support singular max soft computation step loop iterative proximal associate nuclear soft thresholding singular propose another interest evaluate entry actually completion iteration top singular value us advantage loop carry bfgs execution bit ram cache completion extend class consider comparison potential gain belong alphabet report article assessment obtain upon author
recent attempt eliminate variety approach level translate derive term relate language eliminate mt mt method corpora require aligned sentence extract simplify learn representation pair bag informative bag encoder supervise nlp label able reach achieve report bag sentence fix within learn word sentence present nonlinearity representation sum bag word encoder function decoder autoencoder bag encoder decoder must careful certain section reconstruction convert bag representation obtain encoder word linearity sigmoid tangent sum representation least decoder form bias reconstruction sigmoid training reconstruction training mini batch binary correspond vocabulary typically aim reconstruct since million training thus trick bag assume perform mini bag word bag mini batch result descent mini batch still produce reduce stochastic reconstruct bag efficiency training reconstruct whole dimensional previous work binary input bag autoencoder architecture investigate architecture directly firstly encoder frequency notice nonlinearity bag validate moreover assume output bag trial multinomial efficiently numerator normalize sum opt leaf treat internal root root right branch node observe bag decoder tree decoder compute logarithmic course bad decoder assignment finally need parametrized decoder linear branch logistic encoder bag language sentence language align translate similar representation sentence reconstruction language specifically language specific language word word bag w w similarly autoencoder encourage language language decoder language decoder reconstruct form decoder language specifically reconstruct loss follow propose embedding learn correlated term specifically optimize correlation encoder factor ensure either bag reconstruction autoencoder le translation horizontal line across input hide highlight parameter similarly language word tf obtain document describe representation word train simultaneously encourage aligned embedding use form neural network language work investigate rely word learn phrase mention separately skip language align network phrase phrase base phrase level embedding capture cross word follow follow interested corpus importantly corpus learn language successfully language language embedding corpus contain unlike pre sentence align language interaction induce embedding english corpora document pre hierarchy economic market contrast learn interaction learn embedding corpus en de autoencoder embedding reconstruct effect procedure language experiment topic document topic document process use en de de pr pr pr believe office wish report shall month year agree microsoft en en de de microsoft microsoft markets competition competitive exchange business material procedure summarize follow train extract language validate train document language document language represent document averaged perceptron train epoch epoch contain word tf combination early cr error training use reconstruction unlike cr section perform representation epoch use word cr cr merge mini batch adjacent sentence hyperparameter select validation portion discuss like qualitative learn perform english word english word term embedding cr show english word actually also notice semantic word show supplementary visualization embedding language describe compare learn neural encourage aligned embedding mt test document translate mt mt default parameter induce embedding every class summarize result en de en observe tr comparable embedding language rely correlation indeed cr de classification english en en tr cr cr al majority evaluate effect vary amount training classifier either tr en en summarize observe cr remarkably meaningful embedding even size excellent merge mini single significantly word rely essential effect use batch cr cross surprisingly tr decrease use mini batch size even en batches de en cr english
find match summation procedure exhibit bad without generality w unique probably stay dimension dimension unseen kx I use assumption bind depend time oracle combine euclidean constraint minimize n inequality classical nesterov fy fy l x fy x variants b eq fw nf w fw fw simplify lyapunov straight note also step standard q apply n give q nf change w I f nf k nf nf nf nf I simplify change equation product simplify w sn w f sn grouping remain term nf sn f expectation product sn w expectation straight forward n sn w I f w w f w expand term w sn f notice inner product recall simplify expectation sn iw w inner define constant convexity eq sx l l fy fx fy l f note fy f holding
comparison excellent benchmark reinforcement ensemble appendix besides split thus value respective medical drug schedule drug pi despite success drug maintaining associate long attract optimize drug lot structured patient alternate cycle successful regard protocol schedule sequential decision action correspond type simplify formulation pi amount reduce pi load little possible follow describe identify validate clinical variable describe patient problem generate also follow author briefly base approximation greedy merge select action pick experiment varied round simply representative state begin cost kernel drug schedule large appendix state state reason transition simply figure drug schedule show decrease contrast solution case schedule correspond unable reproduce confidence interval substantial associate illustration run require fact transition tree curse hand hand increase gap time complexity average verify implement independently single agent determine representative uniformly random state benefit reduce cost agent vote tie break seem boost show agent confidence fast conclude mention experience confirm stable method solution hand solve transition line line problem involve sample computational prohibitive demand empirical evaluation use structure frequency effectively experiment fix equally effective across system treatment potentially replace regime scheme reinforcement problem generative develop base collect reproduce original dynamical validate policy brain slice associate linear apply electrical total transition result clinical policy apply electrical frequency hz hz hz run sparse modification make representative state time reasonable use varied overall characterization value vary penalty associate electrical fix latter name seem expense occurrence solution policy hz regime know date clinical able time method draw one characteristic apart complexity transition ever access entire sample modification additional reinforcement determine transition set undesirable memory domain impractical incorporate ignore usage inefficient limitation action split improve clarity suppose transition index eq q want know simplify write reasoning derive know update discard transition store vector overhead recursively far subset fully incremental computing step transition drop transition update ij w I instead policy learn transition thus integrate planning algorithm cycle sample transition algorithm counterpart incremental version recover batch transition approximate matrix state execute state update select base ia allow inclusion representative state representative state suffice application update dynamic inclusion state refine think way strategy proposition impose allow sample near representative add experiment incremental version build require tuple keep extend transition reach time value see note current may apply address issue bind computed transition process use add stop algorithm show value space action space discount factor similar apply application base show think matrix iteration solid theoretical state use let encounter q value triangle q value ts slight abuse mdp hand see apply write ts contain incorporate ts ti ti resort step fact action factorization implicitly possibility value either completion apply bellman short restriction somewhat circular look empirical task match store exploit scalability task balance performance reinforcement use model batch task action show result result policy gradually go sample intensity process important policy transition confirm demand memory small size approximation processed show overhead strategy normalize v transition collect see discuss balancing address simultaneously challenging figure balance scalability bar perform adopt control start triple balancing exactly refine incorporate sample transition grow representative state add agent set close representative balance adopt section fix benefit incremental algorithm sample transition problem always subset option limited amount batch transition approximation greedy strategy compute compute show triple balance task amount insufficient describe control transition batch note amount complexity allow large use transition take hour time detail allow success cf triple balance task especially policy direction approximately memory argue fair latter amount former batch exactly process computational cost train phase look opposite trend surprisingly performance computing grow process beginning grow fast visit datum magnitude correspond page cr one fix neighborhood experiment compute close representative guarantee adjust advantage adjust cf experiment compare quality policy broad reinforcement huge body literature broad overview book narrow attention start smooth kernel essentially implement approximation kernel implicitly inner two framework reproducing relate discuss smoothing reproduce roughly rewrite term product properly kernel mapping apply reinforcement value approximation alternative mdp slightly propose applicable weak reproduce attention technique closely function see similarity consistent assume approximate assume show function surprisingly form mdp transition contrast transition numerical demand see build method small bellman transition applicable reinforcement kernel operation grow adapt start exploration completion may computationally feasible resort propagate exploit use whether exploration overcome difficulty underlie link transition use kernel limit case transition subsequent ignore work later guide scalability first simple potentially reduce computational burden user suggest representative state among sample infeasible problem resembles define mdp representative algorithm come hard aggregation row rewrite formalism element infinitely narrow close representative easy aggregation practically place would sample row compute coincide cf property instead factorization trick contain build strength mechanic incremental make transition code line sound theoretical value compute underlie show arbitrarily small desire translate build difficulty arise practice successfully different present reinforcement list believe become valuable resource reinforcement possibility future algorithmic demand principled method procedure solely far think kernel advance elaborate exploration regard integration broad investigation principle amenable algorithm complexity sample see understand however equally ask whether benefit view sample need achieve grow exponentially dimension way avoid exponential dependency sort regularity break curse incorporate think may cast incorporate whether impact sample interesting question investigation one reinforcement build model sample resort potentially useful reinforcement x b satisfie exponentially assume great ensure set sample transition experiment obvious property let w follow suffice z otherwise provide intuition magnitude impose region make possible adjust exploit allow accord factor difference understand small threshold considerably even difference ensure sufficiently small second influence size term plug let strategy q know w k necessarily yield obviously inequality regardless though analyze show sufficient multiplying define apply let true remain show recall eq inequality otherwise rewrite resort guarantee hold guarantee make finite mdps ia column ij p ij ij element analogously resort obtain lemma favor course bind trick property first less depend contraction mapping draw derive theoretical generalize mdps bellman think resort conclude desire contraction map derive upper fix point bind valid think notice bind derive theorem operator vanish trick reduce assumption order whole behind create zero nonzero element small mdp suppose define equality long infinity world world model discount transition reward result near agent reach goal grid compose balance simulator thesis use task angle vertical plane episode reward fall past cart reach track locate step comprise equally correspond hypercube origin cover axis velocity cart angular balancing simulation use adopt version length mass policy equally drug schedule system ordinary euler action discount parameter numerical suggestion existence monitor drug day sample initial day later reward drug select patient report policy value sample day correspond describe dynamic generative develop task label five slice hz hz hz hz manifold turn give rise therefore reward event model policy episode start simulator simulator adjust parameter accurate normally keep agent position dimension break try several pick result task episode start position kernel transition largest large find near outside specialized avoid store list algorithm modify policy compute approximate table across action decomposition ensemble split general increase task consider cut particularly effect experiment varied adopt rl algorithm evenly trial preliminary fix define technical report school university discussion regard relate subject make simulator national da discovery mm mm mm b c j z bt bt bt z z reinforcement conference manuscript substantial approximate learn theoretical statistically construct grow paper turn reinforcement idea transition product swap factor potentially much insight build difficulty transition discard incremental make result reinforcement regime compute resource would set potential difficult world significantly reinforcement reinforcement learn conceptual long artificial intelligence construction interaction among particularly persistent obstacle recognize real must realization come across reinforcement incorporation difficulty last two decade collective rise reliable stand mean add always eventually unfortunately good theoretical property since transition policy become prohibitive burden limit applicability nice present algorithm transition stochastic swap factor obtain potentially small fundamental exploit insight contain word whose construct become structure extra flexibility approximation take account find construct transition memory number property make regime study real never also sound theoretical view bound compute compute also appear bound solution present factorization trick divide theoretical one control bring reinforcement single double balance drug incremental follow extend line triple balancing discuss present guide reinforcement summarize related context conclusion research possibility try maximize reward interaction environment discrete state must choose finite set action move select certain reward agent policy maximize return discount reward future mdp mdp tuple task hand action transition mdp search resort theory dynamic agent policy e notion programming perform worse know policy mdp one action represent become throughout conventional letter capital letter dynamic ia fundamental programming expression give dynamic reinforcement mdp transition environment transition reinforcement learning use finite solve continuous reader think also kernel finite solely give occur occurrence state finite mdp reward eq dynamic mdp width admissible suboptimal discuss dynamic compute transition state bellman computational want much dynamic allow follow section importance objective serve rest factorization mathematical explore briefly slightly modify useful element artificial artificial direction element transition word accumulate artificial similar switching probability compact version idea cccc represent big white circle symbol use immediately surprising fundamental characteristic recurrent irreducible irreducible regular regular insight stochastic factorization transition factorization property strong idea former motivation save resource possible trick reduce summarize idea programming transition matrix obtain mdp solve scenario convenient mathematically obviously apply stochastic factorization trick unfortunately computationally demand approximation mdp stochastic mdp norm ij upper theorem mapping identity write hand say development tight bind factor mdp version classical state deterministic reduce sense factorization trick rise basic mechanism trick number mdp exploit function adopt show trick reinforcement construction main factorization trick mdp long computational impose calculation leverage component trick similarly kernel list assumption suffice assumption mutually build ccccc ccccc b ccccc ccccc b b b ccccc ccccc ccccc b cc b cc b b matrix transition occurrence apply mdp strategy note easily illustration conclude construction expression mdp depend state recall interpretation representative illustrate formal mdp discuss transition define state dynamic degenerate case change change define representative particular rise conversely transition mdps depend representative state understand implement work value dynamic return compute simply input lr return key mechanic require bit version construct cost become instead application bellman complexitie linear computational requirement kernel kernel region avoid store result compute occur become reasoning look thing transition representative computation action transition occur formulation practical would generative transition provide interpretation construct action stochastic continuous focus define knowledge function computation training kernel weight compute
inverse covariance estimation receive community use especially dimensional arguably penalize matrix covariance penalize advance effort literature focus develop principled increasingly simply literature review seminal work recent paper conference maximization restriction address gap pseudo establish sense move practical recent minimize cyclic descent update hold hold wise algorithm converge minima equivalent much ten address important propose rigorous associate method lead massive extremely fast principled solver outside notation diagonal term propose scalable thorough advance optimization derive proximal efficacy model fista treatment investigate popular minimization iterative gain popularity seminal backward splitting nesterov thresholding essence proximal divide objective part nesterov accelerate extension combination momentum accelerate section composite part vector zeros k matrix matrix initialize j j wise soft entry algorithm accelerate constant iteration fista choose search reduce iterate section implement three feasible iteration heuristic proxy information gradient fista multiplication zero extreme use w ij os operation complete operation inversion optimize allow parallelization contrast fista perfectly multiplication dense machine high restrict like provide proof coordinate algorithm argument convergence essential belong base function state bind hence bound start semidefinite matrix simplify arithmetic ignore constant theorem depend ki eq go hence function positivity trace function value element continuity remain sequence generate backtrack k solution backtrack set backtrack fista iteration outline comprehensive numerical give comparison wide result breast make wise implementation algorithms library fista implementation increasingly large scalable library make work eigen library algebra c name various step dataset non edge sample size guess criterion wise implementation highlight comparison supplementary material synthetic two little difference marginally fista two coordinate perform attribute fista fraction propose method fast coordinate time coordinate behavior fista perform axis converge constant fast appear perform fast fista fista rr nz second rr nz c sec sec dataset arise physical science gaussian outlier hence characteristic assess breast et univariate cox patient survival gene breast reduce gene often algorithm wise especially due newly fast fast gaussian graphical estimation advance propose gaussian inverse rate fista thus far compare fista coordinate demonstrate outperform coordinate wise general outperform wise magnitude test set comprehensive examine effort similar appear one several thorough contain relative rgb rr rr second second rr rr second rr rr c c form kkt involve term definition rewrite
em vb miss result conclude method implicit perform outperform miss auc auc auc vb comment vb em vb em em vb set cp miss average run instead pick model negative consist start view global optimal solution factorization advantageous iterate find recently co factorization perspective naturally conditional approach approach lead improved department engineering ci ci university approach aim structure constraint successfully paper bayesian couple even exhibit several world miss factorization modelling advance propose lee one factorization meaningful modelling field include recommender bioinformatic suitable variational bayes vb probabilistic appear incorporate arbitrary rule give individual factor mm indice product collapse index factorization one step multiple suppose distinct index specify attribute variant tensor relational heterogeneous datum large class tensor contribution variational exact characterization conditional richer naive formulate entirely term low tensor predict entity try kullback divergence since analytical intractable bayesian acyclic dependency write px pz leibler symbol gamma factor preserve conjugacy framework degenerate distribution px x miss define mask observe missing handle smoothly follow notation short tensor value refer particular element generative fix update via em convenience representation force sparse representation write mm stand compact kl method em compare vb quantity average q integral several deterministic approximation approximation method introduce attain maximum log set induce approximate become easy approximate hide intractable resort propagation intensity rather formulate posterior distribution mm q variational log indeed regard different computation mm vb resemble large couple observation available couple tensor side sharing method incorporate knowledge additional tensor set configuration distribution couple vb sufficient expand log drop irrelevant z miss link factorization link performance evaluate tensor vb low rank tensor compare vb miss use implement variational equation vb vb scale extract include type entity activity equal user equals collect preference gps trajectory datum location respectively aim link side link contain link may activity number link collect show social news resource user comment news lin action comment explicit contact extract five user topic among illustration study comment r r user tensor relation compare comment comment user comment comment ex ai ai dm k ai dm en tensor array couple update mask sparse storage instead work enable specialized storage cost roughly scalability vb vb tensor sparse value array miss cp cp kl extract reconstruct solve times ten rmse second slow mm mm vb
svm f margin step use svm package include solver follow divide thereby example slack objective package additionally recent reformulate form example elaborate transfer interpretation transfer aim different analogy slack margin classify hyperplane I back sample slack slack svm effort svm effort svm mean ignore hard concentrate satisfy question answer oracle answer first second margin constraint satisfy ignore one return margin transfer low vice lie oracle negative regression slack come model slack function validate space margin transfer explicitly predictor sample hard classify class ignore pair sample amount every consider three different showing handle attribute bound present modality subsection analyze propose margin transfer compare ordinary svm access accuracy report joint choosing cross validate range normalize range fold fold validation multiclass complete training thorough couple modality exploit annotation incorporate description shape concept introduce attribute classified default attribute provide dataset class attribute texture attribute predict binary classifier image statistic time transfer image image cat versus versus versus cat versus versus cat versus versus versus versus cat cat cat cat cat cat versus versus versus versus versus versus total svm highlight blue significant confidence additionally figure utilize object svm transfer information svm able information bar coincide mostly margin transfer high regime problem check consistently high original translate transfer margin hardness modality fulfil box annotation design object image object level know exact annotation available ball activity class image ignore box image group ball ignore uninformative annotation one dimensional bound representation latter accordingly draw amount sample remain similar double statistic repeat pt image green green highlight significant utilize provide margin ball ball highlight blue indicate improvement utilize information transfer pair svm column see utilize bound box grain outperform baseline svm case experiment method exploit margin transfer group margin transform respect standard bad margin rely space ability hard scenario hand complementary turn word image sample symbol pair normalize extract image bag split second introduce variety product group broad binary per contain text description text advance word vector instead term extract neural skip gram codebook word convert sentence normalization normalize descriptor visual codebook c transfer reference text versus versus c c transfer text text versus bag versus bag tie bag tie tie versus utilize equal utilize performance transfer description capture mainly preserve explore utilize setup versus strategy binary sample sample class maximum classifier model selection cross classifier cross validate performance good task l transfer attribute ball additionally reference method outperform dataset transfer svm contrary dataset tendency outperform column reference column collect annotation easy hard reliable rather make objective lack variability distinguish easy annotation study set particular ask select prominent proceed obvious difficult compute score range observe object size human annotation look easy proceed transfer pair hard human identify correlation learn correlation sample hyperplane easy hard score similarly predict train space space complete user easy score hard original entry coefficient c transfer transfer human versus versus cat versus versus versus cat versus versus cat versus versus cat cat cat versus table collect annotation space versus see figure datum space human space closely explain human attribute classification versus strong annotation utilize information suitable explore attribute description compare annotation little blue coincide rather comparison attribute information like performance classifier lot misclassifie wrong hyperplane margin principle human study set handle improve utilize multiclass two approach margin transfer transfer model hardness guide essence annotate study future explore direction european european framework technology introduce learn computer computer recognize want computer fast expense additional image look scenario study vision able binary multiclass interpret hardness object train thorough analysis space incorporate information user study hard learn inspire teacher teacher new teacher explain answer solve teacher generalize machine introduce successfully apply attribute annotation digit description gender resolution source metric back help want category label object plus direct yet representation annotation image useful quality extend publication examine different information semantic specify localization image context modality understand test evaluate core also easy enable define identify sample easy incorporate encode hardness formalize technique svm new contribution transfer analyze predictor report additional handle situation unified contribution section method naturally conduct experiment object utilize method attempt useful problem common access sometimes visual represent feature texture modality assumption extract accordingly space classifier would datum possible vice describe characterize sample space case know towards generalization quality margin interpretation simplicity notation linearly separable support turn linearly separable hard svm call soft fully slack example increase margin svm sharp soft hard answer
distance stem taylor expansion note bregman triangle inequality square dissimilarity divergence risk error choose stress mapping bregman divergence deviation latent usual smoothing spline analysis spline endow situation inner radial metric spline non description manifold due datum hz prominent target ica prominent noise raw remove subspace method follow system perspective scalar observation latent evolve observational dynamical embed extract channel source describe describe use filter outlier highlight note observe point reflect signature original normality hence highlight operator euclidean apparent interesting use dissimilarity segment spectrum channel temporal ica processing remove obvious subspace dissimilarity prototype determine select select dissimilarity psd channel point locate channel locate axis channel present range small dissimilarity target present simulate take dissimilarity view potential improve provide augment automate system normal investigation dissimilarity approach interesting measure distance situation augment pt corollary purpose environment response requirement decision reduce measure water prototype exploit dissimilarity representation mapping system analyse map euclidean non make concept use realistic target sound extensively fundamental commonly ratio surface high produce hundred thousand worth display result conventional observation observation result unknown preserve structure original preserve projection capable mapping subject represent accommodate content ie integrate anomalous behaviour track measure dissimilarity assume isolated entity nonlinear metric represent need even dimensionality reduction seek purpose transformation dissimilarity preserve employ nonlinear network parameter adjust classical scaling method transformation often x prior dimensionality often useful f nonlinear radial weight distance euclidean may
formulation characterization symmetric handle minimum cut undirected half cut denote represent pairwise respectively characterize undirected graph characterize iff label equal constant characterized fig characterize undirected correctness due optimize v x u graph contain indicator represent variable cut otherwise characterize undirected conclusion introduce next fig general unary general besides denote node undirected convert symmetric represent capacity construct characterization minimize transform minimum know minimize go introduce important equivalent transformation study literature convert submodular instance x x unfortunately benefit energy kind characterization switch denote conduct indicator graph characterize exactly indicator set indicator second type mean characterization indicator capacity capacity change uv uv uv uv flip original algorithm existence design minimize submodular function extend general undirected provide convenient sum part undirected check positivity sup optimality efficacy possible minimize influence transformation fig equal dense efficiency equally paper I derive absolute maintain record ratio negative order flip accordingly repeat undirected iteration decrease exhaustive search influence part submodular test guarantee globally could produce labeling well propose greedy call equivalently replace contain propose iterative refinement initial labeling experiment consist modular permutation modular modular function part modular result st computation iteratively refine permutation lead minima avoid permutation situation graph label modular approximation generate large need simplify energy refine variable initial approximate st graph minimization labeling iteratively generate labeling convergence optimum besides automatic submodular enable global optimum except iteration undirected formulation vertice characterization experiment within small fed initialization code publicly due energy study computer vision potential necessarily energie primarily hardness factor hardness combination technique energy hardness future energy use energy factor systematically performance energy large energy efficacy synthetic energy rule purely focus evaluate optimization energy form randomly produce empirically hardness variable node uv uv respectively curve combination minimize variable random initialization initialize bp unlabele lp bp energy bp clearly method bp fast appeal bp attribute unary fig solver configuration maximal iteration solver make I high energy note thus obtain energy minimum comparative unary experiment thorough cover connectivity unary test plot different value vs curve six connectivity unary energy understand computer vision study energy see speed outperform initialization worth bp usually dense energy variable label tell always low much test configuration certainly label worth use label seek labeling preserve cc connectivity c weak unary equally initialization unary usually obtain unary reflect unary pairwise potential computer unary cause unary pairwise potential hardness chinese character specifically image train prior character uv uv x vx pixel labeling either prior average fitting minimize u fig show result bad energy find lp similar thing able energy long time initialization local initialization contrast much slow speed comparative help hardness future direction importantly promise perform energy summarize generally efficiently correspond properly namely dense kind energy great vision enable extend submodular solver labeling thorough comparative find connectivity unary closely hardness reasonable recommendation energie vision several extension optimality still iterative desirable analyze besides combine future furthermore plan finding available energy vision besides plan function computer vision mrfs program excellent national nature foundation china national computer demonstrate dense function solve problem energy none technique cut bp individually efficient mrfs potential comparative recent
life size vi split subset subset present require hidden act bit output require bit relatively reconstruction bit country popular rbms model categorical unit poisson beta rbms categorical employ rbm attempt present ordinal sufficiently especially seminal hand recent numerical modelling variable except treatment variate model previously statistic name mix variable underlie need correlate handle hundred scale single know previous include share capture probabilistic manner time know adopt category category rank introduce mixed variate boltzmann rbms variable multiple modality six information ordinal rank capable handling variety include demonstrate large wide plan multiple relate share posterior interaction rbm architecture able inter without go intermediate plan propose comment day country thing go think health education social know opinion pose greatest great spread nuclear environmental grow old last currently separate never simplification mean field define approximate kullback leibler ik ik complexity fast continuous recognition university modern become heterogeneous modelling modality option choice interpret particular aspect rbms variable naturally task include convert input latent posterior thereby reduction multiclass tool datum completion multimodal model large scale opinion survey feature extraction completion boltzmann attract task include multivariate deep rbm markov visible represent interaction extraction nonetheless rbms visible variable gaussian extension ordinal poisson good rank type investigate modality example survey person ask range statement answer response vs categorical iii choice g iv ordinal assessment vi answer response come person inherently american typical chinese concern child education however model among layer rbm suit undirecte visible thereby introduce variety away nature dimensionality reduction reconstruct predictive learn perform large international opinion involve people section mixed variate machine jointly ease include single visible variable bias variable parameter hide category category visible type continuous singleton energie th visible unit variate markov connectivity must moment g gaussian assume energy bias parameter datum energy decomposition assignment ph ph ph extract similarly follow q subscript emphasize heterogeneous equivalently functional continuous limit gaussian reader image functional provide binary u I I category rank thus simplification describe detail assignment person interested let category subset assignment empty possible moderate assign category indicate category independent indicator indicator receive individual I review expression treat simply ignore consider convert triple proceed variable ordinal lose use specifically define free set completion give parameter application typically perform visible variable belong family ph due space constraint empirical estimate expectation resort approximate eqs carlo sampler run specifically k use hand category I multinomial speed cd mcmc observe stop introduce estimate completion treat hide ignore paper follow condition variable variable translate conditional predictive learn effectively latter typically inherently easy strategy identical except e ordinal categorical exact evaluation optimisation gradient argue preferable effort may actually third generative discriminative objective one hyper control generative component manner ph learn predictive complete miss predictive inference unseen ideally unseen simultaneously resort special predict simplified categorical ordinal type simplification assignment indirect ia I ia I c I aggregate probability pairwise I ic I sort lead optimisation variable simplify q experiment large general world opinion publish period question obtain people subset question ask certain answer ordinal difference response zero variance detail particular user type bin u ordinal mi create baseline baseline ordinal category normalise scale
front box input work assign understanding segmentation build diverse crf train particular task even though nature front segmentation exploitation score crf base sense try segmentation move lie feature label image segmentation tree recently within pixel classification propose pool inter work employ advantageous decision recently slide fashion convolutional spatial issue outline introduction inter refine coarse difference unary focus close direction mechanism node crf cut ignore dependency treat crf drive study recently show efficient segmentation manuscript make densely technical respective focus problem material similarly evaluate semantic inferior interested herein publicly available imagenet dense segmentation spatial evaluation instrumental success dense cnn implement convert connect one original enough pixel score densely variation previously employ subsample modify introduce zero increase three first efficiently sample respectively illustrate wavelet transform framework add patch dense imagenet straightforward fashion last sum term position output weight target optimize respect layer test original image illustrate map correspond simple interpolation increase resolution negligible produce coarse score output force learn increase hour report train gpu ingredient size pre imagenet network typically large field field filter become dense score address spatially fc reduce zero fc gpu dense top testing sec channel fully connect dense rough less suited pointing exact localization deep model layer successful infer work direction localization convolutional employ super essentially segmentation successful recent recognition capacity remarkably address localization challenge produce semantic significantly speed sec recent explore increase boundary localization specifically max pooling layer filter second convolutional feature main feed softmax layer adjust discuss fine resolution layer improve crf benchmark consist foreground object original contain extra annotation term pixel class crf stage unary fine network pixel decay tune validate fully default validation fine parameter matlab refine size around round crf crf x crf crf crf crf crf crf evaluations set augment crf crf boost improvement turn qualitative crf crf features add improve fully connect crf denote crf fig leverage refine employ arbitrarily view adjust several kernel crf modification set slow improved reduce variant different crf crf latter input attain change crf match performance fast run large parameter crf crf crf crf ht quantify segmentation similar annotate usually occur pixel locate narrow fig exploit intermediate segmentation connect crf significantly crf state art paper able object extend excellent file web site vs crf model crf crf outperform art crf crf speed crf attain employ cat train tv crf crf x crf work combine convolutional neural random prediction detailed map advance challenging segmentation refine integrate train fashion plan apply source depth map video weakly annotation box far powerful challenge vision acknowledgment partly cs fp fp acknowledge support anonymous comment constructive feedback present reader crf performance add crf layer crf camera field view token google google california performance classification method probabilistic address task classification semantic segmentation response sufficiently localize invariance overcome poor deep response final layer field crf system beyond previous image task reach test show careful novel community response neural choice become level year computer array fine grained categorization manner result sift partially
point put contingency count cell contingency column denote fit observation column expect cell split fit elastic although default tune minimize observation adjust search response value pure stop run logistic elastic net terminal stop contingency cell fit observation column expect count c split split variable linear logistic space split take ideally candidate minimize logistic sub dataset intensive take distinct exhaustive candidate categorical split form ordinal select sub nominal search computationally cart due manually proportion success case well fail option keep split response split subset summarize sub simple regression l cx ia iteratively apply point depth minimal complexity prune cart terminal pruning tuning parameter tree nest fold subtree sequence predict response dataset probability predict validation calculate rule denote subtree q often determine variable importance term important adopt importance forest give grow accuracy update measure measure obtain testing measure important correction probability bias present allow predictor cart select much selection bias allow split split incorrect take guide separate square exhaustive cart allow split find preference categorical serve regressor frequency contingency categorical value chi bias categorical numerical possess degree employ bootstrap calibration selection towards variable increase numerical split reduce however close problem transform outer loop create multipli make equal detail sample response split candidate draw convert value numerical categorical numerical proportion necessary value value multiply variable great split effect bootstrap independent table predictor skew study categorical exclude fit simulation sample iteration iteration logit nan jump cubic assess simple regression bar regressor regression selection model see predictor categorical chance simulation cubic correct choose selecting quadratic bias apparent multiple bias select linear node unbiased select variable select split possess fit regressor select split much frequently third experiment bootstrap logistic regression option show spaced interval selection tendency variable display multipli always three table p p jump cubic se frequently accurately prediction apply compete various fold validation setting option include correction new stand prune se simple linear tree without correction theoretically recursive partition multiple default argument multiple fold validation se pruning penalty coordinate fold validation parameter categorical linear logistic set testing take categorical serve candidate internal proportion sample outcome predict probability fit index although scale free ratio range ratio prefer matter rest adopt percentile proportion misclassifie straightforward prediction area curve receiver operate roc binary probability assign negative collect dataset uci repository order total categorical cat miss removal categorical l range max private local pay people census observation education level education st th college school education education numerical max status description c armed force op service house sale description gender c capital capital gain min capital capital loss max hour hour work week country country origin c census unbalanced sample observation stack population among group categorical find country visualize united majority observation education group school college education majority employ private self people low relationship suggest majority white white top interaction categorical visualize distribution level combination categorical education degree consistently low education suggest interaction education show population worker among worker although actually high counterpart ht scatter capital capital loss heavily education level increase rest picture seem advanced beneficial explore census classifier use predict misclassification subset naive rate tree cm l error auto cn objective merely algorithm hence analyze census naive bayes dataset employ permutation measure provide cart early cart selection bias lastly offer prediction measure rank subsection analyze census develop modify exclude good simple grow cart partial pruning list validate htb l option terminal display structure root model split age go population send branch sale service provide high school education good chance make year capital regressor capital gain split variable circle indicate ht indicate red otherwise green circle furthermore branch capital capital include close branch draw terminal capital ht independent interaction depth grow tree drop program small se short option short tree regression leave terminal however model interpretation therefore terminal list multiple split status sale service estimate age consistent node mid nonlinear green circle indicate observation red otherwise htb c intercept education capital c ranking explore census display ranking misclassification capital gain education ranking score obtain importance present htb variable capital education age variable capital hour education relationship country tree importance capital gain education term determine capital capital nonlinear age list predict test glm default setting htb se se census glm perform misclassification naive naive solely misclassification propose combine partitioning call flexible allow predictor play role build tree option logistic model fit regression elastic penalty datum control selection apply chi square split exhaustive adopt guide bootstrap bias correction prune cart determine prevent compare compete accurately census dataset miss occur deal miss complete grow miss fitting miss algorithm affect factor predictor study multinomial ordinal logistic tree regression tree decision make logistic tree response nonlinear recursively multiple descent method estimation elastic net allow structure comprise graphical provide apply adjusted chi test exhaustive bias linear square elastic net penalty tree recursive partitioning model traditional latter tool logistic provide however logistic satisfactory fit detect nonlinear pattern space tree regression logistic partition split node interpretability smooth node build tree still tree stand split rest overview regression review describe split importance discuss bias make split correction term accuracy show application census suggest build method review build brief exist logistic regression tree widely response predictor relate code categorical categorical ordinal let ik rewrite mle solution score equation eq nonlinear solve iteratively log term calculate parameter procedure fit logistic coefficient odd nonlinearity interaction select diagnostic tool second check pearson statistic logistic especially follow chi distribution hard interpret disease dataset uci repository absence heart list gender pressure dl dl c achieve exercise yes st induce exercise rest slope segment number color reversible heart eight predictor compare suggest parameter indicator significant introduce transform regression simple numerical transformation selection result final eq significance due hard besides hand extremely inefficient method couple ridge least square introduce stand shrinkage selection lasso similar ridge regression achieve lasso coefficient perform elastic elastic possesse shrink ability like ridge method add elastic net logistic coefficient estimate elastic include ridge penalty penalty situation correlate predictor fast outer update quadratic descent least square programming package efficient coordinate develop computation recursively automatically capture nonlinear interaction response complicated category variable tree provide practice insight datum method automate aid fail rule cart develop cart improve aid introduce prune aid cart search towards follow cart employ split know one guide design cart stand separate guide obtain construct contingency residual discretized group split employ chi square test way contingency guide guide accommodate grow maintain guide heart intermediate branch predict estimate misclassification give heart classic mostly design set response classification build classifier guide se grow use cart find state cart terminal insufficient statistical develop cart logit fit terminal variable terminal node contain cart variable terminal cart logit logistic regression mix q fit hard interpret may predictor inaccurate terminal unbiased analogue guide piecewise regression perform split sign residual test guide regression employ trend allow effect trend comprehensive guide datum logistic option advanced separation separate predictor another logistic contain split nominal build employ regression cart pruning tree computationally intensive due partition algorithm parametric include similar guide separate variable fit apply stability fit variable instability pruning grow logistic tree split split logistic recursive pattern provide
use gradient variational covariance gps concave covariance compute variational requirement parameter compute gradient respect take diag respect compute p diagonal element negativity estimate hyper variational parameter fast variational computed maximize step respect maximum hyper take step logarithm summarize involve dependency variational maximize update maximize variational ascent guarantee complexity algorithm dominate inversion dependency improve performance inner loop considerable speed number dependency approximation output predictive test predictive softmax latent score label predict associate dependency also noisy rely result refine take label value dependency refine account successive use prediction separately assign q l lt ty l l ty l complexity algorithm convergence similar discuss sequence label processing base segmentation feature test compare field elliptical slice loss point hamming output table percentage average hamming various problem crf partition ham c c segmentation crf crf number dependency advantage provide good next give performance marginally compare beyond labeling set consider matlab ghz intel processor crf code language runtime table fast use dependency result slight segmentation base np labeling task arise process output set arise label assign common crowd ability capture large handle labeling miss variation various labeling measure accuracy subtract repeat plot miss label increase show label model learn make neighborhood extremely handle propose sequence likelihood dependency become computationally set scheme neighbor wide processing long handle datum label provide variational machine problem arise natural model gaussian pseudo perform labeling likelihood range component become gaussian inference algorithm neighboring label capture range dependency label label usefulness keyword likelihood classify input output commonly task name entity speech pos label sequence label account among machine community framework labeling fields crf crf markov field prediction pointwise estimate validation bayesian crf crf markov maximum margin solution label kernel crf overcome limitation crf gps learn gps labeling labeling posteriori map instead label gps chain monte field expensive dependency pseudo complexity grid vision effective propose sequence label long range become computationally inference mcmc suffer perform effectively dependency usefulness labeling problem arise natural useful labeling might miss crowdsourcing introduce gaussian classification arise process prediction different form label pos input component discrete component belong represent example n component presentation determinant infinitely function latent kernel smoothness scale commonly square se associated hyper single take every label zero evaluate input class softmax gp multinomial probit likelihood close multi approach approximate inference laplace test approximation yield approximate label classification fail consider entire class number intractable label separately multi gaussian process markov neighboring component capture dependency computationally difficult pl approximation vision sequence label long capture become intractable deal pl define pl address process pl dependency become intractable pl field crf entire capture long dependency expensive sized clique pl entropy markov pl suffer pl cyclic among depict cyclic arise labeling neighboring component cycle make inference pl discuss efficient label output capture output sake clarity presentation component different dependency relation direct component assume depend neighboring dependence component denote dotted dependency cardinality output take account dependency direct graph figure label greatly overhead amenable data label output ignore component label dependency model gp local associate f call define value latent pair input depict family softmax among output define gp gp size associate size collection dependent latent diag function notational simplicity evidence posterior hyper
ba ba bb bb bc rectangle grid node yshift xshift mm yshift mm node gaussian ba bb ab bc yshift cb cd cb cb cc cd rectangle bad slightly hierarchical descriptor exploit ad represent information color kernel specialize color gradient generic formulation se come tool neural scheme approximate om variant shift invariant kernel convolutional full image moderate moderate layer nystr om set grid pointwise pool illustrate layer finite positive control image map respectively act patch quantity admit activation map k follow linearity obtain network comparison cnn benefit involve hyper size later cnn start operation learn follow pointwise linearity point next lemma expansion mapping mc constant front integral need weight arise chapter different cm datum set integral sampling kernel e latter orientation typically explain prevent curse learn training importance unit produce sampling map w therefore dimensional operation follow resemble right exp xshift yshift exp xshift yshift anchor north xshift mm north anchor north cm build present principle datum patch formally patch p z early problem sgd infinite preferred point initialize fast bfgs ensure convergence stationary type convolutional leave experiment matlab bfgs rescale average always cross try always consecutive last layer produce map z prevent unsupervised discover natural patch field priori spatially localize oriented work attention know achieve cnns kernel even image patch learn filter perform obtain display among exhibit interpretable explicit relate wavelet dataset handwritten map patch pm methodology comparison otherwise four architecture gm simple patch filter gm gm use filter work raw pm detail architecture augmentation c tr cnn gm gm pm na na na na na cm cm cifar architecture cifar report gm pm work rgb patch color pm gm co especially despite layer require learn note either augmentation cifar planning investigate gm cifar na na methodology idea near dataset mnist cifar regard consist leverage understand theoretical acknowledgment grant project ce joint centre european research project elementary kernel p show z h pointwise kernel one composition p mnist cifar gm pm subsampling factor map indicate goal devise paper new convolutional neural either represent solve network learn kernel benefit invariant obtain network complex train literature invariance recognition cnns prove recognition cifar dataset accuracy state recently attention convolutional network large visual consist pointwise linearity operation pooling result empirically perturbation encode visual cnns capacity technique cnn understand recently invariance architecture characterize wavelet convolutional adopt traditional indeed descriptor reproduce produce multi representation main scheme convolutional neural interestingly unit kernel believe direction learn supervision vector yet achieve cifar architecture augmentation open author several related kernel cosine successive rely integral kernel arc cosine kernel neighborhood oppose representation manner visual parameterize neural function consist compute response perform kernel contrast propagate layer kernel obtain benchmark produce representation link convolutional neural produce sequence interpret non refer spatial map represent literature follow coordinate hilbert practical image coordinate represent characteristic neighborhood instance size blue map provide pixel value non complex terminology introduce kernel represent pixel convolutional z replace definite spatial correspond closeness place invariant show concrete example map difference characterize angle exactly descriptor introduce set location center patch visual encode color convolutional generalizing convolutional us hilbert build hilbert coordinate location patch word ii definite
correspondence biological typically reproduce particularly early expect increase structure type aggregate summary wide table supplement purpose estimate increase scenario case coarse correlation respect reproduce lead wide slow overview supplement width cm mark dark red triangle provide supplement bayesian inherent model uncertainty parameter fig predictive plot generate calibration bias e display distribution scale prior range forest reach specific width example choose correlation fig inverse metric quantifie model technical limitation choose expert deriving field goodness directly approximation allow likelihood observation give provide connect forest datum forest virtual assess uncertainty fit detailed datum abundance growth unbiased well fig aggregation v correlation indicate trade reproduce fit occur mostly extent produce growth datum growth maintain scenario include also growth growth account uncertainty correlation plot true high correlation aggregate type difficult visualize compare constrain strong correlation mcmc make interpret may opinion would mid specie due specific tree rate full secondly range resolution affect equilibrium slightly temporal output provide good relatively runtime intensive model encourage heterogeneous address technical challenge effort conceptual cost gain probably inversion practical far beyond considerable interest increasingly available practical problem answer source construct type would error challenge provide answer include likelihood weight account correlation conventional see pattern decide base statement include possibility rigorous statistical process testing subsection apply nonparametric widely nonparametric approximation distributional advantage equilibrium prediction parametric start parametric ensure acceptance reach acceptance would accept large correct late number mcmc parametric conjecture favor parametric encountered study promising parameterization traditional approach prediction mechanism mean approximation relative interesting model couple cost factor run difference secondary rigorously test understanding datum likelihood seem run dominant heterogeneous well standard like helpful comment review advanced grant research service open publication research centre department modelling http inverse scientific quantifie prediction express general inversion metric observe computational reason likelihood general assumption recent year likelihood method concentrate application rich forest inference approximation place conventional well virtual data sensitivity commonly result demonstrate simulation offer considerable conceptual particularly heterogeneous datum structure adjust include therefore provide fairly parameter derive physical physical experimentally range restriction measurement need output bayesian increasingly decade addition particularly likely quantify approach metric often bayesian datum current usually application make distributional vary around ad hoc mind variability environmental condition likelihood art limitation predict principle prefer derive understanding process output observer could complicated stochastic limitation reduce technique arguably attract well stochastic inferential likelihood estimate pattern likelihood approximation repeatedly rare apply parameter rich simulate aim base likelihood approximation propose infer virtual identify field examine aggregation statistic inference fit field line refer specie relative year apply leaf area index per area relative diameter height sd growth sd rate relative end growth sd growth diameter diameter diameter growth relative dynamic prominent dynamic lose factor tree gap formation create gap natural forest within gap explanation forest formation history tree light highly diverse specie type specie similar functional per gap formation forest merely predict possible variation composition use forest use base forest forest gap forest various location area cell assign interact position entirely measure diameter breast height tree dimension relationship calculate subsequently light act important describe specific growth dimension description scheduling supplement model gray type rate original uncertainty likelihood one carlo sampling approximation robust bayesian suited deal interaction expect likelihood short introduction brief inference posterior calculate normalization uncertainty compare likelihood obtain broadly speak say quantify chose wide type proportional prior give many uninformative purpose facilitate definition likelihood conventional study quantify model occur uncertainty situation usually interact typically ad hoc variability hence conventional likelihood independent mechanism description virtual field field total forest approach go deviation particularly dominate error forest gap formation explain variability expectation output conditional technique suggest prominent arguably attract recent year discuss currently abc apply similar principle suggest classified parametric approximation estimating obtaining convenient normality fundamental requirement obtain variability correlation output matrix supplement abc achieve efficient unfortunately generally accept good summary statistic modeling fit first aggregation count number individual size growth quantify forest subsequent metropolis check visually supplement detail evaluate parameter estimation burn chain supplement visual virtual three create advantage true
expert well expert output decision rather high representation convolutional layer build work compression great valuable information non neural average prediction expensive evaluate target correct pursuit carefully capacity network arguably complementary ensemble grain convolutional connect relu unit class evaluate million image annotation example ground entry fraction wherein look two expand internal google million span table successfully network multiply add required evaluation well perform imagenet imagenet class thus train comparison work general add train model consider result task assumption underlie extend allocate fixed regardless capacity cardinality group difficulty necessarily either obvious connect layer capacity discriminative contain fully connect appeal augmentation purpose specific augmentation theoretically potentially division capture wherein pathway activate task relevance gain strategy google op universit view google com neural capacity significantly descriptor label visually hide pathway connectivity label report task augment multiply perform exhaustive architecture expensive furthermore satisfactory architecture discover improve fully increase tendency overfitte setting thousand class prove challenge jointly grain entity often kind challenge visually belong two semantic visually traditional building prediction exist additional significant sensitive production must rapidly matter service level way improve performance computational evaluate however immediately compete objective add capacity train neural hold datum demonstrate recognition base less overhead held apply generate possible
spherical benefit ad examine cluster mean deal pg mean mean pg project kolmogorov ks examine hypothesis cluster mean rely split concern single employ accuracy ks modify benchmark uci repository generate exist unimodal sample unimodal acceptance hierarchical rely criterion recognize check criterion split checking continue cluster cluster usually transform section signature illustrate signature generate run figure illustrate top middle sort plot compact form sort small therefore around dense small signature signature sort absolute unimodal suggest signature sort signature expect cdf signature index choose comparison demonstrate step threshold split un ng cluster leave single behavior associate unimodal lie boundary therefore method split e htb two signature signature deal unimodal unimodal family simulation ks cluster default significant ks confidence addition computation ks splitting mean benchmark vi adjust rand vi ari cluster ari optical digits ari vi ari vi ari vi idea signature splitting signature represent advantage propose simulation cluster cluster signature propose specific application hierarchical statistic test cluster group family rely available challenge involve hierarchical cluster splitting criterion nan hypothesis dataset estimate pass improper splitting incorrect cause universal approach kolmogorov splitting criterion also cluster propose
graphic macro ltb lt lt lt ltb lt lt lt package conjunction terminal option explanation load graphic graphic macro ltb lt lt lt lt lt lt ltb lt lt lt lt lt bp cb set instance error etc task examine affect find attribute fr increase requirement complexity frequency certain instance make complexity prevent avoid algorithm add loss boost misclassified often boost instance remove filter instance amount frequently misclassifie heuristic wrong decision machine filtering discarding instance decision instance weight influence discard learn clean train predict attribute predict correct accuracy validation let compose distribution thus feature possibly train factorize figure distribution unobserve variable feasible b cb explicitly possibility I rather focus classification show generative differ mostly concerned training seek probable hypothesis map ignore figure use approach neural network decision possibility ignore generally handle probable simple prefer instance learning seek maximize model observe label motivation pass datum calculate trivial use outside hypothesis induce approximated assuming train hypothesis multiply though trivial distribution generative discriminant asymptotic diverse diversity refer hypothesis diverse create unsupervised algorithm different prediction hierarchical agglomerative default result dendrogram height line connect create representative diverse conjunction terminal option load graphic terminal graphic ltb lt lt lt lt ltb lt lt lt bp r mlp near near na I bayes learner forest prune cross validation induce approximate hypothesis weight mlp random weighted examine train backpropagation derivative activation sum correspond forest select algorithm track represent sum meet mlp forest uniform weight weight biased p algorithm produce kronecker delta paper instead induce feature approximate single hypothesis calculate score mlp backpropagation score large value value class instance set rule score number instance examine divided leaf neighbor agree reach node normalize percentage instance cover true instance em algorithm belong clustered clustering data algorithm cluster filter ensemble examine good would remove misclassified near neighbor repeatedly remove misclassified handle experiment experiment noise introduce label examine handle learn uci repository pair sign rank expert previous improves generally handle artificial add experiment row algorithm subsequent value case higher represent significantly high handling noise handling increase mlp contrast noise handle achieve handle achieve highlight handling noise work set handle beneficial weighting induce instance weight hand forest achieve filtering represent approach increase decrease rand l g examine handling generally achieve accuracy inherent however random forest robust obtain low instance artificial forest average nn noise handling mechanism beneficial instance filter class compare biased accuracy significantly outperform case case compete weighting nine diverse represent suggest show obtain r high ccccc ccccc nn c l weighting weight filter filtering weighting great accuracy estimate compare achieve mlp weight significant algorithm hand achieve accuracy four mlp high overall accuracy level accuracy except examine increase classification forest level achieve high achieve significantly high effect filter choose produce high avoid ensemble individually affect bold ccccc ccccc r f mlp examine learn base examine weight extra select threshold filter instance great effect algorithm instance individually backpropagation filtering artificial weighting use handle future equally outlier equally machine handling
cross early end rich lot error selection selection stability base false set predictor outcome signal set boost integer boost continue select repeat select learner pre value show control predictor assumption exchangeable original boost variable exchangeability see assumption noise assumption restrictive hold sensible stable propose choose little potentially e half fit select signal subset signal minor importance long parameter compute equality assume check resemble overview essential subsample derivation modification stability complementary subsample importantly bound derive assume exchangeability without original selection procedure drawback assumption control per family variable run size error rate unlikely select compute random complementary additionally otherwise subset select procedure selection threshold case tight assume simultaneous unimodal additionally practice third simultaneous r concave convex concavity strong concavity concavity support tight application refine markov distributional assumption inequality derivation aware selection assumption generally concavity may concavity hold distribution unimodal concavity seem wide bad selection low include e control rate case exchangeability special e variable strong correlate select control similar practically identical number positive statistic test hypothesis reject learner commonly per per without adjustment significance conservative control setting gene study use relate false reject give situation fix conservative controlling situation obviously specify g specify choose upper usually per everything adjustment bind upper boost conjunction stability additionally impact characteristic accord regression predictor independently draw design predictor observation variable vary influential influential variable predictor correlate predictor toeplitz setting vary bind complementary improve time figure scheme true positive depict uncorrelated rate rate average replicate circle increase uncorrelated predictor generally natural biased base learner yet error bind truly influential threshold figure enough appendix figure correlate false positive bound conservative median setting overall concavity standard control distributional simultaneous probability general stability conservative assumption simulation open circle positive grey horizontal line increase positive increase appendix influential highly stable number select per number positive contrary false examine control express pathway patient design single ability cell utilize mb media produce end utilize nm indicate peak give measure background replicate patient due supplement package biological replicate I e incidence per incidence one contain annotation include non difference measure patient control pm intercept group effect control irrespective effect replicate effect belong mean code formula group specific either control case constraint code effect coefficient differ specify effect offset contain height respect offset additionally interaction keep learner check pm total learner effect overall group boost differentially stability per boost choose related significance lead cutoff r concavity bind subsequently cross validation stop stability supplement stability find frequency learner base selection assumption term selection dash gray value concavity determine vertical gray line dash line value concavity confirm affect weight patient decrease unit logarithmic rate finding might sequence code code patient paper detect five level report patient notice either affect function reduce collect notice patient whole seven reduce five finding might consequence observation add variable keep conjunction complex generalize additive gene positive per conservative specify sensible control adjustment extreme stability control tight stability suffer different boost package package use store package fit boost additionally specify assumption concavity alternative method fit interface specify index framework function miss stability selection check sensible situation acknowledgment discussion fu chen genetic center analysis conduct present setting independent toeplitz replicate open red represent positive setting toeplitz assumption circle true influential toeplitz observation line together observation replicate red represent positive independent predictor toeplitz compute replicate
high versa balance action small aspect argue bad propose family function find tight exist policy evaluation nonparametric policy space show outperform art powerful policy evaluation fit policy outperform purely approach outperform rl computational depend solve cost batch set problem expensive medical mining complexity surrogate action sampling distribution solve knn sign basis function question loss approach investigate additionally action space effect choose change could restrict impose constraint maximum study loss let n assumption exist unless otherwise bernstein geometric mean least inequality auxiliary x x maximizer action likewise q q optimizer property imply apply apply n use lx q exist q minimizer policy choice r inequality apply inequality l n paragraph v l c k get upper imply convenience bernstein pn class range fr ec loss minimizer loss cause quantify loss suppose action empirical minimizer use mainly estimate regularity factor currently extend note similarity considerably different line unless n geometric probability focus event eq xx maximizer x q use likewise write decomposition optimizer finally use apply inequality imply n probability proof lx q expect least get last inequality property bernstein geometric obtain result get separate infimum c n previous desire analysis modify proof remain follow policy finite vc xx maximizer x inequality eq likewise optimizer inequality apply inequality appendix satisfy state include probability least choose loss policy space choice add lemma probability eq n lemma appendix z inequality apply recall l infimum use confidence get desire main remain vc vc dimension independent main rhs integral cover relate entropy r indicate complexity equation theorem originally chapter lx lx eq bind additional quite flexible derive benchmark car balance standard benchmark reinforcement flexibility choice evaluation tb approach car task value base choose method space policy implemented walk value iteration minimize neighbourhood suppose endowed integer kx tie knn pick knn lead rule implement dynamic car discount initial uniformly random policy collection length episode reach goal go step reach episode gaussian reduce systematically knn bar standard reach return episode knn sample reasonable knn especially benefit purely knn underlie interesting depict function regime small effect optimum choose properly selection beyond sequential tb balance policy pure value fit denote tree fit algorithm evaluation improvement policy action minimum note extra represent trajectory length trajectory choose use amount generalize balance successful show per average tree row especially size try solved perfectly confirm exploit compare exploit tree computationally costly practice choice collect style powerful science engineering claim theorem corollary proposition fix c g p gap indicate inaccurate state space mdp restrict qx hold qx summarize definition reader bound measurable w subset discount mdp r optimal value policy denote ax choose arbitrary manner function optimal introduce complexity control low mdp characterize simplicity naturally mdps mdp actions action understand action informative perform improvement base greedy ideally greedy policy action roughly action less likely action likely wrong characterize summarize action gap gap p q control action gap imply inaccurate measure mdp mdps satisfy mdps almost find mdp form show gap norm qx supremum norm hold qx distribution close policy compute include policy space distribution construct entail improvement iteration gap choose greedy action large policy projection greedy weighted gap e minimize weighted evidence value exploit function impossible optimal ideally use use sample change sampling loss qx qx qx n loss difficult action possibility relax surrogate weighted hinge approximate iteration whose outline present dataset classification compute estimate q bellman residual minimization fit iteration combination td exploiting intuition policy improvement iteration action gap accurate maximizer distribution greedy large measure accord weighted bad policy account region belong dataset simplify discussion weight loss action gap regret greedy flexible policy parametric space finite grow example latter method neighbourhood tree grow reproduce hilbert nonparametric sparsity basis achievable estimate generalize state g policy affect another choice match policy well ideally automatic note dataset transition generate sample generate early iteration independent extend action gap difference action qx qx qx theoretical computational solving problem since loss difficult one relax gap return kx ix influence minimizer derivation action point majority greedy action would base tree represent analyze policy policy distribution e enable specify behaviour algorithm analyze affect early n clutter identically I action define pointwise pointwise expect loss action value empirical w instead close carry analysis behaviour take care main relate loss appendix complexity vc metric localize complexity often use localize rl aspect interested study take care issue cause relate relate make notion choice vc metric etc g localize rademacher since lead moreover vc policy property rademacher quite scope localize rademacher analyze reader p rr ex quantify extent localize rademacher complexity define root subsection fix policy assume consist bind important term rich mainly behaviour point determine global neighbourhood dr cf rate considerably behaviour term result rademacher exist action regularity problem evaluation improve benefit guarantee randomness constant second behaviour point hand side existence uniqueness prove make define error neighbourhood complexity possible extreme everywhere single policy subtle aspect note conservative behave proposition also corollary behaviour estimation last important gap regularity regularity geometrically regularity often assume randomness q c quality evaluation quantify question policy error question proper selection algorithm mdp main greedy error greedy definition induce greedy error quantity dynamical mdps future relate approximate algorithm quantity coefficient integer state supremum derivative r discuss definition ready main paper sub eq k regard approximation error apply early discount implication resource focus later iteration beneficial though apparent value iteration tailor type reader refer discuss early moreover dataset nonetheless might upper g present gap value classification pure designing state consider derive policy report compare pi use example everywhere know uniform run policy vi design performance policy good solution achievable considerably vi search pi comparison vi involve policy space small pi considerably evaluate report use poor approximate greedy attention policy fit belong large region space differ optimal relative region poor belong chain difference everywhere reward belong result pi go quickly however vi pi performance optimal action function report pi anti fall pi pi differently drug despite maintain long attract interest community optimize drug scheduling strategy lot attention recently structure alternate scheduling action patient simplify formulation reduce available combination drug small interaction develop real decision
compact identify space accordingly complement partitioning designing proposal infinite effective simpler yield ensure discretization invariant next context mcmc great inform direction differ method example proposal exploit discretization scheme langevin basis construct adaptive variation hessian use proposal adaptively define gauss newton adaptively refine proceed follow construct use current value gauss newton evaluate operator approximate low terminate adaptation evaluation f distance successive fall prescribed step eigenvalue additive terminate use build theory ergodicity f step covariance leibler hellinger distance symmetric operator differ range finite efficiently project onto reformulate dimensional subspace criterion maintain full hessian high maintain dominant left structure case truncation hessian truncation direction benchmark two representative proposal exploit proposal transformation obtain proposal term numerical langevin proposal rw brevity rw comparison hessian explicit langevin data langevin sde hessian sde form langevin proposal mala newton algorithm instead mala burden hessian principle modify newton h langevin newton langevin employ whereas use pde infer proposal proposal use invariance refinement spatial coordinate term potential field govern pose superposition four weight bilinear uniform endow normal prior operator prior length make field prior field observation potential figure b sensor circle carry affect two magnitude signal ratio noise snr set proposal rw langevin proposal every hessian posterior weighted benchmark proposal half autocorrelation onto eigenfunction operator large proposal bias result burn mix along direction onto prior eigenfunction projection mode evaluate mode yield proposal autocorrelation choose projection onto st expect hessian weight proposal map brevity comparison langevin sampling start produce adapt langevin proposal employing denote account variation dominant dimension truncation threshold relative sampling snr snr show algorithm produce mix mix noise snr due broad lag improvement global snr global refinement global convergence grid diagnostic use adaptive diagnostic inform drop course procedure comparable three yield grid yield effect discretization model refinement also first level noisy path double drive force molecular dynamic govern globally increment brownian motion follow condition sde double model science perhaps notably molecular negligible mass fluctuation map differentiable continuous wiener framework euler dimension h efficiency proposal solution truth path conditioning quantile marginal enough length burn also adaptively construct highlight sampler summarize trace also dramatically improve mix langevin h langevin pde langevin reasonably proposal plot lag autocorrelation figure eigenfunction figure improve trace langevin mark adapt construct global decay autocorrelation mix measure lag general explore proposal discretization dimension early dimension independent algorithm new sampler wherein direction capture global identify strategy proposal offer gain efficiency certainly construction construct incremental updating strategy optimally variation hessian available resort active gradient available approach might covariance regularize estimate second weighted proposal construction approximation posterior difference posterior local value room implicit discretization locally independent provide construct provide proposal mala discretization langevin equation proposal extend development idea acknowledgment acknowledge financial energy office advanced sc mathematics capability law technology inference explore high represent introduce chain monte carlo adapt structure intersect develop introduce general operator proposal mcmc sampler local hessian distribution scheme langevin operator proposal adapt result sampler large pde inform langevin sde high represent discretization example include inverse reconstruction differential markov monte carlo mcmc representation unknown refine present enable build change local information design cast proposal operator proposal allow structure exploit overcome standard particular absolutely choose deriving yield yield performance mesh work integrate design finite carlo use building employ algorithm effort proposal space analysis improve finite setting research local geometry scale include stochastic matrix prior metric langevin implicit hessian log direction differ idea formalize lie involve result great relative also optimal covariance suggest infinite proposal differ preserve independence attempt operator employ version hessian single near noise parameter dominant relatively inform dominant likelihood hessian posterior direction inverse smooth dimensional acceleration mcmc algorithm dimension wherein concentrate goal dimension independent sampling allow gaussian flexible manner local covariance e bound adjoint operator proposal sde incorporate proposal mcmc either metropolis rest review formulation method well previous aim improve sampler enables inform global sampler pde evaluate sampler sampler highlight key difference performance brief remark inverse proposal equip prior geometry proposal separable give assume away unknown element banach assume additive zero space associate norm brevity possible drop assumption forward result locally condition forward hessian prior adjoint class full operator define define covariance refer laplace differential respect measure limit shrink q maximizer vanishing posteriori mh accept hasting define chain otherwise mh continuity mcmc walk adjust langevin mala mh mesh refinement carry reference therein mh langevin continuity consider langevin sde sde see reference therein order implicit parameterized defines proposal langevin walk simulate langevin sde provide approximately technical putting let rewrite invariant remain one employ desire autocorrelation increase
search corpus scheme illustrative pattern trend discover similarity corner topic universe social clique topic discover detection clearly group instance community another life genetic dynamic material research dedicate develop material mutual molecular recognition show research appear example illustrate use outlier htb topic community orient pure separate community couple represent cover range htb discover biology population related htb connect network second corpus comprise english portion contain span different topic execute implementation lda labeling labeling wikipedia topic wikipedia nsf grant conduct user wikipedia agreement perform significantly nsf grant elaborate label topic wikipedia construct node use wikipedia network rather illustrative major trend salient macro visualization figure wikipedia subject music fan basis salient group connection summary green daily medium discover wikipedia music topic plot write daily news medium study label performance great topic nsf grant corpus wikipedia many topic broad general summarize single perform albeit equivalently phrase focus wikipedia expressive produce address find especially mine technical document primary limitation text optimize article summary corpus work shorter handle performance grant extremely text cause difficulty certain wikipedia topic contain article description minor character text twitter lda design short investigation improve lda construct employ explore nsf basic english portion wikipedia text previously develop department analyze refer collection represent label network model base network show powerful characterize collection text connection set insight topic large efficacy nsf grant span year entire portion wikipedia interpretability insight popular topic latent lda discover topic represent explore domain user search unclear begin search user automate organization massive scale quite challenge large trend lda fundamentally return document absence typically probabilitie label explore text raw output challenge insight identify leave numerical trivial implementation train lda massive corpora big discover challenge unclear discover oppose thousand hundred thousand present work investigate challenge graph nod collection similarity topic effective building summary contribution area topic visualization labeling section topic visualization discover employ community network discover macro subtle discover extraction unsupervise employ topic large surprising visualization next represent efficacy automate wave address challenge topic topic light graphical exist several like make towards interpretability exist relationship topic document provide insight topic come subtle connection among topic ill understand document exception correlate topic association association represent incorporation reveal challenge exist base appear extract refer relation view subtle third scalability chen et show unable corpus document finish week limited text show scalable cluster domain cluster machine process million access machine cluster correlation structure among aforementioned issue scalability efficiency association fashion similarity topic exist labeling visualization tool identity node scheme probable word topic derive always expressive fashion unfortunately unable handle text corpus topic gap motivate labeling goal topic method set supervise labeling consume labeling must corpus consideration corpus wikipedia especially deal collection describe art edge technology subject reference corpora requirement prevent utilize employ corpora topic intend interactive utilize text document engine document comprise topic filter domain filter remain document sub labeling cope scenario document collection reason label couple topic filter characterize labeling topic label filter might repeatedly execute document collection labeling work appear easily well long short text aforementioned issue motivate development massive throughout represent collection topic document sequence element return like exploit trend topic employ use insight discover densely topic detect group employ heuristic modularity modularity fraction expect fraction link evenly initially assign topic greedy fashion assign community modularity efficient large topic computational network mark community topic represent powerful tool discovery heterogeneous corpora topic label mechanic dynamic flow dynamic mechanic flow agent equilibrium game graph graph combinatorial theory algebraic human evolution modern early year human modern human evolution water engineering coupling mode modal recognition orient protein protein protein protein protein protein bind research social social social discover nsf rank lda subset corpus greatly facilitate document similarity comprise purely take corpus way topic proportion transform mutually exclusive cluster employ well text htb label return term filter hash hash dy gain sort essentially topic naturally observe topic extent express noun expressive retrieval instance expressive alone expressive denote uniqueness noun report proper noun system technique principle noun test association contingency extract test gram phrase save label individual intuitively term appearance early frequently weight candidate harmonic topic recently document belong entropy measure p proportion document perfectly document gain high simultaneously rare algorithm sort small candidate select sorting might select label frequently albeit slightly latter two indicate specifically sort normalize combine comprise scalability document online collection long combination dependent deal reason straightforward multi core line instance map job identity hand execution oppose pass processor system
function rare occurrence et implicitly matrix shift wise local form term independent term word context objective may perspective wider different similar performance dimensionality enough well choose weighting context unobserved explicitly sampling take corpus rare choice word sake efficiency avoid remain pearson correlation r r iterations pearson depict iteration two band may truncation less see well iteration less weight objective shift like done explicitly shift share objective though completely weight strategy bias tend converge term may approximation optimize future investigation focus choice report also skip implement explain similarity two specialized function differently represent valuable context meet context context word embedding matrix vocabulary similarly context word context count
remove turn leave stage fig result general result stage sift contribute performance take spatial representation selective convnet feedforward mean k k boltzmann machine convolutional network column deep maxout net major challenge dataset lie capture object scale pose accuracy outperform several feature encoding fisher vector field discriminative spatial preserve sift unsupervised learn encoding encode nearly accuracy challenge grey handwritten digit training normalize center pixel use atom block extract sift scale voting field size give result deep learning achieve success digit misclassifie machine network maxout single achieve highly complex architecture misclassifie digit misclassifie human clean beneficial focus detail l algorithm k map good mean query mean employ retrieval feature subsampling break map sift final multi voting e view retrieval train combine stage nearest search fair task table experimental dataset compare sized bank triangle filter bank achieve small combine consistently mean mean k representation view achieve without supervision near copy dataset type image perspective etc type three pooling respectively retrieval robust outperform nearly encode term map bit layer neural term unsupervise one major advantage feature representation classification unlabele sift discriminant experiment challenge classification unsupervised feature learn bag word big dictionary rich significantly last far without train several acknowledgement national science china national foundation education china paper investigate feature unlabele network network attract researcher recently effectiveness feature mapping tend describe robust purpose call ball local density show efficiently programming problem traditional method representation ignore global representation descriptor also field multiple pooling extensive several comparable learning attract attention interest representative deep abstract among one typical dl convnet stack classifier variation convnet different vision great success layer time performance autoencoder rbm restrict boltzmann adjust consume practice motivate among mean cluster commonly unsupervise simply cluster parameter involve base number cluster practice capable superior gmm representation uniform cluster contain influential description issue key definition robust noise outlier commonly encounter application could actually tool minimal describe target class obtain addition cluster surface come support boundary far fig ball may datum feature representation align quadratic need hundred programming save amount extend adopt range part level preliminary feasibility method call verify extensively retrieval benchmark part organize section regard learn investigate performance empirically automatically useful rely learn utilize representation subsequent g object pattern obtain distinguish noise since think variation transfer knowledge relate domain regard kind unsupervise word aggregated descriptor fisher typical include step local filter gmm center cluster filter bank step patch encode vector learn bank input image follow brief feature usually pool local regard encode count bin coarse way encode alternatively count effectively robustness code patch tx tx tx uk tx simplify define signal dim encode rich object method connection bank feature advantage encode even rich whole procedure architecture deal code encoding activation centroid less note triangle essentially complicated g autoencoder hence characteristic different point believe make difference aforementione mean cluster since cluster latter present overview center means triangle encoding discuss extend typical layer component image map set operation contain pool map serve option bank pooling grid major generally speak big filter help representative accurately high pooling need dimensionality architecture aspect local texture actually kind challenge highlight importance encode second spatial name adopt architecture dictionary add sift post pooling processing procedure essentially project response robust representation contain center radius description spherical point avoid outlier face tradeoff goal minimize formulate slack relate th atom simple atom face capability distinguish b patch attract atom see patch attract fourth atom yield atom see potentially could however since actually contain three encode drawback atom corresponding atom major capable characteristic dictionary effective row explain compare counterpart c experimental bag patch preserve huge dimension suppose field input densely encode map filter filter feature pooling map pool p p window information lose sift sift descriptor computer sift representation sift descriptor densely cause dimensional descriptor pixel dimension extract bit histogram dimension dim preserve rich subsequent task exploit scale scale different context since characterize object appearance information capture lost level valuable complementary manually design descriptor sift feature unclear high pattern c layer learn scale information way information training atom correspond size fig learn learn typical convnet window learn example give orient window effective feature big pattern one useful learn train layer view size pool combine output multi let weight versus normalize output decision eq classifier coefficient mention extensive experiment include image retrieval whitening whiten operation linearly transform matrix sphere euclidean use unless note important subsequent default recommend ball q encourage default background ignore use triangle encoding direct counterpart simply furthermore addition method network mean denote l default conduct behavior propose image label object partition remain image pre fold image fold fold report across fold randomly pool fed vote range major et control unsupervised compare particular extraction
na trivial bind assume vanishe enyi surprisingly show specification behave os r enyi er use sufficient statistic entail generality er probability represent os absolute distribution os enyi n ig ci nd np ig ig np mp np cover parameter equal yield close er unless observe isolated node er k f favor isolate order vector n still observation regard identity n k g proposition scale version k n isolate node write n write imply p rewrite exponential family remove without isolated normalizing figure apparent isolate time iid node isolate g existence mle easy bi verify odd element bi length tends tends infinity conduct non ideally compute element vector illustrate department pa family statistic label graph science attempt model parameter variety see form example node expressive power perhaps statistic g literature property statistic edge see whose derive dyadic long know originally originally increasingly property degree summary among statistic investigate offer preliminary trivial complexity statistic case rely base dyadic challenging contribution mle include concerned dense os enyi os enyi appeal possess demonstrate finally define bi undirecte label node iid observation observe ratio number number observation case observation observation available extract observation precisely statistic family scale subgraph subgraphs sufficient configuration edge node configuration triangle subgraph subgraph three node connect adjacent node g connect subgraph e contain order convention ratio edge equivalent os odd reason joint sense uniquely converse move move enable possible graph base practice usually term different labeling well distribution asymptotic prescribe non time different em em verify consider imply pg pg calculation regard degeneracy reason change odd point na point odd even vector lemma possible consist characterize odd project polytope coordinate depict em em em mle mle even e even n odd le n l odd b k k nn l conversely n imply vice imply mle proceed one partially ie appear appear term denominator side assume model node plausible simply polytope would simply subset whose nonzero deal extreme suppose graph start stop
macro corollary california berkeley human comparative measurement study choice selection empirical evidence amazon pairwise comparative fast one less ordinal characterize minimax rate measurement quantify ordinal confirm ordinal ordinal knowledge non expert human domain crowdsource collect human low come price noise crowd response address noise study set answer crowdsource image student approach enter subject ask search internet entry alternatively compare item figure show image ask internet restrict pair item ordinal allow precise value ordinal even measurement convert simply order value suggest original ordinal ordinal encounter measurement inherent possibly conservative evaluation ordinal measurement recognize human allow evaluation perhaps cost lack clarity regard fundamental question gain measurement significantly ordinal reliable pair comparative preferred measurement invoke study widely theory show preferable theory theory tool comparison investigate perspective estimator ordinal also incorporate highlight fit ordinal ordinal choose ordinal ordinal approach enough ordinal topology compare argue always ordinal approach lead processing mean seven amazon show argument insight ordinal subsequent focus experiment subject version question answer subject ordinal version pair present seven experiment broad coverage audio l circle audio relevance ordinal clarity comprise box area circle paragraph people ask ten people audio sound key correspond sound audio internet show subject query ordinal form compare answer subject five seven experiment ground truth solution compute fraction incorrect ordinal convert half error truth ordinal answer process unlikely setting ordinal argument inherent subject human evaluation compare amount ordinal ordinal discrepancy complicate aggregate produce final answer whether address small aggregate popular two comparison presence additive quality pair entry ordinal identifiable shift analogue involve individual remain entry intuition scenario analogous choose suppose enough ordinal item facilitate induced distribution allow measurable sample semi minimax place ask evenly ordinal minimax error provide many regime enough risk f case time characterize treatment strong convexity informally difficulty estimate arise risk ordinal setting dependency ordinal collect overall enough summarize loose tight axis pair comparison treatment allow comparison second comparison simple parameter play comparison carry treatment extend rather provide central role comparison comparison measurement graph laplacian refer laplacian determine canonical example large evenly along align clique disjoint vertex unweighted get unweighted complete nk well possible hand scale bad topology consider error distance frequency execute ordinal uniformly replacement inspire practical pool pool log rest unlike ordinal evaluated element minimum knowing infer scale correlation put rest per ordinal ordinal well identify mistake per hence ordinal ordinal ordinal audio coefficient coefficient compare human system rely non expert human crowdsource mechanism critical argue fundamental ordinal provide choice observation question estimate either bound overall option complex one incorporate worker accurate threshold determine crowdsource setting testing improve collection useful adaptively choose topology aware potentially ordinal analyzing topology review present present collect amazon com amazon platform put individual offer exchange payment task follow experiment comprise task comprise question organized ordinal answer question task worker country allow estimate usa allow american move experiment rating product truth circle question random experiment age connect table section associate standard run ordinal square coefficient coefficient average list present possible distribution kullback leibler pair estimate kl family following give metric induce distribution minimax subsequently positive entry graph topology satisfy semidefinite lemma provide e vector I I vector normal location minimax see instance ordinal scale nd construct j give pack choose bound desire impose constraint hessian function tw derivative l say set hence note put everything together standardized laplacian gaussian distribution divergence p packing furthermore I give pack bound note issue remain verify fact final follow relate loss I scalar log concave inference allow n define minimize k last lemma I I say virtue p e put everything substitute b tn index vector
mean lead sigmoid among gaussian recent phase place learn benefit small batch appear get energy landscape range probit rbm unit generate binary bias threshold field average persistent collect field mnist digits separate digits classification hide unit probit rbm comparable rbm cd probit rbm rbm cd operate representation probit rbm smooth exploration mode main produce rank area hand rating arguably learn directly instead go intermediate ranking per variation item choice share tie necessary rate unseen item person fast maintain estimated markov maintain chain update step spirit divergence away chain ph rank unseen movie rank movie user supplement data movie encourage diversity list remove movie remove less remain movie rate movie hyper rest implement item popularity naive implement recent ranking metric emphasis rank movie rank unseen movie demonstrate winner cccc popularity mf shorthand mostly consist multiple fact g choice choice choice ordinal rank deal heterogeneity call numerical scale star comparison tool lose format convert survey collect centre people explicit interesting gap uk relative separation china rest feed multiclass logistic regression suggest capture variation widely category category gaussian cost invert gibbs dense overcome probabilistic principle main direct undirected rbms system boltzmann machine variable restriction place handle ordinal study difficulty model introduce model nearby correlate situation attribute add combination modelling distinguish thus model row hide row generic boltzmann type inequalities boltzmann machine evidence representation support boltzmann literature specify specific support assignment censor binary rank demonstrate handwritten digit survey specific potential compute define apply let metropolis annealing time posterior express constraint method would deterministic probability concrete kullback entropy decomposable second thus ignore since decomposable e ie k combine completely decompose divergence ie ix divergence know proper multipli ix gradient w finding let truncate recursive ik short hand cumulative function sample straightforwardly true inequality visible threshold active sample use sampling markov propose certain level slowly temperature temperature g collect successive p constant decomposable gradient w pair simplify px h undesirable chain tendency point flip due fortunately enforce node pd gradient p sx phase way way update estimate fashion incorrect exponentially progress error estimate describe start mode taylor distribution replace thus turn per category plays share location th utility th category ensure lx lx I pe px mf lx rewrite change variable pe mf dy categorical category particular assume must word offer stagewise pick also pick subset give pe e pe l l give model survey country people world china second last gaussian approximate univariate assume form equivalently taylor expansion truncate read eq dirac delta truncation recognition university edu introduce wide range view type observation impose inequality type interval censor ordinal without tie three handwritten social survey analysis restrict rbms prove variety original proposal mainly unit hide applicable detector design input continuous multinomial ordinal boltzmann permit type jointly model thus rbm multimodal social survey audio modality idea conceptual drawback layer specific handling order specify main thesis capture key model spirit one variable turn beyond category compete mechanism economic structure learning relaxed assignment indeed cover model boltzmann machine unit rbms bottom contain variable group type assignment bottom point assignment never fully observe support manner ever assignment censor category complete rank tie propose handwritten analysis believe complex capture form cross connection allow boltzmann energy decompose n w ik w column factor rbm boltzmann specific real generalise transform denote element evidence triple px cumulative simple rejection advanced section due transform specify conditionally box suggest abuse notation let mi mi mi need handle direction sign join multiple constraint refer often posterior g unlike rbms unless mean numerically update recursive ik unit activate normal density normal cumulative distribution reader refer supplement estimate pe general constraint however couple
within define l deviation member evaluate bounding bounding follow I bellman error section obtain write place subtract mdp distinguished move challenge expectation bellman mdp return expect law program bellman dynamic operator take difference mirror early bellman mdps infinite nearby might value violate satisfied lipschitz weak help future use sum reduce clutter global upon write dim dim e n shorthand n set reward bound problem remain contribution bound lose dominant together cardinality address posterior effort mdps fundamental reinforcement planning policy intractable approximate mdp require would whether learn complicated planning examine acknowledgment support stanford award foundation elementary martingale random adapt conditional generating guarantee early discussion f gaussian deduce early apply substitute least solution discretization bind bound fx fx sub minimizer obtain least desire result imagine disjoint without show delay update episode dim kx k I dim upon canonical recall q f fx trace pp I ij ij obtain bind q lagrangian v dual objective tr expression linear rewrite tr say mean imagine determinant say c bind eigenvalue equation bx f p say valid identify know let number exactly simple lemma argument function imagine linear controller reward reward eq semi ball semi definite x x equal value exclude outer high inner mean flat outline optimistic use confidence algorithm episode solve optimistic policy attain algorithm tractable possible claim van stanford university unknown parameterize bound scale cardinality characterize explicitly kolmogorov unify base reinforcement learn state several algorithm reinforcement optimize reward unknown markov decision mdp agent sequential decision cumulative environment mdp planning explore poorly policy environment exploit knowledge numerous statistical efficiency include pac approximately mb mistake know know focus attention reward link overview li broadly rl environment pac number plan exploration near bound attain grow cardinality space extremely even reward beyond nothing beyond parameterization degenerate mdp transition arm another assumption state linear quadratic constant grow remove exponential old regret linearity analyse intuitive introduce show finite exploit satisfie cardinality underlie problem extend previously bandit unify reinforcement provide art important consider horizon ap mdp define ms say mdp associate mdp mdp episode prior notation td dim notion specialized parameterized mdps fix state space future per equal shorthand kolmogorov eq definition regret bound parameterize mdps assumption section bound mdps control traditional notion dimensionality equation unconstraine constrain upon outline optimistic variant attain unfortunately approximate optimistic loose complexity potentially infinite mdp function I mean intuitively length know reveal dependent dimension sequence notion reinforcement
dimensional space method use valid many assimilation calculate constant lead power asymptotic measure long smoothness satisfied suggest remove random confirm small error constant exactly proportional converge infinity confirm asymptotic demonstrate noise method perform small organize theoretical technical asymptotic analysis derivation map depend subsection explicit analysis use formula describe theory read without theoretical put introduce scale theoretical rx dx fx q every sampler direct sampler successive independent evaluate may typically perfect would sampler relate weight recursively product grow rapidly convergence consider assume global degenerate unless state generality locate near zero hessian sampler equation relate behaved approximation direct use find monte independent compute estimator minimizer hessian consume lead term expansion depend large odd hope remove classical present remove map draw evaluate consider weighted formula un ensemble map identify use way pdf q see sampler symmetric proposal sampler expect nearly small regime oppose map review completeness name ensure except star shape straight set choose arbitrary jacobian jacobian gives vanish matrix immediately verify respect multiply map adaptation three use argument probability weight q concern recursive application call proposal roughly see theory scaling scale expansion satisfy rest satisfie drop simple constant conclusion small become order method method small corresponding simple map limit method advantage occur apply except must throughout expectation numerator straightforward numerator denominator conclusion expectation homogeneous degree degree give iterate identity derivation change calculation behind calculation justify rapidly expect expect linear map expansion expansion odd variance obtain expansion anti symmetric show computing see follow symmetry similar algebra symmetry lead formula multivariate covariance function value odd formula see e expand denominator euler give method lemma formula give expect expansion result expand map similarly large combination moment result though degree correlation use computational confirm may sampler depend dimension sampler bad assimilation problem later tie implicitly q cubic potential nonlinear part increment standard variable right hand vanish increment simple quality measure straightforward identity number factor possibility add simple calculation c subtract finally coefficient numerical experiment average computation instead logarithm use weight linear map save maximum normalize red line dot circle square slope illustrate line confirm expansion specifically numerical agree relatively measure moreover random map simple observe result illustrate scale tb show predict observe method estimate initial variable ordinary ode matlab ode condition correspond shorthand ode initial numerical sample vary scheme derivative hessian minimum fix collect become mode appear tb vary lm linear dot red slope blue slope six simple version small slope four perform perhaps modal shape however vary pdf functional analogous filter tb linear lm blue dot circle line slope blue slope confirm relatively analysis method random map regime become simplicity map limit noise suggest procedure analogous example emphasize concern sampler even bootstrap propose sample present sampler center posteriori take discussion wish analyze practice roughly require ode numerical hessian either formula difference simple require linear method require evaluation particle filter want cost cost must sample normally solve map exploiting guess
walk subsequence member arcs solid type graph constitute graph parent arc parent point point cycle line contain point preserve path semi direction shall semi point extend moreover path however text let v adjacent inner inner path apparent notice graph consist walk single walk section relevant context arcs section notice consist arc consider edge point point pointing point symmetric arcs line cg graph preserve least imply characterize keep otherwise finally walk identical independence graph disjoint stable conditioning conditioning purpose conditioning start generate line remove point turn line remove inner repeatedly newly subset cg obviously contain semi direction preserve cycle generate generate generate generate seven case except illustrate generate generate follow algorithm type prove preserve cycle contradiction generate preserve cycle unless path previous c cc b summary marginalization conditioning dag act graph parametrization graph parametrization maximal subject acknowledgement author grateful discussion support grant air force office scientific advanced research project deal marginalization markov property purpose chain mix edge class marginalization provide method generate marginalization conditioning structure start play independence arise study graph appear cg generally formal extensively specific come apparent tool capture independence cg exist independence distribution chain cg independence unobserved structure understand acyclic dags study graph order capture summary capture cg independence cg stable marginalization marginalization independence marginalization essential provide structure structure next section define definition section two read markov property stable conditioning cg capture conditional cg define mixed marginal marginalization provide generate marginalization marginalization conditioning graph marginalization marginalization triple edge two
flow cycle determine flow nucleotide flow flow use compact elementary nucleotide extract power similarly bivariate recurrence relation flow number recurrence length nucleotide flow recurrence relation solved solve extract appropriate notation expression nucleotide nucleotide flow cycle symmetric nucleotide parameter involve elementary nucleotide probability inner sum range cycle value normalization factor factor length alone interpret sum flow cycle show normalization factor expression moderate practically cycle small extra place clarity reason cycle availability make derive exact distribution length formula nucleotide flow etc nucleotide flow point calculation calculation precision expansion gp algebra point length cycle discuss scenario flow cycle cycle cycle cycle follow sequence variance flow need sequence mean length equal cycle reach reach determine equal nucleotide sequence nucleotide cycle sequence equal nucleotide show nucleotide nucleotide probability exact also continuous curve normal distribution eqs evident accurately nucleotide distribution variance eqs two equal nucleotide cycle determine flow cycle calculate tail leave compare order fix previous flow nucleotide discuss curve normal eqs nucleotide flow collective behavior deal nucleotide symmetry result eqs nucleotide individual appear symmetric nucleotide affect next section nucleotide flow nucleotide flow finish nucleotide expression respect nucleotide first distribution flow sequence last nucleotide different see mean linearly sequence distribution cycle nucleotide probability sequence nucleotide eqs magnitude probability nucleotide shift slightly confirm show curve mean variance normal multiply cycle previously figure perfect variance calculate mean length determine cycle nucleotide flow cycle extra clarity reason determine flow cycle flow nucleotide exact calculate continuous variance distribution eqs normal multiply normalization cycle cycle approximate accurately slightly long tail slightly tail distribution shift confirm exact calculate normal eqs distribution technique important sequencing technology case cycle cycle give explicit formula distribution accurately formula variance vary flow flow cycle target sequence long length big figure length sequence explicit useful platform guide monitor daily user generation sequence nucleotide number formula accurately distribution software development sequence generation sequence sequencing ever already biology concept sequence technique compare generation sequencing long read length sequence technology development software development flow cycle sequence fix sequence flow cycle approximate distribution probability recurrence length flow remain brief description exact present explicit cycle length accurately formula would useful practitioner protocol synthesis chemical protocol add four kind iteratively nucleotide nucleotide nucleotide complementary nucleotide
predict stock growth month time period availability business stock market throughout year representative operating account among display stock year determine performance market quantify respect yield positive gain respect stock financial individually financial becomes consider flexibility handle capability initially available stock stock currently analyze behavior discard stock financial sake completeness combine limitation comprise stock small stock price model trend often dependent financial far small market financial parameter critical describe use supervise classification datum financial address step eliminate financial stock explain eliminate stock miss threshold financial secondly perform stock year financial aforementioned company feature normalize begin normalize mean across stock year smooth financial smooth frequency filter domain ideal rectangular cosine transform dirac delta rectangular finally dct rectangular discrete width dct rectangular smoothing component robustness compute stock determine relationship financial describe stock vector denote corresponding financial year let relative date financial past year supervision parameter form year end explain stock current performance make year narrow stock subset obtain allow change supervision reasonable fix cardinality cardinality particular initially supervise near neighbor svm technique suggest svm accuracy svm financial capability financial parameter take form high exponential grid search gaussian kernel motivation work involve time series analyse support tucker condition consider acceptable minimal apply partitioned entire stock average average ratio stock stock prediction ratio stock stock find suggest technique stock evaluate portfolio comprise share allocation higher future return portfolio stock return stock market month optimize portfolio market mid long since analysis performance naive near error svms optimize develop mining effectively collect period art tool cosine principal stock dimensionality several space handling capability discriminate stock portfolio optimization svm portfolio month significant algorithm artificial intelligence believe fundamental room machine perspective broad range simulate allow see occur modify close finance perspective impact stock statistic portfolio build work assign important future
b perturb event optima contain thus continuous density distribute class stochastic insight execute version give algorithmic foundation sampling discrete case suppose build node index subtree root node root root max root read yield distribution inductive know max suppose like leave child amongst truncate exceed truncate partitioning strategy allows still make infinite begin respective set truncate yield think perturb energy eventually complete construct b intuition first bound lb lb lb mb efficiently without henceforth add log density draw recover likelihood add decompose tractable rejection procedure refine upper region lower come come eq region term execution illustrate begin dark blue dash split upper compute child medium node put region sample split produce blue bound great queue tree measure exact termination run partition region return maximization use need determine variant computation computation drawing evaluation region lead case unimodal child immediately without queue maintain provably relate global bind runtime sampling parameter appendix termination refine rejection appear discrete domain rejection maintain piecewise target proposal rejection reject choose split computed region process point accept perform identically parent choose draw volume proposal piece would piece rejection return effect refine unlike return refined blind computation ultimately dominate possible persistence inside refinement aim section bound second show line library automatically compare mcmc slice experiment region region side split proposal case unimodal function peak essentially average run clutter mean fix outlier put term log bound bound dimension half half ensure make grow draw grow exponentially reasonably tractable computation run alternative case instance likelihood allow implementation problem mean bound clutter term quadratic quadratic tight vary point important evaluation explain happen show width interval purpose programming choose inspire physics biology automatically sampler legend c letter parameter wish noise reasonable multimodal large difficulty five set likelihood axis difficulty evaluation rejection accept rejection large varie tie uninformative bound also cost evaluation plus evaluation heavy tail log set set equally mode near decompose put per appear compare refinement discuss early refine piece large split region per drawing range along evaluation tradeoff two rough computation cost experiment single variant trade computation computation representative operating refinement ask algorithm across early evaluation expense bind achievable slice computational budget sample slice across mode ever switch answer question case energy develop approximate perturbation think favor efficient computation approximation develop establish hope explore question high uninformative search region little rejection adapt rejection allow leverage particularly several follow estimator would like advantage might branch branch subset support thank early suggestion appendix detail shorthand eq cdf identity throughout independent truncate interested derive gibbs eq axiom theory analysis construction correctness top construction circle center g thick g north east dotted dash thick g color dot west north color blue dot east g north west north blue thick g alg blue indicate truncation run space form queue proceed effect top would distinguish value sort allow choice distribution top index distribution produce set give prof equivalence distribution small tree notice whenever omit induction realization realize max queue come tree parent indicator intersection subset complete get location partition subset independence max get entire partition iff split result section deal correctness termination decompose tractable terminate bound global analyze global closely realization return thing upper search let optimal least visit global terminate function lb lb k lb kx lb termination queue need simplify search alg rejection bound terminate iteration rejection terminate termination iteration terminate termination history nonetheless stream value bind terminate joint proceed first pdf cdf difference independence look eq equal basically infinite function multiply permutation cube complete correctness termination return linearity linearity bound termination return q stream obtain b correctness consistency requirement need verify b partition fine get different set realization split chosen thing optimal suboptimal bind explore produce reach explore region condition region explore proportional thus region
significantly improve partial filter pca pc generalise entropy roc curve pearson thus filter neuron baseline challenge unlike challenge use variable predict neuron method forest constraint potentially tree depth conditioning level pre use correlation average correlation outperform dataset important poorly term well nevertheless partial well limit promise work unable direction seem disadvantage respect opposite neuron interact predict network e already reach curve report imaging consists two I process neural peak activity ii infer statistic outperform method optimize prove simplicity constitute solid fellowship office specifically tune challenge description respect present tune dataset monitoring simplify dataset believe tune clearly simulator provide purpose implementation detailed presentation present experimental proposal section filtering compose signal statistic method filter filter apply difference peak magnitude remain hard complex filter separately method case process application term sensitive value see filter average correlation f display similar resp equal less competitive weight resp combine prove marginally improve statistic beneficial orientation heuristic activate delay activation neuron indicate direct observation could time neuron whose choose neuron difference modify orient association discover provide enough way contain matrix give use use try value pass filter average pass filter slightly ht c c undirected direct report challenge except connection connection per similar fire e highly topology apply confirm average full pc report slightly account average averaging method pc svd determine combination variable combination combine know whose activity obtain sort magnitude component explain neuron exactly challenge seem necessary exploit value immediate carry infer directly team private private winner r yet effective problem imaging consist step raw peak activity infer association neuron partial statistic lead propose compare respect partial human complex biological connect unfortunately direct brain feasible currently activity neuron produce intensity amount neuron time series include device slow decay represent set represent direct connection express edge indicate neuron term version win method correlation describe present property additionally win slightly short period peak long period raw process fluctuation fluctuation activity overall illustrate figure raw light affect quality accordingly filter frequency noise filter furthermore spike indirect indicator mainly filter model exploit although time exploit propose coefficient precision concentration assume I measure dependency therefore naturally detect association neuron spurious indirect partial interaction metric practically speak straightforward obtain improvement replace principal performance infer process
variable characteristic composition child status status education school health interest home missing survey status miss status unlikely variability neither miss solely complete case pairwise miss restrict vector separately avoid continuous spike participant decompose indicator indicator discard first complete status home since miss primarily account miss compare display proportion month also account apparent difference estimate differential display report regardless actually display quantity median accounting substantial notable exception estimate adjacent line tend gain effective show home home complete mi small interval mean complete mi relatively universe previously miss benefit amount size production conceptually suggest adapt incorporate finally model structural zero contingency impossible skip restriction among expect adapt zero adapt mixed depend value categorical vice versa area gibbs draw parameter outline exact collapse sampler adopt simple quite break missing entry full vector element entry equal categorical variable update chain dependency block get miss block first marginally accord drop submatrix q submatrix suitably component column stack vector vector iv v mean sample dx ib ib rt conditionals b bayesian mixed demonstrate repeat use imputation improve compete run counter model imputation appear default implementation method incorporate default difficult translate encouraging field production nonparametric bayesian continuous develop engine imputation mixture multinomial normal incorporate categorical normal component categorical component categorical continuous link complex minimal analyst missing select collect accounting miss dataset similar complete panel default imputation equation program united longitudinal public researcher policy census period four census change hope reduce improve evaluate census field give overlap draw production panel restrict state census construct comprise individual state assess collection result key among status approximately unless missing mechanism completely inaccurate conclusion multiply sample imputation repeatedly analyst combine rule analyst account uncertainty distributional imputation skewness combination categorical desirable use dataset nonparametric categorical flexible coherent engine idea mixture regression variable categorical continuous specify categorical variable induce assignment dependence couple dependence well attractive imputation help preserve suggest miss categorical si si illustrate distributional exist imputation include bayesian present potential improve default mi multiply miss subsample change accounting finally extension future characteristic survey categorical dependence distribution datum panel display work education level vary discrete usual hour eventually skew hour panel education increase education dependence categorical categorical education df df education level way usual child complex distributional sort imputation discrete variable infeasible cell imply impose often vector include certain assumption seem unlikely thus restrictive approach specify variable e fully imputation prove challenge effectively categorical outcome remain challenge continuous predictive specify conditional undesirable relate distribution typical equation however model distributional many additionally conditioning sequence could result sample individual let let use variable respectively valuable categorical categorical model brief exist model follow intuitive beginning multivariate continuous truncated dirichlet mixture assume eq prior break construction dp introduce adopt truncated version mixture ij I assume imputation model mi engine arise h make put burden normality dependence dependence categorical true matrix distinct structure burden somewhat allow possibly interaction survey component much particularly since allow component independent add restrictive example dx unable already code however maintain give assign truncate stick give r induce component indicator probability assign truncate stick eq distribution call infinite shape rate equal convenient truncation level moderate initial paper specify parameter default hyperparameter usually insensitive hierarchical inverse wishart restrictive relatively particular poorly estimate insensitive marginally large insensitive marginally categorical conditional still latent distribution cell table mixture rx normal represent mean encode encode within matrix dp relax within component well low level parsimonious somewhat assume much flexible factorize author grow due assumption formulation force strong assumption number mixture induce distribution incorporate predictor within model discussion suggest discussion overcome cluster coupling model assignment couple mixture variable encode weak within dependence analyst simultaneously avoid center multivariate dp dp separate conditional assign dp difficult interpret margin somewhat arbitrary evaluate imputation study take wave wave exclude record consist census select continuous categorical variable table efficient keep challenging imply table cell status home education worker work increment hour approximately ensure complete create distribution describe material keep conservative half practice carefully relevant parameter mean quantile variance complete run cluster mid sampler minute make much scalability incomplete implement software application logistic conditional procedure mi complete incorporate age year expect rather skewness make normal likely coverage width imputation superior repeat sampling half coverage rate great nominal coverage shorter bias confirm range small driving capture effectively tend additionally year increase function year standardized child child child home appear play home vary correlation age child log old year year regression
confidence inverse difficulty low parameter rely geometry estimation cone capture statistical inverse complexity rest definition review formally program gaussian geometric beyond relation discussion proof result notation introduce show instrumental characterize difficulty statistical estimation later section complexity program convex duality use denote unit euclidean ball confusion also transpose inner matrix k nc na vary place place high much lie sparsity noisy completion linear view without loss generality standardize notion collection atom denote collection basic countable illustrate figure atom geometry functional q element hull atomic norm cauchy schwarz ball atomic hull atom figure collection atomic determine cone cone cone illustration red black blue dash notion look atom atomic recovery atom atomic norm polytope tangent cone ball tangent cone difficulty sparsity geometric cone refine local inverse regression completion matrix atom consist nuclear spectral hull nuclear parameter scale geometry tangent cone recover atom sign vector cardinality hypercube tangent orthogonal recovery constrain case hull local share geometric geometric two width compact standard multivariate quantify orient introduce show noiseless deterministic example explicit p proposition tangent cone use analysis estimate set euclidean capture balance cover radius radius maximize determine measure entropy width geometric quantity convex volume following achieve linear geometric quantity tangent cone isometry constant isometry define cone isometry preserve tangent isometry constant atomic local tangent cone model high respect introduce lead serious directly generic atomic tuning localization depend geometry atom give atomic minimization complexity specify localization generic utilize selector minimization trace regression investigate section set important inferential point inverse consider feasibility satisfie tune solve version convex program estimator parameter normality contrast test normality investigate program unify treatment general tangent cone hypothesis include theory geometric problem let analysis selector nuclear strong tangent cone geometry analyse asymptotic constant capture extend analysis atom atom unit localization radius guarantee feasible tune gaussian atom arbitrarily accord attain choose operator follow gaussian ensemble convergence bound atomic upper bound statistical accuracy drive two dimensional loss quite key lemma first follow gaussian tuning value address gaussian call operator isometry constant isometry constant guarantee capture local isometry bind design around linear like geometric ensemble design bias program empty feasibility q interval low contrast fix nan standard contrast variance nearly information dimension require equation important tangent cone cone affect tangent gaussian design universal stand conditional statement basically two follow cone determine universal b problem quantity width estimate section apply general high illustrate defer assumption sparse recover reflect gaussian follow geometric regression interest sparsity probability inferential guarantee biased selector type asymptotic normality interval length parametric furthermore confidence interval recovery trace regression lead recovery nuclear assume sense examine detailed calculation characterize low recovery assume true constrain nuclear estimation loss inference theorem happen width atom rich use construct confidence interval perform hard sharp theoretically approximation sign sign norm ball program eq general lead sign sign constant recovery please detail minimization program interest orthogonal calculation rate apply geometric constrain orthogonal recovery model high universal formalize framework permutation plus rank completion permutation atomic corresponding atom nuclear norm view see develop section geometric gaussian carry concrete specific distributional essential shall design random control section choose make happen tangent cone rate local isometry prediction regard quantify local tangent isometry condition illustrate behave worth note condition distributional view constant radius calculate distribution feasibility follow eq analogous event choice tune estimator decomposition extend set distribution contain gaussian intrinsic difficulty unify design compact convex lower bind inverse cone q mean condition volume ratio see apply cone cone cone parameter define design theorem low isometry design ensemble design upper link cone compare upper mild upper left give section eq leave see local tangent cone sharp p reverse hold tangent cone vb obtain sharp please reverse paper unified characterization statistical inverse major tool process minimax low dimensional general packing derive generic arbitrary star shape case prove divide lemma easy prove prove bound defer fashion tangent cauchy schwarz atomic early eq complete case least event easy calculation know p monotonic property probability least lemma reduce probability feasibility indeed high probability column lemma establish relationship last consistency key know denote packing respect e recall eq assume standardize linear cone packing cover ball element would like fix think nk trick fix lower modify introduce closed p tangent cone observation q step require standardize linear generality assume standardize inverse cone packing cover eq universal combine plug convex cone corollary application denote refer width cone proposition simplicity rate treat mh sm width bind tangent sparse vector lastly euclidean enough mh h th tangent cone transformation unit frobenius proper enough corollary corollary cone euclidean gaussian front thus tangent cone orthogonal show euclidean preserve v apply definition school unify ill pose linear compress orthogonal computationally feasible program statistical interval framework inference difficulty capture characterization width estimate drive wide high dimensional draw mathematic computer electrical often fashion mainly analyses statistical interest nz n dimensional much commonly assume atom capture true inference
convex dimension small relaxation unfold u ki ks element n u ki k e least cover compact maximizer believe distribute tw ji k moment ki use hoeffding sample replacement entry proof tensor position maximally hoeffde obtain theorem spectral
mis specification secondly structure establish chen estimator four asymptotically numerically asymptotic obtain specification appropriate series expansion enable illustrate difference four asymptotic notably great technique tend replicate asymptotic former extensive mis superior term mean range mis specification produce mis criterion three estimator demonstrate mis specification criterion mis short memory dynamic true form four mis specify pseudo mean mse mis specification also lemma proof derivation text bound specification spectral estimate away set derivative j g mis model lag autoregressive assume root root unit circle assume finite infinite specification process already assume move absolutely slow rather exponential typical process accordingly provide e realization process generally objective mis say explicit mis produce specification could size consequence subsection estimator hereafter outline derive subsection alternative minimize define minimize frequency estimator minimizer subscript limiting follow exist argument imply criterion express explicit covariance derive mis specify log function estimator expand expansion furthermore operator q appendix q vector density satisfie respectively estimator mm I ahead would mean implication mis specify member value close spectral give fit parameter value predictor class square mis applicability distributional however form associated order well relevant derivation specify determine relationship deviation derivation certain deriving expression full model polynomial power expansion investigation normally nonlinear infinite sum determine expansion replace sum magnitude truncation numerical memory order mis ar arbitrary truncation arise computation spectral expression take variance operator ar fractional series expansion gamma numerical alternative formulation expand yield denote use namely rr function mm obtain algebraic desire numerical expansion numerically ni dd q n mis specify sr derivative exact ar derivative readily pseudo evaluate treat similarly depict contour function discussion choose coincide great perturbation version contour indicate limit neighbourhood parameter behaviour elliptical function close globally turn mis show result write critical nature deviation property range chen derivation estimator theorem chen estimator establish three case outline show converge proportional objective see detail present key point pseudo something mis rate great sum case explore sample memory type mis specification estimator exercise obtain method adjustment curve depict statistic bias mse pseudo mis specification value function value two mis result parameter lie interval mse relative computation need specification replication mis correspond value subsection document mis specification report term true form mis specification turn three list ease mis four estimation j k reference scalar mm define give bias adjustment tackle issue turn begin apparent general separate expression e four formulae scalar specification standardize plus truncate replication underlie limit use kernel formulae u expression evaluate formula necessary behind method estimate maintain mm particular table label n specification size sample increase location go limit salient note cluster domain estimator small dd label dd domain discrepancy distribution reasonably limit distribution case provide four plus visual domain domain perhaps produce reasonably supplement four pseudo true formulae evaluate subscript table additional key value set highlight bold table small bias estimator bias mse theoretical tendency sample value addition tendency value two tendency standardized cluster far estimator theoretical mse estimator increase indicate pseudo parameter overall memory increase nevertheless small uniformly preferable mis still perform bad result record table mean severe mse bias tend suggest mm ar cause mse mm due limit experimental suggest dynamic correct form act extreme mis consequence mis specification severe coincide nest match mis estimate example estimate mis mse lack mis bias mse mse parameterize specification superior assess various specification parameterized model fractional relative estimating effect highlight estimator mis specification mis majority superiority robustness square mis incorrect relative estimator estimation estimator combination ratio c mm l c record confirm time efficient exceed estimator across specification formulation obtain dominant paper theoretical relate mis mis specification four pseudo general demonstrate mis specification pseudo fractional finite estimator extent specify extreme mis specification frequency together former distribution closely mse calculation superiority mis specification establish necessity gaussian dependent process straightforward relaxation restriction consider seem desirable true upon interaction mm extension result anti persistent stationary facilitate broad circumstance extent cover prior possibility mis specification explicit offer approach unnecessary latter also relevant boundary practice previous stationary fractional offer sensible investigation conduct smoothed situation fourth relationship bias parametric estimator mse mse express e mis depend magnitude mse bias imply mis point raise question within estimator still value reference practical circumstance remain current proof even value continuously set partition ny ny distribution conclude variance integral g n q em limit third ik ty ny ty therefore uniformly mis specification toeplitz zero reverse respectively deduce tn limit
term account software potential linearly vanish cutoff continue linearly compare force field comprise inter parallel force schedule replica temperature gradually boltzmann ensemble neighboring overlap tail varied varied force field obtain accuracy receive physics ny usa university usa research division institute ny interest include digital brain computer interface image ph university usa currently physics college work employ bayesian characterization institute mathematical university laboratory california technology usa division ny bayesian application present along numerical technique processing reasoning employ bayesian processing physical science perform evidence often overview behind technique great excellent span decade spread great neural nuclear physics engineering statistic general measure quantify within topic logical enable logical context probability consider statement propose particular conjunction write derive basic boolean logic expression result consist evidence think rule knowledge posterior probability must assign may possess instead probability rule leave assignment result inductive logic inference indicate symbol quantifie summarize particular maximize refer probable map estimate value act give prior problem factor sum often model theory compete probable examine rule way respective case prior lead concept equivalently odd odd q ratio two bayesian minor factor selection special demonstrate vary information possess assign effective likelihood around attain product well achievable goodness factor width multiple volume compatible volume knowledge compatible datum new prior pay entity necessity increase model interval beyond achieve upon factor scale increase normalized unity deviation ignore factor involve factor fall outside decrease rapidly actual parameter single computing odd comprise factor ratio classical selection comparison importance analytically technique quite expensive large forward point reader excellent resource review recent find respect zero eq approximated markov chain attain extreme compute ratio evidence write q sample eq avoid cause integration evidence build analogy statistical degree govern energy evaluate compute high dirac delta function know system energy heat degree view evidence write knowledge way canonical evaluate evaluate evidence previous class parallel method fractional power letting vary smoothly bridge recent sampling method focus logarithm connect distribution path define relation canonical constant yield choose constant energy distribution approach sample produce along path evidence sometimes accurate combination compare share odd define share simplify draw mixed log odd integration open intermediate rely stochastic numerically posterior contrast nest aim cumulative mass contain contour evidence likelihood understand nest use prior likelihood since decrease monotonically sort sort mass use increase thus iteration already attain valid randomly state evolve increasingly give sampling focus high constructing nest restrict higher high nest state strongly evidence simultaneously sequence contour typically nest rely k easily serve peak drawback nest facilitate sampling boundary help arise determination expense introduce parameter focus detection analytically focus spatial sensitivity light sensor estimate select molecular mechanic protein evidence ratio analytically correlation detection amplitude filter originally brain computer interface detection record channel couple refer coupling nan noise denote refer signal state q symbol detector record mt relevant coupling latter amplitude care detect represent without refer detector channel deviation signal entropy assignment mean know equal signal assign parameter write gaussian involve restrict amplitude integral odd subscript amplitude odd subscript range expression contain information aid signal filter synthetic eeg eeg response response commonly matlab channel synthetic eeg pz channel epoch ms length comprise hz single remain eeg effect snr filter performance create varied db typical see eeg application illustrate target signal snr snr db curve correlation b roc curve snr detection oppose quite snr db sensitivity specificity mean specificity I correlation detection cutoff curve b signal snr db filter traditional quantify respective roc db filter consistently traditional filter snr quantify area roc curve snr consistently snr demonstrate select order gaussian model sensor accurate pair surface measure intensity reflect light spatially weighted sensitivity make light sensor place surface consider gaussian sensor amplitude center constant unity gaussian model consist gaussians require subscript index gaussian gaussian intensity figure surface illustrate record increment step sensor surface sensor corner break parameter nest nest iterate stop log value less four compete model evidence time probable model compare black make intensity show excellent agreement involve light sensor determination importance effect find currently method characterize involve series observation light come distant star mechanism specific system perspective variation reflect light throughout effective back distance angle observer connect star contribute side spectral response side remain involve star star center center observer star amount star approach decrease velocity two shift due proximity star induce variation twice since star change approximated exponent anomaly total effect bayesian effectively create comprised effect four orientation allow circular evidence determine whether present perform call period temperature sort star emission fact
lot attention devote multimodal consider fusion fusion investigate recent elegant come bayesian value low misclassification propose extension tailor indexing pairwise allow improve precision take diversity provide adapt challenge benchmark multimodal issue multimodal bring complementary improve image analysis survey modality group extract another order discriminate concept scheme one available merge see unimodal however heterogeneous fusion score fusion outperform give base decision min refer stack extra step stack classical majority vote I simplicity scalability due dependency view combination account machine indeed consider diversity illustration adaboost weak distribution introduce adaboost tackle take account diversity strong framework express program learn vote function minimization diversity learn aim usefulness fusion good perform layer positive base extension loss organize follow fusion image presentation quadratic weighted majority vote value machine theory feature label output size define choose majority vote low specific ib mm h introduce margin positive convention author prove inequality minimize counterpart vote justify elegant pac generalization principle sn nm ji nm h mm finally vote n minimize denominator show performance classification stand rank base pairwise preference discuss usefulness diversity key success combination indexing justify vote maximally uncorrelated diversity popular pair classifier correlation disagreement etc diversity disagreement pair rewrite moment mm objective moment imply direct optimization imply diversity maximally appear fusion separately accord hinge relaxation previous hinge hinge loss slack abuse however incorporation hard relax size stand framework map aim majority imply good trade maximally ht car cat c concept svm car cat average empirically usefulness fusion stack approach objective concept less unbalanced carefully classifier reason positive ratio keep could index sift local binary gradient histogram color moment gradient dimension vector image thresholde neighborhood monotonic gray sift codebook semantic sift codebook mm rbf svm classifiers first h unweighted vote weight vote mm increase diversity follow set classical stacking finally fold validation low select lead performance report firstly fusion correlate linearly fusion experiment clearly baseline high student pair confirm p student test statistically produce constraint hinge really helpful diversity expressive hand diversity vary layer achieve concept diversity average preference student pair value preserve keeping note cost lower pairwise approach showing without approach good classifier account retrieval diversity come feature variability classifier error rate majority vote diversity adaptation preference indexing appear naturally prediction train modality fusion set confirm fusion indexing task beyond
unconditional family multiplying preserve parameter condition sum count poisson multinomial regret universal maximize close greatly compression likelihood likelihood small distinguished role compression sequence count extension eq positive q simply maximize poisson value later great string nevertheless approximately approximately bit description beyond know ideal maximize poisson match multinomial alphabet regret bit parameter minimax bit price pay sake code simplification additional know total arise moment depend alphabet count alphabet total count advance na maximize much tail law sort distribution datum within set demonstrate multinomial count strategy minimax alphabet strategy computational normalizing computing conditional require arithmetic code symbol appear large typical mainly give asymptotic pointwise alphabet formulate redundancy symbol find form asymptotic model focus infinite envelope pointwise work envelope provide large alphabet code purpose consider large realistic code prediction unnormalized count match total entropy projection equality approximate conditional investigate unconditional compare gain describe arithmetic code arithmetic count count introduce iii give simulated detail introduction unnormalize maximize likelihood count count prediction reduce sample right hand probability specify count question leave count maximize term restrictive target start look strategy use code count poisson maximize normalized regret product vector count set count follow follow minimizer partial bound derive pick rest leave close list case multiplier numerically regret expression redundancy reduction bit depend expression bind multinomial distribution maximize upper within easily see flexibility parameter introduction uniquely symbol relative portion symbol sort quite skewed string tail occur subset n nf ratio give mainly remainder product define dimensional symbol rest regret denote distribution minimization derive nf treat code alphabet count alphabet regret symbol symbol symbol induce bit contain count extra flexibility achieve piece code adapt count bit work tail count symbol finally suggest replace string envelope suppose envelope I string distribution envelope distribution restrict minimax envelope tail need envelope match regret alphabet datum choice envelope symbol string discuss fast enough symbol arrange realize count maximize approximated method c similar lemma approximately bit average lead achieve minimize account string finite length string look count component maximize poisson condition maximize multinomial count multiply uniform string stochastic confirm horizon sequential log conditional conditional produce cumulative regret sequence accordingly simplify study large jeffreys jeffreys adjust symbol alphabet predictive hence symbol discount predict rule alphabet frequent symbol approximately symbol assign code book translate chinese bc book rarely code character character total character appear small character introduce maximize poisson count performance code close assign optimal family pointwise finding produce early distribution characterize distribution count precisely fact suffice fig solid hence area curve although unnormalized mid stand term portion half step rearrange upper refinement approximation sum integral also due fact moderately large well k aa know sum gamma achieve follow bind upper denominator attribute little algebra bind need lower eq since attribute step approximation pick numerator subtract yield taylor expansion term get large follow equation c definition q therefore distribution condition equal eq say value function redundancy distribution moment alphabet distribution total count redundancy minimizer redundancy therefore minimax redundancy magnitude part second resemble taylor hence easily partial inequality attribute pick envelope class multiplying exponential normalize na minimize depend envelope symbol eq approximation deduce hence
concavity elementary symmetric negative argument consider km ne inversion restrict column set annotation annotate dpp compute theorem theorem theorem corollary pt pt suit prove useful application efficient dpp learn dpp dpp even study diversity process dpp configuration characteristic recently play increasingly role diverse configuration spread kernel define span kernel associate decompose eq interpret point ability maintain via remarkable efficient fix open set dpp arise maximize numerator concave log determinant contribute convex lead non even assumption form parametric use convergence stationary function kernel bayesian dpp capture scale unknown inefficient ascent mle case differentiable occur limited scenario size counterpart dpp kernel mle technique inference sec modification accommodate sec derive dpp assume know explore model checking technique method dpp spatial diversity image discrete set application size elementary recursion continuous naturally operator appeal density give dpp continuous eigenvalue dpp generally include kernel quadratic show approximation finite represent propose sec consist dpp dpp dpp dpp log iteratively shrink ascent ascent guarantee discrete dpp straightforwardly polynomial material optima dpp present sum gradient continuous applicability maximization scenario dpp operator truncation gradient unbiased attractive hold optimize kernel dpp dpp prior neither markov highlight hastings mh slice although method employ walk proposal tune width distribution walk mh posterior efficiency tuning proposal conservative exploration need proposal use first first slice width expand interval accept otherwise become new boundary shrink markov propose accept alg way extend propose expand whether slice direction approach slice illustrative two dpp evenly spaced square dpp posterior mix walk slice sampling position slice indicate mix slice continuous inefficient infeasible even case eigenvalue seem suffer upper accept reject completely necessary immediately immediately far begin iterative exact circuit ratio alg supplementary tf tf apply candidate slice decide case accept slice keep newly reject generate new repeat propose proceed manner interval computation examine slice bound increase decision make adjusted interval procedure supplementary bound incorporate mcmc convenient truncation kernel arbitrarily dpp corresponding explicit truncation show sec dpp contrast mle even know prop combined eqs supplementary dpp eqs sampler challenge additionally tailor inference moment theoretical marginal eq case nm moment form certain case eigenfunction analytically moment define sec polynomial challenge eqs analytically available low quantity estimate numerically moment estimate large provide continuous dpp operator furthermore trace learn gamma supplementary material scenario use dpp scenario first estimate vary sample moment total point stationary allow many many interested quantify datum instead dpp material derive result parameter image study patient cluster diabetes phenomena stage subject average analyze highly parameter dpp quantifying level moderately analyse sec perform parameter leave evaluate hold sample eqs since preferred isotropic specify material slice sec fig clearly separate two class classify six six examine dpp quantifying relate diversity different category image search diverse search retrieve diversity human annotate six image total five dpp sift material via amazon present remaining ask experiment result spread evenly spread evenly aim human annotate differ six google extract type descriptor supplementary material category image dpp human examine consider probability add dpp category parameterize dpp conditional dpp kernel use informative prior annotate partial set different category annotate significantly feature human human human engine put keep highlight diversity ignore application combine relevance diversity top dpp kernel informative diversity tune accordance human annotate versus increasingly popular dpp infer parameter addition characterization continuous show check dpp study human diversity annotate gradient ascent ascent provide parameter dpp theoretical dpp straightforwardly example discrete gradient ascent
previous generate total follow leave ei predictive ir global maximum algorithm stop maximizer mean median ir function interesting correspond average median ir tail exact good es ei show well ei another series experiment optimize well fix ei hyperparameter es work fix use slice produce nb single hyperparameter section hypercube show ir random initialization nb es nb es advantageous approximation perform ei ei appear entropy explore iteration relatively greedy ei consequence multimodal one network optimize iteration return production term ph medium standard return series adjust daily return stock ba student portfolio truth noisy measurement predictive adjusted sample negative must exist fourier dual write consequently stack version result briefly observation stack easily predictive inversion lemma rewrite expectation approximate position optimizer order enforce location stack include note block kernel additional explicitly factor q multivariate non replace z parameterized compute ep start factor factor remove focus eq replace ep kl remove influence cavity set approximate perform form cavity set moment obtain normalize v identity factor constraint hessian contribution cavity q soft moment turn prior concatenation write q assuming covariance covariance associate eq block zero diagonal arrive university propose call gain acquisition term expect reduction approximation alternative bayesian hyperparameter es synthetic application finance show gain optimize find maximizer nonlinear function derivative furthermore evaluation challenge model computation minimize evaluation graphic another application drug chemical treat hyper maximizer provide noisy output likelihood sequential propose condition iteration final recommendation global maximizer latent describe guide gp prior specify jointly likelihood observation location condition past mean bayesian technique guide maximizer acquisition optimize location intuitively acquisition area likely encourage exploration search space recommendation global several acquisition example improvement improvement alternatively acquisition I draw green black thick dotted anchor cm height box box east optimistic posterior explore uncertainty expect global maximizer acquisition empirically evaluate real gain theoretic interested maximize location measure term differential entropy expect reduction acquisition represent predictive exact infeasible main difficulty computation analytical evaluation perform note equivalently since maximizer intuitively unlike previous entropy analytic entropy analytically marginals py ny term approximate sample approximately use code publicly maximizer domain restrict point dimensional probability write return element vector process multi armed could maximizer domain sequentially construct optimize however evaluate ultimately necessary instead sample derive existence dual feature consist stack inner mapping conditioning correspond finite approximately early approximate argument n introduce additional intractable difficulty definite far assume give force negative hard large simplify require multiply specific encode constraint briefly detail f incorporate joint multivariate derivative computation incorporate cdf integral form expression approximation propagation implementation ep gaussian process approximate non whose approximation I I describe constraint concatenation function approximation computation one classifier incorporate multiply give approximated normalization density cdf approximate variance py unstable occur multiply reduce produce prediction actual constraint acquisition acquisition formal treatment variance gp likelihood acquisition respect correspond integral global maximizer draw acquisition note acquisition respect information parameter approximate necessary evaluate independently input cost dominate inversion hyperparameter ignore derivative observation impose constraint less experiment square zero covariance normalize
signal turn dct compute dct exactly meaningful estimation complexity requirement prominent include discrete cosine transform series possess complexity transform imply transformation new dct quantization coefficient expect exact dct attention good dct approximation propose approximation dct scale round let round implement environment matrix shape attractive nan multiplicative shift transpose inversion make coarse dct presence scale scale factor minimize dct range compression ratio drawback orthogonality result method e encourage discuss adjustment computation diagonal therefore dct approximate possess assessment matrix introduce additional overhead dct processing quantization merge quantization procedure ultimately fast assessment addition multiplication bit count less additional good shift comparison numerically evaluate dash line ccccc assess methodology set standard image bank employ imply mathematically domain retain remain adopt reconstruct process image degradation assess peak noise utilize consider average may compression ratio moreover could outperform mid compression expense index square understand assessment employ image compression percentage dct mse lead clearly adequate high application bit additionally operate demand ratio recognition popular compression compression qualitative retain compression reconstruct propose reconstruct image via dct superiority reconstruct image quality correspondence dct low scenario mse possess constructive usual approximate mathematical series approximation take transform offer option circuit acknowledgment support de dct zero multiplication bit spectral dct adopt design superior sign cosine algorithm computational dct complexity transform compression cosine transform video image
account quantify covariance time instant time previous case would reduce sample problematic chapter issue divide segment correlation dft jointly absolute auto cross component dft stationary correlation great insight inference frequency procedure reduce detection detection exchange achieve accommodate frequency dft screening row column specify number characterize value dimensional specifically give expression detect threshold discover bound result phase corollary independent underlie evaluate statistical discovery organize definition multivariate establish spectral value screening characterize discuss complex screening multivariate framework triplet represent event represent letter respectively denote bold bold low letter cumulative distribution pdf pdf follow definition conditional conditional part real part value variable real part gaussian vector entry hermitian write coefficient matrix letter clear context entry x x order random ik ik time dft translation property toeplitz depend represent toeplitz write ik state series dft e asymptotically uncorrelated auto cross toeplitz give jt I n generality time zero ik z functional ik km I km I km j toeplitz tn tn tn tn write toeplitz matrix km n g km lm circular dft lm km lm km lm km lm therefore ik e km equation let give ik km g km km g kn ik km kn I ik km g I magnitude summation km kn I I I lm lm n proof ns therefore n conclude ik j ik use ik n specify real value covariance write ct stationary ar process r conclude dft ar sequel assume series jointly dft functional theorem independence component property absolute asymptotically time screen dft next discuss correlation screen value identify highly method apply discover frequency matrix definition result characterize regime transition refer integrable strictly decrease assumption generalize make screening normalize correlate component sample assume correlation partial correlation matrix matrix h apply discovery threshold I correlation screen number screening level depict match match pt arc cm arc complex case exist h norm represent unit relate section area x onto fix spherical define uniformly fall distribute generality x chi square v u fold average detail I I dependency coefficient fig screening expression probability discovery least dimension use generic partial j represent population matrix distribute cc first prove vertex graph ip inner summation set distinct l ip k follow sequel index index among sufficiently large p expectation p l lm l identity lp summation p p representation e f combining op sum chen define b j ki p k chen stein index care op line round join round pattern pt pt pt pt pt pt pt pattern pattern pattern pt pattern pattern pt pattern pt pattern pattern pattern pt pattern pt pt pattern pattern pt pattern pt pt pattern circle line circle pt width circle circle width cm width pattern width circle cm width pattern circle cm pt circle width circle width cm line pt width circle width pattern circle pt circle circle circle circle red color pt circle color blue color circle pt color blue circle color color node red color circle color blue pt color pt color circle circle color color white circle circle white color circle color pt circle color circle circle white color white color white circle color white circle pt color circle white circle pt circle pt color circle pt white circle white circle white white pt white pt color color white white color circle color white color pt white circle color circle pt color color white circle pt white circle white circle white circle white circle pt color white pt color white circle color white circle pt complement union red index complementary fig inequality term argument max remain multiple integral op apply relation pp op p immediate provide expression goes prescribe weak define corollary depend evaluate argument zero arrange ok discovery depend assign statistical corollary large sample reduce conversely number wise discover least exhibit phase fix sharp phenomenon motivate definition critical threshold approximate j number critical matrix real screening complex value exponent different small threshold degree dimension triplet small example one discover bottom show either reliable discovery increment sufficient critical enough bring value correlation stationary sample divide series frequency available construct partial correlation series magnitude correlation quantify statistical significance significance statistic extreme assuming screen value maintain degree assume illustrate equation help initialization degree degree select value partial significance inference problem perform aggregate inference straightforward manner example easily value series j degree frequency time frequency degree I l simulation confirm
compression ms refer generation initial dm compression cost initialization pseudo quadratic neighborhood alg computational nature sensible presentation effect cause effectiveness calculate dissimilarity describe ds alg set permutation obtain arrange accord achieve consider generic efficiency compression rs asymptotic case compression efficiency implement varie sec notably theorem compression operation alg clusters parameter parameter present estimator synthesis dimensionality sample restrict experiment choice use exponent easily cause representation graph certainly cluster optimize one evaluate input go complexity section update improve subsection dataset adopted discuss well graph repository contains consider finally first character characterize level image brevity essential reader ref reference discussion since dataset contain characterize vertex edge vertex measure none none complex symbol version sec sec version compression theorem variant variant however first one near neighbor nn equipped primarily herein previous work fast train four aforementioned variant therefore straightforwardly v meaning fuzzy nn summarize configuration execute genetic population synthesis fitness setup allow comparison genetic random mutation aforementione code genetic implement consider execute moreover affect h size preliminary test process time seed report report cpu rs synthesis test regular cpu ghz operate measure routine library refer estimator adopt specifie feature base operate ds function adopt system notably operate classification detail aspect become constrained scenario big entropy operate dissimilarity c nn nn nn gmm soft svm fuzzy soft le svm svm bayes svm c c nn report c v c v mm mm set c c c improve discuss characterization dm enyi adopt construct compression directly compression level explicit set proof consider technique proof study cluster generation system known equip arc adopt result fast less parsimonious concern rs nn rule yield accuracy serial cpu focus accuracy confirm effectiveness moreover cpu highly global bring close applicability big large vector dm row know mainly worth performance obtain embed dm could enyi neural straightforwardly develop like generic object focus bad case give th prototype good assume adopt compression bad compression purpose sequential sequence first cluster consider alg well possible preserve bad order instead odd order alg consecutive element maximum consider combine obtain compression claim single cluster radius spherical measurement evaluate define dimension value estimation quantity input maximizer monotonically relevant remain change achieve therefore normalizing compression side hand simplified provide singleton convention express parameter eq evaluate theorems theorem theorem enyi represent label graph become increasingly field intelligence tool procedure test label mining operate computational complexity viewpoint major purpose dissimilarity theoretic evolutionary focus key subroutine system compression effectiveness result variant indicator classification structural classification set considerable concern complexity offer powerful interact static scenario application cite almost scientific circuit network general label graph rapid motivated availability describe complex orient level operate label deal notable map classification enabling possibility adopt recognition cause intrinsic label topological semantic vertex design classification inexact operating prove offer complex dissimilarity embed classifier show art term synthesis classification theoretic interpretation dissimilarity dm practice characterization effective compression deriving demand improve version scheme rely formal computational synthesis maintain art performance elaborate scheme estimate enyi fast technique span formal compression experimentally demonstrate operating estimator comparable result underlie organize scientific section original classification system throughout primarily discuss bad efficiency develop comparison classifier dataset dissimilarity base min max conclusion direction designing concept key estimation mutual information describe quantify model quantification characterize system shannon generalize formulation interested generalization enyi call enyi enyi q subsection enyi technique sec introduce entropy eq eq simplify enyi plug theoretic datum descriptor entropy kernel zero mean kernel rise enable trade dependent random gaussian evaluate assume value input extent notably normalize evaluation cost measurement nd connect mean straight r entropy estimate accord end set span calculate term constant dimensionality approximate enyi entropy suggest parameter perform task definition sensible measurement account respective euclidean edge quantify involve computation well know algorithm cost adopt fast approximation concern dissimilarity representation element pairwise key nonnegative dissimilarity relevant set rs dm property adopt dissimilarity obey common metric say metric dm embed unnecessary computationally apply dm consists call ds embed fast represent dm common perform linear correct dm aim preserve euclidean mapping spatial representation selection play course role technique select essential feature extraction many image verification clustering system attribute tuple edge topology moreover generality cover broad graph order pair undirected generality nonnegative label face difficult provide quantify graph non difficulty proper effective possible vertex mining define mechanism component modern graph embed basic transform np term basically aim assignment vertex graph successively edge proper positive enable hence applicability whole e svm operate embed geometric information divergence distinguish main category define work extract characterize former g accord capability adopt core g restrict variety graph effectively representation adjacency laplacian reader therein represent input dm configuration dissimilarity although heuristic dissimilarity operate assignment vertex graph edge ds dedicate operation quantify dm fall well unique normalize evaluation range yield extent system operate svm assign test fig give rs fig synthesis fundamental parameter entropy entropy fundamental role compression cross learn validation global optimization analytical respect global allow make use hardware software implementation characterize arrange code entropy threshold label dissimilarity measure range characterize synthesis objective dissimilarity representation compress expand evaluate recognition validation account relate characterize capture dm increase spread separability class assume presence class would compression search rs rs compression define dm dm submatrix value vector measurement concentrate joint close systematic compression representative prototype evaluation first address algorithm complexity grow analyze compressed dm denote corresponding prototype concentrate value estimate become degenerate prototype extract graph notably derive search recurrent although try new searching subgraph improve primary computational first present rs advanced compression sec sec rs characterize sec operation synthesis rs characterize binomial operate cause unbalanced entire dataset use mutual avoid burden involve estimation basic algorithmic scheme grouping dissimilarity main sophisticated generate intra low deduce analytically consider certainly proof order radius maximum cluster close representative cluster dx ix th cluster representative together compression
self prediction immediate observation prediction coefficient prediction depict prediction prediction rescale normalised squared measure series approach normally distribute self assume analogously self respective equation denote continuous depict measure substitute approach discrete track cover pair possess title track another comprise cover average track furthermore perform base meta pre audio music use define query track average track seek cover member entire harmonic content audio implementation account deviation hz extraction describe magnitude apply wave across band pass apply window determine autocorrelation maxima centre incorporate bias towards preferred obtain programming average root euclidean component predict note evaluation align predict interval feature account key cover summary vector sequence correspond circular shift denote circular prior discrete aggregate across track codebook execute square method detailed measure list method compute measure matching case parameter computation obtain string map character describe algorithms string complementary string compression loss x average perform self align analogously predict distance string algorithm kl symmetric compute symmetric distance performance divergence jensen divergence symbol normalise evaluate delay embed radius preliminary separate numerator complementary estimate conditional numerator self denominator song consistently track l l eqn string eqn string compression prediction prediction symbol eqn prediction music algorithm scalability work linear query infeasible approach retrieval indexing apply metric retrieval track relative influence performance examine normalise display surprisingly contrary outperform bag approach account temporal structure outperform evaluate compression dataset compare use average codebook outperform although relative performance gain fig b consistently compression obtain compression compare gain respectively tb h cccc h l cccc cccc h h display c comparison cross yield map baseline determine parameter performance examine qualitatively observe interval parameter correspond baseline approach refinement base normalise histogram observe cross discrete value combine string continuous value outperform gain majority approach consistently value value continuous outperform examine disadvantage self base estimate cross relatively suggest limitation prediction however dataset result conditional self facilitate results state improve distance describe combine fig display score mix combination compare observe improve performance dataset gain combine obtain maximal dataset evaluation reveal gain distance combine report rank dataset consistently gain tc high rank display combine tc evaluate measure pairwise song identification consider representation series cross secondly represent prediction mean determine require applicable firstly propose continuous outperform baseline approach secondly draw cross prediction discrete song song identification million song use compression view measure value preferable discrete representation obtain measure argue due work song large scale base value baseline evaluate alternative series far involve reconstruction recurrent network term architecture evaluate detail cover method quantify cover song information theoretic measure series discrete value operating normalise time song comprise million song dataset continuous outperform approach estimate normalised compression string normalised alignment improve performance value distance refine song cover song normalise audio quantify substantial interest music technique track song identification distinguish require audio track query track audio specificity track specificity since track share song previously record piece music cover song mid specificity song correspondingly song challenge song base quantify music significance intrinsic sequence reflect consider shannon theory predictive form modelling expectation unfold stream sequential theoretic approach conceptual quantifying purpose song determine pairwise track work approach quantify shannon kolmogorov normalise quantify string successfully across music choose cover song interpret compare shannon compete concern implementation extend examine value million song use remainder discuss related song determine describe section conclusion similarity audio distinguish temporal discard retain former distribution feature approach unable aspect harmonic structure variation role al limitation audio sequential music repeat piece music song identification determine song song across band song determine similarity dynamic energy propose et compute matrix substitute alternative recurrence precede combine high temporal et al represent techniques cross maxima feature distance sequence propose key value representation et discrete sequence use spectral peak pick extraction lee maxima evaluate pairwise correlation sequence apply symbolic al pairwise piece music perform analysis symbolic audio audio obtain frame perform differential distance track compression propose representation additional hmms pairwise addition representation compressed applie recurrence plot individual track structural similarity piece music base derive representation observe measure al measure similarity video compression alternative measure extend comparison propose audio concern song large scale music collection contain track collection infeasible perform expensive comparison track collection et pairwise quantify threshold combine locality sensitive retrieval sub salient track thresholde encode integer expensive apply approach theoretic measure measure al nucleotide cluster parse compare sequence alternative consider building sequence compress et motivate jointly normalised series represent piece independent distribute process quantify dissimilarity kullback shannon logarithm quantifie represent divergence widely bag specificity music content temporal audio sequence string number encode string similarly denote encode concatenation string denote kolmogorov aic bit output short mean distinguish string aic quantify string maximally string closely examine approximation determine use similarity sequential concatenation approximate heuristic shannon modification similarity detail quantify shannon quantify uncertainty accounting dependency analogously joint quantifie amount pair source addition accounting source entropy eq interpret quantify average emission knowledge observation additive approximated use entropy eq denote shannon account string estimate
become prohibitive take day lack big hierarchy achieve trivial penalty combine tackle challenge novel strict extremely implement organized notation general implementation oracle reveal minimax rate simulation analysis conduct efficiency necessary jj na na e follow symbol g b p j n notational design matrix ba multiplicative numerical equally sized matrix generality center exist intercept q consist describe parameter hierarchical attain penalty impose come second model whenever maintain without symmetry guarantee simple focus work group nonconvex sparsity induce induce vanish without concern challenging computational matter vanish give scheme predictor role identity vanish construct set functional arbitrarily impose apply often predictor around amount type preferable section offer general variable claim idea main paper theoretical challenge arise overlap indeed appear fast scalable cf penalty constraint derive sharp challenging cf section think admm penalty getting find application design track use admm lagrange multiplier refer penalty play lagrangian solve converge value r package augment lagrangian behave well consider general instance abuse suffice necessarily general operator define version sa otherwise extremely hoc algorithmic provide universal choice large I I convergence establish every accumulation conclusion purpose suppose conclusion include net helpful handling favor fusion cause double see bregman generalize take suffice empirically avoid line accelerate method b relaxation convergence error adopt attain p g p e ambiguity investigate typical type penalty give finite apply yield lasso tackle global perspective avoid strength uniqueness sharp kind incoherence e follow hold restrict cone hierarchy parameter play major theorem compare type convenience g j overall control gm vanish bind large achieve error bias applicable signal lot norm design bound true regularization choice literature group size order light size reduce treatment section conclusion grid extended type noise remove version symmetry develop w w j similarly regularity define however e e g type requirement design simulation usually indicate meaningful recommend regularization sense toward consider hierarchy double e assume ij satisfied hierarchy replace give example conclusion indicator occur mild e minimax minimax comparison benefit show g exist existence effect associate situation behave lasso hierarchy neither practically lasso lasso omit rr compare type consistency efficiency design generate correlation follow interaction interaction effect obey j strong regularization large perform theorem experience variable selection model handle get warm recommend implement set otherwise true calibrate ridge fair repeat time square robustness report run fraction j miss alarm rate cost time pc ghz gb bit summarize result fa fa fa fa cm cm ex ex ex hour hour hour contain achieve compare use group behave fa error performance bad reflect varie notice high effect think day path approach offer gain efficient large scale conduct california dataset consist nine characteristic neighborhood california median respectively challenge nuisance gaussian add study full prevent get optimistic hierarchical outer validation cv run variable post calibrate approximately day take scale median variable predictor panel figure original covariate heat restrict panel correspond interaction successfully add feature nuisance heat however interestingly interaction term confirm boost age difficult interpret overall provide cc function b p p globally p tool fix kkt imply sequence satisfied point denote fix tucker exactly global minimizer proof first difficult theorem strict iterate point b ab conclusion universal constant occurrence global minimizer p x column j follow complementary e p carefully ga e ep ep notational index let j j universal brevity manner easy sufficiently convexity stationary lagrangian
unit b recurrent cell content activation activation long short term memory minor modification lstm make recurrent weighted sum apply lstm maintain lstm output gate gate memory add memory forget gate degree content input note traditional content lstm decide keep lstm unit carry information distance capture fig graphical recurrent adaptively capture dependency scale lstm inside unit separate previous activation activation gate gate procedure newly compute candidate traditional unit wise originally activation gate formulation gate read symbol allow forget gate gate fig illustration lstm prominent share unit additive recurrent replace content unit current lstm keep exist content content advantage existence feature stream important decide forget gate lstm unit gate maintain perhaps importantly addition effectively create step vanish nearly pass reduce vanish unit sequence set model modeling speech music music dataset symbol binary vector use sigmoid speech dimensional raw audio design look consecutive sample consecutive version sequence output layer recurrent lstm rnn sec primary compare unit fairly choose make avoid size lstm deviation fix gradient prevent select multipli validation case rnn outperform dataset music rnns rnn lstm clearly outperform traditional lstm rnn perform fig learn curve cpu dataset update progress eventually stop advantage recurrent unit compare lstm heavily epoch ht second dataset dataset empirically rnn widely long memory lstm unit recently focus task raw evaluation superiority traditional evident task modeling could concrete two unit well experiment understand contribution lstm thorough future acknowledgment would acknowledge research universit de cifar recurrent unit sophisticated implement recurrent recurrent music speech modeling reveal advanced recurrent unit traditional unit comparable recurrent learning task book recently report challenging translation interesting recent achieve sophisticated recurrent long memory recurrent interested variant short lstm unit establish long recently two task al dataset sample speech dataset lstm cpu update recurrent network conventional feedforward neural able recurrent nonlinear composition logistic sigmoid length traditionally hide generative output give model output decompose end generative subject observe train rnn capture dependencie time rarely severe gradient base
mc dissimilarity dissimilarity mc proportional logarithm realization realization nan mean realization frequency relational unknown text profile entropy word profile relative undesirable often appearance network write author expression produce author appear entropy contribution text profile belong author situation transition profile rare word explore laplace infinite entropy proceed choice relative entropy text true text function frequency text symbol window length accuracy length connect independent clause entail change generation window length pick appear high computational accuracy since position apart order include different approach methodology adaptive frequently attribute frequent repeat marker experiment select function illustration text length word pool fix profile solid achieve network compose text likely variation choose say reliability large vary text attribute author find adaptive common implement validation author know text break length author randomly pick attribute piece author utilize interval cross validation round text build trial node well text attribute vary depend dash implement adaptive static I text static true example true static suitable method good shorter analyze likewise change adaptive rapid text text profile effective cross validation correct show variation rate text pick corpus profile word description range variation likely randomness approximately profile gain adaptive choose henceforth fix generation sentence discount length pick adaptively perform validation group correspond author span american author average book per book book mark translate author minimum word maximum english span th average play per length author maximum solve author ask ten text five five text book page text generate profile describe ten text result compute entropy multidimensional ten unknown text euclidean metric distortion unknown text depict empty author solid plot half author perfect two circle close circle fall plane blue dissimilarity entropy number distortion minor entropy relative entropy small text text text book text profile entropy text fig profile circle represent text color use distinguish blue profile specify author different text attribute represent green empty appear blue profile besides principal profile text dissimilarity l l l nr profile thousand rand l nr rand l l l profile thousand rand total word profile fix vary length increment text word profile consider profile randomly text piece build profile length text text attribute contiguous length pick write oppose piece different resemble text correspond run randomly text form profile amount choose ensure every text state page book typical play author difference rest carry information useful overall text even reasonably corpora author word accuracy author increase profile binary whereas ten monotonically author text increase long g word text accuracy attain correct vary text text first correctly attribute opinion piece corpora opinion piece corpora candidate reduce word accuracy acceptable binary short text support evidence besides number text profile text similarity write accuracy word exercise yield distinguish dissimilarity write quantify profile entropy relative entropy dissimilarity pair dissimilarity result relative entropy inter dissimilarity accuracy correlation dissimilaritie pool ten text generate profile remain profile rate profile dissimilarity choose ten text repeat accuracy text inter dissimilarity accuracy inter dissimilarity dissimilarity average account result pair dissimilarity small author carry period composition illustrate carry build dissimilarity obtain since dissimilarity fig plot two eight profile build text represent blue star red dot author period heat inter entropy color represent small entropy heat directly relative entropy inter profile dissimilarity correspond author th whereas remain author profile text notice block diagonal perfect entropy th entropy author relative belong time com carry illustrate text play color small entropy along diagonal distinguish sequentially text computing entropy remain piece attribute inter dissimilarity tend dissimilarity two author form pick word inter dissimilarity profile close vice versa inter dissimilarity profile dissimilarity write write contribute text text author two text mid profile mid mid hybrid profile compose c mid hybrid author word usage infer gender author divide author five pick gender author gender profile contain piece text author author text gender choose text gender profile repeat art gender e fact gender cm x cm nr author nn dt dt ce text author early english play entropy six generating profile author entropy one short length text author word close author profile build accept entropy six entropy confirm construction profile profile build contain author correspond hybrid table profile coincide table relative profile accept profile accurate play repeat procedure play pure two entropy achieve achieve profile compose profile achieve rely appearance number time author unlike word common naive support word candidate pick pool author attribute repeat preprocessing minimize minimize consider degree method strategy nn euclidean two dt dt ce low error frequency method naive decision outperform aforementioned author achieve consistent number average compare traditional method tend frequency fact carry increase majority well naive svms e four author majority entail frequency relational normalize word adjacency entropies accuracy vary text profile heterogeneity regard gender corpora know text substantial text attribute long profile page book novel text act text opinion show classify write piece predictive gender applicability multiple collaborative demonstrate appear importantly frequency capture method test introduce part speech express relationship carry lexical stand word probability compare entropy parameter diverse pool vary text length word tend capture summary exceed achieve alone combine source aspect match text unknown one potential candidate quantify traditional apply availability advance interest base least distinguish word length length characterize carry relationship advantage word content carry information function text play another include vocabulary marker stable marker word build focus frequency usage consider encode adjacency network co sentence normalization describe word encounter encounter turn imply transition interpretation dissimilarity text relative associate chain transition letter word reason usage letter approach somewhat positive result accuracy various selection choose develop adaptive compose author implementation method analyze modify length text influence distinguish text incorporate difference time gender classification section author base alone information capture combination increase author text text
comparable histograms cross reject b scatter method contour svm lower negative performance false positive negative detect whereas univariate test make distinction negative generally cc histogram scatter contour fit gaussian overlap univariate test say comparable whereas separated hence multivariate reject reject decision test time cv result multivariate reject find difference percent multivariate reject rarely happen reject reject rf histogram scatter contour gaussians univariate find difference two gaussian multivariate test recall frequently assess univariate test multivariate histogram univariate test look figure scatter plot contour bivariate gaussians high recall nan detect difference univariate reject decision univariate test four comparison though agree majority tend reject find percent multivariate reject rarely reject reject algorithm hyper lead different prior machine bioinformatic case classifier ec set representation protein body among look contact false find kernel machine predefine measure precision measure discuss fold find eigenvector give fp tn see univariate difference projection look look fp tn decision give tn assignment classification variate knowing explain variance histogram well separate univariate pair sum predefine measure confusion eigenvector may multivariate new projection advantage h histogram b contour fit find histogram separate tp tn b cumulative measure behavior propose error repository compare compare machine comparison parameter iii learn rather pre measure real world bioinformatics literature compare lift precision break calibration correlate roc combine rate fitness unbalanced combine give source test measure specificity medical specificity test arbitrary find clique subset significant allow basic statistic way difference behavior univariate cv fold interesting never fold data set distribute use recommend nonparametric counterpart tighter concerned normality outlier use research acknowledgment comparison tr department engineering university keyword statistical design multivariate abstract statistical test classification misclassification test comparison positive negative univariate test distinction source variate similarly precision variate three univariate automatically candidate algorithm domain precision positive misclassification sum false negative source combine plot visually distinction false negative may able detect error check cancer false patient false positive negative calculate risk false positive negative roc compare roc value sum comparison multiple simultaneously without collect dimensional dimensional negative precision variate work type new organize compare generalize seven future statistical testing pair manner calculate pair hypothesis population test compare near neighbor scenario test sometimes different different kernel instance make distribute normal theorem small measure calculate hence normality statistic approximately multivariate mean correspond term covariance correlation hypothesis fold fold calculate tp fp tn positive false negative negative confusion precision population pair calculate nan hypothesis performance degree reject nan fp normalize mahalanobis origin hoc univariate pair source recall difference multivariate difference significant preferred discriminant train fold fold confusion fold population test x validation matrix test statistic freedom
able compute smooth global metric principle generalize dimensionality mahalanobis projection semidefinite cone expensive moderately task vector share subset grey magnitude study new perspective address basis extract fisher learn high learning weight regularizer element framework call compositional projection flexible apply wants learn global metric exploiting make metric share arguably metric take weight smoothly knowledge principled feature use scalable subgradient descent proximal experimental support generalization theoretical combination suggest empirically art strongly describe formulation global local metric support review present present compositional exist formulation lie distances represent weighted rank psd cast set key rank discriminative basis global set metric define infinite local metric smoothly later highlight mahalanobis psd psd enabling want show later together rest start learn discuss formulation proximal metric learning seek may construct unsupervise implicit feedback click formulation combine metric j classic hinge norm encourage sparse allow element linearity minimum paradigm exploit relate successfully build share unclear allow translation formally task constraint extract nonnegative row define column induce overall constraint potentially benefit basis regularize group convex mt address limitation global capture pattern metric vary capture semantic well costly often severe overfitte learn aim smooth function informally metric geodesic riemannian problem simplification metric instance learn mt however appeal computationally heavy overfitte large furthermore give principled propose weight simple learn rbf bandwidth median td ensure valid pseudo combine locally metric feature instance local denote concatenation subset interestingly recover minima nonsmooth induce involve triplet stochastic step proximal induce comparable improvement backward initialize unlike exist projection onto psd scale thereby problem analysis algorithmic metric consider np return non triplet bound learn basis norm asymptotic justification enforce notice appear suggest long remain mt relevant task refer survey detail method directly representative paper subject learner boost clear multi popular task clearly rank achieve high compete especially train fast fast consistently appendix global letter element section main global pick relevant generate new add current report number use overall use use suggest scale well dimensionality nice entire induce regularizer number poorly st books accuracy avg runtime n min min min sentiment dataset multi amazon review book type treat task positive review review large split testing compare euclidean st independently euclidean union tune parameter set union mt average error rate st dataset mt outperform st significantly mt task counterpart demonstrate ability unable capture solution st element element select mt evenly across able exploit meaningful compact metric segment letter avg dataset global learning target local number basis letter table give method enough fast poor metric discriminative local offer training especially train minute fast dataset mm bad global parameter result competitive understanding apply colored vector pca vary smoothly thereby robust unlike mm point metric variation consistently generalize select basis basis element eventually reduce suggest generalization bind theorem generally global notice outperform element multi particular local algorithm instance test principle way support theoretically generalization propose method research contract nf contract ap reproduce view herein interpret express imply c triplet build admissible define generalization pairwise reader briefly algorithmic robustness ability perform similarly test proximity space two example lie compact belong xu metric robust ns sn adapting show algorithm algorithm sample triplet draw least establish easy say learn approximately triplets metric suppose global learn number nonzero definition subset subset either triplet respective deviation admissible robust set training sample tc admissible argument global tt global
reason rbms way gradually specify intermediate p kk kp x assume gibbs leave w importantly hold importance extend k xt represents generate unnormalized begin reverse show importance mathematical intermediate initial q define anneal temperature annealing isolate two rbms bias evenly although evaluate mrfs intractable tend hope provably estimate preferable save report test log section limit infinite unobserved distribution rbms transition visible approximate v w kf ann proposal reverse chain start gradually x xt identity store chain update reverse implement require operator merely reverse chain stochastic low yield conservative mrf low possibility report test log frequently interest additional agree weak assumption tractable operator preserve reverse similarly algorithm h k w kf ann procedure rbm could interpret sigmoid insight greedy belief single transpose compute rbms view belief net proposal transpose perform approximate unit rather use field interpretation suggest rbm direct layer direct green blue speed example prefer number subsampling introduce rbm handwritten digits significantly depend batch variance reduction apply ideally compute highly exact mrf mrf easy total assign small significantly evaluate mrfs estimate estimate obtain exactly log strong handwritten digit long dataset character across many average two distribution base rate distribution visible bias pixel transition probability test sec probability full dataset rbm log chain match number exact gap gap train algorithm cd persistent divergence refer rbms hide train cd mnist rbms evaluate also comparison conservative conservative rbms conservative probability estimate intermediate hand insufficient accuracy often differ give consistently well method plot estimate uniform appear report high obvious discuss necessarily rbm rbm rbm rbm instance rbm rbm rbm base show increase bind inversion outperform rbm rbm model diverse configuration experiment rbm challenging overall case estimate suggest rbm function anneal intermediate distribution sample poor great rbm matching estimate rbm anneal bottom model use estimate two train hidden layer rbms run test unnormalize obtain unnormalize v conditional sum use give however make unnormalized merely variate estimate table mnist figure mnist quite piece evidence contrast give optimistic imply rbm eliminate mrf model gap right hide obtain unnormalized recognition mnist within give conservative rbm experiment model typically conservative suggest reverse log mrf typically yet rbms conjunction one agreement require simple practical test acknowledgement google markov fields mrfs generative importance mrf partition yield quite accurate wrong reverse original experimental result indicate agree rbms typically year representation deep appeal representation partly directly measure assign restrict boltzmann effective various visual rbm intractable mrf widely mrfs generative function perform practice tend optimistic whether log likelihood lead researcher generative rbms art modeling highlight optimistic rbm tend standard benchmark insufficient accuracy compute mrf likelihood similar boltzmann add write rbms unnormalize f h similarly rbms rbms building
use least implication right uncertainty far turn toward probabilistic equation independence element ms mutually appear right term apply hoeffde probability least union l definition individually mutual j use mutual equation mutual j e use law j use identity law conclude present principled approach four drive distributional distributional know paper show statistical learning guarantee quality robustness project come mit nsf grant theorem remark mit edu goal decision datum past create particular handle simply way past tool provide guarantee robustness robust optimization uncertainty work often needs create future good reaction bad uncertain situation question maker interested answer question probability bring useful address important detailed available predictive modeling technique machine uncertainty set uncertain historical accord possibly illustrative portfolio allocation construct return market advance portfolio solve make portfolio wish portfolio portfolio acceptable make portfolio uncertainty exposition reality return solve good outcome many central complex past predict return might include sale complex carefully different priori range portfolio know possible knowledge guide construct return jj ignore past use empirical could uncertainty portfolio allocation problem define past could hull past return ignore linear potentially assumption draw class intermediate set use portfolio determine fit normality additionally normality define robust allocation prediction interval around union good decision realization reasonably make assumption complex assumption limit applicability modern assumption provide way historical two approach approach tool minimal assumption construct set construct robustness guarantee specific conditional quantile prediction prediction I normality give indicator away two illustrate extreme set error single policy effort choose independent manner learn give guarantee intuition set square loss function element interval uncertainty illustrate percentile model percentile interval provide fourth second boundary create method evaluation every plot optimize boundary residual boundary good quantile boundary uncertainty important specialized order daily ice weather much uncertainty weather conservative costly would budget large possible sale middle attempt tackle principled goal like uncertainty machine propose value needs guarantee finite quality robustness use approach approach uncertainty theoretical design handle classification ranking apply section formulate uncertainty use learn theory guarantee guarantee many either make optimization literature interest empirical along priori probability specify drive set guarantee testing use three design ii goal totally cost policy create estimate theory feasibility randomness thus entirely make paradigm portfolio regression big design set historical bit various instance multi unlabeled available work use prior create feasibility describe set historical correspond approach introduction decision denote vector decision variance problem robust portfolio formulation list option mm nice natural transform relaxed robust optimization problem solve ellipsoid box ellipsoid solve nice element walks give feed set function training empirically start discuss dy represent let say pick ni generalize observation consider empirical minimization solution eq set set function simple instance union disjoint interval non way j slightly method estimate functional generic quantile quantile quantile quantile applicable prediction task yy j quantile example typical pair conditional quantile realization quantile j I regression risk quantile aim obtain true predefine parameter loss let construct definition interval involve two quantile conditional conjunction deviation capture use define large deviation capture illustration member figure equation depend residual solution consider general training input output high pick simple empirical rademacher equal average interpretation rademacher come rademacher covering shorthand result performance term result feasibility depend estimate enter well desire proof provide constant shorthand variable use use rademacher result prescribe output minimization risk sa rademacher range rademacher class capture set follow solution feasible ms bound realization ensure ensure belong high tell affect uncertainty set predictive rademacher scale quantitative confidence distributional study rademacher propose define source notion definition conclusion theorem hold equation use pac pac probably seek instead classifier uniformly pick randomize choose classifier risk r sg pac capture p q certain pre minimize normalize good model define big solve way prior pac thus guarantee nonetheless way distributional robust portfolio example distribute tx tx distribute choose base interval ellipsoid probability mass equation obtain interval solve ds us future realization hold fix unbiased tx mass need justify construction contrast make section weak many approach may efficiently option involve directly guarantee similar solution make class bb use corollary relate bb function positive semi support svms hinge apply hinge upper appropriate kernel particular use dot get consider fourth define I sl guarantee optimal hold individually robust equation j ms designing prediction quantile residual realization source insight quantity learn importantly prediction decision e g svms practice suggest term perform type sensitivity analysis vary set assess intuitively function insight construct algorithm drive hand quantile quantile quantile estimate produce quantile estimation simple close loss special differ lipschitz theorem rely mild assumption make model include account set underlie true sense policy uncertainty lead distribution lead small strong centrality construction regardless relate good expand method use proof result rademacher f l sf sl side increase random sf perturb case sf l sf l sf f sf rademacher trick l give
evaluation termination immediate b preferable optimization incorporate mechanism maxima equally apparent search region inefficient therefore uncertainty fraction would maximize typical end step uncertain next iteration region offer immediate begin express incorporate kind exploratory gain set b maximize set criterion great close deviation retrieve efficacy toy example expect terminate maximum whereas opposite score yx x ask find choose search hybrid local optimum evaluate uncertainty us find maxima problem score individual purpose instance incorporate methodology principle high uncertainty unity permit exploration space original yet advantageous closely proportional improvement high propose tuple cm p cm threshold forest optimization threshold selection virtue exploratory decision necessary maximum unnormalized terminate significant complexity table statistic h b framework thresholding find able superior yield compete validation hope algorithms application parameter hybrid mm edu college area regard hyper via work exploit notion confidence uncertainty enable efficacy machine expect distribution parameter tuple tuple induce fit allow behavior point notion arise form fit enable process optimize search framework immediate fold cross validation discrete fail parameter set exploit improvement performance current show interpretable provide alternative option et al implement mat ern typically target efficacy commonly maximize minimized ability generate perhaps network whether continuous use
rest generality uniformly finite weak demonstrate bootstrap main bind moment hold average family process get stochastically result stop give convenience use use increasing imply rest choice arbitrary fact bound parametrize consider proceed expectation pe pd fp pe deviation call state shift nonnegative q since thm pa idea stochastically independently write continue expectation condition prove stochastically analogous proof bootstrap bind stopping rely defer appendix convert choice analogously however space need define stop event eq q stop unbounded simplification thm tool extend obstacle establish easily geometric brownian motion tight unlike discrete merely manuscript totally stop time analogy great effect process metric covering incorporate variation open powerful average regime treat time weak pe emphasis normalization use stop uniform go notably conceptually idea describe game motivate somewhat different taylor expansion acknowledgement grateful stochastically time justify prove walk generalized beyond walk construction leave remainder unchanged determine average distribution averaging construction replace high uniformly e e result examine x yy attain initial condition remove extend hoeffding bernstein style show simultaneously lemma desire bernstein inequality regime account uniformity iterate reason relate appendix first mixed process write integral like peak refined account factor construction rest current lower close change average family distribution index q mix conjunction appropriate iterate furthermore improve decrease tight strictly extend finite time average proof analogous different initial restrictive closely within f decrease first proceed rest proof verify bind particular recover dominate lead define family lemma upper u bound idea stochastically space define product bind respect notation stop convergence restrictive concrete also whose u finally side lipschitz fast fix write stop prove proof fix suffice assumption last u use substitute give lemma event lemma q chebyshev define convenience particular mean correspond confirm q u monotone imply chebyshev know upon simplification thm thm thm thm thm remark main remark give bound uniform hoeffding bernstein limit law iterate finite bound dynamic arise particularly range concentrate side concern behavior finite manuscript address upper half concrete martingale discrete induce repeatedly write I distribute rademacher rademacher walk law iterate logarithm rademacher random absolute interestingly capture tradeoff dominate regime time uniform limit bind rate encounter strong walk time finite maximal exercise hoeffding manuscript fundamentally weak hold uniformly epoch proof view generalize u cumulative straightforward
rna eeg demonstrate propose algorithm single perform case framework imbalance make balanced substantially apply benchmark extremely observe improvement easily employ sn sp clean rna eeg acc sn sp letter yes yes letter eeg complexity consume component ann construction exact algorithm run big beneficial typical linear experiment near neighbor set framework several sensitive stop parameter perform fast rna similar slightly level loss significantly framework large set try small nearly impossible allow level stop parameter serve fine level main drawback search parameter become considerably expensive method hand running pair optimize parameter global simplicity cluster dataset quality classifier letter cc letter rna r dataset paper promise quality successful reduce degree skewness balanced type support computational propose framework scale solving obtained gradually refine multiple include hierarchy coarse representation update hyperplane computational without demonstrate machine balanced keyword support originally nonlinear svms consume task extremely sensitive apply storage rapidly dimensionality tractable svm parameter tune advanced method tune parameter total quality employ optimization focus qp popular qp scale efficiently cache typically still recently parallelization split subset perform assign different subset accelerate qp often problematic implement svms parallelization dataset investigation accelerate training optimum vector shrink early optimization time save substantial successful optimize completion size practice lead poor measure area include diagnosis bioinformatics class technique datum adaptation sensitive learn regular svm term fuzzy heart method lie algorithmic mf inspire multiscale main objective degree introduce local processing global solution data exhibit linear complexity relatively mf heterogeneity external appropriate refinement framework successful algorithmic create coarse training solve refinement easy parallelization superiority computational method less particularly set create balanced coarse effectively hyperplane let set label number feature subset relate determine map slack misclassifie nest manner dataset respect paper three learn support vector create level approximated selecting include result make make robust change refine classifier level input ni give begin neighbor ann ann select size dominate present complementary already begin eliminate next add already skewness balanced representation correspond correspond label computational resource separability fast processing difficult level refinement fine contain coarse extremely consume prohibitive fine much original much selection complexity apply svm near neighbor support run center apply pair directly exactly coarse approximated experiment datum coarse current line direct opposite contribute binary classification true positive negative negative classifier acc common classify proper performance mostly dominate use sensitivity sn specificity sp geometric
ei pi suggest line keep maintain section technical produce monte integration discretization eq leave computing minimizer easily latent datum gaussian idea meta visualization kn py meta criterion upon produce global gain intuition form th element suggest draw return match armed sampling continuous domain construct sequentially optimize would ultimately optimize analytic shift mapping consist stack approximated product allow correspond posterior construct finite parameterization meta use acquisition randomize strategy exactly extend continuously vary function use albeit acquisition summarize meta develop system factor optimization allow learn around choose optimizer ball width proportional smooth difficulty set optimization set due bad quite maxima hyperparameter let hyperparameter p posterior fully integral must sample acquisition internal every hyperparameter sample ei pi randomize hyperparameter hyperparameter selection hyperparameter occur simply add additional loop sample global problem portfolio portfolio rp portfolio acquisition randomly acquisition ei thompson ei three continuous final absolute evaluation pi expect expert outperform acquisition rather imagine parameterized addition two ei clear winner thompson option motivate function synthetic inclusion strategy portfolio rp expert box base method initial stage exploration enough exploitation actually beneficial purely exploratory candidate precisely propose empirically especially reach digit meanwhile rp expert select acquisition due expert horizon reach rely past robust vary single outlier achieve evaluation refer dataset consist finite transform via constant three south gold dimensional outperform ei motivation pi rp particularly albeit acquisition example thompson poor acquisition boost final portfolio eight control fed simulator robot simulator optimisation problem particle drop particle circular placing degree plane report result poorly rp perform meanwhile perform portfolio tie rp add demonstrate rp rp significantly affect introduce theoretic meta criterion expert particularly acquisition robust portfolio furthermore poorly perform portfolio principled slice acquisition thereby extend popular sampling wang w exploration acquisition clear superior principled collection acquisition function often past portfolio portfolio theoretic consideration outperform synthetic simulate offer acquisition surprisingly performance finally wide inclusion poor acquisition popular successful expensive find minimizer non multi modal technique interactive environmental monitoring extraction network adaptive monte carlo experimental reinforcement broad application area tuning query point construct probable procedure must select state knowledge characterize acquisition encode clear optimal acquisition planning often much acquisition long early probability pi improve see point lie pi greedy quantify correspond ei recently variety advanced technique bayesian acquisition strategy provide problem instance empirically prefer strategy stage process ingredient among analogous acquisition assign utility space utility candidate suggest portfolio predict instead unclear also acquisition subroutine sample minimizer thompson also jointly location condition work optimization visualization expert th query early implement past acquisition lower base content initial portfolio collect kt denote meta candidate criterion great reduction arrive
mac mac snp mac mac mac mac k mac k k mac mac mac mac mac mac mac mac k mac mac mac mac k mac k mac mac mac mac genome study steady accumulation imputation genome sequence study need scalable key genetic call phase information variant represent primary format address issue release introduce extensive parallelism equilibrium fisher many algorithmic accelerate handle large fit ram follow efficiently version offer improvement user analyse genetic come finding broad file format trait map genetic year final generation analytical wide heavily processor comprehensive update notable exceed functional unchanged association identity descent drop replacement case require easy like feature format file platform genomic introduce employ expect benefit problem remain core file format second improvement parallelism x bit machine binary file bit datum arithmetic parallelism loop operate element replacement loop bit parallel logic old calculation roughly every marker marker miss call file long marker block single bit population count bit discussion post processor evaluate quantity however thank count quickly hardware take previously refine develop implementation vector even processor might example marker marker denote minor coefficient easily express bit call marker loop dot encode minor call call call name correspondingly memory requirement sum marker specific marker seven distinct allele major minor minor minor genomic per marker increment minor allele final instead partial save seven increment refer sum several point substantially seven distinguished operation seven appropriate manner bit marker bit marker bit describe relation marker representation us processor act simultaneously increment could denote f f exchange four entry mb modern table snp equilibrium al fisher exact snp exploit likelihood contingency expensive ratio table contingency entry could avoid calculation partial sum likelihood super geometrically move away probable likelihood partial digit stop double precision see example mathematical straightforward modify termination snp property likelihood evaluate handle represent et early termination snp fisher test call mid mid discuss algorithm detail explanation extend method restrict briefly variant strong strong historical contiguous base determination analytic hill cubic likelihood correspond check cumulative likelihood rarely extreme full cubic reveal exploit establish cdf evaluate likelihood inspection variant massive table later pair variant process final spend cache large must entirely skip variant idea implement last implement basic coordinate descent newly integrate party upon implementation speedup logistic crowdsource innovation perform hill manuscript author case user seven operate three intel processor gb run mac intel processor os ghz intel processor ram bit ghz intel processor core gb ram bit denote four core processor ghz ram bit sp ghz intel processor core ram refer dataset quantitative synthetic marker resample p prune remain refer variant phase dataset refer variant snp snps prune disk run straightforward indicate runtime several basic table display execution one simple reflect overhead due use run disk complete cluster calculation disk linkage population employ population count clustering furth improvement case table execution three calculation count version estimate information genomic early standardized genomic os windows platform memory requirement linkage pruning frequently use analysis linkage table single heavily fast contain rewrite variant runtime association analysis flexible major improvement bit population solve make fisher test window build bit despite advance recognize still work genomic binary format file format capable represent essentially modern imputation multiple probability limited call amount serious format binary file variant within file representation translation code program term explicitly design library able convert file party library compression demonstrate weak compressive arithmetic computation directly compress genomic exhibit explicit compressed merely pack support true sublinear sublinear compressive reality deviation efficiency available program operate library layer file ignore work exploit compression software plan software handle limitation meet genome association study context genomic system across preferred size enough serious genome sequence detailed study variation clean sometimes type strongly software package seq file less expressive file management genomic subject type type extend interpret availability name code home page operate os bit intel window require restriction compete interest author software manuscript matlab prototype
prior variation perform series test central less series flat flat prior however ht difference mean alternative posterior different dot fairly plot figure plot posterior quite practical semi situation underlie unknown least tend poorly alternative aim demonstrate involve likelihood base outperform adjust implementation parametric package iii iv wavelet package vary early idea residual respectively cc observe positively seem significantly slightly biased method unbiased first full pilot recall case output two generating mechanism relatively b differential plot short long effect opposite end spectrum peak posterior largely uncorrelated process behaviour spectrum effect memory dominate distinguish consequently ccc green line red consider green blue significant arguably although clearly form scheme outline posterior generally roughly around generating probability usually consider context plot unknown solid reversible dotted look method former slightly level posterior appear exhibit present showing mode pick consistently b complicated short tailed series see density cc look complicated wide reasonably finally algorithm wrong marginal application obtain actually homogeneous datum rr ci summary minima plot density minima popular memory effect work handle typically mind accommodate cope heavy tailed potential scope finally miss augmentation relevant gap example datum record fit assume gaussian gamma choose posterior conjugacy demonstrate gx yield follow conditional reveal likewise p classical denote p ar uniqueness firstly pi special equations ik q ar block divide pt ptc also definite block clearly positive definite also positive eq ptc corollary efficient forecasting whether represent long potentially high potential mathematically stochastic exhibit precise able market public effect frequentist autoregressive fractional integrated parameter parameter short memory hypothesis testing ever methodology range auto reversible capable memory long standard definition autocorrelation zero process long rescale retain stationarity essential rigorously memory property statistical span semi bayesian probably difficulty advantage classical flexibility could offer ability model miss effect towards burden class process phenomena emphasis popular statistic connection fractional property separately model long short argue sentiment parametric model work often model similarity bayesian generally careful approximate process series extend rich memory extend unknown work work nonparametric memory contribution aspect present focus selection unknown memory effect focus long memory conditioning emphasis flexible class remainder paper require numerical likelihood classical statistical bayesian propose focus long memory extension additional illustration method potential time record process joint tr normal integer convenience consequently drop depend temporal motivate stationary refer lag normalise power iteratively k say sequence expand power series act white iid noise purely non refer coefficient write power additional identifiability happen two therefore model hence stationary invertible process process ar invertible say move stationary ar auto process order although close arbitrary long causal decay exponentially word correlation nearby distant negligible turn memory extra possess obtain restriction stationarity clearly stationary similar ar one particularly long constant simplest equivalently assume scalar multiple generally call full long solely flat series memory structure determine one f way call integrated move order recover practical utility formally generalise operator call fractional operator degree use domain connect integrate law relationship ar analog short placing parameter likelihood approximate highlight issue prevent context thousand evaluation proceed dd likelihood simplify shorthand whereby log evaluation require encountered toeplitz scalar exploit yield large practice efficient scheme throughout stationary invertible mean location call I classify three innovation short truncate drop x ar ar retain obtain follow possible write convenience appropriate bayesian auxiliary writing ng conditioning write tx return observe conditional past suitable equivalently evaluate depend augment evaluate via cost infeasible however suffice bayesian inferential ready describe help accommodate extension short begin prior choice yet encourage calculation early familiar specifically improper flat prior limit distribution place imply gamma work aware promise extra component subsequently memory ar distinguish create variance unit unit write general explicit calculation fortunately polynomial calculate coefficient yet space simple proposal orthogonal would alternative explicit jointly align correlation structure reasonable mixing describe suitably walk covariance proposal sort arbitrary large truncate hypercube unconstraine rw mode return mh ratio since term initial choose interval systematically although prefer proposal align find pilot proposal rescale matrix pilot expand short indexing adjacent acceptance crucial transform turn shape model difficulty encounter complicated nontrivial require root write unit serious drawback consider natural remove root realistic appear obvious way remove might modulus unclear propose previously recursively fit motivate show lag useful propose structure limit long nevertheless near full value model individually suffer problem simultaneous update perform distribution rectangle detail expression complicate special must model variety prior simple option would restrict say proper formulation lot complicated prefer poisson parameter non possible birth death neighboring value point body boundary corner transition probability abuse notation detail ar almost identical de replace simplify regard choose set match specify either final jacobian unity made prefer decrease scheme move depend choice rw seek pilot extend obvious matrix etc block omitted specify various propose variance dot vector zero conceptually analysis memory ensure unless simulate reasonable use begin bayesian alternative mcmc tune interest value systematically start spaced parameter varied turn sufficiently mcmc efficacy start long simulate deviation extract average rr mean std ci estimate away interval nearly symmetric posterior locally proxy credible test eight hypothesis would six indicate interval approximately sized repeat experiment generate give analog axis estimate I perform show around space whole
fast avoid search publication context guarantee study nuclear reweighte nuclear conduct plain svd valuable insight quantify simplification update analyze svd plain rw plain relative error nn avg trial quantile simplify adaptive relative solution minimum simply singular svd high aa bound svd solution log heuristic tight accuracy even small singular approximation contrast plain bad large consider signal decay power improvement decay error support nuclear plain optimal svd norm algorithm full simplify fix simulate alm plot via divide norm rw exact figure rw nn recover optimal establish simplified good error dramatically plain analysis rw structure practically entry infeasible quantify impose rw accuracy nn far bad infeasible plain frobenius rw solution provide error toeplitz toeplitz toeplitz structure modeling represent multiplication toeplitz toeplitz start diagonal toeplitz however reach rw infeasible plain toeplitz finally nuclear success solver heavily diagonal entry moderate level rw nn approach solution rw nn continue good accuracy good initialization set cosine angle average utility broad biology biological human human community cell interest characterize different state across combine readily cell distinct state phase division growth cell fractional change expensive impossible directly separate know indicator throughput dna rna seq relative different accurately chemical composition rna normalization accounting bias bias positive state different growth rate mathematically extend z enter transpose side move everything stack stand kronecker block diagonal column row block single rank structure trivial nan compound error exactly precisely explore experimentally reweighte heterogeneity quantification cell hoc express across grow growth cell phase agreement expectation physical measurement extend trend asynchronous analysis research heterogeneity obstacle experiment biological comparison rw non outlier low noise solver rw gain nuclear case use notation frobenius case heuristic solve solution local domain linearization minimum contrast soft thresholding phase across growth choose increase order error quickly soft error entry heuristic rich augment lagrangian reweighte nuclear matrix completion robust quantify heterogeneity measurement solution ls square allow field computer vision identification special dependent uncorrelated singular svd many realistic vary measurement know norm convex plain nuclear relaxation incur approximation effective challenge popular convex solver augment lagrangian problem measurement expression measure generalization square l explanatory problem well base measure come additive noise minimize xx try norm reformulate find close rank solve via many practical additional may exhibit toeplitz outlier huber unfortunately none efficient approach method reach principled relaxation rank use relaxation highly toeplitz frobenius relaxation successfully range involve completion robust conceptually seek low contrary theoretically experimentally plain nuclear incur dramatically weight nuclear suggest base multiplier alm update weight nuclear equation task rely pca study biology quantification heterogeneity problem block diagonal recover state notation combine matrix frobenius norm small error range compute small singular allow realistic exactly measurement noise entry toeplitz deconvolution signal unfortunately vector search one sparse general encourage signal weight plain ideally unknown norm large solve define weight see linearization empirical study weight provide tight try singular one introduce linearization nuclear programming sdp p rewrite nuclear concave log small positive relaxation provide approximation norm objective initialize weight nuclear initialize solve svd k plain weight nuclear formulation thresholding next augment lagrangian alm nuclear interior even lagrangian alm alm augment augment lagrangian motivation optimal minimum continuously increase sequence one allow involve
less contain observe user source correspond job traffic post find remove meaningful noisy movement cc analyze movement york city date show date middle color six tweet second cover tweet see fig localize addition temporal see cluster cluster union square ideally separate group related real six cluster cover tweet mean leave right cover tweet contain date summarize third detect area square interval moreover tweet mainly take place mention event distant cache event cluster time spread present example anomaly event twitter certainly explicitly event extraction multiscale recognition community last decade take wavelet transform enable work interest medium wavelet framework solution temporal graph approach account spatial generally handle scale explicitly multiple believe essential generic scale spatial dimension treat separately state good knowledge approach model relationship statistical analysis spatial twitter believe perspective contribute social multiscale medium understand temporal different separate way present statistical twitter insight influence investigation possibility generalize handle multiscale detection ii appropriate social media scalability mod research ed social media approach reality scale social explore property multiscale transform enable automatic handling interaction propose similarity detect event scale simultaneously furthermore present stream novel algorithm help experimental real collect demonstrate approach extend numerous involve decade see rapid development online social media user generate internet huge many detection one important area medium present several real social service public fashion traditional second amount content complete description large scale advantage attract amount mining event detection numerous literature social medium define real world similar location image video text temporal span discussion game month regard concentrate illustrate twitter event scale york city account scale challenge usually detect yet multiple space interact simultaneously two iii contain event interest understand robust multiscale objective cc introduce baseline localize serve towards multiscale detection towards scale relationship property wavelet automatically explicitly algorithm compute scale furthermore present noisy information stream notion statistic multiscale behavior paper noisy propose event datum world experimentally propose spatial hand believe model relationship temporal scale multiscale insight media framework domain involve event medium reflect concentrated dimension usually namely interval twitter great home non tweet constitute input interest data stream stream contain approach event take temporal paper cast graph tweet reflect similarity utilize tweet tweet generate share closely locate way measure approach constraint base effectively handle twitter homogeneous poisson model helpful event previous impose baseline localize understand event together correspond event similarity measure tweet minute spatial threshold locality impose constraint reasonably text tend refer text angle representation weight scheme tweet adjacency tweet define cluster expect tweet furthermore eq event localize space way function repeat procedure modularity attain suitable purpose cluster cluster usually event priori ii unlike normalized spectral clustering favor balanced clustering enable detection cluster relatively also base approach correspond localize fig correspond cube simple post likely meaningful world reflect twitter threshold sect step critical temporal spatial event discover event sufficiently event set break lead cluster cluster already offer grouping hierarchical obtain usually clear scale result cluster process experimental section event scale introduce novel base cc pairwise similarity tweet adjacency recursive retain post localize space novel multiscale event introduce scale wavelet scheme multiscale similarity tweet fundamental question towards multiscale detection properly localization illustration event rectangular span spread tweet spatial event shall spatial dimension compute scale actually span propose temporal follow two tweet share could temporal vice versa resolution essentially say consider fine resolution necessarily time area could span city place power area city relax event detection choose suffer ambiguity twitter stream either incorporate temporal distance might detail tweet tweet time share occurrence tweet appear enable temporal similarity tweet compute similarity time share keyword share use resolution occurrence cell belong illustrated cc propose similarity transform tool consider wavelet haar way scale specifically haar wavelet aggregate scale temporal resolution coarse series temporal measure coefficient level evaluate similarity properly turn determined spatial two introduce predefine scale coarse distant fine initial fine temporal cell temporal temporal compute level illustrate fig similarity tweet follow share time series choose medium piece informative although tweet share popular series pattern fine next remove spread similarity time take similarity helps preserve information recall retrieval tweet favor tweet pattern approach similarity meaningful rely look common offer flexibility event multiscale h tweet extract series spatial similarity correspond series parameter multiscale resolution construct adapt scale various thank adjustment wavelet expect temporal interval span sect number spatial choice full scale might lead unnecessary influence resolution determine along variability implicitly scale meaningful span determine meaningful level computation haar wavelet maximum temporal relationship accordingly observation suggest denote choose ensure design event example could really twitter influence event result employ filter tackle derive working tweet contain wish specific event tweet empirically intuition useful locate tweet day york city contain locate tweet middle tweet middle contain tweet locate albeit frequent collection tweet appear relevant specific illustrate location tweet middle one tweet strong area spatial plot seek statistic lack assess use complete randomness tweet homogeneous poisson overlap within denote intuitively area concentrated assess level tweet term select say tweet test whether tweet tweet follow edge similarity consider identification evaluate employ function assess spatial homogeneous process sd euclidean tweet area spatial poisson evaluate process paper standardize proximity tweet specifically standardized value km homogeneous minimum depict blue dash line respectively location tweet range homogeneous indicate slightly concentrate space homogeneous poisson possibly twitter user different middle lower spatial poisson explain homogeneous extreme tweet achieve term process number tweet distance concentration tweet slightly intensity tweet far homogeneous order whether observation tweet york duration day ten frequent km focus area avoid number computed tweet illustrate variance ten illustrate twitter follow poisson tweet distance twitter datum assess irrelevant tweet serve strong tweet event interval hour pm contain frequent uniform chi goodness interestingly hypothesis case result term analysis distribution conduct filter aspect next event influence present consider tweet time diverse performance choice bottom leave top respectively real span temporal diverse experimental first event random tweet span analysis irrelevant tweet namely distribute content tweet locate tweet york collect reference term refer tweet relevant tweet tweet select depend daily tweet create tweet irrelevant tweet irrelevant tweet scenario event concentrate area temporal choose tweet goal correspond appear least tweet choose unless go resolution method favor consider ensure tweet group cluster term criteria satisfactory link spatial threshold case threshold large chance grouping resolution representation aggregate time appropriately capture link tweet choice sensitive selection space one concentrate spread spatial event concentrate spatial spread handle scale scenario comparable experiment drop significantly lack temporal spatial reasonably well threshold cover scale experiment highlight handle simultaneous event irrelevant event tweet spatial generate tweet sect detect apply tweet measure cluster tweet correspond event want ensure tweet separate base sect propose standardize tweet term generate time different specifically link create separate increase form link tweet contrast recall positive negative
next differentiable intuition state formal globally strongly point sparse particular definition pair rsc constant locally strongly globally rsc rsc impose nonzero direction rsc form convexity vector past work rsc convex arise statistical illustrative x I link goal loss function case th consequently rsc extension linear observe corrupt lasso inconsistent study previously version past show consistent previous paragraph natural choice eq set semidefinite nonconvex concrete condition pair glm glm log family equation et convexity generalized suppose goal inverse arise let refer lasso take derivative condition satisfies convexity primal technique establish recovery previously extend optima norm function generalize gradient guarantee stationary point consistency recall due abuse slightly comprehensive generalized stationary key argument enforce feasible satisfy condition stationary point program support construction implicitly restrict convexity minimize subgradient note conventional constraint omit automatically greatly simplify suitable restriction restrict therefore uniqueness section follow primal support program unique state theorem concern concern rsc regularizer amenable concern guarantee support strict feasibility dual note condition incoherence usual corollary feasibility guarantee amenable regularizer discussion function function amenable eq feasibility condition size stationary primal unless proper choice exist corollary high cause sample rsc know take pn way guarantee also translate large consideration motivate advantage regularizer mcp scad mcp regularizer amenable discuss later remove mcp establish strict feasibility mcp suggest incoherent preferred variable selection incoherent scad mcp may simulation regularizer condition although theorem nonetheless confirm experimentally situation yet stationary appear unique feasible yet sufficiently examine dual establish minimum program support multiple strict zero certainly global diverse convexity concavity possess still simulation stationary global optimum agree result amenable set assume rsc restrict convex notation strict feasibility dt agree appendix strength amenable regularizer indeed beta min bind remove oracle rate corollary hand expression bound consequence concrete first focus ordinary regularizers mcp consequence regularizer result nonconvex scad mcp amenable fairly incoherence nonconvex regularizer recovery absence incoherence regularize square write semidefinite dimensional imply nonconvex program nonconvex family recover limit restriction prove amenable incoherence condition objective stationary optimum furthermore amenable nonconvex min bind b corollary provide comment consequence regularizers include mcp recall mcp translate regularizer choice asymptotically constant past establish consistency distinguish amenable deal past nonconvex regularizer ordinary regression interpret work zhang global optimum mcp regularizer consistent eigenvalue design optimum mcp strong design paper matrix incoherence appear sufficient condition concentration inequality inequality fairly fail condition corollary scad mcp incoherent function define corollary convex involve nonconvex nonconvex illustrate applicability dual nonconvex function recall corrupted covariate response corrupted covariate may constant suppose nonconvex stationary global familiar sample nonconvex lasso corollary nonconvex global optimum result indeed simulation incoherence hold inspection confirm conclusion simulation power regularizer move function likelihood linear condition may remove regularizer amenable regularizers glm composite positive somewhat nonetheless logistic et uniform categorical covariate relax goal r nonconvex stationary et require paper incoherence show properly nonconvex incoherence requirement generality apply case nonconvex stationary evident priori contain finally consequence graphical recall graphical lasso vector covariance subtle seek scale opposed nonzero grow connected graph cone positive definite impose constraint regularizer formulation actually selection consistency rather norm denote nonzero g scenario govern comment note handle symmetry treat program optimization take definite early oracle sub amenable also unique oracle appendix spectral norm restrictive assumption work incoherence graphical lasso restrictive much illustrate another distinct report result run order begin describe program eq stepsize simulation convenient eq efficiency ease analysis satisfie follow convexity smoothness regularizer optimum program amenable composite appendix geometrically tolerance error ensure iterate consistency optimum addition k iterate composite descent proposition satisfie form c n guarantee bind composite support describe simulation incoherence bound matrix diagonal everywhere else prove provide incoherence eigenvalue maximum eigenvalue satisfy easy run come ordinary corrupt scad mcp parameter logistic boundedness assumption impose generate agree qualitatively simulation show scad regularizer situation incoherence selection consistent I obtain incoherent matrix response addition generate corrupt run update panel correct recovery increase scad mcp recover correct remain put nonzero coordinate rescale horizontal axis match prescribe align confirm scad small mcp penalty previous regularizer consistent sufficient consistency parameter share curve agree et depend regularization scad solid mcp regularizers regularizers design incoherent plot error regularizer regularizers set line nearly align regularization set simulation uniqueness stationary objective setting come either ordinary least logistic unique multiple composite observe initialization appear converge correct slightly however violate composite distinct stationary show composite gradient descent come generate scad mcp regularizers panels b distinct incorrect still continue produce panel observe rate scale predict reach increase overall scad mcp axis panel panel ol initialization composite b scad mcp scad mcp distinct stationary finally analogous loss function equal panel plot plot multiple stationary panel scad mcp regularizer contrast run descent converge point panel simulation plot regularization may geometric threshold scad may empirically vertical cc variety regularizer initializations composite gradient mcp however large mcp regularizers panel develop extended framework nonconvex technique regularizer nonconvex significantly previously regularizer mcp consistent nonconvex regularizer recovery design incoherent recover regularizer future theoretical violate simulation justification scad mcp regularizers penalty assumption result convexity whether local rsc condition guarantee good proper initialization finally useful able rsc constant assign nonconvex amount curvature acknowledgment pl partly support foundation fellowship nsf pd fellowship study berkeley pl partially dms grant air force office research proof technical lemma subsection outline step construction appendix sn low interior step ii shift interior point rule subgradient construction iii establish note ensure concavity condition trivially verify contrary inequality assumption thus contradiction equation false prove guarantee convex rsc amenable regularizer program strictly immediately imply minimum prove support condition turn uniqueness assertion satisfy stationary point strictly establish ss prescribe fix furthermore rsc convex strict summing precede inequality interior hence lemma rearranging imply rsc inequality zero subgradient inequality rearrange second come need c note q imply put piece old claim calculus zero subgradient ss imply guarantee inequality establish amenable inequality follow global claim simplify equality strict dual feasibility require derivation corollary subgradient define simple algebraic property vi vanish amenable suppose strict hold combine regularizer impose strong ensure strict cs ss assumption hold strict feasibility provide inequality reveal scad mcp impose corollary appear rsc state tight denote I sub take gaussian conclude equation feasibility property turning write furthermore condition gaussian eq well q combine theorem proof extra assumption q concentration cs ss remainder expression corollary work imply rsc verify bind check parameter condition hold establish feasibility h applicability corrupt suppose satisfie deviation hold least argument union furthermore well note inequality q first second combine bound inequality inequality inequality conclude define strict feasibility succeed turn decompose establish establish rsc terminology particularly derivative verify inequality eq lie px sx cauchy appendix take eq norm unit sphere ss ss appendix show put piece high return ss guarantee triangle inequality strict dual scaling turn q bind term scale finally previous establish rsc framework provide adjustment remainder proceed simple program amenable calculation deterministic quantity depend convex feasible claim particular subgradient let j j redundant ps guarantee invertible point summarize nc parameter least exist p jk imply old plug back inequality desired define third equality bind triangle amenable jk global objective objective must appendix reveal satisfy feasibility inequality fact convexity uniqueness stationary exist straightforward see suitably inequality turn appendix imply bound apply combine primal eq conclude claim note also follow early norm detail n concern rsc relation imply require proposition inequality statistical result consequently write maximize similar argument equality v auxiliary employ lemma regularizer concave everywhere differentiable tt concavity function rsc amenable sample scale minor program q exist isolate local proof objective nonetheless suppose sake contradiction isolate exist sequence feasible extract convergent subsequence region close must together subgradient v kt k equation second inequality conclude well together inequality imply note gx x taylor divide take limit fx isolate collect kronecker norm claim use claim b triangle role cm ex
gpu computer effort ensure allow final require streaming overview imaging methodology parallel result reach conclude maximization current designing certainly seem extensively modern collect ray raw procedure discrete pixel inherent physical limitation ideally copy physical density theory frame sufficiently notably space obtain htb maximum likelihood intensity model record incomplete firstly measurement sphere ray overall noise process data volume noise useful aim produce way basic procedure step assign maximum full fix terminology firstly discretize generally non discretization rotation sampling intensity denote denote produce frame early em algorithm benefit free measurement scale around fourier intensity eq noise parameter keep decrease proceed sum observe frame rotation get require although formula understand likelihood use rotation encode rotation rotation identify discretized purpose cell regular whose boundary compose cell uniformly divide imply adaptively whenever increase go average continuous pair nearly continuous smoothing add step purpose latter say resolution operation working combination compression nearby rotation interpolation smooth value grid compression whereas consistent fall furthermore defer compare distinct parallel character efficiently probabilitie w inspection consideration nonlinear imply common gpu computational master successfully gpu approach section devise finally simple implementation gpu share core currently typical complexity algorithm stand likelihood rotation frame large space gb resolution distribute divide space implementation single node implement use closely logic briefly cpu overall stream gpu iteration intensive gpu discuss step denote contiguous rotation rotation computation e involve slice kernel care rotation step complex latter kernel normalization rotation division final gpu gpu execute gpu transfer final division gpu c htb configuration every portion implement pattern node normalize sum communication aim use increase actual experience highly slowly improve critical combine discretization hence large value sufficiently simple likelihood practice iteration achievable performance one quality process synthetic assess profile possibility large run gpu equip intel cpu core operate ghz I total gb gpu multiplication fourier bandwidth bandwidth cpu gpu connection device device memory copy bandwidth library use inter bandwidth gpu data volume image ray coherent source shape display sample pattern ray logarithmic table experiment reconstruct run consist dataset synthetic gpu implementation provide gpu gpu expensive consume updating distribute dataset synthetic consist explore adaptive single gpu run measure nearly perfect efficiency htb synthetic efficiency run really scale fully attempt distribute belong distribute inspection require table per per gpu remarkable loose fully achieve single gpu respectively benchmark matrix single gpu adaptive simplicity reconstruct criterion display likelihood execution per execution resolution whenever increase sharp peak successfully increase version quite remarkable run factor load execution iteration take cc configuration q execution time configuration scalability mean work program happen different notably case gpu step contrary scalability compare real increase prominent role measure case implement oriented depend linear compare par work well useful implementation site process stream fashion likely adapt dataset substantially dataset improve european become operational hz bottleneck hope keep ray enable biological protein liu european liu suggestion early classical method atomic analyzing pattern development technique extremely ray streaming energy ray machine associate inversion collection hundred expect synthetic ray currently successful quality problematic protein modern protein
method method per singular minimization subproblem accumulation generate addition compare outperform moreover subsection notation introduce study extend proximal establish conduct numerical finally conclude remark resp resp dimensional denote vector hadamard namely entry infinity norm denote denote resp square matrix denote sign operator also early lower stationary stationary point finally local chen recently point eq aforementione point natural extension proceed lemma subsequently post together multiply side finally arbitrarily choose let observe view stationary point give stationary replace otherwise eq b vx follow lemma post multiply obtain diag q fact satisfy submatrix consist row index clearly permutation together definition lead q pre definition conclusion em minimizer minimizer minimizer subdifferential v q multiply obtain v u mx hold hence local minimizer stationary point eq order hard relation section extend problem iterative reweighte throughout establish zhang unitary invariant function invariant set result class solve norm invariant singular optimal consequence subproblem solution offer solve subproblem decomposition hard equivalence extension method vector converge choose arbitrarily apply subproblem go state outer inner iteration fact termination inner method moreover accumulation suppose bound accumulation stationary entry I inductive argument view argument one outer clearly subproblem decomposition singular arrange q accumulation subsequence boundedness generality subsequence observe inequality show imply due along proof consider side use together follow contradiction follow eq relation contradict therefore relation hold argument first termination criterion section solve minimize sum lipschitz continuously function nonsmooth proceeding regularize matrix subproblem solve latter problem separable suitably efficient tool subproblem singular q immediately em proximal method pt end outer iteration termination criterion satisfy inner accumulation point stationary problem let stationary moreover satisfy inductive use use proof show theorem value arrange lemma accumulation point eq similar used termination criterion conduct test apply solve particular onto restricted convenience presentation name matlab matlab intel ghz ram bit windows service terminate accord denote apply process target solution define criterion successfully recover correspond relative subsection conduct numerical solve reweighted variant aim purpose generate matrix entry generate rank addition set figure detail successfully cpu generally three recover almost instance recover instance successfully six instance observe instance comparison datum subsection compare fill pixel pixel rank detail pattern texture decomposition decomposition testing image pattern six testing result cpu report display recover method column achieve three cpu time generally outperform cpu sr e e rest image recover regularize minimization introduce stationary vector minimization regularize establish iterative reweighted value subproblem show accumulation stationary solving establish regularizer nonconvex reference therein optimization moderately regularizer use develop lemma corollary proposition remark assumption zhang general unconstraine minimization first stationary introduce regularize minimization establish minimizer problem must moreover value minimizer minimization iterative reweighted propose subproblem show accumulation proximal solve outperform moreover minimization iterative reweighted iterative reweighte least proximal decade attract engineering
subgraph bound spectral clique limit detection analysis residual may detection subgraph extremely unlikely intractable detection anomalous continue field enyi give alternative vertex still activity background occurring still share subgraph vertex choose signal embed likely q equivalently simplify portion summation complete normalize principal eigenvector decompose component subgraph orthogonal subgraph yield convenience let combine equality use remove norm back provide embed eigenvector concentrate subgraph principal subgraph I subgraph occurs expect rewrite consider bind subgraph subgraph vertex eigenvector small may highly correlate magnitude convenience combination yield equation solve far relatively heavily eigenvector cluster onto yield separation observe expected signal expect result background anomalous subgraph embed change appear degree embed volume difference matrix strength spectral norm numerator ignore denominator slowly signal certain term dominate thus application interest vertex approximately constant growth degree norm approximate q equal increase ratio allow vertex probably occur constant threshold since norm dependent subgraph clique star substitute meaning vanish author technology office support dr thank early work dr dr schmidt anonymous helpful comment wolfe anomalous application connection graph diverse entity anomalous rest detection anomalous subgraphs agnostic graph base call leverage tool metric subgraph adjacency technique great detection bipartite realistic trend verify area great embed portion background detect highly anomalous subgraph internet traffic theory numerous entity datum organization website computer science know reaction varied entity incorporate datum represent relationship set vertex comprise denote theory provide indeed protein interaction represent computer graph focus community influential practitioner derive classify see research processing transform signal propagate along storage structure lack convenient perform analytical available relational derive detector signal subgraph network hard despite desirable notion small context computer discovery activity social recent subgraph subgraph statistic plant clique plant anomaly deviation description remainder detection relate community graph cast community broadly detection varied domain anomalous subgraph goal outli extremely edge insight noise discussion ratio intrinsic framework regression residual graph space eigenvector two subgraph work algebraic many researcher familiar analytically empirically enable algorithm detect anomaly organize subgraph detection brief detection outline present demonstrate finally open work subgraph size set subgraph set edge whose unweighted undirected connect vertex edge graph union graph work spectral adjacency associate order vertex denote edge connect adjacency subgraph symmetric norm unless induce norm eq absolute eigenvalue focus signal base edge deal vertex degree adjacent observe denote note degree typical background activity anomalous subgraph combine union give observation discriminate scenario formally want resolve hypothesis purely noise alternative signal vertex subgraph research assumption paper variety pattern common subgraph research anomaly history would static focus interest operate complementary outline consider optimal foreground embed via occur equal enyi set al possible occur probability graph occurrence applicability therefore require way count subgraphs observation target target sparse require detection random subgraph hard background foreground enyi situation observe edge test simple ratio require know computable asymptotically theoretic dense complicated model subgraph residual graph analyze widely separation community modularity modularity disjoint entire set entirely edge connection denote half half prevent edge proportion degree cut thing maintain vertex total deviation expect community detection literature numerous exist propose modularity maximization modularity adjacency divide modularity maximize solve indicate technique community suggest eigenvector compute community detection graph inspire amount work anomaly novel subgraph background activity subgraph residual observe variation signal anomalous subgraph reduce spectral residual residual establish theory eigenvector matrix see presence certain anomalous work within resolve framework computationally tractable variety scenario modularity baseline residual several advantage know eigenvalue eigenvector expect term computation eigenvector computationally expensive observe degree possible variable inter connection degree add covariate mention aspect processing framework assessment norm quantitie graph several context graph intuitive would detect algebraic quantity frobenius residual matrix square residual separately exactly edge vertex star vertex star concentrate cause stand edge place subgraph stand apart background work norm metric metric power subgraph principal anomaly appear foreground graph bernoulli foreground subgraph embed interaction residual subgraph submatrix background residual rest within complement identify least subgraph also subgraph stand residual matrix extend arbitrary may one detect anomaly rely subgraph eigenvector alone distribution thus subgraph stand eigenvector background small concentrated subset provide serve proxy compressed sense eigenvector subgraph occur relatively discuss eigenvalue eigenvector eigenvector normalize deviation small negative value demonstrate graph skew distribution cl vertex embed large positive deviation index extremely unlikely occur hypothesis commonly create deviation value circumstance occurrence subgraph detection subgraph connectivity background anomaly eigenvector thresholded subgraph eigenvector subgraph residual consecutive eigenvalue together stability unstable rely sufficiently change subgraph detection subgraph large rather eigenvector whose principal find space limit utilize large substantially residual put number nonzero however np hard penalize form equality semidefinite solve semidefinite eigenvector denote return subgraph vertex thresholded rely decomposition leverage number algorithm thus scale edge run control future technique anomaly degree model enyi simple enyi vertice equation vertex expect expect popularity give share yield degree expect degree approximately modularity fit background mat stochastic kronecker base sum fold kronecker ij np vertex add edge generator create graph mild present challenging noisy subgraph model degree complexity modularity matrix match formula mat structure mat generate approximately graph flip diagonal diagonal make cl expect edge mat er complicated note moderate mat cl vertex degree mat lack low corner randomness enyi graph anomalous er combine entry subgraph bipartite split letting er generate bipartite subgraph expect adjacency form demonstrate metric outcome carlo graph subgraph vertex bipartite statistic outline create discriminate mat cl er average cl foreground vertex vertice vertex er always near cl mat phenomena result confirm note cl extremely term bipartite foreground subgraph mat cl likely cause among detection improve chi improve analyze norm subgraph embed statistic spectral chi cluster bipartite foreground make aspect technique non performance improve subgraph improves cause fig histogram mat minus axis eigenvalue bin trial approximately eigenvalue localize around value yield mat rank degradation performance square statistic rapidly embed improve symmetry projection balance bottom similar exception precision foreground vertex identification use cluster vertex improve eigenvector precision performance subgraph separate necessarily via burden limited trial accord mat mat er equal estimate detection mat er fig much
problem brief construct column element column except denote th row ergodic vary norm define mml agent agent comprise represent interaction I affect edge function denote direct path vary topology strongly exist vertex weight degree weighted laplacian associate family model practical graph simulation random construct edge realization decision convex reveal incur difference well eq perform fix move environmental present unit regret decision incur decision cost run tx distribute objective network di decision nesterov dual turn inspire sub centralize step onto set define dual averaging sequence proximal avoid undesirable strongly agent subgradient p attain via local gradient provide compactly preserve laplacian access path agent strongly construct stochastic require associated direct graph selection diffusion communication edge weight expert associate expert select j consequently present applicable loss decision regret sub weight allocation agent associate embed agent ft f jt j ni distribute associated information neighboring information place link self preserve row matrix row every weight eq graph irreducible give decomposable communication switching set change dynamically selection topology topology positive strongly communication communication topology specify j distribution matrix connect matrix employ present remark assumption lipschitz refer factor standard weighted averaging regret sequence weight dual analogous thus follow algorithm present evaluation appendix impose upper associate highlight underlie consensus problem extend optimization provide employ weak ergodicity reason imply one matrix thus suffice negative matrix positive mp specify note must ergodicity markov product converge exponentially rank maximization realization sequence matrix element strongly connect topology node integer represent direct strongly represent row matrix case switch topology state switch ergodicity connectivity proposition present since q therefore test statement theorem convergence highlight topology diameter coefficient ergodic eigenvalue weight network convergence provide ergodic communication matrix construct diameter integer communication round subsequently since f r ik ec kt ks kt base conjunction imply bind algorithm tighter capture ki perform subsequently topology moreover graph regular graph neighbor uncertainty distribute process objective di manner uncertainty environment belong present quantify ergodicity diameter direct topology arbitrary sequence definition right eq thereby express base first side expand hand first g ix deduce projection respectively present impose bind note imply appendix impose last highlight examine agent similar generate tf tx regret imply statement hand adopt estimation aim estimate convex contain origin vector vary sensor unknown environmental factor sensor assume accurately observation topology sensor represent presence indicate sensor agent summarize q assume local reveal allow error uncertainty environment subgradient sensor neighbor cumulative tp offline noisy case centralize error suitable scenario characteristic wireless dynamic effect example dynamic eliminate framework rely datum distribute step local reveal select lipschitz th th h algorithms sensor scalar agent example also qualitative agreement indicate accuracy topology scenario sensor also demonstrate well tp adaptive topology assume sensor addition various figure result without regret node signal standard furthermore connectivity correlate connectivity designing topology operate uncertain metric scale tp kk network operating algorithm evolve use agent highlight measure average well demonstrate range mobile network failure moreover extension examine filter adopt system operate environment embed decision process lemma present u increase follow result decentralized algorithm step eq evolve thus q upper arbitrarily strongly topology node exist matrix entry represent integer path entry aforementioned induction row note positive satisfie adjacency algebraic literature thm edu paper agent presence uncertainty topology inspire advance base dual gradient link adapt reliability agent rate topology
empirical gaussian process popularity decade gp parametrize determine integrate away gaussian would process reflect natural intuition simple parametric form general elliptical student value elliptical place inverse wishart process form student connection similarly utility uncertain example student perhaps might detail question student wishart process elliptical precisely motivate wishart arbitrary analytic predictive distribution inverse wishart wishart inverse prior gp predictive covariance tp even covariance gp contrary analytic separate analytically gp find misspecification improve covariance distant kernel predictive covariance introduce inverse wishart student wishart covariance demonstrate process section section wishart choice matrix arbitrary wishart definite density follow n wishart parameterization marginalization principal submatrix make wishart appear attractive covariance matrix wishart suffer impractical bayesian wish expect semidefinite wishart however almost thus requirement wishart marginal nevertheless suffer inverse wishart inverse wishart place mass wishart submatrix distribution parameter wishart property motivate define wishart k kx wishart h xx grey represent gps popular nonparametric thorough guide gps provide gps practitioner use place student parameterize wishart prior analytically derivation material marginalization multivariate student kk nk kx tp elliptical sampling equivalence model tp student generalize gp limit tp prove control large tail tail prior sample draw extreme behaviour also dependence variable jointly student distribution notice dependency control gp tp multivariate analytic material kn n predictive intuitively infinity predictive depend train somewhat covariance importantly likelihood tp differ predictive tp kernel hyperparameter student explanation square distribution value large vice scaling apparent flexibility wishart student marginally n scale surprising integrate derive student process show integrate scale arrive process insight lead student student student process elliptical overview distribute symmetric unimodal point distance property want encode make elliptical naturally extend countable subset jointly elliptical density alpha process characterize elliptical density variable r elliptical elliptical either generalize process elliptical tp thus expressive nonparametric bayesian analytic expression predictive distribution gaussian process wishart positive matrix offer novel student wishart orthogonal square row orthogonal rotation volume definite matrix let exist iw careful tell uniformly distribute describe exchangeable affect exchangeable draw interpretation scalar variable analogous wishart unit sphere independently marginally exchangeable orthogonal sample symmetric symmetry distribution rescaling variate common practice latent gaussian advantage model sum gaussian analytic unfortunately sum independent analytically parametrize kernel add independent effect propose handle incorrectly function noise gp gaussian various behave tp student attempt tail inference analytic novel analogous process key hyperparameter include tp respect supplementary regression hamiltonian function sum exponential delta mse function noise test tp generalize superior predictive independently show mean since hyperparameter tp superior hyperparameter well amount record year clear tp much measurement dataset due attribute ph quality score learn learn optimize objective powerful optimize expect improvement ei optimum paper ei ard ern wish multivariate mat kernel represent parameter n note x slice similar use acquisition net n descent intuition acquisition fix plot clear tp prior mat ern plus delta bayesian integrate away slice sample dim dim dim aim minimum function minima tp gp tp come take function method x x initialize one minima corner cube behave
unique minimizer differentiable continuously fix l every imply uniformly lipschitz lipschitz uniform boundedness e hand subgradient bound uniformly measurable constant q imply boundedness ready justify quasi martingale follow quasi martingale surely q function converge stochastic process q condition proposition martingale limit almost consequently converge first utilize let another sequence take sequence produce derive convexity derive gradient frobenius uniformly eq fraction plot corruption rank robustness notably result relatively rank extremely simple pca perform corruption typically less largely corrupt corruption propose sample indicate suitable capacity pca corruption hard htb figure low corruption corruption interested tune rank corruption fraction investigate fraction intrinsic vanishe converge observe bottom row notice dimensional drop begin possible basis coefficient inaccurate however become rank exact rank pseudo estimation however strategy aforementione xu li xu li decade batch big due bottleneck regularize scalable consider apply completion factorization consist basis stationary asymptotically demonstrate encourage robustness decade machine community domain include collaborative filter texture name suppose ambient observation aim low problem many variant typically involve residual error rank term speak intractable tackle researcher suggest alternative relaxation surrogate nuclear max like induce nuclear formulate semi definite technique show filter margin program nuclear come compression world sensor background clutter corrupt case tool subspace handle corruption seminal prove mild condition recover corrupt notably guarantee recovery relaxation max norm regularize superiority theoretical collaborative example prove max schema nuclear follow cluster problem another important mathematical sdp scalable gap progress practical applicability bridge gap regularize constrain empirically promise result filter constrain max however require access batch optimization bottleneck utilize factorization max norm advantage online sample online fit interested aim norm fold develop scalable solve prove solution asymptotically work literature work norm superior collaborative filter hamming influence sample uniformly nuclear generalization nuclear norm superior important regularization scheme scale huge scale effective one mini total optimization global example scale dataset machine server collect return worker obtain consideration communication global try give protocol work showing require restricted rip random accelerate establish aforementioned assume aim rank component incoherent sample assumption establish incoherence exhibit thus liu li propose dictionary phenomenon study sparse incoherence place completely remove distribution follow global work decomposition relaxed problem alternatively without intuition benefit rank factorization optima guarantee local factorize problem convex code alternatively schema demonstrate hard learn dictionary analyze progress assume rip initialization provably minimization non sense optima recently pca algorithm comparable nuclear subgradient max insight max example norm factorization couple optimization technical appropriate factorization amenable online part idea factorization robustness contamination convergence magnitude dictionary constrain uniformly naturally begin state manner guarantee mild give norm conclude introduce bold row th entry set online underlie rewrite intuitively coefficient term fortunately dependency constrained aim recover variable replace note see need solution satisfy still contradict equivalent sample equip rewrite fashion combine indeed optimize minimize empirical infinity empirical converge online detailed noise alternate manner iteration give optimize examine tucker kkt condition basis define warm accumulation b alternatively implementation previous small exceeds w computation computing consider sometimes efficiency computed matrix empirically verify decrease increase increase value help next iteratively corruption perform operator coefficient bi directional initialize low search upper q update sample accumulation optimal subgradient row section theoretic assumption generate surrogate strongly particularly small definite small term problem enforce add solution stationary tends infinity essential tool analysis stage bound firstly turn secondly boundedness g main converge almost index p concentrate ball martingale almost central limit almost surely loss imply surely taylor taking vanishe tend conclude focus mc another index observe orthogonal onto span matrix outside th equal otherwise interestingly max cast introduce otherwise reformulate product tends reformulate mc implementation regularize mc difference element either two newly optimize completion describe sample access compute accumulation optimize surrogate trivially justify decomposition clean corruption correct evaluate fitness subspace variance ambient sample algorithm report htb effectiveness measure last intrinsic report observe corruption low pca tight number besides pursuit baseline performance algorithm corruption corruption agree fit much fast ambient dimension difficult dimensional pca achieve sample optimize curve basically pca algorithm example
roughly sample predictive selection small cross choose majority background dataset px px model proportion mix simplex eq since negative detail gradient partial hessian definite long therefore maximum strictly unique maximization concave vertex simplex element vertex simple evaluation iteration element objective vector maximal remain enough property restrict evaluation line f objective quantity notation current regret iteration iteration show suffice background curvature show background show twice differentiable become brevity denote right depend thus long maximal minimal curvature frank wolfe background simple tolerance number depend objective function step j maximal element step px dominate q computed come early train complexity predictive feature gene set complexity likelihood background model reasonably growth public query take iteration second intel core tm cpu ghz weight old tend citation fig normalization technique citation measure infection cell model capability large dataset cancer human cell cancer cell vice versa top weight datum drive direct indirect citation favorable modern metric precision rank engine list strength gold existence edge dataset publication top relevance table size seven clique four breast human stage remain three brain among collection clique profile clique experiment edge sample form lastly study illustrate support manual search topic gene expression database expression eight dataset manually table test well drive retrieve dataset retrieval fig reason annotation vocabulary description level specificity detail individual result investigate retrieval retrieve query false table retrieve dataset follow connection area interestingly e retrieve query outli human finally internal ranking seem health annotate annotate retrieve e partially sample type e sample people half old background know contain sample disease old health states institute science molecular biology laboratory european bioinformatics genome cb sd uk technology university biological pre set sort normalize essentially compute list sum score increase gene final procedure essentially compute weighted kolmogorov divide ks match scoring gene score value successfully use early select set standard produce active gene activity gene express core
measure I confidence add originally continue unless stop reach datum labeling supervise broadly group ii learn unlabele datum inductive contrary evaluate goodness unlabele supervise illustrate learn understand supervise unlabeled negative supervise datum unlabele boundary remain supervise shift supervised shift positive semi green dot green dot one fail label supervise generative ii type two task speech language solve two way classify learner attribute attempt learner language learner mathematically discriminative calculate supervise predict eq q ignore interestingly algorithm use induce class give instance generate semi algorithm reason completely ignore label provided generate key semi understand algorithm give numerator covariance estimate mean tune accord algorithm machine graph self discriminative broadly ii iii initially label classifier generate classifier initially unlabele classification unlabele reach newly classified remove produce second classifier apply continue converge unlabeled classifier stop reach threshold cycle difficult nonetheless self dataset contain number training create label datum generate unlabele confident add confident cycle continue two co attribute naturally however meet training view sufficient make good classification learning unlabele provide human label label remain unlabeled unlabeled oracle natural language base finding classic state literature domain parse text classification mining self improvement report speech successful execution training come source domain besides third linguistic attribute firstly produce secondly experiment article hold million article north american news corpus perform well contribute contribute author section level sentence section label compare baseline neither improvement sentence poorly however supervise sentence pool particularly require memory execute cycle self source limit success language spam form text spam author reproduce first message dataset besides message show training filter supervise fail good aforementioned mind comparative study produce summary author use four classifier relevance event attribute try combination summary surface content attribute combine supervise measure score score gold summary co training I summary although supervise summary na I dataset cluster come summary summary among author score well word summary cluster drawback attribute primary co co view secondly surface consider ignore na I paper significant effort secondary interaction help identify phenotype pathway classify sentence describe protein supervise tool first name brief description dependency name tree analyze path sentence gold measure similarity interaction protein supervise semi generate sentence classifier evaluate gold supervise version classifier near respective harmonic attribute classifier aim cb aim dataset contain protein cb compose protein interaction algorithm experimental outcome aim perform cb dataset classifier base attribute produce result aim performance satisfactory examine effect training aim semi perform poorly available start well cb report distribution aim addition imbalance affect limitation since rest finding parse text mining substantial supervise difficult unlabele behind success semi classification rather surprisingly investigation find classic suggest despite whether really supervise label semi dot classification gray dot old dark dot represent come conclusion much classification old abundance amount unlabele people nevertheless deal point suggestion classification semi supervise algorithm match hand cluster naturally co attribute value tend class paper complicated distribution datum investigate instance poorly unlabele highly assume decision false negative proportion unlabele choose proportion affect overall empirically effect attribute classification keep detect unlabele usually source unlabele rather semi email idea several natural processing supervise exploit difficult expert despite wide use classification empirically supervise study explore possibility limitation classification parse classification method label data model natural language processing text due unlabeled data process time serious effect supervise
compression reconstruction would explore practical consequence viewpoint combine namely learn elementary finer next require transmission stack result bind appendix theorem hessian respect generative derivative term appear diagonal newton algorithm asymptotic neural activate unit newton hessian computing determinant gauss newton hessian consider reconstruct yx layer wise gauss namely initialize output exactly cost backpropagation pass look backpropagation backpropagation derivative network compute backpropagation pass propagation initialize layer backpropagation derivative value activity incorporate acyclic influence model length backpropagation eq layer derivative iw initialization influence influence influence subsequent substituting influence connect condition non vanish v collect backpropagation know influence term derivative avoid full transfer set provide way compute note rate propagation construction summing yield forward sum backpropagation sc sc bx sc definition prop lem example exercise exercise remark similarity difference encoder minimize encoder decode denoise auto particular viewpoint determine noise denoise auto aim datum hide dimensional feature space simpler describe interpretation framework length probabilistic correspondence simple datum auto encoder short auto answer actually encode theoretic auto minimize compress base encode simpler tight minimize refine space sense arbitrarily close generative result appear also illustrate optimize auto encoder look optimize depend upper aim optimize aim make upper connection continuous impossible sensitive quantifying criterion denoise criterion one differently connection derivative somewhat though penalty frobenius norm row theory additional various variance noise level absolute variational already network situation encode situation try encode encode input encoding minimize reproduce cost include generative dimension feature encoder function go generative depend instance represent neural generative law infer thus previous case dirac single base alternatively auto view draw random apply generative viewpoint good generative give distribution point bit generative eq minimize amount minimize datum know encoding optimize work integral feature contribute significantly contribute yx give know conditional know nice obtain guarantee reconstruction feature variational feature quantity autoencoder thus moreover bring priori auto generative explicitly encode encode use define choice discrete sense section q difference substantial relevant two necessary encode feature perspective encode deterministic cost involve decompose empirical kullback leibler probabilistic cover deterministic dirac discrete two part part comment comment error case probabilistic proposition network activity layer probability value section continuous entropy optimisation kullback leibler possible arguably introduce regularization absence front value fix elementary bernoulli optimization elementary fairly tune kullback term close elementary model two encode space generate one expect save differ save encoding generate probabilistic small pick generate accord depend optimize around feature specific bernoulli gaussian lead interesting denoise deterministic feature positive expect elementary read average noisy reconstruction error proposition encoder set carlo ordinary backpropagation activation sample backpropagation layer factorize linearity namely average input explicit incorporate result feature optimal choice around taylor expansions optimization denoise taylor small value let covariance reconstruction error choice noise deterministic use matrix hessian minimize favor error possible reconstruction denoise optimize bind third compute hessian reasonable valid magnitude particular different way denoise criterion diagonal optimize computing theorem compute gradient backpropagation pass diagonal gauss proposition taylor expansion z h yy estimate choice case direct minimization symmetric decompose diagonal z reconstruction hessian respect newton derivative output auto denoise instead jacobian norm quadratic reconstruct elementary variance reconstruction approximate upper newton approximation dominate square derivative compute computable parameter compute costly gauss layer turn instead distinct stack layer start hessian hx yx yx xy get q gauss namely valid summing sign implicitly matrix apply complex involve hessian adapting involve choice elementary maximize distribution adapt denoise criterion consider minimize reconstruction focus divide bit work situation output usual actual reconstruct datum recover likelihood fix fix loss incorporate various jointly give optimize estimate proposition q square additive average dataset focus error
core analyze model much simple dirichlet access accurate need dirichlet place perhaps effort multinomial system calculation mean point gaussian mixture important natural model complex demand estimate dirichlet multinomial globally converge publish dirichlet go implementation simple produce initialization stay newton access mle dirichlet multinomial fix also matlab review new mathematical lead ng general newton provide fast build dirichlet newton method use property propose algorithm particularly count domain positive gamma function equivalent minus input gamma number n input log problem make equation k mutually exclusive outcome vector sum multinomial piece dimensional count vector appear distribution multinomial yield also dirichlet intuitive notion put take account relevant count dirichlet follow want mle parameter access multinomial sum multinomial observe far observable row count multinomial computation multinomial speed multinomial k represent dirichlet dataset maximize build relate sufficient separating sufficient maximize look newton method hessian dataset follow derivative function library link library matlab follow matrix indicator vector compute hessian derive multinomial start except represent time category row start set constant omit unfortunately require full exponential dimension formula dual first summation processor job section update bottleneck propose use write microsoft matlab toolbox base compare time platform grow start round fp solution algorithm eq multiple times recurrence algorithm bottleneck newton computation become less important xlabel ylabel legend style cells east legend west coordinate hinge shape linearly linear constant dominate xlabel ylabel second legend cells anchor legend pos north coordinate negligible add overhead implicit answer process hold run generate dataset xlabel ylabel seconds pos south coordinate eventually bad version circumstance many per multinomial already solution reasonable size high phase ever long constant around xlabel ylabel legend cells east legend pos north west coordinate constant benefit evident dimension run cause allocation structure xlabel ylabel seconds legend anchor legend pos south east suggest newton computing dirichlet row stop add run constant multinomial amount law sample possibility handle mle intuitively valuable away possibility desire split dataset part add hybrid approach would independently may keep track would row go powerful dirichlet multinomial produce multinomial come single look dirichlet allocation row could produce map build accurately large cluster mle dirichlet matlab multinomial repository repository except correction matlab extend library experiment keep repository categorical
despite empirical significant adopting scale problem counterpart albeit usually mathematical constraint slow gain machine paper xu lasso regularize machine many make outlier perturbation case problem complicated uncertainty svm counterpart paper question two meta algorithm guarantee uncertainty regard uncertainty uncertainty allow second allow set oracle term find formally meta robust counterpart np achieve efficient tool algorithm recently sublinear methodology oracle contribute notably give perturb work formulation semidefinite quadratic program build recently linear trust robust reader address several solution cut goal run rest organize present rest section simple problem employ subgradient step assumption latter problem find bad exhibit perhaps programming formulation formulation fix robust formulation vector constrain without assume specific symmetric relaxed standard feasibility binary current course first robust ix derivation online latter reward maker loss metric call give maker select henceforth robustness adversarial reward thereby online perturb maker predict accord euclidean projection onto step current next achieve sublinear reward upper norm gradient diameter algorithm programming assume function online tp uniformly sequence decision magnitude approximate mathematical adapt formally approximate subgradient descent availability oracle either return infeasible oracle saddle point interior solve solver robust make upper u v gx du input target infeasible ex ti definition algorithm comprise dual update update conclude infeasible terminate call oracle first return robust counterpart infeasible otherwise return infeasible online descent combine final hence imply structure mathematical program term oracle approximate u ix procedure compute g linearity seem subproblem give least q conclude inequality convexity imply mention namely variant capable approximate one analyze approximate program domain choose perturbation noisy much produce upper reward vector throughout shorthand follow correctness proof use unobserved imagine round right hand q put thing final old cube identical otherwise cube claim feasibility program efficiently case interest highly efficient lp case combinatorial efficient generic lose robust solve robust favorable uncertainty counterpart possibly lp unit ball notice feasible correspond robust formulation robust program noise control u euclidean program amenable meta linear noise simple call oracle statement robust lp uncertainty also notation hence robust f maximal use see euclidean counterpart semidefinite program order motivate avoid reduce program sdp approximate qp solver qp uncertainty similar albeit general uncertainty r apply certainly fall scope hold program nominal show claim note fx prove claim lemma demonstrate qp uncertainty mathematical also iteration solution f norm accord k imagine transform eq require notice maximize establish robust approximate semidefinite sdp q feasible frobenius matrix norm counterpart sdp general np uncertainty nevertheless use uncertainty eq u apply robust term present ix terminate call sdp allow schwarz therefore finally note factor effectively solve robust transform problem equip robust approximately employ essentially applicable worst feasible accomplished subgradient solve large problem original problem support robust become process interest
strongly length immediately hardness fact could polynomial decide sc proposition polynomial impossible factor hard construction also unless finally noise aware variant treating remain norm variant also complexity remain wish focus design performance threshold notably easy solve would hybrid author thank look attention anonymous valuable manuscript thm thm corollary definition thm plus minus minus computational observation decade automate overcomplete turn processing dictionary iteration complexity actual technical show learn moreover hardness solution give result dictionary sensor result compress compressed complexity cs exact decade denote recovery read error also condition greedy pursuit omp relaxation subsequently numerous tailor dictionary certain class dictionary verify signal admit approximation setup thus achievable dictionary signal sparsity dictionary instead structure successful include audio speech name informally dl vector allow representation formalize way dictionary far incoherence basis g dl encounter sparse recovery subproblem treat classical cs reference therein broad dl concern nature widely challenge formal hardness well hardness dl relate via obtain hardness result recovery exist widely assumption hardness additionally unless pseudo fully solve within thorough treatment complexity mention goal formulation dictionary usually capture priori column function express fidelity penalty regularizer representation usual e seen often require norm cf priori become trivial exactly column also hold variant allow minimize error require large intuitively certain efficient algorithmic atom justify similarly since sparsity coefficient achieve representation basis redundancy expect maintain hardness result show indeed intractable dictionary hard strong restrict ms rank rank elementary reduce submatrix ms hard cf polynomially cut input hardness sense obtain equivalent set minimal dictionary fact rank require discuss dictionary reduction indeed inversion strongly polynomial gaussian strongly extend discrete objective scale dictionary learning ms achieve obey valid know ms contain similarly associate given decide square problem existence efficient polynomial general impossible interpret constraint turn etc even cost know quality recent work start theoretical guarantee dictionary initialization efficient guarantee hardness approximation ratio size usually e admit quasi almost hardness compare present seek full right multiplication infeasible close inspection reveal reduction carry efficiently provably train hard approximate objective objective maintain hardness dictionary rank ms hardness hardness remark new dictionary row achievable e system formally dictionary permutation nice contain strong moreover remain relaxed mn proof strongly cover collection word employ recent exist strongly complete problem cf sc parameter
big gain power discovery pattern coarse often ref far complicated volume ambiguity content character theory decision impossible importantly category problematic many diagnosis disease consider individual type determine survival without take two wish prefer student discrimination must course reject modern causal examine political benefit new resource well aid reject early goal discrimination enter apparent seem acceptable although trade utility value concern account classify contribution bayesian list ref recent example allow combined propagation processing system pass data module final prediction allow track final meanwhile list build call question live south redundant interpret bayesian tree human advance know combine remain uncertain mechanism wish may way solve artificial intelligence upon framework influence learn forest monitor day hope causal provide researcher policy reason causal us progress still make informally system outcome whether desirable mathematical example mathematical algorithmic method propose eliminate possibility output category time preserve method output possibility algorithm recommendation distinct consider ref seek exclude input indeed order effect correlation calculation require category determine odd patient college list list partition list consist track potentially wish avoid minimize impose eq outcome category population knowledge alone allocation accord life surveillance additional minimize leibl decision maximally indistinguishable lagrange enforce enforce imply knowledge correlation kullback leibler divergence become ill lead different opposed care notion suggest accounting accomplish exclude social mathematically strict particular desire population trade group outcomes california information concern college relative question come case hard impossible rely recommendation remarkable power human critical increasingly become infer outcome world advance review recent popular account seem scope powerful partially provide management prediction make hard assess elsewhere provide challenge evident existence challenge challenge correct estimate individual restrict datum participant goal reverse causal present restrict progress algorithmic box upon reveal algorithm innovation method computer change perhaps importantly allow big concern discrimination previously understand domain tie question argument increasingly familiar nature reasoning sphere joint present university discussion fellowship machine public big made absence fail demonstrate loss enforcing near role interpretable machine mechanism reason well mind augment automated reasoning subtle explicit decision political make reasoning cause machine aid decision make make policy involve pose developed program question interference unable resolve public social category make potentially half united states discrimination color national title ix education line insufficient prevent avoid category north american wrong track north south line decision making may prefer physical fitness access status status happen physical fitness character prove base detail know neutral causal role particularly
need rp rx rx ols ridge summarize basic validate orthonormal basis respectively I vector r efficiently parallelism query functional rp function belong ellipsoid shall control allow finite empirical functional j optimally lead dr h regression finite vary mapping f state risk suppose u feature take consistency stem basis stem ol estimation empirically prove effective previous easily work unlike need evaluation make scalable simulation formation particle matter unfortunately simulation multiple magnitude time step single simulation body simulation lagrange simulation order magnitude inaccurate experiment body simulation mapping area come area na mse result cube note set particle cube cube cube coefficient represent density cross validate parameter ridge report truly come body table average fast achieve improvement come improvement widely time prediction like mining miss embed hmms variable train em aim predict attempt audio prediction segment vertical music particularly short audio piece segment music compression music experiment song sound hold prediction take music predict consecutive total datum instance audio ridge quantity bandwidth mse validate hide bandwidth mse achieve lowest apparent audio audio superior predict sound material missing long cross fast speedup occur motion total look position unobserved step series joint perform randomly miss series output response miss joint position correspond spatial position projection consider segment take parameter estimator validate lc good perhaps surprising prediction point emphasis method speed x use prediction conclusion triple perform scalable manner functional capable massive low function assumption order magnitude multidimensional series estimation see eq l function note analogue response function nonparametric function cover application type previous set achieve address issue estimator capable nonparametric massive reduction previous estimator modern instance grow massive number instance function aim functional response function hence immediately unlike typical regression directly infeasible directly function nonparametric set triple framework quite instance may pdfs financial stock output stock another interested inaccurate output pdf expensive essence inaccurate previously accurate foreground intensity take pixel position foreground pose framework frame unit represent interval predict application predict co occur function motion capture interested predict movement movement state previously infinite dimensional output instance instance million infeasible want learn mapping want resolve scalable since shall inexact observation noisy evaluation distribute I take kernel estimator useful clearly produce lead prohibitive set previous nonparametric response scalable estimator functional response response linearity mapping moreover specific output observe evaluation sample distribution short input onto basis rbf function ij fp ix fp wiener omit simplicity input estimate output unseen function tensor serve mention function evaluation index basis index typically
density careful depend prior able obtain give require even leave strategy wish finish introduce significant address open rbm nevertheless recognize theoretical interest one note generate require decide accept reject would unfortunately however cost run investigate example implementation prefer issue future relevant section detail fully drive brownian reflect brownian motion rbm extension motion drift coefficient take counterpart denote brownian major later shall simulation indicate rejection conditioning arrive dt dt three subsection piecewise linear approximate rbm aim obtain rbm topology drive brownian algorithm wavelet drive reflect computable system equation result map equal radius see satisfie interior strictly surely continuity rbm boundary leave match locally drive eq piece wise brownian brownian brownian etc brownian denote shall introduce brownian increment relate sample approximation pointwise perform possess overcome difficulty proposal ratio constant conditional piecewise later rejection continuity k section denominator accomplish involve exact complete explain sampler justify provide brief description piecewise motion sequence process eq brownian dyadic available dyadic iteration dyadic index l j interval brownian collective refer arrive mr generate dominate piecewise form piecewise shall approximation q dyadic interval construction dominate recall lipschitz topology solve reflect n brownian dyadic contain like brownian motion mention rbm stay interior boundary stay dyadic know indeed continuity observation everywhere q recall piecewise rbm initialize nj nb serve reflect find n b independently dimensional brownian motion suitably propose denote brownian algorithm contain brownian motion rbm boundary match say increment return know simulate add sample know law collection ease exposition brownian available index dyadic interval denote motion density close indeed sl brownian bridge interval stay within ready outside support support sample simply uniform make propose eq density ratio explain reject perform rejection ratio lipschitz continuous positive sl explicit expression present variable accept note output follow know mention comparison necessary know left layer generate far refined rbm comparison part z initialize n nj serve piecewise l solution leave increment proposal product ratio acceptance rejection grant explicit proving follow continuous bl immediate consequence lipschitz continuity derivative lipschitz lipschitz lipschitz bound converge lipschitz since immediately fact lipschitz continuity boundedness boundedness lipschitz respect lipschitz constant lipschitz continuous lipschitz reasoning cm cm claim conclusion corollary criterion example theorem convention remark reflect brownian motion rbm develop tolerance allow piece wise approximation rbm conditional acceptance eliminate suitably design information proposal contribution exact simulation multidimensional reflect rbm rbm turn generalized comprise server heavy traffic arrival definition contribution outline underlie tuple give satisfy driving reflect problem drive process brownian motion diffusion c f occur eventually decide reject make sure
dataset describe section gender classification block vs kronecker toeplitz kronecker tn number operator objective rt follow incorporate toeplitz constraint equivalent study norm toeplitz correct next scale improve make dc shrinkage e minimize estimate shrinkage test gender video field distance frame surveillance camera period wide weather wide often video front back view total work evenly gender video remainder testing demonstrating htb attractive invariance property use object detect box relatively uniform lack clutter detector track connect track rare box uniformity position spatial track dyadic successively split array nest covariance generate reduce mean covariance gender learning discuss ratio covariance block positive weight thresholded logistic advantage adaptively block oppose relate quadratic several trial divide track frame track equally disjoint extract dc dc overall svm evaluate relate kronecker window number logistic particularly use svm indicate regularization curse note rgb pixel box neither exceed coin flip rate htb frame track correctly incorrectly train relating specifically crucial head area gender physical size htb c svm htb htb application temporal behavior nonparametric modeling appearance temporal significantly baseline classifier partially support nf fa wish propose application em plus minus height electrical engineering university usa spatio technique temporal adapt spatio characterized mean spatial correct apply temporal feature box gender choose covariance classification performance accurate human characteristic gender etc video surveillance understand spatio apply classify attribute appearance movement gender challenge low resolution surveillance track spatial area covariance address deterministic shrinkage estimate improve spatio matrix reduce discuss predict gain estimator
mean typical purpose crucial take obtain kernel frobenius expansion sign gaussian definite none expensive far argue mathematically strategy invariant kernel use abstract point consider invariance ensure existence algebraic group sign give group group goal k mn make consider act invariant kernel converse true existence state convenient variant symmetric positive kx call universal let act space algebraic invariant space algebraic differentiable group way finite finite manuscript appendix admit variant non group compact exposition characterize act continuous positive kernel symmetric qx qx diagonal invariant part independent resp part invariant statement space explicitly therefore explicitly invariant map canonical intuitively invariant kernel always separate conversely want kernel certain invariance factorization imply diagonal canonical construct variant observe general invariant name mind uniqueness map idea outline time invariant allow avoid associated fulfil may however argue true common kernel kernel rbf polynomial sigmoid kernel maximum spline combinatorial many ii fulfil kernel unitary absence would imply let unitary transform unitary kx proof together isometry kernel basis kernel isometry kernel also unitary fulfil variety proceed exposition explicitly write invariant version slight act unity invariance plane space group v vx phase kernel analogy sign kernel invariance e multiplicative map obtain scalar definite correlation kernel trick yield x leave phase invariance kernel trick either scale invariance sign invariant kernel scale sign invariant apply invariant effectively trick scale plus invariance combine multiple repeat invariant trick compatible presentation point practically group act mr rr unitary act rise representation unitary row act one center characterize long invariant decide adopt closely inspire existence invariant mathematical differ opinion main two ti kernel invariant substitution require group endow haar term invariant haar invariant rbf existence statement describe kernel invariant universal imply potentially avoid integral one behave quite definition refer reference able direct application definition furthermore remainder manuscript relation invariant feature far see invariant invariant invariant kernel outline aim incorporate motivated element invariance sense pixel give rise kernel statement globally kernel simply concept specific without focus particular technique spectral cluster extend g entropy datum eigen identify axes r enyi concept describe sign matlab invariant gaussian invariance handwritten sign pixel spectral successful grouping I group cause panel cluster sparse wavelet overcomplete six correspond six measure panel block invariance matrix ica address study eeg ica ica short fourier approach enable identification recovery solve overcomplete six linear mixture six play mix apply tree select point dataset figure space six cluster equivalent blind sign entry sign invariant continuous decade systematic constructive incorporation algebraic know kernel trick inner invariance kernel negligible gain substantial framework code variety numerous manner limit demonstrate sign sign invariance validate remark sign sign group imaging practical mainly conceptual future devote incorporate invariance broad field bioinformatics engineering acknowledgement acknowledge national research foundation education technology research support fellowship provide statement corpus always contain group act invariant act canonical show contain imply prove claim algebraic act geometric analogue universal property formulate usual algebraic geometry outline algebraic variety act family furthermore countable algebra geometry equivalence act factor uniquely countable sub countable countable finitely happen infinite hilbert universal proceed essentially translate act algebraic dense take intersection algebraic polynomial function categorical equivalence polynomial field w gr z w kind thus obtain analogue pass therefore consider orthogonal unitary resp consist orthogonal matrix act ng normal distinct alternatively orthogonal unitary resp unitary
corruption ability transfer image exhibit scale pose variability reasonably image database imagenet handle extremely learn direction sophisticated accommodate pose alignment normalization bottleneck keep g stage dimension result stage fortunately replace dimensional convolution filter future leave scalable classifier applicable regardless extensive extract face digit texture baseline study advanced architecture image receive electrical engineering ph institute communications national rd technology interest emphasis vision hyperspectral receive china electrical engineering ph degree science k advance science interest machine learn processing technology china ph university currently advance digital sciences center microsoft research fellowship interest computer vision currently research advanced digital science include computer pattern scientific paper review master degree school university interest school science technology university receive mathematics china degree mathematics berkeley associate university microsoft areas david mention award award national foundation office associate information cr pc pc work comprise hash histogram employ filter follow indexing name network easily comparison simple namely topology learn lda basic benchmark different verification extend ar face well write surprisingly model highly carefully learn record task ar public dataset potential simple competitive object network lda handwritten digit object visual content task largely large amount intra numerous counter intra designing task hand representative binary sift feature object recognition great success datum task require since plausible example method attention deep multiple hope abstract deep invariance intra one key ingredient success image architecture deep convnet architecture stack follow supervise comprise convolutional filter bank processing layer bank stage convnet variety boltzmann machines rbm review reference therein ad hoc variation convolutional vision success usually justify arguably justification wavelet convolutional simply hence somewhat surprisingly fix bank utilize convnet convnet task handwritten texture recognition generalize face intra variability illumination change corruption motivation study resolve apparent convnet goal deep trivial adapt basic network serve people justify use advanced architecture deep come basic easy processing mention adapt filter bank stage basic filter nonlinear simple hashing pooling name two stage characteristic challenge convnet operation stage hash histogram extensive experiment simplification performance orient audio lie couple hashing histogram layer gain robustness baseline likely linear discriminant extensive experiment discriminative section somewhat extreme totally filter conduct fair type network study architecture baseline comparing architecture finding think comparison already par state almost face write texture achieve accuracy extend dataset illumination dataset obtain competitive unsupervised mnist state background effectiveness propose task method deep far vast literature thorough encourage one hand serve surprisingly competitive advanced design success confirm certain remarkable even consist amenable justification effectiveness lead insight seem deep htbp image size patch illustrate pca filter input diagram pixel take collect patch I mn k jj patch mn image layer orthonormal matrix map matrix training course stack multiple filter denote boundary like patch mn mn jj I l l k collect remove patch filter pca filter stage obtain output build pca stage deep architecture beneficial input value output l binary every pixel irrelevant distinct word block histogram input histogram either depend empirical experience suggest digit histogram offer extract transform sift histogram orient gradient learn bag model process convnet filter number block size inspire common filter fine performance empirically stage architecture also histogram translation invariance extract convnet patch local convnet operation center origin filter major addition pca simple auto linearity stage build absolute convnet modulus test stage improvement quantization output introduce invariance output one merge stage field stage perform stage face digit single filter single filter stage learn filter whereas filter totally benefit contain object invariance capture comparative hierarchical architecture field stack relate extremely basic us example form patch eigen decomposition convolution bit number histogram assume overall complexity verify computational burden training decomposition whose convnet sgd gradient solver training take hour cnn take hour exclude fine section network require two eliminate necessity replace filter translation translation direction plane rotation e face face illumination test randomly select generic gradually sample keep even filter scale near nn chi classifier since find successfully stage cnn face supervise size convolution max softmax classifier pre cnn cnn epoch method cross test inferior whenever illumination drop offer cross variation final cnn seem face included follow superior filter report network number pca deep issue worth face around pixel take half hour exclude fine tuning pose apply pca learn b extended face image individual laboratory control illumination contiguous percent percent locate square block image overlap block illumination distance table illumination insensitive robust person various difficult condition surprisingly pca detector maximum contribution would ignore pass yield robustness percent learn cope real ar illumination condition consist image convert gray illumination neural testing overlap compare use table illumination variation perfect dataset insensitive robust feature achieve extended test finally popular recognition capture condition year partition probe probe expression fc different take four year apart gray image compare fairly dimension whiten feature cosine train consist people list standard state list learn art accuracy database rich nature learn outperform ii remark recognition prominent message section abstract subject one toward wide deep sufficiently inter intra class variation probe fc avg face evaluate unconstrained verification collect web pose illumination unsupervised evaluate learn discriminative face pixel evaluation split view contain class inter subset block performance feature cosine four descriptor square operation see root boost square quite show propose effective face condition aware convnet verification work achieve set train convnet align face base image face com l method accuracy dim dim le mnist variation widely test list investigate ratio state vary stage stage overlap region block half face almost case number block overlap ratio ratio c convnet scale set number set variation convex overlap region use error fair method augment good result comparable mnist report achieve result remain convex table also outperform draw figure observe vertical pattern attempt capture become background l description test mnist standard mnist rand mnist background mnist background discriminate background discriminate exclude filter unless otherwise method nn k nn convnet stochastic pooling convnet maxout rand convex texture texture class pose illumination surface variation also
traditional obtain pc proof e nx estimator regression x b n l b n ba ba f ba ba ba n ba ba b b b concentration traditional
entry restrict specific occurrence algebraic point point difference prove induce family n n inner sep flat prove module thus briefly filter module irrelevant multiplicative direct extend module module element great ideal module flat prove crucial result induce flat ideal e module way deduce n relation ideal trivial generator define irreducible generator generator four irreducible generator us di two generators module assume dx remark x correspond irreducible point statement suffice repeat n consider surface correspondence know say nothing critical lie understand vector assumption theorem even satisfy weak determine I define lie restrict attention ideal coincide know next question distinct point split factor tuple critical degree coordinate coordinate critical root integer distinct coordinate appear recall time sum weakly decrease translate ml partition integer addition specify ideal equal coordinate rewrite ideal write term depend statement ip partitioning subset need determine ideal instance critical coordinate since look repetition entry action thick thick thin thick thick thin thin thin thick point thick thin thin thin thick thin thin thin thin thin thick critical point thin thin thin thick thick thin thin thin thin thin thick critical try explicit notation later set consider index every univariate automatically vanish therefore proceed iteratively generator polynomial tuple form formula length ideal fill circle dash sep dash w sep pt euclidean unity directly whether root vanish root notice lead root also lie eq obtain ml develop likelihood n ideal table degree determine projection apply critical point implement software find processor range cpu exploit repeat field different rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle thick rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle thick cycle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle thick thick rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle thick thick cycle rectangle affect degree complexity totally rely partition heavily affect ambient perform eventually two partition length ideal differ different degree substantially ml degree slowly depend list ht ml degree computation well complete within scale thick thin thin thin thin thin thin thin black thin thin thin black thin thin black black thin thin thin black thin thin thin black thin black content school exercise go thank point comment suggestion mr corollary theorem de fu critical projective formula curve projective projective ml exploit list introduction explore recently branch mathematic call algebraic statistical result uniqueness mle log model easily algebraic however instance ask critical lie question know ml degree geometry problem instance hide algebraic light degree projective computing degree exist look aid software degree projective point general consider derivative lagrange multiplier computation value answer ml large algorithmic tool geometry exploit explore nevertheless happen apply statistical algebraic concept tool require moreover algorithmic play key construct scheme ideal encode degree particularly define two solve second flat extend look study example dedicated formula application achieve report computational remarkable degree formulation language want multiply similarity need want lie hyperplane define equation hyperplane paper di ki critical look lie distinguished ideal projective consider equation projective standard onto motivate ml degree likelihood correspondence irreducible irreducible variety hyperplane likelihood function finitely degree verify interested thus adopt correspondence factor order determine degree correspondence polynomial polynomial ideal define q compute ml degree degree map computational get surely degree computational degree long mainly gr ideal determine degree resp little elimination critical lie distinguished
standard element element element integer bad case interior variable encode programming able program programming total synthetic section query set describe result absolute precision order bound discretization domain range l spectral conclude event ms ss notice span complement span eq corollary execute execute c dataset ccccc dataset number grid uniformly choose table bad reduce theorem thm proposition section privacy query say smooth specified order develop differentially smooth algorithm real appeal run privacy improve demonstrate mechanism differential database conduct contain sensitive take privacy consideration database differential privacy guarantee almost nothing one without individual output significantly differential attack individual recently learn datum mining privacy answer differentially laplace laplace add sensitivity typical query database mechanism limitation nontrivial user query limit restrict remarkable due mechanism answer query preserve privacy nontrivial specifically refer accuracy improve mechanism powerful answer choose mention different query practical appeal privacy almost preserve certain attack output modify please survey therein preserve privacy answer query compare good answer universe linear generally universe strong hardness differentially output answer general polynomial hardness recently interest differentially private restrict query rich contain develop fast mechanism hardness result serious barrier privacy al consider query set universe constant rectangle query specify discretize efficient mechanism output database query another lot universe record binary attribute individual database series universe general answer mechanism universe mechanism database data universe smooth answer database learn reproduce ability differentially output mechanism accuracy exponentially time size smoothness contrast employ guarantee well large run super size database please generalize preserve privacy relate mechanism private answer query user complicated numerical experiment mechanism medical record achieve good practically size attribute rest organize briefly describe basic section output contain theoretical result performance section universe typically universe call neighbor differ data differential input range neighbor database take preserve differential differentially consider universe mechanism database eq randomness accuracy high query author weak definition query database respect internal randomness mean use laplace add laplace distribute laplace private output datum universe different database number q mechanism query query nonnegative integer contain order norm bound formally function machine smooth popular function machine function characterize follow good query generalize mechanism privacy accuracy compare differentially mechanism main specify smooth function mechanism differential synthetic description mechanism let query universe preserve differential constant hide depend mechanism I l mt km n r n idea look performance first derivative smooth give simplify running time database roughly nearly increase run increase one want database roughly ccc accuracy db explain detail smooth basis radial differential privacy problem requirement quite typical linear combination correspond uniformly reason soon clear mention set f combination polynomial polynomial satisfy coefficient requirement point basis polynomial query query guarantee combination approximate easily private answer small mechanism merely need generate synthetic evaluate answer answer synthetic query synthetic answer first distribution answer must dataset requirement computationally intractable original discretized universe query involve formulate programming lp well distribution synthetic bad interior upper easy control precision round private mechanism private k query universe differentially absolute constant composition omit detail differentially private mechanism private study differentially output accurately answer analyze simple note set universe query contain differential size universe ignore clarity smooth problem universe infinitely many element must universe smooth see universe grid precision datum universe query proposition smooth query achieve super section note importantly mechanism worst differentially nearly query run suffer minor time lp grid polynomial grid restrict basis function preferred degree simple grid suffer formulate differentially small almost every concern requirement privacy dimensional ellipsoid private top eigenvector square radius private mechanism private uniformly ellipsoid database private privacy schmidt set mechanism differentially top tangent angle eigenvector span suffice private ellipsoid converge true column simultaneou increase kk eq q execute privacy execute sample differential appendix mechanism repository community combine economic heart contraction monitoring rate disease breast diagnostic consist characteristic cell cc dataset size summary attribute universe normalize conduct mechanism privacy guarantee experiment privacy employ combination possess property section linear detailed query experiment value smoothness measure comprehensive performance bad error mechanism query kernel infinitely bad query error relative therefore good relative informative necessarily bad performance present intel processor ghz gb em ccccc table linear column respect synthetic output database ten explain experiment except show accuracy differentially compare ccccc time database preserve privacy appeal practical viewpoint mechanism database
likelihood user item profile analyst unfortunately provably item profile moreover user rating analyst contextual dataset user privacy reveal gender age rating information mf suppose gender political incorporate mf adapt user profile modeling estimate profile bias jointly q special case extend profile dimension depend treat another though priori mf rating comprise easy sampling put gender mean among incorporate per simplicity feature discuss multiple scenario analyst perform mf dataset explicitly profile item analyst typically batch rating follow linear extended profile analyst infer profile b analyst infer pick small profile analyst rating analyst classification infer svms alone involve regression separate inference analyst gender separate logistic svms rating reveal albeit meet simultaneously see near front right cube privacy thm corner perfect maximal privacy user receive accurate rating recommendation ask benefit provide surprisingly beyond fact analyst address issue set comprise analyst access user reveal analyst binary perform factorization dataset extend item profile analyst rating parametrize extend follow analyst identify aid analyst privacy private design analyst salient informally conclusion analyst component analyst private feature learn extended profile highlight three protocol protocol exchange analyst rating analyst section b analyst e finally protocol analyst item user rating private feature fix obtain analyst protocol extend protocol solution fact among protocol constrain way completely third conceptual three protocol practical seek protocol satisfy protocol analyst learn non profile existence service ensure privacy benefit extent make publicly item profile construct describe last allow service generally analyst thereby privacy user limit exposure competition ask accurately learn little c definition analyst privacy analyst access profile mf rating private analyst feedback profile formalize privacy analyst fashion extend profile rating profile mf noise set profile analyst rating item clearly uninformative light denote know private know rating nevertheless consistent mf latent easily rating natural opinion item ad news article tweet video etc rating user user website collect sensor response rating notation define analyst profile program execute analyst user item practice user analyst describe reveal set program execute particular r r l combine subsequently reveal analyst profile analyst construct feedback vector execute analyst refer triplet note functional analyst know protocol extract reveal mapping protocol protocol randomization protocol begin intuitively nothing user value variable algorithm determine protocol strictly inequality scenario extend profile metric relate criterion motivation rating product estimator expect also motivate analyst user brevity omit recall analyst predict rating often error preserve protocol precisely hence analyst equivalent minimize partial protocol reveal much v r natural intuitively retrieve mapping differently compute output simple refer remarkable mp analyst shift rate reveal analyst feedback analyst apply estimator mapping bad eq summarize protocol remarkable privacy accurate mp mp less accurate prove third establish formally protocol intuitively statement protocol privacy preserve among scheme statement imply maximal protocol protocol bias note mp appeal analyst gap rating etc enable minimax follow minimax computed know coincide square estimator close trace r jx clearly privacy protocol accurate lower preserve loss first private variable newly output statement definition analyst improve protocol fact analyst choice statistically analyst profile analyst attempt setting square estimator noise natural item analyst feedback extensively context exist item profile user fact particular maximize factor optimality protocol analyst particular service ask rate could user reveal analyst apply mp difference reveal interaction still depend remain maximally express categorical illustrate incorporate approach incorporate category specific bias representation whose observe categorical incorporate analysis first rating non reveal feature construct perform factorization rating bias j jk privacy subsequently analyst analyst bias user r k categorical analyst gender gender dataset scheme mp round ia fa ss sub sampling privacy attack rating gender view gender gender protocol mp user inference non little dense dataset private happen dense nearly evaluate tv rating dataset include star rating tv gender make tv recommender movie gender user rating rate user dataset seek privacy fold cross split user fold fold user privacy profile factorization empirically item use private mf implementation classifier rating randomly select rating first set rating several baseline detail input classifier profile infer classifier square predict rating computing auc time report auc range privacy item bias half rating private factorization compute regularization fold standard method private rating multinomial na I logistic lr svm radial basis rbf lr nb comprise item well movie rate operate rating user number execute mp rating outside range rating recommender therefore variation mp rating integer two round probability expectation rating high truncate process round round protocol sub round scheme replace rating replace e finally ia fa ss also l age gender auc lr multinomial mp avg item avg avg begin evaluate dense user gender political accuracy rmse apply scheme mp privacy rmse illustrate mp attain model little remove strong failure especially add round world mp effect ia fa suboptimal prediction impact prediction rmse little left value item rate mp method item rate four insensitive reason ss feature rating auc excellent across privacy auc contrast ia fa significantly rmse stress dataset rate nevertheless far impact sub sampling number item analyst feedback ratio feedback report ss rating remain exclude rating least accuracy suffer overall mp world dataset privacy risk impact linear eliminate rating private illustrate gender dataset distribution posterior enable successful auc distribution become indistinguishable rating rating rmse auc rmse tradeoff scheme provide curve almost flat consistently auc pt pt factorization raise question extend task perfectly private privacy usual privacy literature protocol differ opposite mp protocol could begin define protocol define tuple function profile rating two step determine rating user output rating item user truly rate feedback user analyst reveal provide determine x r analyst protocol r extend straightforward regard expect preserve depend protocol rate r protocol protocol bernoulli preserve privacy preserve learn protocol begin establish auxiliary r lemma states protocol construction rate user cv cv preserve strict v item probability differ respectively observe v p p identically particular privacy equal construction couple dominate former privacy protocol rate iff obvious yield must property distribution ready privacy conditioned condition monotone protocol lemma protocol coupling pt pt rgb rgb rgb rgb rgb infer attribute privacy protocol improve rate strictly extensively protocol demonstrate gender age political decrease accuracy give star rating book article kind micro service predict user preference prediction g make recommendation ad service accurate fundamental importance service reveal movie news article inherent call rating movie private political analyst use analyst predict user rate item dataset factorization recommendation analyst learn private access analyst private rating reveal relate whether evident report successfully orientation drug ask happen desirable benefit analyst recommendation rating analyst tradeoff distortion introduce first dimension information analyst understand scenario analyst rating analyst execute news article relevant would analyst rating clearly analyst reason analyst service competitive edge trivially attain natural analyst algorithm natural information analyst privacy enable service make follow introduce study analyst predict rating determine analyst privacy specify analyst reveal user rating feedback rating prediction protocol protocol mp remarkable predict yield privacy protocol mp protocol mp bad accuracy extend handle common deal item analyst discuss analyst item repeatedly analyst protocol gender attain excellent wide array inference blind area auc privacy rating take analyst privacy tradeoff proof model factorization validate vast empirical item rating approximately classifier linear validity real world extensively infer tweet facebook friend movie rating lead linkage attack enable close show political view orientation drug use accurately gender political infer rating tv show release analyst context preserve see construct decision task user population factorization attribute sensitive setting keep release apply distortion threshold address work asymptotic broadly cast treat employ specify
limit set need dependence target handwritten digit subject clear behaviour similarity define image hence clear behaviour sample two would highly unlikely bad behaviour limit square error sample restriction dependency converge call q target mean draw variance give see weighting point weight differ optimal weight point translate constraint program reasonable condition show assumption consistency mean fulfil need behaviour exclude fast dimensionality distance datum intuitively strong contribution estimate reliably target linearly combine asymptotic single state condition contribute go consistency fulfil averaging therefore require restriction dimension dispersion grow slow otherwise sequence uncorrelated zero without consider class target possible use overview jj correlation target condition consistent assumption consistency l p target additional set fulfil identifiable see appendix state exclude sequence grow fast product uncorrelate quite weak analogue behaviour covariance limit remain go infinity consistency statistical fulfil consistency eigenvalue covariance uncorrelated identifiability standard shrinkage therefore shrinkage intensity intensity moderate accurately estimate quantity measure value indicate accuracy classification quantity interest serve accuracy estimate lda simulation behaviour large limit target standard sign random four setting set clear behaviour sample tell receive show intensity intensity target go shrinkage intensity reflect zero go constant relevant outperform finite shrinkage intensity range estimator identical accuracy data set eigenvalue log achieve rescale discriminant ignore pool take pool baseline sample scale pooled improve sample pool improve find figure spike multiply discriminative drop target useful original direction spike still discriminative put whitening help give equal superior accuracy interestingly well spike estimation intensity dominate shrinkage estimate intensity evenly average stable whitening leads improvement discriminative subspace high variance covariance normal diagonal shrinkage target differ large eigenvalue analog intensity dimensionality intensity go target asymptotically less result shrinkage intensity remain finite range estimator also bias multiply sample simulation additional observation set constrain angle intensity rotation angle shrinkage small rotation small shrinkage target superior apply preprocesse filter class eq common computer covariance rescale random target target leave show accuracy estimation covariance pool w discriminative optimal set perform well pool multiply exclude rotation strong dominate cause degradation affect world covariance detailed brain discriminant target vs art stimulus stimulus evaluate decision binary lda approach lda stimulus binary stimulus shrinkage distinct stimulus consider independent stimulus pool comprise subject classification accuracy estimator pool estimate stimulus pool class consider superior brain interface subject imagine measure eeg subject trial subject subject band spatial class heuristic apply log approach subject subject dominate shrinkage high spectrum reduce importance noise apply whitening shrinkage intensity principal shrinkage middle show accuracy average additional shrinkage intensity dominate good receive large shrinkage intensity show ten trial widely year analytic formula covariance shrinkage become alternative pointed case usage target formula improvement shrinkage whiten step real propose explore transfer domain knowledge framework label whiten extent shrinkage quadratic decompose directly eq rearrange define behaviour calculate addition rearrange op conclude ii estimate q ii multiplying minus subtract right side rate generality asymptotic behaviour behaviour asymptotic behaviour introduce eq mean ii eq sum sum conclude proof restriction combination conclude generality assume start behaviour obtain consistency type similar eq sum proportional third restriction additional biased consistency follow asymptotic invariance low consistency unbiased variance eq look term eq expression intersection denote take expression term therefore integer disjoint subset set integer bring form consequence take account get total analogue us covariance rotation therefore lead otherwise term ij eq combination conclude behaviour show behaviour compare compare asymptotic use behaviour show ij op op sum sum sum simplify use obtain part hence side restriction step true al tu tu tu de department science tu tu computer science tu stein outperform shrink sample matrix identity shrinkage concept shrinkage target include datum set stationarity datum alternative estimator serve translate estimation shrinkage intensity yield error specific optimality applicable large go large go infinity limit apply stein stein admissible estimator always gain optimize bias trade unbiased high variance bias low analytic covariance allow square serve shrinkage wavelet density approach shrinkage code standard shrinkage line connect intensity optimal estimate two estimation handwritten digit digit person mean shrink towards two difference truth target figure moment impose tail
seq seq person histogram plus accumulation feature evaluate approximately sr low table represent form superior consider endowed vision track replace singular decomposition svd eigenvalue c diagonal partly project grant lp communication centre university school technology advance matrix riemannian manifold increase performance manifold tangent space reproduce hilbert space shape structure effort contrast offer solution preserve specifically hyperplane rkh space lead texture several histogram relational divergence matrix employ image video provide covariance definite riemannian endowed riemannian tangent riemannian solve widely riemannian induce riemannian inversion employ computationally non linear line research embed reproduce hilbert rkhs former approach enable use learn cost manifold mapping manifold consider space geometry exist load impractical scenario contribution combine recent specifically design dataset image map wherein still preserve employ create preserve manifold geometry point stein hyperplane project point appropriate study efficacy completeness generating hyperplane address training datum identification texture continue paper brief overview function visual future sized covariance topological manifold manifold riemannian map locate tangent tangent shortest geodesic curvature connect stein similarity manifold form aim still geometry aid divergence essence generate hyperplane wherein hyperplane map accord wherein preserve upon dimension despite use computer vision distance mahalanobis metric make euclidean space recently distance rkh approach make space manifold preserve manifold via follow rkh randomly generate hyperplane embed end training I distribute variate whitening form ip pca project uv q define matrix one correspond term calculate step offline complexity query point factor project hyperplane operation number class hyperplane define datum relational second represent riemannian vector employ discriminant classifier later experiment fig generate hyperplane limitation synthetic enforcing follow choose eqn around radius matrix radius establish address geodesic riemannian metric tangent whose introduce say normalise geodesic normalised point geodesic tangent preserve normalise map back present arrive prove result geodesic generate person detail experiment set comprehensive space stein feed choose hyperplane empirical sample choose similar eqn e classification several approach tangent rkhs evaluate synthetic synthetic dataset middle texture person identification version capture move camera contain variation appearance sequence create vector colour laplacian colour texture classification protocol present nine splitting scenario times recognition image pose evaluate consist illumination pose test pixel vector response wavelet locality hash riemannian hashing specifically design manifold person identification seq seq sensitive riemannian hashing seq seq texture recognition dataset v b average table report texture identification task hyperplane rkhs offer classification task present follow result still three look notable identification texture dataset synthetic suggest contribute probably represent variation completeness contribution performance face cause skewed add distribution turn affect dataset number generate hyperplane improve graph compare
however correspondingly randomness capable infer indicate complex lemma bayesian factor potentially infinite also non parametric construct make infer experiment learn apply method certain fit structure hierarchical challenge datum thus determination impractical hypothesis structure bayesian assume construct layer cope challenge potentially factor stroke deep graphical permit structure hide influence factor layer distribution factor develop convolutional connect processing set construct build factor prior achieve data ibp handle case application various network unlabele classification bayesian advance researcher challenge still lie construct nonparametric model infinite hierarchical layer factor proposition infinity fashion hasting enable layer make recursively simulate infinite state wireless security mainly lie number greedy algorithm objective discover hidden factor nonparametric result draw conclusion construct component finite extend hide layer infinity generative use model tn dependency successive weight instance cause influence generation component otherwise hide matrix illustrate remain vector weight generate I indicate element product independently construct variable infinity column variance inverse gamma r impose effect layer much influence receive high conditionally verify delta generation variable layer layer datum level feature extract human face second feature possess cause robust binary bring close reasonably weight express recursive infinite number layer matrix ibp distribution q hide layer select select restaurant customer array try customer firstly th customer customer generative intractable inspire initialize express metropolis hasting approximate infer infer hide changed second infer layer iteratively inference converge infer efficient metropolis inference classic problem generate first candidate density state state evolve stay explain specify generalize infinite derive setting connect matrix model propose express completed iteratively check number column propose link sample probability hide approximated generating multiply return factor zero compute ibp proposal link accepted algorithm use infer matrix turn k integrate column independently compute utilize distribution
u k l lt st result another major advantage derive compute linearly substantially exact rate convergence entry resolve fw fw call frank wolfe thresholding proximal compare synthetic clearly efficient highlight fw additional iteration k frank wolfe produce retain rank execute e easily frank wolfe implement line solve crucial convergence depend maintain non scheme convergence weight max initialization u optimizer k u g step easily recognize fw algorithm establish primal independent n e therefore convergence result still upon duality suitable serve terminate algorithm consecutive iteration fw real arise consider separation surveillance video removal application robust slight setting choose fw fista partial svd fw noisy improve recover whenever fw conduct processor core ghz gb ram bit surveillance normally fraction treat stack wolfe iteration updating component computer vision relate great fw acknowledge class fellowship university office award grant dms first useful proof type recursive constant base claim diameter set fourth v convex rearrange lemma notational contrary repeatedly one due assumption fourth ba x contradiction l l f f z g f f l theorem calculation u l k verify therefore combine l lt sg l k r th l argument verify recall x g l r l x x l minimize l hold feasible thus apply lemma define argument use problem data fidelity sl proximal singular x x decomposition explicitly initialization regard fista optimal scheme fista comparable summarize fista h singular utilize burden svd package g specify singular determine th adjust dynamically compressive corrupt observation fundamental rich application vision certain solve polynomial compressive provable iteration large provable combine classical frank wolfe iteration mainly frank update low svd exploit step scalability promise visual matrix dense noise collect parameter nonzero tractable ij tractable sometimes compressive pursuit equivalently since linear subspace span onto rewrite possibly depend specification mainly easily extended operator consider reduce onto j linear subspace work theoretical mild produce accurate even measurement researcher solve include alignment verification variable outli robust big data application involve large provable work develop first closed form nuclear involve singular hence dominant computing substantially scalable interior solver limit practical applicability involve several thousand dimension full serious algorithm solve frank wolfe norm turn precisely scale scalable practice straightforward frank wolfe frank wolfe block use modify frank wolfe solve penalize incorporate proximal numerical frank wolfe fw general differentiable assume x frank wolfe proceed linearization simple step algorithm detail fw method recently yield encouraging matrix singular value substantially exploit comprehensive survey development fw update five decade replace update sophisticated q analyze produce iterate objective frank wolfe frank relationship special crucial handle use lipschitz constant diameter perhaps practice useful precise iterate duality refinement exist match bad case provide stop burden subproblem result inexact calculation ball operation solution describe part involve serve method easy computational cost essentially nuclear optimal value one matrix leave singular lead dual one modify frank wolfe projection onto easily achieve effectively frank wolfe step proximal mapping follow close extension operator scalable algorithm compressive q application frank although converge typical example make sparse problem proximal essentially combine frank wolfe frank wolfe map l f feasible compact feasible gradient compact frank wolfe generate expression linearize v subproblem solve exploit take right summarize result h q v v major advantage derive simplicity close essentially linear input rank disadvantage update I k entry disadvantage nonzero even fraction entry foreground background separation red practical convergence simulate entry matrix show slow k matlab plot fw p efficient recover drawback frank wolfe incorporate gradient projection wolfe additional term variable frank wolfe produce iterate retain sparse call fw frank wolfe projection fw h k l k k analyze frank wolfe regard set point imply iterate objective produce frank wolfe require invoke frank wolfe lemma diameter fw original fw method surrogate gap define q develop compressive
information sentence sentence lose bag model vision understanding pattern apply document learn identify sentence movie level sentence document sentence level convnet transform embed entire sentence convnet single train document embedding entire level sentence tie convnet abstraction operate document transformation convnet transformation architecture force sentence representation appropriate help show carefully type intermediate version convolutional neural similar network vision cascade adapt text model detailed figure overview right section convolutional model embed correspond process level embed build word vocabulary word embed vocabulary word embed sentence produce vector document sentence embedding sentence matrix convolutional filter bank f fw number map embedding convolution operation map embed along row sentence correspond document sentence map generate value feature along output map stack representation feed input feature word one unlike treat dimension generate representation representation generates illustrate substantially approach typical map image spatial read embed dimension leave model map map generate map embedding level make deep document matrix convolutional handle arbitrary problematic want max pooling embed keep discard fix size length max also illustrate fix length next convolution however pool desirable effective fraction pooling help long propagate learn convnet vision demonstrate learn single convnet sentence precede argue convnet substantially convnet scale mobile device ask performance attain tweet classification comparable set contain weak automatically presence tweet test sentiment assign convolution dimensional use operation sum pool close significantly well model error max bt bi paragraph vector error twitter dataset movie review second wang fourth le convnet review convnet sentiment classifier building solve show sentence classifier focus movie sentiment datum originally sentiment movie review divide review binary train label set word map convnet process convnet embedding model feature width max sentence dimension follow pooling width achieve knowledge encouraging convnet achieve good model achieve expensive since must optimisation paragraph embed feed forward train convnet salient document compact summary review activation addition insight use extract automatic summary text interpretable work obtain pass network fact quite carry approach non layer document adopt modify objective document invert feed induce greatest taylor expansion pseudo entry word document perform pass behind magnitude magnitude change affect explain technique clear perform taylor partial level sentence summary rank sentence produce train I movie review sentiment use classify summary tf weight I bayes I summary technique even keep train review drop create sentence review summary lose choose random sentence heuristic summary explain fact review relevant sentiment review summary short sentence summarie opinion movie review ignore proportion margin pick pick pick last thing complicate actually movie nice connect dot million middle truly massive massive character massive graphic graphic good graphic say go say might well people get topic pure sometimes look home camera great make tv movie sequel get background justification mean tag line influence break looks suppose look look help lead act movie inconsistent movie scene unlikely quite finally actor real throughout record never check cm
rough well hmc cc cccc c hmc function fix substitute rate equation point equation transform transition mix hmc length state case outperform hmc hyperparameter factor distribution ill final condition quadratic rough maintain reasonable require sampler majority momentum hmc eliminate hmc search condition demonstrate technique hmc hyperparameter computational figure matlab table running variation standard hmc complementary could present hmc hilbert hmc hamiltonian hamiltonian importance split behavior explore topology state transition though benefit replace identity space auxiliary momentum momentum swap operator allow situation momentum instead momentum randomization rejection though slightly random walk exploration reversible discrete trajectory map could mapping indicate momentum variable direction indicate occur exploration choose sensible acknowledgment thank member center berkeley stanford course like thank anonymous careful reading text grant support mm support nsf grant upon research laboratory office contract nf present hamiltonian rejection trajectory state reach transition detail balance walk rejection great release source code package probabilistic model increase protein structure neuron sampling describe typically bottleneck work require evaluating improves fundamentally sample generate hasting acceptance criterion suffer forward reverse transition occur balance detailed balance go go thus explore distance long draw proposal random walk distance sampler balance mix rapidly distribution space hamiltonian extend include auxiliary variable hamiltonian contour extend hmc able long update hmc detailed balance accept reject long step attempt satisfy detailed turn sampler transition discrete hamiltonian generate candidate hamiltonian dynamic back slice violate uniform candidate acceptance window rather state window boltzmann method balance discrete transition use balance satisfying restrict sampling greatly mix sampler substantially improve concept relate goal assume energy chain carlo commonly draw probability sample since order must space condition act result unchanged markov detailed guarantee every probability identical substitution side appeal distribution proposal metropolis acceptance rejection detailed metropolis markov forward reverse transition primary advance demonstrate balance hamiltonian carlo extend auxiliary momentum hamiltonian along contour expand state momentum physical system momentum mass drawing hamiltonian dynamic imagine trajectory unchanged flip side st discrete hamiltonian dynamic operator integration like exactly reversible preserve volume sign reverse trajectory momentum total energy discretization cause movement figure leave momentum randomization operator velocity operator randomization operator equation momentum randomization operator cause movement discrete transition occur state state represented way state hmc view additionally discrete short hmc typically implement step sample item composition markov move accept metropolis transition probability metropolis rejection discard long flip momentum back prevent trajectory reject already move hmc beyond hmc explore random walk momentum case momentum small improvement probably adjust fix fraction momentum ill condition ill condition surface demonstrate hmc balance still fix momentum hmc greatly walk prevent typically discard rejection hamiltonian correspond rejection section intuition update discover standard modification hmc
depend party party influential direction collaborative dl training generally speak rely quite dl find aware ever representation dl grouped make classification dictionary first classification specific dictionary representation include lc add regression consistency contain call discrimination fidelity term sub besides discriminative utilize fisher discriminative basis dictionary separate dictionary common meanwhile incoherence fidelity dictionaries discriminative despite learn discriminative approach enforce coefficient either feature denote seek reconstruction sparsity reconstruction formulate training class representation residual ir class truly contribute face recognition replace equation benefit close tx computed require optimization regularizer constraint induce competition among coefficient compute ir reconstruction dictionary seek amount reconstruction learn reconstruction discriminative fidelity detail dictionary iteratively optimize coefficient difficulty simultaneously explore dl consume term keep light big exist format dl q sub sample column term ensure reconstruct easy discriminative overall dl simple learn optimization dl iteratively optimize acceptable thank frobenius optimize denote sub respectively rewrite equation still objective dictionary therefore optimize without equation straightforward solution consists however converge soon worked learn sample cover experiment sample follow reconstruct test form td ir eq replace td conduct five visual nine public benchmark cover scenario appearance environment texture texture leaf categorization categorization color texture camera identification vary sample adopt identification investigate influence test totally diverse scope collaborative general recognition classification various literature generate version change feature factor split regularization concerned fair person good choice parameter important recognition acquire item different day instance take protocol third instance dimensional feature whole feature experiment three build aa camera view collect surveillance consist sequence centre image person result unlike dataset detector manual annotation localization aa big sample sample camera person e camera camera version name etc data aa broad pose subject subject either shot identification treat use treat desire correct top cumulative effectiveness exactly texture feature adopt choose dataset aa dl ht fdr fdr fdr ex leaf aa aa aa collaborative none completely outperform answer comparison superiority concern replace therefore indicate perform well large non predict dataset favor know necessary dataset therefore representative class important property influence intuitive may feature enable sample stay relatively thus easy collaborative discrimination minimum point wise randomly distance point treat distance sample belong dissimilarity rely fdr fdr top accuracy fdr however effectiveness fdr coincide enhance involve likely prove negative fdr four special mark variation fdr definite r low sum show learn influential complex dl lc dl dl dataset also outperform person identification boost dl identification probably room dl moreover dl dl dataset generally learn look dl valuable explore collaborative ex aa aa pt ht lc ex leaf convergence converge several consider dl run concerned fast testing comparable fast dictionary promise score dl superiority sparse collaborative art dictionary comprehensive dictionary acknowledgment support anti anti safe integrated education g etc etc et et mm medium intelligence school university mail mm media ac collaborative sparse simple effective solution get necessity enforce come primary show change computationally similar well unclear sparse know study issue classification select strategy feature dimensionality feature superiority understand collaborative motivate activity sparse collaborative call quite error reconstruct limit sparsity tend force large coefficient sample belong discriminative modeling expensive infeasible combinatorial approximate though consume due optimization ensure carefully per lack argue success lie sample norm representation well understanding treat regularization notation respectively regularize regarded birth attractive superiority representation report superiority people lack reliable problem contribute propose analytic feature dataset predict extensive
propose credible global function densely et orthonormal function piecewise present spline al credible band construct bootstrap strategy lee al yield simultaneous connection discovery al fdr generalize growth multiple view induce sample lee estimate across represent smooth estimate noise construct pc fourier basis response curve software require general mind curve ignore technique empirically wise regression second functional coefficient effectively wu iid coefficient spline study response curve grid representation spline penalty square ol compare covariance estimate estimation technique response within covariance step fit represent b curve work pc decomposition spline spline inefficient within ignore lin aforementione experimental design present model multiple pressure spatially correlate unit spatial grid properly error wise band specify function curve deviation assume strategy spatially car zhang car alternatively correlation function add response correspond iid induce design marginal component capturing example compound symmetry within cluster share longitudinal subject slope serial observe index distinction mix I vary functional correlation st n functional specify separate level product al issue way capture effect spline induce chapter spline curve curve curve induce curve capture induce manner combine response involve functional coefficient effect classic random effect like induced inference effect reference integrate variability provide smoothed introduce modeling level spline base nest curve group parameterization random curve sum iid spline level capture coefficient cycle restrict fully spline basis component smoothing introduce fix smoothing spline iid include curve like model determine random intercept slope spline part scalar white fit kalman filter approach pointwise confidence functional induce periodic et group within group random subject diagonal covariance respectively wavelet form prior account multiple function intend functional sample common wavelet project wavelet fit version assumption across wavelet work allow wavelet variance covariance capture degree greatly calculation allow fully quantitative et genome analysis lee adaptively adaptively regularize coefficient behave shrinkage curve deviation covariance tensor covariance spatially zhang even flexibility capture structure model functional index lee add level matrix consist spline basis predictor nonparametric functional additive lee al effect function variability credible band et probability compute fix control fdr miss et introduce use basis subsequent al zhang et lee method spline basis transform modeling function covariance level basis specific joint al special scale tail likelihood functional coefficient inference insensitive analyze spatially correlate al general error use basis point inference effect iid variance along allow apply regularization function functional growth nest function covariate functional covariate fix function cubic spline grid subtract pointwise pc project span overall pc perform pc wang base residual g ar coefficient random residual variance basis penalty update approach accommodate longitudinal index include slope random function focus simply subtract smoothed mean covariance general nest b subject level random effect level spline scalar curve deviation functional response grid within level functional model subject subject within introduce truncate hierarchy hierarchical polynomial plus al alternative basis polynomial group estimate use moment yield assumption update multi involve sequential residual variance yield also spline represent penalty curve score unlike non correlation project functional zhang functional car separable introduce project choose basis car separate car car spatial vary yet strength across correlation determine basis embed functional involve extend additive functional new include nonparametric mixed model effect parametric form random functional effect parametric model utilize model describe response nonparametric introduce elegant notation specify array function include smooth nonparametric predictor effect plus smooth predictor penalty consist smooth primarily penalize iid multiple level completely determine within covariance scalar smoothing entire limit accommodate scale e grid lee show bayesian accommodate coefficient nonparametric component smoothing vary vary coefficient allow joint wise coefficient functional g derivative series domain model project domain involve smoothing partition locally process tensor basis specification domain log wavelet regularization keep wavelet overall al effect domain effect level effect spectra subject spline common estimate band series wavelet et al content functional among function fitting functional discriminant curve functional within spline penalty smoothed group et al score function error discriminant li yu introduce discriminant analysis predictive integrate uncertainty et adjust enable robust version wavelet transform model wavelet inverse wishart allow covariance structure assume wavelet develop majority classic literature method choice function approach model curve deviation response curve suitable choice basis residual curve deviation capture correlation suggest across curve curve yield estimate independence spline accommodate curve curve choice penalty simplify smoothing replicate rich curve deviation accommodate decomposition covariance parsimonious somewhat assume deviation important affect functional perform discriminant grow correlate functional spatially correlated functional effect induce function simple modeling flexible enough grid mass imaging complex curve yet principal reliably response window os properly et response mixed lee version although fit basis transform inverse transform broken post module linear et truncation basis automatically produce include wise credible band point wise specify effect contrast linear apply functional al software scalar package package along fan zhang implement generalize additive fit additive linear mix r date functional function predictor denoise grid interpretable term account additionally curve random effect may take account basis develop approach respectively case contain discuss truncation bivariate penalty decomposition measurement pc functional coefficient amount pc score wu account regression coefficient chen regression observation estimator et et function expansion keep many pc p spline include scalar effect densely issue context two depend vary wu describe iid smoothed estimation spline locally fan zhang regression al measurement error function covariance functional effect represent spline basis penalization selection wu spline truncation inherent selection b spline within al longitudinal b penalty spline score constrain triangular construct iid spline function include truncation decomposition iid coefficient fourier spline pc integration domain model take correlation inference incorporate model structure develop response mix functional predictor additive pc like sense generalize approach predictor grid extend bayesian predictor effect function predictor function fit past decade much continuous domain method grid commonly encounter longitudinal setting broader yield quantitative euclidean surface function numerous feature paper highlight type naive analyse far dealing value year potentially manifold case inherent high functional application grid genomic et lee imaging et local thousand million functional regression software quadratic functional mind regularization use analysis open basis setting size eigenfunction preliminary asymptotic explore pc basis flexibility covariance helpful graphical another statistically principled approach basis benefit understanding adapt incorporation prior lead regularization various way article accurate careful importance functional difference estimation determine setting functional derive benefit inferential important relevant accounting correlation seem address study type functional encounter smooth function difficulty determine position within curve assess publish primarily compute pointwise band band test still need area need various rich work grant definition additive car autoregressive discrete transform additive model estimation fdr false functional linear regression mixed square p functional lasso operator linear p monte multi analysis multidimensional signal ols ordinary square p partial square restrict ratio smoothly space penalize regression search university md tx observation observe discrete development field accelerate past become fast area within function regularization receive attention follow overview response regression scalar development manner historical primary modeling modeling structure end brief area generalize spline wavelet decade datum increasingly structure rapid datum ideal continuous consist function population could fine across domain manifold euclidean largely longitudinal typically grid area produce spatial imaging domain genomic location dimensional single rather simplicity enable handle part seminal book behind describe display observation involve make population involve strength within function interpretability area attention article development role approach overview follow description functional development scalar discussion development manner historical cluster researcher model methodology theory computation employ section contain list publicly fit discuss focus highlight inferential approach discuss selection either modeling independently function advance nonlinear high domain highlight advance functional datum highlight discussion relatively moderately sized grid observation curve third include euclidean partial representative set functional article fractional fa corpus cca ms plot black ms control intensities part spectra control cancer spherical pressure moderate grid fa profiles diffusion institute scan corpus fa cca ms plus serial cognitive relate see b relatively simple moderately sized grid position three paper literature et al al ms fa one ms status fa patients fa cca ms testing whether differ functional mixed account status cca use alternative could association fa position could surface effect fa cca position marker cancer perform university md cancer center describe et mass peak protein molecular unit whose intensity abundance spectrum illustrate functional regression development flexibility efficiency methodology development functional al et frequently analyze perform quantification peak spectra peak algorithm perfect model entire spectrum differentially protein base patient assess cancer fitting spectra cancer partial manifold study able induce amount pressure continuously higher old close right yield principal location fine spherical manifold plot investigate whether coefficient change functional longitudinal measurement pressure linearity functional mixed capture age index lee within euclidean inherently inference either involve commonly replicate predictor information across discover concept accounting correlation design together size example mixed model model unit across make involve strength across regularity observe behind single krige analysis nearby involve smoothing imply observation internal especially similar distant regularization closely tie modeling apply capture functional believe exploit interpretable estimator time calculation potentially stable separately unified common involve coefficient individual separate grid convenient inferior unified jointly lead information across place unified evidence determine define induce correlation among loading magnitude establish strength allow inference dimensional commonly spline wavelet suited characteristic spline suit modeling usual fitting observational fourier function stationary periodic wavelet resolution basis decompose dual frequency make spike possess basis reconstruct suit sample regular grid principal component pc basis empirically eigen decomposition suit decay capture distant consideration development pc kernel weight suitable smooth spatially function lin et equivalence certain accomplish apply penalization basis truncation involve commonly pca keep pc fourier basis apply polynomial polynomial certain spline limit pre specify location truncation penalty involve coefficient smoothing lead ridge convenient penalty involve cubic spline basis knot incorporate penalty penalize spline penalty use knot penalty propose l spline penalization assume identically iid truncate sparsity coefficient induce prior involve stochastic variable al al penalty penalization wavelet regularization remove preserve dominant accommodate vary grid predictor natural cubic common knot zhang al periodic spline basis penalty robust insensitive introduce reproduce regularize penalty smoothing spline explicitly fitting spaced wavelet basis wavelet pyramid discrete wavelet allow grid mention wavelet suit spatially heterogeneous function spike wang al wang response bayesian probit denoise use wavelet adaptively al lag wavelet regularization scalar correlate wavelet basis transform truncation extend fit fourier cox develop group potential functional predictor sample predictor common lee basis outcome probit orthogonal introduce utilize general adaptive scad fan perform selection multiple basis response al introduce encourage accomplish via derivative piecewise constant mention introduce bayesian use field surface cluster pc spline regularization spline pc cubic spline perform decomposition spline transform spline square combine outcome transform spline pc spline replace pls basis response predictor radial spline ensure radial symmetry inferential bootstrap simultaneous confidence alternative pca perform b spline smooth result present walk stochastically al variational pc series penalty truncate spline handle functional score predictor functional predictor truncation keep number make primary handling dimension keep pc series spline present vs select among functional predictor et datum measurement scalar iid among replicate score et extension di al longitudinal separate score pc previously method li like full functional flexibility li cross product decomposition estimate empirical basis generalize al extend quadratic across pc extension estimation pursuit regression family model smooth function use cubic spline regularize include predictor incorporate li predictor complete orthogonal basis use single extend predictor spline use spline penalty single penalization perform across predictor additive al nonlinear additive pc score truncate pc general perform truncation pc instead penalization dimension et generalize framework function surface parameterize spline iteratively introduce grid space pc score within eigenfunction eigenvalue mean curve product walk deal
set conservative choosing extension exist design x show gray indicate reduction maximize away triangle open triangle iteration determine actual segment ray segment five black triangle optima green close element panel ray select candidate ray opposite direction since ten box full allow segment start spaced show xx nx ray search start design near increase successive search create could choice adjust fast fidelity ray design undesirable many context parameter spanning segment derivative leverage speed library routine huge set initialization towards nn mode post optimize exhaustive discrete provide onto minimize explicitly ray start location choose unless agree precisely return prevent location besides modification round combination green ray exhaustive demonstrate sample development utilize core parallelization eight core gpu exhaustive package however new leverage feature implementation describe design local probability design calculate obtain benefit nn limited nn candidate primarily report early limit ray search slightly unless note ray adjust synthetic paper x cm lrr stage rmse sd ray ray ray ray cm rr rr summarize involve location space since space slight computation experiment defer ray design accurate nn option sub poor add space reduce variability great calculate sized magnitude mean calculation stage would require exhaustive feasible via lead overall predictor low nominal poor rmse achieve nominal coverage achieve expense cover sensible relative time utilize core ghz intel core gpu devices exhaustive subroutine experiment one right reveal search competitive exhaustive subroutine gpu big gpu exhaustive ray competitive ray search nearby experiment lift involved dimensional capability library addition local number rr cpu gpu ray exhaustive search fast exhaustive search cycle unable fidelity experiment dimensional detail therein one summarize prediction design design set hypercube increase design sample increase ray search rr rr exhaustive c cpu second mse second second location separate report gpu cpu cores cpu shorthand indicate cpu core mse summarize compare column time slow core amount large run nine million year old look base give nearly identical accurate one relative small searching set intersect candidate reduce find good column show intel bridge run comparable exhaustive parallel exhaustive slow prefer small huge gps show big gain nearly allow hour ultimately accommodate experiment approximation fidelity rise gps rr rr subset sep pre scale c core second mse second mse second second include build subset separable final ray pre globally grateful pointing result global predictor subset datum calculation obtain design estimate separable function organize row surprising fit compare ray alternative uniformly input isotropic show separable along row final separable substantial input subset limit thus globally scale trend estimate exhaustive search hybrid appear partly round partly slowly setup carefully illustrate fidelity approximation size increase versus several function range follow product cause numerical instability benchmark excellent resource benchmark problem involve computer ht six pair top time bottom repeat training via two motivated comparison subsample separable base square common early variability sub design predict good sub version separable global clearly isotropic isotropic good figure average besides scale change vary report repetition number legend indicate gain normalize rmse even grow gp becoming generate paper focus local approach build local design nearby prediction study surface predictive design discover exploit suggest interest design result ray design comparable correlation isotropic locally global may still parameterization discussion highly global scale scale global substantial discussion rich family certainly design material local figure choice qualitatively design mix whether measurement context author argue lead explore example estimate relationship spatially surface appear unstable smoothness visually predictor finally search design design low search sample setup value computational increase improve may moderately problem disadvantage especially distribute search uncertain reduction deterministic runtime operate random balancing challenge try resource ray perhaps might acknowledgment complete resource valuable anonymous improvement neighborhood school particularly massive parallelization mean two previously ease burden study observe exhaustive subroutine building design rather search location alternative work ray yield problem krige regression neighbor learn big data surface reasonably smooth relationship regression tend gps accurately appropriate input pair implementation grow collection modern thousand sized computer decompose due perform hundred much decomposition require limit datum computer canonical model cope modern simulation approximate alternative lead fast capability magnitude prefer example capture match cluster dense led capability hybrid resource design hour literature focus induce first subroutine vast grid identify design identify site regular pattern greedy motivating scope site gpu computing acknowledge inefficient context global software package remainder outline follow regression emphasis explore design brief gp leverage localization literature leverage big method motivate stein pair gaussian noisy salient typical gp comprise scalar response n nf covariate special especially move correlation definite comprise choice determine correlation throughout thereby stationarity isotropic k rapidly experiment noisy robust stationarity deterministic experiment add numerical appropriate methodology herein gp popular reference likelihood analytic fast maximize student vector nx yx v shape element attractive high represent sensible new eqs require dense matrix limit reasonable hour sized say fast design predictive sensible objective optimal minimize complex proceeding sequentially build globally minimize sized depend implicitly equivalent maximize design advantage avoid huge argue sensible proportional dominating apply sequentially approximate huge search near rapidly decay substantially location close away model prefer choice remarkably quadratic correlation potential local suggest maximize maximize inverse compete minimize distance upon tradeoff proximity observation insight nature global optimal design recognize integral thought result aspect design limit define mesh value possible try minimize need predictor site somewhat globally establish criterion prefer
marginal consider posterior prior derivation present posterior b wu b cumulative ba density derivation equal accordance definition end appendix marginal g g wu box york normal utilize letter k short tail interval communication b w bayes estimate possibly sparse statistic w bayes wavelet threshold selection international review interval journal planning journal regression selection regression american review page cm corollary department mathematic la normally distribute specify linearly suppose uncertain prior frequentist confidence utilize compare interval short credible dirac parameter bayesian interval way nonetheless interval keyword interval credible information regression spike prior author mail address nn parameter suppose interest define specify experience opinion scientific background suggest example scenario include factorial replicate uncertain high clarity comparison interval var var concern uncertain utilize two way frequentist credible interval utilize uncertain informative infimum coverage assess length expect utilize uncertain information follow scale substantially expect c confidence frequentist confidence utilize brevity credible uncertain apart uncertain prior precisely credible informative lead credible deal improper infinite rectangular dirac delta spike dirac delta function function prior class end genomic aim outcome consider preliminary estimator estimation make prior density marginal behave proper specifie belief prior tail credible frequentist b information credible may consist interval posterior therefore focus tail credible interval square p frequentist interval scale length eq offset half scale offset extent uncertain prior section credible interval offset figure scale offset half tailed credible graph scale offset half shortest dash word prior tail shortest credible variant informative frequentist estimator similarity offset illustrate nonetheless substantial tailed credible interval conclusion credible uncertain informative I tm bx dd define interval word scale scale offset statistic frequentist testing hypothesis confidence happen strongly uncertain contradiction irrespective tail shortest credible approximate interval tail short credible depend frequentist numerical factorial experiment factorial replicate effect factor uncertain take let factor code factor experiment identically square square figure scale tail credible factorial bayesian credible context factorial density h short factorial express uncertain improper improper plausibility suggest density interestingly scale offset function follow g tail credible identical interval posterior tail credible scale offset scale length tail interval factorial tail credible context factorial credible factorial begin describe utilize sense introduction scale even function minimize look factorial function cubic evenly space subject follow confidence probability parameter square figure utilize uncertain prior introduction gain coincide standard contradict scaled confidence prior consider interpretation offset scale half offset scale half mark particularly scale bs knot spline place frequentist interval vice versa credible examine view frequentist interval examine bayesian posterior coverage likely favor credible interval frequentist coverage favor credible frequentist interval credible frequentist view lead instance difference frequentist statistical var
improve iteration increase historical exclude report minima algorithm vector bi vector explanation produce minimization therefore team decide use method implement run every network run widely year great win division seven straight game lead team second player vary head history school four four separate decade head north win matter lead least time know concerned decide five high year overall rank overall time method instance year top appear top list ranking rank input error record game incorrect entirely game cause incorrect source decide incorporate game regular team top team decrease modification ran maintain incorporate great negative run decrease minor originally behind place rank see model accurately remove add positively competition limited change final indicate separate team accurately player flexible relationship team easily compare effectiveness margin similar computer calculate could eigenvector centrality percentage graph von sometimes yield cost year array minima several array fact improve significantly time high accuracy result ideal analyze nearly determine unbiased rank college constructing analyze comprehensive team centrality margin pattern measure player create determine occur multiply team vector use map combine individual select would like thank mathematical modeling like school science mathematic work thank put competition north mathematic college question player ask college create apply across college offer college need metric reflect accurately college lose describe year property network team team team probability network occur utilize margin determine overall history simplify ultimately team margin assume team player throughout year compare reality change player throughout make game win good team win team intuitive allow eigenvector centrality total team team player player player optimize difference conference team worst likely run bring team match advantageous winning match determine make assumption play particular unable single portion source process aim merge team specific match name ex st manually allow merge college note college exist national current body nearly college establish college college college datum collect final score division college datum game combine match record data college college game name team generate record college interest greatly decade medium game range present accurately create analyze loss team regard make represent play year node college team team draw point information direction margin game associate edge construct winning list associate analyze nearly use visualize team base previous centrality connection centrality graph connection number calculate centrality centrality degree closeness eigenvector centrality centrality simple centrality measure connect node directional edge loss team use metric team however prominent play form eq power adjacency walk power centrality walk length walk intuitive eigenvector centrality utilize library calculate centrality game eigenvector centrality account good team follow team team metric centrality account rank centrality quite rank centrality ap division eigenvector centrality college ten eigenvector centrality north north st st six determined eigenvector centrality also ten usa model create ranking team rank clear easy centrality reasonably rank simple approach various one college calculate ranking wide historical centrality game division large eigenvector centrality low eigenvector value node give team team player player team team think multipli player team relationship factor life work regard separate influence team fundamentally different play arbitrary return win outcome draw win outcome effect enough player evenly match high outcome game curve look outcome call player strategy break otherwise match curve slightly team chance win even likely say express
perhaps typical variant em like unstable assume sample represent whole np computationally besides sensitive improve employ outlier theoretically optima unfortunately solver infeasible hour robust sample worth improvement exist accelerate robust subset hand boost norm robust error dominate generally large speedup acceleration augment lagrange multipli alm via demonstrate encouraging show fig accelerate reduce complexity obtain solver letter vector row problem set goal effectively entire solely could greatly computational motivation come coefficient informative search optimal author solve inefficient algorithm square presence complex accordingly thus select informative accumulate enhance equivalently rewrite format objective convex minimum system objective reduce entry nn simple optimum infeasible demand complexity day selection system solve factorization forward substitution amount total amount cost infeasible computational hour outlier issue result speedup dramatically save computational cost minute boost feature formulate discrepancy norm empirical element dominate eq balancing matrix informative representative obtaining sort decrease index effective highly acceleration alm beneficial lagrangian alm subproblems iteratively subproblem eventually minimum convenient penalty equality require cause bad introduce multipli require alm consist follow highly solver thresholding update lagrangian multiplier predefine remove irrelevant unconstrained subproblem scalar recently generalize shrinkage implement able achieve problem obtain q solve extremely acceleration solver remove zero avoid failure system cholesky factorization mean derivation save linear efficient e q identity side equal sign substitute simplify updating multiplication solver highly method initialize update rule overall summarize solver highly generally e accelerate speedup solver identity follow gap number view datum accelerate solver solver theoretically reduce maintain speedup solver optimum extensive empirical acceleration experimental setting experiment conduct server core intel ghz mb cache ram brief description dataset summarize complexity handle handle table ten candidate sample add method part vary illustrate code verify superiority method speed three sub show benchmark dataset mnist dramatically time consuming grow theoretical analyze compare surprisingly highly alm derivation computational significantly accelerate speed four summarize table show display fast fast acceleration dramatically verify acceleration fast mean take selection day display large algorithm I e surprisingly acceleration robust highly encouraging column table experiment ten benchmark representative accuracy table draw outperform second well compare well last loss suit increase knn svm shall increase consistent common view boost prediction select subset vary vary quality propose accelerate enhance solver via technique alm derivation reduce computational run
complexity heuristic manual gps calculation effectively verification latter automatically extract feed decision paper follow face verification constraint complex exploit discriminative information take domain computationally version gps gps anchor report recognition accurate human environment illumination human illumination conclusion face dataset control change date showing could performance face exhibit variation pose gender work verification apply face verification good vector number li analysis complex besides approach source transfer target transfer bayesian complex moreover transfer domain restrict wide application datum recently pose convolutional deep face utilize face affine transformation nine neural network although method parameter core gps good gps face verification gps vision importance task asymmetric focus task gp cluster discriminative improve net gps architecture consider gaussian latent mainly follow notable previously non method distribution heuristic manual gps hyper learn avoid gps overfitte excellent read classification observation row column impose function neither analytically laplace approximate acquire method unseen membership observation value small dense area variance good support separate explained point region vice versa dynamic equilibrium find equilibrium employ equilibrium point obviously completely determine row correspond datum determine latent position uninformative prior gaussian introduce obtain need automatically discriminative covariance take improve face verification include place position spherical discriminative prior encourage position verification mainly analysis kernel space formulation replace one direction onto formally negative feature positive negative maximize within n however rather position simplify calculation equivalent form maximize equation position write normalization scale prior gps freedom estimate conventional perspective share information task way distribution extend distribution entropy source describe detail source domain dataset source target write optimize accord learn constraint q model amount minimize multi learning expand equation item ip scale conjugate technique covariance derivation depend ignore constant q easily item inference problem store computationally prohibitive anchor speed put simply cloud kernel identity transform compute efficient predictive center identity describe two verification affine transformation corner divide pixel pixel patch descriptor patch image scale patch extract scale descriptor extract patch regard covariance pair person un image representation predict person cumulative solve call model regard extract feature image person regard enhance hyper function learn cluster finally denote variance cluster th number point refer equation variance regard codebook un face image first pair center also w final face new see differ image encode label call section conduct verification introduce source domain source domain contain illumination people age range contain image subject person image illumination condition contain approximately collect benchmark verification face image figure pose gender collect web dataset know benchmark verification variation describe verification conduct follow procedure source web life domain validation protocol test image mutually exclusive overlap two partition feed sd sd automatically learn reflect tradeoff discriminate ability select domain pair pair consideration method space adopt anchor take determine anchor select validation fix tune vary train trade practice anchor conduct five gps fair source domain regard significantly gps superiority obvious source domain since regard paper choose popular lr adaboost table demonstrate performance learn improvement improvement c svm lr adaboost number rp tree gmm also regard codebook compare three rp gmm generate cluster outperform effectiveness multi improve varie appeal bc combine verification bc decision state publish benchmark achieve level incorrectly obviously human emphasize center patch dense like utilize extract easy validity target split exclusive part training protocol mutually subset subset match pair even implicit among computer face verification currently exist computer base face verification however scientific contrast already face verification human moderate really difficult human specific point human verification contrast face verification familiar face relatively illumination pose face drop substantially change besides human face combination drop original test compare human verification human ask match people verification variation improve verification develop verification become useful subtle information slight human significant reality ahead human robustness familiar face develop verification task latent verification computationally multi constraint approximation propose different verification extensive experiment validate efficacy face
true marked factor simulation compare detect scenario suggest true major situation wish decide epidemic transmission event suppose let observe removal sir epidemic previous example add unobserved one total unobserved infection removal infection come assign density describe allow epidemic progress si ir order time exponential rate truncate necessary missing assign uniform density drawback markov chain never leave z st z dt describe update infection form detection daily infection illustrate simplify miss geometric parameter run distribution poisson mcmc appear evidence suggest time well epidemic poisson comparison certainly insufficient evaluate bayes although epidemic clearly setting drawback issue second miss likely insight expect missing distribution ex thank work uk use ij ik kk sufficient equation form column component solve say fm ix jx yield low attain strict simply assume yield follow ij show secondly specifically suppose var ix recall column column determinant p satisfy straightforward kx kx kx kx equation yield school partially avoid reversible key compete component applicability markov observe event time assess whether poisson poisson fit observation alternatively disease know sir remove transmission generate correspond special generic wish point model example bayes factor quantify factor suffer two practical drawback particular difficulty briefly method assessment difficulty neither entirely setting stochastic process involve epidemic criterion nine candidate involve typically involve simple must evaluate numerically reversible jump chain carlo method precise indicator consideration state factor give expression implement jump parameter propose go compete mixture probability proceed define markov parameter space approach parameter choose instead mixture upon difficult practice introduce prior problem computational method model establish typical consideration identically mixture distribution contrast consider partially structure contain framework detail computational contain conclude discussion ease exposition abuse density typical observe model vector common define mixture partial consequence intractable adopt augmentation comprise datum jj iy augment nan share element datum conversely necessary tractable jj iy miss density depend application probability directly summarie mutually priori eq eq define e jx yield divide numerator fraction rearrange obtain remain find matrix remove form suppose either require bayes express summary posterior somewhat solve find bayes factor yield evaluation three remark solely tool bayes assign straightforward describe third mild constraint mcmc repeat exercise yield estimate allocation suitable density however illustrate full implication illustrate marginal density proportional density explore conditioning difference set framework indicator target specify conversely target adopt become existence important potentially although set miss model always always bayes theory classical alternative example illustrate applicability know model assign nest several order method estimate factor describe compete prior distribution paper cite assign mcmc run correct mix serial population certainly serial mcmc population mixture illustrate agreement event birth birth far likelihood reference measure unit require bayes factor relative x gibbs sx gamma density run good b comment reversible jump require propose jump immediately obvious ideally sir epidemic g chapter close population individual initially remain period specify period remove play epidemic contact member time contact occur become process pair assume epidemic distinguish characteristic disease infection observe removal intractable infection process
diagnostic precisely projection assess whereas suggest language like manner datum positivity assess datum positivity constraint study language covariance produce calculate treat score measure covariance projection language project inner language unweighte alternate balanced observational design achieve alternative example unweighted back language indicate positivity capture variability show analyze simulation effectiveness positivity constraint identify vary robustness examine linguistic datum five adapt root combination simulation addition uniform zero node simulate example entry differ depend pairwise variable direction ht circle fill sep circle label label x label h eight node utilize version sign corruption long generate four eigenvalue would non separable assumption thus information gain resource require certainly investigate simulation design language relate note simulated basis retain construction used check positivity repeat explanatory htb notable amenable consistently positivity particularly third explanatory effective satisfy positivity far guide variance positivity constraint become reliable amenable identify range explanatory plotted range display structural similar varied performance notable indicate power positivity positivity still even first furthermore constraint positivity appear tree positivity underlie assess sample variance matrix status perform observation sample assess gaussian constraint assess tree amenable affect scenario amenable tree amenable amenable amenable furthermore satisfied amenable positivity comprise relate choice due diagonal ten amenable amenable provide amenable matrix amenable consider well respectively result across sample component suggest tree retain even tree non distinguish group highlight interpretation notably due express back sound although difficulty invert sound trivial however include acoustic assess functional capability positivity robustness moreover albeit interpretation particularly exploratory linguistic comprise provide result new amenable linguistic tree examine projection tree amenable explanatory distinguish first separate tree indicate broadly begin end effective finding scope small set word describe apply language pathway offer plausibility historical development tree constraint decomposable structure paper usefulness identify variability distinguish group circumstance common separable practice work separate language technique carry across assumption truly assumption nonetheless purpose consequence language covariance acoustic datum unlikely however sufficiently second violate evident copula marginally necessary scope shall describe particularly give nevertheless covariance circumstance violate valid capture separability often separate tool thus deviation efficacy separability perfectly determine order basis intend suboptimal capture proportion three hold may unlikely completely exploratory permit great aside performance selecting outline language covariance ten furthermore include wide analysis word across language tree enhance possible tree constraint point satisfied particular constraint give basis provide distributional exploratory could binary describe bayesian replicate wishart covariance matrix could formal testing potential adjust method tree language could linguistic much semi however even currently despite fundamental constraint mark diagnostic relationship application subject appendix c component explanatory tree amenable b c percentage tree individually q mm mm increase development range application image medical well great availability power analysis statistical attract analysis acoustic english language find amongst principal acoustic word linguistic way speech utilize cross language inference language suggest observation unlikely identically probable relationship historical language acoustic know certain would historical record tree reasonably support language american language relationship language long describe tree linguistic variable feature language quantitative use evolutionary language recently scale attempt reconstruct tree language european language researcher shift away evolutionary relationship toward somewhat structure assess researcher acoustic five language american exploratory adequate relationship question area algebraic statistic literature g particular g much fully characterize case analogue binary apply covariance consider component realistic permit observe linguistic tree amenable tree good explanation set preprocesse audio spirit object application language functional object description form language describe tensor decomposable reduction brief yet exploit acoustic language question section use tree constraint describe construction score projection acoustic section general effectiveness tree constraint investigate simulation assess accept entail tailor relevant address whether evolutionary relationship acoustic describe evolutionary model explore choose tree constraint acoustic linguistic language complex involve language origin component tree comprise audio language american treat ten language resolution bit classify gender share integer ambiguity make language straightforward ten many word language model become increasingly common study involve sound reasonable observe finitely along smooth e duration vary intra adjust known alignment transform take intensity though possible alternative duration dimension measure generic unit point hz hz store broadly indicate standardized period derive word encode word counter gender beyond frequency gender short acoustic seven european language identify suggest acoustic gender european language gender adjust macro gender henceforth gender adjust object interest word area statistic principal component multidimensional functional counterpart also functional multidimensional scale acoustic benefit reduction provide extraction reduce subsequent technique implement straightforwardly estimate produce singular must one approach loss project datum prominent aspect retain remains optimize efficiently mode technique linguistic semantic macro comparison argue construct priori group canonical technique discriminate start maximize subject component uncorrelated efficient fine detail consider within consider aim identify function variation express equation eigenvalue infinite discretized estimation zero number h datum use function maximally retain comprise interact standardize frequency simplification likely consideration say function subsequently challenge covariance separable separately respectively product ff tt solution although implement pca multivariate tool purely technique approximation implement functional setting necessary differently encounter tool frequency time approximation tend question dimension reduction row description independently affect notable concatenation visual find successive uncorrelated combination language language reduce perform eigenvalue find eigenvalue produce great language r theoretically covariance hold mode variability distinguish mention section decompose propose decomposable structure separable product separable structure elsewhere novel though strong basis separable number retain amount purpose decomposable overcome obstacle cause commonly encounter datum numerical theoretically observational set standard kronecker language frequency l direction treat frequency rank require usually relax kronecker eigenvector solve note separability hold less basis still far pca language maximize language approximation densely exclude frequency broadly similar result direction begin particularly corner similarity end word covariance diagonal suggest covariance albeit acoustic e use linguistic study figure show within component take dimension project sub solely notational clarity htb projection mean plot encourage project aspect know language distinguish language dimension separate language acoustic indicate discriminate operate simultaneously manner language close proximity post share particular acoustic examine matrix acoustic interest compatible identify language dimension combination model help assess plausibility algebraic mathematically leave note non vertex various considerable include setting concerned determine unobserve examine
short phrase generality linguistic describe four directly evaluate representation behavior format determine core inferential relationship architecture representation allow us model represent vector work system natural language artificial logical language three training logic define mutually exclusive understand logic cover reason logical reasoning statement construct third cover quantification plain poor strong simulate logical concept neural build semantic representation language sentence reasonably size training whether representation text address question put nn disadvantage competitive performance specific carefully task adequate case yet inferential approach nn tp name set strict strict seven logic set universe straightforwardly pair phrase relation exclusive neural linguistic say via composition phrase merge phrase form fix input tp tree depict process separately structure share fed layer generate two phrase output softmax seven relation sentence powerful nonlinearity apply output column node learn add add full rank dimension multiplicative interaction comparison nn layer layer independently learn nonlinearity use nonlinearity provide valuable study generalize strong structured structure vocabulary embed comparison tree regularization use descent sgd layer tune time use five cross percentage addition balanced report accuracy code generate review period table dot kind logic atomic proposition infer relation theoretic sound inference relation depict basic involve compositional successful reasoning compositional sound inference kind experiment train test boolean set entity domain structure entity proposition fig divide evenly test train ideal correct create function tree structure composition use recursive ensuring relation appear test tp reflect geometric relation recover relation nn fairly well question tune nn label potentially frequent relation example p successful familiar atomic symbol novel recursively show exploit fact testing string artificial system symbol formulae logic operator relational statement data assignment value tree model learn expression pass unlike baseline guarantee ignore relationship statement sum baseline test accuracy formulae correct approximation training cutoff gradually decay model suggest despite sentence quickly suggest tree structure generalization generalization composition many work weak regularization even class rare prove logic language considerably key model representation semantic natural interact lexical quantification functional since form language consist english artificial language contain noun lexical lexical noun noun generate sentence time directly kind describe infer word level complete logic sum baseline lack order information previous experiment potential single token logic almost perfectly sum largely find across suggest fundamental obstacle perfect architecture validate elsewhere experiment handle investigate model ability noisy label natural sentence sentence variant template sentence corpus train learn language label aware nonetheless show learn world adapt word initialize wikipedia since input pass recursive layer vector layer win corpus label train separate softmax classify datum source find quickly technique across replace collapse collapse finally regularization dropout input tp htp cl little play play water water break spread use amount available lack lexical show strong neither win include resource task well result test mutually exclusive pair phrase substitution little overlap annotate use neither sentence ten category syntactic configuration suggest encode relevant also lexical inference address none dramatically impact model like corpus compositional syntactic summing nonetheless substantially remain confident truly quality evaluate task clean artificial algebra logic structure quantification reproduce logical reasonably sized promising semantic remain fall short recursion whether overcome strong encode logical inspection hope reveal learn produce similarly model learn perform complex inference corpus language rapid make provide optimistic meet
baseline use directly pair equality prediction difference summarize pair predict treat inequality equality pair pair use train paper draw training width evaluation metric area roc auc roc calculate evaluate rank threshold rate false negative fp inversion fp negative opposite predict opposite roc visualize nonlinear rank deviation pick train model rank pick level rank clear ranking accurately recover rank contrast exploit pair rank close error equality varied pair general model decrease increase good model train good pt rgb rectangle rectangle rectangle rgb model rgb model rectangle rgb rgb rgb rectangle rgb rectangle rectangle rgb rgb rectangle rectangle rgb rgb rectangle rectangle rectangle rgb rectangle pair rgb inequality rgb rgb rectangle rgb rectangle rgb rectangle truth cm level model equality pattern x pt rectangle rectangle rgb rectangle rectangle rgb rgb cycle rgb cycle rgb rgb rgb rgb rgb rgb rgb cycle rgb cycle rgb rectangle rectangle rgb rectangle rectangle rgb label rgb percent incorrectly predict rectangle rgb rgb rectangle rectangle rgb rgb rectangle rgb rgb rectangle rectangle rgb rgb rectangle rgb pair line band rectangle rgb rectangle rgb rgb rectangle rectangle rgb rgb rgb rgb cycle rgb rectangle rectangle rgb cycle rgb rgb rgb cycle rgb cycle rgb rectangle rgb cycle rgb rgb rgb rgb rgb proportion pair rgb roc rectangle rectangle rectangle rgb rgb rectangle rectangle rectangle rgb rgb rectangle rgb equality auc set figure pair varied simulated maximum area validation roc mostly design equality also clear test auc advantageous optimization different rate people person rate point equality inequality consist person style major minor gender home convert simulation grid select generalization training test performance algorithm simulation rank high varied proportion calculate auc perhaps learn perform error term study machine formulation max margin relationship qp solver future interesting capability slack add function simulate directly model pair show rank pair advantageous pair work scale large smooth gradient acknowledgement ss thanks li helpful discussion definition theorem rank nonlinear algorithm equality label pair feature label element well geometrically segment panel naturally human movie minute actor goal comparison generalize measure one q indicator extensively study interested equality algorithm datum bold letter row propose illustrative simulate real item pair option comparison rank reject discuss connection comparison rgb rectangle rgb original rectangle rgb rgb rectangle label circle circle rgb rectangle feature feature rgb rgb rgb circle circle circle rgb rgb rgb circle circle circle circle rgb rgb rgb difference difference curve grey w classify reject option output tie literature comparison tie model paper guess extensively machine model rating ranking tie input input ordinal margin learn approach input database label learn limit answer model present explain apply linear cost program qp index label rank use yet thresholde comparison thresholded training pair learn threshold suboptimal issue rgb rectangle rgb scale rectangle circle circle circle rgb circle circle circle circle circle circle rgb rectangle circle rectangle inactive qp vector rectangle rectangle rectangle rgb rectangle rgb rectangle difference qp qp learning case consider maximum margin program lp qp difference comparison nonlinear ranking maximize margin define function small lp boundary boundary draw separability datum linearly perform change variable solve qp maximize let datum qp qp significant middle right panel feasible max lp comparison feasible rank consider boundary together margin take margin qp max margin lp general answer middle qp define however solution lp qp panel learn
interestingly minimizer closed appendix z entry wise prop update huber q proposition iteration update lagrange multiplier recovery inexact pricing perform novel topology test next real load benchmark latter comprise generator transmission network consist range generator offer c regard offer quadratic cost piece linear block cost reflect level fluctuation nominal offer table deviation load generator benchmark realization load realistic load competition load demand consumption match demand benchmark perturb fluctuation price solve min separately day ahead market entirely market infeasible occur primarily experimental line collect entire run ghz intel gb ram matlab parameter typically validation assume degree tuned degree give ambiguity normalize diagonal small proportional trade degree degree table recover benchmark evaluate real collect consumption scale competition site benchmark tracking test simulate grid infeasible yield alg initialize solution alg sublinear regret track alg initially yet interestingly available subject upon exploit complementary strength advance compressive proved grid consumption month enhance competitive market rapidly probe program rich could leverage heterogeneous characterize research admit unique belong subdifferential optimality implies trivially minimizer multiply identify prove prop imply q post multiply three depend back remark b edu edu grid major goal market competitive yet reveal information solely available market calculate multiplier economic lp minute matrix successive lp spatio price inverse strength formulate streaming datum input optimization compressive alternate direction multiplier economic price system offer maximize limitation competitive market market participant real delay market involve price demand generation look grid vision call market reach account increase fine design prediction operation grid extend preference attack grid task monitor security operator generalize tracking purpose attack information know transmission line could inform line could electrical cluster reveal influential moreover pricing characterize decentralize although attack topology using readily attack market characterize possibility attack explore attack detect identify attack design attack generally grid topology topological change study transmission reveal overcomplete topology spatio temporal grid rely phase line delay phase could grids topology scheme access quantity delay topology readily system energy lagrange multiplier constrain economic lp typically solve minute primal offer instance quasi underlie lp laplacian line exploit spatio temporal factorize positive novel scheme complementary strength constitute section one regularize thus simplify big alternate version streaming market distinct load validity finding letter vector one zero vector symbol stand respectively semidefinite regard norm norm frobenius flow market correspond topology capture via branch matrix grid matrix express flow approximated phase stack invertible resolve positive readily real market market determine demand price henceforth market minute adapt demand fluctuation collect horizon market grid stack tn respectively factorize product recovery problem find multi generality offer generator offer correspond sum generator handle similarly constraint remain simplify separately noiseless namely eq constitute blind recover rich recall definite grid light property equal grid positive eigenvalue diagonal imply invertible entry far expect prop write line transmission market period day california line overlap zero combination recognize satisfie pair inherent ambiguity unity satisfy leverage matrix counting balance enforce np advance yield replace diagonal value property definite nonetheless guarantee minimizer interior hand admit trivial convexity least low minimize norm constitute np q solver focus henceforth although could solve relatively interior handle hundred main challenge feasible former involve transformation intersection shift albeit relatively project alternate multiplier admm solve coupling admm assign lagrange multipli equality constraint iterate q copy multiplier partition admm admm update whose understood entry wise eq
speed la grant mat de amp set dimensionality accuracy suffer serious convergence context approach amp amp result amp amp cost coefficient signal large powerful two drawback necessity first require pass amp bp utilize sparse amp generalize reconstruction possibly specifically subscript notation individual column zero function compress cs amp efficiency reconstruction properly matrix optimization approach concern amp amp drastically move scenario lead result instability slight variation strict serious main reason issue identify namely sequential bp iteration strategy prohibitive entirely clear convergence modify posteriori favorable situation instance might modify spectrum prohibitive strong limitation third take step bp oppose utilize solve derive amp greatly improve amp careful amp preserve modify amp amp possess derivation next sec sec numerical testing improvement obtain amp cs value term statistical normalization constraint enforce stochastic examine bit cs model eq degree factorize marginal strategy amp marginal bp mmse implement message ultimately allow marginal message send subsequent message send node correspond algorithm current distribution reference parametrization lead follow sometimes bp integral without specify sequential pick update complete sequential bp amp pass send create burden possible term physics spin bp standard expand detail investigate one make close iterate relation attempt key derivation indice old one implication denote later describe point difference amp difference note also generalize la output channel one replace function depend term present demonstrate amp desirable reconstruction conduct computer processor matlab calculate utilize non fail negligible possible use require later effectiveness amp projection control give present fig amp converge solution meaningful namely experiment row element normal experiment robust demonstrate iteration amp comparison obtain minimization experimental molecular rare naive item
efficiently pair hypercube hypercube calculate implementation processor memory application parallelism discover probable probability although problem significant dp bn involve procedure processor set processor straightforward algorithm compute structural separate dp responsible compute different separately computation dp procedure failure greatly dp transform variant processing fill parallelization nearly perfect parallel time processor respectively extension discovery difficulty previously way score processor exchange non neighboring avoid transition spend communication novel fast network rest dp upon parallel feature conduct empirically capability discussion encode variable convenience dag specify edge indicator structural feature compute averaging directly dag demonstrate convenient formally represent specifie say consistent modular structural modularity independence modularity modular feature e indicator either example represent assume node il ig convenience family measure parent note bound need sum evaluate recursively effectively proportional size take therefore compute posterior compute forward backward fix joint eq evaluate correspond change time posterior compute recursively recursively db edge present serve parallelization dp find computation bn parallel hypercube compute difficulty responsible depend need need evaluate subset hypercube processor exchange score new coordinate computation exchange processor way respectively intensive finally integrate nearly load balance parallel facilitate mapping computation hypercube generalize describe parallel parallel discussion score dp operate lattice form set inclusion power direct node incoming receive compute sum assume correspond node send node lattice next processor undirected hypercube variable connect node differ hypercube edge parallelization hypercube hypercube algorithm denote processor processor time processor active time receive inverting compute invert bit step score manner receive neighbor neither processor summation require exchange compute new processor invert bit neighbor invert processor hypercube operate start processor similarly run summation subset processor hypercube independently processor take certainly algorithm score hypercube cluster time truncate formula lattice formula view truncate hypercube computer correctness definition encode variable otherwise hypercube use denote processor take processor responsible transform run line start invert perform encode processor processor dt dt st j ns bit string variable processor processor processor processor dt st mapping hypercube processor neighbor invert iteration necessary computation communication happen kt sr present hypercube proof specify processor compute line processor line subset thus processor among processor thus characterize k j jj k sum limit thus n kn truncate run hypercube compute line total processor processor characterize proportional k n n di di di di di di di responsible partition dp lattice specify hypercube part specify execution processor active except let hypercube lattice hypercube hypercube process string specification order formally lattice map respectively computation figure lattice partition process process complete computation hypercube hypercube processor complete computation hypercube processor complete start processor work feature prevent processor h subset processor assign processor compute processor operate reverse compute processor add score evaluate processor name encode bit string otherwise hypercube processor hypercube processor compute processor processor processor execute processor evaluate characterize run score line computing processor total serial time parallel parallel score evenly processor storage parallel achieve scalability compute intel core processor core gb memory core node use memory core allow regular user maintain core core serial synthetic set compute run serial run serial processor memory per processor collect calculated run speedup value dominate well speedup plot efficiency maintain successfully run second serial memory usage usage per compare show table time much well fast communication maintain core memory core able interesting peak gradually efficiency mathematically minimize solve yield e core consist provide piece evidence e core suggest quite examine respect core usage start increase core range usage core examine usage core core consistent per usage per core allocate storing require memory store program order run overhead usage per core total memory stay usage stay usage memory stay respect examine far observe usage usage per core memory usage test require gb core core gb observe core hour core half hour still far requirement bottleneck determine parallel capable probability efficiency computing probability feature demonstrate capability scalability processor algorithmic way exchange development develop parallel dp involve transform object network experiment limit future less possibility space chen exact discovery bayesian programming dp fast serial parent optimal processor run constitute take coordinate correlate exchange develop parallel capability dataset scalability processor algorithm bayesian graphical direct acyclic dag concern bn observation utilize bn prediction inference interested structural probable discovery aim identification represent structure bn decomposable fitness dag certain search employ minimize application probable bayesian extensively method one space compute average regardless possible super find optimal network np hard constant fast known programming dp likewise posterior probability dp algorithm exact probability space I parent node bound simple fast substantially slow complexity large dp computer
engine observation bandit reward choose tradeoff bandit offline raise relate reward choose evaluate offline evaluation causal research try reward causal policy draw round reward depend select end contain call action indicator otherwise choose q offline word even simulate test fast way guarantee long give reliable definition place probability ahead exploration adopt uniform choice experience know reasonably improve exist reasonably differ reality concern conservative procedure effective exist production randomization randomize precise problem sensible approach work well score meet system drastically variance overall calculation offline score offline sometimes correct verify offline experience verification turn challenge private randomization seed generator generator select multinomial final form contain offline verification verify seed reproduce check alternative pseudo random round statistical detect experience expect occurrence score hoeffde statistically significant collection therefore compare variable close statistical significance harmonic consider offline policy resort whether various concentration inequality confidence interval helpful insight result necessarily use due case suggest reality normal approximation take estimate interval component engine enable translate query error form rank web instant user correction web query absence new entity read person another correct query cnn really noisy channel model cnn user train set query query good rewrite merge rewrite correction loss real predict cnn desirable correction serve behavior furthermore query offline measure engine technique improvement search goodness concern lead offline place terminology candidate decide reward click page business reveal although goodness click process fraction user week yield datum use experience top must candidate send parameter experience metric affect benefit randomization reasonably include datum collection decrease relevance quality candidate higher tend likely choose collection towards promise likely run arithmetic harmonic major help issue lead eventually interval normally bootstrappe sample replacement bootstrapping compute unbiased offline click finally implement multiply interval size confidence therefore include interval offline policy online statistic could validate offline examine estimate day scatter online offline offline center plot bias offline offline large online ground reciprocal metric closely daily vary week offline fraction click position measure receive offline offline substantial fraction subroutine offline policy datum collect fall training label base positively metric rewrite target metric capacity prediction set necessarily correlate optimize eventually fortunately reliable offline select capacity offline week statistically demonstrate demonstrate upon medical include amazon product customer review compare purpose click find baseline appear match correctly appear whose name fail correction contain leave probably intend methodology research retrieval dominant relevance collection ranking mean gain successful scheme however argue addition alternative evaluation engine challenge user center system comparison provide promising engine people measure click serve randomize online recently technique identify winner require run offline approach expensive mention offline technique closely causal aim measurement change often intervention statistic literature recommendation formulate contextual interactive offline knowledge analytic technique head offline engine formulate contextual approach unbiased true real use verify reliability offline evaluation promise number action set list set problem exponentially large would see acknowledgement yu chen microsoft online particularly compute click key nature change engine result query infer reliably page impossible accurately metric feedback new serve baseline b successful expensive consume technique run many test log metric focus promise design information storage service evaluate cumulative ndcg approach highly successful offline metric human give high relevance score website com engine suggest opposite actor website like search sensible third experience engine module engine overall search engine challenge imply consider feedback evaluate user click infer effectively substantial increase optimize online metric
condition x j j constant since w ib last derive formula w n j number mle consistency sufficiently sufficiently eigenfunction et gaussian discretize first class covariance discriminant criterion note accurate propose also depend less investigate radius kernel order penalize discriminant b spline functional pls perform functional pca penalize hereafter method penalize pls hereafter code code spline website simulation category testing display approach svm bad rate regardless large suggest observe author mention article approach sometimes could simulation lda outperform ccccc gp cccc rbf ccccc pca fail normally class orthogonal function pca explain variance project testing display simulate average deviation expect functional discriminative perform sample lda code rate algorithm list table list misclassification lda approach sample become misclassification different observe rbf especially increase kernel significant finally cccc lda gp cccc rbf ccccc size misclassification functional neither lda cccc ccccc rbf misclassification database list table well lda become performance training size large improvement gp lda moderately proper list table rbf kernel kernel rbf order gp svm matter train moderately gp rbf ccccc misclassification table work good gp significantly selection dataset matter kernel share lda suggest dataset happen skewed etc cccc gp lda rbf ccccc fisher discriminant inspire smoothness translate prior mean posteriori probability fold theoretical incorporate correlation tuning parameter outperform multilinear chen song chen gaussian discriminant image extend fisher discriminant explicitly formulate smoothness functional application fisher discriminant pattern vision stand either grow principal pca discriminant lda directly must dimension reduction technique situation dimension pattern empirical fact intuitive theoretically applicable datum pca lda merely large specific obtain randomly sample discretize sampler kind scientific application discretization example digital regard functional include spectrum spatial discretize standard functional canonical discriminant etc vision classical method provide utilize smoothness fisher illustrate reason even violate hence performance evaluate classification fold propose bayesian smoothing accuracy modification utilize datum exist way require tune functional computational effort tune simultaneously less computationally intensive approach argue kernel discriminant vector etc solution idea infinite observe lie project point later empirical matter kernel choose article arrange review brief introduction bayesian smoothing penalize actually framework parameter require section remark draw direction generalize class characterize class fisher usually estimate still normality work assumption rich extend category ridge datum idea approach intuitive function observation lda smoothed smoothing computation effort smoothing filter tune pre classifier datum tune besides smoothed problem name penalize discriminant usually become noisy fisher solution tuning datum matrix audio dimensional manifold example al manifold regularize well regularize regularize outperform filter careful modeling moreover sensitive usually select cross effort assume smooth discretized datum represent project either predefine b etc analysis pca partial pls introduce wavelet classify approach lead optimum predefine observe good basis function remain provide representation hence preferable recent functional pca classify functional pls functional example pls preferable representation discrimination difference group represent well merely guess pls basis group pca basis far discuss fisher extend technique functional example utilize utilize functional pls utilize pca since fisher lda provide survey technique smooth bayesian toy example observe balance distance smoothness smoothness unknown norm order smooth norm norm describe certain assigning smoothness td log irrelevant interest minimize smoothing smooth smooth denoise technique follow technical smoothing expression resolution image th vector noisy phrase article focus interpretation kernel assume apply prior section perform functional knowledge bayesian formulation functional bayesian model functional underlie process unknown assign bayesian smoothing also inspire smoothing prior bayesian well posteriori address simultaneously covariance section suggest show
code fraction retrieve recall top scheme panel retrieve respect scheme panel fraction retrieve recall code retrieve plot optimal fraction retrieve recall respect scheme quantization projection sublinear recently simple quantization random offset lsh simple offset need code hash recommendation bin bin use code quantization offset convenient practical department computer apply sciences university usa department ny usa technical compare code quantization quantization utilize quantization random offset offset depend neighbor importance requires offset needed performance significantly depend build hash width say bit code sublinear time neighbor extensive reasonably theoretical code building table coding near determine code code similarity storage datum twice rank retrieve estimate practically store demonstrate improve similarity base bit code paper focus quantization projection neighbor search identify similar near neighbor search numerous etc neighbor early day compute near extremely text still square appear partly assume marginal machine normalize convenience e effective tool idea multiply projection classification factorization singular neighbor potential benefit bit project real convenient transmission suited indexing approximate sublinear near locality hash two quantization scheme recent intuitive quantization bin operation monotonically monotonically function suitable locality hash lsh well window offset write randomization fix accurate often one separately optimum sublinear neighbor suppose exist target distance neighbor algorithm distance square present presentation target exceed certain value lsh difference lsh characterize gap small difference optimum figure figure range confirm always gap region h gap good gap panel panel gap attain entire gap similarity similarity level high good comparison smoothly plot confirm replace might iii mention panel see gap much decay small code value see always large instead gap similarity gap well gap attain well divided subset another table result average query uci repository build hash table original use lsh independent appropriate hash compute hash datum hash belong repeat retrieve union retrieve hash retrieve datum substantially small fraction retrieve retrieve desirable evaluate dataset number experiment consume way evaluate lsh count retrieve exact positive avoid retrieve top neighbor retrieve datum ideally meanwhile hope keep retrieve present bin width small fraction lsh target basically
define ensure array almost surely compact array measure content theorem I infinite dimension application covariance differential index space tensor krige background tensor product recall infinite tensor range projective product nuclear space infinite banach nuclear fr hilbert dirac mass weight I ie may unit scalar encode ce ce ce demonstrate may covariance integrate tensor ci u closure one e e ni v ni reproduce rkhs tensor continuous definite follow structure value eq map kernel value value sure subtle kernel metric define ce ce f scalar metric f array continuous transform array ol estimator predict output space space map function array section assume describe law array let denote adjoint operator act array w ai ad e w dd tensor mapping orthogonal onto subspace ed define definite ol v possible compact ensure ol suppose strictly ol linear strictly definite strictly dense ol blue virtue gauss dirac let diagram encodes full structure disjoint subset different label indicator function design vector metric ci present operator ol value norm divide zero operator difference joint equality v continuous varie continuously argument omit continuity similar omit inclusion class define dense continuous hilbert choice diagram separate map diagram construction involve dense diagram dense subspace latter separable translation affine construct space construction external identity map prove agree subspace map center point center prove map composition affine restriction hilbert space part residual main translate v bound claim continuity let exist functional expectation unique prove h functional replace continuous respectively markov function lipschitz show e imply apply triangle quantity use inequality quantity vanish complete simplify expression vanish main combine fact equal rest term equal justify ff b b reverse uncorrelated p p array formal equipped map ei formal minimal topology dense tensor therefore ci ei vi ei k negative denote distance equal quadratic positive proving demonstrate convexity prove part hence hyperplane exist solution separate would separate distance prove part write vector compact support compact point convex combination consequently measure adjustment prove theorem thm thm thm axiom thm thm hypothesis consider know parametrize embed hilbert space extend construct ordinary absolute generality version uncorrelated consequence exhibit extend framework tensor ol define krige space construct hope article connection theory serve community goal topological space partial covariance unknown topological largely diagram may subspace restrict ensure projection hilbert inverse general ordinary extend full affine strictly definite dense e hadamard mathematical pose depend linear gauss ols unbiased continuity result joint continuity assumption parameter geometric familiar coordinate free though novel section stochastic add noise demonstrate ol continuous parameter example general satisfy satisfied often value encodes explanatory uncorrelated nb functional form preserve linearity construct unified v use theorem ols estimator b g express adjoint ol xy xy yy x x value mild explanatory law imply sense converge goal explanatory unified x continuous operator form reduce denote adjoint ol xy yx yx number even try simultaneously unified formalism linearity map structure array ensure represent covariance arbitrary construct general mapping krige corollary generalize krige predictor hilbert arrays finite index machine svm reduce present svm arbitrary topological demonstrate structural approach valuable formulate ensure essential consequence consequently unnecessary topological basis metric formal language readily convert program hope tool community author di david david la pe na schmidt david stein l nsf grant denote represent vector scalar equip topology minimal hausdorff space separate continuous separate exist functional ensure topological potentially describe c series topological fr array space part describe additive constraint approximated compact theory ignore exist parameterized space hyperparameter topological topology weak convergence nature parametrization topological manifold often equip encoding parameterization modeling correspond perspective noise hyperparameter dense intersection set full empty practice separability support hypothesis tail care take ensure strong v anti symmetric try classify operator continuity ensure p meaningful sense ensure allow proof define separable completion nan consist functional equal everywhere separability consequence separable hilbert ensure hilbert space encode dense diagram continuous equivalently admit factorization h embed originally motion construction wiener construct separable section trick wiener functional subtract continuously clearly isometry wiener map well measure transform translation deal case wiener deal general recent finite theorem still ensure v pl diagram leave space complete hausdorff separate point discuss introduction p change vs continuous also continuous v act linear adjoint adjoint map transpose purely algebraic satisfie separable hypothesis value finite vector continuous linear generalize hilbert continuous lemma abstract close subspace closure diagram exist identity diagram path imply hilbert respectively restrict remarkable function neither whole inverse exist domain complement similarly denote complement space main particular continuous hyperparameter map equal h composition domain banach hilbert space open space affine adapt banach corollary admit construct ols recall partial inverse ensure theory hyperparameter consider map ols define extension close impose existence ols ordinary least square unbiased map mean composition sequential continuous proof ensure ol across prove continuity ol vary jointly datum short suppose banach equip topology structure demonstrate ol regression ol conditioning uncorrelated problem error parametrize law gauss ol estimator unbiased since ol well define f define affine support gauss correspond hilbert space orthogonality achieve optimality optimality least residual complementary ol right main ol projection estimator consequently adjoint operator l f projection original explain residual variance unbiased theorem ol consequence q functional loss see function nice tradeoff gauss blue error possibly distance onto error arbitrary functional inequality direction functional nonlinear stein total powerful tool give drawback room accommodate continuity ensure robust strong hyperparameter mean p stochastic yy residual bayes agree conditional mean stochastic every abstract ensure p topological purely may stochastic define convolution agree original measure trivially uncorrelated theorem demonstrate admit continuous corollary problem map lebesgue measure reference euclidean generalize dimensional complicated continuous banach space map ol linear banach fr ols certain necessary continuity theorem condition spurious theory see see continuous g ol yy value residual construction define auxiliary v denote auxiliary operator hilbert representation measure p define integrable value yy yy non continuous ols residual p demonstrate uncorrelated next ensure correct every concern domain satisfactory respect proof ensure joint continuity ol mild assumption space banach proposition check integrable however construct continuous checking optimality equal measure convolution follow respect law define convolution measure respect justify prove v measure independent dependence structure equivalently satisfied motivation definition measure distribution law form let ol convolution light ols continuous uncorrelated continuous map found immediately admit continuous linear solve problem use krige representation value array arbitrary index classification spatial array predictor thesis formulate topological space topological space primitive assume locally hausdorff hausdorff array function key pair simply array compact open topology measure array array array fr arrays think array array fr compactly ai fr operator fr value fr r array index set call time space array consist limited time value fr spatio spatial field spatially jointly value medium indexing random admit scalar I definite kolmogorov extension equivalent ci co array compactly cf subtle kernel replace si ie scalar value section value estimate reformulate statistic
minimum criterion tree analyze finite correct model order optimal prediction optimal whereas estimate depth estimate build estimator risk predictor order user use building report report prediction technique solve exploit handle handle report predictive information optimal estimate cell one simulator around distribute physical group sub allocate band resource middle group symbol symbol call physical resource constitute group band band scheduling transmission band frame band allocate allocate whole technique periodic five band rate aggregate band aggregate convert comprise user simulator exponent macro simulator channel generic channel model realistic channel model system angle distance profile delay delay characterization channel extremely even different link make mathematically ease detailed parameter l cell grid inter site km power level hz user distribute direction inter interference explicitly model macro transmission scheme proportional fair full bandwidth band information band user ms measurement none loading effort partial loading exponential arrival macro height ghz inter site interest km total power level hz model fix identical uniformly inter site interference macro fair adaptation band band user ms delay ideal profile loading loading exponential inter arrival simulation request user every frame typically rate table look u main b traffic inter arrival load situation loading vary discrete full loading algorithm band time model characterize mathematically traffic sequence predict algorithm build discrete build propose apply technique appropriate modification build estimate active build variable use slide context current length allow dictionary word assign incoming character add word dictionary contexts repeat generate tree illustrative get incoming character follow step get character step obtain step tree repeat get character whole occur example look bottom see parent occur occur seven suffer difficulty word require ever increase memory store word unnecessary learn furthermore asymptotically due asymptotically save measure cycle long control channel know receive short cycle sense channel feedback channel traffic period sequence length limited require depth modification short recent observe symbol plan depend use depth depth depth model tree assign occurrence recursively recursion even zero might could occurred look example alone value look depth occurrence give frequency state indicate j j sequence use probability next number time occur occur occur future store build frequency evaluate see build user problem build frequency tree tree reasonable must capture tree must reasonably depth particular px uk estimate estimate correspondingly word might large fix sequence possible achieve complex require property explain detail subsection sake notational simplicity henceforth subscript characterize sequence call extensive sequence source hence entropy volume predict future mutual information physics extensive extensive component rhs sub entropy extensive grow extensive joint writing calculate require knowledge sub observe predict compute entire practical system joint good markov predictive varied study nature expect grow linear monotone decrease information go decrease past retain information equally happen interference decrease sequence possibility sub extensive despite simple process immediate q term metric function increase k learn sequence infinite require never form imply predict require user behaviour capture empirically seem logarithmic instead continuously understand looking express apparent argue pick value increase need distribution hence limit despite slowly prediction user significantly compare extensive gain substantial obtain increase order beyond increase increase order use current bind order order sequence sequence model th markov chain description criterion observation estimate observe user build discrete building logic parameter th maximum user use bind determined model hypothesis problem likelihood hypothesis increase fail estimating cost pick maximize try maximize first log penalty covariance ml increase equation ensure optimally trading implement priori try estimate determinant option use model eq model aic however sequence grow nearly sample correct correct aic derive detailed correct aic aic criterion ensure initially usage determination use predict give occur frame look sequence cycle minimize observed vary section fix calculated estimator posteriori pick predict possible since accuracy treat equally rate predict transmission loss throughput map pick wrong result rate choose propose throughput event pick numerous cost enable pick fail transmission assignment rate difference rate rate denote give observe calculate expect minimize transmission successful incur hand high pick bias map thus low loading load simulator sub frame generate understand behaviour loading generate extent variation greater adjacent value load user full loading variability loading hence issue loading procedure implement simulator frequency one access tree early see probability instantaneous estimate curve stop user repeat reasonably tree reach period sequence length truncate use truncate predictor henceforth predictor fm probability fm fm fm nine median median user table partial scheme median naive prediction technique fraction predict efficiency percentage obtain user reduce rate scheme improve fact predict complexity mechanism partial loading cumulative distribution mention cdf load predictor outperform fail fm fm users loss fm fm map nearly achieve cdf partial loading compare outperform user rate fm user fm user map technique scheme look graph percentage map percentage map predictor error treat especially predict feed high full scenario variation likely sometimes feed work well compare fm choose performance load partial loading loading require adapt loading traffic frame presence partial loading investigate propose prediction need cast aic correct aic model sequence provide minimization rate implement simulation substantial level gain
dataset make idea notice independent subspace independent vector specifie belong dataset class independent subspace concrete specifically subspace originally subsequently independent class class project projection belong linear cost case linearly separable datum dimensional compact preserve separability specific manifold structure derive result margin subspace application root equal vector element indeed random element aside relation adopt template projection initial template new template feature preserve template study require multiclass first cosine angle vector approximately preserve two present draw gaussian true cosine angle preserve additive significantly angle empirically value close cosine serve preserve empirically otherwise notice hence preserve preserve irrespective angle eq high term vector preserve less preserved term representation examine preserve preserve dataset need hold simultaneously subspace continue belong projection preserve denote continue span column span straight remark need hold structure preserve random error subspace margin close principal maximally circumstance project subspace separate let well relate subspace disjoint subspace subspace pairwise sparse sr various recognition security sr compress sensing equation overcomplete dictionary solution achieve overcomplete property overcomplete dictionary reconstruction advantage represent zero coefficient training term class sample ssc subspace subspace cluster datum recognition projection however cosine close preserve vector angle verify setting random dimension space empirical rejection vary indicator operator rp c rp pca arbitrary cosine vector empirical absolute cosine angle probability rejection similar evident preserve equation length arbitrarily great dimension rp see dimension rp hold suffice trade goal report comparative accuracy pca mainly technique less extended initially locality preserve projection embed reduce technique yield accuracy claim subspace rather separability projection well classification exploit well classification random projection illumination independent per person take illumination image intensity vector evaluation illumination database pose illumination dataset split pixel pixel intensity projection table dataset result times accuracy dimensionality significantly fast preserve random use projection explain low I lie surprising even major projection occur streaming change data lie preserve still hold stream template attack original template construct inverse recover approximation ii yield accuracy par original become essential highly paper initial step life security require dimensionality ensure subspace preserve argument hold show cosine projection preserve evaluation present irrespective hand hold simultaneously r r get direction least vector get rx rx similarly hence hold similarly union bind hold probability n concern security system problem template template many quality early processing present formal projection essential dataset preserve generate derive physical etc gain popularity decade systematic assign resource template physical template attempt security research area deal trade security state successful far generate course generate highly template transform template increase volume drive security since computational increase system employ volume vector highly degradation employ perform task generate template many fail
generate perform virtual alarm set able presence case resp identify case high snr due converge section performance hyperspectral university image pixel pixel nm dimension randomly available run select whole abundance matrix row point redundant row reveal thus redundant spectra determine algorithm several subset mutual redundant discard give assumption surface square abundance map scene cast n determine angle obtain outperform c c avg angle approach blind model allow determine abundance consider synthetic real demonstrate include adaptive change environmental optimality detail every prove address hyperspectral without assume scene signature satisfy increase depend hyperspectral multiplier address reweighte synthetic demonstrate hyperspectral imaging continuously grow area receive decade hundred narrow adjacent couple surface chemical image application include surveillance hyperspectral image pixel distinguish mixed pure signature material mixed mixed pixel abundance determine number extract pixel scene separately virtual among alternative part separation consider express linear combination fractional matrix abundance constraint place symbol column study assume scene index least generality mix reformulate represent abundance admit row zero identify exploit turn situation observation handle cardinality unknown dramatically affect mix valid snr shall represent whole scene blind self estimate abundance prior impose negativity order cardinality candidate model multiplier admm solve square study literature three free author norm rather pursuit order scene predefine negativity organize describe result admm namely minimizer usual model f projection operator find lagrange multiplier carry represent run tend converge dual q stopping criterion must call main admm splitting subproblem square positive lead addition constraint turn realistic restriction index respectively family depend kronecker presence consequence l approximation constraint term objective reweighte algorithm update calculation update weight proceed calculate follow reduce norm solve optimization augment respect lagrange multiplier carry exactly hereafter obtain set amount analytic alternate likely converge minima reason warm initialize loop initialize j spectral extract library mutual coherence spectra coherence eight generate negativity three
explain component much incorporate smoothness technique frequently subject worker focus baseline brain store array correspond brain begin template image number map template record subject specific template voxel represent image process use volume examine visit create image voxel subject contain result require pc material central pc show figure correspond grey primary tend differ grey approximately remainder primarily feasibility pca interpretation pcs section confidence bootstrap distribution pc overall computational bootstrap propose notation typically bootstrap matrix vector store solution rank ultimately arbitrary pc adjust section sample although svd fail handle svd appropriately adjust find detail material complexity sufficient lead bootstrap bootstrap pcs bootstrap express bootstrap b distribution pc variance create interval additional subspace option additional scale project onto pc vector create percentile pc hold work block break calculation low operation material bootstrap pc require roughly equivalent however sign know across variability principal pc cause sampling decompose interval include absolute variation pc adjust sign change column reason sign column equivalent sign require calculation element calculation simplification note whenever occur adjust multiply result equal cosine angle dot adjust angle range interpretation pc dot product adjustment choose sign method previously sign pc column approach rarely different shift material dot product intuitive find take bootstrap characterization property store b eq operator bootstrap complexity combination multiplication oppose traditional multiplication computation bootstrap transforming allow calculation calculations parametric bootstrap translate dimensional component construct pc specifically pointwise confidence pc pc interval calculation full bootstrap matrix simple confidence component moment interval ci e percentile normal across bootstrap attain calculate store pointwise percentile ci define percentile unlike moment percentile calculation percentile tend bootstrap sample moment interval e quantile tail moment interpretation quickly calculate bias accelerate suggest norm pc unit p pc function percentile note calculation geometrically dot condition cr angle note cr coverage create band band contain exceed absolute contain component pc subspace combine pc pc influence rotation whose depend pc rotation characterize variability principal equal pc manifold kp create norm operation refer suggest form variability cr also principal pc simplification adjust rotation principal orthonormal covariance pc pc approximate rotation pc parameter interest pc rotation lead suggest rotation pc sample confidence interval pc however adjust rotation might apparent coverage problem estimate clearly eigenvalue less clearly rate pc curve panel variance result alternate residual spaced pc simulation residual median across moment percentile alternate value generally scenario median coverage pc spaced eigenvalue slightly coverage plot row correspond right plot show pc row rate confidence component well spaced increase appear affect either fast bootstrap eeg estimate first pc exhibit minimal variability rotation pc column pointwise three pc percentile agree although percentile interval skewness bootstrap pointwise interval show band around pc band calibrate pointwise coverage simultaneously contain statement population pc interval somewhat ad hoc furthermore contain within band norm low boundary pc tight imply nd wide hour reader think artificial pattern variability third bootstrap central pc pc little bootstrap pc spike hour shift spike population pc plot us peak peak bootstrap shift tend third component pc variation second noting bootstrap due rotation second pc bootstrap variance display vector panel see pc pc place place weight ci plot figure variability pc exceed absolute surely pc note percentile rarely thought percentile create bootstrap first bootstrap see simulated draw eigenvalue know percent bias across bootstrap covariance eeg slight bootstrap percent bias pc show computational feasibility deep interpretation pc provide pc estimate variability variability fit pc pc magnitude pc fit pointwise ratio nan display rotation pc truncate analogous substantial pc rotation panel notable percent pc base percentile test measurement subject calculation standard ghz intel memory parallelization calculation attain pc conservative loading bootstrap offer speed improvement approximate force test demand percentile pc minute minute error force high cluster reduce parallelization attempt file file relevant block sequentially material bootstrap need sample show time dimensionality vertical axis log time show pcs standard bootstrap require calculate bootstrap outline fast bootstrap lie feasibility eeg standard component usefulness demonstrate ability bootstrap well bootstrap rarely theoretical well study study pca research pca verify bootstrap useful generate display direction bootstrap variability pc calculate also beyond alternative method describe pc variability elliptical pc rank result region describe sampling variability span observe b cr region q elliptical fully define direction primary axis fact precede want first pc lead residual onto resample alternatively remain pc resample residual project vector approach source acknowledgement national institute health grant number national imaging grant stroke ns dataset article solely package estimating sampling variability large storing bootstrap computationally infeasible fast calculation component eigenvalue leverage coordinate metric compute solely bootstrap storing component fast brain allow standard bootstrap minute approximately image analysis dimension reduction field image multidimensional dataset pca identify projection maximally call component pc subject basis pc pc population pc low expand discussion sparse sparse dimensional demonstrate variability pc variance confidence pc typically existence order interval bootstrap confidence pca bootstrap context determine challenge calculate store bootstrap exact order magnitude bootstrap lead variability pc low little pca largely drastically reduce dimension far bootstrap scientific application method directly determine bootstrap principal regression quadratic penalty remainder organize section basic pca fast discuss motivate eeg fast computation final pointwise pc bootstrap use coverage pc pca eeg present denote denote block notation denote element denote vector identity generally rectangular highly pca subject basis vector maximally basis principal pc subject respect call pc matrix subject center singular decomposition denote contain singular order value principal vector diagonal contain variance know variance pc construct scalable calculation estimate draw replacement sample pca bootstrap calculate variability bootstrap computation infeasible important pc depend coordinate measure coordinate along number exceed involve reduce parsimonious vector span still include transformation first step parsimonious computationally demanding number coordinate equal improve efficiency pca contain low subspace span principal observation sample span sample skip demand reduction represent parsimonious orthonormal lie loop translate calculation bootstrap bootstrap svd step resample b b b directly score value sufficient b bootstrap apply decomposition setting result use generalize suggest decompose lead project however approximation demand decomposition score imply align coordinate coordinate orthogonal rotation pc vary bootstrap unlike estimate variability pc pc variability magnitude pc rotation pc pc weight majority bootstrap variability rotation pc parsimonious dominant component drastically storage memory requirement
restrict radial simplify fix drop lead formula simply otherwise build x numerical building depend operation inversion dimensionality add term preprocesse relatively asymptotically rbf kernel metric modification repeat exploit calculation arrive kernel kernel build unlabele consequently fold validation separately separate feature apply active unlabele perform nine dataset uci repository briefly scale fair regular rbf almost dataset geometry detect k algorithm bank cancer heart experiment perform small include use fold mode start good accuracy table rbf kernel mahalanobis diabetes breast cancer heart cluster empirical rbf data scheme separate base meaning run train separate htb dataset rbf bank breast cancer diabetes heart clustering behave gmm vary difference perform advanced cluster second rbf conclusion cluster various choice technique internal validation part applicability reader mind different method gmm gmm gmm bank breast cancer diabetes rbf easy selection practice exist library table narrow rbf achieve rbf also behave simply base htb rbf bank cancer diabetes heart technique perform limited table well small notice range outperform great importance resource internal application active rbf rbf rbf bank breast diabetes heart rbf achieve kernel big htb rbf mean regular htb dataset rbf bank breast cancer perform parameter increase resolution opinion reliable finding well rbf consider eq measure result exploit ability limited grid applicability training time great ensemble repeatedly pair hard behave yield high justification phenomenon us question conduct experiment improvement competitive base simplify regard view lead correct transform multivariate asymptotically rbf obtain behave rbf however select typical svm empirically fundamentally subproblem reduce amount show conceptual distinction approach model worth though experiment see general framework section section example edu pl gaussian transform dataset mahalanobis paper gaussian consider local geometry dependent feature projection emphasize construction new divide represent construct empirically nine uci repository stability find learn property without exact geometry density method conceptually often lead decision combine form complex algorithm introduce building exploit geometry construct feature part give see change problem solve implementation svm give less complex compare rbf mahalanobis grid crucial classification scenario ignore importance index evaluation measure tune require evaluate visualization characteristic roc visualize structure describe issue connect comparative dataset uci year grow among mahalanobis calculate disjoint element still define thing construct infinite ii fix iii iv projection dataset restrict first worth geometry try incorporate inside numerically remain question evaluation benefit problem focus contrary modify gaussians contrary rbf mahalanobis rbf variation gaussians projection thing point projection gaussian sake
convolutional layer kernel third convolutional layer neuron layer previous layer fully layer follow relu layer relu pooling pooling unit space apart center location response lm layer normalize adjacent kernel constant layer dropout size data conv conv pool conv conv conv conv fc conv fc fc fc layer shape prevent apply augmentation extract patch horizontal image patch first convolutional filter kernel layer pool convolutional layer convolutional one convolutional size pool convolutional neuron stochastic example weight small decay regularizer insufficient sentiment training suffer overfitte imagenet promise fine initialize except layer regard forward pass divide epoch fine initialize deviation initialize neuron bias fourth fully connect layer remain initialize forward train model softmax server core e processor gpu training day test store gb disk space annotation pseudo ground truth top annotation percentage image pseudo truth label k accuracie tune cnns cnns fine tuning accuracy visual sentiment concept strong community usually mid sentiment performance acceptable rank per cnn greatly base gain accuracy top without top detect tune serious incomplete incorrect detect concept accurate ground correct accuracy top important boost base train multi binary suitable retrieval instead annotation section model rank precision noun show design cnn performance object detection candidate concept classification convolutional network cnn train newly framework deal biased training datum prevent initialize weight show newly cnns annotation compare localization cnns improve leverage high boost help improve build comment sentiment sentiment top tune em edu berkeley sentiment classification method deep convolutional cnns visual noun discover tag web utilize nearly million concept popular show great imagenet cnns train newly deal contain strong prevent overfitte model train imagenet train deep cnns annotation accuracy retrieval storage index social medium social among research effort sentiment visual medium attract video opinion greatly media communication education concept study extensively computer visual difficult impossible big sentiment tractable sentiment mid fill concept noun strength discover occurrence relationship tag image chen leverage sentiment concept thousand category million image deep network cnn able achieve classification performance improvement efficiency similar imagenet much capacity control compare stationarity locality dependency mostly cnn easy feedforward connection parameter slightly theoretic cnns capability imagenet specialized sentiment concept train gpu successful back propagation achieve large benchmark dataset imagenet competition win result learn deep convolutional small dataset mnist achieve success et unsupervised follow supervised fine tune insufficient concept bank paradigm prove successful computer learn label recently et domain paradigm briefly review sentiment define affect monitoring opinion corpus sentiment visual issue et sentiment consist noun detector level drive video retrieve extract million sentiment image ensure prevent suffer
planted apply three synthetic form hard condition network commonly bipartite community broadly say except genetic vertex internet movie actor movie actor examine level spurious detection sized non type community community degree community heterogeneous resolve community vary amount specify create whether preserve degree correct expect illustrate induce bipartite mode projection network performance projection unweighted projection bipartite obtain let share weight adjacency matrix diagonal size correspond projection type adjacency entry weight path vertex draw ability recover partition vertex sbm partition type vertex measure normalize infer correct treat observe group partition group group partition hx I degree matrix identifiable community produce community divide evenly bipartite sbm unweighted projection amenable method mutual information vertex partition vertex recover plant structure less easily identifiable community create overlap broad uninformative third community connect onto overlapping type community difficult community divide evenly impose give prefer plant bipartite adjacency normalize infer correct exhibit classic phase find plant partition extremely extremely inaccurate mode community sort sbm weight unweighted sbm without correction unable plant partition community sbm unable lead degree correct remain correct fail partition optimum sbm modularity unweighted projection fast modularity able projection variability projection limit difficult projection outperform design produce relatively high arise find frequently topic co word removal mask noise stop priori correspond extract actor movie broad actor interpret language movie match actor movie bipartite adjacency sort correct define characteristic movie group movie group consist language correspondence language infer responsible b correspond stochastic block create community regime show bipartite efficiently sbm bipartite community mode one mode community avoid projection discard create clique assumption problematic fail bad scoring either outcome correctly suggest whenever projection avoid bipartite evident substantially outperform method result general many system word contain hand exist contrast vertex produce despite connectivity sort language indicate movie language group movie per actor dense show similar separation edge existence brief aside mode commonly yet implicitly projection subsequently network raise question due bipartite whether measure correct sbm two sbm increase apply bipartite must learn bipartite cause bipartite known information utilize accurate subtle point use choice choose selection burden model bipartite network also selection compare likelihood choice control extra sbm distinct community generative remain area difficulty limit assumption bic produce incorrect decision promise result recently broadly principled advantage interpretability infer process ref membership community bipartite future hierarchical adapt explore beyond community inter centrality bipartite form structure sophisticated project award gm ac sciences fa ac air office scientific research project solely represent view health design collection manuscript source implementation appendix implementation author find set x green partition size plot type highlight correct show efficiently infer structure green bipartite common connect like counterpart bipartite choice information interpretability solve community detection bipartite bipartite include vertex may trivially bipartite block statistically choice interpretable synthetic structure bipartite bipartite vertex vertex bipartite appear specialized remarkably document genetic actor movie user mobile author common way find subgraph definition vertex likely two vertex never definition community suit block community bipartite pattern detection bipartite apply community mode share neighbor type reduce implicitly without construct bipartite scientific ever mode large bipartite measure number path length projection bipartite projection modularity moreover information bipartite identical mode projection even highly extension modularity propose broadly speak model vertex connect bipartite express implicit restriction maximize modularity partition independent yield type consist pure sometimes call co partition elegant structure network system network capable bipartite sbm develop specific overlap edge structure partition specify among generate sbm parametric give explain configuration advantage explicitly state fact bipartite directly apply sbm bipartite also sbm bipartite exhibit quality community formulate bipartite search network community plant background uninformative bipartite bipartite block hereafter correct vertex extend include correction begin vertex group group community bipartite adjacency instead connect thereby notation work sbm vertex type must pure develop correction expect adjacency let number world allow poisson calculation easy unlikely correction bernoulli enforce bipartite constraint restriction q model bipartite generation bipartite network lack lack subset sbm discover bipartite structure bipartite discuss generate symmetry group bipartite network seek practice easy logarithm parameter denote sum drop maximize assignment constraint pure community motivation derivation correct degree correct tend broad structure sort degree model denote parameter edge number enforce eqs probability observe adjacency normalization connect belong rewrite maximize drop constant multiply partial lagrange multiplier yield likelihood network dropping maximize correct respectively substitution heterogeneous pure vertex type partition classic lin input adjacency vertex assign type vertex index group propose type propose move move possible choose move decrease move help optima pass objective optimization many score among independent well degree correct model demonstrate community bipartite sbm naturally clear specialized model perform sbm mix vertex community nearly less vertex type eqs posteriori sbm constrain indeed know bipartite network complicated rule connect generative correct numerically counterpart provide mix ii vertex reproduce adjacency sbm remain numerically bipartite network imply identical principled algorithmic key understanding make vertice uninformative give hand sbm lack informative density observe community whenever sbm bipartite group sbm network show bipartite
setting capture position pair likelihood item order intractable contribution key collaborative ranking contribute suppose list item drop mention tractable scoring let ij scoring function agree relative simplicity thus suggest specific rank permutation probability first position suitable rank community extension model propose make collaborative interested choice user rank item rank position begin end employ introduce permutation adaptation trick unseen log verify either item rank item interpretation user principled item ahead choose item respect community user permutation sum denominator take recursive array start pz step fix lagrangian pz multipliers lagrangian zero maintain lead pz u close apply resort u z z want output rank unseen finding rank unseen item sort order new item order position item old repeat new item position rank likelihood list introduce new item new item place th item old list naive right recursive assume position odd notice z z odd permutation cost per pick permutation cost case item hasting method move learn maximum computation generate sample accord significantly per propose cd work instead run step enough relax distribution cd stress pass cd model one chain move graphical efficient regularity application attempt consider pseudo abstract pseudo mcmc technique mcmc configuration configuration log unchanged position require energy would denominator current eq q energy pass local order item new search position low energy computationally choose probable pass special factored parameter let start score factor ignore position monotonically attractive constant position prediction case nice interpretation u basically say subject much occurrence potential item distribution use dependent item user swap item change gm collaborative rating well recent author pairwise preference none attempt rank pairwise discrete often limited goal deal item recently attract retrieval setting item image compute collaborative ranking study assumption introduce community user base inference recommendation tie rank correlation collaborative filter information community permutation rank make successive item manner introduce account specific generative second rely assumption make learn inference method time collaborative service community service community exploit rank list research preference rating rate star user scoring assign qualitatively carry evaluation limit intuitive way express preference easy place visit assign importantly recommendation core recommend unseen address open preference require intermediate filter system item decrease complete user collaborative
share herein principle science aspect publicly reproduce mark anonymous comment work co national foundation university systematic sigma biology mark technology center center conduct herein cm htbp three differ prior model probability pm analysis retain prior sample population plot glm replicate analyze proportion simulation replicate divergence population adjust analyze strong true parameter exclude divergence e support extreme divergence simulate population distribution top column unit assume site six please summary dataset pair population u indicate distribution million assume site please support cm analyze average place prior eight analysis mean implement scale relative size term upper population population different time unit model hoc adjustment possible alternative prior sample efficiency support asynchronous nonetheless conclude divergence divergence informative dispersion statistic see plot perform single divergence thus utility whether share one infinity numerically reliable zero easier less interpretable one index divergence estimate code mixture analysis grey black model b exclude divergence appear valid cover plot project summary statistic two dot summary empirically inform pair population draw indicate plot distribution million per htbp cluster divergence simulate divergence distribution divergence give unit million htbp pair population draw u column time unit million site generation dispersion accumulate retain glm htbp correct simulated analysis relationship glm adjust prior setting replicate include population bp probability divergence proportion bin consistent divergence event population simulate setting black black com word support author date establish population occur process assess bayesian method pattern simultaneous paper agree prior assumption support distribution divergence lead marginal likelihood probability model alternative posterior analyse rejection unable rich reasonable averaging empirically inform prior analyse tendency cluster exclude true tendency analysis share divergence primarily hypothesis explanation bias paper demonstrate unlikely region support wrong history computation fortunately predict principle flexible accommodate place vast region frequently seek phenomenon split potentially event probability choice study method support divergence broad period agree divergence uniform spurious sensitive prior mechanism behavior ref align prior likelihood posterior divergence prior rejection space within computational toward sample insufficient exact support approximate exact posterior support simultaneous divergence across divergence wrong perspective bayesian employ seek phenomenon exclusive distinguish ability share correct sound improve markov chain sequential carlo rejection implement try narrow informed uniform sample rejection would produce estimate posterior consideration empirically inform prior bayesian evaluate approach potential analysis error mix unit divergence exclude posterior reach evolutionary accommodate uncertainty divergence without likelihood model temporal divergence surprise rich show jeffreys also use analysis make hypothesis hypothesis suggest share divergence insufficient likelihood time suggest narrow highly inform divergence preference raise paragraph prior sensitivity expand method inductive belief become parameter value p describe belief value parameter define belief collect bayes calculate eq likelihood elegant belief accumulate dataset belief datum uncertain valid bayes say empirical empirical study obtain favorable bayesian justification parameter estimate candidate bayes calculate estimation choice see update normalizing denominator long probable strongly informative prior incorrect look entire parameter normalize posterior distribution average likelihood bayesian much bayesian value integrate marginal posterior measure uncertainty justification twice peak justification inherently belief prior analyze rule analyze fail uncertainty belief compare confident confident nonetheless bayesian perform well particularly aggregate parallel information prior lead favorable group wise frequentist coverage across far support example estimation concern practical narrow empirically inform uniform strongly model inform thus average estimate time match mean divergence divergence across limit furthermore prior see preference consequence posterior exclude explore possibility way average narrow likelihood prefer less yield dominate model generate reduce mixing unit time si prior sample prior model retain euclidean deviation remaining retain euclidean summary statistic across twice replace differ likely divergence nearly prior exclude time strongly prior distribution analysis model time sample behavior clearly insufficient computation hypothesis straightforward prior via summary sample plot orthogonal axis analysis indicate model problematic model provide narrow graph sa figure sd well exclude simulate randomly identically average dataset simulation replicate bayes odd exclude model exclude exclude value replicate glm adjustment exclude importantly exclude figure factor glm adjustment figure result simulation analysis narrow prior obtain toward model exclude elegant bayesian inference narrow prior density dominate marginal likelihood integrate capture uncertainty exclude fortunately flexible greatly spurious allow uncertainty clearly risk inherent bayesian justify risk power detect temporal pair distribute prior approximate demonstrate consistently simulate sa extreme divergence h sg variance evaluate sa glm adjust sg overall average highly cluster divergence per site mutation translate million power analysis simulate replicate exclude proportion divergence sa importantly exclude one quite many analysis narrow prior per detect among random decide address rare enough certain unlikely divergence provide much insight infer confirm work generate random divergence g interesting spurious important assessment power inference event versus look information determine mechanism prior cause divergence establish integrate produce likelihood employ parameter average light fundamental surprising tend cause insufficient would produce sample divergence time estimate present rough arbitrary strict furthermore branch tree million whereas prior implicit generation year importantly arbitrary estimate without year actually densely large number narrow toward clustered magnitude case perform slightly panel correct panel hypothesis cause look make phenomenon example insufficient sampling hypothesis create variance analysis number draw furthermore insufficient prior toward less increase see sensitivity repeat analysis dispersion index figure trend error bias present strongly preference model generate simulation choice confirm model confirm model si
robot collaborative call robot switching demonstrate sequence training learn update reward markov process preference task predefine role human task execution human action place gradually choose step part simulated take accumulate propose framework plot accumulate robot demonstrate place action deviation increase policy human demonstrate sequence therefore expert manually reward function policy use code plotted line denote accumulate validation action informative robot online capability two box box robot box axis goal robot preference rotation position position horizontal vertical box angle demonstrate subject output user vs total human human observation plane human specify team execution human team person move robot human information box robot robot type leave surface behavior human introduce stand worker associate action execution automatically learn type subject task demonstrate accumulate reward compute take collaborative offline execution show performance deviation algorithm reason policy agent compute code expert model task make enable policy human user robot robust collaborative completely without intervention unsupervise robot representative inverse reinforcement learn wherein human variable offline include human validate collect subject person collaborative robot system operate people need human group adapt style adaptation human integrate robot principle present enable robot collaborative access take completely intervention robustness deviation previously behavior base human preference actual human preference human constrain choose formulation collaborative task much partially collaborative preference action human participant infer demonstrate action human unsupervised demonstrate sequence robot reward type learn wherein type infer offline online user set compute robot align new action validate human conduct person collaborative learn human explicitly manually human perform manual work people although aspect work use type unsupervise pose kalman method ng solve program attempt match policy approach involve expert optimize margin compete game theoretic art preserve theoretical performance improve type user include robot user policy human scalar robot action encode reward decision model improve interaction rather human consume infer dominant associate researcher employ tree search likelihood behavior estimate automatically human train play human set human learn work collaborative ai player framework partially observable system human use black simulator partially decision human driving task variable human model matrix specific rule interactive recently framework learn system yet plan efficiency plan uncertainty aforementioned task automatically framework enable estimation user offline priori unsupervise dominant differ previous parameter limitation require amount robot uncertainty formulation work formulation collaborative game ai automatically human partially observable allow computation accordance preference partially observable represent preference rather partially validate human experiment policy perform code significantly collaborative describe framework stage show stage stage human preference align preference human robot reason type assume access set sequence human collaborative use cluster dominate human partially observable human framework learn human type task robot compute reason human accumulate ask execute human alternatively informative human robot block propose dominate human collaborative demonstrate robot ability human initialize assignment repeatedly execute e step converge stable complete line transition update sequence line two assignment use require select ideal use bic range line multiple initialization often consistent likelihood penalty parameter since transition iterations bic determine mixed treat value associate correspond describe reward function optimal different human robot introduce align preference serve partially learn reward preference human reason uncertainty describe learn approximately human mixed robot accord factorization observable variable load describe set observable variable task step progress toward task completion observable human observable task observable human discrete task level action x fully robot r observable state fully give immediate action human robot receive action observation receive eq reward robot want robot choose action align must manually specify consume applicability learn human belong associated type fix human mdp tuple section demonstrate action reward decision human compute represent discount accumulation feature q expectation type demonstrate trajectory type empirical begin attempt mixture expectation expert human type terminate implement monte new guess solve subject I terminate use compute back result list count accumulate function algorithm second ignore first half generate human reinforcement demonstrate sequence calculate function partially type policy human structure expensive improved planning updating belief solver scale thousand algorithm belief allow satisfactory solution applicability human robot hand task collect human share task place available position phase human give robot task execute leave demonstrate subject
cause spurious digit image maximally choose vision theory visual feasible advanced vision minor induce bar bar restrict observational image pixel increase minor nature cause behavior top label observational train fully layer create iteratively difference time green indicate successful switching variable bar h bar fail red column cause induce horizontal percentage training fail time iteration six irrelevant image pixel train bar bar behavior plot ten error whereas stay constant text detail handwritten digit terminology dataset nature contrast change part visualization digits detail binary human observer contain observer contain either image modify become originally conduct mnist digit amazon error progress successive show neural close row digits plot successive remove target image link recently field encourage image grain classification network easily adversarial attempt causal purpose extract truly causal feature causal causal discovery meaningful entail unclear counter distinct problem account causal macro micro set well macro basic graphical methodology clearly candidate cause micro science economics specific acknowledgement fellowship pp support would prove causal complex part use simple technique subset distribution proper improper observational inspire compatible causal strategy observational refine polynomial compatible equation root usage constraint additional create useful class still vary assume simplify parametrize arrange create joint n n dimensional simplex proceed proof value distribution lebesgue constraint first note fix define say image hold hold write equation term define polynomial constraint omit equation simple constraint hold equation contradict consist measure since finitely lebesgue remainder direct causal prove indicator property application theorem equal restrict joint distribution among generative form fig main text induce observational induce appear observational fixing observational express apply auxiliary observational hold observational observational full rest proof observational refinement example variable model induce causal observational agree observational causal case fix observational partition align neither zero verify induce last induce observational shannon construction note pair correspondence far small entropy causal follow causal consist binary white analogous article observational namely observational spurious class would causal e causal variable causal must information component separate experiment start ten net validation architecture layer layer layer maxout activation training use batch dropout momentum adjustment iteration decay decay column stop validation network mnist machine digit one digit set machine algorithm maximally image start ten contain zero digit zero contain create mix five image ten digit question mark digit digit target majority vote annotate digit neural completion pt minus plus minus pt axiom computation systems california institute ca electrical engineering california institute usa sciences california institute technology usa rigorous visually human neuron construct micro causal observational minimal effort provide finally image identify visual human behavior traffic subtle increase face scientific economic communication cause constitutes cause compose million variable pixel rather present theoretical framework visual image cause pixel advance target behavior macro construct micro visual cause distinguished macro variable contain cause causal thereby variable prove causal visual cause effort connect method automatically cause illustrate synthetic code implement available online cause set definition intuitive equally extract aggregate micro market finance causal learning automatically framework causal consist raw pixel micro macro specify causal micro causal latter choose criterion plausible macro variable efficiently search whole macro approach derive mechanic support incorporate cause feature behavior plain observe distribution vs investigate micro macro variable avoid distinction observational literature learn datum generally aggregation pixel causal macro instead annotate pre feature case use external visual contain vertical bar bar bar bar whenever bar construct visual cause identifiable bar image influence call presence bar causal presence bar bar value cause presence bar image introduce bar occur presence bar bar cause identify cause ability distinguish among causal causal possibly example stand illustration general spurious identification bar model provide theoretical pixel configuration account feature micro visual principle identifying cause behavior image pixel cause pixel atom table constitute difference causal relation possibility without pixel probability pixel stand relation without throughout almost except lebesgue relation causal imply provide appendix appendix extend restrict distribution partition put distribution require false put trivial agree observational proof indicate visual cause image problem observational assume causal change within inference observational region say observational arbitrarily within causal belong observational contain construction explain observational within spurious variable whose causal spurious well range spurious macro construct visual contain causal macro shannon theorem constitute small macro fig relationship observational causal due unobserved cause irrelevant structural treatment causal describe set desire directly relationship causal separate variable pixel may whereas presence absence shape image intervention use underlying variable causal intervention macro macro intervention image intervention keep impossible value change special macro causal would physical space specification develop visual behavior knowledge allow return maximally desire causal function causal ci k di search close desire causal close note candidate image causal check spurious close desire causal heuristic want offer automate produce causal effect predictive learning pose 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identifiability subset restrict stay split problem j j j mean conclude second equation hold hence set supremum collection method several distribution distribution abc bernoulli reference move integration average sample inference gaussian mean generate prior gives mean see success beta q standard deviation sample poisson distribution shape mode eq unobserve derivation helpful form follow zero since analytically integrate unobserved vector gaussian uniform normalizing integration matlab integral standard variable call consist column diagonal determinant integration integral uniform posterior normalizing matlab sequential abc get threshold choose schedule abc purely quantile thereby posterior informative posterior acceptance hybrid blue circle purely benefit hybrid yield quickly polynomial brief care center day care sample represent infect thus chain transmission dynamic care care center assume review transmission dynamic care infect evolve h small negligible small equation describes infect previously infect infection co infection parameter probability infection model static infection day care infection determine yield uniformly contact transmission number carry transmission individual care care center stationarity exact long regime model three internal infection infection infection carlo abc distribution involve review simulate care center infect individual proportion empirical cumulative summary day center form empirical summary accept distance threshold threshold adapt straight ex plus minus ex part increasingly generation difficulty latent find simulated computation indirect address choose statistic enable objective discrepancy thereby validity theoretically bayesian inference real individual day care likelihood computation modern ingredient statistical evaluating model prevent use computationally principle rely particularly suited model unobserve likelihood independence easily model unnormalized unobserved unnormalized exist alternative simulation generally simulate moment indirect approximate propose idea vector observe critical rely expert make benefit easy statistic include construction inference basic observation distinguishing set panel easy discrimination classification accuracy discrepancy appropriate leverage solution similarity become thus point bayesian continuous series htb discrepancy parameter datum circle easily distinguished rule indicate area distinguish task solve chance chance performance classification parameter chance accuracy trivially obtain employ yield probable unknown analysis regularize combination material method approximation parametric kind size move see material far set average lag coefficient quantity independent affect correlation point single bernoulli bayes generate identify average exception lda reason sensitive correlation bayes outside learn overfitte actually fold fold observe datum simulate set measure union typically call example pair validate accuracy q use discrepancy htb learn bayes rule sample size accuracy curve indistinguishable svm max perform bayes classification assume true distribution discriminant lda exception move estimation interested simulate minimize next assume parameter condition give size motivate data parametrization compact follow consistency frequency law learn classification accuracy bayes h n generate classification mean set must equally fast condition bayes condition condition solve available good rule suggest abc comprise several posterior major simulated show discrepancy abc materials algorithm empirical density function count gaussian show contain red curve approximated numerical integration supplementary match supplementary regard time tend fast max evolution appendix analysis report iteration datum series occur histogram scatter plot abc deviation posterior posterior infer binary count series result abc pdfs bivariate scatter obtain result contour autoregressive indicate stochastic epidemic center observe matrix day care indicate particular infect day care infect multiple perform statistic hand appendix brief hence classifier abc transform reflect nature well standard deviation subset material applicability summary assess reveal infer epidemic expert average generation lda second second spend epidemic infer result four classifier similar expert simulate classifier appendix datum drive compatibility abc discrepancy able incorporate expert covariate classification expert abc expert insufficient expert statistic epidemic simulate datum dash curve pdfs histograms abc expert black plus marker circle marker use generate free methodology limit classification problem whenever classify offer reveal free appear actually connect assess knowledge problem incorporate specify lead consecutive application choice automatically propose allow even random inference infect contain relate property rank fraction variability subset choose contain one size row subset dimension extract matrix run abc random rd wish thank computer epidemic model centre coin author contribution research rd rd research write text attain parameter example variance series parameter mark contour panel minimal equation unknown variance compute mean squared plotted sample decay square size line circle outcome dash bernoulli report degenerate standard lda omit since decay estimator property material provide main posterior accord abc monte univariate empirical pdf red pdf dash scatter reference contour solid contour red dash binary infer count infer poisson posterior infer posterior series distribution lag zero move infer monte carlo together iteratively prior movie process max rule bernoulli mp mp mp mp mp gauss mp mp mp mp gauss var mp mp mp mp mp mp mp mp quantitative relative deviation infer posterior distribution comparison integration supplementary method relative rule abc posterior occur plot posterior spread posterior error deviation surprising estimate variance twice estimate quantitative curve deviation average detailed material provide investigate applicability lda always since consume applicability point two keep value eliminate seed simulate seed show subset row diagram bottom discriminate localize run abc usual seed knowledge black point marker qualitatively even tail result infer infer posterior still shrink simulation iteration pdfs abc concentrated expert abc projection diagram random applicability lda time mean cross mark location produce localize region l st rd posterior evolution pdfs scale histogram solution expert knowledge code blue circle red abc subset result abc sample datum fourth generation posterior matlab default kernel bandwidth subset circle simulate generation visualization generation random circle yield square pdfs posterior qualitatively blue subset red expert black marker show compare fourth generation classifier concentrated abc red qualitatively shift abc subset work real st nd rd pdf evolution pdfs scale black expert classifier abc datum fourth bandwidth posterior qualitatively similar
diversity start eqn measurement representation survival share enable eq conditional multivariate facilitate monte data augmentation eq density posterior distribution weakly mean tn posterior avoid affected individual jeffreys half cauchy nuisance seem jeffreys prior implicitly complete whereas last undesirable scaling prior coefficient seem flat introduce weakly informative jeffreys prior hyperparameter due notion logit implicitly scale follow demonstrate small show latent identify autocorrelation accurately behind true c sim sim sim b affect basic covariance simple comparison useful progress hand martingale always reflect gain covariance association response strength eqn assess interference add vector exhibit presence start noise yet association remain magnitude reach model robust detect association response signal true conduct sensitivity survival figure large curve support well favorable treat ht record censor performance table bias adequate future besides ar capture lastly sensitivity roc clear traditional logistic nonparametric consideration also sensitivity aim subject forecasting traditional method notably use automatic widely learn spatial hierarchy process describe overall capture fine variation hierarchical conceptually goal combine forecasting incorporate longitudinal time ar complex structure possible restriction definite magnitude joint fluctuation share view time perturbation increase specificity bayesian area dimension efficient especially mixture process hierarchical instead mixture scale condition shrinkage coupling noise enable estimation extension improvement longitudinal exhibit distinct hierarchy gaussian mean hazard away skewed generalize may incorporate direction improve acknowledgement development program number partial grateful foundation patient comment h se se age infection bc infection infection diabete covariate white infection infection bc infection infection cf diabetes pt pt novel forecasting combine nonlinear past error minimize survival event hazard bayesian objective shrinkage detection accuracy forecasting function key longitudinal forecasting need clinical without researcher property fluctuation development study longitudinal past help prediction latter provide vary behind spline limitation allocation knot trace difficult avoid bar address quite smooth knot keep fast robustness form increment differentiable major progress krige predictor successful magnitude usually measure interpolation inside monotonically prediction improve forecasting gaussian process longitudinal describe batch spline batch mean individual approach greatly forecast longitudinal collect survival commonly adopt inference longitudinal song recent development survival online recurrence taylor nevertheless remain joint baseline full recurrent survival function cox eqn accommodate event collect forecast multiple correspond adopt discrete cox relative cox prediction tractable describe late approach process longitudinal survival longitudinal enable share trend among capture deviation survival smooth baseline serve share set nonlinear population autocorrelation straightforward completely fully maximization remainder survival section simulation performance apply patient remark assume process notation worth multiple independent copy copy word process subject span longitudinal censoring begin individual share correlation covariance generate example exponential replace role knot avoid ar resemble forecast obtain conditioning eqn difference process benefit illustrate figure generate subject remove ar line adequate autocorrelation cause heterogeneity fit subject common prediction prediction trajectory subject figure slow result improve h b covariance use eqn recursive forecast series analysis estimator forecast cox cox widely adopt
space integer isometry rip rank rip rank approximate constant constant desire economic leave reader pursuit calculate singular iterative practice obtain retain rate singular pair iteration tolerance matrix pursuit achieve problem denote mp mp obtain closed form solution formulation k property get complete algorithm mp completion compete algorithm singular projection fast fitting accelerated boost atomic minimum greedy three solve low problem comparison coordinate method include boost method available online http www edu thresholding http stanford edu http www stanford software http boost boost http boost greedy component http www cs il image collaborative problem compete matlab external package system intel ghz g ram follow compete recommend candidate validation mp mp stopping criterion ground result pursuit mp mp pursuit fr mp evaluation conduct mp pursuit mp mp computational inexact experiment iteration singular pair iteration curve mp preserve practice run away especially mp c mp couple crowd couple crowd exclude pixel guarantee rank mp stop iteration control list numerical couple much use need high fail htp boost mp boost mp mp matrix include dataset collect recommendation anonymous rating user rating range collect website contain anonymous rating score step size range rating rating algorithm value six dataset time mp fast among compete satisfactory scalable matrix storage rank computationally satisfactory advantage easy understand scale rigorously linear rate compare art completion compete generalize function mp weight one k inverse property remark scalable pursuit economic introduce weight updating reduce time storage complexity satisfactory advantage easy learning rigorously version significantly well real world netflix compete achieve low attract significant machine mining reference form value approximate span vanish outside matrix attack cf widely relaxation rank trace generality sorted base singular al develop algorithm penalize ji improve number iteration accurate singular propose investigate property addition trace low rank computationally expensive singular svd truncate svd large accurately attention recent solve wu question solve progress ik trace problem et improve efficiency although scalable singular efficiently iteration algorithm et solve problem reduce without theoretical involve compute lie refine apply frank size refinement information slow update constrain optimize solve consuming sophisticated refinement lead computational cost algorithm type weight lead boost learn alternate drawback inefficient slow rate refinement iteration heuristic affect speed improper even extend orthogonal omp residual standard update matrix consume algorithm operation appeal propose orthogonal pursuit orthogonal sub extensive rate iteration achieve drawback store rank current weight update storage tackle economic version iteration restrict observation keep economic retain convergence knowledge fast among verify empirically problem netflix main contribution paper computationally orthogonal matching pursuit theoretically accuracy solution need singular pair storage economic linear rate version storage scale matrix version rank converge operator onto span outside equal zero observe keep inner matrix equal frobenius economic linear extend sensing present isometry property empirical evaluation verify effectiveness matrix unit choice original approximation zero eq nonzero solve orthogonal pursuit algorithm coordinate achieve unit current observe follow notice rank unit frobenius product see reformulate eq singular computed basis matrix available least vector row entry multiplication simplify incremental implementation desire stop terminate method rank prefer alternatively residual orthogonal matrix mp stop pair right residual output pursuit construct onto subspace obtain current lee et step first choose svd use svd rank expensive difference omp recovery property assumption I e involve isometry convergent achieve orthogonal one proving say one independent formula stop suppose full statement indeed conclusion induction hypothesis column follow fact rank next relationship convenience define view k l r property fact uk observe eq last triangular inequality easily conclude equal index noisy mp help remove study orthogonal matching omp right singular matrix well mp basis save storage adapt need weight matrix square weight basis ki procedure simplify
locate view view anomaly single view machine anomalous develop view anomaly detection cc dd view anomaly detection application behavior aggregation database information management multiple view language wikipedia view anomaly find document language item anomaly movie movie probabilistic latent anomaly latent share anomalous generate use anomalous assume latent latent view space share view projection view view anomaly consistent projection infer give propose step use collapse step matrix map estimate likelihood iterate vector use instance robust assume latent suffer anomaly contrast anomaly latent matrix properly anomalie security analysis exist anomaly detection two view horizontal anomaly view simultaneously low embed close together spectral location view hyperparameter constraint label anomalous sensitive label extend principled handling combine multi method anomaly filtering instance corrupt view anomaly instance different foreground view instance inconsistent cluster propose probabilistic canonical cca view anomaly share view maximize minimize reconstruction anomaly score non anomalous anomaly parameter anomaly factorize assume every shared instance anomaly detect anomalous latent anomaly anomaly anomaly view observation th observation vector view anomalous inconsistent across multiple variable instance generate view projection select latent anomalous view consistent assignment view anomaly view inconsistent propose infinite view th mixture weight denote dirichlet prior weight use automatically infer instance generative propose precision draw mixture nj assignment observation nd stick break weight precision share integrate indicate latent analytically integrate vector precision multinomial prior factor view integrate calculate q q q either principal view different every view view spherical describe collapse latent analytically integrate weight integrate need notational current exclude th exist latent indicator subscript indicate latent eq intuitively latent view inconsistent view projection logarithm maximize quasi iterate e assignment view latent assignment matrix score use use sample inference iteration cross dimensionality space cross validation predict randomly anomaly evaluate create anomaly anomaly consensus anomaly model latent anomaly score base reconstruction require value control whereby different view embed together cluster view cluster score remove representative anomaly anomaly multiple kernel gibbs sampling area auc anomaly average propose performance datum anomaly infer latent instance view anomaly auc detection instance anomalous anomaly instance inconsistent inconsistent view need datum view case view propose anomaly auc vector dimensionality ccc breast diabetes heart breast diabetes anomaly vector anomaly anomalous anomaly multi anomalous non anomalous respectively observe latent three anomaly average different anomaly anomaly auc class anomaly lead anomaly propose propose imputation multi view anomaly estimate average feature instance randomly imputation whose datum effectiveness propose anomaly breast cancer diabete propose evaluate propose model view anomalous anomaly latent dimensionality generate mean vector show error imputation synthetic decrease rapidly anomaly indicate dimensionality square anomaly anomaly anomaly show model carlo latent high rate anomaly anomaly rating movie view represent movie user second movie
roc replace loss pc optimization could motivate manifold equip roc arbitrarily wise tv tv generalize universal choose deal outlier type basis roc differ way introduction reveal outlier inherently cut residual choose pattern roc pca loss asymptotic bad study point bias trade tune non quadratic optimization roc pca easier expensive utilize loss sparse shift outli loss bernoulli hinge visualization sometimes pc order importance pure simply complete robust roc benefit free robust reduce roc roc pc yield roc pca repeatedly roc challenge orthogonality constraint alternate manifold iterative nonlinear reduce way instead treat introduce lagrangian multiplier view problem manifold manifold orthogonality update tangent p manifold begin riemannian f difficult show valid trial determine trial due lie pd lemma fast formula turn batch low proper size bb search result quick nonlinear scheme backtrack aware trial point bb solve f f solution behavior bb ensure stepsize small criterion recent recommend important start run convergence type roc ps involve sparsity induce algorithmic solve denote ds thresholding let satisfy justification continuity uniqueness various thresholding cover penalty couple general summary complete roc pca e roc output line bb monotone k bb else see k wants directly form roc similarly e roc unless compare number sensitive subspace recovery long supplement ridge grid simply say constrain roc share fortunately estimator subproblem thresholde q constrain pca problem gs gs monotone integer decrease empirically pca great establish difficult asymptotic wish robust challenge modern tool approximation roc pca r supplement ignore intercept outlier formulate eq ambiguity always evaluate assumption globally oracle inequality p q hold gaussian include dependency expectation high probability obtain see supplementary detail nature apply incoherence commonly assume sharp infimum en minimax conclusion sparse contain reduce attain roc outlier great give estimator p furthermore show essentially ij p rp multiplicative therefore roc pca essentially tend real life roc sparsity algorithm dimension review material burden fail principle widely perform svd reduction outlier occur dramatically long value exceed true outlier outlier relatively greatly course effect removed estimation roc well slightly acceptable robust seem balance c c c focus subspace roc pca plain pca review spherical pca already integrate inexact augment lagrange multiplier test presence outlier class approximation concern subspace estimation conservative h c c outlier skew direction handle outlier behave level occur roc pca behave extremely perfectly pc except give extremely pc affinity observe dataset due sampling trial fall chance computational report table roc procedure computational robust great outli affinity run although roc pca run superior especially c roc performance type fail c give three randomly entry otherwise outli pca good job affinity set careful result check subspace propagation much try ij section implementation take threshold pc affinity increase roc pca sample way four outli batch try follow pca standard roc computational roc roc pca segmentation collected extract seven hand window contrast adjacent pixel outlier assess goodness call adjusted account top mark first run principal seem panel relatively change yield intuition also plot e observation find class roc automatic intervention roc pca segmentation among detect scatter help separate majority pc roc clean detection outli pc pcs pca pca l sensitive may reveal observation mathematically rise roc pca come guarantee robust computation regularization reveal roc pca minimax topic include optimization high direction jointly investigate observation anomaly independent outlier skew subspace let wise roc pca fix remain minimizer manifold canonical necessarily however detail omit proof universal constant occurrence space stand onto abuse notation dimension j j p term general sub gaussian dimensional random bound bernoulli assume mean r cs submatrix r r j p p j lemma follow let p dd c follow satisfy np sr sr q clearly b q applying row follow cf exist b universal model kullback ip mn apply rr pn r cn r line omit advantage instead repeatedly rank loading vector lemma recently attract attention computer study apparent original novel orthogonal complement analysis pca combine enforce rank element guarantee roc tackle basis optimization iterative efficiency real supplementary big arise pose challenge computation structure tool pca characterize rank solution value tr principal function know call life statistical inference break result subspace estimate pca extensively robust statistic lot researcher statistic outlier norm zero facilitate wise variant application video e outlier original call major htbp analysis attention affect subspace toy outlier identification short interestingly subspace outlier though possibly pc handle later stage raw coordinate space panel sample point whether show outlier skew pc unfortunately check coordinate offer help reveal propose complement roc recovery exist approach roc pca aim involve enforce establish roc review pca robust section thresholding roc pca roc pca extensive numerical datum conclude detail issue pca notice statistic introduction five covariance method limitation review representative work complete pca robust loading eigen dispersion monotone function gain large robust give repeatedly estimation run largely keep value cost matrix dimensional matrix accommodate portion observation contradict robust achieved simultaneously identify worth pc estimating unnecessary repeatedly direction project obtain idea systematically pursuit algorithm computer c direction center recommend make reduction svd reduction text unfortunately serious time new pc projection estimation apply run fast class un moderately high trial rely may acceptable application first deal element outlier observe dimensionality subset direction pass two datum point far reweighte share property excellent simulation yet suffer aforementioned drawback robust handle wise anomaly may efficient svd may problematic lie restriction difficult accurately trial direction subspace contrast purpose direction direction fail spherical elliptical pca relatively center onto plain elliptical spherical preferable recover simultaneously spirit step toy example intersection angle pc however spherical direction bad pca
result obtain pa query complexity far uniform condition achieve condition super speedup open stochastic complicated circumstance appeal evolutionary include genetic algorithm evolutionary cover nature inspire heuristic estimation etc study rapidly decade analysis development theoretically investigate combinatorial measure develop cover budget analysis case general even application nearly general give conclusion domain run exponential bound general derive performance learn analysis notice share consist cycle building commonly reproduce solution quality guide sampling portion capture framework simulate sampling strategy evaluate probable pa pa count fitness reach solution close pa upper bound version pa incorporate compare polynomially search polynomially far allow super polynomial notice necessary version iv search paper euclidean bound close continuous exist sake convenience generality besides mean polynomial grow minimization continuous compact assume loss domain implement every achieve good enough quite correspond two pa fitness evaluation take reach quality reflect intuitive evaluation probable approximate pa call find em generate solution new short global heuristic solution framework start record far solution search cycle learn hypothesis mapping via iteration empty solution well space balanced region balance uniform tt ts tt ti tx x fx note concrete summary stage accurate could explanation rigorous illustration various genetic deal solution vocabulary element mutation select vocabulary operation mutation mx change commonly otherwise solution search x simulate ga ga circle area hx mx mx use approximate polynomially way behavior ga simplify ga probabilistic argue base optimization entropy unify building framework correspond sampling step particle particularly simulation perhaps sophisticated set particle velocity determine current velocity globally good particle initial distribution velocity contain globally learn utilize set hypothesis globally capture search leave heuristic general output eq xx sampling hypothesis need size implement sample naturally bind iteration solution sample event whole failure sampling learn expand overall failure belong tm furth term simplify employ stage transform set current putting priori knowledge performance meanwhile small performance usually baseline serve search word uniform search pa query much average rely learn investigate accelerate search approximation event event condition expect error rate numerator second equality error average learn uniform within optimistic plug u kl u lemma ignore logarithmic find exponentially use bad search ask sample worst keep query meanwhile n super optimistic note obtain face barrier search still sphere sphere volume c f n straightforward pa probability search meanwhile consistent feasible note td ti td td tx x x prove obtain pa approximation query use sphere volume algorithm simplicity error obtain letting algorithms iteration eq n accelerate uniform close acceleration complex sphere algorithms optima show spike spike spike differentiable optima pa algorithm search sphere cover sample label dimension cover thus member pa proposition approximation achieve query error follow uniform algorithm every query complexity proof error convexity severe like spike function significantly affect still show algorithm super polynomially improve question therefore proof way powerful improve I super error circumstance modification nevertheless achieve learn proposition super acceleration sphere complexity choose use sphere affect error require tm obtain letting meanwhile explore require powerful condition imply error inside cost sensitive mis algorithm note label negative false negative error side control refine eq since algorithm behavior result use complexity probability sphere result affect iteration want produce
refer probability refer unseen datum type iii gaussian mixture assign calculation error know select method convergence normalize marginal likelihood direct relation asymptotic expansion marginal allow error resolution asymptotic study likelihood variable probability probability bayesian observable analyze kullback generating analyze selection regular expansion maximum regular case method case derive type iii regular analysis estimation expansion reveal order iii regular result determine advantageous estimation type remainder organize estimation result estimation formulate latent variable express learn hierarchical true express refer maximum estimator respectively maximum unseen posterior posterior normalize true include define latent type estimation define bayes estimation q likelihood eq estimation kullback true error expectation ii error eq present publish expansion estimation method expansion estimation difference information converge variable true two refer since provide label converge analysis label avoid analyze prove expansion notation ii indicate respectively corollary compare I method denote nn saddle taylor obtain average last eq form eq eq base rewritten let confirm relation prove find asymptotically advantageous type estimation error bayes accurate estimation advantageous estimating latent observable estimation bayes method variant type ii target constant integer q q numerator denominator type ii let observable let target q panel show type maximum likelihood respectively advantageous lemma type function follow bayes type iii follow advantageous target previous subsection advantageous estimation follow prediction observable estimation likelihood estimation let give leave panel target target show obtain error bayes estimate observable expansion parameter proof obtain relation estimation use bayes numerator maximum accurate mathematically explain comparing find I accurate prediction variable write iii rewrite eq formal form type ignore likelihood method estimation bayes kullback decompose thus target term variable estimation target confirm accuracy method present accuracy latent type iii variable estimation indicate equivalent advantageous estimation type iii research foundation equation omit equation corollary theorem datum hierarchical often kind part measure investigate latent asymptotically maximum bayes variable estimation regularity satisfied indicate accuracy equivalent advantageous estimation keyword machine learn datum science represent underlying example observable consider unsupervised observable component unknown often way likelihood method
g semi separability important computational matrix operation energy separable term particular notation hamiltonian h well energy energy similarly h energy energy term key exploit separability describe invariant alternate simulate hamiltonian dynamic crucially though separable return per step discretization correction simulate preserve need mh correction preserve reversible reversible see reversible transformation emphasize discretized semi hamiltonian start joint hamiltonian point discretize small seem actually omit hmc important auxiliary hamiltonian one step auxiliary accelerate see word auxiliary potential hessian hessian approximate hessian hessian distribution rough main difficulty correlation bottleneck hmc target computational cost iteration beneficial cause bottleneck gradient aforementioned curvature strongly give nearly mix separable ess mse hyperparameter notice efficient level adaptive illustrate hmc hmc suffer hyperparameter narrow contrast hyperparameter variation soft effective per ess mix neither hierarchical hyperparameter ij th dataset logistic benchmark hmc group dataset hyperparameter gibbs fisher use I g much high stochastic ar observation px update sample approximately almost many ess another benchmark prior function j di mx j posterior update iteration general step update update extremely slowly sample latent hyperparameter consistent method effective hour tb tb ess min ess min ess latent hyperparameter ess ess hour version riemannian manifold hamiltonian monte retain flexibility difficult several model outperform hmc computation hmc minus plus pt pt minus pt plus minus pt hierarchical hamiltonian advantage introduce use design allow hamiltonian exploit mix fast simple sampling instance natural complexity control overfitte complicated allow pooling dataset allow little chain hamiltonian hmc hmc note riemannian hamiltonian aim exploit property computationally problem simplify decompose allow large move hyperparameter term previous sampler practical datum datum distribution ii draw hyperparameter group allow statistical strength cause hyperparameter usually control cause difficulty sampler illustrative distribution nx illustrate generate hmc ergodic invariant auxiliary gaussian three step simulate hamiltonian metropolis mh correction define h hamiltonian dynamic physics decompose discretize discretized identity often outperform popular slowly correlation distribution mix act recent varie position result call hamiltonian al fisher manifold hamiltonian dynamic long non separable system hmc simulate hmc hmc within gibbs hmc mix slowly huge variation easily separable pose challenge hyperparameter jointly joint
selection generalization experiment synthetic previously high dimensionality pose computational challenge fortunately expect sparse small feature weight classification classifier phase solve elastic doubly elastic regularization perform feature selection grouping correlation microarray expression fmri algorithm take terminate hybrid algorithm proceed phase use svm average second classical add regularize regression elastic net select e grouping enforce attractive text hierarchical regularization benefit svm history back inexact lagrangian function equivalent constrain call variable splitting iteration augment lagrangian lagrange multiplier converge optimal subproblem minimize augment view subproblem solve minimize split admm auxiliary objective optimize sequentially iteration follow update lagrange multipliers original decompose subproblem hinge loss soft thresholding lack associate interior method class qp need define c iy consider kkt svm svm optimal solution index vector term compute give svm likely dense whereas svm identity primal large argue svm p identify realize implement qp mind objective admm illustrative non converge approximately plot early remain offer sign terminate adopt relative surrogate evolution index warm admm solve decision although reformulate qp norm already enforce reduce helpful form comprehensive unknown population result unseen arise abundance feature produce accuracy likely identify domain knowledge feature include relevant feature explore associate information equivalent qp implication show utilize homogeneous implication represent add constraint p number linear constraint feature classification want undesirable increase convex qp hence incorporation ready novel combination combined framework motivation exploit good knowledge kind elastic net svm incorporation admm method unconstrained inequality hinge n tw norm obtain p p minimize individually constraint augment involve solve q ty ty ty iteration fact minimize augment lagrangian convex crucial efficiency novel introduce slack part augment subproblem system q kb full reasonable constraint solve cholesky factorization observe element constraint k p k yx subproblem subproblem solution lack summarize admm appear additional parameter four admm svm practice one computational experience fairly insensitive phase solve base decide primal dual reveal formulation slack transform equality qp formally experience knowledge counterpart e admm admm k test nine publicly two class article auto real seven set subset sample instance test instance profile breast cancer cancer fmri brain activity except fmri website summarize experiment clearly produce test significance time outperform size competitive able original c c support cpu cpu synthetic presented begin dimensional specifically variate matrix diagonal else training contain four entry sample generate block two close entire population test know value block membership l denote information precise confident negative mean exceed distribution domain tends often exact two train train feature want achieve explain expert knowledge hyperplane generalize entire significance accuracy support cpu phase solve extension solve demonstrate advantage prediction general enough classification cc x tx x ty ty ty k yx k b k k b k n yx yx k
invoke primal lp dual eq probability since return rewrite gs uncertainty generalize note cone satisfy either give hence empty interior pick enough hence equal counterpart measure conclude present payoff shall proposition regard one round game risk neutral operator rewrite primal equivalent redundant gs gs immediately trivially round amount return lemma proposition game uncertainty g p f unique neutral value arbitrary arbitrary statement linearity expectation write give q price prove statement obvious induction straightforward gs set u lemma round track backward exponentially dynamic programming computation option corollary uncertainty binomial go number interval decrease imply uncertainty black convergence price european option round game lipschitz continuous european option geometric drift positive motion recover price convergence condition guarantee keep corollary upper tp term fr weak triangular hold lastly condition f gs gs r integrable end f simple game equilibrium general round concave uncertainty u gs gs concavity preserve induction get bound follow analogous initial price payoff hand measure convex payoff option game concave gs te prove backward obviously contradiction existence immediately next u continuous neutral contradiction exist say exactly bind three satisfie impossible let move infeasible infeasible infeasible imply finite pp trivial see bind uniquely define exactly bind constraint lp infeasible contradiction may lp compute either furthermore focus follow length exist ir ir g tx program lp solved analytically let decide consider similarly continue get mx r r r r tx small g r r lx lx lx use r lx r lx tx lx lx l european options convex induction corollary neutral measure th second convex conclude induction p tx r induction define backward induction note proposition I I rewrite let x integrable x gs uniformly integrable provide price black model equation pde characterize demonstrate pde rigorously pde model bellman informally denote use control assume moment xu x follow pde pde sense coincide solution verification pde optimal control application eq denote expectation satisfy denote take control black write boundary reduce ordinary pde whose lie theorem payoff natural allow movement community jump motion asset adopt adversary jump control jump extend adversary magnitude jump price payoff adversary jump assume adversary small modification present need model function perform occur move return freedom choose sufficiently small sequel single round asset formulation q write measure concentrate enough w w round analogously dynamic replace jump diffusion study demonstrate appropriate return consider jump option price jump diffusion poisson adversary jump throughout round adversary jump uncertainty set ordinary assume polynomial require upper next choose lp formulation round similar neutral characterize feature neutral measure comprise ordinary uncertainty whether jump dual occurrence whether reduce merely dynamic define approximation scheme length backward induction polynomial achieve pay run enumeration set adapt easily highlight first manner continuous binding reveal tx tx xt convention similar use probabilistic arrive section prove uncertainty payoff subset problem integer count integer equal reduction proceed count sum european option uncertainty payoff ordinary neutral round price factor move let measure coupling tree trajectory round next express price equation different let corresponding price similarly reduction lower justify necessity price single round query payoff monotonic lipschitz difference price word explain purpose least point easily relax j u dual neutral complete argument section property invariance proposition definition liu department science pa address address need financial option game adversary price finance european payoff option construct pricing demonstrate introduction artificial neutral measure finance extension incorporate jump payoff price type option limit european american type substantially explicit trading replicate option american design option payoff regression non pricing degenerate dynamic contract asset european call contract stock york stock exchange price european call option stock price price stock abuse price stock convenience exceed rational exercise stock immediately market payoff single european option payoff asset depend contract american option exercise price option core economic market practitioner trading area discovery expand order option movement asset european option choose portfolio consist asset portfolio portfolio must european otherwise positive risk fair option original price underlie asset brownian may unique example stochastic option pricing control movement stock motivation probabilistic description always market see capability broadly speak systematic context answer stochastic finance community question question option european payoff acknowledge literature limited example et al et convergence european adversary convex payoff line limited option al consider option european american wide model price movement non imagine show case secondly non convexity american pricing algorithm major contribution whose lipschitz involve construction artificial measure error neutral commonly use kind appear structural american option pricing thought adversary allow online contribution constructive pricing analyze option payoff monotonic nature artificial measure way deterministic extend convergence american whose pricing thought adversary online besides contribution extension include convex black option adapt algorithmic rare jump financial market important smooth price movement financial market sort pricing work chen formulate pricing robust asset dimensional typically theorem assume computational feasibility issue al recently price constraint variation asset et asset adversarial maximize gain round price movement european option price black price geometric round infinity call market study leave replicate option pricing unless american option payoff call option option payoff algorithmic show major market cm cm cm p european yes hard new european yes american jump yes yes function result pricing general discuss price jump generalize american option present hardness section model equilibrium convex pricing payoff address generalization extension american option jump present hardness discuss spirit discrete chance specifically consider option day transaction option total soon decide value game limit cost market always asset market price asset round round freedom return notational convenience begin asset position impose capital shall soon describe option price illustrate interest result zero rate option game worth asset payoff gs option get option rational adversary outcome gs option price strictly gs option give gain arise time obtain asset th gs argue option price interpretation fall strategy adversarial payoff strictly price option option adversary payoff positive price option trade adversarial strictly wrong option trading strategy adversary payoff access payoff leave set later allow dependent simplicity highlight merely equilibrium contribution come center start give shall adversary scenario characterization general low merely objective analyze equilibrium game induction benefit evident linear allow american options european american option limit derive guide borel gs maximization probability satisfy neutral widely contain interval cover risk construct minimax rp see detailed proof despite result case payoff convex uncertainty able characterize hand risk neutral generalize result game coincide suitably complexity extend option analyze pricing algorithm result correspond price geometric rather present payoff function omit end full payoff decrease use fairly allow adversary arbitrary allow discrete multinomial recursion th round let b adversary lp option price discretization backward induction stochastic area financial latter regard adversary choose way move long round track round price could multinomial exponential overall due discretization grow algorithm portion multinomial additive recursion interpretation formulation elaborate resemble mesh option notably american asset movement risk neutral mesh backward monte replace lp explain multinomial tree price give formalize building block continuity prove suppose decrease lipschitz building follow assume view hybrid discretization characterization bind lipschitz satisfie rely neutral measure several backward induction measure correspondence artificial martingale asset movement see effective propagation first error second come future handle second characterization proposition share write become recursively asset expand martingale bind thus arbitrary multinomial price bind shall appendix long run essentially tight set multinomial uncertainty though parameter remain unchanged still generalize price american include neutral algorithmic european single round upper american option make adversary exercise otherwise remark adversary right exercise upper upper argument nature hence early exercise gs gs focus gs gs gs r gs single depict significance characterize solution term neutral european option maximization expression characterization round consider american option option european counterpart exercise detail appendix use payoff recursion
pair event use denote query point belong range strictly r write use put everything follow hand finish proof get suppose sample use condition depend return begin note continue argue occurrence choose equal one function least return ds free get suppose uniformly random mean exist exercise second claim argue check orthonormal written follow I lemma corollary proposition support grant science united science foundation grant center program center foundation grant recognize limitation size training example cost mostly lead non trivial loss parameter desire case kernel learn might consider label domain kernel minimize possibly reproduce hilbert vector machine hinge hilbert space employ hard polynomial efficiently insight span reduce define q convex polynomial implicitly predictor hilbert size lead learning attempt know algorithm fall specific dominant learn entry matrix attempt number require read instead full map kernel see technique exist question surprisingly knowledge learning example kernel maintain learn always pay finite match study give type matrix generally assume much small number low make method previously although assume conclusion informally impossible make kernel learn suppose evaluation sub method uniformly away away datum substantially rank low impact go kernel predictor nature particular evaluation constraint soft regularization attain corollary loss attain although identify evaluation budget hinge loss square unless ridge regularization attain role loss recognize key role discuss section appear loss highlight kernel recognize relate sparse nystr om row budget perceptron sequential early algebraic rank work kernel work access interested predictor learn study complexity e support focus organization organize class constrain term consider constraint regularization without discuss different type bound consider conclude open appendix utilize essentially permutation formally block entry column immediate hence diagonal mean unit ball hilbert focus generic hardness still induce quantify follow permutation linear also approximately kernel close inspection truly distant fulfil fulfil presentation boolean although proof contain technical intuition approximate suitable approximation relatively budget detect block integer return approach reduce evaluation need standard require learn demonstrate absolute average equal parameter coefficient require subsection satisfy bind parameter size budget different way phrase find formally exist universal kernel return vector give budget fast set uniformly replacement get j j sample set vector coefficient essentially original drawing enough solve guarantee loss error match corollary number away moreover small degradation ask study use reduce main interested draw corollary negative parameter exist target appear roughly change optimum thus seem domain matrix find eq partially get thus optimum regardless show loss learn trivial examine detail get corollary absolute loss exist universal constant target lower verify bind theorem former match sub sample minimizer convex note unlike corollary whether technique corollary move change regularization loss location may hinge loss location universal verify long assume get dd quantify constant error establishe range without since strongly scale lower emphasize attain hinge regime consequence budget norm want algorithmic require attain hinge pick corollary different evidence hinge natural consider smooth differentiable square universal target low attain absolute essentially kernel evaluation sub error learn budget portion readily verify lead lower consider pick mean equivalently b latter former complete discuss explain early nystr om approximation feature typically depend result low low ridge soft ridge operate dm rank exist return sub require sample scheme tight theoretic focus use various loss conclusion case attain well trivial small optimistic bound substantially although know weak low tight exploiting result stochastic believe applicable optimize risk respect underlie obstacle basic subsection close sample instance training sample question tight know rank extend possible extend query respect randomization loss discover research program machine centre de support consider define certain uniformly begin equal diagonal block one compose block correspond sized block proof intuition entry block randomly algorithm
paradigm likelihood enjoy mle intractable common heuristic unfortunately iterative minima optimum achieve common alternative heuristic minus optimization feasible take extra lie round find often focus namely program sdp pose orthogonal cloud orthogonal ia gaussian e mle minimizer mild orthogonal refer ta unfortunately exponential intractable note c dc semidefinite relaxation drop outli recovery nature infinitely impossible even remarkably even often recover mle camera motion understanding hold distribution analyze lie fact mle dual carry various plot correspond reference alignment treat certain observe recovery formulate support conjecture ia gaussian entry solution alignment signal observe copy sake reader happen level mle considerably positivity also recovery vanish
art ap contour visualization work contour detection fine technique effectively overcome limitation sensitive pixel require local interesting direction examine contour acknowledgment go edu contour detection classification facilitate efficient cnns contour challenge per per base patch contour effectiveness performance fundamental vision recognition explore intensity texture take structured forest efficiency boost effort pixel perform contour construct pixel learner adopt convolutional cnns approach subtle deviation cnns feature image distinction perspective pre per image cnn imagenet adapt new model pixel edge classification convolutional ensemble contour follow contour section detailed technique research effort achieve survey present picture progress two area emphasis early contour detection image amongst orthogonal contour detailed discussion subsequent identify increase apart detect local technique notable address independently patch analyze combination mid contour sketch token forest classifier pointwise mutual contour clean contour current contour deep architecture encode contour restrict machine neural fine mechanism adapt imagenet pre convolutional produce detection benchmark cnns cnns cnns feature exhibit hierarchical perhaps implementation cnns generic computer vision problem cnn image cnns cnns cnns sliding cnn deep art cnns recognition demand contour detection exploit contour explore per pixel independently generate cnn feature employ architecture effective technique focus yield image patch design appropriate investigate characteristic detection point convenient multiscale contour detection feed machine classifier multiscale pyramid extraction neural extract per pixel feature feature convolutional conv convolutional conv stack convolutional pixel convolutional level pixel illustrate pixel pixel contour feed include local contour structure well distinguished neighboring pixel test correspond image patch conv add softmax edge image train cnn conv cnn except plane pixel edge conv properly contour propagation label patch training edge patch relatively alone usually still address edge evident certain distinguish learn database softmax cost cnn prediction layer bias edge reduce log fine sensitive fine rather directly back convenient strategy create bias positive twice non vice versa cost sensitive fine ap train baseline traditional pixel tune tune tune ap conv conv conv conv imagenet negative fine tune heuristic branch bind coefficient capture aspect fusion various learn test berkeley assess tuning technique detection competitive also demonstrate de contour detection contain validation image worker average contour f threshold precision ap non tuning server gpu set imagenet pre softmax ten modification pixel tuning finish pixel require fine fine tuning pixel train positive fine boundary per negative various tuning conv carry classification softmax observe fine experiment pre fine architecture traditional image fine tuning pixel fine tuning improve sensitive fine fine negative possible boundary boundary
mmd valid unbiased mmd basis function size method repeat mmd permutation consistent test htbp mmd mmd deviation approximate unbiased mmd mmd correct evaluate efficiency varied fig number varied compete except method mmd gb ram gradually become efficient mmd exact mmd extreme subsampling mmd increase slowly trend validate efficiency vision contain object set toolbox extract construct bag word pyramid codebook spatial pyramid feature image mmd discrepancy data pc cpu ram compare mmd mmd repeat standard execution meanwhile speedup linear mmd mmd mmd std second speedup mmd several strategy employ strategy utilize integrate task distribution describe bandwidth mmd bandwidth approximately match scale variance distribution method mmd level bootstrap type error drop quickly demonstrate empirically bias mmd eigenvalue vary four exact gram matrix two former latter utilize execution comparable efficient except mmd method bootstrap mmd determine statistic base calculate invariant kernel advantage method theoretical explanation one side intrinsic mmd mechanism side metric include find threshold thank valuable anonymous constructive comment support national china program cb china project china mmd unbiased mmd reformulate proof equivalent tool mmd linear df distribution bounded f j claim substitution side function fx minimum iii finally ga ga verify fourier use mmd procedure mmd equivalent geometric explanation mmd mmd calculate fourier pi I lk eqn eqn convergence prove mmd virtue certain eq uniform theorem dr jk net hoeffding net ii accord literature eq claim reformulate combine mmd unbiased mmd pdfs random maximum frequency accord hold cm circular discrepancy institute usa university china discrepancy mmd abstract maximum discrepancy mmd recently statistic sample quadratic however scale accelerate mmd calculation mmd take sampling fouri time mmd approximate determine accuracy convergence theoretically unbiased namely circular understand mmd extensive metric assess experimental mmd fast mmd test range use accept reject hypothesis since unknown mmd design measure distribution embed reproduce kernel hilbert recent test successful application biological datum attribute mmd family supremum kernel selection albeit various mmd pair assess greatly speedup mmd year mmd subsample extremely mmd possibly mmd bring high variance mmd recently split correspondence exact mmd omit inter block change smoothly mmd experience actually coming efficiency throughout effort accelerate development speed datum development mmd constitute branch research attain task utilize subset however accuracy mmd paper mmd implement summary fold employ theorem mmd mmd scale moreover sequentially utilize sample mmd mmd result mmd variance theoretically prove unbiased mmd kernel viewpoint extensive metric available mmd shift mmd consider nx mean discrepancy usually ball rkhs laplacian characteristic p prove empirical x I space empirical mmd mmd accelerate especially invariant kernel classical underlie approximation borel fourier definite value proper gaussian p view multivariate identity measure substitute thing harmonic theorem amplitude combination amplitude calculate time fig combining calculate mmd also unbiased mmd uniform bias mmd mmd spirit kernel dirac delta function calculation still statistic quantile approximate pearson first three moment mmd geometric circle circular explanation extensive metric mmd eqn project determine variable kernel investigate circular fix sample distribution sample circle pdfs two mathematically modular arithmetic dirac delta measure closely mmd circular assess circular claim dirac delta circular discrepancy provide circular close mmd claim amplitude combine mmd claim definition circular dirac delta eqn sign clear two maximize two construct project unit explanation circle angle circular discrepancy aim largely separate sample show diameter zero circular diameter maximize htbp angle light dark color circular discrepancy spread gaussian uniform intuition suggest tend dirac point ensemble circular gaussian circular unit shift invariant eqn circular discrepancy approximation circular eqn define kp normalize eqn
fast frequent evaluation make bad period slow lack long sufficient stage implement compare prox sg proximal dual power prox proximal prox accelerate prox method fista search scheme prox sag version sag prox sag demonstrate prox sdca dual ascent complexity need show prox prox sag prox three perform well prox sag prox prox sdca prox sdca complexity analysis complexity sag obtaining iterate regularization prox sdca prox sag quickly follow full prox sg show list include prox hybrid prox sg switch prox scheme improve substantially similar hybrid sag behavior method prox prox perform parameter worse much slow sag prox proximal prox two average large general average structure extend reduction compute modify gradient reduce prox enjoy complexity sdca sag recent achieve improve component substantially write proximal associate combine fy fx rx py tx rx fx tx use collect product tx ty ty ty put together arrive prox define mapping k rearrange inequality drop nonnegative dropping left minimizing convex problem arise machine know method use multi scheme gradient converge rate gradient problem many simple interested case advantageous incremental stochastic operate single know least component choice include net nonnegative regularization class close formulate set e mixture soft hard possible present base rx gradient overall I exist large may come either although must step specify definition proximal gradient compactly view case accelerate variant component component also average prox randomly take kx fx prox sg evaluate prox introduce sg much prox fair iteration step iterate satisfy see interesting ratio prox need prox gradient evaluate find accurate accelerate prox complexity hand prox sg sublinear prox sg expectation prox sg efficient low solution vast sum prox randomize incremental algorithm typically require survey prox sg prox prox sg structure room several work special develop component evaluation significantly superior prox sg zhang conjugate exponentially size overall computational cost increase batch gradually reduction technique batch maintain prox sg iteration whenever update gradient modify index pick replace prox sg kx f fx I variance use prox imply eq optimality I fx rx rx last convexity prove respect I q fx k k f k kx k f kx f q ix last inequality apply twice convenience need lemma well close q low slight completeness lipschitz continuous define g x fx step prove g x k note prox still proximal independent k first schwarz inequality eq q sides inequality independent addition q sum l
large wide variety machine finding validate importance rare reasonable priori use training cross resample separate fitness repeat final overall efficacy resample cross bias variance bias traditional fold cross small research repeat fold comparison denote induce resample complete optimize resample split fitness relationship characterize final setting entire approach optimize generate determine historical assessment goal choose model e neighbor versus near focus accurate assessment manuscript focus model acceptable precision illustrate predict compound molecular compound predictor descriptor analysis count molecular surface area size assess svm model radial tuning radial resample use tune hold perform resample iteration roc curve roc fitness figure cost fitting peak reach begin become winner strategy sub area curve determine treat equal substantial highly unlikely yield parameter lead computation computationally drastically tune predictor resample step efficiency major time manuscript acceptable value parameter fit analyze simulation characterize efficacy efficiency adaptive resampling resample value unlikely choose avoid clinical trial clinical trial objective unlikely clinical involve pre trial substantial detect specify effect well perform fitness value particular concept applicable assessment tuning fitness use value unlikely consideration long setting resample continue maximum resample process would continue fitness precision nominal still consideration algorithm predict break consider detail fitness well describe assess resample resample contrast treat error interest statistical test strong fitness correlation fitness split tend another compare fitness likely inferential statement inaccurate estimating parameter model comparison adaptive manuscript side current comparison secondly attempt positive context correction nominal manner instead directly correlation iteration diagonal covariance compound within effect equation slope parameter cell current condition parameterization performance numerically equivalent reject rejection bad resample tune whose remove subsequent return sub occur roc performance fix compute whose resample model remove remove usual winner select quantify time adaptive procedure speed execution process fit svm result multiple tuning distinct drive linear assumption since curve unity resample leave skewed generalize multiple normal residual develop consensus characterize tune across set purpose tune decompose comparison compare setting converse true number tie handle team set loss usual manner tune interpret odd approach use associate reference consequence large may consequence estimate magnitude remove model resample use tuning side ability asymptotic interval generalize least quantile great eliminate resample iteration single reach svm ten play pair roc curve computed estimate good average area roc ability fitness model filtering eliminate value estimate largely bar confidence effect inferential fitness roc near unity rmse skewed linear assumption residual may skewness statistic understand study system simulate nonlinear model independent add set efficacy use feed architecture tune decay repeat resampling varied minimum confidence interval create hardware failure simulate model adaptive efficacy quantify speed procedure version relationship tune six validation setting decay competitive eliminate quickly depend adaptive procedure use nominal simulated condition match however resample resample choose base overall fully first small size efficacy set discard decrease range effect compute comparable value figure median least total occur speed drive training surrogate tune parallel many computer architecture loop line serial computation loop model configuration logical computation lead substantial parallel speed fold sequentially parallel offer parallel benefit processing parallel additional require svm worker process resample speed adaptive parallel study use combination processor task process previously sequential median adaptive degree parallel eliminate technology
noise realistic inherent dataset google house consist google million image x natural label dataset imagenet step imagenet augmentation x probability cifar architecture configuration file implement layer keep imagenet model architecture noisy datum stochastically change label change use generate training five epoch figure contrast operate beyond label one addition noise consistently achieve well also rate train perform show effective negligible tb cccc cifar label flip detail color mean indicate incorrect increase convnet greater incorrect cifar training level identity decay b noise especially show scalability noise imagenet imagenet scalability noise explore adversarial imagenet random case label zero location correct change confidence size consistently superior imagenet add outli imagenet category use imagenet fall release fix imagenet outlier add outlier train normal version outli use run training run perturb amount robust effect h image image realistic label internet outli cifar mix totally outli image fraction know cifar convnet error second model train layer give gain web image challenge around imagenet web category internet search imagenet dataset example image highly rank precise outli train imagenet domain reduce add imagenet consistent three ii flip large training matrix flip convnet flip noisy label focus type large scale former gain gain however minimal deep implementation little facebook ai com availability label allow convolutional manual datum impractical noisy e available image may accurate noisy layer process simple modification deep demonstrate scale imagenet image availability large amount imagenet label image impractical tag keyword contain training abundance understand consequence convnet contribution classification noise dominate explore expectation convnet significant degradation modification convnet enable level simply layer softmax softmax model handle flip outli noise conventional distribution supervision convnet library readily imagenet classification degradation noise noise input noise several type source noise cause fast amazon www privacy handle label incorrect remove correct difficulty distinguish informative hard effect knn cost deep incorporate noise single tune cross realistic label make adjust class parameter deep particularly relevant layer pre purely convnet train availability clean provide supervise either one datum require external force pick inefficient resource fraction near decision sample unlikely many even pick informative challenging rank search engine likely regard train convnet difficult human drawback impractical many method problematic dataset consider may drastically result structure problem class aggregate denote parametrize label capacity asymmetric oppose example cat likely tree data b initially base green start updating datum prediction combine parameterize train maximize noisy eqn label true quantify confusion sample true make identity perfectly predict noise confusion eqn measure reality label objective force label eqn p equality ik confusion noisy noise singular force identity combine parameterized force model convnet softmax layer softmax role softmax linear modification perform back propagation noise model accurately predict unknown us infer noisy noise constrain network update back entropy base gradient weight project subspace unfortunately follow confusion alone give base actually infinitely force force noise base encourage blind deconvolution act base necessary ill pose take hold hold minimize sensible although
truth multiple experiment heavily class never mechanism gold long question require worker neither depict baseline mechanism answer gold skip base figure require worker worker choose computer object task depict worker gold question show short text worker movie order worker search entire search worker question baseline skip confidence present evaluate worker answer convert upper answer error match short worker paragraph play prevent search obtain text searching line task depict gold question baseline correct gold standard skip base mechanism present worker solution answer true audio audio second comprise topic movie speech depicted amount worker gold compare correct gold skip base mechanism c figure mechanism crowdsource axiom mathematically surprisingly mechanism feasible mechanism preliminary baseline mechanism suggest additional benefit pattern worker difficulty question worker secondly mechanism post process incorporate information overall accuracy simplicity facilitate easy worker conclusion mechanism practitioner researcher engineer crowdsource acknowledgement thank many discussion thank read part manuscript first author microsoft paper although skip case skip simple offer valuable recall assume worker confidence mind formally worker worker answer correct payment wrong answer remain mechanism first question skip skip confidence greater indeed answer likely skip let order permutation question mechanism compatible payment maximized worker answer algorithm payment payment strictly consequence distribute question relation associate condition payment maximize confidence finally see every question worker choose payment worker probable lemma theorem begin piece represent expect payment give answer identity random choice gold question worker expect payment function must compatible compatible mechanism simple element small mechanism skip last question must worker indeed worker answer question compatibility payment choose answer payment choose value side represent evaluate ball thus coefficient must constant appear desire argument permutation question complete element element value worker skip worker attempt last x worker answer quantity compatibility apply value constant get proceed induction begin simplify apply variable side polynomial coefficient particular polynomial order simplify hold answer evaluate worker remain question get necessity use remain constant pick pick side get result function type gold standard gold comprise evaluation argument question gold question worker evaluate divide desire complete restriction threshold observe employ free axiom neither confidence worker worker skip question worker likely eq q desire compatible payment question incorrect question incorrectly question proof induction incorrect zero payment base induction hypothesis assumption consequence incorrectly payment induction question prove necessity payment gold incorrect confidence payment denote payment gold e eq fy l l linear lemma must also comprise fy g meet specify give finally budget requirement instance payment payment worker question payment payment strictly zero incorrectly remain worker question payment confidence first worker attempt answer desire skip payment gold first include gold function must worker confidence question worker choose skip question order worker answer free prove compatible strong free even worker show work first p payment payment pattern strong worker answer expect payment worker payment worker question worker p gm payment happen answer payment question question p payment result payment maximize desire finally strictly hence worker restrict applicable avoid worker question payment incorrect answer proceed answer strong payment payment answer answer induction fy allow payment statement proposition proof proceed induction quantity payment repeatedly unlike hold quantity payment rest payment correct rest answer gold question alone originally question argument constraint thereby complete proposition skip strong answer incorrect hold even incorrect worker level correct irrespective question contradict requirement payment free axiom axiom payment u zero function payment prove payment observe propose payment function behave compatibility prove uniqueness replace payment eq evaluation proof proof payment replace payment theorem proposition pt axiom theorem california berkeley microsoft berkeley edu microsoft com field range predict structure large labeling task traditionally perform expert expensive rapidly increase interest worker crowdsource time typically fundamental crowdsource novel quality worker proof desirable free mechanism additional involve observe necessity engineering dataset need build speech label th task expert student pool student limit labeling perform worker internet know crowdsourcing generating label crowdsource crowdsource overhead crowdsource minimal crowdsourcing gain popularity bioinformatics environmental management computer crowdsource create require large amount label crowdsource often supplement automate task difficult alone worker crowdsource expert label crowdsource typically focus post quality input process reliable engineering crowdsource service crowdsource amazon gain popularity task image answer worker show worker identify depict bridge typical crowdsource worker complete worker attempt question encourage worker skip reward worker confident feasible requirement crowdsource job comprise question payment mechanism existence gold standard question question perspective randomly worker confidence answer answer belief correct worker likely assume question worker aim maximize respect gold question worker note payment solely competitive market worker call payment mechanism payment crowdsource threshold wish worker skip smaller great select likely possible mechanism indeed wide narrow impose natural requirement practical consideration worker wrong question attempt payment compatible mechanism satisfie axiom amount worker total gold mechanism among gold question correct answer axiom requirement weak payment answer question would half axiom none strict impose gold incorrectly prove surprisingly mechanism natural mechanism satisfie skip discuss one require offer minimum payment worker denote payment free axiom make payment answer gold wrong free axiom amount addition equation illustrate amazon platform display ask gate bridge figure gold image question mechanism payment additional answer gold payment take worker aside mechanism pay gold pay answer nothing worth note equation conservative compatible worker payment impossible compatible free ask worker explicitly fine g level low moderate high reveal confidence payment mechanism generalize axiom indicate high confidence question attempt turn payment worker exist compatible mechanism free axiom present low confidence gold provide worker question incorrectly level question mechanism pre mechanism compatible payment mechanism generalize free axiom offer payment worker hold amount amount conduct experiment amazon worker aforementione among expect also observe expect answer higher correct payment take present result formally theoretical claim appendix crowdsource consider worker task correct multiple choice figure question audio etc question accord correct mind possible answer answer probability answer correct shorthand define confident every skip pre confident ask worker either skip answer question either skip high skip phrase moderately absolutely sure skip skip correspond mechanism skip appropriate fall corresponding question assume question answer evaluate gold question gold question question question payment payment worker gold question payment depend e amount individual worker positive non evaluation answer worker determine payment worker evaluation determine setup crowdsource non negative skip question incorrect payment confidence worker question incorrectly confidence payment mechanism attempt maximize payment sequel payment refer expect payment worker e answer uniformly question worker question worker perspective payment level average arise gold summation regard correctness every choose answer skip information ask worker payment mechanism worker question moreover question select likely correct simple requirement axiom answer worker gold standard payment zero would payment binary half answer incorrect weak payment incorrect payment answer gold set mechanism present indeed worker skip confidence answer confidence great correct mechanism compatible mechanism certain minimum payment make modify axiom accommodate necessity payment answer worker incorrect pay additional worker payment whose answer compatibility uniqueness payment worker describe mechanism discuss worker ask select worker ask indicate range answer skip level worker skip worker make skip consider special axiom answer gold high formally require specification threshold worker indicate skip skip set specified skip attempt set every fix specifie level option question worker skip skip choose confidence confidence worker confidence indicate high include level select call payment list worker answer correct set restriction worker threshold must coincide additionally compatible threshold post constraint etc paper propose payment input threshold evaluation payment eq show worker select level also give early mechanism worker else whenever confidence confidence mechanism payment mechanism algorithm compatible satisfy generalize payment worker minimum payment modify free axiom accommodate necessity minimum payment answer worker confidence level modify pay payment worker answer x compatibility uniqueness mechanism worker mechanism compatible axiom skip zero payment incorrect receive payment generalization mechanism impose zero answer gold incorrect question confidence even question wish impose strong requirement payment worker primary focus section skip axiom strong axiom g axiom axiom propose event payment axiom add extra worker unfortunately minimal requirement satisfy compatible exception gold worker least impractical crowdsource free axiom compatible strong free axiom skip list worker worker payment free worker mechanism satisfy free axiom try worker act mechanism worker strong exist compatible condition answer nevertheless mathematically interesting belief event prove uniqueness worker skip payment next thing answer confident worker answer question confidence answer gold payment worker answer worker answer confidence greater skip confidence small obey free free condition I none answer mechanism strong compatible even section consider set worker payment payment increase example expect payment aim utility payment make worker answer gold question payment aim belief regard correctness answer gold question recall free axiom payment gold question high question worker
dependence decay linear em stream competitive erm averaging independently provide regularity herein decay strictly range tuning decay undesirable start seek drive arbitrarily remark regard parallelization stochastic see subsequent quantify procedure implication streaming work algorithm iterate sgd unclear erm dependency specify essentially variance iterate distribution erm paper provide average erm herein work erm initial low rather initial special least guarantee global super comparable adaptation would alone suffice identical wide need decay many iterate competitive variety reason lead dependency error convexity solely term argue estimator applicable mis model asymptotic argument make certain restriction variance erm finite stochastic convex smoothness case bound square estimator mis convex converge notably develop become loss near sufficiently attempt directly erm pose linear new generalization order exist generalization demand pass scale immediately observe datum sum store erm mean convergent algorithm increasingly hope cutting fraction erm streaming would eventually essentially hold entire second proving succeed aforementioned conjecture tight regression suffice generally believe erm herein statistical pass algorithm contain erm fact avoid streaming erm polynomial consider special least focused solely computational use state art sum function convergent obtain generalization approach comparable streaming setting slowly strictly yet summarize algorithm erm corollary summarize corollary theorem provide provide performance guarantee provide throughout dependent erm restriction asymptotic instance whereas section size analogous rao approximation problem set benchmark dependency rapidly decay assumption analyze assumption low approximation differentiable strong q weak assumption number definition assumption imply convexity instance whereas ratio namely derivative self self weak condition equation condition hold eq standard assumption phase aside close minimizer quickly global obtain curvature fast stream progress away size random streaming provide sum smooth proceed stage stage draw draw sample opposite show achieve aforementioned generalize particular choose coincide use assumption oracle streaming compute draw one point guarantee assumption fix denote upper regression achieve erm furthermore competitive ratio near follow upper drawn iteration p parallelization implement linear count stage theorem geometrically remains spend average gradient algorithm fully enter phase enjoy even dependency emphasis flexible driving erm super competitive drive erm polynomial know approximately multiplicative let number iteration decrease know super polynomial rate convergence set specify drive ratio competitive ensure adaptively fast fast vanish quickly characterization numerator erm streaming focus erm follow regularity condition third derivative problem polynomial specify application benchmark widely study least square upper low erm extend generalized problem huber regularize illustrate erm performance streaming erm distribution streaming bound become meaningful large initial decrease square word recover mis specify aforementione model last ridge necessarily specify follow provide statistical erm erm suppose remark appropriately universal x bind erm suppose equation erm comparison lie set allow erm upper bind least define regularize regularize w analogous interpret mis fit rate erm straightforward self include completeness logistic self consequence smoothness instead assume smooth smoothness second norm suppose definition follow step function suppose independently note assumption eq q recall schwarz yield multiply yield finally bind progress conditioning take sum strong little term yield assumption progress assumption jensen result case proof utilize follow lemma convexity self self self convert another self restrict property monotone small assumption hold eq solving recurrence bind high lemma suppose hold eq recall random hand last balance lemma ready stage positive lemma q eq q erm sense third exist one regularity compact interior invertible exist constant neighborhood bound appropriately choose also universal bl infinite theorem lower erm interior along argument erm bound less function eigenvalue appendix universal small term arbitrarily taylor p p w w w w right side boundary inside enough bl lemma diameter convexity enough erm taylor enough thus taylor inequalities q use perturbation n w w w erm great define pz less term complete constant acknowledgment thank lee rf microsoft done institute support nsf fellowship grant instead hessian assumption upper convexity fix
validity however concentrate therefore generalization property forest first assumption feature proposition pf pf pf fp ff pf pf fp ff set follow basic selection forest case build randomly choose node independently base forest follow internal forest internal internal extend give reason empirically moreover might applicability sake strategy choose use subset strategy application high utilize image computer vision slightly principle event feature tree assumption proposition know get feature node total binomial probability compute select tree entire count clear count partition partition denote partition partition two list element partition compute compute partition compute need count e number return compute tree know probability get time compute multinomial internal tree strategy internal simple give component need determine frequency relevant non relevant nan hypothesis determine threshold get time provide hypothesis specifically choose false positive desire negative empirically much equation threshold rely already discuss important presence feature relevant expect drop non gets select determined predict thorough synthetic quantitative analysis map propose depend cluster bag subsampling replacement subsampling bag publicly software neighbourhood forest problem zero binomial ensure correlation different size balanced set problem spurious close reality false estimate nan dimension per setting record selection threshold prediction experiment strategy htb ccc strategy strategy dimensional positive rate capture various different strategy plot threshold observe positive rate gray line observe accurately capture substantial subtle surprising effect depend quantity demonstrate frequency hypothesis article problem htb ccc false model black curve probable relevant statistical label assumption relevant feature evaluate false rate observe bag addition experiment false vary present expect size make relevant importantly prediction observe prediction strategy threshold limit false rate situation determine equation false rate false power summarize training column code false negative mark fashion false positive give false negative false rate desire I proportion relevant right show false negative training strategy high increase feature dimension change sample one hand spurious become node higher false positive desire change seem observe increase due increase relevant feature false also seem believe importantly show share drawback statistical see trend change increase second increase believe competition increase correlation false rate expect false change part analyze statistical relationship theoretical feature per set train forest false repeat experiment time present simply false positive versus relevance forest compose cc strategy fix rate relevance tree set desire false relevant rate increase increase tree surprising tree weak chance tree get number increase third number cause decrease positive rate observe get furthermore large subset decrease false large precisely observe graph number surprising tree feature weak spurious high chance trade detect characteristic forest comparative study determine threshold testing aim significance permutation yield applied frequency rate bag train desire base marked permutation base use permutation relevant positive permutation adequate time value ii selection frequency comparison propose selection permutation provide setting false permutation limit desired propose slightly false desire propose false permutation testing testing computationally parameter permutation hour ghz gb ram forest optimize though testing load backward computation frequency cost propose integrate permutation comparison problem rarely depend complex approach derive type brain software software discretize study understand disease map false control map map matrix publicly open access htb generate mean correspond dataset discretized vertex extraction image denote mean decompose denote matrix generator easy verify estimate one create produce display map synthetic similar fully formulate disease motivate relationship label map local region denote vector figure region region entire set generation give diagonal pearson label assume equally experiment sample number size strategy forest determine repeat c strategy ii positive synthetic size leave column false high desire plot sharp negative competition explain previous overall threshold split relevant relevant especially threshold experimental demonstrate false positive feature positive topic spurious assumption actually sort get high spurious relationship feature tackle subsample bagging bootstrappe bag ratio ratio improve spurious statistic enough bag lower bootstrappe prominent cause rate competition hence selection rank make algorithm backward combine bootstrapping scheme rather non straightforward backward bootstrapping stability determine unlike integration difficult propose burden negligible forest segmentation medical problem compute space think feature apply challenge dimensional nature bin come effectively model forest principle false heuristic threshold computationally keep rate great modeling bagging bootstrapping effect provide approximation threshold author thank comment read carry whole resource center resource national institute national health share program grant number rr rr rr also support foundation well foundation datum acknowledge grant mh mh r ns ns ns medical foundation mh project process brain subject surface measurement subject surface efficiency consist map width well software request remark convert forest tool ability bioinformatics popularity forest still crucial ingredient efficient computationally researcher ranking article build feature process forest threshold additional synthetic positive false presence light need approach applicable selection datum growth almost scientific machine substantially main determine sense maintain resource allow constrain certain scenario selection eliminate entire set relate refer label trait particularly biology part routine great potential feature wide biology univariate forest certain advantage contrary detect et study second deal feature often need logistic problem computationally tractable phenomenon explicitly raw require dimensionality transform interpretability analysis fourth modification regression multi unsupervised burden variety phenomenon despite popularity success forest importance still lack ingredient false produce frequency permutation importance last propose variation ranking identify irrelevant determine feature relevant assignment forward principled threshold random forest heuristic know interpretation natural determine quantify number false positive threshold quantification desire level threshold quantification allow informed algorithm eliminate irrelevant build principled efficient satisfactory drawback motivation attempt determine threshold quantify score permutation label significance lastly propose permutation testing variable burden limit quickly infeasible drawback reason feature relationship variable irrelevant time hypothesis give rise principled threshold computational ii tree biology forest especially advantageous truth available approach set assume complex computed measurement share drawback yield false article overview forest different importance motivate approach theoretical conclusion forest rf technique year vision bioinformatic due property beyond capability predictor rf overview rf feature adopt covariate denote describe importance bag permutation importance rate across break prediction forest quality quantify importance measure statistical undesirable decrease score importance increase drawback test determine measure make remark definition study present yield similar ranking show basic form category eliminate et measure variation suffer drawback date determine threshold integrate simple show threshold positive would like frequency use empirical measure ranking surprising permutation selection selection frequency random contrary possible need bootstrappe desirable limit question answer select label answer question estimate selection principle relevant relevant feature probabilistic training subsampling scheme subset node optimize false positive threshold determine model node randomness
state maximize among covariance lag maximize satisfy independence far admit close start submatrix next give specify extension factor q let partially specify band extension extension correspond band central partially matrix admit factorization lower triangular main diagonal provide dc extension dc representation partially matrix n dc entropy dc completion start nest principal dimension completion extension claim statement k band equivalently find inductive eq adjoint k k k equivalence maximum ij dc among covariance independence solve problem toeplitz eq factorization inverse take dc kernel determinant third right hand side triangular diagonal equal definite tc special expression determinant particular tc kernel tc mention tc spline entropy stable spline family dc interpretation spline stable spline admit kernel matrix exponentially decay poorly make exploit proceeding recall factorization qr thin qr factorization assumption uniqueness thin qr factorization triangular thin qr easy cholesky q thin factorization orthogonal unit vector triangular partition function rely qr thin factorization whose unit vector thin qr orthogonal triangular note assumption thin way qr suppose hold thin factorization moreover impulse thin factorization compute create step qr factorization compare complexity algorithm part require accord create cholesky create scalar scalar computation qr require step require computation iteration solution marginal maximization storage negligible particular stable depend cholesky conditioning return cholesky b hessian marginal th finally computation make use possible adapt stable approximately large recently introduction family kernel correlate dc kernel entropy interpretation exploit completion problem graphical dc dc admit determinant factorization property dc highlight dc exploit associated algorithm section chen approach impulse response process covariance depend parametrization identification framework correlate kernel entropy tune correlate tc point paper property indeed extend whole family kernel maximum exploit conjunction dc particular dc close inverse determinant kernel highlight dc estimator system rely error finite model select trading selection goodness criterion validation understand nevertheless sample equip cv model identification recently propose impulse see learn e maximization order parametric paradigm case prove robust aic crucially process variety function literature straight system identification recently correlate assess bank kernel admit interpretation entropy unknown zero represent impulse impulse response model covariance call practice impossible impulse impulse system decay exponentially impulse response certain become turn hyperparameter estimate maximize marginalization impulse assigning crucially quality recently identification covariance class diagonal account impulse impulse response tune
intrinsic suggest autoencoder reasonable indeed dot train frame dot figure filter learn phase shift fourier discuss bi able exist lc precision activity movie probably lower random dot still propose consist test video action train video feature sub block concatenation super descriptor layer spatio plug performance autoencoder outperform localize surprisingly outperform unbiased linear inference experimentally scheme unit linear without arguably represent intrinsic summarize region response invariant invariance perspective increasingly increasingly dimensional affine response invariance superposition region reconstruction square define response reconstruct input multiply bi reconstruction define feature active suggest high linear hide acknowledgment award education research university david university autoencoder typically hide natural act negative role representation intrinsic autoencoder activation like regularizer regularization autoencoder deep network simplest minimize observation eq q activation sigmoid relu prevent solution large input contraction force hide activation penalty across tend autoencoder mention scheme show negative bias desirable learn autoencoder consequence autoencoder weight vector take part reconstruct represent select combine cf bias allow role yield increasingly outperform autoencoder regularization autoencoder additional show without negative bias state cifar explain hide optimal unit tend force align dropout hide activation reconstruction autoencoder shrink autoencoder regularization type rbms common autoencoder yield negative effect color bias rbm contraction strength relu cifar result histogram cifar view arguably autoencoder constrain feature undesirable consequence encode effect bias relu sigmoid act activity inner weight ie yield value significantly weight bias activation spherical effectively define radial basis datum effect activation gets overlap cluster autoencoder however autoencoder relu even region merge word able define multidimensional autoencoder function add sigmoid autoencoder autoencoder manifold restrict active fix relu autoencoder write equation solution solution nan space eigenvector unit unit eigenvalue although w active reconstruction sigmoid zero active autoencoder thereby density reconstruction superposition activation value come probabilistic support one propagate learn dropout back prop boolean figure product derivative relu continuity common minibatch non optimization truncate follow unlikely work well often par well autoencoder denoise truncation contrast negative bias relu view inversion well activation activation function square drop cc autoencoder variant reconstruction either active cone thresholded thresholde literature plot illustration activation unit encoding shall autoencoder long hold span weight relate multiplication operator operator may tie weight one tie minimize frame identity hold choose cifar ability various contain color pixel class sample consider invariant recognition autoencoder mean evaluation base whitening whitening normalizing divide eigenvalue train choose initial epoch train total momentum representation weight validation training threshold parameter try regularization strength observable discussion early autoencoder increase input space tend ht input number patch cifar evaluate
eq depict measure provide fine uncertainty call improvement strategy spirit associate ei select ei ei sampling use criterion sequential therein reduction acceptable evaluation dependency function bivariate improvement equation ei assume computation independent accord minimize carry simple advanced monte spirit describe batch optimization define krige toolbox qualitatively behaviour criterion ei depict search situation sampling region thereby induce isotropic mat ern mat ern may gamma kind optimization figure error distance small c line posterior solid credible vertical bc bc bc bc evaluation sample mat ern simulate pt real value bayesian approach optimization usually formulate consider denote good put emphasis expense maximizer classical ei associated loss assume tractable call numerical value location maximizer ei l g real continuous set choice sampling point strategy explain motivation proceed present implementation qualitatively numerical criterion view path rewrite uncertainty inequality
truncate relevant estimate produce produce posterior filter brief represent filter solution integration relevant integral specification filtering applicability method illustration density instance ng current directly integral description preliminary final likelihood evaluate multiply impose restriction marginal produce fine grid compare abc choice well compute estimate use likelihood associate discretize normalizing produce generator uniform euler ng filter method filtering view typical equivalent gaussian transition exploit conclude computational kf euler base ng technique produce main slow infeasible score order initially impact abc first report remain abc auxiliary match summary abc via report square posterior give estimate posterior c panel one approximate c euler score marginal abc ss c fp panel rmse match multiple prior panel highlight parameter score accurate four marginalization marginalization minimal summary statistic four estimate inferior euler inferior four panel three superior notably improvement euler explore application certain reduction consistency auxiliary equivalence maximum summary also dimensionality application abc problem integrate extremely comprehensive model marginalization integrate auxiliary model common relate frequentist indirect inference efficient number exercise principle simple approximation approach success statistic success subject go also front despite static nothing state accept filter smoothing exploit see use state smoothed posterior state average burden develop herein base approach axiom conclusion condition exercise problem early new approach state abc summary data exact mle auxiliary auxiliary achieve precise sense yield mle separate integrate curse drive intractable abc fast produce volatility illustration free space kalman filter volatility increasingly integration fan review technique evaluation statistic calculate statistic simulate generate information summary statistic use technique determine devoted way ensure maximized contribution statistic approximate observe indirect approximate produce intractable frequentist model partial paper continue spirit demonstrate fix size key motivate decision seek state space setting auxiliary model abc matching focus qualitatively auxiliary achievable technique exact likelihood function sometimes ii literature well true efficiency investigation allow via abc alternative summarie space force adopt inexact auxiliary emphasis possibly drive least discretization usefulness apply illustration reduction result motivation note asymptotic auxiliary typical quasi mle establish technique allow tolerance criterion mle define measure proximity yield mle abc achieve setting block marginal integrate principle avoid outline paragraph applicable auxiliary base right particular already thereby exploit whereby kalman kf achievable illustrate via approximate discretization true augment kalman evaluate likelihood general applicability implement particular approach number principle parameter yield inaccurate non posterior marginalization meaningful basic would statistic reduction illustration score marginal section feasible repeat sample accuracy exact posterior assess abc weight autoregressive iv reduction conduct assessment firstly key reproduce overall superiority remarkable yield score particular exercise concept abc albeit exact available proceed diffusion accuracy approximate role volatility adopt linear central chi exact purpose deterministic base filter ng produce posterior associate euler model score great case gain still gain certainly mark linear simulation accurate euler approximation draw distribution draw posterior moment accept rp criterion sort tolerance arbitrarily small proportion approximate use practice model draw correction draw markov carlo sequential carlo smc improve give choose close b biological comprise may information spirit attention feasible may characterize consider scalar financial drive computationally euler measurement apply base particular discretized model see recent express initially financial adopt e smc inferential infeasible contrast continuous constitute augment vector comprise unobserved define x x period simulate simulate follow draw subsequent conditional crucially retain criterion view relate reference content importance proximity comment highlight key motivate sample illustrate form expression highlight applicable useful inference cardinality abc arise distribution ef possess achieve reduction relative member ef due vast possible either member ef reduction ef even gaussian necessarily reduction familiar move simple dependence marginally normal ef sn familiar toeplitz matrix autoregressive ar order construct structure accumulate achievable straightforward increase accumulation across successive reduce ultimately need attain determine sn determine approximate sufficient sn reasonable approximation sn ignore qualitative characterize nest link sn lack accurate base arbitrary turn motivate asymptotic mle asymptotic mle model cox algorithm would draw posterior mirror demonstrate chosen see also likely enough parametric abc knowledge appropriate analytically tractable computationally enough yield matching show certain regularity condition auxiliary log positive assume form coincide true analysis abc evaluate frequentist whereby degenerate consistency appendix q generic draw choice approach limit irrespective select hence degenerate parameter consistency fact frequentist obtain e avoid specifically abc property approximation go decrease unless increase solution within drive look abc posterior le large gain score close choice definite weighting implementation magnitude approach approximate produce simulate technique specify consistency frequentist consistency limit irrespective maintain remain impact mle draw yield criterion yes enough equivalent estimate course parameter expand point scale criterion subject condition regard affect value equal irrespective weighting matrix mle matter true exactly go comparable regard estimator weight form preliminary abc estimate posterior yield negligible proceed operate solely score extract set level survey appear path tolerance within stay away give addition constraint impose necessity reasonable recommendation namely technique perspective value induce form error firstly fundamentally summary selection posterior exercise equivalent exact otherwise posterior necessarily error analytical exact density density error highlight less thing curse increase firstly bring need secondly bring certain full statistic technique estimate retrieve remain still dimension set see approach framework match diffusion natural discretize auxiliary mle construction select estimate posterior marginal produce statistic translate function coherent full vector technique et joint estimate marginal marginalization yield non analytically approximate score base abc example begin formulation random variable include jump exclude discrete produce either notational continue true discretization affect functional section illustrate true e error process initially diffusion use represent evaluate g include simulation filter smc computation technique numerically brief transformation calculate variable involve select accord deterministic sigma yield turn cloud moment implement kf weight sum sigma excellent introduction generalize non burden comprise update sigma point sigma estimate score method non set accuracy discretization accuracy function aspect discretized model order accuracy regard likelihood reference relevant g document order kf filter measurement approximation beyond embed specific euler discretization exercise abc method set summary statistic conduct firstly within kf evidence two increase exact mle abc space compare summary whether curse marginalization section volatility use score produce function discretized result relate particular nevertheless serve illustrate dominate summary overall approximate particularly accurate simulate sample setting sn summary statistic may sensible observable distance firstly euclidean across draw statistic secondly briefly produce summary statistic step procedure scalar subsequently simulate conditional calculate r denote vector statistic evaluate kf weighting mle evaluate mle score estimate integrate numerically kf logarithm performance figure abc statistic abc euclidean fp produce initially reject algorithm draw process sn marginal posterior normalize likelihood evaluate kf multiply fine likelihood posterior produce panel summarize produce kf percentile dot short per text panel highlight abc true notable remarkable marginal poor statistic base abc figure produce percentile extremely accurate fp parameter produce accurate still fp tend approach yield score
respect property magnitude greedy counterpart set statement introduce prove rsc low regression section extend broad hard compressive well q thought e f n see need property rsc rsc satisfy convexity rsc constraint satisfy convexity constraint except rsc define except replace project gradient descent select magnitude projection significantly strong thresholding formalize index oracle sparsity step p combine rsc rate rsc sketch critical condition f f third small element ts hard thresholding insight see gradient hard I singular vector matrix top singular pm diag von trace follow rank rsc replace projection operator fw satisfie secondly place consider restrict span variety datum result differentiable rsc respectively incur crucially scenario z label hold nc l universal constant put incur look specifically corrupt obtain replace case additive miss n rsc sparsity hold small hold high error fully keep iterate prove fundamental rip consider rsc fs input tf concentrate fully popular compressive pursuit far analyze rip least use section rsc base arbitrary observation thresholding objective function hold observe property rsc satisfy sparsity tf ts thresholding family iterative introduce rip sensing member replacement backward method backward similar fully partial hard thresholding projection partial hard project still perform small add rsc either suppose added thresholding element magnitude observing provide proof rsc iterate ex dimensionality variation running times level increase htp clearly scalable l different increase verify project recovery offer scalable choosing recovery run time keep one independent datum hard style sp implementation l project scale lasso within consequently shift time experiment record support describe sake clarity htp result sp indicate equally recovery htp gap whereas unable respectively slow htp moreover nature sparsity level figure htp take less converge offer htp large run verify high keep select coordinate outside support heavy draw level increase see remarkable ill size study hard differentiable nevertheless convexity rsc sp require rip restrict large universal insight relax support show follow rsc already establish literature variety put relaxation term order competitive arguably dimensional learning generalize algorithm analyse unify probably create atomic norm claim microsoft com edu linear project known thresholding offer fast solution extremely statistical tight match lower enable analyze hard thresholding htp sp setting extend fully statistical consequently structural work feasibility issue often end np problem sparsity constraint rank require deal demonstrate avoid strong rsc restrict smoothness relaxation suffer despite
calculated mind reconstruct reconstruct systematic calculation affect library unfold iterative start histogram inverse many reconstruct represent presence intermediate dimensional bin measure solving simply bin approximation connect measure preliminary true calculate usage certain systematic principal guess connection create measure guess comparison candidate attack unfold sample test measure equal bin distribution chance true relatively big stage try add another entry require good value time influence make add match sufficient growing illustrate correspond systematic translation monte carlo assume true candidate ensure come ill fluctuation illustrate add regularization term role regularization projection guess coefficient dominate bin range bin prefer constrain complex bin fluctuation illustrate smooth
visible reconstruct rbm term update become decay bring improve performance avoid interpretable large value reason decay find illustrate benchmark user movie mainly item movie appear come internet movie http www distinguish different recommendation application depend class explicit implicit convert integer value binary rating big rate big implicit prediction rating feedback etc predict observed rating take rating imputation implicit usually hold predict rating observe evaluate take recommendation miss zero become dense rating recommendation distinction notable analysis mean rmse evaluation know usually rate fraction trivial besides suffer rating mean rating either indicate item paper receiver operate characteristic performance roc classifier insensitive reduce roc roc auc mainly technique recommender three typical baseline aspect tie factor select discrete latent latent undirected graphical start problem belong aspect mixture rating pair fill movie generality suppose movie well movie seem movie one put rank really want movie miss cause come prediction situation difference need test movie know answer ignore prediction rating prediction replace suitable feedback item situation technique understand four know difference negative feedback task though give perfect predict rating miss value include uncertainty deal class sample feedback class prediction ignore test could deal problem ignore feedback train phase rating indeed u could movie movie correspondence actor representation actor actor similar movie actor distance correspond actor mean actor illustrate experiment actor movie example etc partition c actor lee berkeley david new could easily situation typical rating task comparable imputation give rating imputation task superiority give future firstly good secondly kind application combine deep recommendation want layer undirected variable energy number visible unit bias joint configuration give sum configuration visible unit unit q rbm represent rate rbm represent movie rbms unit different unit rate bias tie rbms movie rate people rbm bias
locate operation identifie plant surely subsequently discover known detect hardness work programming relaxation method also fail hardness plant clique widely theoretical plant clique sense identify plant tend researcher hardness plant clique problem prove approximate nash similar hardness clique principal submatrix detection note section impossible achieve statistically sparse significant polynomial semidefinite programming assume pn parameter clutter theorem instance conclusion hold attain minimax time plant clique principal different idea submatrix uniformly replace flip sign row independently obtain joint distribution identically vector scale suitable false exist convergence pn plant plant clique could identify also college second air force scientific fa mathematics information california research let final apply take deduce take set need immediate independent deduce use result set proof notational closely zero proof appendix cardinality ham every distinct observe u u l generalise kullback eq curvature q denote index zero deduce conclude require assumption theorem n consequently risk follow sequence polynomial polynomial algorithm identify planted plant tend contradict n u w rademacher entry small subset large coordinate absolute u u w let clique matrix diagonal adjacency associate bipartite let convenient variate rademacher independent rademacher let ny iy shorthand elementary variation initially algorithm observe p pn mn n mn bernstein belong hypothesis large let sufficiently reverse conclude hoeffde large eq conclude sufficiently contradict rademacher rademacher coordinate four subgaussian deduce hand binomial right similar fourth final hoeffding final inequality result eq exist u require write measurable kullback leibler divergence denote derivative generalise measurable function denote basis require r variation proof take value element arbitrary b measurable inequality subset disjoint set write b g eq I na mb f I f c empty f intend short introduction notion refer reference information much follow generate desire string problem denote acceptable string solution string collection perform task mathematic history notion define notational system call calculus finite distinguished transition machine access consist label component square string start head th operate machine move square stop symbol terminate finitely step say terminate computational solve notion calculus modern input string terminate exist solve class notation configuration replicate branch far replicate continue replicate machine replicate output computation string terminate replicate make parallel step widely computational relate algorithmic complexity problem exist g section machine non situation problem triple distinguished state transition head state probabilistic exist terminate say string however independent fs randomness problem suitably construct polynomial thm lemma thm dimensional simple lead eigenvector fast fundamental trade satisfy concentration time minimax polynomial time variant semidefinite relaxation show essentially rate algorithm project multivariate onto span lead covariance device effective small size pca encounter diverse modern area vector arbitrary unit case eigenvector classical would unit lead eigenvector covariance inconsistent asymptotic call design interpretability eigenvector belong give p remarkable author attain minimax treat sample particular subgaussian infimum moreover show rate lead cite principal estimation subgaussian neither polynomial compute naive search become infeasible moderately whether computable polynomial attain progress nan ensure asymptotic problem plant distribution minimal level polynomial test arbitrary test tailor sufficient thesis principal rather distinguished fundamental phenomenon occur formula statistical section precise satisfied subgaussian certain key importance subject mild restriction place semidefinite computable plant clique result fundamental computation section estimator recovery motivate step class sparsity therefore semidefinite relaxation drop fm mf fm u complexity implement interior show solver rewrite make amenable possible matrix onto decompose orthogonal diagonal projection j optimisation htbp pn nu point certain particular step number often considerably costly compute operation take complexity algorithm operation use algorithm lemma incur population see projection describe property
cell lstm slice pass lstm tb lstm start lstm start net feed feedforward net time slice show tb gate forget gate gate input gate cell notation gate adapt consist secondary annotate literature see use output hard encode description full far filter identity cb divide n cb tb h loop bend bridge model layer lstm relu unit skip pass relu concatenation use weight sample bias layer initialize lstm norm divide exceed gradient scale correct field lstm perform rnn use correct tb ensemble prediction secondary cb currently show lstm conditional neural field inspire recurrent show feedforward backward net structure future include architecture ss supervision ss read final version article acknowledge support gpu acknowledge bioinformatics centre apply computer science technical university secondary bioinformatics svm slide sequential network feed recurrent memory secondary use cb secondary problem feed short memory explore recently memory lstm network include machine recognition lstm protein secondary neural specific improve performance conditional approach secondary structure prediction non typically feed classifying naturally present slide prediction backward network include feed neural recurrent feed forward inside primarily introduction large model lstm paper predict past secondary elegant train separate rnn start recursion
variation recursive compositional rnn vector vector fix word embedding treat give compute role e rnn table discuss bag word standard bag yield model exist automatic regard tree proper value important sentiment automatic scenario result sophisticated bag model weight capture hold generate set sentiment dataset contain label phrase task could way fine grain task negative neutral coarse label embedding fix embedding label full sentence neural mention recently paragraph obtain sentence paragraph vector produce recursive performance paragraph sentiment force semantic mean syntactic orient decide coherence sentence corpus contain report pair article assume coherent example permutation protocol consider window concatenation representation adjacent classify either coherent make get score random permutation state art regard use illustrate model entity art model task feature obtain weight art task c recursive entity propose version neural feature token weight dl architecture newly paragraph model model neural additional extend deep minor adjustment leave emphasize evidence tuning acquisition propose neural automatically contribute compositional acquisition recursive demonstrate significant neural one type obtain representation sentence long dependency extent capture brief short sentence embedding token parent layer involve activation nlp obtain embedding feed optimize sentiment feed aforementione classify negative embedding sometimes capture semantic syntactic manually feature nlp benefit c model neural train implementation recursive please report search parameter optimum batch mini batch embedding word suffer intrinsic drawback us token movie contribute sentiment network flexible less token sentence consequence representation trivial refer specific architecture neural svm notable big word come advantage evidence brief svm sentiment classification constitute supervision detail learn embedding less influence zero compositional former mostly specific word embedding neutral phrase regard issue compositional enable composition rnn interaction vector g pos tag tag compositional approach extent compositional idea weight svm try incorporate neural neural binary idea involve associate final token movie impact token like sentiment task optimize propose capability compositional approach undesirable information nlp task brief distribute framework beyond token gram phrase recursive recurrent constitute type framework acquisition recursive g include sentence compositional within sometimes input token explore embedding manner capability capture depend architecture work short lstm first back determine information memory lstm partially address recurrent model widely translation sequence token phrase sentence denote word vector stanford representation idea associate additional weight importance current towards retain relatively information expect importance current whether sentiment embed representation convolution function enable compositional bias intermediate view use three neuron output project parent current importance embed concatenation lead around
yet matrix functional matrix lead guide select real carry sparse sequence belong ball arbitrarily actually accurate signal encounter g estimator fully functional estimation exhibit structure functional thresholde penalize likelihood optimal rate class functional intuitively estimate since one estimate together matrix estimate rest organize section motivating efficiency market capital asset pricing prove minimax large matrix extend estimate performance two let integer space contiguous integer write identity denote coordinate iff trace operator element square norm real define associate rectangular determinant variable may change motivation functional test first gene validity financial economic dimensionality large statistical due limitation well illustrated suppose canonical statistic eq power avoid large lead provably powerful statistic implement dependence mild condition hypothesis asymptotically moreover depend unknown statistic estimator suffer power rather may prefer thresholded estimator empirical could covariance estimator chen take relationship need statistic exclude th exclude leverage reasoning thresholde hard quadratic beyond scope observe difficulty take hybrid consist put scale let q hard asset pricing asset risk specifically asset express factor return portfolio see market correspond let follow q factor pricing risk return possible market efficient traditionally problem point estimate residual ordinary define statistic moreover expression provide estimation impossible motivate case functional rapidly norm diagonal unknown situation remove diagonal implie measurable appendix limitation class covariance normalization frequently obtain sample simplified assumption accurately regime diagonal element could vanish employ estimator covariance empirical eq j diagonal element rest establish interestingly quadratic threshold plug present usually functional optimal minimax measurable proof subsection remark theorem tradeoff keep trivially match present arguably nature give minimax keep light effect actually boundedness diagonal noticed meanwhile omit information present phenomenon functional fan setup low model control decay coefficient inherent functional clean logarithm functional matrix functional matrix maximal row functional quadratic functional natural quadratic outside wise q boundedness space low assume exist define infimum appendix write bind estimator naturally technique minimax plug constant proof indeed theorem yet interest prove actually taylor expansion term parametric second contribute rate taylor stem estimation remark combination minimax convergence make enough maxima sum quadratic price extra term present phenomenon wise functional functional simulation conduct evaluate apply dimensional simulate financial study estimator part end simulation auto ar matrix otherwise vary quadratic bs chen time base omit bs estimation estimator b bs dash log size top bottom leave htbp plot aforementione estimation use naive plug obvious four b dash well solid small estimation improve capture quadratic cause eliminate come work well proper threshold simulation choose employ cross proper thresholding consist sample thresholding estimator threshold test construct thresholded consistent estimator candidate threshold choose take apply matrix suggest functional apply splitting study test functional dimensional two group gaussian consider correlation rest functional increase mean vector choose percentage identical represent situation equally comparable prop empirical six repetition bs chen chen modify estimate thresholde empirical counterpart dimensional univariate nan employ estimating splitting evaluate correction correction family fdr list estimation method also take error bs first ignore well combine individual aggregate signal together statistic outperform identical individual combine estimator functional among correct bs bs indeed bs claim chen performance leverage sparsity capital asset pricing standard return index keep change stock adjust service database series rate factor return portfolio test model window previous suggest efficiency dependent reject less factor except financial reject decompose q beginning observe cauchy schwarz simple together jensen inequality separately lead theorem eq old proceed get next eq lemma inequality conclude complete eq right decompose bound rx side taylor lies represent equal ij ij summation q eq q I low order exist order due utilizes dominate since bind remain similar obviously far show random constant give bind facilitate technical proof ij ki q marginal q prove get q easily convex monotonically increase employ maxima last jensen side show choose due actually sub omit follow ij claim minimax detail sequel leibl divergence distribution go sample eq di inequality end together square employ jensen inequality prove bound employ reduce begin prove constant leibler indicate zero mean obvious degenerate comes perfectly correlate since use instead lead use technique display imply reduce testing matrix yield enough low fix recall support give follow resp collection
weak exist hold speak optimization iterate process induce rigorous pd proving conclude contradict indeed suffice independent express convenient note correspond vanish vector independent set vanish show inequality state summarize initialization iteration large among optimization radius absolutely adequate linear quadratic convergence generality motivate conventional performance denote complexity relate radius iteration apply arguably property know iteration see particular radius highly inversion iteration matrix inversion sum sequel extremely fact derive inversion function combine equation must invertible yield us initialization point coefficient matrix eq shall omit inversion coefficient optimization partition readily complement verify plug implication polynomial quadratic polynomial technique goal last chapter show polynomial degree definition degree yet polynomial much small much must lead inversion newton complexity invert regular read thus say approximate radius iteration would reason decide inversion fact balance many algorithm limitation induce imply spectral radius scalar one chapter obtain wide family inversion whose depend derive iteration lastly address sequel spectral evaluation real q chapter simple designing efficient attractive minimizer strong coefficient obvious fundamental algebra root denote root therefore complex eq first equality line employ triangle economic indeed equal polynomial big summarize coefficient bound tight requirement coefficient real polynomial although notice claim regardless scalar value spectral positive us eq polynomial establish get thus plug inequality yield likewise range regard assume rest tight monotonically argument harmonic plug inequality yield thus rate inversion matrix big analogous sequel convergence inversion well rate inversion matrix positive definite follow argument assume convergent eigenvalue thus accord scalar q prove embed sub previous section tight heavily rich coefficient attain thing get radius ball although class disadvantage general strongly disadvantage priori convergence factor contrast attain factor furthermore ball scalar inversion surprisingly match range chapter ball perhaps algorithm carefully remark mention well importantly latter oppose low iteration say prove restrict differently show iteration relatively section coefficient scalar inversion disadvantage ideal algorithm execute spectrum finding optimization inversion whose answering efficiently computed coefficient conjecture tight state optimization matrix coefficient matrix scalar pure polynomial observation base mention vast conjecture polynomial expression coefficient straightforward convergence question root polynomial polynomial meet prescribe derive correspond remarkably enough adjust function discover systematic turn particularly due nature let iteration say admit say say scalar inversion efficiency constraint obtain consequently iteration efficiently coefficient lastly inversion sequel shall closely admit denote iteration matrix inversion coefficient respectively lastly give show polynomial lemma attain radius polynomial enough nothing eigenvalue order arbitrarily coefficient perfectly match polynomial latter unitary diagonal diagonal eigenvalue equation correspondingly apply equation converge choose equation grow linearly properly major drawback impose task inverse impractical spectrum algorithm similar fairly kind correspond various include versus present previous chapter demonstrate superiority present section suppose comprise coefficient exist apart inversion exhibit appeal canonical recall consistency rule express definition substitute counterpart obtains analyze matter future designing optimization matrix scalar bad principle maintain analyze satisfy verify iteration exactly see rule eq bound preserve path polynomial achieve economic section namely hope root yield equation q eq multiply remarkably extract accordingly yield lastly see interval ensure eq heavy spectral exceed convex combination straightforward verify conjecture look optimization convergence rate bad barrier focus analogous state polynomial tune spread present demonstrate idea let q result optimization specify illustrated expect counter decrease along algorithm initialize iteration contradict exist suboptimal contradict upper readily complexity eigenvalue avoid later sort use nesterov sake densely discrete approximation decomposition chebyshev kind nesterov dimensions degree exploit shape spectra contrast clear optimization strongly lastly applicability heavily frequent application form world iterative perspective derive one algorithm mathematical generalizing question radius answering question unify many optimization outline smooth strongly design polynomial improvement spectrum gap adjacent spectrum easy vector might applicable essence one analytically quadratic function technique counterpart quadratic beneficial unbiased estimator inversion give rise whose preserve expand scope method regard inversion matrix scope sag particular replace less batch simultaneous really lastly characterization modern implementation refer enable algorithm minimization get field learn understand convergence task say inversion believe likely entry dependent quadratic unclear light one execute optimization part adapt part eq denote scalar assign notion science many brief elementary terminology result thorough value exist continuously mainly twice differentiable convex follow useful characterization kind twice continuously differentiable smooth assign positive definite matrix strongly appendix condition coincide surprisingly shall soon scalar convex characterize optimization process presentation sag paper strongly convex optimization however directly would involve expression apply equivalently rewrite transformation change let compactly define define consider asymptotic certain scalar scalar thus inequality logarithm side rearrange similar proposition exercise remark thesis develop convex optimization focus examine application turn powerful analytic whereby particular new natural nesterov accelerate employ economic rather interpretation early solid iteration low regime contribution summarize operation optimization iteration mention early attain convergence state algorithm consequently restrict accelerate heavy potentially natural present new scheme offer exploit exist bound new presence huge lastly tight suggest extension framework analysis stochastic sag accelerate sdca etc thesis dedicate whose would thank remark great unconditional year real value problem science economic express minimization significant hard structural well algorithm hilbert wide range make fast kind say solve function accuracy arguably prove attain answer question exact widely accept science analyze optimization part issue box assume acquire round carry thus show obtain smooth algorithm smooth employ order receive oracle number query minimizer exactly bind seem mechanism oppose huge admit structural hold contrast exploit assume structure consideration technique approach modern certain class optimization reveal number design concern optimization algorithm closely know gradient descent gd optimal suffice researcher lead discovery nesterov see slight gd q unfortunately strong gd primarily base sophisticated researcher field e scenario task admit remarkably way derivation idea present ascent sdca solve minimization great sdca enable derive rate thorough sdca repeatedly pick denote refer careful convert quality n q obviously take step coordinate variable process govern eigenvalue straightforward calculation normalize eigenvalue plug bind eq inequality distance less must sdca tight remark loss smooth require motivated major part generalize aspect precede argument work appearance chapter introduce terminology convergence low bound bounds radius corresponding elementary theory appear form chapter framework essence quadratic bind plug derive bind bit entry denote column last furthermore q compact conclude sufficiently yield grow linearly spectral asymptotic bit grow scope corollary statement statement readily theorem endow induce notion series specify establish property let iterative vector pd z fix convention product multiplication carry I linearity inversion accordingly omit take z convergent conclude convergence may together property q naturally radius say derive equally question arise convergent differently vanish interesting vanish denote hold km recall conclude access iterative method stationarity iteration matrix lead theoretical radius nonlinear section evident method determine rate spectral symbol order worth point manner perhaps direct proof eq equality let follow matrix q conventional see matrix satisfied assume form simultaneously invertible invertible eigenvalue arbitrary denote eigenvalue order consequence remark simultaneously invertible importance derive understand may
atomic programming treat symbol mean representation character mark character nlp improper nlp character modeling program example token c code refer double share character return code nlp representation token nlp type double etc representation severe language word token improper level problem token token indicate express explicitly compressed abstract node constant state compress token furthermore node program fact distinguish structural semantic example see nested loop inside comparison follow assignment implementation sort program code detection program task theoretically map directly composition nlp state compositional semantic roughly capture semantic barrier overcome theoretical formalize code build representation learning symbol similar symbol aspect refer factor define intuition symbol reference programming criterion via layer node experimental leaf p primary leave notice number hard dynamic pooling take maximal mathematically continuous bag pool totally satisfactory assign parameter position regardless treat child gray bar position model information code network closeness euclidean likely representation sample negative pairwise representation symbol training substitute symbol least large often set error prevent fitting add weight overall objective compound constant cast compound compound assignment compound compound break break switch constant root weight hyperparameter code gradient momentum l randomly sample formula derivative accordingly code error propagate learn node speed adopt momentum derivative current first epoch axis epoch cross compute q sample cv respectively label actual coding indicate effective program deep blue demonstrate initialize architecture gradient vanish poorly contrary representation serve initialize coding criterion cv decrease drastically epoch high performance unsupervise report rbms autoencoder generic mnist handwritten digit explore give train result report field evaluate analysis classification adopt logistic regression accuracy support machine radial rbf kernel explore linearity improve explore underlie improve accuracy experiment cc random guess logistic rbf learn evaluate empirically near neighbor program experiment beneficial supervise propose result confirm program evidence literature make artificial intelligence believe become various promising tree extent possible perspective treat adopt traditional usage representation atomic symbol nod statement lose neighboring pattern dimensional meaningful code program perspective feature suggest technique computer foundation cognitive interestingly inspire deep achieve high architecture network high human cnn specify explicitly neighborhood neighboring circle line detect convolution kernel feed layer abstract abstract beneficial task object domain acoustic add cost function slow local neural reasoning base though trivial property mathematical give false important program beneficial point include code learning question address remain unknown literature perspective program integrate program question apply field besides novel code build deep analysis feed successfully relationship term criterion building confirm feasibility program primary success address problem program source consider field intelligence primary learning near liu xu zhang edu cn com make field artificial intelligence ability complicated feature engineering etc impossible deep program architecture pure back paper criterion program representation learn reality qualitatively experiment code building program evaluate beneficial feed representation achieve support confirm feasibility analyze program also give primary success new analysis language conclude program rich capture justify learn become one machine variety processing speech recognition compare traditional deep major architecture capture highly complicated non efficiently real world human engineering interestingly unify deep achieve application program deep practically infeasible analyze neural back propagation gradient vanish architecture extract poor vector abstract architecture directly reality node value element certain analyze representation qualitatively criterion successful analyze feed accuracy feasibility program light code dataset project website future analysis source code contain feed network neural architecture good first program first analyze program deep field contribution include introduce technique field propose explore program neural code motivate explain experimental analysis draw section traditional engineering g consume specific evidence suggest may et nlp application construct expert automate human deep neural easy example deep extract organize local category human engineering moreover decision classification task approach automatically automatically recommendation predict code nlp short neural network capable capture complicated feature program interesting promise deep neural network capture complicated feature practically analyze program symbol discrete symbol symbol feed possible symbol characterize symbol also one code mean cluster direct pure back poor optimization poor deep alternative learn representation regardless detection fed benefit focus language programming language nlp improper language one flow always code indicate branch loop extremely read language programming notion concrete program source build node nlp algorithm flat build fact motivate research representation program eventually make analyze program consider literature deep progress deep neural widely technique artificial comprehensive neuron build neural network input figure typically compute activation non etc power insufficient world back propagation stack multi neural power sufficient boolean inefficient grow exponentially order complicated layer raise generalization single neuron circuit architecture deep organize abstract high layer architecture capture abstract efficiently make train year architecture al stack layer stack rbm feature energy stack autoencoder minimize stack rbms autoencoder neuron initialize learn initialization back specific learn result meaningful
cloud transformation desirable property consistency relative say follow transform relative transformation consistent invertible transformation sake completeness consistency ij attention invertible definition ii transformation ji reference coordinate tool derive frame eq linear transformation set note due hold dimensionality k make multiplication right see contain express requirement suppose empty contain decomposition rewrite state contradiction state invertible retrieve find let td give retrieve block create call identity point dimensional instead square optimisation rank nan retrieve svd invertible affine invertible affine invertible represent similar construct proper affine transformation simple transformation homogeneous row similarity transformation transformation isotropic rotation retrieve estimate affine extract factor use I isotropic solution onto orthogonal matrix scale retrieve align onto arithmetic mean jj similarity isotropic transformation transformation extract transformation ensure determinant equal generate compare error solve generalise problem correspondence assignment ground truth generate eventually transformation generate turn transformation j j truth dot transformation entirely particular factor translation part arbitrary allow factor da dd uniform interval random singular dimension pairwise solve one wrong correspondence simulation shape noise shape comprise dimension find similarity transformation good shape subroutine reference perform symmetric factor one shape randomly select shape reference shape reference select shape update solution first follow aggregate contain propose transformation enable simply miss wrong experiment shape miss point point discard wrong randomly wrong using investigate original shape shape average shape set j every additionally common occur frequently must order determined point shape base mean missing see increase amount slightly miss point marginally however reference base mean runtime pdf pdf pdf correspondence shape shape comprise point dimension column correspondence assignment shape keep coherent wrong green additionally point wrong assignment wrong wrong shape aware result transformation extent transformation shape reconstruct additionally shape assignment align consider correspond previously correspondence iterative wrong shape level different level wrong marginally method result use alignment iteratively bias reference completely depend underlie transformation retrieve approach permutation generalise transformation experimentally effectively reduce set pairwise generalise approach correspondence remark frank centre de frank sciences de alignment object play object solve globally practice well relative transformation free however wrong may fail observation free retrieve nan matrix directly present transformation demonstrate noisy transformation even wrong correspondence encourage alignment object transformation play recognition statistical shape shape remove difference order common shape cloud shape vast research field establish remove scale base unit dual reveal robustness method object computationally expensive align object object induce align object reference nature constitute noisy wrong transformation observe
optimize pairwise mapping membership within variance minimize identity dissimilarity dissimilarity som problem triangular major inequality q som optimize upper cluster orient quality dissimilarity practice quantization give dissimilarity triangular one prototype far thing prototype som quantization rather way make explicit way suboptimal point dissimilarity prototype display put som particular quantization quantification induce restrict point quite describe way address observation let point equip inner squared distance matrix classical combination sum directly keep coefficient perform dissimilarity square variant som batch pseudo euclidean derivation amount etc relational proceed iterate batch som main prototype represent combination euclidean prototype update online som equal notice preserve update tend less sensitive relational som computational cost relational som incorrect gain som solve som quantization induce point interpolation som availability bad relational cost operation dissimilarity coefficient cost batch stochastic som som approach approximate explore pointed relie provide quantization cluster try solve without rely scheme dissimilarity deterministic annealing introduce cluster update anneal role gradually increase practice soft mapping procedure annealing iterate evaluate reached increase evolve som implementation obtain dependent anneal scheme miss al analyze temperature dominant dissimilarity minimal critical temperature addition full update intensive note update reduce relational som clearly try criterion som proxy relational som easy deal enable machine leveraging allow implement method som fundamental enable batch som value combination som hilbert implement som mean map become trick solely knowledge trick som notice use need coefficient som trick enable som som equation sake completeness som propose vector kernel som suffer constrain make dissimilarity guarantee equivalence find compact som limitation som distances scheme propose relational som som median som som relational som som even relational one numerous dissimilarity som section opinion possibly burden relational som dominate median som suffer performance som truth capability prevent display gap natural cluster matrix visual increase som median som see well remain test nystr approximation study extensively relational neural systematic som som som converge lead note analysis random initialization well e initialization case batch som point anneal schedule strong batch som som obvious online som batch som epoch online som presentation point roughly som online small epoch complete cost som contrary relational som per prototype lead per report prototype place presentation epoch online som cost batch relational som careful relational outperform online initialization properly comparison variant anneal missing simulation conduct one sophisticated anneal deterministic anneal solution critical largely increase really comprise loop loop give outer anneal iteration order magnitude batch algorithms som also adapt effect final remain summarize opinion prefer careful som pair like validate explain dissimilarity attempt minimize prototype criterion classical shown directly problem k mean simplify give relational burden dissimilarity sophisticated combine art refinement could like would empty organize dimension obtain som interpolation median opinion main som provide rich yet possibility dissimilarity limitation space visually som variant result case dissimilarity u matrix display numerical cluster etc type visualization interesting mainly som dominating anneal version som soft membership situation visualization build som node dissimilarity cluster show som simple graph visualization result generic dissimilarity som somewhat compare som work need visualization interesting display review som adapt dissimilarity follow differ strategy identical relational computational aspect experimental opinion relational som couple nystr dissimilarity som discuss usefulness som representation som dissimilarity improvement output som serve practical purpose beyond elegant paris france numerous represent machine technique really dissimilarity kernel measure object self som review use set outline difference discuss advantage drawback variant actual dissimilarity som practical frequently elaborate form attribute relational different type instance online customer product customer several copy leave review say adapt machine consume theoretical level euclidean view need build generic share datum successfully fairly give either dissimilarity dissimilarity kernel type see mining dissimilarity generic dissimilarity generic typical near dissimilarity self som research operate helpful organize describe general dissimilarity dedicated som som focus modern som presents som extension annealing describe variant som provide som property remark insight som specify dissimilarity kernel classical som dissimilarity matrix dissimilarity point convention number negativity also order som notice function satisfy mx aspect theorem reproduce hilbert mapping hilbert structure build machine rely hilbert algebraic trick construction dissimilarity always dissimilarity som relate counterpart definite dissimilarity dissimilarity introduce som principle som clustered datum specify induce prototype maximally numerous possible function lattice point reduce influence som prototype som algorithm adapt close close goal prototype reflect vice neuron associate cluster essentially major variant vector proceed strategy stochastic online som select randomly update batch som unit obviously neither prototype step som turn operation involve dissimilarity
bad case arise ground state least still solver find low find solver make instance perhaps instance degeneracy gs tw tw tw tw time slightly gradient tw tw interestingly tw gs tw meta technique gs tw gs tw easier moderately gs tw conjunction explanation load color graphic explanation need graphic macro ltb lt lt ltb lt spin configuration construct intermediate series function depend spin configuration adjacent dependency arise precision store spin h hz know combination subgraph sufficient require size representation depend mean grow face relative spin z rescale fortunately range limited locally computable grow energy spin configuration along spin note also h choice subgraph calculation subgraph exponent actual quite omit example temperature certain typically subgraphs method find energy ground spin energy sum imagine e loop arise spin convenient description generally individual function purpose run find minimum energy give spin usual include ground know moderately search low dynamic define apply subgraph define spin state vertex range spin compatible spin configuration spin standard tree make partition choice devote family assume desire subgraph type update spin dependent update appropriate version independently term belief propagation description give cover provide produce I subgraph describe evidence section de observation consider restriction necessary purpose balance rigorous efficacy subgraph induce subgraph induce subgraph reason primary specific reference cpu ghz something platform dependent counting spin spin take fact spin flip spin comparison tend least concerned find execute subgraph sampling involve track combine exchange compare site update combine parallel well significantly good subgraph recall subgraph contain require vertex induce subgraph vertex ignore restriction induced spin pre update subgraph however monotonicity lose induced subgraph guarantee subgraph induce subgraph obtain guide subgraph try good easy solve exactly calculate gibb give subgraph trivial spin usual burn independent operation item demonstrate evaluate configuration sum whole detailed balance clarity tree take instance tp illustration show work though proceed vertex dynamic collection connect example vertex many order process successive vertex process note graph picture inductive contribute tree interaction quantity maintain vertex tree case vertex maintain add contribution vertex edge join meet construction move vertex boundary new choice old picture build practice compactly spin represent difference reverse pass subgraph care fast implementation straightforward method deal numerical computing generating loop rather slow avoid rescale storing aspect produce carry choose originally aim optimisation wave hardware quantum contain wave hardware aim graph arrange write bipartite throughout collapse highly fact embed applicable hand simply fully dimensional behave critical odd temperature complete effectively fact translate vertex practice temperature rapidly branch method increase explore choose randomly behaviour spin prediction show expectation ratio suitably temperature upper standard well tune space sort meta carlo monte subroutine use top iid assume hamiltonian intend undirected range choice decide upon probability adjacent require probability exchange take random spin overlap excess average average unique might expect take close decrease smoothly lack analogue tw collapse aggregate vertex spin relevant step process reach something close distribution spin exchange step temperature equilibrium carlo increase appear acceptable margin turn far elementary operation approximately compare discussion color conjunction terminal option package graphic terminal graphic macro ltb lt lt lt lt ltb lt lt lt lt bp ltb see load package terminal need graphic macro ltb lt lt lt lt lt ltb lt lt lt lt lt ltb describe problem clearly see fig uncertainty serve compare return assumption equivalence termination comparable parallel tuned way difference negligible fair careful control subgraph site try simulate identical shall mean example vertex shall single site whereby spin immediate though operate one spin outcome operating individually update immediate vertex size possibility external field state small change energy change rescale interpret number mention specific make interesting fair make currently stand method reasonable accuracy respective may fix match underlie try majority accurate optimisation technique great eigenvalue monte sharp inaccurate characteristic much would inaccurate considerably hard still proportion fix rate class consider negligible chance describe general simulation average heat capacity actual simulation lie maximum remain decide trial slightly low justification assume lie couple easy full determined set temperature perform temperature perform energy average whole form choose small time rely purpose provide state assumed estimate week side variance reason break parallel include exchange spin flip device process run computer stage measure consider time respective fairly robust vary much optimisation respective level flip measure intel flip tw tw degree optimisation tw far arithmetic operation simple tw necessary arithmetic loop due level optimisation fast implementation describe enough well tw r r size advantage tw tw useful implementation tw face large four name gs tw tw pt tw tw gs tw tw state finding
resemble pick subset stability correct high taking illustrate simulation section number score interval blind obviously pick asymptotically assumption concern score grow weak theorem apply true sequence unconditional interval law number well denote well must attain unimodal bring advantage covariate optimal behave randomized independence arguably simple setting might somewhat scope belong roughly mean heavy cauchy pareto reasonable instance consider distribute identically orthogonal eventually recovery much weak covariate proportion vanish illustrate simulation error simulate accord cauchy student degree freedom count uninformative score uninformative covariate reach large score joint distribution emphasize direct criterion criterion covariate truth know performance covariate q vector denote additional tailed material coefficient zero entry choose control situation specify covariate covariate correlation among covariate toeplitz correlation close covariate multivariate toeplitz grouped predictor toeplitz indice informative consist index interval informative covariate covariate follow loading coefficient realization covariate follow elsewhere target break dependence covariate adjust ratio section truly informative rank retrieval compare choice lasso method consider subsample randomization base covariate regularization method specifically grid regularization equation pick fair covariate l informative amongst rank rank estimate report average repetition systematically outperform lasso reason difference importantly evaluate lasso single selection coefficient separately successfully select informative coefficient coefficient noisy true informative covariate systematic appear set material possibility modify rank magnitude thresholded show property precision remarkably stability plain qualitative stability limited range toeplitz false positive average positive preprocesse step investigate whether variable method use image handwritten digits dataset external interface heterogeneous come generally contain goal complexity experiment art wish extension base mutual information select covariate iteratively select iteration computationally costly part replace select result updating score r adaboost mh report result repetition covariate covariate covariate adaboost mh number boost lead significant main conclusion small subsample final subsample less memory furthermore base observation cost parallelization easier disjoint small covariate prediction comparison stability overall relevance support stability concern subsample small generalize error insight subsample practitioner version need obtain false regime result positive appear loose far refined concern procedure covariate theoretical toy certain circumstance heavy tailed search increase particularly appeal practitioner largely experimental precision show whenever precise method analysis give dependence precise rule choice subsample size far hold choice place b numerous discussion author notational shorthand x ratio repetition simultaneously relation repetition observation disjoint joint thus variable binomial theorem relate upper bounding upper xx lead desire follow completely random would covariate conversely one wherein minimum conclusion establish end imply distribution slowly lx belong real deduce rescale slow variation negligible disjoint marginally converge distribution imply continuity value low eq indeed remain conclusion still similar argument extreme introduce follow disjoint purely formal sequel bind note inequality hold second factor term enough converge variable arbitrary conclusion eps fill line microsoft ec base repeatedly subset covariate effect benefit validate method covariate carry information selection aim goal identify informative insight outcome identify focus informative variable science broad field share drawback unstable different come vary significantly prediction stability selection half final picking exceeds expect positive guarantee choose remainder paper variable selection base box propose stability direction half instead draw precisely randomly observation extend approach stability theoretically simulation compare semi secondly empirical comparison randomization covariate base full generality covariate interest covariate false influence besides toy perform one investigation subsampling combine stability apply observation goal reduce reduce complexity linearly particularly base load small furthermore allow parallelization processed expect generalize suggest improve subsample large independent finding insight subsampling method empirical comparison subsample stability randomization base take covariate theoretical certain covariate subset randomization improve description algorithm present motivate observation investigate randomization give select informative work dataset contain observation covariate time random partitioning threshold need enter observation subset size base covariate stand stability pseudo note precisely covariate subset per initialization frequency lt various way idea subsampling replacement strongly relate developed combine subsample investigate bootstrappe variable cox data final machine apply data importance apply lasso bootstrap selection include measure combine give simplify assumption study practitioner complementary variant selection bound selection false negative complementary genome discovery covariate investigate decision build regard training classification test drastically cluster selection selection method give method use stability variable usually method might outcome subset statistical typically discrimination covariate perform simultaneously assess conditional base original aim covariate much fast selection globally maximize select covariate large mutual already latter simple namely minimal candidate covariate add covariate individually rather simple information estimator speed find solver variable zero additionally relevant lasso outcome solver analyse random section observation full unknown subset section draw disjoint equal denote subsample index obtain define performance need set covariate exclude false since box average sample method frequently irrelevant one draw independent random define use rank covariate relevance define uninformative selection definition base reflect subsample theorem positive uninformative false expect positive base equation correspond negative dp leibler bernoulli depend choice denote choose equation corollary formulate suppose noise covariate select base large informative l bind well bind allow merely specific allow
beta interval epoch frequency expectation turn truncate posterior component must around establish quantity hence truncate normalize sub interval around chebyshev iid property ergodicity stochastically exponentially decay give along recurrence cycle maximum reward rs final coupling section event c occur precede paragraph enough parameterize parameterization induce across reward observe transition yield useful unobserved mdp thompson parameterize derive frequentist general parameter space result suboptimal probability encodes term kullback learn interaction environment act notion environment mdp comprise state difficulty reinforcement learning stem primarily uncertainty essentially plan face environment structure g reward transition need accumulate current influence exploration exploitation efficiently modern underlie maintain confidence transition instant action environment mdps freedom mdps observe instant transition motivate control queue queue instant either fast rate govern service arrival queue cost service hold alone mdp example arrival conceptually learn structure posterior thompson sampling impose prior uncertain mdps parameter prior compute reward state transition prior rule main thompson mdps reinforcement parameterized mdps operate posterior cycle throughout cycle sufficiently space initial prior sufficiently large neighborhood parameter logarithmic thompson without closed path scaling admit rl constant involve kullback leibler geometry weight divergence true mdp mdps discuss detail encode thompson bandit set significantly improve space advantage possibility state queue dependent parameterize arrival appear significantly distribution like thompson difficulty encounter algorithm latter algorithm theoretically construct optimistic tracking confidence thompson analytically tailor often complicated exercise posterior quantify thompson evolve parameter potentially convenient manner pose exist thompson bandit degenerate special mdps rely property action close conjugate additional arise generalize bandit iid reinforcement evolution couple evolve make evolution especially challenge thompson scheme thompson study purely mdp completely former work establish parameterization parameter latter mdps continuous frequentist regret role merely use depend explicit mdp parameterization work overcome derive directly normalize kullback accounting adaptively self normalize concentration sum cycle interest mdp help cycle measure transition mdp initially parameterization factor action reward mdp extension control play every serve policy kind threshold mdp discrete stochastic process ta tr denote htbp space output action time stop epoch probability c repeat bayes end horizon mdp denote assume broken fashion correspondingly term reward random contiguous epoch turn version use time marker maintain denote time epoch use sample mdp update via bayes effectively epoch begin state need hold recurrence mdp ergodic chain mdp recurrence time log ratio upper primarily control divergence kl divergence employ average mdp reward merely ease exposition concern behavior state immediately follow policy apply mdp correspondingly important leibl bind parameterized mdps kl divergence convex appropriate mdps policy across set mdps average policy fix comprise mdps average resp correspondingly parameter resp mdps region time belong occur et n playing policy begin epoch instant measure express solely count weight frequency correspond policy ideal trajectory scalar truth true specifically decay weight kullback leibl condition use consistency think neighborhood exist decay mass around neighborhood play policy finite section space satisfy typical root top type assumption policy dependent constant interpret r main leave reader interpret game policy negative epoch play policy coordinate policy irrelevant far throughout eliminate sense record impose growth occur dimension vector time policy eliminate maximize final optimization thompson square scaling usual regret mdps hypothesis least rs compare diameter mdp follow mdps gain obtain mdp finitely finitely bind suppose assumption assumption mdp quantity generality main thompson absolutely continuous lebesgue ease exposition mdp theory finite mdps irrespective action transition mdp parameterized take mdp optimal imagine recurrence density cube mdp loss policy check setup establish mdp small hold consequently conclusion hold directly kullback measure policy improvement due marginal kl divergence factor additive suboptimal apart kl divergence nearly significantly simultaneously term differently bad like uncertain thompson counterpart like force explore essence thompson lie evolve exposition finitely parameter expression ct ei ei depend evolution replace empirical average respective policy tv ct ct simply environment approximation insight shrinkage horizon property mass truth imply suboptimal mass less pick thompson scale insight estimate time bad end c c interesting sample count policy zero ct ts path across arrive though argument coarse quantity rigorous indeed technical tool tailor thompson mdps include inequality iid quantity using leibl establish frequentist line example buffer customer queue action action slow fast service slow resp service queue probability empty bernoulli type instant cost hold gain queue service however cost restaurant order hold model importantly arrival parameterize use confusion dimensional rate curve constant valid arrival depict policy parameterize arrival simulation parameterize fix dependent uniform discretized result time regret across horizon increase advantage confidence visit htbp single queue mdp regret line demonstrate thompson enjoy regret bandit et al relevant study mdps true mdp space w prior useful arguably weak notion influence performance moreover episode oppose set treat sampling mdps investigate nature prior deterministic uncertainty rl frequentist setup algorithm maintain interval optimistic adaptively shrink confidence interval time state occur inefficient mdps parameterize al mdps discount set pac different notion equivalence approach learn mdps plausible model available suffer serious lack adequate low seminal paper lower parameterize reinforcement sense finite space though crucially sharp thompson favorable continuous bandit parameterize rl derive bound pseudo reinforcement would performance happen feedback delay application reinforcement particular regret thompson function mdp term correspond could variant thompson path material thompson decision express iterate weight simply mdps ct tc dynamic follow index policy generate stationary policy initial c cs thus early simulate mdp round epoch index next resp resp record henceforth ease concentration define empirical j transition u marginal frequency u conditional whenever virtual deviation mean logarithm empirical constant probability argument finite irreducible chain recurrence expect negative iid markov cycle satisfy ergodicity conclusion appeal maximal concentration inequality lemma constant sample iid analog iterate maximal concentration increment event moment neighborhood taylor around consequence martingale turn tail half fashion henceforth parameter x typical trajectory previous q kind regret sample constant time sample exist denote log write assertion log universal eq see complete proof n constant guarantee proceed I instant optimal result policy integrating give decay exponentially expect also proposition begin iterate estimate far conditional sum finitely take complete help refine compare hold usefulness stem hand help kl cs cs derivation right side inequality ax c x step find maximize far
replication correct rmse stein wrong pool seem somewhat simulated make rarely underlie effect size misspecification appeal among group gene level gene control gene cell control cancer gene large correct generate I performance size difference estimate adjusted large quantity correct average training split use c stein htp stein positively contrast bias numerator adjust k contrast numerator rmse uncorrelated positively rmse rmse oracle statistic effect statistic oracle rewrite show next assertion definition differ location index statistic lemma loss generality argument calculate frequentist q converge zero notation use throughout th statistic size obtain false oracle estimator equal theorem corollary remark interest gene dna sequence datum naive suffer chance alone size independent poor bias without simulation context thousand measure group reject nan goal feature effect study hypothesis practical true throughout manuscript hypothesis effect chance large effect size among setting dependency among expression approach case build bootstrap study among inaccurate one strength largely assumption proposal proposal broad normality make consider statistic manuscript paper review selection introduction marginal density refer correspondence bayesian estimation refer notation manuscript denote tie e index intuitively quantify small negative effect oracle estimator bias practice parametric make estimator dependency among later failure dependency inaccurate exist principle could exist accommodate dependence immediately obvious tractable account dependency provide framework k approach involve large tb datum bias small bootstrap bootstrap bias apparent adjust accurate data htp size size code red estimate code size bootstrap estimate stack along estimate histogram set difference effect parametric briefly assume availability generate htp estimate calculate observation generate step algorithm multivariate generate draw heavy rarely datum challenge approach perform know generate repeat dependency implicitly independent observation replacement calculate estimate base generate favorable genome wide association study nucleotide regression new resample observation replacement logistic compute snp properties oracle equation simplicity estimate normally relate bias among bias amount mse relate simple mse estimate mse square bias hold among correlation assumption throughout normally various emphasize covariance correspond order statistic frequentist statistic bias estimate consider scenario equal scenario variance explore size increase bias since increase adjust advantage approximately approximately adjust frequentist selection unnecessary proposal relative exist intermediate assume two cluster increase bias estimate b direct size within truly jk q uncorrelated lemma dominate assumption motivate correlation among correlate statistic study account correlation among statistic inaccurate effect simulation study show statistic generative datum compare proposal study bias know use equation correlation bias equation spline five nonparametric stein positive stein consider version generate generate datum correlate specific present context correlation among test adjust effect mse estimate numerator rmse value large one indicate adjust perform estimate respectively generate n p ar block ar correlation study htp leave middle ar block correlation contain element equal ignore correlation test lead inaccurate effect normal distribution use report large average replication ar uncorrelated outperform account bias around keep decrease oracle oracle rmse similar oracle lower spread true end bias less oracle tend rmse htp small replication simulation section generate set cc stein htp block ar cc stein
account bundle expectation brain choice already mention one illustrate study study activity decision activity area could difference event reveal model optimal exist rather reinforcement brain serve policy target depict brain put along mostly td actor value td latter actor algorithm tackle belief decision shape state much reward belief attempt novel actor release basically set space mathematically support choice instance could comprise ultimately draw actor since activity direct understand utility consequently evaluate end many remain ahead static essence section type stay pass base prediction calculation one future process tree aim study modify general actor future agent model action depend sort action outcome deal amount feedback controller design operator monitor actor choice experiment actor promising situation exploratory actor actor composite actor action transition current environment state reward account immediate future receive composite recursively next state although generalize represent temporal amount eq parametrize error modify take grid take move towards goal side shape associate action ask reward movement way come pathway bold straight line policy first time reward find pathway mid execution te algorithm inspire assign behaviour importantly transform action worth actor function execute make interaction distinct trend assign actor mechanism incoming represent process way probabilistic structure wish framework element find prior come pathway trial author comment aim find behaviour actor ever process concern finding suggest existence belief execute pathway sub offer modify actor light importantly resolve refer challenge actor lack task actor keep encouraging study particularly affect great deal field experimental economic cognitive social brain multiple understand optimal course mention reciprocal process cognitive provide great insight determine economic group uk
addition solution common correlation magnitude select covariate scale sense covariance residual sum significance newly constrain fix adaptively appealing impose reduce issue affect relatively regularization parameter suitable lead oppose covariance covariate model appeal covariate enter covariate establish appeal verify covariate condition require sure insight consider true sample row I copy error deviation range lar generate screening screen event strength sure estimator large probability increase factor screening generalize characterize consistency lasso model asymptotic intuitively put noise one example condition wide regularization lasso event implicit strong contain sparse lead misspecification apart misspecification misspecification inference recently reveal true play characterize impact misspecification misspecification ingredient admit nice asymptotic penalty lasso variability selection include scad mcp question statistic testing would possible extend lasso corresponding regularize question generalize gain insight let admits bind false vanish panel df example penalty combine model choose tighter sure seven enter generalized slight abuse reduce constrained case compare chi df combination combine become top panel case show fit panel conservative interesting seem latter combine interesting transition phenomenon interestingly phenomenon relate model high dimension remain development comment nsf dms valuable covariate selection huge grow devote different study
come np ic np ic x ic jt rt lie impose utilize global color carefully choice always segment coarse heuristic provide parameterized eigenvector neighborhood scalar dependent represent exclusive point merge tend member converge mode share bandwidth iteration trajectory get cluster initialization principle member trajectory trajectory might merge start trivial trajectory else current point include trajectory c basically member trajectory location location member comprise mode position indicate arbitrary neighborhood u refer member u u x u u eq u ne xu ne xu continue u u u pt mm subspace extended equation kernel domain need set shift sec analogue g ne ne xu u likewise mm agglomerative iteration shift compute neighbor iteration merge criterion cluster within merge merge merge bandwidth converge perturb bit equation neighboring point point reformulate local trivial neighborhood cluster bandwidth neighborhood particularly likelihood denominator normalize fix come u update bandwidth eq shift update bandwidth fix utilize fix begin employ shift moving mode soon trajectory significance weighted point indicate confident use cluster confident choose base evolve scale drive adapt local orientation become oppose scalar partition bandwidth bandwidth member point subset consider representative joint density represent structure asymmetric arise shift mode converging immediate evaluate use across basically localize fit numerical stability update decomposition fall trajectory mean trajectory concentrate conservative also h effect color plot varied effect cluster plot image ms color domain image quantitative vary except enforce see smoothing level intersect datum proximity trajectory could cluster eventually end converge local basically merge high mode cluster mode help function return implement space stability work basically distance contain direction divergence measure metric information theoretic like partitioned post step mode ensure conventional shift post additionally could adjacency add naturally mode mode ensure divergence merging variance mode proximity somewhat specify adjacency density location form additional separate adjacency close adjacent cluster remain representative influence cluster mm ref ref ref ref ref use image maintain segmentation segmentation method color ahead close operate merge operating domain indicate quantitative qualitative effect variation although show value give monotonically cluster remain couple rapidly iteration net happen point iteration serves offset computational bandwidth straight implementation shift scalar improvement near search exploit domain hash delta end merge fine tune shift point perturbation robustness salient result perturbation result label label label affect salient keep still place varied typical method reduce break boundary break boundary maintain segment simulate gaussian respectively position ms comparison show domain segmentation training image together completeness segmentation also indicate search segmentation well low salient test mixture vary reasonable local mode salient adapting set comparison domain isotropic bandwidth near neighbor kernel mean shift indicate lack dataset feature uncorrelated uninformative processing attain ms qualitative quantitative efficacy scheme varied leverage adaptive shift unsupervise adaptation shift cluster point normal video indicate issue growth shift focus agglomerative edu com adaptive shift rgb font color font font color font color font plus minus pt pt widely detection due adaptive methodology allow evolve adapt turn methodology space preserve due issue though allow effective introduction shift unsupervise reference establish utility low level feature popular cluster segmentation high video scene parse segmentation improve utilize fix scalar bandwidth proper heuristic flexible lack cluster offline automatic smooth affect
mind derivative entity intend proof physical find well ht considerably come baseline operating protein protein less classified classification mark tendency protein large discrimination correspond contact discover protein locate peak specifically address protein analyze completely extreme increase high density preference give simple raw observe contact scaling perspective topological contact graph structure perspective adopt discrimination let move obtain preliminary aforementioned ds analysis satisfactory weighting indicate substitution perform substitution ds denote rate ds drop ds dataset induce pure discrimination viewpoint ds stress obtain representation could technique ds test set drop consider system easy justify operate space worse easy operate basically physical arrange different pointed contact adjacency sound descriptor structure worth make contact justify also cm cm cm ds ds ds g regardless protein entropy deviation ambiguity g analogously protein although well discrimination fact protein ambiguity herein interpretation toward protein topological architecture represent ds dataset value show operate aim quantify discrimination full protein structural complexity descriptor achieve training set instance split ds nonetheless consider structural descriptor representation complex characteristic ht derivation insight protein aggregation drive paradigm protein pure experimental base pattern recognition capable geometric recognition pure correlation substitution cost existence largely physical code protein interpret transfer transfer demonstrate combination mutually root contextual balance aggregation mode dramatically shift single mutation ref parsimonious force represent component angle preference represent confirm picture protein upon balance contact recognize importance vs note demonstrate baseline discrimination domain organization describe constitute topological consideration among accuracy protein structural constitute evident error ds achieve operate discrimination ds obtain weak conclusion regard ds operate context use protein spectral moreover achieve ds tell interesting ds reasonable accuracy protein topology topological effective ec relate contact ec allow modularity ec align ec similarity complement pure topological ec information show significant reverse predict ec compatible result proposal refine physical pca impose evolution obtain discrimination power hard task odd et adopt contact adjacency obtain characterize structure markov complex reach stable preliminary similarity protein regardless degree devoted purpose discrimination degree point contact play protein play formula loss atom yes alpha end focus actually analogously happen structural formula much sophisticated play physical description opinion protein integrate heterogeneous derive valuable investigate phenomenon different pattern device gap orient classical approach architecture vertex quantification build relative entire cell standardize hardness superposition drive force intra inter interaction shift phenotype point come aim compare different protein contact able identify interesting general notably consistent contact discrimination present paper prove relationship description develop discrimination structure protein encode structure paradigm unfold experiment certainly paradigm principle matter fact protein specie require intervention protein correct drive force aggregation formation within interaction shift equilibrium correct interaction balance subtle single point mutation important stress protein physical aggregation disease aggregation protein great upon explanatory study author fair reading gene genome sequence translate protein cell free condition protein experimental intrinsic toward aggregation intrinsic give feature ph keep concentrate solely devise matter protein amenable al aggregate clearly aggregation aggregation estimate relative aggregated author bi modal consistent protein protein intervention avoid aggregation bi modal character perfect peak fuzzy boundary behavior trend protein predict sequence structure protein one frequent population fold beyond obtain good physical representation possibility physical obtain alternative protein protein analyze recognition operating pattern representation conventional mechanism input offer interpretable exploit infer drive possibility reach aggregation constitute progress evidence sketch hypothesis model computational successful vs able single important dynamical threshold ii highlight relevance contact discrimination contact spectral structured dataset protein representation start introduce basis describe framework experimentally evaluate graph represent protein present discuss statistical analysis demonstrate iii focus point bi modal largely protein report protein protein dataset unbalanced bi modal character make selection threshold protein interval protein dataset would protein respectively provide basically symbolic e character identify ht experiment organize consider balanced dataset protein high protein low degree simplification original analyze achieve elaborate paper protein respectively place protein considerably large protein consequently denote ds largely unbalanced concern datum protein suitably represent pattern protein ds character show ability sequence symbolic successively substitution sequence matching component derive descriptor dissimilarity protein assess mean stre minimum alignment operate globally operation quantify involve transform usually dynamic programming quadratic process e string general cost overall distance worth odd protein comparison limit successively try structural side similarity recognition system algebraic use calculate deal inner construct eq gram pairwise euclidean center squared evaluation evaluation high determine similarity input pattern converse obviously length use sequence e predefine hoc substitution characterize operation aforementione mathematical correction make sure regardless character process base different weighting scheme ds unitary cost substitution cost substitution cost negative position determine search specifically tune hand encode lying mutation genetic vector solution instance evolve determine genetic algorithm implement check fitness change ds process graph kernel coverage classification system ds ds sequence use among substitution huge difference contact map share modular pattern protein existence suggest common possibility rely homogeneity consider pure topological widely scientific measure ds complementary way enyi chain stationary ii computing recently ambiguity root fuzzy set interpretation concept provide interpretation state ambiguity instead chain completely equilibrium see stationary always unique chain define order enyi order enyi know upper distribution maximally tend converge degenerate associate visit ambiguity uncertainty fuzzy ambiguity fuzzy hypercube encode map type fuzzy membership consider basically vertex union disjoint among fuzzy fuzzy q membership fuzzy generate notably account give term centrality fuzzy uncertainty ambiguity compute monotonic non representation ambiguity value calculate combinatorial problem assume maximally maximally accordingly descriptor topology characterize topology classification accuracy result obtain herein assess relevance check possibility learn physical reach canonical describe physical solution globally variation pc pc
effect bar privacy increase add regression select correct plot analyze specificity increase become additional effect also observe decrease specificity explain candidate namely allow effect term perform regular sparse private penalize penalty small regularization bound observe specificity e amount regularization begin sensitivity specificity decrease correct differentially regression show condition snps bar recover term middle bar correspond bar main specificity simulation elastic respectively sensitivity specificity aggregate genomic collect attack propose differential privacy come privacy arbitrary extend end end differentially procedure convex include parameter noise sample regression focus penalize widely analysis identify disease show applicable datum privacy utility risk tradeoff identifying decide guarantee allow release information concern decade function strongly subgradient w notational extend include v mh h ng lemma differentially private successive private even necessarily function dt dl l singular value strongly nuclear f therefore define namely distribution sphere absolutely variable freedom unit sphere science technology publication attack considerable develop method datum preserve end private regression focus penalize widely identify disease show private elastic genome wide genetic disease typical information thousand nucleotide snps associations snps certain phenotype phenotype relationship multiple snps problem approach dimensional popular select phenotype snps association impose penalty competitive researcher statistic snps individuals participants privacy challenge publication publication attention snp database elaborate genetic turn interest development privacy database external selecting snps approach enable step participant involve elastic snps differentially perturbation mechanism penalty satisfie net way logistic heavily regularization differentially differentiable net extend stability select function penalize validation mechanism slightly thereby accurate penalize elastic differential differ individual differential private q differentially validation let also draw used loss differentially private budget function stability validation give I differentially incorporate perturbation procedure describe regularizer convex maximal singular differentially q differentially private noise always section compare follow function noise j ng invoke obtain ne bb therefore produce result previous derivative iii differentially logistic elastic regularization logistic exist well differentially private section phenotype snp represent minor snp contain snp case control simulate linkage correlation minor allele real snps minor allele frequency analyze allele frequency multiplicative e describe odd q log choose result size snps high association
global feature part clique tree return obviously opinion pool special model indeed kullback divergence approach close expert see weight offer interesting discussion desirable sub aggregated belong author model independently weight fine product unity deal interest manner abstraction model two layer dynamic field movement trajectory record annotated present primitive atomic activity label partially semantic rich interaction ph ph learn unseen adaboost ph marginal perform deep bp method network state node mrf instance mrf total take base fully grid take tree latter per bp adaboost mrf complexities htb adaboost mrf collect scene acquire background primitive activity complex activity comprise primitive activity state generally miss map result htb c tv sequence time specify type correspond transition potential level correspond define ahead look back choose primitive activity correlate five top however instant offer complex activity often period square room window avoid simple indicator overall perform mle method slightly cost slow adaboost guarantee generally mrf bp show generative model recognition justify bad flat bottom layer variant consistently accurate flat encode mrf bp flat adaboost detail randomly adaboost mrf boost capacity boost learner strong reach log iteration condition meet learner previously tree structure span tree attractive guide mle diverse mle share relax share tree rewrite parallel adaboost mrf update time version property wish provide express log pool create mrf perform capacity poor novel boost network adaboost offer efficient maximum tractable network apply new activity recently stress readily appear exhibit tree tree automatically tree plan aspect adaboost mrf wide sign iff inequality induction obtain iff scalar word sign iff proceed prove part trivially ii assume apply rhs yield substitute notation hold complete pair graph q q select q eq previous learner structure except structure boost mrf idea handle thus practice reliable often work adaboost mrf surveillance activity model grow temporal activity address aspect level abstraction popular model level discriminative infer model suffer error propagate direct hmms dynamic discriminative crf general useful activity potentially achieve generative provide surveillance end propose activity problem know difficult crf expressive representation encode complex however great challenge generally intractable exactly mrfs deal efficiency limit happen situation mcmc attractive impractical extremely involve efficient addition learn mrfs explore markov adaboost operate markovian optimum mild discriminative include adaboost mrf label application essence adaboost mrf multiclass boosting call adaboost mr round tree weight error hard span tree combine parameter select tree since method work inference guarantee mild adaboost mrf reach unique adaboost consider variable mrfs label handle evaluate adaboost monitor adaboost mrf maximum likelihood hide markov handle comparable flat paper continue discuss conclusion boost undirected graph simplicity assignment discrete observation specify clique support state crf define family weight vector inner function goal concave global often expectation respect intractable except tree belief take quantity need tree extract intractable bp bind log bp bp guarantee converge formal extensive converge besides mrfs activity recognition promise due structure chain field output structured principle example feature expectation past decade supervision indirect supervision arise crf conditional incomplete show associated clique equation inference clique review boost develop adopt boost give boost strong boost literature expect assertion suffer possibility rank smooth verify indeed upper assumption identical posteriori appear exponential boost approximate estimation mle another related boost boost seek address structure apply bp inference variable limited field term adaboost mrf label visible formulate objective adaboost mr summing visible numerical l l weak learner learner interest sensible general network use computation learner become span weak learner thus visible span incorporation h collection mrf markov forest example simple spanning later shall continue derivation loss method unfortunately intractable alone propose loss tractable use tree recall boost sum numerator special select span fortunately around summation numerator apply inequality numerator mild assumption rhs far simplify I see tractable evaluate
iv iv poor prediction crucially train satisfy iv expert put ensemble bag due voting stacking require sequential training meta predictor satisfy iv stack limited ability iv process enjoy prediction fusion weak uncertain prediction yield sharp prediction together close another start expert model conditional model expert sense expert assign pointed training general expert special gaussian q qualitatively confident less confident correct would slight misspecification expert biased combination rule confidence expert desire behavior expert predictive necessarily measure measure ignore bad generalize distribution focus supervise measure widely anneal balance degree freedom case cause mode dominate product cause expert small combine ignore gaussian suffice show power essentially th gaussian cm control individual reliably change posterior expert variance expert come shall cover variance half change physics result variance true carry potentially effective explore kl divergence cm bag training uk dataset gp way gp tree construction ball recursively subset build expert expert gp expert expert sum ard learn conjugate jointly change normalize sum expert combination scheme learn independently core independent second one time tree test commonly metric standardize log square bagging combination large score explore variant expert path point could define expert boost consistently empirically previous expert correction cm c c c expressive give expert superior sophisticated variational inference gp uk parallelization take although time long similar competitive superior sophisticated well extend emphasize benchmark naive result potential use automatic h rmse method name gp expert take expert gaussian expert many combination thing
outline green operation training user text rank retrieval network gpu three retrieval requirement dataset image gpu able matter second dataset million image live web system input rank recent convnet contributions convnet architecture gpu computation stage parallel architecture budget limit image every architecture begin comprehensive convnet category evaluation scale retrieval take medium large dataset evaluate two image equip pc performance dimensional convnet scenario hundred thousand less fast cover spectrum descriptor namely reduce last layer ii product quantization convnet convnet feature dimensional convnet quantization produce architecture retrieval scalable capable adapt vary paper evaluation protocol assess representation architecture describe dataset subsequent condition world object category lack scale retrieval annotation use common large annotation measure annotation category additionally imagenet train convnet evaluation carefully tailor collection base assessment exclude reason split comprise aside noisy tag represent snapshot popularity sharing site confirm contain image svm evaluate retrieval measuring goodness page retrieve critical evaluate basis proportion positive list class page retrieve large object category rank list evaluate adopt protocol annotation rank class dataset rank list generate complete annotation detail procedure avoid set annotation image object category evaluation train people rank precision evaluate different scenario remove occurrence dataset use annotation remove annotated occurrence rank list retrieve purpose retrieve exclude image precision rank dataset easy since contain instance class test google search image tolerance differ google image centre world retrieval practice sub different image google pool web query google pool describe annotation apart tag annotation contain ground truth rank rank paper raw rank fall within top image combine image annotate positive top annotation store annotated facilitate annotation share scenario annotate class annotation scenario convnet base basis describe image benchmark imagenet focus evaluation baseline traditional encoding fisher detail reduce convnet consist connect layer cnn dimensional feature layer factor use compression feature sub vocabulary cluster learn use successfully descriptor sign iff tight qr experimental table result major challenge decade perform even class produce top dimensional convnet feature retrieve positive constitute scenario convnet precision particularly challenging observation method cnn nonetheless challenging class rank performance drop convnet compare negative appear image comprise repeat evident explain particularly severe appear convnet base much image starts cnn cnn cnn cnn cnn bin retrieved convnet indicate whereas retrieve query track retrieve road several compression convnet appear rank convnet compress appear exhibit similar drop gmm codebook gaussians intra fisher convnet publicly toolbox convnet configurations paper imagenet configuration contain convolutional fully way classifier remove turn convnet imagenet image descriptor cnn setup linear learn hinge machine experiment aside gpu store gpu memory outline green iterative query ii periodic model category retrieval fully exploit advantage convnet experience requirement instant repository follow choice internet model image fashion current I allocate memory storing rank efficiently sect convnet indeed even accurate convnet gpu hardware without illustrate cpu front user interface internet gpu trains repository follow convnet image line use pool google search return feature compute store mb dataset mb pool feature experiment equip gb convnet feature quantization without significant degradation performance make gpu gb aside fit amount single storing repository preferable ranking place cpu memory typically user front end sample feed regular front rank list back end interface gpu back end front responsible mini batch batch equal size end pool positive training front diversity quick every mention store gpu compute gpu top pass back end within dynamically expand pool top return image first four step right outline red blue moderately challenging performance run measure simulate internet typical image return experiment four class challenge performance converge final evolve time order diversity convergence occur fast real term typical rank often architecture query return result background expand pool web present performance true rank position head image change feed supplement motivation role end setup analyse training final rank list rank image bias diverse jump performance image feed deal challenge degree intra appearance image suffice rank close retrieve fed positive retrieve rank positive third ranking suggest coarse train relatively tail ranked suggest image available interface list available continue refine tail rank restriction mention false positive outline limitation impose classifier train demand google live additional figure query novel much query hierarchy query challenge representation repeat thick convnet base robust effect return abstract concrete figure return positive appearance object retrieval build upon advance convolutional representation compress incremental learning retrieval second entirely single gpu cpu gpu employ learn diversity rank acknowledgement support grant acknowledge support use research attempt resolve object collection query bootstrap source latter extract identify analysis pre computation classifier dataset effectively bite
green gibbs black dot green top bottom gibbs static parameter computationally inefficient degeneracy approach deal fix lag work practice impossible quantify realistic free recently appear literature perform batch ml forward smooth filter recommend per implementation besides memory gain particle ml situation whereas time similarly particle crucial line ascent establish admit empirically slowly useful conduct computationally expensive computational available particle move degeneracy recent paper dynamic propose estimation inference suggest well move might engineering sciences research grant ep j energy ep research partly ep g research part nonlinear de g sequential model particle chain carlo discussion l e chains york propagation particle en york maximization em markov smooth asymptotic unnormalize chen liu kalman filter b method carlo particle markov monte update b al hmms maximization hasting hessian evaluation backward formulae technical stability optimal derivative markov condition general state hide approach method j f new york filter year efficient markov monte appear york consistent technical report department sequential smoothing augmentation model inference base dynamic economic filter smoothly economic working papers university department economics thesis department national http edu file pdf via simulate normalizing sampling path w target monte dynamic west nonlinear estimation self series space l dynamical system iterate particle filter comparison smooth smooth new york computational particle million particle lee computer science lee resample sequential monte carlo lee le maximization base markov g le recursive hide chen liu particle liu j west combine filtering liu york liu chen f et al university particle filter financial particle approximation information parameter e smoothing application j paris nonlinear non ergodic l wu neural particle filter property simulation r p g score en recursive system filter model static wang online bayesian estimation markov west new markov nd particle markov chain monte n switch state discrete carlo lee gradient nonlinear non state signal also monte smc reliable numerical state static sophisticated present comprehensive review limitation popular environmental formally process latent markov density markov q parameter component space euclidean space well popularity stem model asset return specie neuron response spike train datum simulation technique illustrate inference space line observation gaussian scenario kalman filter call filter term deterministic implement markov obviously successful popularity easy implement parallel implementation numerous g practical model depend line infer often specie infer extended particle parameter recognize naive problematic past year development estimation recently paper overview differ paper focus e comment attempt make discuss implementation reader original reference broadly classify maximum bayesian characterize datum line specifically framework sequentially organize challenge review unknown dedicated filter ml simple summarize main open ingredient q density equation score fisher associate inference intractable inference reasonably carlo score carry involve integral approximated challenge approximated want sequentially suit integration estimate sequentially line outline particle ingredient numerous particle denote task posterior filter filtering easy likelihood q class class kalman space example intractable technique approximate numerically auxiliary special bootstrap filter sequential resample let density necessarily negative nn rely follow importance eq notational omit weight remainder filter h n I n nx x recover far recover bootstrap filter filter density n intuition recommendation make particular e np nx practice one mostly filter resample introduce replicate particle discard particle weight serve computational effort promise region present simplest resample scheme resample scheme propose advanced particle sake present operate iteration regard degeneracy weight meet particle result give insight difficulty particle test respect reveal nothing regard behavior space typically target n successive population eventually particle literature fundamental particle accurately fortunately possess property optimal condition uniformly chain observation informative informative effectively hide find hold possible strong recent integer explain tool many one approximation unbiased variance great constant standard design recall additive critical implement parameter substitute dx np recursively time g hold grow least degeneracy still particle require point suffer degeneracy yield potentially variance development alternative marginal large collect information approximate expectation x particle mean resample component particle suggest practical choice take tp particle vanishing bias since joint p filtering transition backward recursion tx x x procedure generalize tn dx dx tn dx filtering operation note possible operation unweighted hence cost rejection one trajectory average approximate expectation costly acceptance small particle hybrid procedure combine rejection marginal n obtain filter stand approximate path lag high backward estimate fast approximation procedure interested computing recursively backward require perform pass step backward procedure simple propose easily eq give approximate explain provide access I I directly require consist approximate dx n perform much naive forward trajectory many lag vanish constant exponentially fast forward backward procedure actually regularity describe particle section introduce maximize ascent technique approach initialization get maximum see approximated wish ml carlo evaluation optimize calculate monte ease helpful distribution cdf cause small change result importance introduce computational suitably elegant method number result cdf due sort particle maximize log estimate sequence sequentially receive compute batch recursive subscript sequentially advanced convergence em establish state remain evidence version easy usually procedure approximate term form posterior framework lag smoothing approximation batch benefit smoothing estimate limited experimentally demonstrate line ascent algorithm particle significantly em counterpart particle estimate yield particle uniformly regularity ascent procedure justify prove alternative set assign density joint line approximate unfortunately design linear variable gibbs strategy see particle mcmc class mcmc method build high proposal manner particle metropolis acceptance sampler eq implement appear probability sampler particle particle run tp dx tn particles particle accept probability ideal acceptance remarkable admit performance large variance increase likelihood ideal sampler target component general theoretical analysis select variance typically linearly mean algorithm upon improvement sampler particle unclear emphasize contrary estimation procedure none degeneracy particle variance increase favorable approximate particle approximate density n particle merely introduce possess discuss bind degenerate successive resample diversity approximation clearly simple asymptotic polynomial approach artificial parameter approximate density propose estimate static transform slowly whose bandwidth artificial correction introduced improve recently much result artificial require practical filtering bring information process chain run sample approximately px ix increment run n l lag introduce vanishing present avoid introduction model lag originally propose consist auxiliary approximately distribute p n n n add particle ii mcmc step nn contrary ergodic ergodicity increase iteration would prevent use suggest practice exponential family type exponential scenario n extension conditionally integrate propose artificial dynamic advantage add diversity rely implicitly even necessary statistic degeneracy notice conclusion practical observe estimate lot run demonstrate come accumulation particle mcmc sequentially reweighted substitute running degeneracy reach step update truly approximate posterior instance assess focus numerically impact degeneracy particle degeneracy practitioner valuable simple exact use hence dimensional effect degeneracy panel variance panel panel column method particle particle middle investigate bias rely n detail second per sake approximations numerical simulate replication compute empirical particle separately variance mse rescale growth particle roughly mse particular low superior agreement result variance growth appear sharp h realization algorithm panel panel dot horizontal ml kalman affect see previous every case replication size estimation statistic fairly particle I achieve use say
induce v notational work motivate rank lipschitz dominant term right notation result never section popular constant offer factor proof obvious get first force contraction principle cover argument r good key theorem rely inequality prediction case build define constant loss consider well online guarantee recall refer algorithm gradient set exponent first thus encouraging approach generalize apply erm possibility derive tight bound principle scalar equal bind clear take aware contraction principle pose apply context allow lipschitz involve number loss bound scale scalar vector plug prove recall rademacher convenient refined version rademacher bernoulli uniformly complexity follow plug optimally uniform erm logarithmic computable least hold large great building intuition mention additionally suffer serious loss base begin need essentially provide constant basic class function smooth cover class cover appeal result rademacher bound place smooth constant bound probability upper negative decrease increase satisfie least prove minimizer elementary calculation loss constant loss factor aware bound document particular dependence since top version variant technique error bound optimistic derive smoothness loss lipschitz improve pass argument rademacher logarithmic thing argument much apply situation multi acknowledge nsf expression q yield shorthand denote arithmetic mean geometric give final appealing give prove proposition w give put bound lemma smoothness function play role generalization bound risk minimization affect error rank continuity become continuity enable rate give art loss contraction rademacher complexity turn contraction refer generalization bound smoothness optimistic learn argument question smoothness consider use default lead suboptimal well regret guide average expectation establish error uniform loss norm rank expect right optimistic result generalization erm prove convergence convex illustration literature discover lipschitz constant number document novel theoretical insight retrieval lot recent research sometimes call subset bipartite training
programming formulation execution ad hoc negativity penalty nmf focus popular lee present speed result rather implementation algorithm dataset library science multiplicative provide fast continue hold collection collection use roughly report error svd nmf come also nmf iterative know sensitive initialization old four choose factor algorithm dense random random generally centroid decomposition expensive nmf compute quickly least software svd factor text event svd dimensional centroid consume centroid decomposition fast random averaging vector completely initialization initialization centroid initialization inspire term initialization choose column generally column text idea centroid center co occurrence method co occurrence matrix form co expensive dataset expensive impractical summarize name random lee intuitive basis centroid nmf must run intuitive foundation centroid random slight occurrence large co occurrence expensive subset classify frequently occur document website number report relative computed storage centroid random occurrence sec already column average column quantitative qualitative report mining application essential rank precise solution table easy give report basis error global minimum remarkable closeness subsequent table initialization start table random maintain occurrence initialization suffer diversity topic cover vector easy nonnegative factorization basis classification classification two category classify previously report c c offer bank rate company attack stock bank exclude market c pac induce country fed aid create c c bank company offer bank stock create bank trade offer attack net net country trade bank interest company feed gpu share initialization c analyst bank analyst analyst analyst bank market bank market analyst trade market price bank bank mark national price trade bank price market nearly nmf simple run fix iteration expense convergence number appropriate frobenius angular describe paragraph frobenius exploit norm compute trace angular moderately storage intuitively appeal angle successive I iteration vector converge show similar simultaneously objective clearly angular convergence measure maintain descent require maintain angular measure angular expensive require note regardless choose criterion wise compute iteration period htbp recent lin criterion issue lin stationarity check stop instance stationarity lin propose criterion stationarity fit agree conduct termination present truncate however guarantee minimum must recommend type one paper poor lastly stop replace application iterate reach unnecessary especially case one interested qualitative produce cm good nonnegative factorization nmf algorithm sensitive initialization alternate square include present six four algorithm lastly appropriate criterion key word alternate text classification b department mathematics college usa phone department institute advanced north university usa phone nsf institute nc david follow collection store count time document stack document create document intensity create recommendation customer observe gene sequence condition many interesting create goal mining automatically retrieve interpretable within decade call information retrieval extend mining popular achieve use thereby create factorization decomposition linear svd work point svd interpretation work common user next however know nmf nearly goal nmf familiar phrase term frequently replace observation etc application processing store huge matrix sparse nonnegative decade realize replace matrix several advantage denote rank much element identify essential component attribute low svd pca ica etc nmf problem use svd retrieval k v call indexing approximation reveal svd particularly possible approximation nice fast algorithm uniqueness concern successive comparison need available truncate svd dominate mining appear storage subset matrix svd produce always dense many sometimes svd truncate understand statement say document vector basis sign interpretation interpretability issue svd basis sign vector maintain mean local practice nmf prohibitive sized problem different nmf nmf approximation distance unique orthogonality orthogonal space hand nmf factorization maintain factor easy application basis naturally correspond conceptual strength one become severe nmf issue different nmf local minima minima guarantee algorithm become application help guide initialization issue currently create sparse desire accuracy improve undesirable nmf algorithm avoid alternate least square wherein least use problem problem restriction add user however parameter user resort advanced provide intuitive bound alternate differently alternate solve must algorithm design least appear function matlab active swap basis still remain bottleneck result make ad hoc appealing well practical algorithm initialize w solve
importance discuss focus kolmogorov connect rate distortion information theoretic rate lie direction communication relate communication sensor step treat adaptive source code method mean zhang et minimax machine machine construct constraint place treat nonparametric method divide smaller interesting promising adapt al estimation trading bad exponent minimax risk complexity denote monotonicity satisfy equality q mutual obtain follow concavity since calculation joint eq conclude suppose prior q desire low remain complement cauchy formula therefore write decomposition characterize quantization combine together involve codebook stein satisfy geometry orthogonal third scaling factor front inner space furthermore symmetry spherical q pn analyse us b px previously deviation tail dimensional sphere fix unit logarithm exponent nm pn sphere let vector universal acknowledgement nsf grant fa amazon education grant author comment lemma remark pt university central characterize normal mean storage particular limit encode excess risk sharp establish pareto tradeoff storage estimation present storage constraint particular bit excess risk noise ball pareto tradeoff storage methodology decrease asymptotically together minimax bad risk estimation infimum estimator setting increasingly error computationally prohibitive heuristic efficiency incur place estimator eq require within minimax bound feasible estimation pareto tradeoff versus operation computational mean nature establish tight computation apart concern wish question risk budget bit encode motivated star statistical task send limit estimate cloud environment amazon ec estimate store amazon cost dominate cost much lose principle quantization motivation risk storage nonparametric model orthogonal sharp characterization pareto tradeoff curve euclidean ball radius rich stein phenomenon apparent reader consider require minimax distortion source practical advance efficient compression use perhaps relevant excess analysis relate work briefly result distortion produce identically realization distortion allow budget store datum describe bit datum image cardinality refer quantization join tag encoder auto font pt node decoder every join source code write decode risk many normal term function excess minimax pt pareto tradeoff versus bit bit curve pt bind exact define limit view usual quantization excess quantization zero mean illustrate technical supplementary material sketch estimation code follow fact n accord optimization iii prior independent low problem eq n dp combine prior source coding attain give observation encoder radius vector method outline decoder codebook nb sphere encode eq store b nb bits b n xx make several remark code asymptotically codebook mapping uniform idea behind encode magnitude procedure norm computational shannon codebook implement attain desire vector could possibly code desire low probability codebook distribution magnitude agree intuition ball base uniform ball supplementary material
presence use approach probabilistic approximation provide pr example fast true rate efficient efficient open clean efficiently exist dy dy dy p dy dy dy dy x far form target conditional probability p dy dy dy dy dy thus follow b u composition prove proposition feature feature converge rkhs rkhs e ne weak large target rkh e n properly modify map constant k hoeffding equation rkh nx I inequality kx kx f theorem first estimate discrepancy density ratio bregman fr subgradient sample numerator rx fr fr class fr fr fr fr fr fr fr fr fr asymmetric rademacher complexity exploit bregman divergence rate noisy ratio exploit bregman divergence consistent approximation target ratio approach sufficiently pr theorem bregman divergence distance induce estimate square rate bregman x combine equation iw compare hinge non convexity dataset mean var var e conduct randomly split training accuracy dependent develop empirically perform well often implement uci asymmetric baseline estimate use iw synthetic estimate estimate true c heart e iw image iw synthetic comparison tuning even need estimate datum show hinge bad uci occur two space inaccurate difficulty noise valuable approximately efficiency importance test gaussian performance show bold iw comparison uci benchmark iw loss iw perform comparison c iw iw breast heart importance learn optimal one empirical rate density ratio numerator denominator space hard depend feature centre computation engineering technology technology liu com label observe independently fundamental scenario surrogate design label classification noisy noise show consequently bind reach confirm efficiency crucially rely situation corrupt therefore inaccurate attract significant amount machine random independently prove pac learnable soon pac introduce follow query restriction statistical approach learn intuitively bayesian support multiple reader survey algorithm free important investigate classification loss pac learnable class vc dimension analyze risk surrogate report asymmetric model class loss unbiased estimator risk unbiased surrogate loss use dependent cost latter notable calibrate surrogate calibrate improve benefit classification design noisy weight convexity different focus open limit practical consuming good try estimator design optimize asymmetric flip probability probability exist clean likely noise minimal training mn adversary adversarial target class learn algorithm margin smooth unknown summarize loss neither surrogate calibrate logistic world apart calibrate empirically cauchy loss show surrogate directly presence unknown risk risk df f classifier minimization erm design surrogate algorithms df df df df hand algorithms erm manifold slow guarantee deal derive erm r df f presence still asymmetric address learn presence asymmetric minimizing reweighte ix ip x ip weight rate upper complexity rademacher complexity rademacher
application sense noise note trace dataset low embed high dataset view point locally parametrize nonlinear denote dim boundary parametrize illustration mention kind take direction work point uniformly due analyze clean contaminate tx critical beyond connection common neighbor near complete denote quantile trivial connection dim trivial laplacian remove spectral evaluate notation eigenvalue set sample eq eigenvector embed different parametrization dataset idea new coordinate dataset clearly matter meaningful detail might discuss embed dark red mean norm datum map remove remove shaped recover figure refer explain seem emphasize relationship surrogate norm dark red mean large vector connection choose circle connection remove outlier remove note outlier parametrization figure dimensional scale presence datum average directly diffusion map task need point diffusion base determine near since ground may check distance preserve near rank measure quantify rank estimate plot cdf variant benefit dataset clean perform predict performance well non negligible portion well proved respectively large worse well even investigate influence connection function discuss connection dataset contain randomly version set align classify kind shape etc solve coordinate general rotation resample interpolation numerical numerical dataset circle equally spaced radial group circle equally since image equally rotation act exactly choose surrogate kf kn clean contaminate sample build connection next assign connection neighbor denote w recover rotation connection remove diagonal entry top build slight description describe rotation could rotation complex th complex difference estimate rotation ground observe rotation visualize angle piecewise exist clearly entry mention connection way could dataset graph semidefinite component interested etc diagonal remove h build clean presence obviously bad graph laplacian suppose convex dependent satisfied independent interval width map indeed give lipschitz inequality second immediately find invariance gaussian simple clear display imply see naturally discretized hausdorff choose affinity might discuss symmetric complete eigenvector eigenvalue note act view walk affinity status move modify accord encode connection might give regard synchronization example imaging obtain scale macro scale imaging frame least synchronization problem realization synchronization group intuition synchronization notion vector status encode top eigenvector view status vertex status value connection define notice walk confusion eigenvector decrease special spectral find visualize want first eigenvector mention comment specific allow study system understand bundle simplify exploration refer reader compact dim smooth x dx uniformly simplify constant uniform find assumption collect independently geodesic practice access bundle value function assumption present field associate bundle function discussion note tangent evaluate comparison work indeed tangent mapping field evaluate generalization map geometrically map coordinate field function connection frame encode geometry underlie viewpoint angular replace function laplacian operator derivative convergence operator heat operator onto mention eigenvalue also eigenvalue associate eigen decrease increase able discuss function theorem status trivial function act independent manifold status figure root theory refer trivial bundle eigenvalue matrix asymptotically constant curvature diameter hold laplacian diameter combine subsection application geodesic diffusion dim hilbert schmidt become product interval guarantee finite map abuse approximate riemannian manifold eq q diffusion time tu li tm laplacian map dm diffusion choose yet point propose li variation mention global signature diffusion although different connection dm dm sample refer diffusion eq map define pair dm dd discretization resp abuse notation dm dd dm suppose spectral dd accurate allow geodesic eigen statement dim satisfy g ed tn truncate map show embed embed uniformly uniformly want visualize circle topology visualization guarantee topology counting eigenvector analysis preserve visualization berkeley edu nsf dms wu stanford fa matrices diffusion geometry f analytic technique laplacian idea appear study additive product modification algorithm increase illustrate simulation mathematic idea type object measurement datum wide problem image technique channel etc far us method depend undirected vertex affinity function metric lie sample nuisance component connection denote analyst image establish experiment application metric dataset present information see example laplacian lie focus give motivate projection macro molecular collect model ray macro molecular special ray transform ray treat analyst reader canonical vector therefore inverse increase play particular ray transform direction know would properly align snr projection image proceed common image ray unit conceptually ray rotation unchanged rotation rotation angle stand rotation angle rotation clearly nuisance want measure e j distance ray align think vector like estimate element equip macro interest turn diffusion produce good information rotation ray imply analytic em understand corrupted noise concept geometry image connection sphere encode graph geometric derive estimate understanding impact free aim impact interesting practically useful procedure concern impact way may impact three building first em near determine enough existence likely create near graph build clean projection instead neighbor noise although affinity invariant noisy quite affinity determine experimental setup free corrupt additive follow scheme graph affinity connection connection connection intrinsic draw turn estimation geodesic manifold refer scalar denote entry denote block entry block matrix give connection graph affinity define associate connection hermitian interest large equivalently correspond synchronization property study impact two main explain affinity noise suggestion method graph commonly one also main work matrix theory show notation repeatedly transform stand hadamard entry matrix explain presence apply algorithm give subsection general subsection study affinity connection put subsection subsection modification apply generally matrix q matrix well data analytic well spectral graph idea affinity think corrupt follow dramatically spectral eq suppose exist remarkably condition could completely magnitude purpose eigenvector noisy call scalar entry assumption lemma since interest however eq approximation turn significant consequence lemma standard computing handle work weight suppose furthermore exist lemma computation e significance recent mathematic unknown ratio allow developed tool much since dataset block diagonal q imply imply q far trivially turn average broadly accept contamination demonstrate assume observe version image interested unobserved clean pure object datum dimension dim domain denote accordingly transform mean transform act setup replace invariant transform take grid angle equally spaced point compactly support inside disk angle exact hand discretize act discretized view dim act object act mean act simplify discretized object argument q approximation conclude circumstance transform p n lemma get fact semi also eq argument case light follow p contains follow assumption natural refer reader page immediate consequence gx gx gx fx prove get weak study word furthermore translate example eigenvalue equal eigenvalue might due discretization pixel discretize pixel reasonable interesting namely situation grid depend evident mostly correspond commonly polynomial q ij notation complicate clean behave rotation minimizer another contain exact rotation ij ij ij canonical metric orthogonal group positive independently proof theorem suppose underlying rotation go infinity g tend infinity regularity clean computed rotation compute step ij minimizer hence indeed plugging imply conclude assumption near minimizer minimizer robustness tie clean image number reflect difficult reconstruct manifold precisely reach small normal highlight clean distant distant geodesic fine fine subsection since purpose current far refer interested subsection argument discretization would argument operation map grow polynomially quadratic form gaussian extend hold example entry particularly contamination corruption pixel variable standard argument lemma slightly thing large replace whereas large eigenvalue weak instance bound would change dependence
robust outlier relevant contamination subset implementation resource explain replicate figure display monotonically hard little lose discrete configuration bias hard nearby outlier distant outlier increase colored dotted percentile point leave regardless spatial outlier fit simulation poor increasingly increase show almost bias curve gap increase still influence correctly correctly qp across comparable bias setting qp despite configuration curve correspond fits th percentile fit median finding performance method outlier contamination resource plot outli misclassification principal focus loading result outcome nearly separate outli pca character scale set remove choice add use section package resource explain provide replicate implementation seed result seed outlier observation put scale since outli curse dimensionality relatively balance analyze wish online resource use parametrize parsimonious eigenvalue criterion minor model contain replication hand extract nine replication coefficient shape manually combine replication digit digit plot main outli panel dark blue light curve member visually vertical loading outlier majority assign exclude space examine set dna collect separated value henceforth non comprise measurement application measurement light non dark visually group appear distinguish vertical group difference variability reveal fit detect none outlier additionally identifies fit robust meet robust method outcome deal outlier fail change pursuit fit index pursuit criterion pursuit thus example arise nearly pursuit contamination easy real field widely establish situation enough art outlier prefer inference plan bad distribution corrupt consist unknown contamination denote entry finite context define point estimate sample original whereby cloud form lie definition simplex determinant index shift member depend zero capable satisfy outlier pca estimate break contradiction contaminate must satisfy index unbounded unbounded subset pp population index index position I numerator exist orthogonal projection pursuit approach fix q exist fix conversely equation equation side high give lemma n h em analyze pca fit therefore reveal pca exceed carry extensive systematically outlier keyword outli exploratory pca used explore new account variation need criterion like robust pca case exceeds shift estimate computable accurately observation heavily contaminate insensitive pca relate robustness concern surprisingly instances fail criterion introduce robust meet criterion several search free outlier minimal clean compute loading principal component loss robustness retain component begin draw integer specify least one default offer possibility diagonal eigenvector first loading next compute loading measure member use square direction remove denominator direction increase subset contain index observation value number set eq cm evaluate pca give direction optimal index convention measure member share direction member decrease denominator numerator increase overall grow equation consider direction select candidate subset give denote full space accuracy summarize I member subspace numerator denominator member member call experiment index rarely projection pursuit pp proceed direction pp computationally pp outlier fact select h pp appendix similar also select select configuration contamination small eigenvalue end criterion use scatter subset bias towards outlier criterion favor cause assess pca classify outlier observation assume normality normally one freedom index second pp subset base dominate grow discuss comparable around overall practice impractical nevertheless suitable application class number processor suit enhance ex package evaluate pca although pca
set generate randomly pick strong weak variance compute calculate weak strong agreement theoretical offer formula full scheme however thing exposure exposure full match study fix variable relationship fix effect strength base choose respectively correspond regression intercept poisson estimator rely form episode exposure median absolute proxy deviation max max size full efficiency match size value standardize standardized bias instrumental full matching low absolute little large full main manuscript aspect median absolute method tend higher expect nonparametric parametric concentration gap quickly bias covariate expect however match cubic exponential log root low logistic high variability method use measure type size design type error I type notable exception function contrast maintain correct linearity compare another parametric covariate design generate similar study manuscript look different strength code author unfortunately code contain upon author issue future manuscript ij ij ij become main manuscript within variance var proposition medical centre institute study control regression instrumental iv offer trait important instrumental stage parametric assumption account additionally square covariate balance blind outcome drawback estimation matching simulate concern child trait decrease per episode cell trait substantial ci clinical treatment age two every height list characteristic public health question clinical cause growth among child estimate occur child child key child growth intervention strategy implement net incidence surveillance body exposure fundamental limitation observational concern important example limitation control child impact child well episode suggest control future study likely child family randomize clinical trial aforementioned study account instrumental exposure outcome instrumental core instrumental associated exposure pathway associate see detailed covariate available plausibility conditioning affect outcome exposure write restriction effectively randomly set correspond exposure pathway pass trait substantial trait provide violate cell trait cell trait carry trait region violate cell study american child child cell trait area trait affect development support although cell trait direct united concern study consider receive risk potential trait induce child thereby increase go pathway tend child trait restriction tend increase assumption condition likely trait mechanism table provide baseline child besides affect plausible observe birth match match among plausible unobserved role match match assignment probability probability unit match set sensitivity possibility influence instrumental monotonicity potential outcome affect individual mr restrict severe weak disease evidence therefore unlikely influence transmission monotonicity individual exposure bring effect exposure plausible risk characterize cause exposure cause iv iv monotonicity whereby episode material q iv monotonicity effect change exposure individual exposure nan thus deviation material regularity condition normal interval estimate maximize specifically give response exposure match solve get ratio material supplementary meet proposition material effect effect binary outcome datum whole attempt unobserve instrumental analysis would impact free condition match latter ij influence study birth covariate pz ij pz ik th unit respectively match set child th difference could trait randomize match odd receive unit odd q child presence hypothesis effect inference significance reject quantify addition interpretation understand material match iv traditional iv conventional assumption covariate conventional outcome iv put outcome generate simulation exposure outcome supplementary detail quantifie produce affect outcome simulation five f ij indicator adopt five normal identity multivariate generate I predict simulate procedure two compute absolute absolute median respect simulation exposure write manuscript section provide statistic nan every ii ij moment eq nan hypothesis statistic derive bias proof supplementary material expect statistic notation lemma mainly I fourth moment growth get rewrite central limit distribution converge high entire normal interval spirit interval corollary present q rewrite rearrange get equation coefficient
stability include follow result inequality ensure exist denote lie rectangle leave hand bound conclude minimax sense recall focus online estimation generally decade knowledge minimax estimation spread study establish moment could localize clearly meet depend present estimator algorithm x minimax rate bound meet condition impose predictor define require burn assume start recall rate technique follow lemma provide matrix spectral radius norm corresponding modulus let integer appear hand modulus lipschitz adopt convention integer division middle jt derive provide compactly iterate recursive vary polynomial constant also positive dd result r tt define uniquely soon remainder independently hence appear respectively choice prediction remain setting give analogously respect get positive introduce jensen get eq xx xx xx xy small index decompose exponent integer get inequality consecutive hence ratio give obtain equality center variance imply q use hand bound center inequality suffice define assumption check index include monotonicity least statistic admit number value proof improve either satisfie adapt ty n aggregate plugging obtain minimal net choice improve condition stronger impose noise aggregated predictor aggregate predictor p pm line obtain observe improve cardinality net aggregate prohibitive acknowledgement acknowledge de en de de france period parameter initialization convention study aggregate rely essentially three uniform time vary linear condition linear moment sharp inequality adaptive time autoregressive aggregate predictor minimax rate condition aggregate adapt rate study aggregate applicable application high stationarity smooth environment model time sequentially track operation low storage prediction online normalise kalman rely smoothly statistical evolve along adapt practice exponentially aggregation tuning emphasize meet online computation validation weighting aggregation develop parallel learn seminal statistical game community prediction survey statistical set stochastic setting design exponential weighting investigate stationary recently inspire sequence contribution exponential compute aggregate possibly aggregate past feature therein recent view inference stationary application locally vary autoregressive index propose provide solution rise minimax paper organize provide oracle aggregate predictor stationary process section base set aggregation method proof oracle risk build predictor appendix proof additional aggregation serie non coefficient deduce instance guarantee infinite almost moment context representation admit decomposition say often existence often use vary extend stationary vary sequence suppose general process come sequence additional sequence derive appropriate sensible introduce sequence introduce approximate sequence rescale inference assume smoothness reference therein white see suitable derive predictor smoothness extension sublinear since representation iterate model aggregation additional negative negative mild basic stationary usually rely weak recent work assumption usual predictor evaluate period predictor wish sequentially derive new predict accurately accurately well present aggregate predictor simplex recent contribution find sequential label recursively remain predictor follow building convention later element ti n tt x k compute brevity switch distinguish strategy whole oracle error aggregate convex remain term aggregate control negative predictor l satisfy number context autoregressive generally tx lipschitz satisfy stationary hold satisfying label aggregated noise aggregated predictor obtain find section sense directly appear assumption assumption aggregate give aggregate use assume aggregate directly since require remain risk compare explain well remain inequality page prohibitive optimal order choose provide hence noise observe influence risk choice follow trust existence autoregressive let define similar vary ar coefficient continuous vary unique eq moreover proposition kind initial variation smoothness representation seminal consequence bound sequence vary ar coefficient denote expectation assumption admit standard lower satisfied set determine index refer time rather correspond study statistical prediction predict sense sequence predictor definition sup smoothness subtracting exactly observe addition accordingly provide rely bind bind exhibit smoothness unknown smoothness locally clearly control use provide combine show predictor automatically minimax hold see seem minimax predictor sufficiently minimax section aggregate adaptive assumption seminal minimax nonparametric presentation model maximal process natural practical predictor behave aggregated predictor behave aggregated predictor ingredient minimax refer fact build lipschitz depend predictor possibly predictor say predictor follow locally minimax predictor achieve rate proof say minimax predictor smoothness sufficient minimax rate aggregate one locally bound predictor satisfy aggregate follow assumption satisfy section limitation obtain optimize account factor see drop oracle condition require minimax predictor roughly require operation restriction strong correspond appear eq exp indeed case front remain three equation aggregation deterministic aggregate adapting n adapt unbounded convexity probability lemma distribution ta get log multiply develop successively x lipschitz plug hold get inequality independent apply take expectation jensen give appear remains decrease simplify simply set autoregressive density convention nonzero write henceforth proof
goal environmental science collection scientific reality stochastic may understand biology environmental equation notable infect sir commonly dynamical ode transmission disease population allow simplicity dynamical disease two deterministic describe however stochastic dynamical select observe context via criterion suitable dynamical biology methodology procedure datum abc abc monte sequential smc smc drawback abc intermediate detailed incorporate abc smc hierarchical perform estimation credible type appear dynamical e framework become remainder organize provide use section describe smc simulate dataset population state understand transmission several deterministic deterministic sde study division epidemic epidemic occur dataset include key direct transmission indirect transmission infect mass environment characteristic population model stage specify account several adopt transmission number infect total accumulate accumulate discrete generality consider model transmission infect transmission sde ode direct transmission add population unknown transmission per markov epidemic direct transmission ode equation direct transmission ode sde ode may simple transition interact direct transmission wiener square derivation transmission sde derivation complex sde indirect transmission ode sde let environment indirect indirect transmission coefficient infect specific rate correspond sde model transmission let denote increment length length add infect infection death small furthermore square identity euler sde wiener converge square system initial condition note choose beta discrete gamma describe section discrete ct depend integration infeasible carry monte unobserve approach consecutive observation environmental epidemic record simulate straightforward many numerical euler simulate simple numerical euler monte algorithm suitable choice model build basic datum accept simulated data rejection propose smc proposal avoids rate algorithm white tolerance index index sample else sample candidate normalize go smc proportion select abc context simultaneously bayes commonly uniform normal euclidean smc simulate include death epidemic indirect model st simulated observe smc five describe abc smc algorithm set describe record indirect transmission five indirect transmission sde compute factor indirect transmission sde model high bayes factor indirect transmission sde figure favor indirect sde indirect transmission sde high dataset factor bayes although always pick indirect transmission sde apparent favor model weak h dataset note indirect sde smc epidemic parameter abc smc distribution ode function simulate sde euler one month epidemic h figure favor indirect compare significant factor favor indirect transmission strong average forecast predict epidemic bayes favor epidemic smc l ode direct sde indirect ode mode interval table fairly uncertainty surprising transmission mechanism quantify still try mechanism transmission epidemic super epidemic indirect transmission rate agent assess cumulative indirect transmission sde epidemic trajectory predict reasonable datum count smc environmental process continue
choose motivating give point lattice pseudo study sufficient table report initial also table run modification different past sample subsequent stochastic probably instead however analysis modify becomes decide include modify research mathematic mathematic approximation estimator intractable adaptive monte adjust control examine asymptotic new algorithm resample reduce degeneracy combined resample small estimator impossible intractable spatial statistic wide range constant overcome model include paper prove normality adjust generalize version importance complicated use markov asymptotic motivating statistic family unnormalized density dominate maximizer constant estimate form instrumental density maximizer guess behind instrumental density defer subsection far parametric instrumental value update history algorithm put measurable c continuous constant trivially uniformly insufficient likelihood approximation converge converge iff maximizer sequence converge gx mx g mx mx gx converge surely prove inequality cb step sure convergence introduce maximizer maximize surely already pointwise accumulation unique maximizer conclusion immediately easy boundedness concrete log maximizer assume maximizer normality derivative continuous inequality fulfil asymptotically stochastically begin regularity also consistency automatically fulfil concave adaptation appropriately family particular motivate play similar together appendix position well need show q denominator enough asymptotic use notation martingale dominate assumption exponent lyapunov last formula sufficient taylor expansion consequently zero hold adaptation improve instrumental use instrumental maximizer multidimensional dispersion various determinant eigenvalue trace covariance trace equal asymptotic tr schwarz optimality theoretical might show exist mc illustration distribution odd ratio family scalar omit optimum instrumental version suffer degeneracy instrumental instrumental density degenerate effectively impractical practically eq computed mc may change adaptive algorithm compute history special letting exploit derivative conditionally efficient degeneracy instrumental parameter markov kernel preserve compute put ik ty uniformly family var pt close look reveal indeed q within produce complicated maximizer ty surely boundednes martingale consequently pointwise convergence compact resample purely instrumental evaluated variance also ty value statistic key estimator fulfil difference therein run concavity hold apply difference give discussion precede order derivative
exploit effectively intersect introduce directional space avoid neighbor cluster establish variant namely use framework hybrid model directional local sparse coding improve underlying structure corruption geodesic practice validate art euclidean notable test synthetic sequence brain image texture seed digital technology university fellowship institute review describe implementation compete compete sparse smc embed sphere adapt current smc map tangent map task spectral matrix riemannian metric apply whose page replace euclidean metric standard riemannian third compete scheme embed lie riemannian apply classical manifold embed embed euclidean space manifold pd sphere already embed cluster radius distance angle threshold notice change local tangent space large covariance precisely number gap smc accept publication code code comparison number smc remark weight ji ij ji weight unstable experiment collapse experiment ij suggest try notice ji accurate though advantage overall ii vi exponential use rest experiment vi level phenomenon figure set analogous computing pd logarithm map sample general riemannian manifold slow logarithm require column orthonormal comprise subspace need singular equivalently pd compute logarithm complexity major cholesky decomposition matrix multiplication equivalently find eq dot take gx riemannian image computation map computational point involve distance riemannian complexity riemannian cr cl computational sphere pd major occur near associate facilitate use neighbor assume effort small one second product necessary entail nn product task solve solver popular alternate direction method multiplier covariance compute operation knn nd nk affinity burden identify entail order nd nk zero geodesic angle reduce moreover affinity thus order become kn nk nn complexity sphere approximate neighbor total special kn nk preprocessing near consider specific avoid leverage elementary algebra eigenvector xx tx coincide small cost pick subsequence convergent approach infinity gx rr origin angle arbitrarily contradict measure inequality application bind er gx depend depend riemannian apply orthogonal appropriate change matrix word th entry satisfy rhs hard see constant later perform schmidt orthogonal use preserve sign induction sign assume rectangular express eq upper triangular estimate change row rotation next sign lipschitz function rhs result denote tw I far finally side simplify equation sign induction b ix r definition h direct detail exclude imply riemannian claim follow lemma work tangent space q minimizer map map via new express rhs bounded rhs follow argument plug fact let follow prove generalize subspace fix concave fact imply long lemma proposition theorem definition axiom edu electrical digital technology university mn usa riemannian manifold lie important cluster algorithm matrix already texture medical imaging clustering construct affinity exploit geometry intrinsic geometry encode code importantly directional tangent space establish intersect geodesic extensive validation real theoretical state modern moderate quantitative framework study manifold hybrid linear framework dataset model union whereas propose underlying try propose cluster apply euclidean space nevertheless domain riemannian manifold sphere manifold symmetric semidefinite psd moving model utilize extract low identify spatio filter pd manifold texture nevertheless strategy handle application advance representation dimensionality development modern rely assumption exhibit subspace embed cluster manifold generalization develop example illustrate nonnegative distance ms motion ms analytic manifold promise geodesic distance manifold solve cluster dimensionality work quite convex separate intersect closely locate accommodate subspace restrict manifold embed suggest also strategy include inspire algebraic method strategy code sphere coding encode subspace energy minimization local order pca guarantee multiscale strategy riemannian orthogonal group pd manifold address spirit tangent space logarithm transform tangent neighborhood integrate infer despite popularity generic scheme low embed space end provide euclidean intersect nontrivial clearly modeling paradigm generalization superior performance extensive synthetic dataset efficiently restrict applicability work analogy nevertheless apply since neighborhood furthermore model distinguished previous multi manifold careful incorporation directional local intersection ii neighboring cluster different algorithm neighborhood include multi careful choice formulate preliminary background riemannian assume dataset lie geodesic riemannian gs mx subspace w generate uniform two htb justification setting include kind noise review riemannian review recommend riemannian tensor geodesic minimize among curve tangent exponential coordinate around image cf maps htb sx ix ix jt problem support define key quantity quantify directional geodesic angle present solution well concept geodesic geodesic logarithm radius neighborhood sample rt I nd intrinsic tangent form bottom cf tangent subspace span top exceed see top top occur short geodesic connect tangent empirical cf euclidean motivation propose practical discuss geodesic tangent spectral carefully represent locally come sake underlie assume accord connect clearly intersection figure intersection demonstrate able tangent geodesic angle beneficial neighboring belong red local disk exclude red done angle w point figure arbitrary point estimate difference estimate geodesic angle reliable dimension neighborhood close intersection neighborhood away intersection cf connect neighbor linear algebraic intersection filtering procedure guarantee proof cluster geodesic tangent neighborhood radius threshold estimate tangent distance threshold set label assign geometric point f matrix eigenvector whose exceed angle affinity apply affinity follow achieve correct statement rely underlying simplicity geodesic geodesic smooth compact geodesic generate satisfy inequality correctly sufficiently relative geodesic tangent practical geodesic tangent choice differ threshold one drop affinity indeed indicate portion point intersection sense connect point intersection pairwise coding produce large weight point come point radius default angle default compute among ij x nn cluster affinity algorithm task use code multiply latter increase nearby local explanation cluster code tb detect use avoid assign far blue briefly leave many neighborhood choice r compute cr cl depend riemannian cl symmetric pd matrix cl dimension cl riemannian modeling logarithm discuss rather computational particular sphere complexity assess performance follow adapt smc riemannian metric review appendix labeling label maximal label label q six generate dataset iii iv pd vi white construction non draw comprise random dataset subspace comprise two pair group model symmetric random group generate entry normal construct dimensional dataset comprise lie parallel arc draw vector since bar blue accurate bar one mean step diffusion direction baseline dimensionality generate direction noise detail level brain carry riemannian pair randomly perform report accuracy clearly suggest cluster demonstrate among compete assess structure various image goal independently pixel e capture apply setting obtained obtain transformation database covariance texture patch size feature specific patch carry transform patch covariance belong texture pattern patch demonstrate database angle shift image shift patch draw affine affine transform image plot dataset onto euclidean demonstrate transformation average cluster rate report transformation horizontal spatio dynamic video action leverage average experiment spatio temporal database employ associate spatio temporal patch subspace cluster distinguish spatio study govern temporal rule frame video eq latent vector distribution respectively explain subspace spatio accord procedure arbitrarily mp dm th conduct subspace give rise dataset rise video database video randomly distinct per frame patch patch reduce ft estimate underlie consequently category cluster lie pattern look shift version visualize subspace project cluster contain video speed video associate video spatio database patch spatio patch previous set associate consequently subspace create random video represent motion procedure choose video database repeat clustering table achieve highest clearly sample nonempty form remain whose connect exactly appropriate parameter specify organization describe present undesirable event negligible theorem briefly simplicity sufficiently end set instrumental fix later angle second within estimate noiseless discuss generalization idea consider modeling euclidean apply general lie euclidean manifold basic infer ambient space riemannian local different system local directional tangent space uniform assumption ambient care map e logarithm refer complete introduction riemannian geometry map tangent space coordinate endow riemannian choose metric north straight shortest short geodesic connect blue connect origin point measure n give throughout noisy thus denote support expect covariance matrix compact w bx r z bx xt xt mt principal recall apply geodesic lastly stand th review detail geodesic definition geodesic riemannian short geodesic connect let compact riemannian neighborhood example gx gx xx local matrix neighborhood define event define cover sample fraction proportional concentration covariance ensure assume establish geometric property constant appear angle intersection last claim gx imply constant transform third expect equality slight depend riemannian metric arise observation conclude triangle inequality thus next exist define order follow distance span eigenvector st st
impact make represent view well assumption particle move correspond detection detection accord predict dependent weight denote modify express inside outside keep probability within camera term dash equip estimate multiple camera pair proceed describe camera calibration reliable estimation require knowledge central object derive objective section multi previous camera joint origin system align camera position orientation calibrate camera pair space camera camera position camera orientation velocity principal distortion camera camera space jointly state object scene well right left distribution camera q one impractical prefer single variate describe detailed relate generate express camera assess either spurious confirm status interestingly track camera group track similarity hierarchical estimation single object exhibit linearity conclude wish possibility right infinitely camera representation compose particle express dirac density camera gm particle implementation sensitive curse dimensionality need maintain grow exponentially calibration particle approximation make distribution observation update estimation target degeneracy particle update camera end experiment away camera pair resample frequently cope observation approach filter particle particle assess representation single evaluate track performance commonly employ problem return match assignment cutoff determine shift emphasis latter influence sensitivity outli mahalanobis locate plane axis gm detail six velocity metric appear increase induce merge prune false alarm propose show depth particle moving target object track camera configuration one consider estimation position orientation calibration particle calibration figure demonstrate object cover whereas distance axis component would allow high address camera image propose novel camera calibration relate move target sensor exploit space camera track scenario well estimate filter object correspondence object correct automatic develop advance sensor calibration compare scenario fellowship science grant ep camera calibration camera address unify joint object state problematic exploit tracking version kalman correspondence hypothesis density filter ability update explicit measurement association target discriminate clutter camera static move object determine sensor kalman statistically sensor away sensor passive alternative observation case position method datum consider joint geometry make proxy express enable filter logical address object multi object track camera statistical follow measurement simple single state estimation calibrate follow calibrate camera since address independently turn concept object correspondence estimation calibration image vision tracking majority algorithm interested object co nonlinear dependent inherently estimation statistical finding rao previously investigate though researcher transform estimate uncertainty thereby consensus uncertainty non complementary instrumental reducing despite reliable camera measurement belief camera measurement direct define reflect separation computer extract attention focus space euclidean projection two noise range consequently object assumption kalman example know mean minimum establish practical theory extend tracking refer position move tracking sequence measurement filter sensor due comprises step describe update bayes sensor filter dynamic noise kalman long valid deal mild kalman kalman motion case vast majority ultimately know velocity world system history kalman filter presence observation kalman poor particularly depth kalman variant almost track target description state bias filtering specifically design sensor characteristic successfully integrate vision track camera relate sensor come sensor system environment know false sensor rely measurement identify vision suffer robustness rely measurement frame image optimally development sensor practitioner overcome association integrate estimation object filter develop set develop multi location multiple sensor discard confirm sensor image sensor rate measurement necessary sensor object attract international moment approximation filter density object statistical filter provide mathematical foundation datum assign measurement correspond ahead object robust scenario filter complexity target static recursively use make calibration refer imaging view scene solve inverse accurately scene reconstruct ground knowledge calibration euclidean perform directly projective practice calibration hence extract appropriate calibration assume calibration turn projective update measurement certain correspondence possibility incorrect consider consequence input begin calibration perfect input formulate extension object fact calibration multi object inherently camera address problem condition turn camera simultaneous contribute self underlie pair illustrate real onto respective recover purpose relation real formulate projective geometry triple represent projective henceforth projective perspective relate homogeneous refer equality perspective projection concept purpose perspective must namely express coordinate resp plane link camera setup cf non camera assume camera form baseline henceforth respective u camera projective geometry linear projective camera camera pair express mean proxy convert equivalent via inverse allow camera right camera plane respective projection associate maintain one correspondence purpose state state object intrinsic behaviour tracking justify compare resolution position sensor extent observable shape camera calibration extent increase difficulty convergence model extend observation camera affect spread resort distribution another stem know noisy nature image generally space non raise although object circumstance approximate however serious limitation infinitely far camera dotted curve contrast dash highlight usual representation space easily previous camera via gaussian uncertainty back transform make uncertainty kalman demonstrate inverse inverse suitable tracking modelling instance velocity object coordinate velocity space velocity step model object dynamic raise relate decompose step sample particle result recover gaussian mean kalman approximate enable nonlinearity reason kalman filter non linearity observation fairly represent recover particle optimistic yet objective aspect distribution several case overall yet evolution case appear though shift evolution well particle particle capture motion associate object handle kalman observation camera covariance suitably augment velocity unobserved specify camera say camera consider receive observation velocity compute distance camera sufficiently enough whenever infinitely away camera consequence negative represent behind must issue general determined carry uncertainty challenge camera geometry available nonlinearity camera yet detailed beneficial prediction handle camera pair similar space camera camera property strong introduce left respectively say abstract hence produce camera projective predict space resp resp call particle mapping represent object another onto camera approach object together observation dynamical object two camera right camera velocity distribution
p concrete section time checking theorem depend start vertex write I p direct calculation side c condition negative bipartite generality imply contradiction verify state conclude first claim markov field definition proceeding condition see minimize neighbor equivalent constant next polynomial calculation already omit finally j gm total let let identity follow note eqs therefore condition c compute consistent indeed fairly easy want estimate formulae immediate generalization obviously asymptotic desire markov exist note standard tail yield different choose model logistic fail unless satisfie efficiently develop consistency assumption distribution show establish result use ellipsoid project related round various task I aware statistic acknowledgement partially grant fa fa grateful discuss prior publication self proof yield convenient extend simple next eq induction dimension claim prove boundary let exercise transform claim simplify assume return claim correct replace polynomial oracle point replace query polynomial call sequence project gradient drop definition intermediate I q hence approximation error term soon guarantee already require approximation guarantee oracle conclude mr claim conjecture claim proposition remark task pre straightforward advantageous simplify analysis contrary reduce theoretical implication technique dataset statistic representation subsequently widely adopt turn carry replace solely phenomenon entirely concrete shall da ib clearly notation nn standard prove entail information already n fairly sufficient conditional word able distribution simple example give section word argument base organize computationally efficient statistic approximate polynomial theorem estimator unless remarkably indeed discuss denote possible follow denote deterministic variable hessian add subscript parameter sufficient law denote expectation use subscript emphasize consistent description description two terminology motivated remark terminology discuss indeed consistent surely hence finally consider q polynomial estimator exist return devoted implication negative approximate intractable consistent fairly class remarkably consistent direct consequence polynomial sufficient unless property specialize present consequence present self reader convenience standard statistic notation positive following recall pp pc p inverse mapping concave position version second maximize set modulus gradient work reality oracle assumption efficient replace give project indicate orthogonal
deterministic particle many alternative usual proposal gradient involve particle represent show theoretically experimentally particle capacity validity mnist reasonable network part digit could individual expression base hope insight stochastic feedforward apply context dropout like acknowledge source classification huge variety class complex useful test gradient pixel epoch epoch give see exclude comparison bias follow estimator c test unbiased deterministic na deterministic hybrid computer universit e perceptron mlp give mlp type structured make general activation per lead fundamentally try estimator test compare training estimator feedforward multi mlp model mapping hide typically unimodal isotropic factorial oppose conditional simple distribution advantage conditional configuration produce output approximate multimodal mapping empty add introduce noise hide improve additional setting decision solution cognitive early mix poorly mean inefficient optimize work propose draw feedforward guarantee propagation unit additive dropout long way gradient estimator bias approximate back unbiased network unit provide show demonstrate face person mapping object base hide configuration offer alternate extreme unit back rigorously training method less estimator benchmark mnist study stochastic binary hide layer extension original activation activation perceptron mlp denote row sigmoid softmax probability product q probabilistic deterministic back bring criterion summation derivative directly problem enough increase sigmoid unit behave nonlinearity maximize expectation deterministic maximize output q achievable distribution deterministic game achieve performance look situation divergence negative conditional h dy bad job maximize deterministic ph simply explore share auxiliary network line paper simplify use generalize give rather role mixture notation unbiased estimator new estimator technique draw c empirically sufficiently close estimator hide layer average mini batch epoch train top bottom appendix estimator correspond mnist task train experiment benchmark feedforward network base handwritten database multimodal first digit preprocesse sampling independently grey different individual mean expression subject training datum thus gradient keep perform significantly experiment comparison feedforward deterministic network weight deterministic use way hybrid consist neuron incoming connection neuron use gradient neuron stochastic gradient mini batch momentum epoch epoch good model test exclude comparison notable significantly particle infinite computational performing task binary especially interesting also outperform hybrid output probability neuron learn propagate hybrid easier full
h exist onto q eq norm choice solve subproblem spectral singular theoretically bind norm full slow consider firstly project norm ball project row onto norm ball al logarithmic length matrix preserve sparsity dual variable improve tradeoff mix convenience approximate project descent discuss projection divergence formulate tb initialize kk divergence evaluation parameter divergence px arguably preserve marginal kl intractable tractable surrogate divergence tractable divergence minimize mean inferior support implementation pool step iteration project divergence result marginal focus aspect computation accuracy extra roughly perform accuracy mrf rapid compare marginal absolute marginal mf step wish compare represent norm grid test intractable reference evaluate various denoise berkeley gray purpose xy iy encourage ise wang rapidly truth decrease error compare run time approximately less condition mrf fast project competitive method well sampling mixing set include research centre program give dependency state start distribution sigmoid index influence great change definition previous inside previous dependency convenient form series relaxation convenient rather mrf result matrix mrf prove mrf section give result dual representation ij c lagrangian dual independent straightforward eq eq eq extra thm proof corollary algorithm extremely practice amount sufficient univariate mrfs project project various compare univariate marginal mrfs intractable motivate markov among variational field approximation factorize distribution find tractable kl propagation propagation propagation use approximate kl divergence approximation correspondingly simulate target inference draw markov chain principle mcmc accurate difficulty markov converge mixing inspire model minimize sampling converge distribution ise model mix spectral strength project mix project budget limit ise project first condition univariate mrf euclidean somewhat computationally previous validate divergence project parameter interest gradient pairwise mrfs potential jj x ix parametrization pairwise mrf convert include sometimes convenient statistic indicator configuration representation equivalent review qx qx stationary state markov always distance irrespective start run gibbs sample central dependency one informally change matrix except central rapid mixing theorem generalize informally multiplicative mix multiplicative gibbs n size variety natural ask give computationally norm additional adjacency regular induced p norm rx gx rd use tend dependency large mrf mix necessary tighter convenient field indicate include impact regardless univariate self zero one distance mrfs parameterize project distance apply dependency projection closeness
separately control reveal motivation put error jointly constraint satisfying put seem lead penalty lead return eigenvector outside diagonal entry elsewhere false construct sparse h enough primal dual pair kkt solution optimizer kkt careful analysis second construct unique make elastic fact essentially elastic control establish negative control signal carry frobenius bind false eigenvector sign nonzero actually generalize direction subspace dimension whereas assume require second satisfie diag bind hold claim consistent solution rank unless increase oracle operator error factor small lead dominate gap selection depend condition result insight therefore remove assumption identifiability assumption interpretation assume identifiability long valid true subspace develop free interpretation let interpret maximization technique seek dimension transformation maximize replace sense correspond versa corresponding word sum residual satisfying projection smooth shrink reduction notion sparsity measure persistence eq well imply principal assumption subspace principal subspace principal simple uniqueness point interpretation predictive covariance stable perturbation pca linear establish rate suitably identical sparse roughly speak pca assume q orthogonal correlate term thresholding usually penalize relevant eigenvalue rate selection open time consistently range testing least solve plant clique beyond barrier interpretation lead collection shrink effective give reduce tractable nearly predictive covariance validation lead drive tuning property technical appendix proof section primal start form write tucker problem additionally solution let variable support need kkt feasibility check feasibility nonzero feasibility diagonal agree principal subspace q establish primal elastic net max max z h kkt optimality dual version existence imply consequence uniqueness unique small enough contradiction control obvious view norm wise principal dual sufficient eigenvector entry imply assumption sign tb bb hence frobenius pt lead orthonormal span principal orthonormal f argue condition orthonormal apply proposition rotation cauchy schwarz jointly choose theorem constrain lemma unique verify principal part gap next principal subspace unchanged matrix twice operator noise small th exceed jj z jj come contain perturbation theory dimensional exceed part strict condition dimensional claim inequality invoke comment style graphic label http edu support nsf nsf dms theoretical sparse often implicit raise sparse variable select consistently say result truth investigate general condition fail sparsity dimension technique reduction variable combine central classic seek linear retain much variation set correspond span interpretability yield consistent truly decade pca low convex relaxation two algorithmic development include detection various explicit often implicit truth lead subspace raise sparse say essentially nothing independence second selection agnostic semidefinite program whose sparse estimate formulation perspective rather appeal subspace time develop method multiplier consistency population low plus identity wide include estimator select assume hold important dimensional theoretical mainly exception eigenvector leave open also address broad roughly variable correlate large interestingly min generalize direction paper interpretation beyond independence literature concern main set symmetric assume quantify may helpful think vector necessary theoretical assume semidefinite two probabilistic tail sample q bernstein sub tail constant absolute subsequent introduce tail proof edge adjacency absence pair plant clique everywhere find clique polynomial time plant clique give reduction plant clique pca simplicity presentation focus applicable graph follow parameter efficiently alternate multiplier principal encourage moreover norm assumption mild condition
limitation locality problematic observational uncertainty outlier interest locally graphical ii issue locally model graph improve inference utilize becomes computationally scale fully address crf specific function zhang constraint position encoding al potential pairwise al fully connect crf target output aforementione fully graphical define specific effectiveness key deep take different inference intermediate manner et introduce train conditional factor conditional field sum yu introduce conditional consist layer layer input result observational characterize implicitly limited aspect connect graphical deep structured graphical limitation two type model explore leave exploration unify could benefit improve boundary boundary observational uncertainty outlier fundamental fully due structure study investigate feasibility connect graphical encoding layer maintain deep graphical methodology behind inference segmentation problem section present conclusion observation conditional goal propose incorporate interaction preserve advantage long interaction connect interaction impose high make intractable issue introduce auto encoding make structure measure represent essence variable extract parameter reformulate represent auto encoding characterize auto encoding principle add encoding base determine structure model auto sparsity characterize high number structure auto encoding modeling structure random show configuration fine coarse fine depend configuration utilize next section tp represent mathematically formulate label auto layer multiplied represent label auto layer intra inter layer interaction fully among computed encoding graphical interactive structured computationally tractable interactive image segmentation foreground object background annotated fig tp region foreground background unary gmm gmm parametric histogram apply account tackle train unary pairwise mrf consistency utilize background foreground use layer deep eq train mixture annotate result auto auto layer maximize joint auto encoding represent structure layer characterize image certain specify auto encoding auto hand random adjacent auto encoding fully implicitly variable give encoding layer previous layer compute encode foreground state energy minimize try base datum pass summary encoding py interactive segmentation evaluation database scene use database manual ground single consist image manual ground truth image contaminate range total seed foreground seed q positive false implementation method structured connect also comparison perform crf comparison component unary number encode iii image segmentation encoding encoding layer node annotate sample layer matlab colour total configuration cpu ghz gb ram classify colour tp image ground truth unary f level single dataset table segmentation object noise scenario slightly scenario free noisy noisy noisy object preserve unary appearance fully j light post ground truth unary handle slight model seed water variation illumination texture capture water foreground well handle
often one option expect translation reward bad lead upper confidence scaling need range feed kb select exploit cc maximize confidence sensitive ucb dominate small ucb need negligible even possibility ucb kl advantageous range become minimization subgaussian subgaussian coefficient tc weak bandit finitely arm analyze moment know moment must replace mean modify bound condition moment finite denominator calculate n general cc fast gets get sublinear term note put thing get prove similar simplify inequality b develop every well bernoulli variable transformation scale conduct keep variance shape unable instance shape negligible cox denote differential equation fix brownian motion price option price option monte method simulate trajectory value option monte carlo standard variant use derive w ip multivariate py ix I specifically feature denote correspond accord logit parameter I train result posterior normalization integrate integral monte drawing target posterior suggest step additionally mcmc anneal step move metropolis hasting langevin hamiltonian sampling operator generate use remark via consider sequentially carlo estimator minimize strategy development estimator outcome strategy provide strong practical advantage monte approximate lead task difficult know call unbiased expect error formalize scenario interest practice unbiased method rejection allocation output produce combine mse analyze meta strategy formalize excess mse combine produce contribution reduce bandit identify arm square mse correspond bandit conclude carlo reduction adaptive allocation alternative provide finite cost estimate suitably bandit formulation yield aware estimation generalize monte bound regret performance art base select discover computation complementary adaptive design fix bound proportion optimize differentiable broadly base estimation even approach bandit sequential allocation agent choose step long term payoff payoff action unknown give th payoff action action kt kt maximize rearrange mab show payoff polynomially consistent policy optimal action polynomially payoff upper confidence achieve regret asymptotically ucb drop requirement fit weak finite action ucb regret bound j jt kt kt jt jt kx kt kt improvements ucb ucb v variance construct bind variance arm payoff substitute bind scale relate worst bad ucb practice recent ucb bind chosen increase efficiently smooth kl ucb achieve q term low ucb expect large ucb additional regret another approach receive significant interest ts randomly proportion posterior ts outperform ucb bernoulli indeed time ts payoff problem distribute payoff variance set thompson value payoff resample step obtain kt kt kk kt kt kt formalize finite monte variable converge unknown target different assume estimator previous procedure whose monte evaluate excess implicit excess multiplied sublinear asymptotically match adopt value estimator unweighted sophisticated approach weight sample respective ignore highly weighted variance sample come estimator tv kl regret arbitrary allocation mse estimating satisfie cm algebra reveal bandit versa furthermore allocation obtain regret problem conversely arbitrary observe choice mse bandit observation low x choose k let v dd distribution fraction establish mse regret mention simplicity assume mse use feed modification effect regret additionally regret ucb bandit factor immediately imply regret handle still payoff unbounded take time definition modify accordingly estimator produce identical notion time update estimate assume observation etc round etc time cm kt kt kt kt kt kt choose sublinear note value produce stochastic use correlate biased sum kt kt kt l kt ft kt average compete draw k use aim time suboptimal generalize let let estimate mean assume appropriate problem upper comment logarithmic attain concentrated sample restrictive estimator rejection sampler geometric furthermore moment sufficient variance simplification cost since x follow difficulty reward construction use use ensure number round dependence nevertheless need last remain small follow concentrate c regret allocation also estimator example show ucb suboptimal c ucb problem event time show achieve assumption concentrate subgaussian tail thus last rest conduct experimental effectiveness multi allocation differ evaluation bandit payoff identical variance detail ucb payoff low regret independent run scale optimal second plot suboptimal highlight dash red circle bar indicate ht particular standard separate permit bound four bandit ucb kl ucb prior relative approach ts scenario whereas v effective kl perform ucb monte carlo pricing financial pricing evolve accord cox finance detail assume follow price european option give naive estimating simulate independent simulation strategy introduce drift encourage simulation importantly importance space restriction prevent description mixture drift consider drift mixture coefficient drift proportional new generate next approximated price parameter price wide proposal result effective bandit suit allocation ts winner likely level option pricing surprisingly bandit allocation believe stem entire mixture remain area future allocation continuously setting important estimation challenge comparison training evaluation desire quantity difficult popular dimensional chain monte generally sequential carlo sampler combine transition slowly offer advantage considerable effort set step anneal rate schedule mcmc execute anneal appropriately tune choice effective large bayesian logistic sized consider estimator anneal anneal schedule suggest slice use entail dimension detail differ slice internal
limit proven overfitte yield solution result improve search small strength include directly need put density concept use non require linear direct possible possible apply map rbf preprocesse cluster seed position rbf network beyond scope research help library intel ghz well uci briefly breast diabetes heart focus regard chemical prediction implementation implement acc coefficient accuracy average four center one obviously linearly term symmetry maxima entropy result margin rule flat method achieve almost kernel benefit show lead however nature bad measure perfect fitting capability support regularization entropy histogram number threshold purely cauchy lead model capability dataset capability cv dataset dot maxima suggest pearson seem overfitte decrease acc breast diabetes heart capability across fold correlation confirm aim balanced suited balanced greedy optimization start consist point sphere balanced high summarize protein classifier contrary class active importance lrr rr svm ht ht examine score fold similarity svm classify fold one dataset could rbf although lead construct big slow number require thousand evaluation build ht speed aspect actual huge database check build good consider one place middle perform five lead exactly score achieve model protein ht follow regard result complexity magnitude one internal geometry chemical well might detection active inactive family method actual paper entropy linear term projection justification property invariance svm propose constrain open issue optimize outline also classifier study behave uci well behave balanced balanced correlation propose partially centre work discussion suggestion institute sharing regard would access make proposition section edu pl classifiers hyperplane method separate hyperplane model concept namely quadratic cauchy schwarz general invariance margin analogy broad aim balanced quality directly far confirm uci appear real data classification linear perceptron logistic decision class idea behind neural modification like extreme machine activation neuron play decision base set classification lead htb sigmoid polynomial second good ask answer lie way boundary thus question directly fact split proper linear decision division allow reasonable decide entropy divergence cauchy schwarz denote schwarz divergence computable mixture optimization consideration cauchy schwarz translation maximization boundary consist entropy add view project data subscript denote width rule denote schwarz perform common reason classification rarely particular simplest interested good sample obtain precision support linear density final formulate estimation contrary well occur threshold fact kernel window width single nine ten uci worth solution different svm fundamentally different knowledge detailed insight geometry activity protein describe internal structure capture group encounter show exploit simple represent specific group active positive sample worth linear advantage linear classifier maximize balanced imbalance require datum directly scaling try margin build significantly model svm parametrize free drawback complexity describe schwarz divergence maximize margin play practical consideration drawback conclude long receive much hardness answer basic question et open related support concept generally present svm conceptually show span classification criterion self cauchy author broad tree recently deep architecture aspect cauchy schwarz insensitive restrict search unit sphere next discuss go schwarz divergence cauchy schwarz normalize density density correspond rescale easily r rp since schwarz projection rescale maximization restrict finally invertible put assertion inequality notice analogously datum generate gaussian let multivariate eq denote consideration well formula observe quadratic us attain eigenvector density q therefore entropy vector eigenvalue one multiplier attain say minimal maximally intuition schwarz information class say crucial onto line coincide discrimination next analogous limiting set maximum tm tm tm tm minimum equivalent cauchy schwarz attain q interpret discrimination view know span construct maximize close sample opposite class simple linearly vector express reformulate class unit generalization concept lie maximize threshold size even four require modification removal separation formulation lead threshold dataset would threshold number threshold appear probably subsection large maximization margin try threshold typically often window distribution although choice nontrivial atom local lead limit linearly separate information arrive large whose start potentially overfitte maximize margin formally show behave gaussians arrive formula analogue result prevent threshold support thesis divergence contain information entropy class support section empirical evaluation often unnecessary high consequence show cross minima put start gradient sufficiently linearly separate discriminate separate eq linearly suppose assertion mean mean assumption obvious contradiction suppose separate unique normalize svm going limit svm maximization margin arbitrary eq proposition choose separate cross eq directly maximization let denote possible margin along margin point q result margin big lead conclusion potential additional construction classifier result threshold result construct lead life uci repository versus schwarz introduction entropy cause reduction analogue hold limit reduction number threshold put width begin window eq denote element equality apply obvious calculation p v q v v trivially apply twice yield objective serve purpose optimal discrimination classifying tries maximize margin minimize result threshold show generalization dependent threshold probability point training lead choice minimization
idea design kernel encounter walk coincide walk termination relaxation labeling approach proximity walk similarity constitute contact level contact whereas build propagation common induce graph string stream unlabele propagation hash structure approximation fed base kernel propagation propagation concept utilize walk graph graph random walk kernel graph attribute appropriate review propagation commonly level kernel endow possibly partially observe adjacency matrix k represent markov state indicate walk current probability represent normalize partially graph attribute unlabele fully encountered leave row kronecker simple never leave encounter encounter induction iterate map initialize distribution unlabele distribution introduce simulate transition assign probable scheme discuss label extension naturally employ propagation suitable type basic scheme attribute steady prediction however converge steady kernel obtained provide kernel entire encounter represent accomplish sum iteration rather next v I ie edge weight label graph determine node label graph graph structure node evolve propagation kernel contribution important feature propagation node update maintain throughout attribute correspond attribute propagation kernel semidefinite propagation valid valid semidefinite positive semidefinite number convolution semidefinite kernel propagation circle style circle draw width cm bin cm thick draw node white fill circle fill circle circle circle bin bin bin circle fill bin fill fill fill b node circle white circle circle fill white bin fill bin bin bin bin bin circle fill bin b font b font g circle style draw width circle style circle draw parent thick circle node circle fill node fill circle edge bin circle fill fill bin fill white bin node node fill fill circle fill node fill b bin circle fill circle bin bin bin bin font light assume graph perform graph node bin strength compute count plug base simple outer product count th graph count value exploit exploit computation graph summarize kernel eqs design propagation kernel compare scheme suggestion kernel briefly discuss runtime runtime count strength computation note aim basically along operation information time usually label propagation kernel introduce distribution attribute propagation restrict represent iteration quantization locality sensitive detail next row attribute kernel attribute attribute thresholded commonly node deal attribute product kernel respective dimension kernel attribute distribution node explain locality sensitive hashing use derive way propagate attribute propagation kernel panel b thresholded quantization approach implement kernel distribution attribute inspire locality seek quantization space space probably bin vector discrete appropriate attribute simply locality hash give function family real independently know standard draw attribute integer kernel case decrease choose hyperplane hyperplane hash hx intuition behind expression hash attribute element endow distances hellinger scale locality family direct eq square root point introduction vary insight powerful propagation scheme utilize algorithm kernel appropriate unlabele attribute specific part change marked label propagation database graph total label efficiently sparse efficiently due sparsity general propagation graph green fully label bin tw diffusion node differ iteration label originally graph propagation label partially essentially unlabele adapt label add relevant large label iterative goal pixel among grid kernel value arrange node capture kernel spread grid simply goal complexity edge medium sized texture patch consider neighborhood complexity would million grid number fortunately exploit flexibility early discrete convolution denoise processing image isotropic invariant propagation adjacent lattice node regular neighborhood ignore neighborhood ignore boundary graph regular note neighborhood common grid actually grid form regular square derive line two grid define node note specify structure center carry value neighborhood neighborhood neighborhood illustrate gray sep count count count count inner count count neighborhood derive define operation modify graph interpret discrete variable g computation fact process operation digital resort highly develop example compute fast bin iw convolution simplify notation grid unlike natural notation represent third tensor make exposition enable efficient convolution label probability kronecker delta propagation appropriate matrix circular introduce pixel neighbor equally space approximated filter grid graph algorithm propagation highlight green fast fourier time efficient adapt tensor virtue grid use circular neighbor rotation make attractive implement extension propagation ask sensitive respect propagation use propagation kernel compute classification answer diverse graph include chemical texture flexibility kernel diverse database attribute attribute image pixel use total node node dim bioinformatics label anti cancer cancer protein world semantic originally introduce image represent random illustrate b quick among connect adjacent semantic label human small dataset annotate ground label mode truth pixel semantic object fall remove solely class thick white inner version color cm matlab dataset except evaluate run classification experimental exist learn iteration split protocol introduce validation learn enhance continuous encode bin chose evaluate propose propagation kernel choice analyze respect accuracy randomly propagation propagation first learn full combination far repeat normalization normalize large value ex yes yes yes yes yes yes yes yes yes attribute graph indicate actual propagation quickly perform bad computation label art comparable classification give split finish method pair test l h assess partially graph remove accuracy subtree design partially graph variant unlabele additional another miss obviously baseline slightly bad propagation might beneficial large missing method compute via string implementation kernel scalability property begin calculate intermediate classification kernel successfully partially question derive propagation bold significantly pair l accuracy split finish within protocol parameter learn full dataset learn quantization neighborhood performance color color neighborhood baseline label gray occurrence matrix compare intensity label pixel graph show sophisticated art computer vision texture feasible feature commonly computer ensemble conclusion mining computer accuracy standard error fold quantization min degree version additionally claim table support propagation prove flexible thus ultimately base principled uncertain unlabeled walk discover share construction namely propagation kernel propagation kernel common induce iteration graph kernel close experimental accuracy attribute moreover tie propagation adapt development directly message probabilistic deal closely derivation propagation ex kernel computation perform machine ghz intel processor compare computation extent mean computation finish h h average accuracy error fold run randomization parameter attribute hash per attribute computed version perform unnormalized mark memory perform learn training unnormalized kernel memory memory lemma kernel monitor graph leverage early propagation scheme walk capture label attribute benefit many label unlabele direct attribute leverage informative propagation kernel considerably regular modeling video database exhaustive structured area research domain become diverse situation example annotate document content modeling goal represent structure efficiently popular kernel similarity classification several literature strong assumption information proposal encode encounter real world rich challenge information lead partially uncertain aggregate source semantic annotation partially available collect sensor consist coordinate detector possibly semantic annotation entity document entire providing amount surprisingly broadly challenge design unlabeled label kernel attribute gain drawback handle graph attribute flexible consume overcome problem relational supervised aforementione efficiently initialize uncertain partial
degenerate distribution evaluate filter hmm forward marginal exactly marginal demonstrate fairly narrow performance practice successful application particle comparable implicit instead relative bind data base parameter express distribution assignment chinese restaurant number prior assign number ht cluster color treat mode structure particle c c synthetic gaussian table simulate increasingly build prior clustering marginal accuracy cluster overlap variance greedy cognitive particle outperform particle particle ht accord infer model use sort important experimental researcher amount extract spike attribute neuron spike belong naturally motivate sort use particle filtering filtering achieve particle choice dimension wishart material detail well filtering use parameter comparison qualitative demonstrate spike despite calculate hold likelihood spike quantitative summarize particle filter particle hold smc smc l particles smc bl particles smc generate induce assignment assignment restaurant probability probability select assign probability proportional visit hmms matrix use concentration b hamming hide multinomial resample log bl particles smc illustrate filter multinomial resample quantify compute hide show show particle next world taken begin apply character chapter book calculate predictive log character compute state sample particle hyperparameter show predictive outperform total outperform field study apply illustrate fix study lattice spin vary variation run field result principle randomness introduce vary initialization field particle particle trade achieve increase accuracy examine field achieve number field large particle importantly actually achieve iteration use large particle introduce framework particle discrete practical optimizing empirically algorithm particle optimize kl select optimally performance particle monte carlo efficiency advantage deterministic error sequential resample conditionally particle diversity degeneracy show ess performance relatively narrow avoid parameter worth note diverse unique particle conditionally unlikely avoid happen without resample combination key particle limit distribution important future require proposal question incorporate proposal monte carlo identify proposal distribution model combinatorial factorial completely may desirable proposal monte carlo achieve superior widely particle filtering wide method framework stochastically particle play deterministic variational approximations monte particle synthetic monte stochastically sample method make success sampler often paper approximation suppose place intuitively cover target variational treat problem ascent particle minimize divergence particle introduce filtering experimentally overcome problem able produce sometimes degenerate eq index clique stochastically generate proposal weight important particle approximation converge sequential filtering apply dynamical index sequentially conditionally probability step replicate particle particle degeneracy concentrated particle parametrize variational relate normalizing identity thus equivalent unlike monte converge variational improve bound iteration unlike likewise potential update filter helpful particle single particle particle set associate resample illustration different space circle indicate indicate cell high time iteratively rest conceptually approximate continuous use particle approximation approximation sense use pass attempt capture
due develop accurately solve heuristic orthogonal orthogonal matching pursuit subspace pursuit pursuit etc support component iteratively current update solve also greedy accelerate thresholding pursuit primal dual basis pursuit find variety apply comprehensive overview greedy relaxation counterpart read control evident certain properly coincide minimizer however challenge convergent nonetheless broad iterative backward splitting convergence characterization global minimizer novel primal active set develop study note minimizer mild contain true cf coincide parameter extend sense mutual incoherence isometry rely evolution dual global organized collect provide minimizer convergence estimate essential analysis minimizer datum noise column appear last submatrix invertible mutual incoherence isometry rip rely mutual coherence sense small mutual coherence mc say satisfy exist constant small mutual rip nontrivial basic disjoint subset hence nonempty assertion follow apply first note unit diagonal satisfy identity list product column column coherence matrix series immediately disjoint give set iteration active variable active estimate rip estimate rip condition follow hold update triangle dual e hold characterize minimizer multipli lagrange counterpart nonetheless aim recover expect shall oracle derive directly equivalence minimizer nonconvex global minimizer minimizer minimizer thresholding equivalently active minimizer analyze small active contrary nonempty b hold separately deduce contradiction minimizer minimizer see minimizer perturbation local deduce alternatively minimizer hold level solution formulation minimizer minimizer deduce contradict minimize minimizer nonempty global lemma eq rip I I assumption contradict optimality assumption monotonicity rip deduce view appeal assumption together yield contradiction moreover support problem minimizer argument theorem deduce oracle due equivalence lagrange clear equivalence cf problem active property algorithm determine active solving square newton convergence guess parameter active minimizer active minimizer empty max inactive check assertion inequality eq converge finite inner lemma index stop reach mathematical induction iteration lie large reach terminate stop satisfied analogue g kb deduce relation lemma rip similar omit step converge terminate monotonicity line reach proof imply criterion algorithm pursuit omp pursuit htp rip sense analyze omp appear rip appear omp rip omp require active lie iteration move inside outside confirm make omp htp due primal set method active component primal dual component iteration naturally illustrate efficiency sense signal mean take specify three setting approximate mild reasonably value recover exact active greatly insufficient practice rough study variation observe c unless estimate htb cc gain insight evolution inside set contrast omp iteration flexible valid bernoulli dct htb b e dct partial p nj guess observe dct matrix attribute hence cc bernoulli dct six art literature pursuit omp accelerate iterative compressive homotopy algorithm e percentage reconstruction whose agree realization setup value numerical observe largely ht illustrate greedy cpu result dct table compute setup observe reconstruction computing scale omp htp omp e htp omp e omp omp e e error omp e omp htp e omp e omp e e htp p rr omp htp omp htp omp htp omp e htp lastly signal matrix square update solve cg guess cg stop cg residual one signal sampling apply inverse transform transformation nonzero entry reconstruction visually appeal excellent exact reconstruction far confirm reconstruction true reconstruction partial nonzero table remain largely almost reconstruction effort competitive b omp htp omp ccccc method cpu omp htp ccccc omp
net overlap make voxel similarity minor difference subject overlap voxel person voxel force voxel encodes picture voxel force encode rise large argue account similarity difference across subject code interpretable drawback glasso voxel rise cluster sparsity negligible voxel encode picture sentence consider select voxel encode picture primary setting look simulated function toy overlap vary level corrupt white deviation average group retain fraction retain regularization minimize latent lasso group matrix plot account inter group glasso active non pattern color within active reduce overlap group overlap group however account hence glasso lasso account structure poorly perform bad glasso explain introduction motivating arise particular tumor gene cancer gene pathway also pathway replicate data balance dataset overlap group standard ill h compare lasso one perform enforce use constrain solution generalize group arbitrary result overlap outline fmri biology minimal generate result general light overlap paper designing allow cognitive view grouping voxel spatially co locate significantly remains see whether motivated way voxel functional take since index q corollary sub index singular prove gaussians product equivalently write define multiply mean width argument index mean g follow similarly prove time quantity sake lemma play dimensional feature group subset many however structured selection group comprise set rich lasso framework generalize conventional allowing present challenge paper overlap automatically feature classification establish bound classification bound classification lasso group fmri voxel source localization activation microarray synthetic demonstrate play role machine learning application feature far number search sparse notion prevent interpretable sparse case group lasso group coefficient group overlap penalty one success structured selection arrange prior feature specific propose overlap group reflect example relevant play role gene pathway pathway pathway bold letter matrix sparse simplifie allow tool correlation realistic focus classification randomly true inner satisfie euclidean norm normalization quantifie label inner product error bound consideration yield constant fact consider product measurement model model somewhat often correlate compute enter bound follow structured user define depend assume feature hand group structure wherein group relevant feature localize union group localize subset armed state theoretical later within measurement solve statement explanation parameter suffice looking amount group overlap program succeed group hold fmri nonetheless belong propose overlap group sequel lasso lasso tool encourage less encourage identical accomplished define group pattern glasso consider interested paper motivated fmri goal cognitive activity feature shape neural structure neural somewhat across individual guide locate useful general useful voxel voxel regularization lasso across penalty account coefficient account common across motivation grouping overlap classification pattern arrange select also extension lasso group interested apply regression sparsity analyze feature arbitrarily accord also restrictive introduce activate coefficient contribution theoretical analysis consistency reduce know sparse high recovery extend arbitrarily overlap regression work overlap group logistic far rich knowledge provide unified bound non overlap group method mention suffer task undesirable effect many version toy experiment advantage group especially image gene biology regularizer encourage coefficient overlap generalize two scope level consistency lasso group overlap group make minimal applicable setting correlated design turn translate unified theory structure motivating fmri fmri yield low hold test interpretable domain breast cancer breast motivate fmri author consistency give derive biology notion result potentially theory overlap propose sparse lasso character recognition gene handle show admit proximal operator exclusive modification express strategy problem solve coefficient bound induce sample possibly overlap group overlap characterize program solve extensively characterize correct author generalize glm classification minimize maximization linear subject organize structure sparse selection regularizer recover pattern across group overlap result toy datum conclude future return main unit motivates optimize natural positively natural thing maximize quantity term define difference corollary group overlap sparse lasso course effect parameter g g get pattern prefer tend prefer value function two sparse select take account account group group c g show indeed exhibit group overlap consider see list consider instance solution zero group localize sparse finally fourth correspond lastly imply hence convex optimization convex program positive homogeneity lead homogeneity possibility optimal representation triangle inequality lagrangian version structure prevent value method expand progress take gradient shrinkage proximal overlap special case general structured point composition obtain final correlate vector note definition group setup ideally solve instead overlap group lead admit relaxation tight relaxation vector overlap follow iii follow ensure tight first non crucial consistency width theorem lemma completeness behind ideal interested contain scaled hull non scale outer end ideal follow fix magnitude manner ij fix small average exact substitute give construction nc ij conv nc width result ideal scale result width constraint side jensen iv follow note lemma away treat us inequality statement proof lead light overlap group complexity want would bound regularize correlate interested correlation fmri major motivate voxel brain exhibit amongst entail number matrix measurement matrix obtain constraint reduce variable enforce constraint lead generalization theorem correlate entry sample wish group vector condition matrix along prove perform toy recover sparsity interested experiment toy datum yield optimization cognitive biology group commonly encourage set select wish focus less restrictive encourage restriction correspond task relate accomplished define subset search solution common across figure interested ability well interested arrange column correspond g coefficient standard result glasso task overlap star plus involve sentence half fmri retain stimulus yield distinguish point stimulus exist expert partition discard average bold voxel within subject kind encode assess aid discovery voxel expert pre
method conjunction example belong discussion goal study possibly turn inference reality collection recognition concern classify category independent category member define approach category share available insufficient quantify object category major literature subject dedicated numerical challenge problem histogram interest category model empirical identify relevant intensive conjunction derive mixture briefly discuss quantity also overview challenge article quantitie category option belong end uniquely determine observation height certain country category belong consider try approach density density single mixture parameter describe member emphasis belong component represent guess respective category independent category adopt know problem determine quantitative state knowledge category similarity notion consider category coarse key group category replace represent iterate coarse category characterize distinguished intensive relevant intensive often challenge less determine extensive element moreover coarse homogeneity among category complexity hand choice describe find invariant coarse mass particle intensive shall intensive expectation law allow represent scaling encounter constitute harmonic precisely geometric harmonic regard although nonetheless might infinitely interested quantity maximum among define mean short find lagrangian correspond matter convenience calculus belong note improper non compact support functional uniquely moreover narrow broad influence uniform distribution consequence distribution line modify literature well distributions eps familiar arise example drop list demonstrate abundance sake clarity discussion law quantity nonetheless recommend one conduct assessment start outcome along functional density aspect often dependent get statement quite simple density determine become randomly h p rough knowledge
select summarize equivalent define confidence test framework screen respect algebra measurable distribute element distribute rank computation screen apply theorem statistic v partition sum unconditional conservative sense type exactly select subset sign specify hypothesis compute via previously f discuss fact decrease need smooth monotonicity truncate distribution family hence likelihood formalize immediate consequence unique interval relation e hold boundary confidence interval however interval correct whereas response see confidence exactly unknown diabetes diabetes patient age body pressure measurement quantitative baseline disease year statistically predict tx I assess unknown use fit adjust adjust cover nominal always broad propose hypothesis test screen particularly apply match pursuit square exhaustive selection procedure condition incomplete list ease construct interval fan recommend algorithm follow lasso select two selection follow stage event sign sign lasso screen encode test algorithm valid marginal omp commonly omp correlate residual residual omp selection encodes omp choose screen variable confidence omp non eq eigenvalue several sign negativity factorization network interval estimate coefficient primal pair iff complementary choose give kkt q encodes omp marginal screening hypothesis linear important selection framework screening apply marginal screening part nsf dms grant lee nsf stanford fellowship theorem definition theorem proposition remark framework marginal characterize exact model allow coefficient contrast statistic exact eigenvalue like assumption marginal regression negligible particularly suitable focus applicability framework broadly propose procedure include match pursuit non inference estimator commonly distribution test however square perform variable aic match marginal select response screen selection procedure marginal screening dataset intractable comparable lasso screening screen select truly screen combine selection lasso far extend screening utilize response confidence nominal coverage may long difficulty post estimate sense counterpart post selection operate subset distribution marginal represent event construct truncate develop although marginal propose clean illustration applicability discuss extension apply cover clean screening omp negative square high focus establish restrictive select correct refer theoretical property interval subsample interval pursuit extension leave thorough investigation position mean distributional sample high requirement test compute regression propose compute method paper matrix response tx screening choose high absolute notation distributional long test similarly hypothesis empirically test screen h vector select construct interval independent snr interval drastically nominal model equation hold conditional screening event selection select sign partition q previous constrain partition possible subset sign event set tool
build response prediction amount response three classifier building vector implement comparison obtain net rmse acc bayes auc acc auc acc different calibration single observe obtain table amount datum possible leave building calibration model save one calibration repeat calibration measure prove three justify post histogram discrimination measure extension dirichlet simulate show compare work investigate histogram mini mini experimental histogram width provide similar another paper calibration problem auc theorem theorem helpful reading expect bin histogram theorem rewrite recall identity apply identity write finally combination histogram rate auc calibration would recall hoeffding inequality amount high concentrate concentrated auc follow auc nice possible show empirical concentrated output transform non overlap bin base two use assumption auc method partitioning simplify calculation part summation convergence summation recall method sure fact rewrite inside equal notation term true concentration second summation part iid concentrate around empirical estimate construction frequency bin rewrite last inequality fact order completely reverse apply b k histogram calibration notice proof show auc point even auc histogram dataset department university computer department abstract calibration critical category classifier method model calibrate transform well histogram scale use paper introduce measure calibrated three use calibrate discrimination capability parametric extend parametric demonstrate outperform comparable accurate probabilistic datum decision task unfortunately optimize task generally calibrate predict fraction concept readily outcome reliability display calibration curve predict outcome calibrate tend probability low calibration line calibrate represent make uncertainty calibrate produce calibrate critical determining business decision area study nearly extensively level develop calibrate machine traditionally focus development model discrimination exist potential well calibrate learn simplify linearity another scaling improve intend calibrate show calibration objective perform solely post objective theoretically justified limit post process perfectly calibrate discrimination roc least good two exist post processing apply predictive calibrate use limitation rarely prediction parametric histogram sort partition bin find contain return fraction bin calibration increase calibration algorithm position classifier variation predict introduce measure evaluate classifier prove three use calibration capability measure discrimination introduce method describe section present classifier mapping instance classifier calibration intend calibrate follow notation remainder predict locate inside predict locate inside predict go infinity empirical capability calibration reliability diagram reliability measure calibration prediction sort partition ten bin calibration calibration error estimate pi instance post calibrate bin bin prove max classifier lipschitz decision boundary histogram bin classifier behave histogram frequency near interesting open histogram condition bin non histogram calibration method plug bayes rule follow prior dataset term histogram estimate bayes iy iy iy empirical estimate bin algebra calibrate plug classifier likelihood advanced density simple density kde bandwidth bandwidth optimize use validation bandwidth unbiased however notice kde report estimator risk smoothness target rate practical fortunately binary classifier calibration prediction classifier calibration kde frequentist approach frequentist model mixture bayesian building calibration lack choose collapse collapse implement refer kde auc acc kde rmse auc acc describe calibration evaluate performance run acc roc curve auc discrimination three root square error calibration outcome separable scatter plot train quadratic method simplify discrimination figure intuitively ideal quadratic allow classification auc c auc kde method perform svm poor surprising parametric parameter make violate perform relatively poorly improve
slack pos train slack negative example balance tradeoff pl explore al modify balance occur svm handle imbalance use address imbalance resample address imbalance resample analog imbalance train pl approach overall imbalance imbalance use tc svms aim train extraction aim perform system aim tc belong treat separate ten category hyperplane decrease pa name pa al svm hyperplane high hyperplane pa imbalance pa corpus imbalance drastically imbalance figure pa example label get skewed corpus systematically example random loop criterion meet x f hyperplane point pa al guide sample require confidence many college computation aim enable confident proportion proportion size implement point graph show highest aim perform begin area want stop observe identical al reveal nearly category slack pos classify situation model achieve train outperform hyperplane two annotation stop learn hyperplane classifying round approach hyperplane hyperplane train vice hyperplane predict svm scenario call pl address imbalance estimate overall imbalance imbalance utility demonstrate situation help show sampling pl instead modify characteristic exploration investigation center university md usa edu sciences sample data characteristic sample datum use passive pl explore case al weighted svms arise address imbalance cost estimate corpus imbalance imbalance pl recently interest reduce annotation sample characteristic modify passive learn pl case svms factor imbalance pl showing improved substantially address imbalance pl show prevent degradation date relatively imbalance scenario bring pl approach imbalance
batch occur time annotate batch label predict q otherwise batch evaluation tb batch label prediction multiclass explore size stream characteristic performance need procedure carefully seem provide repeat test across attempt run dataset hypothesis verify classifier effect baseline logistic look propose batch classifier evaluate run datum available confident measure expensive annotation cost however visual arithmetic modify select informative batch try answer give robust high show experiment good discriminative work complexity framework feature video stream surveillance scenario grant surveillance query place human resource limitation way software video reduce impose herein incremental evolve visual stream develop track consecutive update try balance restriction evolve camera drift address well non stationary environment show little decade surveillance spread rapidly area hour day year massive amount come evolve method reflect environment underlie datum refer change movement change dynamic camera angle etc model update concept new enter machine model need surveillance angle camera surveillance camera view often disjoint due constraint enter surveillance system track object capture camera around stream record various associate detect consider span person person side capture camera person person b person capture camera switching position simple typical tracking occur b identity person movement track identity suppose global necessary capture identity scene label annotation consume impractical annotation extensively explore un label visual environment concept drift evolution researcher mostly address active lead evolve video stream surveillance scenario body camera surveillance adjacent overlap whereas require overlapping view herein continuously stream label still setting focus tracking receive directly track object framework camera axis indicate period discuss limitation former incremental provide future work bp cm cm cm cm stream stream concept table sl constrained md sl unconstrained md sl unconstrained md md constrain assessment method denote dimensional sl semi recent surveillance object change video receive despite abundance concerned environment recognize resource obtain label approach video promising scenario identification semi label track limited environment classification address label perform fully method track therefore stream challenge real stream mostly appear mining constitute handle drift recent evolution learn generate dynamic weighting heavily cluster employ active although considerable stream view explore area stream come thus track appropriate require stream start stream drift accommodate partially review qualitative review scenario stream visual learning framework label obtain carefully point ensemble incoming batch combine weighted voting sketch tracking analysis frame movement generate stream environmental challenge illumination lack bad acquisition device motion noisy miss address gap cause tracking provide batch index th stream present slot stream track camera stream frame start potentially batch align stream frame correspond object representation bag word scope obtain slot name composite arrival batch try predict kind work suffer assign view object help interact human stay track initially composite yield probability batch slot computing batch batch estimate predict accept correct otherwise label batch multiclass integrated yield batch composite c slot composite update prediction section classifier design composite equal many numerical problem logarithm monotonic k pc batch decision composite approximation posteriori build batch prediction frame arithmetic prefer author arithmetic geometric presence experimentally option decide automatic prediction rather manual criterion uncertainty perhaps rely confident class define confidence however consider away probable margin since little batch margin label would help discriminate effectively relative confidence exact involve put forward probable modify pc pc product issue alternative confident write follow comparison denominator simplification strong independence aggregate end side rewrite trivial obtain characteristic four confidence every corner indicate herein class corner move inside decrease linearly increase corner drop like composition class least would triangle ensemble informative lie corner b slot obtain batch manually belong tracking mix identity slot batch observation batch slot batch posteriori given predict option find discriminant probability play design discriminant still estimate experimentally challenge automatically resort class also present slot class previous incoming describe individual dynamically update respect multiclass ts adjust update normalise slot choose weight normalise q conduct capability order stream scenario test drift evaluated cover scenario camera surveillance conduct experiment dataset synthetic dataset label suffer simulate situation change parametric table present equation generate gradually also process use scenario complexity stream drift drift appearance drift scenario gradually stream class stream class appearance environment evolution depict occurrence drift number enter enter stream illumination well capture employ automatic scene position perfectly stream object object descriptor vector curse system suffer set
inspire component precision feature community split classifier generic classification problem goal build analysis normal population boundary separate poorly pose encourage employ already number proposal lda ridge ridge ridge penalty interaction discriminant independence naive alternative feature presence strong paper span naive use across force precision place share sparse make classifier interpret refer within model idea identify applicable likelihood bayes partitioning several community community improve interpretability classifier quadratic discriminant discuss idea real conclude training observation set independent quadratic precision ki jk maximize q last penalty force pattern kx important feature tune shrinkage matrix rise naive classifier degree severe naive eliminate quite instability superior strong group force share lead produce sub potentially massive quickly solve purpose et maximize q additional lasso purpose goal interaction instead whether precision component similar rule optimization speed solution namely admit l entry skeleton symmetric symmetric thresholded admit component estimate induce exactly vertex induce nest component vertex theorem quickly check connected component rule different block simply solve block make certain impossible operate machine block feature mutually generalize set split estimate connect sample involve work class admit kk kx lx posterior cx serve equation fit classification posterior adjust intercept normalize consequence tractable infeasible scale modular naturally multinomial logistic regression generalize thus variance conditionally independent community apply average complete linkage correlation matrix procedure conditionally community univariate strictly kx kx different make identifiable preserve x ef j gaussian monotone differentiable denote define l exactly conditionally nx correlation exploit directly base base base class define define n validation community correspond apply linkage agglomerative cluster cut dendrogram linkage cluster merge one time knowledge true agglomerative cluster consistently vertex component ij ij partition result proof appendix community use element absolute estimate single linkage ij pg l pick study example achieve misclassification interpretability tuning result diagonal correspond covariance refer version regularize misclassification standardized tuning four performance follow validation respectively minimize misclassification case class class precision result deviation replication interaction situation favor give small deviation naive standardized decrease model perform class table summarize result situation term favor level increase increase strongly dominate misclassification capture interaction need cm cm cm al illustrate handwritten digit handwritten le normalize sample smoothing help filter select misclassification standardize parameter naive decrease dramatically naive keep range include noisy term unit standardize correlation diagonal band use regression regression logistic regression compare logistic lr logistic show prediction training tune experiment identity employ transformation definition lr use linkage cluster community transformation population class identity cdf cdf transformation skew bi summarize misclassification perform one introduce class conditional distribution cm email email spam
relationship linguistic word embed example architecture network language learn unlabeled deal several language al skip skip learn text close position embed base similar context although word become public introduce benchmark name source performance moreover take discussion research organize create report art word consist task generate evaluation tuple word whose close bc question correctly answer two category task include e syntactic nine give evaluation semantic rest nine syntactic question tuple word tuple unique word pair small dataset check tuple pair tuple question tuple publish reason tuple pair word tuple capital capital city california apparent rapid comparative think read past merge pair extract new pair english derive pairwise word evaluation introduce scope vocabulary extract knowledge cover vocabulary corpus snapshot wikipedia corpus million token vocabulary size besides phrase token release leave phrase pair leverage wikipedia semantic merge wikipedia page area find u wikipedia city accordingly name entity remove name take advantage semantic new rest extract candidate filter vocabulary statistic wikipedia word gold semantic syntactic task base merge filtered vocabulary much well word pair tuple http microsoft com en performance state distribute representation public public public site representation representation demonstrate reasoning table yield reasoning task training dimension likely well analogous different diverse note experiment question vocabulary regard incorrectly measure use recently embed relation relation type regard pair researcher list nlp list syntactic list new word build collection art word research topic future work plan phrase consider web knowledge basis thank xu evaluation state word embedding c pair capital city california rapid comparative greater easy reading past l word word tuple water exercise face city attribute depth sound entail pay c dim dim dim dim dim dim capital city comparative
determine movie positive rating full popularity keep high tf tf tf document term occur keep consist set world extract stock year label label use day one ignore minor price result website review mark label positive review label negative assign normalise appear appear tf rating popularity description website associate vote normalise apply dataset vocabulary document collapse variational implementation collapse collapse run relative change fold calculated response response proportion often goodness obtain perfectly calculate full prediction fold relative deviation fold small choose validation hdp place every iteration residual converge iteration number topic perform show use computation coefficient algorithm sample binomial take step glm learn label way jointly glm compare show well variational variational clutter perform movie outperform pick right pick drop yield good pick posterior make little additionally glm significantly jointly competitive dataset partly model increase number document one remove since newly empty easy change allocate topic number word allocate make difficult change smoothing contribution fact low predict movie review deviation score imply noisy influence stock stock change stock price movement market pick subtle sample maximum fold method fold plot chain calculate compare chain chain converge chain indistinguishable give plot significantly indicate well experiment directly linear regularization validation fold accuracy movie dataset document dataset marginally outperform movie review outperform popularity benefit l glm usage dataset coherent usage powerful extra topic enable top negative frequent movie review topic contain name actor flexibility nonparametric mean consist term allocate actor review poorly flexibility result group around consistently actor coherent even small topic term spread topic negative actor name topic see specific actor associate movie topic seem name focus contribution movie rating learn divide content topic concentrated actor topic algorithm train learn hdp like sentiment term regression rating cccc party instead call topic six company teacher nature winner onto topic suppose unfortunately lot flat air break ten cccc political bank serious usual face want topic cat top frequent term topic movie review term good topic consistently stock negative topic involved stock price cccc topic record publicly water hold c whole american china effect account c drop account range among topic stock price demand indicate grain actor specific trend possibility topic supervise nonparametric datum document popularity supervise hdp topic document choose model overfitte occur learn grain learn classification experiment world movie dataset experiment also jointly learn glm model hdp inference improve outcome extra add sentiment propose general typical sentiment dictionary wide response generative restrict kind topic previously patch patch similarly paper keyword google view complete research interest include reader school mathematics networks associate school microsoft fellowship research focus e continuous time image medical associate nonparametric joint method group world dirichlet generalise dp allow flexibility nonlinear dirichlet hdp mixture seen supervise hdp learn predictive solve allow learn structure allow model adapt data nonparametric frequently new increase group item analyse performance involve label input predict response dirichlet allocation group text document vocabulary e topic think infer topic mode membership document mixture successful collection citation database corpus wide include information retrieval latent learn model document dimension turn model regression collection possess categorical order type sentiment modelling topic dimensionality topic topic ignore response learn response response cause end assign supervised topic predictor document lda learn unsupervised topic orient predictive example topic learn consist term sentiment topic contrast topic line document prediction make unsupervised topic infer supervise perform must topic unseen observation opposite model topic relative contribution typically dominate capture leave component number run difficult method naturally handle flexible nonparametric hdp generative predict model contribution topic overfitte hdp number g rest briefly review exist work group introduction generalise later paper describe use real dataset consist response outline group amount exchangeability predictor response response predictor corpus encoding vocabulary document rating category kind group outline previous paper nonparametric gain flexibility flexibility process gps supervise logit generative covariate response model jointly mixture assume cluster multinomial logit response conditionally covariate logit protein fold classification machine mixture generalise glm explicitly dp glm continuous use generalise regression generalise prior coefficient gaussian neither dp glm predict prediction group learn however flexibility topic leave work regression label dirichlet process hdp nonparametric analog lda allow flexible modelling though variational bayes significant suffer lda paper extend hdp learn stochastic thought fact distribute wide flexibility posterior important problem dp glm document exponential family dispersion predictor glm document exchangeability assignment imply glm map inferential possibility symmetry break exchangeability topic assignment use sensitive broken process generation label choose mean alternative proportion document response constant estimation expectation method lda collapse use require topic topic consume propose necessary response infinite prediction extend cluster align response response model generalise cluster allocate need advance beneficial supervise unclear model model generalise linear condition number vary coefficient generative sample effect regression treat coefficient treat topic also coefficient categorical model dispersion range range document dp concentration document level topic corpus act base document prior density document consist word dirichlet likelihood observation conjugate topic integrate collapse keep track model response categorical glm coefficient process draw concentration topic iw draw I choose topic supervision easily aid response document document infer group publish group previously could set previously pick topic allow topic give control type learn topic particular allow use allow topic document entire label document dp approximation collapse common technique
kx rx I h hx kx h mx x kx bind together corollary imply complexity logarithmic minimax section express term exponential label speedup agnostic active gaussian learn study long allow hx guaranteed exist satisfy result hold allow generalization key definition facilitate agnostic hypothesis agnostic compression small vs f extend version define agnostic disagreement coefficient except respect hypothesis case satisfy case begin extension agnostic pf p agnostic x b finally set f vs c vs vs purpose agnostic passive original agnostic specialized exponential speedup low accuracy regime passive sphere additionally provide express term disagreement translate agnostic active algorithm improve still universal budget label request request label satisfie exist class linear request corollary complexity passive low bad long satisfied hypothesis flip intel collaborative research institute intelligence page version also completeness present formal proof trivially union probability least right let work exist active active simple showing sometimes aside interval z I mx mi mx km match low aside disagreement technique exist literature vc active immediately vs I vs vs vs vs vs give ii j h jx w w jx w jx h completely letting vs w w vs x ds quite strong gap require therefore include distribution unlike disagreement potentially eliminate replace measure g vc vs xx wiener run improved label disagreement lead quantity small tight refined linear mixture gaussians density compression characterization agnostic active selective sequential pac learning paradigm sequentially request pool stream active label request active perhaps advanced article base sequentially request classification attractive thorough numerous guarantee literature thesis label exception root relate selective wiener wiener instead measure learn article characterization disagreement lead wiener correspond induce set special improve upon technique factor either measure sometimes vc wiener active characterize interestingly also express specifically vc relate compression disagreement coefficient show always logarithmic factor axis align product compression relate compression arrive base represent result setting naturally extend new agnostic well observation wiener disagreement applicability disagreement formulate result disagreement general wiener denote arbitrary call aside set disagreement pm throughout discussion vs hx satisfy vs two name exact query idea mf x selective terminology terminology article formal notion minimal specifically implicitly remain exist disagreement active coefficient classifier ball center coefficient introduce also root disagreement active disagreement specific show asymptotically provide passive interested even passive disagreement date sufficient work show bound target pass continuity dependence little big vary also several disagreement perhaps unit within though disagreement coefficient disagreement term disagreement vs bf clearly x chernoff since prof imply converse show constant algorithm else attractive maintain obtain f count first follow provide bound establish bound allow dependent direct compression rely imply bind frequency specify completeness result begin line purpose compression collection measurable set invariant value hold remainder case let distinct n index I n suffice n total q due e expression follow provide result selective improve selective classification directly compression ms ns vs ms furthermore vs probability vs nm mt monotonicity union least q bernstein imply let also imply query let integer result n nm maximizing imply furthermore union nonnegative nm u nm nm bound primarily interested guarantee final element mn nm mn nm er er minimization large e relating quantity definition fix vs universal large monotonicity instance implication monotonicity imply corollary sufficient condition improvement passive minimization typically versa factor validity interpretation imply validity valid regardless interpretation choose stick throughout decompose implication implication strongly connect direct equivalence statement therefore mi x mi mi plugging reveal implie imply imply union probability fact appendix study hypothesis conjunction smaller know result specify classifier include member normal distribution covariance combine constant particular label complexity bind result factor reduce rank describe problem product x j b classify distribution align base slight cdf gx kx kx gx kx ki ix g equality monotonicity intermediate let hz ia g ib axis rectangle furthermore every monotonicity hx combine set point monotonicity former ia ig ig ig ib ib b ig ib ig ib hz ki ig kolmogorov monotonicity cdf second union imply probability realize uniform right side without toward denote ix ik x j event probability px ij b probability
ability since also non generate nn compound depict average shrinkage case vertical value regularize dot depict oracle bottom sample average dot vertical resp shrinkage set twice view generalization tend quite case similar play primary interest oracle scatter fairly scatter case shrinkage outperform estimator close identity towards identity detect signal receive represent unobserved clutter v signal signal receive datum variate example unknown parameter accounting channel vs signal problem follow scatter consider refer alarm rate match fact great clutter model belong alarm equal rejection threshold however replace require fix become scatter clutter rarely case retain although consistent inaccurate affect regularize estimator provide property detection pd nmf scatter investigate detector study target signal trial datum detector secondary estimate detector scatter clutter note figure reflect pd nmf numerically integral signal clutter db observe pd mc detector able pd nmf detector length secondary regularize natural scatter suitable penalize estimation uniqueness establish regularize estimator sufficient uniqueness choice matrix estimate provide study illustrate method express go infinity readily definite hermitian sufficient singular semi hermitian trace one j assume z converge need show nj pa l b exist readily continuously differentiable non unique ng k v proof iv contradiction show sequence next convergent ii observe equivalent find next show identity proof rely representation r possess derivation therefore omit eq gives resp resp expression complex real similarly real definition david department nj usa mail scatter lemma theorem insufficient constitute generalization scatter penalize pair derive uniqueness concept geodesic include uniqueness establish solution shape match sense iterative scatter compare counterpart support match nmf adaptive maintain time nmf detector convexity distribution scatter normalize match classic multivariate analysis technique covariance eigenvalue variate sample z I many simply completely inaccurate example occur medical imaging insufficient support normalize filter realize require key partly difficult conventional environment well inefficient estimator draw non datum pose difficulty scatter scenario estimator scatter covariance problematic propose constitute generalization cost include uniqueness utilize study regularize focused regularize scatter variate note problem correspond treat sufficient establish ensure cost regularize strict derive regularization matching sense usefulness scatter application match although generalize real review complex symmetric ml introduce penalize stationary solution type give uniqueness uniqueness estimator proof hermitian resp resp ij symmetric f unknown parameter scatter generator ensure tp cn reader denote variate maximum scatter divide emphasize estimate scatter generalization estimator elliptical necessarily relate elliptical function estimate ml denote minimizer covariance estimator function relate elliptical nevertheless estimator scale mild take estimator numerous interpretation huber tuning f chi square freedom usually result estimator huber estimator estimator additive introduce cost denote regularization enforce matrix real case penalty though enforce precision dependent scatter paper grow eq parameter well describe naturally notational convenience equation solution satisfy regularize equation rise iterate convergent proof regularize continuously differentiable estimating uniqueness follow huber correspond huber detailed subtle gaussian form n z show estimator differ motivation estimator view ridge singular large spherical towards scale regularize weight p hereafter use minimizer hereafter assume continuous readily huber play key role uniqueness previously utilize uniqueness scatter geodesic convexity positive aforementione paper wherein treat class notion complex differentiable riemannian convex say strictly geodesic thus concept geodesic enjoy convexity euclidean minimum furthermore set subset minimum matrix omit complex analogous cost addition span cost whenever include geodesic corollary see convex convex convex penalize proceeding review hermitian respectively view case equality hold readily follow consequently geodesic convexity side strictly geodesic strict imply desire scale invariant regularize admit interior show need establish penalize minimum estimating place sample require proportion penalize condition sample extend condition occur continuous multivariate condition equality cm inequality
principal evaluation analogue lie neither algorithm involve matrix validate claim experimentally principal principal vanishing feature use kernel vanish kernel scenario synthetic perturb uniformly iii handwritten digit pca circle span degenerate principal square correspond runtime order htbp manifold manifold circle circle example approximated feature see near quantile example rest pool one vs follow evaluation correctness follow feature generating sample degenerate repetition sub degenerate good advantage dimension many repetition grow distinguished priori degenerate random feature extract generate fast principal comparable outperform experiment show capable extract competitive considerably scale comparable performance european european framework fp grant research ex thm matrix span fact scale kernel usual framework duality vanishing indicate cross novel algorithm pca pca real synthetic extract novel empirically validate trick costly implicit efficient huge learning task overview many make kernel source efficient linearization computational bottleneck part kernel kernel singular subsampling sub art write seem consensus subsample kernel unseen kernel output propose algebraic potential explanation issue suitably empirically practically able address analogous opposed subsampling prescribe thus address advantageous statement matrix manifold obtain lie invariant coordinate consider help span span reconstruct data point contain span back membership work reasoning hold large kernel cross approximate potentially k z kernel assume span conjecture approximately kernel subsample method discussion introduction kernel consider lie generalization carry goal relate image row span identify polynomial degree functional cut polynomial vanish informally identify vanish ideal relate duality algebraic geometric duality set vanish introduce scalar product slightly arbitrarily convention natural well satisfie n feature dx yx rkh duality algebra feature vector subspace polynomial fx vanish vanishing ideal sense dx usual duality result section kernel adapt exact approximate save polynomial kernel discuss algebraic statement prove latter spectral throughout manifold sample span cut denote sample row introduce kernel mi jk dx dx duality statement vanish appropriate span ns abuse notation say matrix independent notice elementary z analyse choose characterize dx xx dx kernel span imply I mf statement turn statement let denote matrix dx z xx nystr type demonstrate ideal duality employ datum manifold precede collect observation arithmetic decomposition require linear exactly svd via cost recover common subsampling completely need subsample contain subsample singular characterize
approximation read summation take grid form numerically discrete fouri dft slice restricted estimation memory series use g similarly excellent fit high frequency together summation correspond approximation final likelihood restriction image scale concern evaluation numerically scale invariant process size historical construction multiply cascade two multiplier multiplier poisson multiplier lp lp result mark bold la lf mmse lf lf lf mmse lb lf mmse lf mmse lf mmse lf mmse lf lf mmse r cm mmse lf mmse mmse mmse lf mmse lf certain limitation multiply construction localization multiplier specific normal poisson multiplier vanishing ensure consider estimator usual reflect linear standard sampler iteration burn period preliminary simulation illustrate fig estimation plot red classical bias lead vice apply lf mmse estimator realization process strong discard yield commonly improper wavelet transform deviation root realization mark c r lf mmse mmse lf mmse lf mmse lf mmse lf lb lf mmse lf mmse lf lf mmse lf mmse lf mmse well bold r lf mmse lf mmse lf mmse hyperspectral overlap pixel mmse c lf indicate red b center white dot original dot half patch histogram discriminant lf lf mmse lp size mmse therefore due result systematically outperform lf second deviation regression important remains directly reflect overall fit poisson multiplier find slightly inferior log multiplier due arguably slightly size force choice gain lf term bias rmse particular lf yield world notably reliably detect lf sufficiently enable estimation parameter patch come increase computational cost computation time image large cost lf belong design wavelet coefficient study g go fast wavelet plane indicate lf report practically rmse outperform lf factor likely lf real world fig pixel channel hyperspectral acquire project pixel mmse c lf red frame inspection indicate reproduce structure texture spatially homogeneous visually texture fig bottom leave corner corner spatially coherent consistent strong display variability texture lf spatially estimate strongly mmse lf corner truth illustration estimate image quality indicator quantify image measure coherence fig index mmse lf visual inspection spatial coherence fig mmse lf conclusion variability lf mmse lf yield portion since necessarily fisher discriminant criterion mmse lf separate bottom superior lf bayesian procedure quantity analysis I wavelet yet generic parametric statistical logarithm account impose theory design process approximate metropolis wherein infeasible constitute operational applicable size assess numerically process improvement rmse factor enable reliable pixel patch estimate future notably hyperspectral image rgb rgb member member characterization many application useful texture current two dyadic scale image difficulty estimation construction wavelet exploitation suitable within model enable infeasible assess several range significant benchmark discriminate commonly use gain notably enable image patch recognize common texture texture invariance literature scale image invariance tie pointwise image long recognize multiscale study regularity play central contour characterization contour texture consist densely strength provide texture tool regularity location description fluctuation collection value standard image tool include texture classification art capture image spatial location scale increment wavelet wavelet formalism theoretically summary would handle entire signature fluctuation regularity primary identification stochastic self former class tie construction fundamentally principle magnitude quantify image class process refer reference seminal tie j c lead estimator suitably relatively pixel sufficient encounter severe transform logarithm number pixel scale pixel consequence analysis severe first analyze yield make discriminate goal propose validate address overcome limitation e parametric formulate lie exhibit decay remain date wavelet exception brownian jointly parametric propose specific however method rely strongly easily formulated estimation procedure parameter recently employ heuristic parametric univariate univariate parametric relation actually image develop require mean logarithm process multivariate inspire covariance induce nature cascade construction new radial parametrize pass pass filter define wavelet characterize vanish filter low filter g purpose discard normalize reproduce similarity formal detail wavelet transform dyadic center cube eight neighbor within fine wavelet reproduce old exponent follow cube theoretical require construct formalism detail class extensively theoretically cf strictly regularity assess practice number wavelet choose sufficiently hold imply meaningful relevant novel statistical logarithm wavelet estimate bayesian numerically selection process lp stand normal log analyze range plot standard log associate within marginal wavelet member self member note wavelet process confirm formulate trivial reason property even log wavelet marginal significantly cf row mm standard normal lp center logarithm f wavelet cascade covariance logarithm wavelet suggest indicate decay r
produce infeasible iterate optimization parameter default exploit gradient option correspond bfgs hessian interior trust admit instead automatically indicate hessian solver brevity table report case costly reduction exact second order evaluation interior point minimum also recently practice solve subproblem concave subproblem formulate order tailor stationary existence adopt matlab module carry library initialize intuitive comparison among solver report give set solver solver performance ps occur solver mm ip tr nf fit nf fit nf nf nf tc ip ip tr ip tr nf number evaluation nf second cc table comparable fit difference attract overall satisfactory suitable scaling sophisticated flexibility scheme identification lead nonconvex nonlinear bind optimization problem analyze propose especially solution present compare art issue application projection scaling result depict obtain summarize view propose constrain performance nonconvex obtain respect technique order view hessian matlab convenient strength capability good impulse response order especially costly good performance rely scale constraint issue address wider arise machine identification sparse idea need context learn rgb rgb l lemma di di via b di crucial address criterion asymptotic hyperparameter marginal maximization primary importance impulse stable strategy projection play role computational order design presence box extensive effective flexibility wide arise processing maximization identification concern automatic dynamic model building measure field broad spectrum topic hybrid continuous tool system attract considerable automatic parametric maximum pe whose attribute mainly asymptotic property restrict understand see fair modeling consider instance advance consume costly demand fast reliable automate procedure identification identification inspire jointly computationally control face strategy main induce smoothness early reference field system regularization convex relaxation variation norm induce framework regularization prior bring encounter automatic determination make shall address impulse single describe convolution unknown white see maintain herein framework thus model impulse infinite prior hyperparameter prior flexibility encode impulse hyperparameter mention variety impulse response hyperparameter nonconvex handle large even matrix costly extremely ill design feature simple structure usually negativity exploit projection whose basic stepsize scaling computed scale technique convergence classical convergent combination choice robust signal problem scale scale projection apply impulse plan derive optimization present focus split define matrix presence negativity box present effectiveness identification respect solver conclusion symbol trace possibly absolutely lebesgue lebesgue shall shall record estimator impulse clearly pose dimensional literature problem follow marginal therein introduce density arbitrarily impulse decay impulse rewrite impulse ill condition stress truncation problem en condition e typical example say vector integrate q hyperparameter uninformative paper call estimate estimate follow map compute symmetric unless strong available impulse solve impulse coefficient introduce year impulse system structural dynamical system impulse combination exponentially decay seminal kernel paper appear family discuss alone impulse response exponentially decay rate semidefinite role alphabet grid list treat noise use problem k fx solution view compute lead negative value formula occur iterate objective product go point well good performance adaptively keep stepsize strategy cm cm cm describe show bb spectral successively rule different nonlinear rule unlike stepsize review idea bind extend approach split whose equality gradient q formulation relate method well study capability positivity whenever several processing lee negative exactly multiplicative algebra multiplicative correspond scale consideration motivate scale stepsize ill ill nonconvex recall objective scaling driven consideration constraint consist find feasible e eq iterate devise violate direction end define index I u define consequence x k diagonal eq lead choice scale relevant burden satisfied evaluation algorithm rely implementation gradient side ill implicitly devise detailed gradient cholesky factorization factorization ks finally objective formula cholesky simplicity formula q account ill previous formulae computation reduce matrix latter case occur iteration sake completeness report compute cholesky factorization compute cholesky compute matrix without need additional product detailed develop formulae ij moreover g ij x I k require kernel task worth need stack element output snr snr ss ill concern perform fine grid impulse coefficient consider kernel tc tc ss ss condition quality evaluate impulse
encode understand three medium handle rely ideal randomization randomization technique stand enhance scalability partial gradient proximal calculation algebra randomization iii first flexible framework surprisingly approximation centralized communication communication concept offer benefit exhibit significant acceleration counterpart quality inherently approximate speed processor motivation linear observation encode perturbation noise entry basic law physic image nonlinear phenomenon recommender retrieval model dimensional process first study convex ls vector multiplication absolute control ls producing mostly hard turn signal critical lasso denoise geometry readily low r r n n dct key lasso dimension exploit implicit operator dct large dependence multiplication cholesky finding newton surprisingly possess structure provably enhance competitive accuracy point normalize standard normal dimension noise fractional access sg optimization choice exploit within simple namely replace gradient gradient obtain strong access conjugate require full highlight role solution quality solved nearly understand mid within sequel reader basic notion convexity exposition gradient several fast reach target information expensive non smooth fortunately cost drawback dominate matrix multiplication l single potentially short reach surprisingly simple analyze need lipschitz continuous twice method lipschitz iterate bad case convergence attain q lipschitz case iterative function evaluation hope accuracy nesterov momentum achieve typically optimization provably offer key benefit unique improved efficiency obvious definition e strongly term estimator statistical differentiable hessian l strong simply lipschitz minimizer improve instead obvious highlight guarantees lipschitz imply guarantee turn convexity accelerate obtain gradient exploit convexity ccc proximal denote convexity summarize discuss section numerous momentum parameter computational trade rigorously bad rate lead cf similarly kk solve proximal accelerate proximal descent accelerate proximal l formulation basic method behave dramatically improve accelerate allow accelerate describe smooth minimization nesterov smoothing continuity consider smooth function naturally image graph quantum seem efficiency first generic require reach slow smooth fortunately composite objective far retain approximation non upper proximal gradient x x interesting proximal elegant incorporate classic method constrain preserve map operator offer flexible signal prior combination atom representation example atomic structured sign facilitate perfect proximal quadratic explicitly calculate connection efficient formulation soft infinite number admit rank whose atom whose operator numerous exist feature obtains simultaneously estimate smoothness calculate step order accelerate take method accuracy rely adaptation compact wolfe since element method optimally exploit f interestingly function tractable prove useful many application cover far composite q proximity operator efficient enhance smooth lipschitz commonly poisson objective cover apply call alternate multiplier augment h h u distribute turn closely relate bregman penalty input iterate feasibility admm criterion admm step fortunately support periodic admm long h k z k drawback proximal difficult surprisingly inexact admm certain reader generalization issue multiple problem solve simple infeasible optimization calculation randomize example convex objective extension notable graph via incidence aim solve find goal square relax q include positivity reality simpler preferable obvious operation calculate coordinate modify idea capture essence history classic gauss cyclic linear systems coordinate illustrate coordinate key descent iteration amenable pick lead configuration seek hope magnitude effort justify convergence provably slow coordinate hope slow surprisingly randomization choose coordinate random surprisingly rate randomize salient descent objective necessarily smooth cost incremental update importance coordinate randomize uniform strategy improve algorithm accelerate composite version explore often preserve accelerate version contrast randomize minimize include elaborate choose j kx p coordinate problem analogously contrast unbiased empirical observation expectation sampling index indeed optimize minimization provable capability enable beyond decomposable sg decrease step unfortunately lead constant stochastic quickly non vanish gradient indeed stochastic tune recent result size optimal convergence set show lipschitz another stochastic convergence algebra decomposition singular value multiplication due relevant representation efficiency method uniformly instance value svd idea behind randomized representation fashion generalize benefit nearly hence modern describe accelerate proximity operator depend nuclear traditionally easily synchronization however error bound rigorous secondly gradient finally obtain first objective integer draw r r tr classical qr surprisingly nearly approximation small provide randomization rank keep rest deviation around spectrum zero rapidly algebra iteration reader benefit randomization method routine multiplication block benefit parallelization ht proximity factorization core importantly randomize parallelism accuracy indistinguishable raw computational throughput storage capacity mid law consumption massive storage resource cost seem ideally heterogeneous computer within broadly drawbacks specifically communication master machine lead consensus synchronization activity computer procedure synchronization version development first practical scheme parallel proximity reliable communication refer ideal parallelization split job calculation compute great computer machine directly processor location form final ideal x nx k k f beyond parallel smooth artificial technique eq form basis extremely optimization far
label truth label independent run bp initial know well model sbm node core degree instead political line leave show correct bp fall instead sufficiently division succeed division political minima free energy correct global large therefore accurate partition determine size break move core divide political analogous learn giving panel algorithm structure propagation transition fraction agreement temperature large transition critical create line qualitatively behavior two overlap critical advance learn algorithm energy model heavy tailed formalism learn node node use future grant fa grateful de france membership application recently discover phase transition put access fraction well diagram cavity find agreement hard easy hard algorithm transition jump end qualitatively network detection network many include generative variety establish cavity analyze recently rigorously group size node chance distinguish propagation factorize likely globally factorize point locally lie transition factorize locally unstable efficient achieve reconstruction threshold transition first know reconstruction phase bethe point exponential would accurate compete point correct three transition dynamical regime community principle exhaustive energy likelihood enyi exponentially unless label shift transition essence factorize value accuracy nod certain belief cause throughout terminate phase beyond important setting label consume conclusion later replica lead result many investigate cavity belief physic happen easy mathematic calculation trivial rigorous calculation methodology carry property inference distinguish configuration posterior equilibrium configuration formally understand formation belief solver analyze variable affect algorithm picture hope organize section block section stochastic find transition accuracy qualitatively block split group assign edge parameter label via bayes rule label message neighbor estimate marginal node equivalently ignore loop message bp reach node accord marginal bethe optimal since likelihood however comment optimal exhaustive instability factorize phase straightforward adapt formalism except node reveal message external replace know plant ratio node connect equally infer partial interesting label label expectation maximization minimize bethe initialize em learning phase transition factorize fix go stable unstable overlap chance order overlap jump factorize accurate convergence number iteration block leave hard overlap critical heat map smooth logarithm base showing overlap various transition picture agree qualitatively analytical show experimental overlap value overlap peak thus phase transition critical overlap note author predict survival strength phase position predict picture happen regime transition focus plant graph qualitatively graph phenomenon easy numerically color
difference post content learner within enable profile discussion implication education community principle model community advance insight improve design equip diverse body like enable well department education help insight share massive international web conference community mat neural schmidt negative signal computer science piece massive rd knowledge conference w student self program stochastic gibbs transaction machine intelligence nonparametric poisson count data st international conference science city automatic relevance determination matrix prior ik ik nh ki ki control importance observe row lie irrelevant community place standard follow kb u equation fast et convergence cd ik k row column soft membership desire rmse feature treat leverage matrix iff matrix negative hadamard scalability issue ibp subset real world root mean negative log determine sampling row entry remainder via burn I hold rmse na I benchmark predict hold compute I repeatedly evaluate ultimately yield reveal great predictive I computational second run offer model extract feature avg student business interact multiple sub aim interaction final project sub company business challenge google think google competitive technical google internet project sub question assignment technical explore characteristic another assign community posteriori learner belong procedure community learner initialization affect group step run high extract community outcome unique create comment communitie participant framework dimension select reveal contain community crowd discussion individual respectively act statement similar knowledge group participant participant pass member proportion albeit degree pass course notable level act participant involve group group group large group pass explain high people final project similar member view discussion fewer locate suggest play role motivate discussion may difference response topic rarely learn pass second pass similarly nearly characteristic student case people statement proportion high read group final participant people attain master p indicator suggest number possibility course limit simply preferred individually act location member htp project sub place discuss final topic help careful community project support participant proportion act view relatively low pass significantly likely group trend suggest community still pass participant act group view seem participant group answer learner pass course assessment process sub participant discussion project review likely sub group p partly group opposed recognition pass group act review learning post word p interestingly proportion project course yet proportion fail furthermore comprise focus participant goal course outcome participant distinguish virtual meet discussion relatively group participant group albeit significance suggest participant support people group people world participant pass group participant project outcome htp extract subsequently composition reveal offer way sub degree sub social instrumental characterize learner need truly open identify relate discussion participant enhance discuss participant construct adopt reflect cognitive observe learner collaborative crowd see leverage crowd way information group instrumental project support crowd learner enhance idea resource confusion expectation learner mix outcome outcome clear kind achieve learn goal prefer learn individually group learn group relative pass project instrumental ask project yet people pass request help trend consideration case contain
action add remove action easy number number neuron output action denote connection connection round backpropagation layer sigmoid connection network neuron matrix reward reward action phase play model phase use exploration update play action know high adapt stationarity achieve expect train exploration k w w choice adversarial justify stationarity coordinate choose q consider exploration list initialize exp arm play action action reveal choose action predict network action exp km kx reveal update exp weight expression instance less possibility run layer achieve computed prediction outperform contextual bandit similar tend fast rate regret class circular every iteration increase low non stationarity outperform first drift worst well seem robust one neural stay bad entire model adversarial bandit network non stationary trivially show empirically drift serious candidate address issue contextual linearity regret contextual present contextual bandit stationarity reward neural network reward know expert choose successfully stationarity reward formulation variant appearance begin solution contextual bandit structured variant tree contextual leave reward combinatorial limit et al problem without per equivalent naive approach ik elimination policy exploration exploration unbiased train contextual thompson achieves performance point stationarity landscape landscape change reasonable continue main various hide neuron advance bandit well compare
focused point trick fuse detect lagrangian consistent give first add converse subtract consecutive conclude obtain conclude perform change variable put eq quadratic simplify explicitly show notice lemma prove extend k instant quantity sa inspire fourth conclude min mean absolute carry conclude proof express notice include lagrange multiplier otherwise infinite simplified last three pose subject proof opposite optimality conclude proof theorem definition remark work partially lead receive european european framework program fp asymptotic variance focus stationary series parameter detection mean establish fuse consecutive otherwise correctly detect key optimality technique class estimating mean area stationary datum detect parameter segment diagnosis noise removal piecewise signal propose know special refer recent survey tv denoise current tv fuse basis pursuit denoise trend regularize vast relevant focus location signal recovery fuse lasso study literature next sample lasso focus particular circumstance interesting fuse duality convex structure give study improve include variance non stochastic assume relaxed often specify value function piecewise variability total variation difference fit square likelihood tv fuse eq tv choice regularization parameter balance fit case value replace norm q known know question efficiently wide interior software moderate sized size alternate multiplier nice property theory sufficiently follow linear reduce large neither sign change prove reformulate standard know recover pattern hold lasso asymptotic however recover hold consistency exact change zero tucker kkt first add respect optimality condition namely unconstraine point stre key view bridge discrete brownian walk change drift insight analyze property start dual define provide intuitive explanation use times transition subtracting expression q term analysis follow signal satisfy piece package specify convex program call ht estimate detect change correctly reasonable replace q shown explain optimal explain bias term change appear walk problem study proper stochastic see kkt kkt condition solution derive solution give optimality condition give location notation sign know transition establishe decrease sign appear reference transition notation value nice estimate sketch piece wise condition drift end increase take specific bias whether respective minimum case segment part segment lie bias increase segment close beyond point consecutive together location merely bridge intuitive consistency view call approach consistency problem know since formulate lasso reformulate convert satisfies equation x n noise vector construction exist body exact variant construction lemma normal formulation interpolation normal matrix lemma x ai ai condition recovery k piece wise signal notice basically linear spline knot change condition k every shape k bad two k necessary reader necessity lasso consider lk n x simultaneously lasso noise almost interesting distribution achieve exact focus possibility achieve matter change change respectively change na notice consistency change detection impossible hand difficult slack sign relation regressor equivalent regressor basic pp exchangeable need establish low probability low ingredient walk end take sign slack add equation follow pair sign sign inconsistent sufficiently choose small change since lie determine point fourth I provide slack dual segment min consistency n suppose satisfied n n ks n nm simplify presentation consider accord start change consistency level within constant interval level neighborhood reach require reach course accord equivalent without generality take sufficient due enough require q event next condition reach give intermediate reach former reach eq initial segment give event therefore establish proof convergence sign condition
tx product sign recall higher vary compare recommended sign well difficult get norm simplify optimal value space optimization efficiently space depend radius top recommend lsh product focus comparison base use collaborative filter netflix rate user movie procedure generate item involve characteristic correspond outperform recommendation item netflix rank top user gold base inner different hash item q subscript ideally item generate item consideration recommend sort list recall start top rank item rank top gold standard increment see precision recall balance relevant report netflix figure indicate inner product sign product implement algorithms hash return item query lsh point fraction inner number product hashing repository thus recall aggregate query operate advance unfortunately top near neighbor fix threshold ratio scheme minimize evaluation perform rigorous choice threshold scheme select run thousand choose choose average query target produce low compute recall curve compare summarize computation numerous database provide scheme sublinear present asymmetric sign inner maximum subsequently solve projection demonstrate part nsf nsf sign conduct especially lsh complete demand number implement lsh thank computing team well department server provably hash author transformation approximate near hashing provide approximate sign theoretical well experimental evaluation paper problem technical size relate three equivalent value norm many arise element review recommender detection locality sensitive hashing lsh popular efficiently solve asymmetric lsh transform query collection transformation develop hash popular hashing etc another solve show lsh advantageous approximate neighbor bottleneck approximate near neighbor nn dimensional construct near report near neighbor lsh lsh time extra preprocessing existence lsh translate provably sublinear locality sensitive hashing lsh call neighbor need sensitive construct lsh generic framework lsh concrete hash present lsh random scalar xy popular sign random projection choose vector hashing show locality restrictive hash transformation unnecessary lsh main provable lsh lsh requirement incorporate figure fix panel transformation norm inner provided convert search hash sign projection popular hash adopt cosine transformation convert search transformation hashing function new suited exist code projection already outperform lsh get assumption generality q term approximately another method show problem define give collection define hash noting attain z lsh terminology construct guarantee algorithm query time minimize convenience see plot compare sign present value parameter fix would practitioner burden change norm affect top
expression fit er network gene table er confirm share across network gene across er recover run differential gene sub gene er er er quantify level gene gene b indeed group distinct er status differential er gene co expression correspond centrality level er annotate tumor two encode er rank centrality activity tumor interaction encode member factor highly er breast tumor lead death breast three role marker tumor express triple breast control pathway response primary breast cancer develop include capture markov feature node undirected matrix gene expression share across across apply methodology recover co expression er tumor factor method include approach tool exploratory analysis loading type observe categorical integer g status age load identify individual specific use extract gene greatly specific use necessarily extract matrix perform linear category network recover gene unique specific tumor exist differentially co restrictive covariate interest projection sample projection rely post hoc covariate interested see uniquely extend approach projection inform extend parameter shrinkage induce shrinkage loading substitution write row multivariate jointly sparse generate dense variable loading dense bernoulli chain appendix follow load residual beta three shrinkage specifically induce layer factor loading describe beta making equivalent indicates loading beta bernoulli hierarchical component follow extend work specific em posterior probability log estimate form form conjugacy prior beta expect log take eq simplify related finally conjugate description update sample I equation k manuscript warm start simulation parameter load component element parameter sparse dense specific conjugacy bernoulli z pz z k across th th eq dense parameter gamma conditional conjugacy q proportion beta parameter sample equation sample accord nk k accord equation accord accord accord process comparative breast gene datum cancer maintain cancer institute gene expression five run setting correct run default produce close match zero true iteration cc set accept delta row run set normalization construct minimum set sim component run redundancy statistic check redundancy across run follow number non group loading loading vector transform non gene number identify co prior gene matrix identity prior explicitly quantify value sample discard variance significance store count number find time draw structure explore co expression gene encode locally model infer gene jointly relate recover across diverse simulation gene expression breast cancer gene expression recover network mechanism necessity gene mechanism consist within expression undirecte across constitute gene co construct undirected module detail pairwise gene relationship rich computationally intractable gene describe compute gene cluster gene algorithmic partition gene module undirecte partitioning create imply gene gene module hold impact gene disjoint connect characterize group co gene expression decompose gene factor loading assume costly level gene limit loading encourage loading sparse enable extract sparse counterpart limit robustness system thousand mechanism create cluster prove without careful recover factor robust proportion expression besides encourage vary induce sparsity non subset uniquely variation correspond loading identify exclusive address subset exhibit latent loading vector orthogonality loading environmental technical status heterogeneity adjust remove covariate signal univariate testing estimate expression expression level module capture control exploratory association testing base recover co expression signal essential effect cluster number call behind develop gene unique next simple network sparse covariance matrix reconstruct recover quantifying sample gene express network specific iii co apply approach without effect level measure express er er similar gene co successfully sparse expression collaborative comprehensive fall category category assume approach category small category identify sample hierarchical cluster third category build iteratively group identifying group gene induce laplace impose loading specifically bayesian build induce prior gene error priori error across gene sample mean multivariate gene variance k latent remove factor factor factor loading parameter flexible modeling layer loading sparsity pc run initial correction dense initialize sparse software recommend quantile match set quantify proportion recall refer recover cluster precision score loading model calculate recover invariant switching sim loading score figure truth although relevance sim show recover sim expense sim dense loading sim sim pc simulation recovery score relevance sim inferior cc gene ability cluster gene heterogeneous e expression level orthogonality design gene row bottom row sim column sim recovery axis axis legend b recover co undirecte build gaussian recover appendix estimate analysis rank estimation component regularize highly approach matrix quantifie pair correlation specify submatrix correspond loading component edge gene assume edge nan edge distribution edge practically network semantic subset carefully component component covariate rank covariate breast cancer tumor filtering miss patient stage breast old among patient disease signature distant survival focus network er er er negative er patient er breast patient patient patient er er mutation mutation start random factor change recent stability run factor gene
start distance population first measure dt ds kernel pairwise correspond center sum matrix covariance simple standardized covariance discussion invariant application li population function identically copy introduce real value cdf center distance matrix distance I I always effort implementation correspond correlation value step newly unbiased sample publish let follow center name unbiased follow q distance I I k need inner q statistic apply subset notation statistic size element fact arithmetic prove similar must statistic lemma later statistic statistic removing order hold still define statistic counterpart sum entry remove identical lemma statistic correspond inner see degenerate moment nan I normal consistent independence difficult similar deep value argue univariate start lemma counterpart denote formula sign proof x I sequence modification use fast coefficient adopt spirit result estimator distance theorem term per lemma compute right note compute algorithm hand reader description idea subroutine describe propose fast run simulation distance screen experiment regard evident fast implement dyadic matlab go solution fast matlab desirable call low language theory implement htbp rr sample fast visual core cpu ghz mb r size direct generate message compute pairwise memory method require memory illustration run minute trend method linearly size experiment size rr zero pearson advantage wikipedia direct implementation pearson correlation representative bivariate moderate bivariate curve rotation rectangle aforementione uniformly rectangle curve size case observe get zero correlation significance pattern evident htbp size large fig sample method pearson correlation zero even distance clearly reader particular interest clearly pearson converge fast converge counterpart correlation note case stay previous large li analysis dc sis study use direct li originally advantageous size screening covariate marginal utility utility distance correlation dependence measure sir study li screen magnitude marginal function forward backward potential research name independent rank utility indicator appendix bivariate compute algorithm average assess covariance enjoy cholesky know minimum active predictor quantile replication proportion replication replication close minimum well screening sure ensure close different simulation empirically examine effect cutoff integer sis require cover dc sis sis c indicator model interaction fan consistent challenging utility predictor sis model replication sis c htbp sis present sis sis linear model far underlie dc sis outperform little chance predictor counterpart clearly advantage evident observe dc find application lead armed find adopt correlation evident certainly make need em proof subroutine call input observation sort indice th similarly partial use follow recursive definition compute step algorithm partial input quantity compute triplet stay e th among recursive enable partial sum compute subroutine eq subroutine dyadic input assume dyadic interval assign fall integer compute simplification similarly q simplify nn j nn n n nn nn n nn ib nn ib ij ib nn q rx rx r write evident verify ik k n n b kn kn ik b compare indicate proof x verify proof verify j j j c rewrite
lp statistic satisfactory job deviation range desirable strength application idea discovery sparsity density estimation note copula copula define copula du discrete continuous lp copula square integrable copula density admit x equivalently proof expectation lp copula function maximum entropy product score recommend copula well nature function slice describe conditional dependence utilize lp nonparametric data us v fisher contingency result smooth copula v wise bivariate expansion expansion show dependence contingency formulation integrable singular u k appear decomposition comment interpret transform lp canonical contingency maximum number estimate v th dark black light medium dark contingency seek display column category correspondence lp canonical lp define lp profile column profile lp correspondence table lp dependence use aforementioned fig bivariate correspondence display association scoring alternatively copula profile display weight spectral sm lp correspondence multivariate usual normal analysis use lp theory discrete density correspondence copula col construction piece copula correspondence piece correspondence compare shape function row category fisher know ratio storage requirement numerically table lp correspondence lp provide large contingency spectrum provide gain establish connection lp statistical derivation contingency pearson odd first verify p x prove j decomposition fine understanding dependence dependence contingency comparative simulation measure second follow identity significant index present belong simultaneously correlation whose uniformity monotonic transformation representation contingency next property em bivariate form contingency table choice orthonormal switch orthogonal orthonormal polynomial follow polynomial gaussian copula note verify complete py lp significant element evidence coefficient linearity investigate em fisher smooth compute divergence number computation freedom degree dramatically drop applicability sparse component quantify effect lp coordinate correspondence insight density change level covariate illustrate understand age accuracy chi depend classical association contingency like chi sparse table many small chi statistic yield df associate group em em em high linearity also great regard age agree age increase find transformation contingency q power testing restrict attention vary contamination vary iii bad leverage point vary tail scale em shape winner pearson ex winner ex ex winner winner ex carry rejection significance power setting finding remarkably wide setting em consideration investigate scalability detector distance generate report run three write evident size almost infeasible computationally efficient r ex ex ht lp discrete marginal demonstrate conditional approximated jx hilbert theory express fx fy fy jx fy jx alternative parametric wrong specification notice flexible function lp nonparametric yield easy misspecification integrated aic coefficient generalize major definition version discrete variable chapter extend incorporate algorithm provide generalization classical order lp function investigate useful lp correlation quick purpose follow relationship bivariate appear rank jx special corollary bivariate interest comprehensive insight follow lp fy fy fx fy lp simultaneous estimation slice densitie various conditional quantile accept reject comparison approach seminal linear nonlinear produce curve goal child lp discuss spline display b density slice result conditional quantile curve recall marginal age highly skewed prominent tail area algorithm satisfactory job estimating argue regression scientific level c component show fig describe conditional rapidly density bi quantile also evident shift go beyond description modeling tackle tail polynomial interval u u probable satisfying verify probable value probable table moment lp moment discrete distribution poisson geometric ex student df chi df lp lp matrix distribution em em thm statistical university pa college tx abstract approach exploratory lp statistical science statistical exploratory tool concept along real illustrate statistical systematically tackle fundamental orthonormal hilbert exploratory big lp orthonormal estimation lp smoothing correspondence analysis contingency goodness fit height recent availability type nonlinear univariate statistical technology substantially execution get easy become availability isolate seek systematic automatic permit easy establish relationship various statistical attempt em unify fundamental quantile lp skew density lp broad range goodness categorical modeling correspondence coherent hope lp design outline analysis illustrate use example nonparametric general orthonormal precisely mid lp lp orthonormal unit mid px mid lp score power note lp score function respect continuous recover polynomial orthonormal shift give model score interpret statistic score function form orthonormal score always derivation continuous hoc avoid power express transform discrete continuous introduce concept generalization lp compact efficient compute know challenging problem define lp moment unit function hilbert identification treat lp illustrate datum rating skew lead goodness fit discrete scale multiple density lp skew choose quantile density mass call comparison du du lp skew gx em family estimator aic type du theory equal equal provide goodness two skew figure previously analyze select baseline density drive aic select skew suggest model drive aic skew mode reflect mixed review sharp probability review
score law investigate theorem instead hard example projective cluster cc compute om nc kk k algorithm compute nk nc observe residual theorem old simple explain attractive practitioner understanding via appropriate see effective rigorously effective power law corollary axiom approximate select correspond large leverage score surrogate provable proportional leverage deterministic provably leverage moderately power law provide evidence decay sampling match state art lot qualitatively reveal component pca sharp actual form low surrogate interpretability make attractive practitioner rigorously matrix utilize deterministic technique discuss literature suffice highlight contribution score define leverage first back propose sample successful theoretical lack theoretical utilize randomization identically pass replacement constant ok k two ii version implement admit open provably tight leverage sampling provably counterpart law decay provably obtain approximation literature decay theoretical decay leverage score state art deterministic low rank well art nm whose clear equivalently write define spectral ij respectively describe leverage section approximation wish deterministic leverage compute singular leverage score simplicity sort well score ensure column accumulate energy carefully control parameter choose prevent approximation ki generality let sort require arithmetic operation discuss modification innovation let algorithm least column immediate column extract requirement leverage decay fast intuitively leverage score extremely fast decay output score come expect leverage decay column offer subtle point leverage score imply trivial argue leverage good make intuition consider score find relative score follow require absolute law algorithm art expectation constant indicate bound apply power theorem spectral deterministic column notice relative column fix norm power decay world investigate amazon citation soft google ct slice image citation work leverage sharp decay deterministic leverage score particularly htb plot power law curve exponent list leverage plot red marker power good true leverage exhibit decay decay correspond enforce sample offer good leverage set co bag word row row take finding achieve k indicate sample plot move achieve agree early however moderately case column suffice indicate loose law decay power optimality leverage decay vs plot relative vs fast surprising rank confirm compare error contain exhaustive approximation l nucleotide email work c work c c c c c c c c c c c c c c c author near deterministic theorem author test build third leverage score replacement tool matlab approach execute repetition observe appeal almost leverage randomize well digit show set law competitive case fit profile near quick overview deterministic deterministic go seminal qr develop strong require arithmetic column score column sampling euclidean view seminal idea randomly simple column euclidean hold improved accuracy distribution volume fast another include row sampling orthonormal matrix result essentially kind svd mention dimension thin cc cc k vector respectively well k k I use eqn nk novel deterministic score theorem use perturbation eigenvalue cc conclude proof straightforward picking imply lemma let decay column convergence positive monotone eq leverage score collect leverage score hand first side hence q q eq preserve immediate implication highlight nk frobenius unitary transformation zero conclude step compute top describe improve maintain guarantee
variant r cost long approach run complexity condition large rather arise independent dependency hard nevertheless practitioner address propose call stagewise completion exactly complexity desire complexity block project gradient descent context precisely iterate singular projection matrix step onto set solve unknown result extend correctness leave preliminary showing correctness matrix albeit number desire wise try recover get finally stage iteration remain novel understand project consider completion general hard obtain svd spectral bound eigenvector norm analyze prove natural extension bound perturbation enough eigen gap due consider interest organization overview section warm technical capital letter denote entry respectively stand singular tail result satisfy n c argument symmetric pp update sample select decay completion input continue n n k step routine kx x next p desire issue desire stagewise prove st present run projection onto basic avoid step ensure decay obtain pp k n iteration invariant k stage wish frobenius update k invariant present proof prove every follow induction clearly suppose outline q hold n hypothesis n argument tell tell stage hand use sample establish invariant completion complexity good completion behind entail term taylor series perturbation good eigen technique context still suboptimal interesting handle definition pt matrix completion recent convex paper iterative desire knowledge complexity project projection potential bound extension good eigen may independent low
observe zero loss round lf mf leave try sub similar adaboost produce indeed possible network linear key assume implie tell since say small observe q mini batch goal sure batch eq function satisfy constraint exponent go go suppose convexity advantage ability boost plan challenging task plan generalize binary claim example unlike like adaboost artificial layer recent development machine show variety successful gradient sgd ability major disadvantage sgd attempt momentum reduce natural weak learner boost adaboost example guarantee boost weak guess iteration ensemble train major disadvantage adaboost ensemble classifier mean predictor paper sgd rate sgd constant boost set network adaboost maintain proportional focus mistake example intermediate intermediate output achieve prediction edge sgd choose random sgd finding tx gx tx next try define mistake crucial number iteration example
comparison nc comparison functional commonly ad region brain average voxel preserve patient pool fdr voxel organize provide drive conventional fdr procedure finite lattice voxel usually hypothesis ise normalize ts possesse ise therefore neighbor odd reflect together odd account nonnegative dependency voxel reflect ratio conditional bayesian mrfs index mrf know mc prove et compound decision theoretic framework fdr imaging nan hypothesis operate fdr drive replace partitioned procedure base fdr fdr follow pool reduce order homogeneity fail model distribution state voxel six boundary test individual probability oracle drive procedure imaging model estimate model maximize introduce categorical k function p separately constraint method lagrange multipliers w sl lx lf take second derivative equivalent solve nonlinear nr nr rather search together maximization term em backtrack decrease definite function recommend n sampler constant approximated reader normalize refer work back number voxel avoid degeneracy likelihood penalize et gamma finite distribution penalty consistency normality chen think inverse would formal auxiliary em equation remain unchanged pool set update estimate stop consecutive specify stop monte three carlo recommend practice satisfied step stop criterion dimensional data datum drive procedure compare group globally procedure ise drive procedure sampler et burn simulation cubic lattice replication ise parameter comparison fix average positive yield fdr fix fdr level control case low generally fdr lattice additional procedure control fdr initial distribution claim fdr set embed generate burn five ht voxel noise signal signal result summarize summarize claim procedure detect procedure find area ad surprising decrease cluster affect include al temporal gray ad et al proportion rarely disease high proportion pre claim group mix progress subject et et al surprising chen p longitudinal treatment false spatial signal false discovery powerful discovery multiple multiple testing dependency j scan disease interaction maximize generalized likelihood automate b york burden mm testing group iterative image mm r random chain topological mm chen mean false discovery fdr fdr mm b series large false dependence la longitudinal brain mild cognitive york simulate normalizing bridge relaxation thresholding functional rate false control p value weight l operate discovery procedure mm discovery thompson monte likelihood dependent datum gray r longitudinal clinical emission diagnosis approach diagnosis online lee patients g relative capability mr imaging associated disease w j association cognitive chen lee e categorical analysis emission image random field association smooth gaussians hide measure field image transaction machine intelligence disease diagnostic approach mm early diagnosis disease l de li reduce automate standardized diagnosis mild cognitive p comparative nd ed york li gender mm hide master thesis di rd ed york oracle compound discovery multiple mm x imaging ph department mm population nd ed york wu false discovery test diagnosis nuclear modality ed pp zhang field effect nan mail edu department ann mi email department university ann mi nsf grant edu analysis write complete find http pdf disease human brain change consider voxel emission imaging study purpose voxel multiple testing procedure mostly ignore neighboring voxel substantial significance minimize subject false three expectation powerful conventional comparison mild cognitive disease status increase normal imaging false maximization ise mild ad predict progress diagnosis ad extensively emission imaging et demonstrate patient nc early voxel common difference seminal false discovery rate fdr error control fdr fdr false accept hypothesis error fdr fdr fdr traditional rely heavily assumption independent rarely wu suffer severe ignore al independence procedure procedure modeling dependence chain hmc oracle drive follow al pool different prove procedure high li et implement genome study analyze et among know software parametric topological fdr al fdr cluster way voxel level
recommendation combine complementary effect another regularization soft constraint impose remain prediction change incorporate uncertain leverage auxiliary rating incorporate factorization correspond auxiliary uncertain rating recommendation requirement need recommendation incorporating constraint collective e fm g summarize work collaborative recommendation auxiliary knowledge transfer via collective constraint attention art rich collective enable also adaptive cause sophisticated table datum base user categorization table recent filtering perspective domain domain classic transfer transfer practical recommendation cross domain heterogeneous technique effectiveness natural heterogeneous heterogeneous transfer flexibility heterogeneous simple exist integration typical recommendation recommendation application type social mobile useful need learn rating item recommendation evaluation motivate sophisticated objective metric exploit recommendation source explanation generation recommend item even leverage auxiliary collaborative recommendation learn categorization recommendation auxiliary recommendation collaborative research quite big ai especially variability heterogeneity recommendation auxiliary different technique exist transfer propose categorization knowledge generic describe framework finally thank technology play role internet display rating usually exploit user circle preference behind collaborative recommendation service practitioner extract auxiliary order especially transfer extend exist categorization technique collective transfer rule regularization fourth representative knowledge strategy expect collaborative recommendation recommendation standard component internet provide service use recommendation content recommendation collaborative recommendation content recommendation focus collective intelligence prefer similar explicit due fortunately additionally user rating recommendation content time contextual information potential help reduce improve recommendation static static user content etc l user health dynamic context remain quantity item context etc user item relevance network share friend etc feedback rating item user etc item specifically target datum question extend categorization transfer answer dimension transfer collective transfer rule knowledge detail particular may finally conclude paper summary discussion target target feedback content predict auxiliary left user denote htb fundamental transfer transfer transfer collective transfer transfer type study specific via transfer regularization categorization collective transfer later expand category expand strategy style mainly transfer recommendation factorization mostly successful rating incorporate different mathematically regularization contains explicitly show representative work framework eq aim adapt extract auxiliary target transfer similar adaptation section adaptive transfer strategy eq regularization transfer implicit record rating extract factorization feature factorization work auxiliary dense classical learning particular note compare collective codebook transfer early regard far transfer behavior book level rating pattern auxiliary co entry denote rating corresponding codebook codebook constraint rating share target indicator codebook codebook expansion membership codebook recent generalize codebook include share supervise label information effectiveness document categorization rating user target rating pattern kind stable high thus available auxiliary collective learn share effect bi transfer richer similar representative collective transfer note knowledge learn simultaneously knowledge collective factorization rating idea item share bridge use link factorize variable model propose matrix factorization item rating user factor universal auxiliary feature topic item specific feature item item item movie description lf gp nb multiple user auxiliary learn similarity target auxiliary effective compare alone item user weight similarity share feature selective align collective knowledge star numerical item besides matrix inner share share transfer balance rule effectiveness heterogeneous datum transfer latent incorporate raw learning constraint transfer mathematically rule include raw auxiliary knowledge knowledge model rule b ii fm fm represent user novel vector rating triple item entry rating pairwise latent via prediction basic factorization inner vector fm
develop thresholded assume accumulate past eq thresholded simply converge eq stage accumulate per gradient accumulate rate determine determined behavior tend modification rate begin g dropout relu cause pseudo draw minibatch correction g I memory outlier update mnist maxout compare algorithm test sgd momentum decay summarize converge technique include use learn momentum sgd analyze use minibatch compared minibatch able almost log local htbp htp gradient insensitive tuning hyper tune sgd improve provide preliminary deep able comprehensive empirically mm thank computational provide partially support cifar research grant discussion read maxout use parameter learn momentum momentum sample maxout experiment implementation experiment lead careful learning amount noise gradient utilize tuning element curvature far new automatically preliminary neural well stochastic search propose exploit curvature learning rate deep paper use hessian cost parameter gradient dimensional proposal estimate option gauss newton estimations hessian suggest way curvature diagonal inverting operation track order variance reliably root mean statistic reduce keep newton order method transformation gradient quasi transformation basically quasi newton scale affine directional newton propose solve directional require inversion maintain quadratic order directional update unit vector algorithm element size option hessian write directional unit want estimate curvature estimator consider visit bias optimally trade bias incur use bias expect estimator gradient unbiased bias base parameter make average next minibatch find close strictly real basically correction reduce stochastic step directional compute numerically unstable stochastic decide numerically taylor equation always practice work average also decide step take large step move rule assign element gradient update scalar multiplication perform detect outli detect
ensemble tailor rsc ensemble classifier art gene filter set decomposition analyse provide motivation rsc overview ensemble literature technique rsc ensemble scheme classifier constructs represent explanatory call possible value explanatory response measure define attribute sphere stem sphere centre practice sphere tuple centre within distance centre close class union call contain example class pure cover problem involve find cover proper cover graph case np cover pure requirement introduce parameter proper cover requirement contain case retain sphere admit infeasible datum classifier parameter section rather constructive whose decision classify new key diversity ensemble broadly speak diversity ensemble employ heterogeneous sampling weight different modify classifier randomization method follow ensemble bag bootstrappe member boost weighting create diversity classifier boost bag forest bootstrap optimal searching subset candidate replacement attribute forest combine sampling rotation forest involve partition transform component entire set train technique ensemble form sort scheme analyse diversity link decomposition framework applicable simplicity restrict class expect entire attribute actual response expect classifier explanatory denote equally set loss bias cause error result capture underlie decision describe prediction I variability cause finite component loss example call average bias decomposed express variance variance incorrect net error variance decrease net principle benefit variance ensemble address biased combination without generally estimate perform ensemble conjunction designing ensemble design diversity ensemble produce interpretation single sphere informally repeat discard datum cover center find find sphere sphere great add covered case save sphere centre class description distance normalise onto htp sphere classifier create center case store store boundary effect sphere cover cover cover sphere target sphere target cover classifier class sphere close centre cover select close cover outli preferable area decision specifically illustration figure area dense area cover smoothing boundary set competitive classifier use base basic rsc could create diversity majority rsc ensemble member assess diversity make rsc cover classifier parameter sphere use number misclassifie within parameter advance cross however impractical automatic implicitly sphere close class dataset growth sphere hence rsc principle ensemble informally current rsc classifier case list add htp lb e gd j formal description give vote continue focus misclassifie weight previously misclassifie drive constructive outline member different attribute sphere build classifier attribute replacement original attribute set majority vote employ final hypothesis input lk j base rsc generalise hyper rectangle classifier wish rsc ensemble technique aim confirm rsc rsc base rsc ensemble except rotation forest rsc ensemble outperform dimensional assess performance adopt describe base nan hypothesis classifier alternative result nan reject post pairwise lie different rank classifier hoc diagram introduce bar clique powerful adjustment nan follow standard distribution control classifier simply perform lc dataset diabetes heart tumor uci repository six benchmark table example class attribute diverse continuous normalised use six evaluate subspace ensemble feature feature base classifier rsc classification measure four ensemble pattern addition notice accuracy similar classifier separate rsc table adaboost train adaboost bag setting train training quick quick selection rsc impact overall describe rank difference average rank level htb c bag image diabetes cancer rank bag diabetes heart firstly high nan hypothesis significant performance majority comparable bag decision base rsc bootstrappe support base set tie basis classifier classifier comparison difference indicate misclassification direct resample perform adaboost well bag experiment widely weight rotation subspace ensemble ht c c diabetes cancer rank rank diabetes heart train optimal value entire show critical diagram significant difference clear outperform clique forest eps difference subspace rotation rank forest reduce base cv ensemble size however many demonstrate improve classifier indicate cr cr cr cr diabetes heart cancer show combine difference diagram ensemble increase ensemble mean critical detect similar apparent htp rsc classifiers eps diagram ensemble dominant classification instance domain research area database offer instance ensemble think outperform help responsible number case learner attribute demonstrate overcome inherent fact htp l bag multi bc cn lc pr htp ab bc cn lc pr rank ab bag lc pr avg htp l dataset bag bc ct lc avg f broadly speak use scoring rank produce information ig select training adaboost bag subspace rotation well conduct describe ensemble adaboost bag default parameter eight rank high rank conjunction filter space produce ensemble diabetes decomposition diabetes heart heart image error var var rsc vs diabetes rsc vs heart rsc vs rsc
expression decompose inspire communication analyze amount require reproduce observe quantify content send shannon character entropy fortunately equivalent identify without generality input line constrain simply correspond computing problem allow simulate namely pm v assumption implication relevance minimum causal influence require explain achieve quantifie measurement go determine measurement via solve consider display involve bipartite scenario produce introduce hidden serve decompose generality similarly suffice influence nature identify real likewise distribution pa px py x usefulness take close indeed vx observable uniformly e measure measurement dependence ref namely measure reason numerically lp stem proportional distance converse obtain mutual measure bipartite outcome maximal impose ad constraint obtain relation conversely realization projective correspond upper depicted text insight pure projective scenario share protocol receive particle produce quantum focus reproduce independence particle weaker arbitrarily involve receive dag source assumption usual correlation extremely characterize despite non nature input symbol input function analogous usual equivalent analogy measure measurement dependence degree measure eq fulfil measure obvious marginal impose observe distribution reproduce write distribution determine previously step write entry pa b represent correlation matrix encode reproduce entry equivalent eq find must also optimisation minimum equivalently find sufficient product decomposition v maximum compatible observe specific input state basis assign marginal thus full determine range linear minimum depend correspond must relax reproduce amount source sketch could measurement parameter q optimize four free look free remain free probability discretization continuous confident non conclude test maximally state distribution prop observation prop prop theorem prop definition prop prop example prop prop problem prop correlation causal experiment impose classical explanation natural ask locality independence conceptual treat systematically alternative causal mathematical causality express variety range relaxation novel causal interpretation distant experiment share observe detailed physical setup arise consistently property language parent causal assumption choose observer influence ideally constraint verify commonly ask quantum locality relaxation observe question great relevance locality protocol constrain structure security often important theory several question attract considerable example measurement dependence strong resource impose possible correlation measurement source measurement reproduce projective relaxation locality bit distant correlation framework treat relaxation measurement locality several concept mathematical aim describe causal correlation community develop quantitative structure systematically represent dependency degree influence explanation observable cast computationally program demonstrate operational minimum reproduce observe measurement define cast quantify relationship discrete dag graph parent unobserve identity encode relationship dag scenario perform input outcome causal measurement generality encode together hide causal mechanism locality measurement independence express network relaxation need devise way quantify sensible introduce concept causality bipartite locality causal causal communication relaxation measurement independence input correlate common observable observable intervention value place influence keep intervention decomposition parent probability notion coincide maximally correlation consider locality relaxation shift cause fig situation highlight relevance measure note use quantify drug interested case bipartite illustrate easily understand much e independence probability relaxation compute program long variable variant usual quantity carry restrict explicit component component expectation modify arise application constrain objective appendix constrain minimization variable reproduce reformulate primal lp dual highlight another nice framework close valid detailed section proof first minimal influence require require simulate intuition precise output eight quantity last term represent regardless particular quantify communication quantify locality require shannon send message framework binary maximal produce bit communication protocol reproduce correlation locality interpretation exist set two fig compatible form polytope polytope pa quantum matter generate correlation locality outcome dependence resource simulate mutual small number measurement fundamental implication increase requirement find result leave regard either maximally state two state unable mutual bipartite input assume numerically relation minimum mutual I ref ref quantum require tight leave room detailed one go via dimensional primal dual lp formalism optimization framework programming reformulate sign respectively concrete consider call transpose program call obeys call hold lp admit primal least crucial feature programming duality feasible feasible problem eq convex convert lp arbitrary implicitly obey yet however convert straightforward procedure kind combine yield lp crucial primal albeit implicitly auxiliary replace norm non unconstrained yield desire statement reformulate simply include lp clearly lp non programming formalism particular relevant direct dependence cast associate equivalent value primal I I ki argument constrain non negative order convert lp space lk implicit negativity guarantee due relevant eq extend primal simply simplify variable u variable suggest replace constrain non negative bound inequality everything value value primal proceeding along primal resemble extend q form decompose upon correspond dual lp simplify normalization drop go infinity turn desire simplification proof accordingly go present derivation namely solve primal form dual finitely feasible lp bound vertex feasible unbounded direction ignore text negative calculate optimize optimal applicable provide least hidden part contain geometry program explicitly dual duality dual feasible inequality establish see hold simplify consequently fully denote aim contribute ignore problem constraint constraint guarantee contribute thus characterize convex hull however well concave polytope vertice relaxation dag fig minimal direct influence simulate lp determine analyze depicted fig analogously variation briefly end assume causal consequently possible hide observe decompose way define quantify observation move characterizes last equality b compatible corollary problem translate consideration b state
map element mean point phase point measure great circle distance rational project velocity video salient motion region scene map map video near knn undirected network comprise stability connect matrix self reinforcement point laplacian typical dominate scene select query scene perform detect rank query label successively positive label assign precisely individually c identity parameter range initial assignment iy jj final vector evaluate diverse crowd public view exhibit motion subtle comprise use test capability propose detect instability region scene detect detect scene corner scene accurately able salient interesting motion dynamic category motion cause individual crowd google aim individual crowd motion anomaly effectively sequence manually label public merely publicly difficulty perform comprehensive determined region score region salient clarity different interesting perform scene feature inaccurate due illumination rank produce lead mis demonstrate low level stability phase similarity indicator salient detecting source surveillance scenario importantly track model capable discover optical flow drawback optical extremely motion source grant fp h education pt chen signal edu chinese email ie edu place salient crowd attention surveillance framework salient crowd scene transform feature global global discovery intrinsic dynamic level identify eliminate public demonstrate effectiveness salient various area security growth public recent automate video law technology event hundred thousand monitor due crowd severe attention span manual monitor task demand cognitive attention therefore effort towards solution identify interesting salient region ultimately event security deviation ordinary anomaly rare interesting scene generally accomplish firstly activity scene identify anomaly region contrast exist motion regular region due scene crowd motion dynamic detect accomplish unsupervised contrast crowd motion feature global crowd motion plane allow intrinsic motion dynamic manifold perform iterated laplacian approach indicator salient motion dynamic unstable crowd aforementione purely unsupervised requirement stage experimental public dataset capability propose crowd salient source local affect build motion amongst source individual crowd enter leave instability dominant flow scene crowd behavior motion tracking trajectory require commonly track motion trajectory geometric structure source semantic detect anomaly principle enter scene semantic tracking tends fail aforementioned method certain extent crowd tend fail dense track behavior individual build crowd entire flow field hide inherent motion lagrangian crowd flow stability unstable motion discover field level optical identify interesting region use direction feature scenario occur motion move opposite direction cope motion area source localization salient dominant flow motion unstable crowd limited detection instability real world public crowd behavior show classify bottleneck sensitive detect salient accurately summary propose low contrary require salient region indicator represent crowd optical flow crowd velocity dense optical vertical flow optical field accumulate comprise obtain inconsistent velocity average crowd motion field broad crowd denote low level computation dominant flow crowd capture subtle quantification dynamic particle particle apply track velocity initial position unlike optical representation
feed mf unfold framework base model still feedforward potentially elimination incoming express imply incoming belief message aside compare message mf differently yield flexible see question affect unfold generalization similar bp mf consider interesting maintain objective edge weight implement deep maintain general activation rather optimize test time parameter discriminative objective form raise possibility train sigmoid change tie investigate performance architecture instability example interpolation sigmoid generalize easily straightforward optimize related computing message formulae schedule accomplish propagation add parameter could necessary check might level training connection unit reasonable sigmoid simple activation vanish activation generalization especially appendix even complicated sigmoid activation unnormalize bp reciprocal uniform update appear comparable commonly spirit spirit maxout similarity whether difference generic mrfs mrfs incorporate novel deep unfold nmf nmf apply many aim spectra together basis operate usually magnitude fourier domain source column time basic appropriate generalize divergence yield active subject negativity multiplication initialize reconstruct mixture general nmf basis combine train separation discriminative base task basis term bottom control objective account de reconstruct speech part activation basis uniquely nonetheless bi basis bi directly derivative convergence basis reconstruction train reconstruction basis incorporate criterion toward across call cast identify iterative train network recursively multiplicative split nmf part update variable eq propagate negative activation time th layer recognition challenge stationary child home development journal speech room impulse six db test set consist total noise training room impulse complexity evaluation target spectral magnitude window window feature slide window consecutive frames slide window reconstruct line nmf vector speech iterations nmf base room network use feed hide layer tangent activation denote activation index dnn element nmf experiment frame vector logarithmic magnitude nmf compression linearity clean take direct consider speech experience train function speech amount objective magnitude speech magnitude sequence mask spectral nmf mask estimation come dnn comparable solely context deep architecture db snr db topology x minimize momentum early stop cross development set prevent set use nmf nevertheless perform investigate different dnn topology hide per development nmf optimize shall multiplicative training condition basis solution basis sample basis determine deep nmf kl divergence update equation layer use evaluation combination perform initialize layer basis layer mean final basis describe special layer basis context layer train frame reconstruction consist row nmf whereas train parameter rr snr db avg k avg experiment table nmf topology deep strong improvement relative compare dnn nmf deep dnn result deep nmf versus dnn least performance nmf discriminative training layer consistently much gain one huge conservative time come increase intermediate topology need speed accuracy work factorial research unfold unfold algorithm propagation markov field variational conclusion general framework exploration space deep architecture difficult sigmoid could see unfold architecture approach difficult hope novel insight show sigmoid mrfs sigmoid mrf n ba affect posterior easy mrf sigmoid use drop normalize sum constant assign add form rather separate activation look index message belief uniform leave eq verify become sigmoid numerator denominator back term fact exponent exponent give rise various c geometric arithmetic limit improper geometric derive calculus power jensen assume continuity limiting complete proof form pass clarity exponent mf bp q mf source stack activation stack able stacked source reconstruction function rule get rewrite forward pass start layer give start train basis activation nothing prevent spectra magnitude oracle thus optimize actual also typically speech speech thus wiener filter include optimize reconstruct speech measure source l b w h l h h accordingly kk use notice kl simplify final analysis avoid obtain rescale normalize version determine intermediate layer derivative intermediate split h tr positive part follow k see never store quantity update heuristic split respect part use positive multiplication factor eq operation need respect split positive recursively set negative apply though nmf split eq eq carefully storage tensor reformulate operation matlab create negative part eq eq rewrite similar everything source reconstruction wiener kl nmf update without normalization normalization perform part respect recursively gradient use respect layer tie computation usa deep network successful knowledge build constraint expense deep way architecture unclear aim advantage deep optimize parameter powerful architecture within show allow field new architecture inference algorithm negative incorporate sound source unfold yield train multiplicative style speech show parameter advantage goal strategy avoid iterative analogous layer architecture use expressive inference advantage intuition david computational analysis incorporate base straightforward include world visual geometry subtle latent constraint insight gain modeling model mathematically intractable belief variational approximation derive latent despite greatly slow challenging deterministic define expression execute discriminative speed versus trade produce system well know conventional mechanism box method work dnn system clear discover modify consider art methodology address difficultie task design deep step derive probabilistic tool unfold gradient relatively straightforward case field mrfs mrfs unfold belief show architecture unify formulation generative level mix power despite non form unfold basic multiplicative preserve non conventional sigmoid require unfold code unfold back belief markov field reweighte belief bp train implement inference predict unfold original critical architecture network pass mrfs input propagation replace conventional sigmoid novel work also mrfs network schedule unfold inference sigmoid result sigmoid belief mrfs rely contribution deep architecture base unfold mrf network parameter propagation showing benefit speech discuss mrfs sigmoid pairwise vertex v n variable identify pair I h I abuse indexing mrf write function typically represent scalar argument function divergence true equivalently maximize fully variable product posterior preserve form sigmoid message comparison message note must maintain schedule avoid maintain update rate implement arbitrary schedule compute keep eq schedule implement message arithmetic message optimize update schedule message convenient log take inside example sigmoid mrf write fact twice notation
association rule side rule support association support frequent situation combination variable variable association issue one treat tree base refer simplified rule tree rule ensemble summarize order build list order rule satisfied outcome new frequent classification mean outcome variable default rule avoid overfitte add preferred preference shorter break tie data rule default metric update base default rule rule assign frequent original default rule add algorithm rule tree ensemble summarize rule rule rule variable transform rule ensemble secondly replace discretized version discretization deal rule rule classification without ensemble function example team choose team game team team player play team team ccccc data randomly select assign variable framework build regularize maximum extract extraction rule provide alternatively extract forest outcome assignment rf sub rule consist assignment rule metric r format execute predictor return n x c rule short rule error x rule select rule metric condition n furthermore simplify tree condition x else extract interaction remove prune top frequent condition essentially condition rule forest contain outcome therefore lead assignment frequency consider variable interaction sup c n c cl r auto breast heart simplify learner package use forest condition length forest also extract condition randomly sample sample testing perform randomness average error rate difference large lower divide pair test rate run two well table statistically difference statistically outperform difference great great may classify table high c aa ff c c account good none heart x good yes c b x x x framework algorithm measure pruning condition rule extract frequent forest generalize r ensemble process conclusion area leverage classification numerous propose summarize extract build extract build categorical idea ensemble forest accurate understand tree framework measure learner ensemble learner also form framework regression many random forest tree rule rule forest predictor outcome predict ensemble learner information hard particularly model background understanding software refer hard discover potentially huge particularly ensemble refer interpretable rule frequent prediction interpretable insight ensemble irrelevant redundant set redundant rule discover frequent new function framework extract prune individual one summarize rule rule extraction ensemble decision tree split internal ensemble e random forest ensemble long transform package first index leave right column split value current leaf leaf node assignment package version currently environment status point conjunction aggregate root split current node leaf ensemble multiple leaf node refer conjunction rule rule extract rule ensemble often random reliable assign stop extraction reach also frequent extract tree ensemble value benefit ensemble want interpret error remove variable decay value consider rule prune leave pruning apply sequentially decay decay threshold last last remain remain next thus remove rule rule consider large frequency however rule similar would desirable derive rule redundant selection select relevant redundant condition create let whether target relevant redundant variable essentially present detail furthermore condition assign condition length process consider condition regularize forest node empty root ordinary gain decision call regularize forest add
derive bound approximate atom dictionary approximate operate pre power accurately relevance approximate dictionary criterion feature examine feature mean principal axis investigate instant criterion measure relevance compare dictionary dissimilarity construct mutually distant atom similarity investigate outline propose construct entry discard belong kernel dictionary predefine otherwise efficiently projection spirit criterion distant pair atom condition use name drop error mechanism see criterion compress sense particular kernel formalism coherence large cosine angle coherence criterion restrict orthogonal include function note denominator expression atom correlation kernel include eq analogy deal j two definition atom notion gram elementary issue approximate span fold one discard hand approximate approximate atom latter duality discard approximation onto subspace span dictionary criterion projection onto latter correspond maximum inner get moreover projection therefore approximation investigate expression order proper discard sample discard low threshold discard approximate get firstly derive expression follow secondly bit give projection onto atom upper compare discard coherence coherence linear norm coherence bind discard give quadratic atom unit thank min theorem turn aforementioned eigenvalue gram dictionary conclude atom bind approximation atom span follow derivation empirical project onto span give quadratic approximation upper span dictionary due triangular cauchy schwarz summation summation term belong discard thank contribute latter discard sample derive bind criterion threshold atom fundamental use method visualization nonnegative base empirical onto span ni eq schwarz thank constant consequence n follow discard contribute light term threshold criterion follow study theorem identify relevant unsupervised connect sake component seek principal axis correspond eigenvector large norm axis expression eigenvalue call highlight connection criterion th principal associate gram approximation upper kernel axis large small say principal axis moreover criterion use n tight one indeed derive coherence dealing criterion roughly discard explore atom namely lower beyond describe detail centroid axis devise criterion argue criterion behave interesting desirable without note machine approximation initially provide eigenvalue dictionary completeness put well gram associate eigenvalue center absolute gram follow measure distant bound distant substitute gram coherent atom coherent bound expression measure receive degree university ph degree system optimisation security technology research associate system model laboratory technology nonlinear processing representation wireless signal nonlinear adaptive system award publish review paper many framework radial network order address several operation efficient connect criterion theoretical approximation approximate propose bound criterion class mean axis resource gram pattern bring new demanding address conventional indeed underlie available stay computationally tractable model subset contribute reduce bottleneck scheme aforementione machine instant discard criterion widely investigate literature gaussian least relevance discard predefine residual representation crucial issue model literature investigate distance distance network radial function distant advance compress mutually least extension coherence criterion criterion study conduct analysis cost criterion favor criterion coherence extended criterion unit criterion cross previously derive tight bound extend result criterion bridge provide approximate already retain discard secondly provide bound approximate approximate sparse aforementioned criterion include approximation feature empirical centroid principal axis picture illustrate remainder follow present present aforementioned criterion approximate distance reference atom gray color new generalize machine present study seek feature connect desire one empirical feature control fitness regularity solution increase quadratic loss logistic reproduce inner reproduce state moreover reproduce property unit norm kernel deal unit restrict principal classification unsupervised form q derive sketch monotonically increase available constitute bottleneck indeed set instant instant thus parameter consequence continuously increase overcome need order fraction form predefine expression atom paper analogy stress fold instant dictionary challenge arise determine optimal instant reduce solution elegant overcome intractable recursive determining discard since already belong
semi bandit open adversarial sufficiently happen often item increase divide regret associate event prior tight summarize prove gap evaluate gap section discuss tuple item distribution combinatorial optimization item ground th entry item number observe item agent environment goal cumulative regret feasible et nu te c item te te feasible fa tw pe te propose algorithm semi bandit compute item around se se call combinatorial observe weight th marginal second initialization compute follow item one guarantee terminate iteration one entry optimization oracle regret gap eq dependent upper asymptotically tight latter bandit definition efficiently offline problem solve computationally efficiently optimal semi bandit efficiently operation computationally regret factor regret match free upper bound gap suboptimal theorem solution present proof yet proof lemma initialization suboptimal hard bound claim event item suboptimal sufficiently later event happen least mutually exclusive claim item happen happen exhaustive exclusive show happen follow eq contradiction happen ready detailed time event happen bound gap minimum suboptimal q stochastic eq detailed item item order event total regret substitute gap suboptimal identical relax step define many mutually exclusive suboptimal decreasing establishe key define respectively happen happen sufficiently happen happen happen definition furthermore must ie happen inequality assumption contradiction result happen appendix number happen step finally apply gap bound suboptimal associate regret event large small regret combinatorial bandit regret bound appendix key decompose gap separately step bandit one gap dependent gap start point node mark path weight weight note design key equivalent arm return scale know path bandit gap dependent step inconsistent logarithmic semi bandit integer path proposition divergence bernoulli bernoulli low due qp gap integer semi regret path equivalent bernoulli bandit payoff bind adversarial environment problem synthetic demonstrate suggest bind experiment path problem ground feasible grid upper corner bottom right corner bernoulli mean gap number validate experimental report trend item chen upper upper upper nearly mirror perturb geometric resample recently adversarial combinatorial computationally efficient open combinatorial bandit efficiently bandit instance upper bind apply indicator learn observe feedback clearly bandit observe low combinatorial adversarial nevertheless set stochastic van semi bound upper differ frequentist computational efficiency offline inefficient straightforwardly instead respect thompson often perform practice straightforward variant thompson weight thompson resemble regret main work derive novel gap dependent ucb semi bandit achieve near computationally efficient implement offline efficiently combinatorial semi bandit efficiently quite mild choose sufficient purpose may large well leave derive suboptimal speaking match factor eliminate modify leave stochastic c outside terminate finally rewrite regret equality history regret condition base te te ec happen conclude remain suboptimal follow quantity sufficient number event happen count magnitude happen n suboptimal item item increase one suboptimal guarantee observe happen happen ng te tt happen counter item event bound trivially regret introduce event regret definition event suboptimal gap solution order
schmidt preserve column spline qr decomposition invertible solving expansion span qr decomposition illustrate dataset fista fast algorithm size solve exactly avoid theorem assumption flexible interpretable model combine allow nonlinear know challenge treat combine selection lasso additive outperform across broad spectrum high apply real half million excellent attractive generalize additive datum relate q fashion glm present extreme treat incur unnecessary popular multiple field economic major obstacle categorization rarely know challenge exclude entirely overcome challenge automatically feature relevant take empty spam spam sparse partially fine grain spam provide bridge penalize glm spam interpretability motivating assume spam interpretability manually reveal appear exactly variance feature memory speed make aspect bootstrap linearity author decide versus nonlinear exclude developed optimization alternative spam smoothing operator penalty smoothness scad purpose datum toward perform perform feature contrast scale formulate theoretical introduction additive development inequality without demonstrate setting spam thorough often regularize spam relaxation main linear nonlinear denote I coefficient basis regularization loss n convex hierarchical choose sufficiently increase get feature none set problem described idea modify proximal suitable step size reciprocal lipschitz convergence sparsity proximal dual descent pair value decrease suitably choose seek deep understanding section regression set reliable factor regime highlight spam oracle guarantee wide decomposable regularizer kind potentially strong design compatibility etc see difficult different inequality rate make despite name inequality rate standard assumption particularly difficult slow slow matrix expansion assume emphasize concrete choose control different feature relevant letting denote present slow prediction suitably choose grow like show grow linear implication performance achievable reduce slow constant incorporation develop error presence nonlinear spam special follow easily statistical reason prefer spam aside truly spam incur estimating term bias sufficiently penalty intuition whereas spam linear note orthogonal spam line supplementary material establish thus basis vector find prediction away regardless group parameter apparent correct grow spam grow serve verify intuition regard incorrectly investigation various scenario spam lasso experimental use cubic knot bx x choose hold report generate x x lasso spam build additive consider evenly optimal thus pure pure parameter deviation show c lasso spam ccccc spam illustrate spam component ground truth spam toward summarize validation nonlinear ground model spam outperform carefully control high spam spam level accuracy priori accuracy improvement compare spam spam reliably linear counterpart spam addition spam correct support cross mistake plot b visualize shape close perfectly comparison visualize spam red plot recover internal permit future spam deep spam generate point train three choose set term win show advantage spam lot since lead parameter trade lasso regime reliable spam lasso nonlinear component linear still lot nonlinear since regime effect surprisingly dominate linear nonlinear unable effect spam less biased pos spam real size summarize characteristic logistic spam deviation spam character email length letter letter allow error spam substantially compare regularize logistic stay spam problem digits challenge middle create force dataset remain spam spam logistic regression select nonlinear confirm small act performance yet effectively avoid overfitte categorization application corpus volume predictive although popular obtain transform suboptimal computer vision utilize dataset place web image match central test whether two geometrically match expensive pair filter image likely pass verification vocabulary visual word observe carefully control regularize logistic select sparse partially perform regularization formulation make practical oracle advantage thorough experiment demonstrate find additive improve popular
reduction common setup pca mutually retain projection perspective variable formulation axis formulation factor probabilistic component pca projection rank reduce large eigenvalue eigenvalue sample reduce condition invertible exceed sample discuss reduce rank rank vary dimension become covariance isotropic reduce covariance balance control estimate use information criterion determine bic maximize free control diagnostic assumption diagonal covariance correlation estimate latent proceed set var reduce estimator residual order covariance ar scenarios ar coefficients scenario occur ar constrain arise autoregressive see ar scenario ar express stack ar constant constraint reduce estimator p fit ar become ar involve rank covariance var follow unconstrained step estimate ar show ar coefficient reduce rank noise iterative iteration current repeat step replace covariance compute residual conclude setup interpretation rank setup useful explore dependence impact ik marginal marginal weight help unobserved characteristic characteristic series associate characterize position setup adopt row space find interpretation interpretation unobserve form behind multidimensional see e group model construct representation ad hoc space lead representation dependence via rank var mention estimator shrinkage reduce rr structure estimator three estimator structural compare reduce performance dimensional attempt shrinkage shrinkage propose see e shrink sample balance control tuning shrinkage ss two choice target dimensional covariance first remain increment serve case accord minimum bic shrinkage ss intensity analytically infer reduce covariance admit summarize frequency replication reduce small covariance estimator reduce correct reduce rank probability select rr reduce result rr ss metrics stein sl z dimensional metric square define z stein estimator standard stein covariance covariance mark bold stein loss mse simple estimator improvement sample stein mse covariance satisfied stein loss various size medium size stein estimator stein estimator significant see improvement mse stein rr rr shrinkage much significant improvement rank percentage sl reduction rr ss reduce concerned stock scenario constraint china scenario e zero ar coefficient setup interpret stock stock finance technology ratio consecutive daily trading display return black red technology pattern dependence stock purpose estimator modeling series first fit coefficient fit unconstrained return bic fit minimum panel display vary minimum word dependence stock represent display stock dimensional stock panel dimension phenomenon stock stock close far observe stock opposite along stock finance technology energy stock stock hand finance stock stock dimension provide separate distinguish among stock exception technology separation panel vertical separate technology stock distinguish finance diagnostic check display cross estimate variable exhibit auto correlation auto cross dimension consistent black red finance green reduced covariance estimator large lead scenario estimate refer covariance residual correspond ar coefficient zero reduce model fitting return reduce impact model aspect interval ar estimate forecast error ar coefficient ar estimate indicate time interval temporal var stable unconstraine forecast mse forecast var ar matrix ar panel compare mse sparse rank estimator step forecast estimator p concern china var temperature series scenario section var sparse ar stage introduce order choose minimum bic obtain non coefficient reduce latent insight actual position finding summarize display panel dimension compare pairwise rank finding factor dependence temperature since neighboring condition impact temperature emphasize position purely reduce estimation var temperature panel display estimate computed auto correlation cross correlation like provide temperature nsf grant research section proposition give isotropic analytical form plug column consist regardless eigenvalue additionally complete mse forecast forecast approximate approximate forecast part come var parameter mse plug estimation dimensional q noise zero upper sub equal replace thereby forecast rank large dimensional reduce estimator outperform compete covariance estimator large var fitting reduce estimator interpretable description var autoregressive k time vector autoregressive vector
identical iterate run also verify leave proposition unchanged bregman divergence hand lemma check condition coordinate outside zero satisfy u term result divergence know strongly lemma strongly imply inequality throughout turn implicit eq claim adaptive variant denote logarithm think sum random bind measurable measurable finish optimize need union start fix lead total desire finish combine union proof use find tw tw invoke b kb follow note schedule check cauchy schwarz eq convexity plug bind case depend let denote want fact root appear meanwhile eq gm th sigma take direction get simplify desire proof tw tu tw sx walk drift standard lipschitz next going stop markov never adaptive regularizer q tw subtract yield slightly condition eq establish probability prove state technical lead give analogue give sequence loss run mirror descent loss regularizer notation weight extend control assumption ensure excess adaptive mirror descent learn need analogue term regularization risk also strongly result proposition proposition combine eq begin argument st apply regularizers eq yield regularization schedule meanwhile probability apply gm inequality proof invoke give lemma meanwhile directly inductive start note immediately prove inequality lemma respectively fact provide together induction know inequality need b suffice theorems probability attain rgb proposition condition support foundation fellowship nsf bc stanford fellowship grateful descent condition regression setup recover achieve statistically quasi optimal feature meanwhile computational resource descent streaming sensor click throughput streaming store obtain parameter article procedure exploit streaming formally prediction maintain iii interested q square loss second classic note closely observe ambient population lasso pursuit weight encourage literature show lasso attain restrict van van require solve global streaming kind optimize database test finite online stochastic remarkably ignore sparsity regret convexity dependence update gradually informative word spam intercept select analysis run feature example contribution stream lasso batch one algorithm take soft thresholding require carefully support different epoch attain conceptually also empirical goal spam spam mail weight gets gradually variable example weight path move look brownian motion et size result involve gradient sgd streaming sequence response loss condition generalize limit x output tw algorithmic easy convenient exploit rewrite equivalent mirror mirror usually close form encourage advantage descent way induce update efficiently stream online also aim classic see proposal convex trade dual describe proximal make condition guarantee advantage bad exist batch restrict van van even assumption guarantee performance stream tt theorem restrict orthogonality feature uncorrelated noise suppose sequence point tf tw w ct match bound since regime online algorithm optimization lasso stream data simple classical stochastic descent comprise achieve replace long statistical excess exist sgd like composite stochastic similar namely regularize mirror like streaming algorithm performance derive heavy prior carlo beyond streaming set none large dataset pre remove dramatically decrease memory meanwhile show locally regression compare streaming algorithm screen subsample investigation start define theoretical mirror framework leverage section result obtain parameter assumption long defer assume draw sequence depend main depend statistical expect weight available gradient simple zero relax condition condition well understanding recall sequence meanwhile quadratic lead long next f tw b uncorrelated relaxed respect evaluate ii parameter control showing succeed uncorrelated second weighted average naive online algorithm loose algorithm output q strong feature exactly relaxed establish design van van main assumption make design similar condition minimax sufficient essentially condition strong convexity loss fact weak stem achieve see guarantee overview relate assumption algorithm yield condition assumption batch big example cross hand allow analogue hold weak resemble regularizer adaptive mirror descent result form regularizer loss immediately upper emphasize proposition turn many originally use hoc follow corollary linearize main perform ensure sparsity final term strong goal show overview indicate detail remainder enforcing scale inconsistent us sparsity restrict bregman divergence noise pair scale bind standard inequality result bind cost penalization meanwhile give adequate time none method optimize runtime taken rough measurement runtime expensive operation th power basic multiplication size step control control dataset collect trust genome association single nucleotide snps case diabetes population code snp allele else compare random dataset compute plot average slide length streaming describe mirror take advantage intuitively mirror sparsity result become mirror divergence small measure belong rest coordinate proof mirror adaptive relie use size apply depend sequence mirror descent regularizers replace statement helpful suppose determine st coordinate simplification regret optimize bind equal convexity tw tu w satisfy get rhs inequality u ambient dimension large analysis dual regret could worse work loss improve regret rather ambient regularizer oppose convexity advantageous common strong convex remove like remove entirely decomposition sparsity mirror regularizer tw remove still simplification sparsity condition one state explicit forget mean noise preserve unbiased random walk still although main absence acceptable term grow dependence hard restrict attention become specifically term regret exploit order achieve impose introduce statistical thus need scale ensure cut hold square indeed appear inside exactly provide transform excess cost error notice depend implicitly specifically excess risk signal pure independent response cost sparse could potentially thus give orthogonal formalize relax uncorrelated hold long thus main identify logarithmic sparsity loss assumption moreover combine control obtain desire convexity individual convex strong adaptive mirror present section strongly losse key technical strong convexity necessarily almost lipschitz respect get mirror loss yield analogue convexity convexity hold invoke simplification expect strong convexity mirror descent regularizer necessary main result namely risk proof far excess stochastic algorithm main implicit logarithmic bound must parallel convexity idea discuss extend analysis provide analogue extend correlated bound know give guess transform standard generalization batch bind
certain sparsity variational variation signal derivative variational view two different tool regularization mainly additive unknown signal version filter return vertical filter gradient square root directional second consider derivative mention derivative calculate pixel version column stack minimization relaxation approximate signal derivative signal tv one variational framework signal function introduce coefficient parameterization I represent length parameterization addition denoise case optical flow estimation force behind naturally describe piecewise impose piecewise tool therefore already literature commonly refer look vector block sparsity handle enable recover usage motivate elegant efficiency dimensional recover organization synthesis propose recovery polynomial technique stable adversarial guarantee mean white block present propose direction research relationship note know subspace intersect select close pseudo norm count synthesis span np solution property relaxation pursuit omp compressive pursuit pursuit sp pursuit htp framework subspace consider correspond span sub correspond estimating zero synthesis problem approximation analysis gap sp htp piecewise change use observation wise synthesis atom function know represent atom represent plus dc therefore coefficient sparse approximated sparsity omp extension recovery hard generalize order e ambiguity fail recover representation contribution treat approximate guarantee though piecewise turn piecewise degree employ assume find signal model projection constant jump appendix htb theoretical guarantee two lead bound adversarial denoise rely property case rip matrix isometry rip piecewise polynomial function jump rip treat corollary recovery optimal project optimal yield depend may polynomial perfect noiseless also subgaussian compress sense even adversarial case exist vector finite proof jump parameterization form energy common application synthesis model generalization trivial operator high space coefficient parameter coefficient block aim greedy therefore start gradually element find current guarantee appendix advantage theoretical guarantee high advantage relative continuous continuity absence jump important add jump impose continuity solve may add edge demonstrate order polynomial continue compressed sense piecewise polynomial compare use start linear compare outcome tv denoise draw connection piecewise jump dynamic white gaussian without continuity synthesis without continuity similar focus analysis straightforwardly fig result continuity essential correctness recovery indeed jump segment recover jump signal preliminary parameterization provide may good denoise mean square approximated piecewise reference therein piecewise polynomial note recover measurement close piecewise point problem figure continuous constraint well due restrict initial thus achieve recovery reason number location though gets happen also order piecewise linear impact continuity significant line clearly strategy without tv effect appear tv reference htb tv present tv reconstruction recover texture thus slightly inferior tv cubic tv recover perform compare polynomial function jump small column norm function difference omit normalize signal fig present recovery behave achieve nonetheless move say likely suppose significantly performance model texture e plane horizontal vertical discrete apply turn vertical one tv denoise fig contaminate additive recovery result suffer image optimize quality setup time provide average use result well outcome new tv notice tv act form add direction one apply also diagonal derivative focus task together segmentation compare cut htb cut segmentation htb segmentation cut segmentation texture continue use house demonstrate result suffer tv loose texture inferior tv removal texture denoise salient edge recover image texture segment minimize segmentation polynomial instead segmentation display piecewise image together cut comparable place behave though segmentation room filter parameter suppose truly leave idea suggest improvement segmentation scheme reasonable great framework novel representation thing solve problem
work derive rate al achieve parametric demanding large offline divergence difficult get knowledge density contrast require support implementation estimator divergence functional index functional divergence whereas divergence specify computationally currently asymptotic appropriately true density density unknown smooth show average density accomplish concentration inequality taylor expansion random uncorrelated derive central entropy empirically distribution estimate classifier bold face type paper density give divergence divergence nn estimator ik ik plug randomly divide part n common mse convergence exponentially convergence optimally sample key idea exchange constant index set basis parameter zero optimization trade use mse entropy obtain rate kl estimate ensemble theorem summarize function kl n I ng normalize assume let asymptotic variance construct sequence uncorrelated estimator ensemble complicated deal prove central ensemble unit variance random eq variable extension et unit I sufficient condition central asymptotically uncorrelated define lemma necessary denominator slowly numerator require l il il taylor expansion density sufficiently smooth power product density bounding power schwarz inequality lemma arbitrary let realization independent eq require grow functional case bound eight previous result cauchy schwarz sum use combine eq bind apply mse number non enable divergence strictly finite task confidence classification problem increase utility focus show uncorrelated theorem assumption smooth strictly also neighbor qualitatively convex divergence simulate uci repository central kl truncated density cube normalize linear relationship quantile normalize bayes class decision error average coefficient chernoff upper eq include optimally error estimate minimum three class bootstrappe discriminant fold cross validation interval rate class fact class compare linearly measure distinguish c misclassification rate paper establish ensemble estimator truncate nn give result divergence mse sample include simplify derive distributional convergence extend central acknowledgment partially nsf nsf fellowship grant density assume ix l ds assume ii iv include beta lemma prove unit variance simplicity denominator converge nonzero slowly denominator preliminary directly tackle quantity il il l l let l l q binomial series expansion density show uniform kernel estimator il f om truncate uniform nn well eqs l l l value density inequality truncate nn lx kl lx lx pr lx lx uniform lx lx kl inequalitie pr cx il eqs provide along let fixed realization throughout first complete proof define way truncate distribute jointly establish event relate relationship cauchy eqs prove splitting covariance case fall second hold dy r xy eqs combine om hand side result eqs note g surely apply functional assumption expectation z ik I ia q z c q z respectively condition g result independent since I give g lx om om cauchy schwarz get imply om om complete om consider numerator previously covariance general independence remain require follow partial let
htp colour smoothed fusion result fusion approach also smoothed region noise false e classified region positive htp apart frame detector problem fusion smooth automatic detect illumination show apply analysis drawback rely algorithm however researcher focus well gate hardware human method human illumination essential well colour solution apply false able cope moreover novel combine colour detector refine specific person wide illumination purpose qualitative quantitative dynamic colour play role wide processing human computer interest detection colour multiple threshold colour component pixel fall pixel colour large region aforementioned solution although successfully suffer false common variety complex illumination image achieve colour change narrow require stage understand performance require pixel collect web framework use rule histogram automatic stage secondly histogram density distribution product automatic strategy detect colour suitable colour space segmentation colour space normalise rgb colour segmentation al log opponent colour human visual opponent colour encoding colour illumination claim illumination factor detection system remainder give work derive fusion result conclude process pixel video colour human medium detect colour internet early tv news sake video automatic annotation retrieval interested reader detailed colour leave appropriate colour use classifier pixel classifier decision colour colour database colour colour belong false complex illumination colour people another dark robustness use invariant colour cope colour within narrow approach non et detector plus detector manually label et bayesian model bayes bayesian decision minimum pattern use build collect web reader encourage suffer tradeoff propose comparison solution employ eliminate secondly detector attempt employ fusion colour show first et adopt secondly employ calculate face histogram smoothed density respectively rgb colour space convert opponent pre elliptical mask illustrate elliptical face image centre minor axis reader htbp detect include non etc interested remove edge detection due computational detect smooth region image useful colour available image choose space model colour opponent colour colour human visual use opponent colour secondly colour illumination theory study certain never human opponent colour illumination illumination log code mean human colour vary appearance colour image affect illumination image camera characteristic learn boundary solution channel channel approach approach person share smooth smoothed histogram histogram capable describe shaped colour model elliptical define colour matrix mix weight satisfy htp gaussian detection angle position red dot angle eq center axis axis boundary axis axis therefore increase robustness integrate single vote pixel produce smoothed histogram result fusion eq fusion make tractable section space quantitative analysis conduct google colour consist dataset difficult popular web top
pixel noise dictionary penalty obtain clean expect impulse quality model peak ratio image recover huber increase impulse huber quality descriptor tag scale typical retrieve wide range difficult use tag furthermore automatic image noisy tag visually noisy tag image image tag indicate semantic error prediction system could include description topic learn visual tag semantic describe visual visual store novel image tag refined code tag cluster discover refined tag estimate assume descriptor measure reconstruction tag vector tag penalty penalty mixed obtain comparative annotation image keyword vocabulary varied tag randomly refined tag use refined tag figure entire residual huber penalty penalty provide recovery furthermore corrupted tag penalty lie union allow use laplacian code contribute provide opposed locality graph furthermore possible confidence measure cluster huber overcome clustering quantile allow gradually quantile previous study huber quantile model robust outlier order reliability clustering cluster uci breast illustrate cluster dataset flexible function enable drop move away perturbation dataset behavior attribute availability choose generative analyze complex require code use challenge perform warm interior enable fast warm start dictionary inference incorporate another graph penalty believe scalable expand applicability important topic model large content figure lemma new york city prior learn simple interpretable discovery recover assume availability sufficient datum sparse fidelity reconstruction euclidean domain motivate look conventional loss huber outli quantile discover structure representative consider linear huber loss learn dictionary fidelity algorithm convergence behavior experiment study function robust image tag refinement annotation confidence generate classical linear response residual shrink improve assume refer control trade regularization deviation observe sparse model speech blind separation supervise semi assume dictionary observation sufficiently objective may code robust modeling detection develop flexible dictionary code function enough challenge penalty penalty treatment act specify penalty penalty finitely general penaltie huber penalty quantile huber classic penalty section regularization estimate non parametric viewpoint follow useful squared penalty arguably outlier robust impose proper improve example company accounting year company event economic b pixel due residual differently well extensively response various quantile vary company predict various quantile planning management quantile along quantile particular company figure statistical dispersion free dictionary use heterogeneous may noise apply tag width height marker leave middle axis thick cm marker axis marker axis line axis thick height marker middle axis coordinate width height cm marker sample axis middle width marker middle x class measurement recently generalize approach full prove assumption classic alternate code dictionary bfgs length important method enable practitioner test kind penalty interface residual automatic penalty ensures coordinate potentially lemma calculus illustrate utility apply experimental evaluation huber reconstruct penalty case annotate tag since tag joint tag huber penalty piecewise structure calculus affine composition use underlie useful addition specify combine measurement affine composition building linear map action vector addition affine penalty coordinate wise different penalty minimization variable constraint obtain function respect seem complicated keep mind conjugate purpose conjugate write tucker system optimality condition kkt optimality advantage characterize wide nonsmooth automatically kkt nonnegative slack equation dual equation full present include code directly solve kkt direct optimality namely kkt system guarantee discuss nonconvex dictionary approach code penalty descent e dictionary smooth addition interested apply entire block descent accommodate sharp coordinate fail convex coordinate minimization generate coordinate point limit penalty influence penalty key contrast residual mean lot application demonstrate smoothed huber outperform standard quantile penalty turn smoothed envelope calculus envelope convex clear always minimizer prox f converge salient penalty close envelope capture convenience also member amount idea envelope huber threshold smooth huber idea capture conjugate calculus b claim solve implement block column close column pose row decompose residual square close structure
study sampling noise multiscale random technique carlo deals example want estimate give unlikely standard technique unbiased rare achieve relative see fix relative rapidly event standard monte carlo rare lead reduce popular method sampling estimator address process random environment process stochastic sde dimensional wiener gx stationary field interpret perturbation slow see rigorous mathematical provably carlo medium form eq estimator provably asymptotic interested hope formula deviation since important become motivated relate molecular simplified preserve deviation order magnitude deviation event dominate event issue poorly paper provably importance quantity deviation deviation principle particular provably importance scheme rare author work address design logarithmic multiscale environment absence fast importance scheme design asymptotic efficient noise without scale find presence asymptotic importance scheme construct paper see periodic fast motion inspire sampling investigate regime paper nevertheless able importance asymptotically optimal scheme rigorous bound ingredient gradient bellman modification account multiscale periodic medium purely partial differential sampling ingredient remark theory optimality motivated deviation fast sampling scheme model classical interpret fast moreover example variable asymptotic sg cx vanish relation interest ergodic simply rough see gaussian specific correlation structure assumption environment result review concept classical paper main randomness environment review example environment need environment necessary ergodic preserve acting preserve action unitary generator densely generator stationary define measurable consider locally field define ergodic relation cx x fy make assumption environment diffusion uniformly cx fy gx derivative globally operator literature canonical lebesgue relatively close unique ergodic environment useful reason write operator follow divergence form j almost eq hilbert equip expectation measure particular unique measure process tool cell literature consider lemma weak abstract equivalently average integrate average invariant section even prove allow known q conclude representative first case essentially correspond ergodic periodic say period lebesgue shift operator obtain periodic special deviation relate scheme periodic environment space equip fr borel algebra borel invariant shift particular dynamic ergodic invariant via wiener measure x strong performance characterize moment sampling scheme appropriately choose control behavior shall need ingredient equation crucial establish efficiency case lead actually performance standard clearly multiscale environment random expect something ingredient shall solution associate measure notion appropriate equation nearly complicated may scheme guarantee performance rare scheme difficulty precise way introduce recall continuously differentiable xt mt illustration several condition consider nevertheless condition expense hard establish second small sx continuous control x estimate since function neither approximate argument establish condition set point infimum closure infimum everywhere logarithmic choosing measure elaborate issue mention completeness merge paper multiscale measure good performance asymptotically plan multiscale control simplify recall unique process explicitly q limit state infimum satisfy q allow result control random infimum particular define display limiting result computation deviation lower ergodic control diffusion appropriately combine particular immediately moreover subsequently fact equation definition definition example slow component z explicitly unique agreement function eq effective drift moreover old asymptotically theorem depend field case immediately cover since significant simulation differentiable moreover numerical effective simplifie become explicitly interface change measure carlo independent control simulation realization field zero fine grid simulate construct euler sn respectively small practice deviation measurement estimate well pair simulation c change standard carlo computing generator discretization presence fast scale rare significant discretization weak precede euler decrease decrease quantify significantly computational highlight fast multiscale environment table simulate turn imply significant sde fast small diffusion ergodic interest monte carlo estimation event functional rare provably result scheme optimality gradient theory framework rigorously
individual follow give close two induce work accordingly try try find metric label take accordingly inspire deep deep embed approach triplet argue triplet strong approach obvious inspire comprised instance feed share fed intermediate distance denote embed word encode network sample different class task express objective correctly classify objective learn closeness label comparison softmax apply effectively create measure traditional sgd examine replace q regard triplet implement train dataset consist image handwritten digits correspond house digit fourth class cifar big augmentation apply zero unit variance instance image third epoch epoch fix instance image follow fourth linearity consecutive network configuration order representation augmentation similar work similar svm knn affect notice later feature respect know augmentation mnist examine image euclidean significant semantic space measuring embed three use dataset gain try similar unfortunately use network relate context leave conjecture h triplet net patches spatially perspective could patches rough unsupervise applicable consecutive frame expect frame take minute net may provide well past classification environment human well accurately provide comparative easier collect triplet attain annotation introduce explicitly way classify model triplet classification consider know insight deep leverage new acknowledgement acknowledge z gpu use computer science institute technology prove successful model mostly implicitly part learn comparison wang ranking retrieval learn immediate possible usage learn deep learning extensively deep require feature
cluster galaxy dark survey boltzmann ordinary early galaxy iii training recommend name ball sample top recommend get might dynamic medical capture achieve examine rate return simulate couple boltzmann element article like article like formation star galaxy formation deep promise thing explicitly take boost research partially treatment turn back propagation derivation u ip I rejection version show red draw region probabilistic version curve tangent line black dashed correspond line search optima gradient take consideration generalize back propagation collaborative commonly recommender system conventional recommendation rating sparse many application significantly recommendation content utilize collaborative topic appeal take source nevertheless auxiliary information advance input propose hierarchical jointly collaborative rating extensive three world advance art principle behavioral science service recommender rs increasingly individual effective many g amazon netflix rs extensively service rs roughly three collaborative filtering method hybrid content use description recommendation activity seek get content method concern generally difficult activity nevertheless cf method prediction product receive information user hybrid gain popularity year rate auxiliary divide category couple coupled process feature guide manual contrary couple allow interaction hand guide feature power cf balance influence auxiliary couple often outperform couple collaborative probabilistic lda factorization pmf interpretable result nevertheless auxiliary problem focus recently potential vision language although appeal content inferior capture integrate deep unfortunately model boltzmann machine instead perform extend item item incorporate crucial significantly model recommendation music recommendation cnn belief model boost exploit content rating cnn poorly rating challenge develop collaborative deep learning couple rs bayesian formulation call rating feedback allow although admit boltzmann recurrent summarize deep extract effective content unlike simple probabilistic framework besides derive propagation hierarchical bridge state rs nature incorporate auxiliary boost world advance art recommendation take implicit test item movie bag vocabulary size article library rating although movie recommendation plot movie consider handle recommendation tag recommendation play corrupt output weight matrix bias vector convenience collection bias correspond ready detail give bayesian rating feedforward neural clean input output usually I bottleneck input layer clean solve regularization tb clean generative network draw draw bias k l clean process generation input feature gaussian dirac sigmoid degenerate formulation layer act encoder maximization minimization layer column weight matrix l clean j draw user item r n j layer bridge rating representation capture similarity user computational graphical notational use tb could carlo typically incur primary fair consequently devise style obtain maximize posterior joint likelihood become content encode item take encoding reconstruct example layer perspective perceptron fourth term infinity degenerate two common corrupt layer positive learn directly happen go decoder graphical experiment greatly coordinate ascent leading rule ij rating user control learn layer propagation gradient find optimum several term completeness appendix let observe denote operation offset tb sparse tb layer extensive real domain demonstrate effectiveness qualitatively real domain netflix way different practical situation dataset mostly independently manually seed article tag user result contain rating rating entry two rating movie netflix movie first extract rating rate positive plot procedure text item extract article stop tf vocabulary item user rest setting repeat select average aware precision performance recommender system sort rating recommend item recall report recall list collective factorization incorporate source simultaneously factorize raw feed music recommendation mention section collaborative perform collaborative simultaneously collaborative vary use hyperparameter perform rating well achieve hyperparameter achieve cnn also good tune determine directly grid tb cm level corrupt adaptive regularization word representation show compare sparse baseline sometimes achieve perform due rating overfitte dense inferior reason specifically sparse dense setting go outperform sparse dense increase go integration rs careful boost number setting number layer exceed layer deviation table tb cm layer result somewhat two discuss information rating word integrate content handle sparse much tb dense omit previous item learn directly interaction hence suffer bayesian component latent item factorization performance get already close even bad pmf gain take user dataset profile match topic article return train might tag topic one correctly article system network rate article article
train model negative skip gram framework actually finish table performance respectively default predict embed embed order mean dimension c model semantic syntactic accuracy word skip gram relation relation fix relation relation relation matrix relation either single skip type knowledge word skip group five accuracy root basic unit word composition syntactic besides recall truly word precision truly similar letter even though recall limitation truly task skip gram similarity relation predict almost increment indicate effectively impact natural language high leverage especially rare since context information rare building provide effective rare rare knowledge away frequent word balance branch add conversely input fix skip gram matrix fixing skip gram input skip relation fix worse skip gram well sharing bring effectiveness propose branch branch aspect leverage keep perspective easily understand language noisy additional may leverage avoid knowledge coherent cognitive information branch help insufficient refine experiment claim effective robust contextual specifically two branch branch balance update branch united analyze learn embed especially word far fair snapshot wikipedia corpus denote process similar occur less million token vocabulary experimental frequent baseline demonstrate power see skip fail relate rare corpus near neighbor random see relation leverage enhance embed rare similar knowledge suggest share neither similar contextual refine tradeoff coefficient generation similar word final embed example table indicate rare word great favor successfully bring contextual distinguish useful manner framework contextual influence balance branch branch framework great e rely contextual analyze draw frequent word contextual rare word rely compose rely rely contextual task mainly rare give discussion two conclusion gram illustrate phenomenon carefully frequency contain word show take fix ratio gray represent fall index frequent word word word frequent rely information rare rely surprise ht embed setting conclusion skip gram combination matrix train frequent word frequent rare sum like overall ratio middle drop besides ratio skip matrix skip gram trend probably small size conclusion middle skip gram relation model rely strictly opposite achieve besides skip always skip gram combination rely knowledge rare word word rare update consider initialization middle novel neural high leverage occurrence knowledge branch enhance huge work plan leverage basis like representation bin liu quality representation e embedding address mining information retrieval processing method context semantic syntactic challenge handle rare insufficient word cognitive take essentially address particular novel call build word meanwhile refine accurate experiment reason word similarity enhance effectiveness author department mathematical sciences china liu microsoft st china china mining retrieval ir language nlp obtain representation embedding bag continuous skip gram skip gram leverage document transform syntactic lie context various couple include insufficient context adopt extract space look matter rare effective way instance sometimes build new root guess probably henceforth similarity act bridge rare word inspire recognition enhance word beyond contextual already skip take learn might word kind guess counter inconsistent similarity since clear stick effectiveness word embedding tackle issue finding cognitive reveal human contextual connection mind trust knowledge similarity discussion actually neural architecture embed consist branch contextual branch efficiency similarity similarity branch branch share word feed forward layer modify embedding demonstrate representation task similarity contribution include neural framework learn learn embedding rare knowledge noisy contextual rest review knowledge section embed time semantic allocation approach yield limitation scalability obtain embedding variety natural process unified architecture representation amount process bag word model skip gram skip gram amount text embed intuition context word skip gram slide stream sample slide word format train vocabulary feed map vector predict word back matrix train word embedding skip gram perform nlp quality embedding rich extra word embedding word knowledge quality embedding recent effort explore introduce framework yu et objective semantic semantic take empirical enhance work syntactic semantic valuable word obtain high embedding rare word leverage since unknown word popular one previous especially recognition letter dnn first work letter replace word input gram order small rich select feature tag generalize vocabulary nlp neural combine neural word representation contextual representation text leveraging describe neural architecture word recognition cognitive learn language usually gradually build base try several channel recognize sound guess study something know word guess know article quickly mobile device compose letter guess word retrieve association current guess mean context different context g historical hard meaning decode manner sometimes error share rely knowledge bring lot distinguish help contextual avoid please human powerful leverage contextual embedding novel knowledge contextual next architecture embed skip representation similar context gram follow denote slide window sum vocabulary impractical directly proportional vocabulary art aim probabilistic discriminate generate bilinear generate noise use include embedding represent distribution set power computing summing vocabulary become vocabulary knowledge beyond predict new introduce context leverage representation contextual refer want prediction softmax branch branch detail ht accord obtain branch necessary similar knowledge forward layer method actually th leverage top word high score connect connection update huge contextual branch branch I yield dependency frequency frequent easy collect rich contextual might insufficient knowledge little correlation word word knowledge even though contextual divide frequency way interpret embed embed couple extract similar take original overall weight back propagation process pair frequency update process take skip uniqueness branch branch four complete four knowledge quantify two string one represent word root stem split calculate eq denote four separately combine together experimental experimental regard effectiveness mainly compose part baseline skip effectiveness mainly rare quality word embed rare gain noisy word rare also give study gain insight balance contextual branch question denote suppose answer question find bc regard correctly syntactic similarity associate receive receive average
nlp amenable counterpart follow computer vision successful application convolutional neural nlp hierarchical convnet support document show adapt vision document evaluation automatic extraction document label simple classify extract system alone preserve extraction internet restaurant review site front page neural method compare convnet sentence sentence convnet transform embedding sentence entire sentence convnet sentence embedding model train embedding softmax train sentence level sentence tie sentence embed produce convnet pass show learn relevant essential extraction application specific sentence model interaction embed handle explain nlp form document sentence follow detail level sentence correspond sentence process embedding produce sentence build vocabulary embed use together sentence produce embed sentence document embedding embedding sentence convolutional bank w fw maps layer dimensional align axis level obtain location sentence stack new matrix feed wide feature sentence include sentence document embed issue convolutional handle problematic interface sentence max pooling apply pool along row discard length max pooling sentence single convolutional sentence ie tie sentence document convolution pool weight level model model create allow extraction salient contribution network activation deep computer work fact formally carry net extraction create map assign sentence give objective perform pass class invert feed great infer first order expansion function formally approximate entry easily perform single pass intuition behind indicate change embed write sentence whose identifying appear respect correspond dot th vector document sentence implicitly sentence huge write level huge indicating appear respect backward rank extract sentence rank use qualitatively however compare lack gold extraction reference document create extracting extraction choose subset sentence hand irrelevant movie sentiment originally demonstrate movie review label divide label review experiment label review break review sentence break sentence perform task map number generic symbol times leave word vocabulary proportion rand fix convnet word rand pick pick full percentage sentence label convnet show produce create show reference full review use heuristic sentence convnet movie data sentence map width nonlinearity weight tie document document input bank width follow pool nonlinearity fed predict positive extract salient review review I full several baseline embedding embedding apply predict document technique convnet model assign sentence heuristic last sentence informative content sentence review drop
remove cifar name optimization prove effective critical effectiveness process provide flexible various remain frequently machine often optimize spatially develop beta cumulative bayesian multiple stationary inclusion greatly improve state produce reliably goal estimate proxy perform promising ability routine advance ability class express accurate value input crucially main exploration exploitation bayesian limitation output assumption simplify regression realistic stationary present many inherently non optimize bad g classify expect generalization performance gaussian variety stationary particularly suited learn remove shape optimization straightforward idea space another advantage share structure experimental empirical bayesian benchmark method outperform parameter challenge consistently converge methodology involve model stationarity fundamentally component effective powerful distribution widely linear bayesian attractive bayesian uncertainty unobserve positive set predictive cross apply common choice automatic determination ard eq ard mat ern space model gps covariance function propose project multidimensional thin spline flexible gp spatial extensively statistic perform simple yet flexible address effective gain elaborate technique problem multiple henceforth refer domain input predictive surprisingly function transform produce task index eq directly parametrization covariance noisy expensive explanation strategy probabilistic expensive bayes observation surrogate determine proxy query posterior probabilistic acquisition tradeoff acquisition combination define surrogate acquisition marginal improvement criterion normal propose independent acquisition analytic utilizing train collect scenario task predict sequence sharing find function auxiliary task project hypercube hyperparameter researcher often space monotonic perform grid transform space take stationarity inherent stationary property input ideal unknown evaluation use transformation transform specify value th cumulative distribution shape determine cdf close non accurate statistical software package alternatively stationary function stationary choice beta capable variety monotonic choice slight approximately approximately contract outer expand center logit shape single explicit transformation hierarchical place integrate treat collection hyperparameter monte slice e arise various median zero identity center empirical analysis expect trained task account allow inferring effectively try task suitably model task versa rr grid base protocol benchmark run evaluation low benchmark far evaluation comprise distinct experiment compare benchmark show task evaluate standard expect ei follow treatment ern use involve tune deep cifar layer layer optimize two six regularization weight norm pool logistic show stationary improve convergence dimensional convolutional network bayesian optimization different dimension evolves become observation intuition transform weight figure weight connect input weight transformation logarithmic transformation effectively mean variation occur medium scale especially variety learn different dropout confirm set learn non trivial rate agree dropout surprising give hyperparameter unlikely expert highlight utility subset benchmark benchmark design strength hyperparameter perform bayesian differ use forest package input result improve well standard decrease worth note function drastically many interestingly approach naturally deal albeit fundamentally uniform explain discrepancy unlike forest smooth input mean ei locally via method select random define function absence marginalization apart deep convolutional average learn regression annotate layer apply
introduction expectation necessarily write expectation probability write respectively mathematical refer size oppose emphasize aforementioned terminology level indicate less throughout paper make evidence size h force reject decision via ix h reject retain statistic reject key u detail general refer specify say variable yet mid mid goal make retain may corresponding relationship randomize correspond formally independent every state report reporting implication theorem equal entire derive closed form observe important reject abstract randomized value avoid usual understood step decision bring discrepancy randomize characterize nature discussion additional contain usual skip uniformly u e x viewpoint careful inspection indicate lose report whether equal randomize reflect lose report alone information specification equal information procedure conservative otherwise minimized bias whose minimize additionally near bias indicate conservative simply rule additionally report rule slightly moderately conservative etc take define adjust randomize especially countable goal decide reject retain procedure mx section formally multiple function potentially xu retain reject typically false discovery fdr family reject false reject expectation respect u single one procedure definition ip mx hypothesis equivalently decision well define randomize wise procedure dominate single counterpart extend wise expand providing specify rank procedure u xu independently relax conclude remark assume satisfied xu right x integration could generate uniform compute record xu b binomial wish hypothesis far generate compute xu observe xu indicate reject apply formalize allowing reject xu size reject adjust xu decision xu refer adjust x adjust abstract randomized generate compute xu histogram adjust randomize xu computed appear fall formalize link xu xu x I independent practical implication adjust abstract reject hypothesis fdr equal hypothesis depend nan fdr adjust decision demonstrate choose fdr perform generate gene hereafter step abstract adjust group adjust gene summary step step u analysis step histogram reveal wu additionally allow mid natural section decision randomly discover discover specify mid indicate additional unbiased natural reject take previous statistic identical test traditional testing procedure automatically whenever approach outline generate mean portion utilize computed idea retain nan small value reject usual step apply collection remain refer step method table nan hypothese h sort result reject retain nan additionally hypothesis due abstract summarize low group adjust randomized expect randomly mid nan p n implement mid automatically hypothesis step hence nan reject computation microarray choose retain nan microarray address well discussion choice supremum bias ideal variable multiple testing define define unbiased xu randomized sense adjust variable additionally adjust adjust mid microarray via wu yield last row adjust reject nan enjoy nice though less arbitrary drawback vary adjust mid abstract randomized hypothesis show testing view randomized approach abstract randomize understanding procedure consequently report assume generate compute consequently analytically fdr procedure value independent ensure remain valid approach may mention recommend goal illustrate opt significant provide tool quantify decision impractical single practical complicate similar additionally multiple testing long microarray support value setting adjust generally adjust mid ensure value statistic identical function usual mid may lead test may histogram abstract randomize whenever upper histogram adequate mid decision fact certainly report randomize ultimately certainly choose natural mid randomize reject understand verify px follow claim definition xu xu definition second law independent claim c mid abstract statistic distribution provide extend testing version aforementione usual randomized decision nan hypothesis differ adjust abstract dominate variability tool adjust abstract example type adjust mid value abstract adjust randomized throughput technology force microarray nan hypothesis test widely accept hundred procedure recent year book review ultimately employ nan hypothesis arise utilize challenge test evident testing nan vs alternative equal reject retain take testing strategy probability impractical decision depend generate reject mid natural less equal advantage neither mathematical generating variate abstract example capital uniform variable interpretation mathematical detail focus aforementioned abstract information likely randomized yield decision consider binomial consider suppose randomize randomize
stand c fourth permutation loading pc z mix fix sense ip ip ip r ip converge distribution tell limit pc classification ip directly geometric representation normality simplify situation pairwise leading set resp far asymptotic condition generality correctly classify set misclassifie converge identifiability notion identifiability consistency dimensional context state na z order q second definite b last sensible compare plot classifier line color report comparison consider statistic r r rt ratio group distance method error size c lda lda cccc lda lead nan figure figure instance grow htb panel figure show theoretical quantile uniform inside conclude uniform distribution htb underlie group permutation test greatly compare update rate lda classifier exhibit notably decrease apply permutation test experiment performance statistic table test different test statistic c usual low performance classification pool covariance shrinkage estimator positive modify anonymous suggest compare modify linear discriminant analysis appear give classification pool fisher discriminant discuss discriminant discriminant distance bayes n thresholded covariance eigen decomposition e thresholded estimate majority voting denote majority voting classifier lda lda lda lda lda lda svm lda svm svm lda lda svm remark problem classification set observation classification rule label label perform set motivate free empirical employ datum variation principal component extract major multidimensional scaling conventional well result demonstrate related cancer bioinformatics principal multidimensional scaling em em advance ease semi automate cell image discriminate cell belong set right task image rule predict study much precisely problem suppose consist iy cell predict sample contain image characteristic datum make may may characteristic rule base vector x I regard empirical contour estimate normal label observation set distribution whose mean covariance represent contour distributional lead believe idea image mostly focus extract texture first simple model set propose context free method statistical principal transform empirical direction subspace classifier scale extensively subspace real classifier discriminant label observation extract multidimensional scaling formulate relevant conclude arise label class incorporate characteristic level membership hierarchical observation consider variable dependence membership hierarchical classification absolutely visually evident correspond matrix ii k k pa parameter covariance vice versa panel example utilize separate discriminant clearly distributional hierarchical long hierarchical bottom panel common play role voting web appendix experience vote show framework transform extract appropriate set cell problematic overfitte location distribution principal visually separate location extract extract orientation dimensional space value th ij ti pi orthonormal variance situation major direction use lead pc direction essentially eigen covariance large pc bad indistinguishable understand whole precise subspace web subspace important matter model drive argument lie manifold element conventional pca first mapping form intuitive canonical angle subspace span unit form call angle angle without generality matrix orthonormal l modify measure choose variance good b I multidimensional nm j euclidean geometry extract value point multidimensional give require optimization detailed discussion configuration write euclidean z fa solution minimize n gram consist product nonnegative coincide map pc classifier extend map pc fix pairwise distance I n represent point configuration point increase eq work th permutation large th pc web discriminant extract obtain permutation consider study discriminant vector machine discrimination minimum distance competing include classifier class size membership majority call weighted hand classifier lda classifier precise definition classification summary svm lda lda denote zero ij p p u highly auto ij von distribution parameter performance section misclassification rate test equal independently summarize repetition dimension exhibit misclassification rate compete surprising pc good covariance comparable voting method pc feature effectively discard hand pc relate result successfully pc feature correlate show comparable correlate simulation confirm lda hierarchical appendix cccc ccc lda tumor remove process analyze normal contain datum available http www edu user software image different leave fixing misclassifie together misclassifie dimension inspection orientation right point north east close orientation north cccc lda th svm th th technology processing cost expensive clear training choose set classifier summarize exception original set raw texture introduce classification selection procedure model classification method orientation possess adaptively use classification multidimensional modify nonlinear solution mapping observation multidimensional regression mapping modify solve extend independence relaxed dependency observation especially cell close high correlation greatly accuracy classification focus hierarchical classifier greatly vary greatly feature exclude set sensible properly accommodate classifier conventional sophisticated machine machine choose selection work classifier present potential advanced kernel web section code acknowledgement discussion relate set associate anonymous valuable partially nsf dms partially foundation
recommendation datum netflix million distribute movie million star increment movie netflix user item million fm netflix performance topic vector user fm fm fm tool default comparable fm netflix well baseline quickly conclusion vector perform since exploit history validate topic traditional fm fm feedback update fm result performance result confirm future com zhang liu recommender system widely factorization factorization fail benefit feedback rating movie rating prediction history influence rating dirichlet word feature fm implicit fm exploit result demonstrate fm collaborative gain netflix score score rate reasonably element problem focus factorization dyadic fail role factorization take long memory factorization fm engineering fm specialized svd work fm implicit user call topic base fm model fm model topic model natural language nlp area classic dirichlet allocation text corpus express several document use latent lda rating want user history degree movie document movie obtain topic document interest movie draw latent topic train fm fm build efficient implementation continuous bag skip gram simple learning huge big vocabulary million train latent vocabulary represent word instead word fashion fm fm follow fm fm experimental collaborative latent approach conclude introduce feature character item fm fm previous latent history consume contrary latent time necessary explain fm algorithm make model gibbs firstly parameter introduce element rate latent user item dot product vector correspond item model topic index cross dot product exist firstly train user secondly item item tell topic item topic fm besides cross term item item step fm history datum show htb uk il model method baseline topic base fm item belong exploit user extent observation word utilize rating regard
constant effective obtain motivate follow heuristic pass select among positive choose compute quantity b ng ta ta start small iteration surrogate function perform simplicity extend individually heterogeneous dominate updating surrogate simple several surrogate time compute code matlab source software package core ghz gb ram six publicly dataset consist challenge website pre standardize zero unit name storage size gb dense sparse consider q magnitude would use learn impact output additional present limitation toolbox adjust sgd schmidt et size finding sgd g accelerate propose automatically zhang language mark mark schmidt similar search upper heuristic describe suffer update storing result mb mb mb second fit extra scalar x duality experiment conclusion empirical sag sdca fast variant predict convergence significantly well good dataset satisfied mini sag sdca mini batch gradient descent sgd compare pass preliminary present show solver coordinate appeal smooth sparse present scalar vector stationary way approach p whereas surrogate five algorithm lipschitz scheme adjust initialize x natural section bad objective search significantly adjust asymptotically l five seem converge substantially point minimize asset probably applicability include non smooth asymptotic point incremental competitive state art solver large scale logistic store iterate function publish alternate multiplier future currently scheme acceleration direction result inexact proximal acknowledgment author would mark schmidt associate comment sake directional directional direction directional direction convex directional stationary convex present characterize surrogate function find appendix regularity strongly strongly smooth regularity function differentiable l p l successively widely applicable popular various scientific especially process importance machine precisely asymptotic stationary one expect apply composite obtain proximal rate experiment machine sample large enough usefulness penalty convex convex incremental minimization successively objective locally decrease principle theoretical popular various approach maximization statistic bayes mode surrogate inverse image factorization scalable minimizing function exactly intractable find stationary difficult motivated represent index context amount explain become machine large point inherent sublinear enable solution incremental propose minimize algorithm incremental method per cost problem soon appropriately result upper class surrogate lipschitz obtain sure convergence stationary guarantee provide remarkably composite function method sag schmidt schmidt zhang two work different sag sdca useful cut solver scale schmidt show incremental dc nonconvex penalties mi ni mi devote present basic minimization exploit function n mul exp add x mul mul nh intuition quality key arise require definition analyze variant hold error surrogate surrogate surrogate difference state analysis surrogate define h obtain classical bind proceed function precisely section iterate asymptotically assumption surrogate composition show asymptotic adapt proof gradient nesterov set sublinear convex minimum impossible instead condition mild directional exist stationary appendix condition see point assess stationary differentiable converge note critical convex surrogate strongly asymptotic stationary increase equality denote limit sequence h n non negative necessarily two accord define l sum converge h derivative direction therefore cauchy infimum guarantee exist algorithm composition composition monotonically grow infinity note make write proceed z universal smooth relation directional upper part describe note nature nesterov originally proximal adapt regardless nesterov segment monotonically simplify introduce consider lr recursively subsequently lr lr bound nesterov convergence convexity point show minimize make strong surrogate show slightly convexity assumption surrogate f convex successively prove classical convex though obtain well one example algorithm exist though bring new asset smooth natural surrogate smooth sum remark ni mi f let composite surrogate function smooth surrogate follow analysis admit definite surrogate surrogate appear learn objective without newton though quadratic surrogate rate practice introduce incremental dealing probably sgd update n consider smooth admit order surrogate schmidt randomize incremental sag smooth variant incremental sdca composite optimization incremental update sgd sag sdca storing iterate context incremental propose bound iteration approximate surrogate al surrogate initial number iteration initialization surrogate randomly pick update section study specifically start non stationary essentially surrogate sublinear case one rate theoretical surprising surrogate use directional surrogate proposition hold proceed tn obtain inequality g surrogate surely monotonically e evy representing result g f let h since converge also analysis composition ng nf remark approximation error almost surely asymptotic effect g proposition easy te h n conclude notably surrogate strongly minimizer proceed point conditional iteration incremental relation similar convexity la l summing e n f convergence rate strongly convexity convex batch time obtain share sag natural make section use store vector z z sag suggest even behave guarantee converge sag sag substantially sag decrease large unless condition convex sdca resemble offer sag procedure sdca involve step update appeal strongly surrogate within upper amount perform sdca resemble convergence independently refine
instance bag kb ik feature assumption implicitly bag define hausdorff also certain ik instance contribute bag single bag represent bag standard representation convert bag strong make object originally hide positive bag negative instance goal classify bag instance focus instance classifier combine noisy relaxed formulation bag specific instance still apply bag bag product combine posterior probability away relationship bag bag whole directly define bag instance contribute survey concern label bag instance desire phase bag mi si category empty something label possibility bag goal train classifier require goal classify classify bag many vice versa important reason prediction bag false misclassifie negative bag correct bag misclassifie similar reason bag classifier bag bag instance goal optimize optimize bag match mention annotate link deal weakly annotate annotate bag annotation bag classification process elaborate object bag label bag see multiple instance bag fraction fraction learn bag estimating first turn scenario instance bag however si si assumption section inside exploit improve difference describe bag name set object mi label straightforward bag instance versa done supervise near mean vote majority pooling distance convert scheme nearest similar obtain instance patch image level bag level build bag beneficial several test I good mi mi kernel subset bag subset share object bag instance bag output bag classifier class label permutation find instance jointly guarantee perform instance label bag instance diagram deal vector bag instance popularity motivating reason classification firstly great power logical entity secondly label bag costly bag lastly advantageous bag whole illustrate bag si mi si si accord whether instance si popularity scenario si mi base several problem incorrectly mi mi progress connection make category diverse many classifier extend direct type input si mi originally indirect bag unlabele instance unlabele bag mi direct bag indirect instance bag instance test mi si heterogeneity object mi si train happen label bag instance discuss section bag bag classify bag phase bag bag integrate perhaps si mi si attract attention strong raise minimal need situation scenario review bag learning gap easily context collection label several scenario adopt enable suitable thank kind lead anonymous suggestion classification formulate directly traditional set training vector vector label set well extension either training propose mutual difference map relationship field pattern problem difficult regular available classifier previously unseen learning scenario include instance base review scenario reason might choose reason feature restrictive object example drug classify effect shape activity therefore label costly consume computer diagnosis application voxel tag image region predict patient grain g voxel label bag instance face verification video consider video confident prediction label neighboring video classify goal possibility show fig si multiple instance mi si si instance predict activity mi mi object bag mi mi scenario mi si scenario face verification represent mi face bag si objects bag si object bag classifier necessarily relationship lead poor happen type field finding therefore believe understanding scenario field overview bag stage process provide insight scenario often intend survey classifier particular exist survey type furthermore focus cover extend multi multi begin application bag representation instance bag label associate category conclude discussion prediction activity whether influence shape fold bind number possibility set however active available possible assumption label positive cloud atom shape compare unseen logical atom inactive combination contribute bag instance pixel object medical contribute several inexact algorithm bag article email website histogram application assign document bag instance seem classify relevant go paragraph relevant classify document email situation social page security classify describe website feature label relationship bag goal unlabeled bag individual neighboring link detect failure account music retrieval spam filtering motivate failure bag occur correspond failure frame instances example spam spam normal help later proportion would discount proportion circumstance proportion instance receive discount rather address store problematic fraction label assess customer bag represent ik label instance bag interested find learn characteristic bag bag individual classified bag bag label bag example assumption instance bag lead subsection organize type l cm si si instance supervise collective mi bag bag bag bag mi
mild gain sublinear regime allow characterization perspective many show flexible practical compare adaptive perform former well problem analyze specific allow consider framework low bind characterize assumption salient give sequentially observation framework interest compressive sense cs extension nonlinear formula type inspire capacity channel similar adaptive look nonlinear adaptive method extension lower help match bound happen cs necessary sublinear sparsity mild gain body however cs little ref bit low bound extent knowledge generality look gain sum variable difficulty problem stem observation increase observation may uniformly simple cs another testing item item observation pair use integer decoder index present analogue generate bind away zero proper subset upper upper generate iid error sufficient recovery tight iid mild scenario possibly function function sample asymptotically define mutual term maximization sample bind variable choose specific testing bit upper cs index salient reveal element binary probability event start write give uniformly identity analyze identity completely binary expand remove eq follow second putting consider proper recovery inspire feedback fundamentally extra overlap term explicitly past output discuss implication adaptive testing bit testing item large outcome item include formally boolean test item test denote outcome outcome formula arbitrarily error testing kt bind author assumption identically author argument asymptotically match low outcome adaptive testing cs row support bit nonnegative measurement measurement noisy variant snr iid measurement increase asymptotically snr look length variance w equal iid constrain total power noise ix tx satisfy simultaneously ix x k due maximization since maximize subject power jensen submatrix dft conjugate transpose easy independent maximize constrain entry maximization exact adaptive kt I let infinity independent relationship characterize linear
operation transform frequency domain fourier fourier transform respectively find reduce find noise pdf obtaining pdf using state ip di calculus lagrange differential give prior find pdf bayesian favorable minimax without assume boundary obtain q chen use problem loss become performance limit favorable bayesian criterion estimation widely give limit observation noiseless discussion condition satisfy cumulative give pdf threshold pdf simulation terminology shall call value assume condition e fx proof require minimize expression conditional I condition set u u u proposition condition arbitrary conditionally independent dependent dependent identical follow sensor observation fc sensor single correlate sensor receive say design provide split region whether lie sec correlate design dependent consider work show sensor rate identical location prior differential condition threshold assumption conditionally observation future distribute plus minus height em chen consider estimation conditionally fusion center sensor sensor identical sensor capacity wireless sensor observation certain observation present location condition widely achieve relax conditionally counter distribute er rao parameter estimation distribute local sensor fusion apply sensor however energy traditionally simplify relatively little decentralized optimality identical identically distortion criterion fusion true goal bandwidth scalability robustness change distribute quantization besides aware limited bit center channel throughput address decentralized sensor constraint address wireless sensor work consider estimate optimality average distortion build major contribution summarize derive conditionally unbiased estimator fusion sensor identical design bit rate sensor condition sensor use gaussian condition satisfy regime binary calculus attain hierarchical dependence organize model paper problem conditionally sec prove sensor attention wireless sec sec location binary relax conditionally independent sec optimality condition sec fc pdf sensor play fc sensor sensor sensor receive local observation realization set assume distribution fc paper conditionally identically likelihood realization fc free quantization sensor fc along observation realization refer estimator possible strategy view similarly fc true argument expression herein optimality cost assumption independent use optimality know terminology fix mapping require write proposition condition strategy minimize give variable u form consider random solve person rule remain remainder specific mean q find conditionally conditionally motivation behind widely estimator posteriori asymptotically unbiased er fixed conditionally optimization estimator achieve apply therein necessarily surrogate identical conditionally strategy exist fusion mse restrict strategy strategy wherein sensor bit identical quantization fisher sensor contribution datum since fc give sensor conditionally independent output eq observe identical group solution unbiased incur sensor bit sensor quantization without conditionally solve constraint sensor fc instant answer whether sensor less sensor sensor previous sensor question address sensor bit channel sensor scenario capacity wireless data channel ideal allocation sensor local value channel carry bit bit constraint admissible capacity sec conditionally unbiased estimator sensor strategy yy fisher quantization binary optimal quantization identical bit optimal divide rule first contain compose quantization integer rule decision notice imply sensor sensor capacity sensor decision fix sensor rate constraint equality conclude sensor bit quantization rule condition example gaussian sensor sensor govern statistic single fisher distribution contribution single straightforward calculation give proposition exist candidate binary sensor binary fisher cumulative derivative fisher desired result necessary optimality binary snr bit
current health extraction extraction period code code hierarchy period code intervention procedure character digit use diagnosis episode would network entire rare fig display health summarize cumulative stagewise classifier sec risk eqs huber otherwise possible fast optimization bfgs converge unique patient divide subset remove potential patient specific outcome correctly identify portion harmonic appropriate macro mae level adjust imbalance behavior classifier hyperparameter regularization purpose regularization investigate sensitivity hyperparameter fig measure high outcome former overfitte sparsity reach right peak surprising force largely unless otherwise cccc cccc share moderate risk risk precision assessment risk risk month health moderate score machine roughly improvement accounting macro average mae resource resource case moderate high negative classify risk relative figure significance remarkable management average basic resource slightly less false negative table stagewise share resource negative significance social negative serious detect case examine whether machine improve run also alone predict extra lc cccc share moderate high cumulative risk cccc share cumulative record stagewise different size large framework examine stability sec result record fold fig index function ordinal instability issue drop include laplacian instability stagewise stagewise stagewise separate snr moderate attempt moderate attempt attempt due drug history number code stagewise sec width delay diagnosis classifier distinguish class top stagewise share recent recent moderate drug abuse home moderate attempt moderate attempt moderate attempt code history moderate abuse moderate abuse abuse code high attempt self student attempt activity self produce stagewise share sec delay diagnosis stagewise classifier offer rank separately rank stagewise aspect association outcome medical machine prior chance factor discover one well clinical economic issue positively attempt ed physical report hypothesis result automated study depend discover totally universal fast tool clinical work risk although gain study literature new also statistic statistic largely ignore mining relation prior result relation realize regularization generalization place predictive surprising lead condition suggest may correlate tree ensemble instability end stable predict difficulty impossible partly conjecture perspective rare event thus clinical health patient impractical concentrated attempt differ latter chance death contribute literature separate medical system useful comprehensive sensitive relevant distant history limitation framework external variety medical result far match exceed state clinical discover resemble factor perfect label collect health alone track ii diagnosis may perfectly due suggest conservative derive predictive health record explanation central criterion similar similar lasso framework information resource validate challenging predict risk discover health knowledge collect exploit predictive diabetes stroke health heart attack pose research deal situation modify outcome ann collections management helpful wide record present great challenge datum largely temporal novel regression conceptual temporal construct extract diverse ordinal weakly predictive stable employ domain specific feature introduce index generate prior sparse random framework margin risk health produce enhance medical offer great useful support patient predict clinical outcome moderate construct automate historical medical record patient within challenge effective interpretable noisy modality mixture static moderate dimension event disease approximately episode dimensional call unstable instability highly pick see next weakly limit unstable paper framework address challenge number extraction novel patient medical record temporal extract diverse class ordinal without specific risk cumulative stagewise equip sparse selection smoothness interaction include diagnosis branch evaluate feature rank account thousand health assessment community ten develop year response health service assessment population traditional risk provide benefit factor criterion predictive feature clinical stability health prediction improve high class case detect agree previously report decade extensive readily collect demonstrate efficacy contribution generic conceptual view temporal temporal extract risk ordinal introduction stagewise risk formulation relational ordinal list feature signal weight contribution stability comprehensive demonstrate method demonstrate data problem knowledge task adopt type comprise clinical history rating automatically relevant classifier paper organize present feature relational section section provide far follow conclusion two type disease present predict outcome current intervention plan plan allocation medical knowing help trial assess construction hand pick highly previous free mining utilize diverse type fast grow still limit thousand signal fashion stability keep interpretability model stable another stability could goal especially model logistic produce unstable highly correlate relate distinct change output condition feature issue produce similar consider weight stability index popular set represent use discrete problem include weak subset exploit aggregated reference quantify redundancy e exploit exchangeability group suggest context interpretability sparsity recent network primarily stability ordinal since frequently use ordinal odd odd ratio risk factor special natural consecutive separate idea study study generalization however could scale vector case whose linear predictor scale phase train medical often choose clinical quantifying factor rather build assessment quantify aspect relate attempt multiple risk assessment technique clinical aim interpretability application item attempt prediction record study limit poorly complete analyze use nlp technique event event gender patient several clinical serve evaluation outcome generate make use international scheme intervention process condition type diabetes hyper define response drop drastically beyond specify width event time wavelet like kernel detect trend describe ordinal model outcome risk lead risk share share relaxation discrete underlie random l l say outcome coarse divide determine py cumulative usually convenience unobserved logistic interpretation odd cumulative risk reach immediately progress stagewise outcome may attain level attain stagewise level current pass probability py pz py z pz accept level accept eq step distribution discrete nice odd fail subsection treat outcome risk risk qualitatively people never treat stagewise sec instead distribution training select unstable variation medical redundant lasso tend feature feature vary unstable feature change instability problematic clinical another relational model quantify draw x suggest ranking length select list term feature select feature regularization impose snr since draw set way subsample snr stay criterion probability individual snr natural criteria health behavioral due rare feature wherein scale parameterize delay code diagnostic structure distinguish time consider correlated fig network code use experiment let nonnegative encode share modify correlation semi compound laplace minimizing transform diagonal translate encourage paired feature hand go special rewrite regularizer treat relation probabilistic identity equally laplacian encourage prevent toward weight frequently weak without effect flexible extension specific e diabetes detail world propose risk health strong enough recent health service region ed patient period record case record increase year every care item cover aspect abuse service history record
base mutation fusion event interface h commonly tumor fusion copy mutation label leave survey distinct label red green paper major confirm et moreover al namely even isolate justify al al neither et pca ar pathway mutation al ar pathway lastly several propose assignment gene respective test gene region relate machine discover causal build causality model network extensively community researcher dedicate develop powerful tool tool open science derive abundance graphical ill task monotonic one solve devise fully noise particular address biological realistic intend efficiently quantify efficacy material contain convergence moreover variety level size closely behind virtue solely system causality disease experiment molecular interpret rigorously develop causality technology genomic future several robust infer interaction drug well drug comparison metric respectively separate recall highlight considerably slightly high experimentally edge converge almost bic recall left panel show panel performance realistic term recall panel panel panel include realistic recall panel plot size include realistic demonstrate efficacy correctness hypothesis type reject converge medium pruning effective optimization dominate parametric exponential complexity hypothesis multiply total cost determine help avoid nearly hypothesis effort probability encode counting correspond maximum likelihood ml score compute ml require iterate sample local score computation hypothese one node node parent size parent dominate asymptotically one important asymptotic define structure learnable learnable sample negative monotonic number true filter negative show strictly may appropriately type use fact row conditional follow never parent summation refer exploit fact row I reasoning behind summation parent denote parent parent parent n pa pa pa xx ix ix pa pa pa ix pa pa ix pa ix pa ix statistically generate maximize consist bic monotonicity grow rate grow number monotonicity grow score score bic know perturbation skeleton undirected skeleton structure relation parent optimize let optimize structure child correctly parent parent two undirected skeleton orient incorrectly acyclic create parent consist parent contradict consistency edge parent consistent correctly size relation structure remove include true graph still return structure filter convert computer room york ny universit di di mathematical algorithm genomic compute causal logical relation initially leave tumor tumor genomic gene may outlier gray project co occurrence encode datum causal relation bottom encodes causal among graphical outlier often spectrum cancer patient cause average collapse biological model genomic mutation seem occurrence example allow logical background efficient learn observational noise bic learn discrete usually use integer score bic nice mathematical score belong bic insufficient exact structured possible score et consistently modify program relaxation modification ml conditional great modification monotonic probability author mathematical empirically network improve rely priori available fact know limitation learn bic statistically consistent correctly edge monotonicity know behind score likelihood reflect course compute rely explanation conclude convergence portion causal causality distinguishing cause two event must statistical infer appear vice temporal several parent parent present score temporal structure parent negative true must e means mat three minor modification depend constraint main positive row parent event treat define singleton temporal number zero model fix parameter thus temporal causal necessary specify correspondingly low uniquely singleton row work connection exactly indistinguishable analyze structure parent spurious parent score parent create mathematically asymptotically mistake filter negative mat hypothesis filter free publicly bn network combination depth parent filter possible conduct experiment datum ten topology monotonic type sample topology ten optimization bic across realistic sample evaluate measured fraction
polynomial cell decomposition bivariate take produce root proceed way decomposition identical semi polynomial polynomial truth cell free semi define publication improvement summary year great proved refinement refinement could remove sign invariance polynomial invariance present imply formula extend formulae partial cell exist determine truth note symbolic alternative decomposition technology discussion eliminate projection en application project quantify one far quantify project unless successive may big program construct relationship complex categorization binary output main classification refine non linearly decade perceptron give robust regression assignment output supervise technology flexible modelling predict class belong assign new example class example concept map separate function ever instead computation experiment describe software consist two program fit parameter classify sample transform affine divide two far separate hyperplane margin measure margin correct dependent kernel decide note implementation interactive competitive execution partial projection operator conjunction constraint exist order heuristic choose starting eliminate variable occur eliminate small label construct projection polynomial select low degree polynomial label suggest heuristic construct select low root fail act simple expensive explicitly geometry rather complex suitable take order heuristic convention right project introduce opposite heuristic broken convention pick first reverse convention heuristic break convention fairly choose traditional number machine extract restrict reason feasible clearly engineer increasingly outside perform since quantify problem partial technique stop problem free experiment input problem course quantify building use problem availability split validation problem ht pt minus input ex ex go finish input polynomial go go three six variable admissible measure radial automate radial rbf algebraic heuristic rbf feature rbf besides svm correct imbalance set often classification account negative tn negative fp positive negative denominator sum attain grid search optimisation find maximize commonly value varied varied completion kernel give heuristic perform parameter score margin ideal result select heuristic case observe return result practice classifier return instead magnitude classifier use efficacy heuristic exclusive possibility heuristic select learn test heuristic ask heuristic variable yes indicate list cover least definition fail case occur quantify quantify problem heuristic select order many one heuristic heuristic pick win machine succeed pair compare selection selection approximately repeat calculation chance success pick heuristic machine one bad quantify n number quantify learn seem albeit margin performance surprising well never formally measurement machine quantify heuristic pick two pick successful quantify figure significantly heuristic heuristic free quantify choice although wide benefit machine superiority initial relax select dataset restriction block follow availability limitation machine randomly apply problem real overall work would see improvement selection polynomial connect beneficial allow key heuristic heuristic order variable polynomial combine narrow breaking besides order implementation interesting investigate order elimination thesis draw experience superior problem depend superior yield choose random individual algebraic could development heuristic certainly aware know heuristic involve certainly certainly amongst algebra algorithm algebra solve algorithm rarely entire make decision numerical rather primitive algebra encourage symbolic acknowledgement support china useful comment improve ac ac uk david bridge algebraic geometry elimination field choice place infeasible another fitting model measure select heuristic order heuristic geometry implement elimination robot motion programming optimisation field prove special use rational using often choice ordering dramatically affect feasibility class give number
metric build mahalanobis metric supervision encode attribute describe herein multi version mt share intermediate mt carry encode information preserve individual task mt network challenge mt two citation wikipedia articles circle google mt significant database mining problem model jointly compare learn use data task thereby well generalization benefit learn single popularity develop recognition correlate source new wherein similarly correlate scenario important beneficial setting world citation either predictive citation paper since content citation vary different former methodology sense citation pattern latter methodology electrical article classify utilize member enable friend fine specific circle induce subgraph contain social circle circle relate friend informative building suggest leverage correlation task significant attribute entity rich structural encode learn originally mahalanobis metric structure supervision distance attribute inspire task version mahalanobi jointly learn task share task specific information preserve structure thus prediction far mt optimize via stochastic mt world social yu nonparametric text categorization follow et version near neighbor speech recognition wang et help recently et help researcher study learn multiple graph datum various purpose document find embed multiple al jointly area usually apply multi relational great relevance wherein way improve specific second mt essentially learn attribute topological combine feature local share pair nod another improve predict attribute incoming node connectivity sparse modeling power feature lack difficult link snapshot predict link observe nevertheless well attribute naturally relational datum handling heterogeneity propose mt variation differ depend technical preserve derivation mt model represent attribute adjacency linkage information node mahalanobis semidefinite psd matrix define j structure neighbor denote metric mathematically preserve q simplicity represent refer iteration process formulation subject regularizer set constraint distance nod large distance node satisfy exactly slack variable many optimize network hundred allow I subgradient calculate triplet hinge loss triplets subgradient triplets complexity training iteration reach final update article make challenge little marginal drastically area pose word informative statistic detail bag dimensionality learn diagonal far carry cross validation stage article citation receiver area auc entire svm network structure encode existence absence represent score proportional distance simple auc include single train task train test pool capital letter naive pooling train test use learn task common decision boundary final task exploit intermediate sharing use explicitly mahalanobis metric include learn correlation pooling pool together simply st naive share inferior mt network engine direct use use existence prediction predict heavily snapshot dense link target fundamentally unobserved thing base example structure mt exploit improvement observation align intuition relate paper paper violate constraint violate time fig number violate quickly first iteration cm member network force relationship family member college friend friend associate formalize structure google circle friend profile friend entire attribute social assign manner social social circle exploit circle mt jointly circle social largely strong correlation achieve circle include gender job title last bag type adopt start st circle jointly mt circle exponential begin circle result combination use relevant task compare st mt circle inferior wikipedia entirely social social circle slight social circle number case circle mt combination mt consistently social percentage overlap circle overlap union see circle node circle quantitative social circle common node semantic relationship statistic mt show choose circle mt jointly st performance task show mt get st circle simple pooling bar
sf aic bic perform intensive validation aic bic aic aic comparable validation mse bic aic select bic selection exactly simulation recognize aic tight correlation half validation therefore intensive cross reasonable bic time parameter choose graphical correlation analysis x nx rest determine regularize liu collect regression edge drawback cross find fold detect dependency hour regularize aic bic gene database straight forward study positive dependency structure zhang rl size aic use roc curve positive rate fdr false ccc auc fdr auc perform auc band bic especially band addition bic band aic bic well aic bic respectively discovery rate crucial association bioinformatic general auc fdr decrease large bic increase purpose potential cancer source cancer com searching expression http www cancer genome http survival dataset probe common gene gene patient cox construct co expression gene regularize meaningful module patient survival correlation bic second identify htb gene onto http string predict include physical indirect association string expression co occurrence gene fusion neighborhood gene compare together biological association survival interaction predict gene interaction gene large six link col direct gene protein col col remain link col addition gene gene confirm include remain gene indicate degree col col col gene col involve biological go involved pathway protein pathway poor overall survival os induce cancer express recurrent col tumor col associate et al yu contribute may far explore biological clinical propose propose solution base ridge regression guarantee mild establish fix regularize outperform substantial margin regularize accuracy test especially computation intensive consuming fortunately regularize regression directly aic bic appropriate well validation therefore discovery fdr aic fdr sample aic bic discovery candidate costly source gene expression demonstrate pathway efficiently expression rna appendix naturally mathematically eq ideas manuscript general em em reference look identification sparsity j model ann ms gene signature survival chi nonlinear structure lin ss fan li selection ann tu sc sr molecular mechanism cancer implement genome validation cancer microarray patient lee j j human microarray genome lin ratio regression liu stability advance liu free reweighted liu wu selection via journal graphical statistics liu lin machines lp sparse penalty journal penaltie wang ac prediction meta patient cancer schwarz ahead thompson ms beta cancer tumor yu pn md hc lin lt lin contribute cancer zhang v wu base survival reveal signature predict journal chen j cells h oracle zhang elastic net zhang concave ann regularize liu li comprehensive cancer center school public health university california gene high bioinformatics computational engineering appeal essential sparsity measure nice natural expect np resemble propose efficient em natural lasso elastic combination cross aic mild method simulation dimensional aic identify zero pathway topic regularize regression associate small bioinformatic engineering irrelevant prediction selection regression essential include aic schwarz penalty extensively challenge exhaustive aic bic combination computationally infeasible convex relaxation regularize gradient equivalent minimize asymptotically model consistently experimental pose additional equivalence moreover lin regularize regression regularize lin solution classic solution effectively aim scad fan li liu et mc zhang though zhang continuous smooth optimize solve regression effectively deal natural elastic zhang liu wu time regularize aic bic method gene eq parameter nonzero relevant irrelevant coefficient equation reach vector equation optimize derivative wise division x r nr combine together approach liu calculate sd nonzero sd sd validation number indicate mse consuming fortunately lasso identify cross criterion directly pick aic criterion know advantage matrix five fold cross optimal range regularization log intervals j toolbox www com report ccc sf average bias estimate outperform select close structure bias maximal small lasso result never choose necessary estimation though bad predict regularize minimal mse ss ccc sf mse standard sf compare true
subsequently round penalty programming update equip correctness nearly separable restrict parameter runtime reason dimension implementation normalize solve fitting need tune specify grid always recover match perform compare benchmark profile interest dna chemical dna occur specific dna affect gene site measure site row site cell proportion take represent probability simplex recover cancer shift proportion frequently experimentally costly without recover challenging study small cell compose major sample denote f n outline quadratic average indicate qualitative recover cell fix compute column obtain subsequently linearly row r r affine arbitrary iff unique turn observe exist unique b p repeating precede generate b r b conversely equality hold paper factorization constraint follow yield line imply cf subsequently linearly obtain return solve output linear counterpart regard approximate eliminate vector always column place singular select linearly obtain u sketch form noisy suppose follow solve return corollary follow recall paper permutation separable exist mr fulfil canonical conclude rely seminal inclusion moreover suppose write canonical imply contradict fulfilled iff assertion natural ask bernoulli far away whose answer experiment matrix I trial observe except set vertex hand bad tt draw entry bernoulli present present entire concern two probability draw simplex standard vary draw entry half I bernoulli rest projection aa datum potentially h regard comparison report display well theorem theorem section theorem assumption remark de second addition convexity share scheme complicate combinatorial datum despite apparent et algorithm recover factorization use yield compact representation basis element depend application negativity I blind wireless signal inference gene encode signature case absence overlap assignment involve factorization discuss restrictive model present matrix important research fundamentally boolean factorization binary factorization scheme convexity factorization coordinate commonly employ lack guarantee beyond convergence progress regard factorization nmf al nmf non far complicated impose optimization appear computationally dimension obvious hardness contribution provably factorization linearly remain tractable long extend algorithm show superior heuristic uniqueness nmf alternatively draw suggest continue negativity factor usefulness signature submatrix form row wise concatenation affine hull symbol vector identity present uniqueness connect problem follow hypercube entail affine instead combination essential reason order treat differently otherwise drop factorization unchanged e column linearly submatrix affine independence column affine dimension reconstruct obvious solve vertex contain independent dimension check solve remarkably check irrespective vertex provide subsequently obtain return obtain system solve illustration geometry dot area right row crucial compact system substitute instead pool filter yield determine aim find possibly without handle right dominate cost check form nmf permutation matrix raise whether uniqueness fail broad insight question bernoulli question essentially study improvement crucially theory whose positive uniqueness pose may conjecture affine hull vertex exponentially empirical continue boundary cf reduce cf vertex vertex seem impossible cost indicate candidate column coordinate vast discard column coordinate rapidly candidate check substantial portion theoretical extension form I refer number continuous upper bound weight sum contain theorem third reduction check cf vertex condition apply entry yield r reduction weight pick successively row lemma suffice identify evidence continuous derivation tackle view lemma achieve requirement candidate satisfy feasibility program solve g branch solve check impose recover experiment pool could achieve small pool sequel extension handle particular mind noise change e distance positive singular r compute set
variable fmri bold different scientific question term whole brain usually hundred method scale brain regard big novel estimation fmri fmri come perform cognitive make publicly available detail bold due stop fmri focus voxel activation phenotype behavior stop voxel activation study causal activity predict study direction major analysis simple infer probably correlation indirect alternative lack identification indirect connectivity seeks infer directional brain region approach connectivity dynamic causal modeling voxel region criterion include annotation voxel despite attempt scale infer hereafter type probabilistic direct indirect connection variate represent relationship node parsimonious connection inference inverse zero connection fmri interpretable indirect connectivity reasonably region heuristic approach roughly divide accord major penalize full lasso fast polynomial hundred penalize network instance hierarchical bold hundred thousand area g system interact understanding nod unclear structure include leverage scientific finding advance unify conceptual hide share topological structure incorporate smoothness fmri goal voxel alternate simultaneous advantage demonstrate fmri simulate conclude discussion technical plot biological assume node include subscript index collect notation stand norm pm introduce formulation mean subtract variable center usually column observe disjoint variable relate variable independent introduction assume inverse precision represent precision exist challenge storage hundred could small advantage outline interpret functional voxel form commonly estimate hierarchical also topological role group observation eq extract signal fmri incorporate glasso penalize hierarchical ignore scale via objective objective np conditionally via update update conditional minimization assignment effective incorporate minimization glasso conditional conditional minimization eq minimizer conditional glasso minimizer purpose precision glasso exactly algebraic property fast enforce positive small summarize stop iterative update tolerance iteration way rule usually good alternating point close suffer converge optima suffer select meet precision glasso kt alternate parameter exist scientific knowledge employ scientific knowledge negative log converge probability estimate tuning search small parameter control group compare minimal choice choose bic fmri publicly open fmri consist subject kind stop go illustration author employ implement library http uk include slice correction alignment average template smooth half filter preprocesse linear voxel remove motion standardize glm residual retain analysis fmri residual activity voxel comparable unit variance choice roughly match usual result interpretation goal finer yield network vice versa recover study avoid minima assignment random start compute resource evaluation figure extent voxel symmetry though impose coincide classic grouping visualize voxel examine edge voxel color template connection row color show r region supplementary play stop arbitrary voxel cluster voxel contain cluster together locate brain partly perform brain connectivity inferior brain direct whole activation previous group term connection cause furthermore finding distant region area motion visual cluster ica study coherent investigation study possible connection voxel group average template number arbitrary supplementary middle temporal area inferior sc superior area assess simulation iid precision block column normal vector similar fmri repeat time initialize glasso sparsity grid assess receiver operating characteristic roc overall name clearly glasso sensitivity specificity average receiver operate characteristic embed solid glasso blue line specificity bic select circle triangle glasso estimator glasso idea consider result connection moderate use different retain large likelihood simulate coherence across run measure stable equal truth exactly rate run exactly connection motivate problem fmri interpretable
nonnegative way important estimation exist construct show perform select j k satisfy bound two consequence follow first large model allow define estimator ensure mean infinite give avoid allow constraint use least understand estimator estimate resolve value nuisance devise consistently nuisance remove estimate eliminate nuisance define subsequently show display property nonconvex show write relationship parametrization moreover exponential idea lp lift estimator minimizer result partition mapping invertible linear argue objective equivalence define lead mu mm equivalence constraint use counting inequality contribute last give count thus give optimization constraint polynomial mix formulation gp lp significance solve nonconvex polynomial also introduction easy restrict tensor satisfie risk favorable hierarchical contingency table tensor wrong continuous interpret count bregman leibler divergence bregman risk capable estimate expectation oracle partition function risk convex show optimality optimization space lead strongly q apply lower bind key lead equivalence convexity related show equivalent empirical risk sense eq q square risk function empirical noting show risk promise computational turn attention identify risk key interpret dimensional lipschitz loss respect predictor partition necessarily constant supremum easier bound deviation fix respect lipschitz structural class compose interpret still vector combine fix partition depend triangle need focus combine encouraging need rank oppose exist exploit multilinear np base approach need essentially quadratic away np need measurement lastly loss partition necessarily partition express respective parametrization partition ideal partition express ideal unique could use ideal partition strictly partition partition risk property minimum whenever parameter satisfy property possible partition imply almost ready iid make completion typical sample equivalent emphasize assumption observe assumption without loss q strictly entry write outer nonzero useful say parameter directly make instead use easier additionally statistic must characterization result proposition strictly also proposition decomposition converse q measure uniformly despite fact fix tensor condition incoherence inequality incoherence represents couple versus issue incoherence existence tensor condition apparent look quite incoherence tensor specifically write vector whose entry uniformly within sample order tensor construction incoherence condition incoherence tensor jointly belong entry tensor change variable incoherence property hold satisfied variable ideal consistent significantly combinatorial step subsequently indicate partition gap partition least type mistake index constant depend expression partition event low lie constant proof difference assume next occur type error study approximate procedure estimate ideal need approximate partition discrete clear low estimate tensor tradeoff small analytically tradeoff describe validation accurate nest u set pick threshold e leave suppose compute gap error select optimal follow leave cross assume apply term term term use twice return triangle last focus minimize similarly union union combine success cardinality use overfitte thing ensure sufficiently lastly validation slowly sparse structure selection low involve free use exploit equal match move formulation additional number inequality lp lift still low proposition analogous risk exact low partition partition proof highlight key refer pseudo case modify immediately point note soft thresholding tensor rate imply count rather expand lc none pn pn pn tensor estimate unfold along first second unfold nuclear norm slightly nuclear paper variant norm specialized numerical implementation estimator package implementation worth fast norm informally benchmark implementation code nuclear example present synthetic estimation amount pathway author leave tensor measure jointly choose ensure require though unbounded noise early error essentially reweighte version indicate tensor completion log error nuclear nuclear square nuclear partition production combinatorial nature maximize production pathway relate pathway completion pathway either validation respective element datum measure log show identical construct measure fig measure prediction small qualitatively closely value code describe model nuclear norm coefficient design amount threshold partition log nuclear entry specific hard select achieve low cross numerical data pathway low expressive algebraic definition element inclusion face hierarchical table extend partition complex give estimator directly similar measure risk result measure structure belong sub partition partition test limited instance complex valid extend principal pca consist matrix successively termination criterion additional assumption decomposition approximation tensor hard allow approximation tensor numerical mix perform completion tensor completion identify work challenge towards method tensor tensor negative result semidefinite challenge define generalization correction step positive tensor may study broad question predictor require specific predictor linear orthogonality poor conditioning sharp regard ensure study independent sum distribution distribution acknowledgement author thank lee corollary supported award interpret noisy tensor exist convert completion unable possible tensor parametrize linear amenable choice function thresholding estimate unable exploit structure hard computationally rank completion datum refer model purely predictor different th response specify possible define predictor belong exponentially slow surprising combinatorial curse try estimate value approach combinatorial convert numerical difficulty restrictive impact completely combinatorial principal define combinatorial model interpretation index combinatorial completion possible extension hierarchical rely upon generalize discuss
competitive study subsequently detail scalable thompson appeal thompson bandit pay online action specific ad multiple ad website regard ad uncertain payoff ad pay exploitation present ad increase likely overall exploitation balance course interaction formally choose reward policy reward bandit substantial theoretical thompson thompson bandit asymptotically thompson bandit competitive basic thompson select thompson sampling within past reward reward model parametric parameter action necessary suffice draw action high thompson practically scalable large complex thompson computationally might logit use thus require property scalability thompson limit thompson compute thompson form misspecification address robustness thompson thompson replace appeal randomly resample conduct replicate bootstrap replicate numerical replicate bootstrappe bootstrap distribution bayesian use bootstrap posterior robust misspecification prefer remainder simple competitive thompson also discuss subsequently analyze computational misspecification commonly bandit bandit action select time arm straightforward set round obtain beta posterior ir beta beta play arm illustrate present bootstrap true increase number observation take center true row theoretical thompson armed bandit start initial replicate decide arm play arm update bootstrap replicate thompson sampling replicate uniform bootstrap replicate retrieve arm j replicate could exploit large costly allow exploration armed bandit simulation arm reward arm examine empirical thompson replicate simulation greedy comparison new replicate construct closely comparable thompson thompson empirical regret replicate thompson comparison thompson examine replicate cumulative regret arm easily separable albeit replicate practical small replicate suffice become thompson tune distribution analytical show bernoulli highlight however easily example motivate partly motivation generalization bandit triplet assign become reward would general would resort computationally costly produce give thompson use conjugacy relationship formulation getting update contextual thompson scalable alternative besides kind misspecification liu examine setup simulation thompson error coefficient error include mean incorrect datum factor thus configuration denote vary create optimal arm arm thompson sampling thompson thompson summation ridge penalty compute select replicate select random play arm maximize simulation bootstrap examine full ignore flexible present reward thompson vary degree relatively misspecification significantly cumulative produce large difference thompson confidence interval alternative substitute idea thompson optimize exploitation competitive play thompson substitute double nothing point online parameter perspective matter policy bandit observation series effect analysis
determine period fisher minibatch gpu note actually store compute equation despite sometimes minibatch update factored fisher otherwise two use differ update product ti ti ti enforce minibatch fisher operation compute minibatch place otherwise eq gpu transfer derive quantity diagonal point decomposition compute compute scaling factor implementation save let later compute multiply gpu working factor diagonal exceed orthogonality aspect neural net parallel include text discuss implementation store disk enforce parameter minibatch generalized model explain introduce initialize dnn implement discuss adaptation machine sgd one gpu computation minibatch example minibatch job typically advantage core processor share refer asynchronous prevent relatively gradient minibatch averaging ensure make progress regardless minibatch minibatch limit minibatch understand follow minibatch size minibatch size minibatch sgd become gpu fast cpu aside give result hard access order access access try sequential read file neural sequential pre randomized disk temporal randomization order probably view sgd amount access break training block per epoch process per ensure randomized give nothing particularly disk disk compression prevent instability divergence change minibatch explain completeness single formulate thing minibatch multiplied enforce implement involve store product norm penalty enforce minibatch range exceed sum right eq minibatch tend initialization discriminative layer backpropagation publish reference something hour less dataset eventually become impractical beyond scope instead layer mean hide train report remove add hide short repeat hidden fan softmax layer find essential discard add prescribe versus layer bp another possibly language otherwise notice outer iteration immediately outer average parallel averaging compute train extra meaning collective objective function make whole entropy fix viterbi alignment easily frame speech term train dnn speech maximum objective properly variant inspire minimum minimum phone expectation compute derivative posterior parallel standard ng generally minimum phone procedure net setup exist setup item training describe epoch machine average lattice piece part lattice derivative order ensure modify ensure change geometric fix rate mention generate consist lattice something connection scale parameter combination output advance rate randomly training frame minibatch ng sgd apply sometimes continuously necessary neural e report mean adaptation know constrain gmm system although online thing gmm decode complex ideal order turn addition range characteristic gaussian mean parameter supervision audio case dimension vector train switch I current take input normally consist plus frame normalization energy energy transform transform feature order match statistic include per generally give train frame consist process prefer application convenience cross transfer framework speech amount gpu hardware agnostic way parameter machine method efficient ng seem improve training neural combination parallelism parallelism parallelism minibatch net parallelism sgd speech work combine ng attempt explain parameter ng empirically significance speedup neural speech introduce behind gradient discuss parallelism ng sgd background dnn classify vector time acoustic cluster per duration contain several ultimately top log cost viterbi likely probability course slight supervision label viterbi alignment word aspect machine randomized subset gradually machine across job epoch job outer epoch training learn individual rate proportional rate stay sgd job get average summing concern hessian reach close equilibrium opposite time away relevant sgd namely schedule less relevant issue appendix cpu gpu randomization generalize decode speech recognition decrease thing training mention report start end specify epoch range epoch tune ng sgd ng extensive schedule tune circumstance ng helpful experiment common parameter getting modify enforce maximum parameter minibatch tend early gradient positive matrix fisher speak riemannian surface path conventional compute inverse follow rate scalar decrease something g training sample minibatch replace instead proof keep eigenvalue rate matrix prevent systematically particular clearly idea learn oppose suppose continuous fisher outer log justification fisher hessian hessian gradient direction datum fisher hessian transform parameterization easy fisher qx qx still generally quantity analogous hessian change matrix speech recognition million inversion fisher impractical deal factored form approximate rank block explore per material analytically neural form kronecker natural accept newton inverse factor fisher weight fisher weight consider th block separate include bias term kronecker symmetric positive definite row whose plus multiple approximated factorize kronecker weight row ever kronecker show factor way factored minibatch hold surprisingly online distant significantly fast probably help kronecker training weight matrix act quantity modify factor matrix multiply fisher form processing training derivative minibatch gradient minibatch easy minibatch eq indicate modify quantity natural eq matrix minibatch separate term interface natural gradient minibatch minibatch fisher hold multiplication return minibatch ti ti ti gradient interface minibatch twice want prevent early huge inverse fisher technique rate un fisher slight proof scalar view practical minibatch ng estimate invert minibatch full rank strictly necessary contain multiple unit find follow quantity fisher use stop ever exactly relatively large suitable wide circumstance simple big minibatch size interpretation fairly direction quantity direction gaussian hard fisher correspond replace course practical believe change make fisher try co transform fisher around motivation work allocate effort regard factor think easily provable converge factor use together rescale rescale multiply inverse fisher fisher randomly identity quite true due minibatch easy rescale minibatch back instead sense objective minibatch thing natural describe estimate factor involve multiply weighted covariance current minibatch factored method probably analysis initially steady state minibatch equal confident stable give effort converge minibatch method would kind something modification sgd experiment frequently sgd momentum effective parameter reason momentum quite successfully momentum cpu method likely momentum prevent instability limit per layer minibatch another popular modification sgd parameter schedule theory reason unlikely speech firstly inferior decay learn believe true use essentially direction concern interesting type diagonal diagonal speech recognition english english quick hour half hold hour long cc dnn dnn word rate side difficulty quick method use gmm process space normal build state gmm dnn adapt gmm across context input dnn dnn dnn hide layer nonlinearity reduce explanation number train parallel ng dnn dnn online decode audio datum process input equivalent un normalize frame dimensional include show include compute intend decode cpu million sgd epoch exponentially hardware fairly r core ghz gpu single notation becomes locate gpu reporting take slightly optimistic figure fast figure color plot versus process final proportional learning job natural help ng plain sgd using show amount job slow get speed take epoch get epoch get speedup show simple job word ng plot time simulate multiplying take outer outer depend queue load outer plain sgd ng sgd second ng sgd circle mark c plain ng sgd ng describe gradient ng sgd experimentally versus plain possible parallelization across sgd enable train parallel even one confident experience true relu sigmoid activation final rate speed except prevent parameter training acknowledgement neural code kernel numerous mention aspect setup contract contract acknowledge grant award speech cloud development reproduce purpose annotation conclusion contain herein interpret represent express imply element minibatch interface element minibatch inverse fisher row core inverse multiplication row minibatch smoothing compute gradient compute efficiently setting column respectively smoothed matrix row result rescale intend sure denominator scalar hold rule randomly believe contamination bias turn modify properly sample purpose believe efficiently smoothed holding differ hold row version expand simplify minibatch great formula derive correction correction row differ scalar factor use scale work minibatch g overall large typically considerably overall inversion compute gpu online interface simple multiply fisher rescale correspond minibatch row simple run minibatch minibatch copy twice matrix net input minibatch column update subscript minibatch row online estimate normally compute ensure denominator rank approximation row introduce specify describe make top slow inspired compute think eigenvector scale precise sense actually square root eigenvalue put diagonal put eigenvector add reader seem straightforward decomposition way speed symmetric diagonal definite reduce orthonormal unknown desire covariance ensure dimension correspond inner order ensure f tr work zero representation describe computing multiplication typically expression implement equation scalar whole expression identity part store factor write involve factorization go appear convenience expand
f bx decrease conclude intersection increase bx b bx bx bb bf bx bb b bx bf x get q accord intermediate b dx bx f bb bx get intermediate exist g b f b bx dx bx complete give restrict bb point intersection f bx two x dx bx bx b bf x bx bb bx f get exist bx bf bx bx bf bx f bx bb bf bx bx eq g c bx f bx bf bx bb b gb intersection gb monotone solution give eq prove point start converge prove exist assumption easy nonnegative bb g continuous concave proximal gradient q sequence monotonically decrease stationary point since lead exist subsequence subdifferential exist engineering school technology science key laboratory school university edu sg com edu sg edu cn study singular thresholding nonconvex obtained denote bound nonconvex surrogate solver generalize singular basic subroutine low solve rank attention application background achieve use suboptimal loose approximation bring attention surrogate smoothly scad concave penalty mcp show nonconvex usually scad structure extension sparsity nuclear however suffer nonconvex different minimization reweighte nuclear minimization q singular continuous concave nonconvex continuous objective surrogate construct simultaneously guarantee decrease surrogate quite loose minimizing possible tight surrogate relax name method later operator associate nuclear know value follow perform gx gx x singular thresholding nonconvex still operator open whether monotone nonconvex simply perform singular unique need I p challenging solver nonconvex worth nonconvex none proof rigorous ignore prove monotone detailed work general solution correct exist reason behind hold optimal give rigorous bound compute type special show general solver reweighte nuclear synthesis experiment solve equivalent von simultaneously optimal denote von trace equality hold decomposition reduce eq conditionally share thresholding associate monotone optimality monotonicity nonconvex special choice rigorous since monotone prove find intersection bx popular denote x corollary minimum objective lead solve solve give find local minimum start search fix iteration nonsmooth nonconvex local candidate nonconvex surrogate scad shrinkage effect shrinkage operator proximal nonconvex nearly unbiased norm proximal norm norm necessity nonconvex singular function get solver name proximal nonconvex compute convex method sequence property point expect decrease since tighter verify experiment conduct note test logarithm suggest compare nonconvex enhance logarithm set dynamically decrease conduct two miss evaluate regard repeat lagrange multipli alm plot outperform alm nonconvex logarithm approximate datum outperform surrogate loose c plot curve fast dominate pixel matrix completion blue recovery recover achieve large relative collaborative filter preference similar movie movie movie entry normalize absolute
form augment multiplier parameter equal size scheme subproblem value depend proper reformulate speed calculation qr formulate orthogonal although optimal iterative scheme equivalent highly modern introduce definition whose understand wise r mm minimization close norm rewrite optimality condition give update soft thresholding lr subproblems respect respect formulate subproblem gradient admm scheme essentially gauss admm compute particularly attractive accelerate adaptively change iteratively algorithm easily directly moreover adjust k solve complex operator outline optimize augment identity update norm similarly converge k f stop rbf employ criterion stop current satisfied tolerance u v dd cause overfitte rank fortunately work rank relatively addition provide dynamically adjust parameter scheme I rank detect eigenvalue usually specifically v satisfied big jump adjustment present relationship k k k tm u u k lagrange meanwhile replace computation constraint algorithm guarantee theorem generate ks cauchy proof find feasibility optimality solution stop k addition multipli e k conclusion v uv mn feasible uv please theorems f objective generate algorithm mn f parameter mn value arbitrarily hence rbf main run svd qr multiplication qr multiplication rbf problem usually iteration outline methodology consider space describe try auxiliary completion write admm scheme completion problem solve effectiveness removal background collaborative run ghz pc windows gb conduct task remove generate image image text form outlier mask time true pixel bf alm rbf tolerance detection area characteristic curve auc image bf outperform visually detection significantly perform short bf alm rank recovery moreover run bf alm conduct component recovery rbf chosen grid bf alm achieve auc run fig rank rbf test rbf surveillance detection background modeling segmentation surveillance video video satisfy background frame control hence exhibit low property foreground spatially sparse rbf surveillance video bootstrap consist frame video first column collect background bootstrap input extract rbf experimental database run recover slightly see rbf times show rbf address large ht ccc bootstrap ht reconstruction incomplete face often decompose capture illumination image randomly miss entry reconstruct people present rbf perform visually complete miss word rbf regardless corrupt implement incomplete burden projection raw pixel randomly respectively clear rbf successfully collaborative filter technique predict user evaluate completion experiment conduct widely recommendation k rating randomly test testing experimental report compare rbf soft two manifold optimization root rmse define ij rating denote predict rating rmse three report independent see fix except norm moreover bilinear factorization method trace consistently factorization minimization soft bilinear ccc ccc c rank regularization rbf robust method vary regularization increase become increase slightly confirm bilinear factorization respectively formulation robust rbf side rbf unlike low address scale matrix incomplete corrupted real superior l solution svd accord uv know solution l find uv sd accord lemma uv uv proof mathematical induction q svd algorithm verify v kp nu k ks kp k k complete appendix sketch multiplier boundedness endow subgradient k sequences respect v furthermore u k procedure u k k k I next subproblem optimality lemma complement satisfy ty k ty ty k bound boundedness u k k k k feasible cauchy sequence lemma give obeys qx calculate calculate k k k k k k convexity calculate uv u uv u uv uv uv u uv uv uv lead uv uv uv mn worth nothing globally v g mn please matrix g g u u mn mn feasible naturally follow svd v feasible mn l singular singular resp singular small calculate mn f f theorem theorem rank incomplete corrupt statistic bioinformatic well solved relaxation norm certain applicability paper scalable provable rank miss corrupt incomplete corrupted compressive specifically trace bilinear structured apply alternate multiplier admm finally provide compressive pursuit rank recent recover receive broad different bioinformatics area bring great challenge analysis digital surveillance video text web fortunately intrinsic ambient pca popular tool face pca contaminate outlier miss set address compressive rank base rank filtering recommender miss entry arbitrary measurement collaborative motion click tag recover corrupted image corrupt classical address issue sensitive outlier outlier extensively semantic indexing video surveillance simultaneously recover globally involve suffer effort towards svd svd provable bilinear factorization corrupt
rank consistency separability yes provide separability nd order ex model share motivation extensively last decade yield review ref dominant trend heuristic mcmc demonstrate problem provably structural form rating method considerable ref ranking preference explore combine topic review star scope key summarize dataset formalize universe item share across population user pair specific weight ranking generative comparison user token n htb generative user weight ranking convenience nonnegative ranking row order kk dimensional column finally matrix item principal algorithmic ranking component follow ex similarly probabilistic word draw matrix topic prior ref form ex distinct vocabulary stochastic propose statistically topic prior note matrix infer directly solve approach recent work topic efficiency exploit moment occurrence comparison establish particular establish splitting row make ex item geometry solid approach geometric define illustrated fig separability rank separable order rank least item uniquely rank ranking separable order separability identify real world modeling ranking appear albeit implicitly seminal separability sample uniformly estimate factorization separable hull identify find novel identify estimate exclude ranking rank leverage solid angle novel pair angle indicate distribute direction topic solid use detect separable full rank motivate angle select pair large estimate estimate modeling ranking infer preference predict new see ex outline expand detail algorithm novel identify constrain regression follow column estimate round element binary satisfy jk htb projection matrix ranking projection ranking I ks k ranking I I b k j topic derive approach upper technical hold isotropic alg result specific gaussian separable user furthermore draw parameter b solid angle extreme hull proof material alg alg order compare ex synthetic satisfied demonstrate variability collaborative filtering application heterogeneity well ref movie rate widely public availability comparison focus ranking viewpoint literature ref ref ranking ground adopt ranking use hold likelihood topic consider optimize root rmse ref netflix ref specifically projection tolerance alg alg ex ex semi dataset validate match dimensionality characteristic world comparison benchmark movie star rating million rating user star follow rate factor star art filtering task movie sort score column movie matrix obtain ranking set suggest separability model follow prior dirichlet distribution comparison comparison align column base truth due ranking pair normalize error rp propose estimate recent consistency guarantee vary truth ranking rp ex rp world setting evaluate rating star focus rate obtain star user rating pair user rate star movie comparison movie rate randomly select movie rate movie rate user star rating tie ignore select set rating use training split convert rating testing testing independently log e test train figure result pairwise comparison fix total test high likelihood validate htb htb ex user original dataset training hold use gibb compare rp number comparison fix ranking summarize agree task ex ex behavior prediction recommendation train model comparison comparison objective comparable art rating achieve rating rate convert training convert ignore dirichlet predict predict rating movie rate rate movie star predict rating movie maximize square rmse rating factorization factorization pmf factor latent factor similar pmf rp come perspective rmse rating pmf rating integer real prediction near integer observe rp algorithm pmf fitting match demonstrate behavior design well generate strategy ranking aggregation distribute database web statistical efficiency centralize retain communication upon nsf award fa view conclusion contain interpret necessarily policy imply material largely track methodology new ref handle type setting complexity analyse algorithm account valid rank guarantee distinct note present angle extend direction scope column stochastic noting detail view special prior pmf user type user first indistinguishable comparison main fair appendix single main result almost generic establish individual convergence constant ready splitting comparison large j I apply proposition optimize set union claim distribution tie special isotropic nature ref special type spherical consistently identify ranking give success consistently novel novel prove follow th follow matrix rank distant lemma similarly q span proposition statistic latter algorithm j ranking asymptotically main detect pair start straightforward separable solid novel q solid angle alg vanish iid direction draw novel let intermediate convergence solid novel j norm j j implication ip consistency identify distinct ranking pe p w mn distribute first decompose union sort ki accord pe p proposition two term therefore pe consistently rank success loss generality ranking constrain row k surely row second rest j k round put remain post consistently moreover separable recover column furthermore eq fail normalize solid angle hull combine alg lead complexity
weight bit possibly example bit look index simply bit interaction bit recurrent current parametrization require actual multiplication next aim generalization scheme obvious regularization success smoothed gram maintain index sequence update example require index make correction simply automatically often activate activate regularizer towards examine achieve structure bit thereby return empty return unit understand binary tree weight leave use sum node involve per reasonable overhead multiply add number vector serve help regularization degree freedom computation increase rapidly grow much weight regularize keep interval weight lose decay coefficient correspond multiplying towards issue decision assignment decision decision may prop usual ignore unit training case accord long job space exponentially store hypothesis evaluate provide training gradient heuristic obtain back loss noisy sum select weight index unit active contribution control create outside length one great challenge expand applicability ability require future validate valuable language dataset universit art deep large computation available able exploit large improve generalization decision favorable computation deep increase neural novel parametrization exponentially parameter weight matrix bit pattern activation parametrization tree sign unit hide deep learning abstract supervise unsupervised review application computer possible feasible report experiment factor recent availability net handwritten digits traffic sign face point achieve human far general scene understand speech could correspondingly correspondingly cover category modality concept current still multiply much favorable ratio select pool unfortunately prevent like technique rely generalize way region mathematical cover svms theoretical suggest deep advantageous characteristic learnable computation well aim deep computational efficiency computation objective mind way discriminative ratio exponentially computation
every iteration equation policy turn description conservative discount stepsize return successive greedy operator way step policy distribution article stop continue stop derive analysis latter mention variation approximate step eq simple implement require explain trajectory start stop require underlie mdp implement step new evolve slowly natural policy originally problem variation infinite stationary policy make action r horizon begin build sequence non iteratively policy return approximate horizon stop step one may return start policy good horizon policy practice horizon policy loop shall algorithmic may prohibitive aim next present article describe issue simplify variation grow approximate greedy value loop infinitely kk consider suffer drawback interestingly another parameter policy store similarly stationary set policy shall run formally periodic infinite stationary loop iteration respect compound operator forward eq process horizon form forget policy keep step stop loop infinite stop guarantee algorithm done see set consider go v policy policy order introduce relate wants guarantee fine lack space set deterministic policy p ci coefficient finally small c thorough bound thing dependence discount quality match parent good comparative hierarchy direct implication important mean parent finite infinite guarantee make picture complete mdp though coefficient derivation done guarantee analysis though require scale relaxed improve still slow technique sufficiently enjoy rate express guarantee extra nice hold respect instead theoretically mdps give estimate grey region dark mdps error mdps deviation mdps error ease comparison display practical store proportional may require memory explain make nice guarantee performance identical control increasing confirm well becomes discuss baseline slow progress take million stop except make assess variation differ addition compute problem randomly finite mdps correspond application remain kind mdp encounter brief branch greedy exact noisy mdps run compute e display variable display variability supplementary series much slow naive conservative bad overall provide bridge close relative tend vanish dynamic branching lot difference side fact use k two bound write multiply side observe start assume simplicity get back equation conservative address deferred supplementary cd ic reasoning rewrite follow use begin prove result v matrix ir subset multiply definition show nice fast get corollary small satisfie turn straightforwardly precise greedy show ht top middle pdf middle experiment parameterize branching uniform give reward uniformly sample feature factor
mdp discrete parameterized receive initial reach let trajectory mdp trajectory assumption part quantity trajectory maximize ascent work risk spirit policy detail transition note simulate trajectory together correspond single trajectory realization variable estimate use eq update maker gradient criterion policy necessarily augmentation current accumulate simulation policy still sensible h policy satisfy h kk rl take covered paper add negligible vs policy return return leave tail final parameter encourage take return examine rl benchmark study extensively technique among gradient modify policy expect line game performance versus different characterize behavior well modify emphasize whether extend handle currently standard shape induce sensitive game score line addition limited step modification risk policy induce tradeoff line less frequent batch policy standard warm policy reward average return rx I algorithm converge return return mc high final observe low policy decision maker contain finance understand difference cell controller weight controller may version describe converge full supplementary lr style descent optimum domain beyond reach motivate simulation acknowledgment helpful discussion research european european agreement n finite therefore well know empirical write follow empirical bound integrable q observe thus crucial proposition gradient term even direction quantile problem importance general reduce variance monte estimate estimator wish random mc sampling dominate r ix heart parameterize aim f typically sample gx gx gx estimate sensitivity recall addition outline proposition since know advance eq empirical sample estimator var q need modify scalar assume estimation I suitable reward nr li monte suitable update parameterize obtain exponential rl select deal scheme help reduce estimator difficulty find modify trajectory modify mdp however require simulator modify mdp modify rl set mdp specify give yx mdp heuristic rl trajectory weight bad outcome difficulty define bad note reward action fundamental elegant selection term outcome current policy know greedy selection rule behind high preferred encouraging transition produce bad encourage value trajectory obtain policy extensively rl literature many task td result mention choose naive sort choose use trajectory policy significantly due estimation return assumption ac il ac il risk prominent various domain new form propose gradient spirit estimator local domain reinforcement learn controller extensive finance among payoff define optimization find structure deterministic solve various payoff generally asset many resource allocation suitable case recently optimization derive expectation stochastic analyze allow domain reinforcement sensitive beyond reach consider interpret sensible remark often maximize extend problem straightforward method another incorporate importance important capture lr gradient payoff lr financial rl commonly method extend lr perturbation style estimator formulae lr utility event risk interest return sensitive scale accumulate suitable horizon problem function mention work rate decrease determined single formula version scale stochastic time algorithm subsequent random c convenience everywhere var denote outcome change express calculate analytically express expectation convenience make smoothness gradient note technical detail standard lr lr gradient expectation eq rule z z finally trick multiply inside integral justified obtain lr formula see proof account baseline turn elegant portfolio rl order use gradient need sensitivity performance complicate variable rl system stochastic dynamical calculate probability usually sensitivity trajectory typically trajectory shall generalize rl let support value countable reward formula variable disjoint close l sensitivity proof spirit difficulty apply future next effective sensitivity describe reward yy r proof supplementary r bias continuous assumption satisfy supplementary separate bias use quantile may lr work
efficiency solver second example severe main capture capture meaningful social network theorem jj jj dt dt fan jj jj check directly classical multidimensional work distance dimensional euclidean distance advance unfold semi definite sdp representation sdp capable produce numerically suffer drawback slow datum euclidean establish asymptotic produce uniform sample logarithmic explain develop inexact accelerate proximal configuration cope distance optimization rank purpose euclidean representation high model strongly inspire several sdp achieve suffer drawback distance limit practical resolve theoretical uniform model social network solve accelerate proximal distance briefly discuss relevant bind subsection serve contribution notation social tie actor analysis end actor actor relationship role etc observe relational composition structural social e presence absence similarity actor concerned relationship relationship actor actor way euclidean distance see information matrix often vertice bernoulli obtain pair feature bernoulli random therein mainly bernoulli model measurement social network produce preserve highlight image located strength actor tie preserve reduction fit distance social space reduce low classical scaling section provide one often embed produce satisfactory profile dimension dissimilarity matrix close embed embed euclidean space create embed motivated number try propose use distance manifold unfold maximize variance minimum embedding aim eigen gap distance either sdp enjoy elegant short distance accurately geodesic convex follow density near ball exist guarantee obtain distance highlight short distance distance derive social mail social network depend obtain rely sdp limitation inspire euclidean attract group relate natural interpretation tool excellent seem put category attract community recover via sdp approach guarantee probability theoretical property rip matrix completion sample satisfy rip prove rank recover noiseless observation near adapt short recently group researcher include setting propose estimator norm establish sdp sensor localization incomplete distance positive define matrix aim minimize nuclear equivalently objective minimize embed assume center obviously contradict possible seem bind read undesirable ball embed noise point theoretical result difficulty face matrix additional error derive euclidean nuclear equivalent embed contradict variance possible hence excellent straightforwardly useful learn dimensional may huge difficulty extension distance space theoretically guarantee spirit theoretical briefly describe contribution major reduction building bound building point sdp vs semidefinite sdp primary nonconvex driving sense three first distance minimization argue contradict maximize term third axis approximate embedding analysis accommodate initial valuable available embed illustrate situation lead part combine make guarantee mild control estimator roughly freedom symmetric choice reduce subproblem explain lead treat object benefit lead difficulty difficultie hundred slack explain use cpu contrary fast inexact proximal thousand method advance develop develop theoretical good problem manifold learn embedding provide cast score viewpoint benefit model detailed interpretation report list inexact accelerate extensive experiment paper real trace symmetric positive cone vector vector diagonal orthogonal remove column obtain product entry single sum value contain three part brief review key go describe explain work summarize key embed small definition embed well center known center semidefinite write eigenvalue submatrix q upon distance point first eigenvalue absolute mis eigenvalue satisfactory embed e aim score use interpret lead justification score root show matrix center role orthogonal projection onto geometric onto onto result distance point geometrically center remove embed gram encourage reduction maximize slack trade maximize preserve satisfie empirical total seek improve try eigen eigenvalue rise eq deal correspond interested slack model sdp rule element identically nn distance preserve embed point employ matrix encourage score eigenvalue remain good result moreover deriving solution follow interpretation facilitate description platform subsequent sampling basis sample adjoint vector assume jj estimator always embed give orthogonal write learning use deriving model task task try nothing quadratic slack essentially embed come spectral term small dimension argue minimize principal maximize third initial span coincide term maximize optimization third force control controlling penalty heuristic extensive experiment model sdp subproblem combine sdp keep update solve sdp subproblem therefore justify go bind summarize rather sdp exist use notation fourth positive semidefinite orthogonal subspace q yield bernstein matrix spectral norm mean exist ready following result major bind second nuclear random sampling strong convexity suppose least nc establish explicit depend sampling tail bernstein magnitude magnitude nc q minimization choice fact choice subsection choice major message usually therefore rank rank model tailor estimate serve parameter particular minimization part propose inexact accelerate proximal model easy inequality minimum operator order extensively correspond come diag ap scalar twice continuously twice continuously jj jj nearly estimation error nuclear without loss consider convex quadratic q cone ji equivalent way experiment inexact proximal study sdp suitable fx ax ax start iteration eq fy x kx adjoint major solution fact want form problem compute near type efficiently third approximately meet criterion formulation inexact fista inexact similar omit save demonstrate effectiveness real visualization model graph manifold physical capable generating configuration quality extract physical raise also sdp solver allow test issue subsection four sn communication social communication th fix mail social distance dissimilaritie user communication count imply social employ measure social email network facebook social top actually much high dimensional much explain performance train visualize social individual train connection record actually place indicate circle person place embedding capture lead eigenvector demonstrate sn visualize consideration social distance measure dissimilarity figure mean top top two lead eigenvector capture sn link political around part use visualize social communication widely use social generality isolate remain concentrate near zero correspond gram high however capture two leave blue circle set initial nn describe different depth near neighbor accurately gram matrix indicate capture first mention return eigenvalue lead
angle stimulus part explain dependent interaction subtract term marginalization g dimensional manuscript activity vary stimulus neural activity vary interaction stimulus general decomposition material separate period separate stimulus decision time period trial reasonably period aim separate x st st st st interval stimulus interaction component figure angle axis orthogonal mean neuron component splitting component largely figure delay period two water without period hand interpret stimulus active period clear highlight stimulus fire capture come cell tend dot axis dot absolute due ease axis decode stimulus interaction stimulus line figure time period significant carlo hold one trial neuron condition trial remain decode axis stimulus stimulus stimulus stimulus value time project trial decode close trial classify result accuracy iteration procedure reality trial classify pseudo pool different accuracy accuracie iteration neuron number randomly assign cross apply classification stimulus result chance actual decode accuracy consecutive bin figure period accuracy dataset monte computation hour intel processor centre sup paris forest university school nc usa laboratory ny de usa neurons tune variety tuning introduce dimensionality technique component analysis automatically highlight essential complex population area population decompose component capture highlight dynamic reward etc activity behavioral activity hundred technique common study external action internal conclusion neuron hundred neuron complexity pose fundamental challenge severe area neural response heterogeneity traditionally heterogeneity ignore cell criterion stimulus fire pre activity fmri area ignore higher simultaneously therefore display mixed look single neural analyze use reduction specifically datum account nature train dynamical population response dimensionality parameter control account mixed interpretation solve inform adopt e fire task parameter help sort mixed activity dimensionality latent component easily interpretable control task preserve ensure valuable away principal unnecessary orthogonality recently fire focus guarantee neural activity linearly individual spike train obtain remarkably complex population similarity difference pca largely ignore activity relate activity task extract orthogonal shift around space component fire colour stimulus decision population fire fire trial experimental trajectory indicate dot size visual clarity analysis pca firing rate neuron decoder principal second reconstruct fire principal decoder encoder grey neuron proportion variance stimulus component interaction clarity behind first axis long correspond axis case analysis fire compress two reconstruction enforce two stimulus colour dot decode axis axis encode axis expand reconstruct standard stimulus fire neuron fire stimulus trial trial fire also decision stimulus trial joint trace fire clarity fire population neuron moment reduce dataset pca pca fire rate neuron linearly transform principal neuron interpret population decoder compression latent compress another reconstruct geometrically cloud axis maximize variance projection axis decoder fire task condition find encoder eq like component stimulus information enter new additional constraint optimally compression achieve mapping pca axis orthogonal intuition b label stimulus decode preserve stimulus lose vice versa stimulus decode axis projection decode axis project de axes axis pseudo novel detailed briefly toy matrix stimulus part separate decoder encoder stimulus case paradigm adapt principal component stimulus component third row last row line legend thick interval respective reliably trial activity vertical almost bar variance compose four stack bar different colour stimulus variance stimulus bar colored chart split triangle dot star mark non leave task require discriminate separate neuron method fire stimulus stimulus frequency pair decision trial always trial neuron distinct many mixed pattern analyze period period neural activity across single neural activity activity stimulus row depend row three category stress method quantification activity component principle retrieve system weight neuron fire non activity component amount chart capture trial irrespective independent show cell delay period activity activity full trial activity complex varied indeed spread capture previously persistence period dynamic delay activity due slight variation fourth stimulus monotonic three period period period period stimulus shift fire stimulus encode single stimulus component short slide stimulus remain several technical overall c capture impose activity note nonetheless notable zero represent signal work work memory centre location figure delay second square appear match delay target appear report format figure paradigm adapt principal line correspond condition legend colour correspond stimulus stimulus stimulus line colour correspond match explain individual principal split dot product first axis leave first analyze task proceeding time fire study eliminate trivial eight preferred prefer analyze trial fall stimulus stimulus interaction similarity work first separate easily chart stimulus rotation stimulus decision fire activity stimulus identical task case condition independent stimulus component period stimulus activity delay period component period one notable figure presence interaction stimulus match trial opposite stimulus capture seem existence second delay work memory allocate stimulus whereas stimulus component overall surprisingly summarize obtain decision tuning non achieve validate art classification report recorded activity start exactly display required pre stimulus interaction come one memory figure specific activity extensive discrimination behavioral crucial storage stimulus trial uniquely water locate water choose water paradigm adapt legend four variance chart right show dot product pair axis show correlation activity neuron leave decision trial two self trial align across fire rate trial see lead neuron exhibit fire mixed activity stimulus thick thin bottom neuron tune absence similarity large part fall first component distinct mean respective shift firing pattern neural across epoch agree finding study far prominent component absence decision come reflect movement position stimulus prominent nevertheless clear stimulus demonstrate even activity neuron fire could categorization use pure vary trial different figure choice otherwise furthermore reward format adapt component legend show thick line pure blue period cumulative principal chart dot triangle principal corresponding mixture presentation mistake exclude datum combination present additional component present b component correspond prominent somewhat throughout stimulus especially already period separate incorrect stimulus characteristic correct trial agree predict corresponding confidence summary point pick population population neuron correspond encoder component neuron strongly unimodal distinct neuron component neuron component confirm apply population neuron one discussion response researcher resolve summarize extract latent component cell critical pca fa greatly activity report compare neuron conventional wide spread count response percentage use original neuron show response monotonic tuning delay percentage neuron increase fire period increase tuned match match difference significantly tune neuron level conclusion qualitatively perfectly sound limitation conventional focus bin firing rate tuning highlight neural frequency work report significant tuning tuning curve dependent component curve tuning imagine neuron choose cutoff g false neuron distinct stimulus dataset component albeit vary strength analyze test unbalanced experimental analyze version fire window role first limitation limitation remain many vary firing simply neuron interest highlight full heterogeneity averaged thereby fail neural false response introduce perform fire interest axis purpose neural tuning achieve deal extension replace non fire approach classifier population fire rate cross validate accuracy stimulus match condition work analyze classification shape curve match match true important removed firing reduction instance lda dimensionality account whereas look label lda look projection maximize separation within related concerned reconstruction ill purpose material extend four interesting outcome concern strength fire rate likely overall firing task period influence instance body visualize period method stimulus component active period stimulus figure possibility separate component task trial encoding pair mark star figure though unlike pca enforce orthogonality orthogonality neuron tend one example orthogonal fall orthogonality dependent condition independent first mean absence tend course orthogonality pair component figure many express component end special give figure stimulus interaction particular tuned fall list stimulus case dependent turn represent independently randomly map space encode axis nearly case orthogonality population arguably ensure limitation must need find neuron work trial average differently neuron record across multiple therefore specifically properly trial trial future argue exploratory analysis suit overview population important exploratory nature enable analysis analysis code matlab datum brief description experimental result reader detail trial manuscript neural obtain record simultaneously rr rr frequency hz
appeal sufficiently since lemma arbitrarily sufficiently construct alternate direction method admm rewrite constraint lead recently direction admm study admm solve lagrange note explicitly two subproblem subproblem positive semidefinite cone second therefore initialize section alternate multipliers optimal presentation let stand present global constraint convergent derive admm tackle population utilize actually recover initialize unlike stop criterion set analogous population block size matlab code definite obtain fortunately addition say sparsity semidefinite error admm false significant entry ratio entire plot covariance recover population covariance stand right recovered stand quite close place none deep green recover quite dependent ghz computation correspond ia produce give obviously value desirable method relatively addition decreasing spend addition small recover propose fix leave generate method bad undesirable behave variable initialized table time basically perform stable need regardless time structure exact population j q take dimension c c table distinct recover desirable cpu time identically even start recover admm initialize fix point produce point method increase basically start create rank solution compare reason probably low approximately semidefinite desire c acquire semidefinite utilize nuclear property illustrate numerical simulation time national basic research china cb national china remark proposition construction claim department mathematics state laboratory p china achieve convex encourage favor estimator large mild sample alternate tackle illustrate estimation multiplier population covariance matrix multivariate many decade advance technology year field brain imaging imaging deal massive setting covariance poorly fortunately covariance overcome traditional research matrix follow viewpoint determinant wise constraint utilize alternate challenging problem covariance essence contrast newly simultaneously matrix theoretical model multiplier admm go numerical experiment section last part hereafter denote occur covariance say write goal estimate unknown fundamental population xx denote support
use construct estimate regression task arrive sequentially carry update recursively extend primary local message become processor execute speed advantage communication delay substantial message pass offer easier include failure uncertainty stress vary datum costly scalable request whole dataset already organize regression simulate ease exposition proof sequence distribute generic e form estimate chapter return entity spread processor type moreover besides computation processor conclusion computation nonnegative coefficient constraint introduction stand recent assume represent type bandwidth computation processor update perform effective accordingly processor compute network communication vary advance agent processor counter need analysis may processor major asynchronous consensus bound come demand communication processor communication delay requirement delay adapt agent avoid agent account processor ia ij ti ti operation r requirement particularly restrictive effect update access assumption require communication delay prevent processor sometimes refer network topology direct describe reach entail communication hold vertex finite interval time infinitely nature processor mainly refined requirement processor I update processor computation position exist nonnegative assume addition bound avoid ambiguity comment apart distribution moreover type message asynchronous nice consistency centralized counterpart worker new arrive worker queue perform average worker perform worker value queue turn mechanism define queue worker call queue memory message happen next queue queue memory queue process start decentralized architecture communication dataset worker notice significantly procedure hour calculation processor minute overhead finally compute remark typical er ed worker theory detail relative compare basic estimate online estimate second eq gain non term estimation degradation negligible peak design gaussian design model asynchronous present ts hold initial specify case delay zero delay exploit linearity scalar determined general nevertheless property array assumption assume true tend exist exercise context agreement asymptotic agreement value instant assume processor stop asymptotically reach consensus depend imply agreement recursion stop start observation study ik ii recursion function role ensure bound r boundedness establish iteration observe obeys convention product view divide firstly secondly show induction reveal processor instant convention nonnegative condition recall news constant nonnegative real number nonnegative adapt offer version check consistency nonnegative integrable j eq therefore take side ft satisfy iteration proves secondly set recall positive eq hence lebesgue put piece together toeplitz lemma aim almost fact toeplitz identity sequel letter generic change universal constant fact conclude constant accordance series vanishe assumption provide proof thus deduce constant technical universal dominate aim condition exist time part proof almost enough upon recall euler depend tend right technical prove one vanish immediate consequence toeplitz similarly write follow let grow infinity section corollary proposition gray paris paris france universit es paris paris france com flexibility accommodate ever massive issue computation develop type asynchronous distribute involve processor excellent method computation prediction online pass parallel distribute currently area activity motivate example massive fit computer
natural proximity measure partly artificial setting forest classification many illustrate performance realistic especially introduce methodology forest adapt argue favor replace probability loss example conclude intractable issue thorough review highlight issue difficulty unable overcome alternative compare massive collection pseudo avoid name turn generic form require divergence exact parametric argument outcome approximation even interpret randomized version h accept n prior distance distance generate practice rarely version rarely make budget directly realistic low thing dataset force resort large tolerance proposal fact counterpart factor summary generally quantile simulated abc software pair generate distance function say modelling adaptation abc choice approximately frequency abc posterior summary want produce straight explain summary statistic subset evaluate criterion measure entropy bic criterion characterize introduction regression abc square distinct per se aim bayes abc goal abc type summary characterize selection crucial technique bayes acceptable one extend model pilot adopt compete linear combination summary consider separate select model versus time fit namely choice model stationarity numerical posterior summary statistic derive density estimation nn tolerance quantile deviation distance whole summary simulate distribution show hyperplane separation since model axis second replication time dot dot model parameter generate prior choice statistic produce approximation loss produce choice summary purpose powerful discriminate database learn interpretable nn observation state induce consequence namely framework sample abc table paradigm still prior abc computing posterior probability implement machine equally strategy methodology machine forest bag aggregate scheme classification mostly insensitive presence noisy large exploit towards selection principle behind rely extract maximal selecting rely relevant collection strongly correlate interest performance evidence forest forest establish intrinsic original additive notably forest adapt without description random forest randomness resample cart subsampling predictor node cart cart separation index contrary cart pruning tree random forest make category specific predictor default sub bag instance prove sub implement feature subsampling induce forest aggregate drastically abc number simulate instance computer allocate statistic select probability compute respective frequency variability often loose constitute thus widely forest abc autocorrelation summary replication simulated summary quantity agree rough still forest qualitative part contain evaluation quantitative vs insufficient produce remain far guarantee deal zero worse assess variability require one instead distinct confidence model abc bias induce insufficient face opinion endow variability well report method classifier whole prior easily evaluate bag forest parameter performance parameter wrong overall though irrelevant point replace unstable pair predictive index answer center area interest nan prior distribution observation x eq thus represent jeffreys decrease maximal strongly assess posteriori rather solely integrate average model rather completely intuitive quantity remain paradigm frequent population study random forest approximation core table already produce abc indeed easily rely produce near neighbor distance proximity random forest overall constitute distribution predict true since discriminant lda I marginal na I marginal abc forest error rate associate summary table provide size independent seven figure separation discriminate properly reflect abc really reach optimize performance move two seven statistic degradation justify achieve comparison performance error abc reference construct predictor nn procedure forest rate boundary model posterior evaluation bottom knn estimation forest abc posterior scheme new dataset display demonstrate predictive rate slight occurrence optimistic report rate point compare abc establish historical population population synthetic nucleotide snp respectively discriminate split change tree figure rely version candidate old population third illustration aim make consider simultaneously population range introduction release abc process massive snp production population compete scenario provide figure parameter choice identically model three population scenario leave model center split six population population correspond source prior associate population individual number additional population offer embed ht reference axis evaluate top meaning appendix illustrate first green triangle figure situation recent balanced strongly source population red figure unbalanced source population sample neighbor provide forest reference contain near neighbor index new pseudo random actual derive posterior rate green ever allocate wrong triangle rate simulation pseudo observe considerably case green triangle figure favorable case illustration snp per produce appendix table structure summarie ccc I gaussian summary abc axes forest initial forest summary two lda axis I marginal discriminant abc summarie axes axis forest summary forest axis I discriminant lda summarie abc axes summarie random forest axis estimate rate snp rate typical quite triangle snp abc neighbors forest summary error rate favorable ht lda axes color correspond index black model test additional green red triangle contribute lda occur role lda axis build nevertheless contribution important variable summary statistic lda meaning provide appendix power rf methodology challenging realistic aim pathway record north analyze include five natural individual marker compete introduction abc rf rf computation discriminate scenario dataset software marker plus nine axis additional summary statistic gap section summary model forest evaluate summary bottom add lda provide forest determine relevant statistic summary dimensional setting nn show summary strongly outperform classifier good axis able separate standard abc axes low forest summary axis ccc na I bayes linear discriminant lda abc summarie abc axis regression axis I discriminant abc summarie abc use lda axes local regression axis forest use summary forest summary I summary standard abc lda axes local forest summary summarie axis three evolutionary scenario forest strategy agree solely high respective extract forest show forest certainly another evolutionary assess posterior rate greatly assessment posterior rate equal new near forest summary lda axes table totally choose totally raise confidence probability close choose ht reference lda axis color index analyze snp population public genome database www project population sequence individual snp deviation snp discovery use discovery simulator snp include encode genome chinese china east encode representative encode population usa encode snp individual snp iii distance kb order snp iv snp equilibrium threshold population remove snp median two snps single population european east population independent give european one give second suggest study e possibility genetic usa european east rate origin stable bottleneck effective population size introduce scenario bring insight human population history study genetic illustrate snp context evolutionary discriminate among implement different reference table report summarie axis well present snp seven hour intel x context observe random construct summary table reference table forest report importance confirm lda able discriminate na I bayes marginal summary axis local axis summary summary lda c I marginal standard axis logistic lda summary I lda initial summary local summary random forest axis classification size table forest selecting consider population field surprising scenario population european east genetic origin european select high indicate rate equal forest replicate neighbor summary lda contain argue forest range evaluate single population population china six differ historical event single give european east population two event give possibility european scenario contribution forest important index summary axis mean reference color correspond index black red gold dataset regard model summary vs summary imply consistency discrepancy remain eventually focused abc concluding rely well propose posterior estimate average posteriori space weight solely integrate conclude forest forest substantially large summary suffer curse performance intrinsic require much effort abc allow reliable early argue statistical massive snp dataset production argue considerable massive production increase within use bin yu visit paris grateful former feedback support department writing help asymptotic forest part conduct research environment acknowledge conduct appendix summary snp proportion gene two nan fp population
map fully fashion state noun phrase sentence feature whose direction follow technique rate gradient learn real application depth edu intelligence international usa deep architecture nested datum cubic time sequence length prohibitive length propose cubic linear rao time semi chain crf suitable nest markovian child theory model semantic model free bound depth main drawback formulation inference outside cubic sequence appropriate nlp limit say slice technique handle contribution approximation gibbs cubic time linear depth idea nest across secondly step trick rao course price pay degradation give observation admit markovian simplified assumption widely transition markovian tend persistent assume transition element necessarily idea semi markov complexity segment semi element state detail exactly like chain phenomena nlp character phrase sentence paragraph chapter price pay complexity property parent new parent child child chain must text noun phrase say phrase noun terminate noun phrase belong parent child relation state topology depict topology respectively child parent parent topology exactly parent htb c review idea g rao see method conditional drawback converge rao rao possibly easy suppose sampling yield specify topological length dynamic multiple I start ii role iii child return parent begin emission activate bottom end termination occur bottom level continue top possible fortunately nest explicitly specific e stay suppose time collapse efficient implication exploit complexity length cubic let transition state observational q joint potential rao sampling time since essentially estimation p suggest gibbs rao integration without expensive fortunately omit reader result obtain full passing duration htb cc generate parameter topology model length semantic bottom level symbol generative first learn generate perform task rao introduce inside outside burn discard sampler forget burn want examine accuracy occur decode first estimation measure kullback marginal difference marginal decode use maximal measure kl decrease slow word mode cc htb c experiment run time iteration total datum many linear run depict figure divergence obtain quadratic cubic time log kl
symbol obtain solve formulation obtain separate hyperplane near follow hyperplane choose moreover hyperplane multiply software package quadratic efficiently extremely set moreover optimal hyperplane early illustrate figure reason offer attractive finding situation width pt pt pt circle pt circle circle pt circle pt circle circle circle pt unfortunately reverse vector separability mean discriminate undesirable introduce modification trading error linearly naturally raise appeal impractical alternate approach introduce slack norm vector slack distance feature call define x feature distance dual observation w convert z I problem suppose let replace w feasible reduction achieve separation rather optimization meaningful linearly aspect false positive false negative connection accuracy etc discriminant fx thus assign sample assign ccc c stand sensitivity specificity three lie convex specificity ac se se ac letter lead star star decide star star patient cancer elsewhere brief patient cancer spread group node size tumor possibility half half star micro star micro independent specificity extremely precisely want contingency exact less machine biology vast besides model tumor branching emphasis topic well biology propose induce select set lasso point section available penalty overlap biological master advance immediate application biology base assumption isometry perhaps something order technique compress sense cancer biology modify measurement rip choice orthogonal actual author handle cluster theory attempt promise lead order far development besides orthogonal measurement matrix connection pointing rip achieve compare whether matrix rip sparse classification data justification norm svm guarantee certainly star would medical tx introduce biology turn student support report perspective well sense advance recurrence present briefly open objective advance discuss biology cancer cancer lead death united uk case figure lead million develop usa uk challenge researcher one cell cell come cancer broadly class optimistic accurate patient patient within group cancer consideration appearance tumor measure tumor decade attempt collect central cancer collect massive public project vast amount tumor standardized protocol amongst international country molecular almost clinical annotation become useful contribution cancer need course would picture advance mathematically couple problem facilitate machine biology understand computational aspect molecular biology analyze hundred genome study gene probe sometimes gene correspond amount rna raw transform take logarithm divide subtract divide number sometimes good gene expression feature micro level variation value presence absence mutation specific molecular tumor depend tumor recurrence drug patient beyond site addition ordinal medium ordinal value attribute previous correspond tumor recurrence treat one simplicity real value binary row th throughout handle two namely binary belong take label simplify sequel whereas approximate would prediction tumor consist level thousand around together tumor value lead reliable recurrence cancer disease highly particular key constructing use scope end tumor recurrence throughout regressor weight bias restrict attention regressor regressor understand regressor biological feature possibility still process explain early process raw regression eq gauss everywhere rank ls context matrix less least impose weight constraint lead problem formulation treatment topic least norm unique ridge however would ridge reformulate become lagrange multiplier block equal definite lagrange major ridge every component nonzero function make undesirable nonzero regressor familiar quantity component common even though find minimize analyze square interest simplicity discuss generality alone weight minimize multipli shrinkage depend multiplier technical moreover correlate biological pathway vary sample therefore column correlate case ridge assign nearly extreme amongst discard choose biological level gene pathway correlate undesirable discard rest undesirable would column call elastic net penalty choose elastic net whereas become ridge elastic provide bridge elastic elastic elastic minimizer center suppose always nearly many net number number often see desirable feature correlate biology open biological pure try choose distinct group regressor select element possible group determine clear depend relative size component yet group limit consist singleton group reduce standard variation group group formulation overlap biological decomposition gene acyclic wherein master pathway seek choose gene rather pathway master master gene namely ii pathway g ideally feature intersect pathway example present circle minimum size draw mm draw circle circle minimum minimum draw circle circle circle size draw mm circle mm draw node mm draw size draw date sparse pairwise penalty augment refer suggest sparse overlap continue penalty date address see group obvious hold namely retain g modifying hold reason nod clearly ii thought reveal structured group really truly overlap sg maximal together set biological sense every pair define induce type consistent new application elsewhere choose nothing difference reason unlike en several example en lasso relatively algorithm unlike en assign weight far remain carry cancer predict tumor recurrence tumor recurrence well patient day exclude analyze feature elimination figure percentage error value h elastic year group sense compressive determine take area broad decade compressed find paper summarize I paper randomness motivation discuss sense biology search cancer stand apply fundamental compressed sensing call paper note refer biological biological area ignore hope argument application treatment compress sense development area name several make contribution paper survey reference reader begin introduce give n k kk nk x z quantity underlie large component replace kx c isometry rip introduce rip property easy exposition rip suppose isometry rip hold column row column integer sufficiently rip suppose state know compressed compare rip z write differently rip corollary follow c corollary simplify near ideal lasso oracle compute oracle appropriate reduce constant mean universal bind achieve rip randomized sample construct advantage ensure namely simple proof rip construction rip result fail nothing rip measurement hold tail tail nothing hold essentially objective constraint raise question whether example place result display ideal behavior lasso analog replace joint author effect decomposable rip describe imply
c cauchy schwarz inequality satisfie restrict event combine result tucker example follow simplification restrict event probability lasso consistent theory always case event instance get prediction q last fact u j cf f minimum evaluate independent j indeed symmetry inclusion inclusion yield hand variable f j j entail view cauchy schwarz complete proof remain bind resort ct n j jj j proposition infer case x triangle n x hence put together eq case conjunction least replace replace achieve f k observe close f k j j l conjunction piecewise combine l v inequality far get relation x j imply complete appropriate j v infer u u belong supremum right side q foundation support grant corollary red corner corner pt france ny understand insight incorporation measure reveal moderately correlate performance irrespective illustration deduce considerable effort devote guarantee performance understand common already setting prediction understand gain broad avoid prediction identically read noise matrix exposition simple restrict gaussian noise fix extend conditionally covariate even magnitude tuning amount penalization crucial influence typically unique irrelevant follow strict convexity minima numerous lasso universal conjecture correlate span relevant lasso one correlate suggest number convex hull difficult incorporate computable covariate really small fixed rate irrespective correlation covariate rely assumption covariate low isometry compatibility bound irrespective tuning depend true vector follow extreme covariate mutually extreme permit mutually covariate attention topic loss explore throughout every respectively subset removing set belong transpose u equal subset denote span orthogonal onto n n asymptotic write sequence present show particularly impossible rate even small devoted sparsity inequality compatibility imply total establish quantity accounting within developed allow second fourth question introduction discussion place outline question state concern relationship covariate fast fast begin discuss literature relevant x impossible much small may signal part inequality inequality error loss value vector consequence note appear stochastic least square covariate vector lower usually recommend choose prescribe tolerance demonstrates state cardinality covariate span trivially follow third chi conjunction tail latter fact unfortunately oracle assumption surprising formally integer less canonical avoid unnecessary rademacher reason show covariate rate correlation prevent third fail counter analytically clearly fast without assumption rate note advantage complexity oracle inequality literature state discuss learn combine improvement compatibility fix parameter satisfie reflect front infimum equal refined lasso particularly consequence value get mention intermediate parameter satisfie np inequality frequently refer covariate provide rate order situation available design subset vanish span compatibility every entry compatibility weight compatibility factor measure relax previously lead fast fix tolerance main difference presence replace compatibility factor always quantity think covariate justification tune cross validation choose denoise processing penalty employ similarity problem grid noisy unknown value grid tv f penalize square hereafter tv despite popularity surprising tv asymptotic setting establish n f optimal piecewise achieve instance penalty vector reference whether rate eventually minimax notice tv jump satisfactory tv differently assess jump raise question oracle small achieve c jumps allow increase almost exhaustive entry penalization completely application example fix cardinality expensive risk circumstance index least adapt proposition fast strongly tuning order small deduce rate however proposition considerably signal denoise total penalty previous study tv piecewise tv monotone signal review topic context find find monotone tv risk bind apply estimator mention early tv assume span n nh nh assume interest gaussian k f vector tv crucial minimax tv dominate minimax logarithmic conjecture logarithmic require tuning parameter proposition avoided avoid measure total roughly mis specification monotone hold old function almost minimax particularly old theorem exploit know achieve minimax logarithmic set old f fx tv theorem improve achieve set n l estimator proposition tv risk nearly differ important benefit finite sample inclusion model specification risk mis specification constant use trick mean stochastic satisfying include identical column vanish column vice versa compatibility factor well idea tight theorem less however really substantially one factor remark lead produce mis specify u matrix proposition tell mentioned proportional slightly differ c generally contain look characteristic supremum outside amount c repeat satisfy ct spirit consequence rate design comparison b compatibility furthermore advantage risk bound lead choice rate prediction compatibility factor replace compatibility factor compatibility vanish moreover benefit necessarily rely covariate reasonably small necessarily small expect refined respect potentially flexibility parameter factor front another explore discussion replace correlation achieve replace strongly show lasso competitive procedure explain loss design contain column furthermore believe strategy may conjunction form theoretical section suggest set tight aforementioned consume alternative replace sparse indeed close span find therein understand compatibility factor factor nonconvex guarantee subset moderately large remain interval program parallel involve belong solve set note right gap divide least problem
function initialize smooth since give boundedness result vi ix great x c initialize approximate specifically greatest bind prove prove desire solution approximation establish boundedness resemble convergence behind restrictive prove remain bound presence solution control aim pursuit tolerance positive achieve vi limit arbitrary exact vi cite proof pointwise convergence small achieved denote online lead considerable load online operation use approximate minimization operation denote actor introduce next provide asymptotic subset region approximate approximation bound actor hold asymptotically compact r x n ix appendix actor definite numerator right hand positive require q positive function great inequality lead definite determine actor practice validity capability upper example checking validity lipschitz constant examine train actor find unless pointwise value actor former utilize union find stability admit termination approximation numerical analysis around circular state stay real frame motion field xu total force reference angular velocity control time step velocity velocity force dynamic discretize utility hand x ref prove successive complete iteration enough converge actor due smooth basis find integer actor respectively iteration denote relate policy denote actor repeat make multiplying element follow function random vector I learn select tolerance guess converge inner respective element versus take ghz gb memory windows actor shoot select may evaluate state function goal converge respective pointwise cx utilize generalization per boundedness converge exist reliability stability controller verification approximation vector quantify p appendix respective accurate consider actor approximation control turn negligible evaluating turn exceed bind remain upper stability initial result trajectory comparison loop direct see controller accurate besides go go loop turn cost go approximation suppose upper go optimality simulate ever therefore control however obvious desire find analyze select evaluating converge therefore utilize guarantee stability select belong vanish origin conclusion appendix controlling need pick sample state select discuss observe value lead actor expand origin long guarantee still get good result fig present improve network option rich rich basis turn condition extremely fig similarity conservative condition bound probably loop controller analytical dynamic observe finite stability comprehensive numerical fourth problem foundation improve mathematical field inspire value cost define k trajectory respect associate function otherwise sum trajectorie summation hand control summation control eq let prove right convergence summation evaluate trajectory max sup side note reason finite finite qx along idea boundedness per respective consider q note equation actor follow smoothness compact side asymptotic stability holding hold equality origin one consider hold positive serve lyapunov asymptotic follow long trajectory remain inside leave relation stability concern estimation definition inequality trajectory also set close namely continuity upper boundedness engineer south rapid city email aim answer consider next deterministic investigate error boundedness control assumption approximation optimality condition system result obtain iteration along derive approximation process make development verify stability dynamic rl researcher approximate mathematically great potential popular crucial cite nonlinear due phenomenon happen regardless would lead vi challenging appear good author function factor becomes investigate interesting utilize restrictive easily error write e uniformly denote approximated involve reader study success value include analysis aim pursuit mathematical control develop boundedness approximation check investigate stability term parameter presence error possibly iteration deviation instability outcome system derive concern trajectory controller note train guarantee inside however remain domain controller remain valid use reader vi guess stability lemma vi th iteration denote side approximate represent value hand difference cumulative iteration approximate vi mention train actor remove conclusion result actor actor control directly minimization apply actor even though actor simultaneously learn offline system result stability system due convergence investigate suitable operation minimize
interval motivation weight constant empirical distribution statistic cdf theoretical term kolmogorov distribution case approximated remark tend infinity instead appropriately center objective deduce mathematical practical output primitive distribute weakly without change used reference observe depend ks close monte generalize classical since statistic cdf value depend weight evaluation course gene pose false discovery fdr correction test test asymptotic apply carlo cumulative conclusion gene beyond necessary output test gene set database value gene set initial tumor typical make encouraging set whole mathematical algorithmic advantage test could informative organize devoted begin description monte test comparison notation throughout convergence bridge q law variable ex primitive obviously converge weakly brownian supremum continuous p index induce apply yield q distribution term motion primitive deterministic brownian gaussian easily covariance explain process notation let standard conclusion code implement make manual regard essential distribution else first review integral differential interval hence integral sum increment simulate gaussian simulate trajectory absolute return write simulate increment motion sequence discretize trajectory proportion propose return sided estimate conservative state cdf weight relation validation kolmogorov explicit estimate close curve discretization turn difference classical sample test ks panel discretization simulation value uniform gene logarithm ks plot p value stay test theoretical asymptotic cdf tend conservative set clear theoretical simulation case replacement negligible simulated extracting replacement agreement asymptotic precise gene repository test set use test dataset annotate give name apply gene gene symbol common vector two tumor red triangle base display comparison sake raw fdr adjustment mark dash vertical horizontal line significant cdf simulation monte limited line due monte carlo observe globally coherent corner graphic never converse point corner correspond gene set analysis gene gene significant gene compare center appear gene significant step function curve far enough global significant moreover already deal significance conversely gene significant clearly compare gene tend name letter gene cancer test gene state observe type detect among new testing proportion center convergence correspond kolmogorov kolmogorov major depend gene therefore calculate gene value conservative increase set compare test whole tumor gene encourage raw good precision rank agreement yet version gene proportion ks side test sign difference contain small conversely positive together manual following encourage tool biological pt treatment center derivation center carlo simulation kolmogorov fact asymptotic serve short repository comparison test new conduct mathematical informative test empirical weak
laplacian correspond connect minimizer eq nk achieve solution algebra nonnegative multiplier nk active first kkt inactive last correspond one n z kp graph tp separate component computationally separately condition mean practice category small enough arise entire never attract dominate take assignment practically strongly mean assignment towards simplex solution practical seem let substitute kkt trivially condition th item category expert assignment correspond category similarity qp condition kkt condition lagrange equality constraint kkt linear multiply q respectively column multiply left matrix inversion solution transpose verify formula simplify index number describe guarantee take direction multiplier combine alternate direction qp q positive semidefinite admm new replace indicator function q eq constraint lagrangian multipli constraint apply immediately admm iteration simpler obtain combine indicator function nonnegative entry take part update apply modify euclidean penalty lagrange estimate equality qp kkt condition lagrange multiplier iteration update crucially convenient direct complement leave multiply equation eliminate substitute first admm linear primal guarantee admm qp identify q single simplify considerably basic reason identical copy graph sparse constraint matrix follow factorization reduce fill much iterative solver initialize warm fast inexact write laplacian cholesky factor system convergence primal multipli triangular cholesky factor conjugate gradient available implement require line search converge affect iterate update eqs upon multiplier kkt upon k kkt eqs finally kkt change sign lagrangian subtracting stop estimate respectively particular project inactive weakly value except make add setup also sufficiently practice since high accuracy runtime use graph cholesky factorization take factor solver research subproblem cluster warm outer loop iteration initialize eqs k k initialization simplex next fact arbitrarily try system q since simplex project independently stop last iteration fall tolerance relative update stop runtime criterion test condition satisfy lagrange multiplier iterate need satisfy iterate infeasible project assignment simplex category iteration select fast eigenvalue laplacian relative asymptotically long short small c relative l provide assignment error scale iteration implement assume update nu li n r k nu minus nu break end z find assignment item category assignment reasoning could augment consume assignment point practical natural mapping augment free point dropping reduce quadratic program w point thus probability finite computationally sparse neighbor large use hash retrieve neighbor receive simply reflect rather training describe may relation different constraint unlike prediction solve fundamental supervision actual label item give assignment would valid assignment constraint redundant particular label item label first nonzero multiple category tag category partial category rest simplex give entry implicitly forces summary close assignment directly able transform assignment use perhaps smooth user force category obviously force wrong poorly would entry leave free similarity mapping coincide mapping addition give similarity coincide particularly informative nonzero small item learn nontrivial achieve indeed constraint semantic frequentist assignment proportion belong portfolio portion example include classification restrict belong category exclusive assignment may interpret assignment long history operation economic portfolio seek qp however laplacian model different apply assignment training laplacian item centroid entry w nk g k cluster point fix particular mark define similarity location label ccc positive label cccc cccc diagram ground label relate diagram category contain category intersect quite usual graph denote partially circle unlabele true give crucial propagate unlabele partially exactly minimize fig bottom unlabele obtain unlabele assignment smooth assignment sometimes assign wrong assignment ground truth easily label task sample digit similarity image category give near nn machine width select code assignment unnormalize improve valid assignment lie simplex category respective optimal leave unlabeled avoid clutter point template method use outperform runtime produce assignment topic manually topic pc hardware mac hardware topic belong construct occur extract inverse feature document randomly one topic five parameter optimally mnist document topic high assignment actual predict truth fully label penalty classification improve outperform nearly assignment benefit handle predict full assignment category tag test fill assignment propagate unlabeled sample assignment provide demonstrate select category word non empty category category image build partial give category randomly frequent providing stop unlabeled categorization select base precision f sample although tag similarity tag fig show versus svms sample category nearly always see improve precision recall white green white black draw white old computer window window ex draw old white coin white precision vs fix annotation sample tag combine complementary source crowd expert attractive impractical category complex structure item category incorporate supervision laplacian similarity direction multiplier iteration make factorization laplacian implement search rate represent similarity item since exist application mapping nonparametric affinity accelerate train cm prop thm observation example wang electrical science give encourage similar assign category item item encourage intuition give unique effective multiplier reasonable generalization learn particularly naturally belong multiple keyword tag major many rely label unlabeled laplacian construct nonnegative measure graph exist base formulation conceptually solid foundation graph laplacian widely vision area mention image spectral use surface iterate product laplacian concern soft assignment give take tag annotation consider conference keyword likely extent keyword tag paper paper science redundant large include biology paper keyword besides category assignment although extent inclusion tag item certain patient code category indicate degree item also consider source example bag capture laplacian example source imagine conference paper author author bioinformatics send word predict
bilinear operation vector multiplication slice tensor entry include add tensor part fine grain sentiment intensity class ordinal nature ordinal network intuitively sentiment belong sentiment belong belong class therefore otherwise vector cumulative nonlinearity traditionally multiclass firstly entry assign entry low whenever proper assign purely unit hidden tensor two denote result eq unfold layer hide individual simplification extract vector rather greatly instead word vocabulary extract matrix recurrent neural recurrent tensor handling correctly shift sentiment interpretation follow instead learn operator word word one operator natural word low besides space one learn vocabulary unsupervised would allow use manually annotate level extract convert annotation integer ordinal label denote sentiment multiplicative improve development suggest indeed powerful careful early initialize initialize vs share parameter word limit preliminary word vector embedding yield difference test experiment identity consistent improvement extra nonlinearity identity significant however suggest boost nonetheless nonlinearity marginally explanation nonlinearity nonlinearity crucial stanford sentiment baseline bottom show rnn conventional baseline network structural far recursive bad matrix recursive tree unlike variant explore recurrent compositional interpretation fine grain sentiment ordinal previous previous stanford sentiment tree effectively separation representation vector extend supervised slice interpret simplified word mean share space affect word matrix hand sentence tensor well update towards drawback rnn increase explore word vector would every operator act acknowledgment work grant fa view conclusion herein author interpret imply present multiplicative compositional meaning fine grain sentiment investigate case recurrent well recurrent outperform fine grain sentiment recently publish stanford sentiment generate recent network deep nlp numerous nlp natural properly large phrase dense nlp require properly approach compositional vector act assign representation long phrase simply matrix english parameter vocabulary phrase semantic structural composition operate child sentence represent different way lead recursive neural network nonlinearity use aforementioned matrix composition recently bilinear tensor multiplication composition capture recurrent network rnns neural capability phrase act memory result composition layer next phrase sentence accommodate suggest neural additive theoretically nonlinearity end compositional semantic level capacity sentiment sentence represent space rnn recurrent nn successful multiplicative sequential computation replace rnn compositional sentiment space multiplicative rnn represent instead dense network representation suited scheme recurrent semantic act view recurrent neural network show rnns comparable performance net fine grained sentiment detection absence tree put burden multiplicative rnn reach variant single token per token vocabulary high dimensional equal syntactic semantic similarity representation token value dense usually dimension generally representation learn unsupervised corpus wikipedia embedding capability might employ vector representation input word semantic transform apply partially idea act noun vector word representation long phrase compute generalize space plausible fine grained sentiment vector capture transform apply bag word wise multiplication composition ignore compositional distributional semantic semantic argument composition structural particular recurrent neural consider hidden phrase one determine tree space sentiment active nlp various word level try formulate grain approach explore ultimately task addition bag incorporate account contextual transform represent successive inside phrase word representation representation model constitute multiplication score assign representative degree freedom recurrent rnn network recurrent connection
require alternative representation let bag representation centroid centroid codebook bag bag centroid arguably distance use achieve histogram bin specifically two solution describe mass distance bag word euclidean centroid distance metric limitation computational al regularize magnitude al simple iterative iteration algorithm scale multiple histogram efficiently compute hardware covariance use neighborhood covariance classification compress drastically md asymptotic eq training copy input neighborhood assignment radial basis randomly near cross inspire input classify compress I I imply compressed prediction kl ideal minimize divergence compress set ensure cholesky j j substitute single j I mp gradient matlab http histogram descriptor aim compressed histogram dd neighborhood compress correctly via compressed share label kl perfect histogram r introduce challenge optimization remain optimization nest problem formulation eq histogram occur correspond dual strong duality primal dual formulation identical optimum dual consider fix simultaneously descent ensure simplex normalize change histogram positive jk complexity p gradient update select compressed error person object face material categorization scene real nn six benchmark image background object camera place descriptor texture resolution image surveillance individual therefore wide ratio filter class result image face gray face individual orient angle image popular ratio covariance cloud view object covariance information white set sift descriptor sift bin horizontal direction orientation bin produce descriptor via transformation select size hyperparameter dataset material material material pose condition cnn rnn allow surprisingly covariance reduce amount subsampling essentially learn versus label agnostic set descriptor discard speedup various ratio dataset mark learn exceed match nn remove neighbor gain increase error compression speedup describe compression minute compress get amount compression compression contribution potentially parallelization ccccc recognition texture classification technique histogram histogram compare accuracy achieve describe consist object background black take histogram follow extract context sample histogram log shape white follow extract shape histogram ground bin texture dataset surface feature extraction use bag representation sense distance bin pair error neighbor similarly parameter definition initialize appear outperform subsample baseline dissimilarity compute centroid centroid accelerate centroid compression descriptor text detail average deviation compression ratio dataset outperform match compression speedup classification compression possibly match throughout final compressed set rnn compression ratio lead arguably interesting baseline full magnitude bad time table minute compression especially solve histogram believe hill set reduction consider nn primary sampling iteratively add reduce perfectly technique neighbor notably rnn search result cnn additionally find cubic prototype generation create training set cluster prototype appropriate propose compression finally al algorithms network set histogram descriptor nn speed distance computation bregman ball tree euclidean tree bregman divergence technique onto bregman ball compression devote toward bind efficiently distance bin reformulate add unconstrained amount previously compress drastically neighbor large corollary university st abstract absence sufficient datum descriptor influential descriptor ii descriptor histogram computer gradient image may result visual descriptor definite diffusion suit task nn histogram neighbor histogram descriptor individually bin euclidean distance bin wise dissimilarity descriptor lie convex half embed straight euclidean underlie systematically descriptor incorporate distance improvement nn versus distance costly cubic specialized distance operate constraint require prediction geodesic manifold eigen decomposition need repeat classify test ball dimensionality near many method match classifier original contain explicitly
activation write weight via multiplicative activation accomplished drop individual unit denote scale gaussian test require scale bernoulli sub sub connection exponential visit however extensive sharing make regardless explicitly dropout inspire variety subsequent dropout version weight axis align take mask recognition maxout design new type activation function exploit dropout procedure order interaction among input hide variable connect multiplicative structure relationship input structure recent include network update input output fix formally activation factor input vector activation special z e en feed forward factor neural output multiplicative decompose multiplicative formally feed forward instead l n l hide activation bernoulli comparison feed weight dropout unique represent dropout test time mean secondary input define mean n feed effect learn learning order unnecessary base method neural convolution convolution decompose convolution operation multiplicative noise convolution filter network find restrict restrict collapse single layer maxout million free million cccc constraint relu et bernoulli dropout maxout dropout relu unit gaussian cifar dataset contain color training remain dataset cifar contain mnist cifar convolutional layer layer image preprocesse global contrast whitening maxout result cifar cifar parameter small consider mnist expect optimize tool hyper conv dropout fully conv conv net maxout conv conduct separate dynamic hide factor prevent overfitte continue decrease test linearity experimentally verify testing visit test output procedure demonstrate geometric though procedure relu arithmetic mean appear geometric investigation link weight figure depict evolution norm imply weight factored actually learn penalty dropout seem impose implicit essential network traditional form capacity early decay currently mean amount aggregate apply interpret allow testing make dropout order magnitude additionally restrict loading control recent closely redundancy parameterization architecture weight matrix way parameter c regularization neural network sample poor unseen partially responsible win entry university currently method efficiently number aggregate learn noise especially win entry imagenet challenge use network competition top partially attribute regularization dropout crucial dropout elegant solution number model predictor generalize test relu percent zero activation dropout although training multiplicative activation add relu activation distinction similar dropout method ensemble robust visit briefly review generalization discuss advance dropout connect introduce multiplicative
structure eigenvalue external external exponentially structure detail leave search use estimate remain fixing adopt estimate scalar argue search direction make sense prefer choice regularity already currently try describe span precede side explore confirm tend spectrum show generative process haar measure figure uniform exponential pd draw top eigenvalue elaborate construct estimate middle row figure stationary multiply estimate freedom nonparametric square scale error external external external external eigenvalue external external cg plot eigenvalue external exponentially eigenvalue external standardized norm exponentially external structured external true gray error estimate black bfgs standardize cg demonstrate example gaussian ph w mh bfgs cg method show row figure row far equation marginal gaussian show uniformly quantity estimation rule bottom bfgs standardize prior construct cg arise structural overhead much cg implication method cg remain open question member inconsistent nonparametric formulation nonlinear question regard covariance finitely result remain early appendix map see observe bs prior establish linearly element q element search rectangular generalization lyapunov apply present show constructive part full invertible write q write note eq get bs sx equation clearly similarity covariance establish simply ns algebra complete eq notation element gram matrix ib simplifie similarly simplify equal choice bfgs h search search search estimate irrespective invert inverse last symmetry I sum eq j complete acknowledgment like discussion manuscript particularly grateful discussion basis addition author comment anonymous particularly point ab da draw black white rectangle draw fill black white draw black fill minimum pt sep draw manuscript propose framework estimate gaussian belief conjugate gradient conjugate gradient novel quasi foundation optimization quasi probabilistic solver unconstraine computational equivalent minimize bx iterative solver address step big bar put estimate quasi newton widely derivation method extend maxima probability covariance offer interest estimate entire space family direction evolution occur widely bfgs rule part optimization include early less rule broad among subsequently refine update formulate cite update current estimate hessian call newton equation rule update inversion rule estimate interestingly bfgs update exchange inverse bfgs rule bfgs confusion text sense inverse sense update exchange make explicitly bfgs probabilistic objective ask kind assumption rise contain derivation rule symmetric rule natural perspective another extend problem cg idea closely cg bfgs estimate contain list search method class identical provide variant future aspect newton evaluation gradient certain structural restriction observe statistic probabilistic encoding assumption measure newton method new method map arise inverse point introduction author rarely attract numerical mathematic argument sometimes numerical randomness distinction lack arise precisely apply probability deterministic prefer another point practitioner apply budget numerical helpful point instability save say exact answer ask hypothesis far attempt answer linear problem point popular extend construct posterior construct element interpretation provide independent nonparametric particularly elegant provide prior symmetric frobenius matrix norm member bfgs member consistent probabilistic rule lemma sr give calibration bfgs cg lemma gradient computationally convenient parameterization uncertainty picture particularly calibrate family possible covariance construct finitely motivated regularity overhead conjugate gaussian probabilistic area brief text hypothesis real linear bayes posterior distribution prior dirac crucial operation bar mean link maximum quadratic probabilistic quantifie specific iterative solver shall maintain direct access inference kronecker matrix kronecker element observation solver gaussian quadratic kronecker update search probabilistic much frobenius norm frobenius w w match central role call weight rule family show frobenius regularizer base domain noise gradient dimensionality introduce update restrict class matrix involve derivation consistent framework aspect essence insight previously begin use linear act appendix ij e g element act effect carry symmetric example inversion considerably appendix linearly q posterior step member consistent unchanged transformation turn central define kronecker rise popular method argument favor applicable family rule solver mass theorems consistency solver arise gaussian general structure perfect correct search remain linear within direction hold choice long course good crucially definite desirable cone normal distribution prior cone statistic irrelevant normalization constant conjunction various sided find linearization wishart match second wishart hypothesis individual desirable choice equation one implicit give globally posterior choice manner sr bfgs implicit w ss analogous circular try part sr exception involve estimation adaptation data apply update ignore old direction inverse obviously include cost show though simplification external scale experiment thin line experiment thick spike cause ill condition arise full update sequence conceptual experiment definite generate exponential plot haar measure give projection draw uniformly symmetric bfgs equal one frobenius normalise little intuition exact application rule track consecutive make big relate full dominate hessian apply estimator construct show track gram probabilistic step perform qualitatively posterior covariance use gray line frobenius conceptual uncertainty close solid gray scale dimensional property bfgs uncertainty understand sum inference ratio error calibrate achieve apparent calibrate diagonal confident estimating confident element unit course still vanish e matrix fix degree prior covariance smallest sr observation subsequent one exactly away suggest investigation address possible construct calibrate thus error ideally major increase rank proof gram repeat update classic implementation result equal exact equivalent statement inverse update conjugacy analogue optimizer choose citation put bfgs definite search inverse gaussian belief analogously bfgs posterior establish inference cg cg starting choose pre define bfgs inference mean conjugate bfgs intuitive probabilistic framework bx mm cg bfgs gaussian probabilistic problem cg compact iterate inference b conceptual open extension cg thing direction gradient scalar prior bfgs property conjugate gradient converge fx x orthogonal gradient span span establish cg extend multiple rule bfgs cg establish interpretation cg solver collect extremely popular look calibrate obvious uncertain linear solver reasonably addition scale bfgs implicit cone matrix additionally
obtain marginalization accord recursive degeneracy obtain collection proceed alternatively operation weight degeneracy online nmf model smc sample bb calculate q smc see current make model contribute calculate exact online horizontal correlation need algorithm horizontal line present online em nmf observation exact smc separate process reflect generality I e formulate dimension realistic particle smc implementation sophisticated proposal smc improvement handle dimension formulate markov nmf sequential monte approximation sensor drop collect record amount raw arguably computation effective set meaningful scientific financial political purpose unfortunately classical deal restriction slow large give matrix computation factor make inferential goal precise analysis ica nonnegative matrix semantic indexing understand problem understand probabilistic interpretation probabilistic generative derive posteriori advantage interpretation enable incorporate consistent building fit probabilistic natural online pass algorithmic idea generalise tensor see formulate nmf maximum hide model hmms asymptotic hmm nmf online nmf smc decrease particle smc algorithm propose computation approximation convergence present formulation notably nmf dirichlet allocation view incremental learning empirical th element multiplication element division set nonnegative capital letter letter hmm comprise parameter divergence formulate formulation problem static random constitute therefore formulation mle change expect w law necessary first online sequence maximal surface choose calculate intermediate distribution respect estimate intermediate update find nmf calculate expectation belong family update calculate updating depend law write sufficient statistic form recursion expectation sufficient explain sense assume eq intermediate filtering regardless reason first z z sum recursion nonnegative value recursion claim verify induction start tx z kt z eq bm derivation estimate average sufficient
choice parameter succeed success repeat random iteration suffice terminate uniformly ball isotropic finally termination algorithm unclear excess extend tight differentially loss processing generic private carry output ensure differentially private run next exponential mechanism excess technique pure extend naturally gradient achieve excess carry localization perturbation algorithm htb perturbation parameter parameter algorithm give differentially let generic optimize theorem strong convexity convex run input radius output input output differentially differentially differentially private generic expect norm noise accord gamma satisfie hence hand private efficient version risk formally bind follow guarantee suppose replace efficient output derive low decomposable p lipschitz whereas lipschitz useful incur differentially marginal sake appendix marginal nd take differentially private clearly linear nd I nd excess low private algorithm must nd give differentially private let nd nd sake every contradict lemma exist paragraph p bind differentially algorithm random nd nd counterpart factor loss big nd observe hence lipschitz computation lipschitz lipschitz convex restrict existence clarity completeness sake closure literature know extension suffice remain show minimizer w convexity probabilistic discuss htb log concave guarantee cube fu u pf cube moreover conditional distribution p output vertex locate origin inside isotropic position cone cone cone integral less integral region inequality second prove output least denote event induce good good note efficiently computable exist membership membership efficiently lipschitz enable polynomial run perform step structure run directly p use boost multiplicative concave sampling bound multiplicative output unit since isotropic output denote condition I e q linear use finally plug expression give fairly version algorithm run algorithm decomposable loss isotropic namely differentially private input exp efficient concave convex dataset privacy guarantee yield differentially let let denote differentially output differentially lemma straightforward differential privacy hence private respect factor hence follow finally observe put complete grateful concave particular penalty reduce cube lem lem lem lem lem lem edu support systematic investigation private matching erm contribution lipschitz bound bound provide match lower know strongly convex separate differential surprisingly technique algorithm simple work contribution smooth apply previous previously commonly empirical erm function datum erm information record motivation bound erm contribution build start draw universe close goal q map define obtain variant restriction end collection datum point example svm formulation capture erm add solution fold regularizer dependent function replace come generality lipschitz affect ex know glm statement success namely output expectation convert technique see appendix another excess measure imply upper convergence idea range statement appendix know erm fitting square significant actual loss point minimize helpful median reveal one subtle svm whose consist erm attack notion work theoretical differentially output motivation discussion role several basic differentially private dimension constant diameter constraint rescale rescale replace risk rescale always convert excess bound general multiply replace p c assumption rescale get multiply simplify art loss asymptotically principle technique purpose section factor achieve match factor excess always match convex previously well know risk general function strongly convex several different restriction perturbation tight apply include case smoothing function expect risk descent well variant data compute estimate update appeared previously investigate noisy variant lie randomness follow without privacy step desire excess base gradient measurement risk bound knowledge excess risk privacy obstacle privacy step use net probability op low privacy inefficient since achieve excess excess risk efficiently continuous log technique require provide privacy issue work define appropriately cube output sample multiplicative use subroutine correct statement base technique function however technique mechanism yield strongly sensitivity minimizer hence release strongly euclidean strongly convex first estimate output optimal roughly define running mechanism improve factor mechanism n localization privacy take convexity quickly develop bind way privacy essentially convexity constraint ball hypercube objective bind nonsmooth smoothness privacy set nonsmooth effort design nonsmooth generalization comment implication unseen generalization c necessary necessity generalization error lipschitz modification privacy error roughly root however special generalize match generalization polynomial gap mention work rich seek characterize private bound regularizer unconstrained paper orthogonal though implementation mechanism domain vector obviously differentially study tailor also role development query release completeness additional vector bound tangent smoothness denote vector efficient differentially loss lipschitz discuss localization derive efficient sampling distance arbitrary convex bound private detail supplementary nonsmooth loss technique appendix localization modification guarantee generalization differentially private gradient descent literature utility rest randomness failure analysis guarantee run instead use complete guarantee factor gd lipschitz tb output step randomness variable randomness condition differential privacy addition see randomness randomness randomness least ensure privacy differentially ensure probability least privacy loss differentially private te gd expectation line notice randomness randomness guarantee descent tt excess bind strongly convex let gradient e assume computation take current idea batch sample oppose excess guarantee remove constraint tight privacy use empirical loss section mechanism show exponential major mechanism efficient private excess base guarantee differentially notice utility guarantee risk expectation risk every outside analyze individually argument risk machine allow extra risk boundary figure
sentence assume terminate token system become sentence way convolutional network long architecture elaborate neural work encode input produce sentence initialize generate neural translation rare problem accommodate word however name find similar beneficial despite machine translation address notable exception address rare track unknown sentence source word responsible word could source dictionary know unknown token word unknown token identity apply token easily annotate one unsupervised next link dictionary post translation pair en fr en annotate example token multiple target language oppose token assign repeat annotation elaborate word align source token model target word alignment align token annotation translate token translate word former sentence happen model tend vocabulary limitation motivated annotation include sentence single universal token token indicate denote align source annotate token en fr token sentence speed post sensible word motivate token simultaneously align source alignment universal token language annotate en de annotated token slow effectiveness task quality sentence comparable model corpus use intensive naive softmax vocabulary target side vocabulary use frequent english model treat alignment berkeley setting discard sentence exceed token hyperparameter cell embedding word source lstm memory summarize rate begin mini normalize gradient exceed gpu parallelization allow achieve source training corpus neural neural feature list end rnn k rnns end single lstm single lstm lstm lstm k ensemble lstm lstm ensemble system differ architecture size corpus either sentence text handle rare vocabulary improvement process accurate model origin word report score consistent work english language art system include art neural phrase translation technique translation individual point large performance gain still provide nontrivial useful ensemble compare usefulness depend correctly word source identify great processing employ strategy use one track treat input level result system translation rare rare examine lstm architecture strong correlation highlight translation rare follow et sentence average inverse frequency sentence within rare system standard mt system mt describe rare translation curve produce frequent word part bad sentence many word proportion score rare word win sentence frequent word frequency sentence sentence comparable rare score group examine different copy predict align predict align position train hyperparameter vocabulary good performance still analyze include baseline assume align predict monotone offer slightly english similar word language pair chinese word monotonic imply gain gain model word accurate translation prove limitation force align considerable contrast align unknown target word source post process strong translation suggest easy lstm short sequence increase consistent source provide roughly improvement layer stage include clear op la er en op la la de l er op dans de la le leave european head trade e en est en du des point et du ce e en et mis de du trading pour les la se la de show human translation process lastly strong quality measure performance layer lstm compute good translate word observe accurately predict distance highlight reasonably translate end source example reveal entry b alignment incorrectly align result sentence overcome main current system translate vocabulary applicable deep lstm technique likely necessary achieve demonstrate english translation task substantial system architecture importantly outperform mt acknowledgment member brain team discussion insight stanford
hamiltonian ref wiener filter uncertainty exactly extensive want ref htp color signal reconstruction terminology wiener eqs wiener calibration mcmc monte gray method derive address illustrative ref scheme ref device perfect field dependent exhibit sampling position measurement process align data calibration assume spectra ref length implementation get calibration white relate ref measurement calibration beneficial break degeneracy calibration variation switch calibration strength measurement absolute calibration measurement ordinary eq version eqs version deal differential fig reconstruction ht online calibration reconstruction relate error terminology use reconstruction perfect eq wiener filter assume correct calibration naive pure average calibration signal terminology wiener calibration scheme htp calibration bad result method advanced mcmc similar average classic improvement realization naive classic naive realization mcmc fig classic wiener naive still sufficient sample converge increase increase effort iterative involve ordinary differential ode ode exist stepsize control save significant case one could cope reduce stepsize elaborate suppose representation require bottleneck within due correlation structure contrary comparison naturally calculation derive signal consequently calibration calibration covariance priori knowledge successively account uncertainty solve iterative turn system ordinary equation example calibration perform extremely wiener well well favor new classical sec serve realize package find find ref read pt diagram correspond external coordinate depend external summing close diagram mx sx high diagram end line represent connect internal integrated represent vertex compare non locality coordinate integrate whereas diagram divide leave topology book theory derive generalization flow calibration include ad calibration become calibration replace calibration measurement hamiltonian max universit device crucial estimator reconstruct know calibration starting calibration signal inference equation correction solution thereby differential verify self scheme wiener serve keyword analysis field wherein infer response understand response independent calibration environmental time influence exactly result affect signal take calibration iteratively reconstruction improve reconstruction scheme systematic bias calibration partly degenerate guarantee improve uncertainty present ref theory end signal successively flow approach non review interact latter problem introduce derive main formulae sec reconstruction toy example summarize calibration familiar interact brief helpful typically infer challenge reasonably answer agree within express tuple signal linear response operation act signal contrary manifold scalar linearity transform observation scale medical imaging assume uncorrelated relate denote product bx condition covariance meet relate physics hamiltonian consider denote determinant whole physics permit correlation density pdf gaussian mean wiener vanish gaussian free signal meet dependent scenario treatment hamiltonian compose interact deviation term fully nm expand work wiener expansion expansion analogously theory eqs correction signal mean order diagram boundary equation obtain wiener address infer physical field external calibrate absolutely response measurement device transform signal statistic two point framework aim optimal datum without challenge unknown calibration nuisance calibration coefficient gaussian appropriate higher consider first eq assumption eq correlation result ss calculation analytically adapt carlo increase analytical concept auxiliary r b hamiltonian expansion small whereby eqs permutation fourth signal drop flow reason priori signal center prior interaction source correction quantity read mm dot external expansion place dependency correction diagram also break set accumulate information thereby formulate operator boundary couple valid lead solve simplify residual field field wiener hamiltonian hamiltonian
fill bic size draw color bic style mm minimum thick mm thick black bic black circle cm white fill color bic style color bic sep black circle size cm black fill color text n style circle inner mm black fill bic black style rectangle size thick black bic bic minimum cm white bic text bic sep mm fill black style thick fill inner sep cm thick color bic bic inner sep thick fill bic sep size thick black bic style minimum cm fill bic text bic circle thick fill color bic text bic inner mm minimum fill circle minimum draw black thick fill black style sep mm text black bic style thick white bic bic circle inner sep cm draw fill text sep thick fill style circle mm draw thick text black bic thick bic bic style mm fill color bic bic fill circle thick white fill text circle sep bic rgb rgb rgb rgb rgb rgb rgb rgb inner sep thick black sep thick black size thick color text thick fill black circle inner sep mm white fill circle sep thick black color text black sep cm thick fill inner mm white fill color black mm thick circle white fill color black style draw thick color black sep draw thick thick white inner sep fill color white fill color text black style circle mm fill text inner sep mm cm white fill text black style sep minimum draw thick white fill text minimum cm thick white color black sep size draw white style cm white fill style draw thick white black sep mm size cm thick circle draw fill color text sep draw thick white fill style size thick white black style sep thick fill color color circle sep draw white black thick circle minimum draw thick circle black color rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb inner sep size thick color text style inner thick style circle sep mm thick color draw black fill color bic bic circle sep draw thick fill color bic sep minimum thick black color inner bic inner sep minimum draw bic text bic inner sep mm bic inner minimum white circle sep white black thick color black circle sep bic style inner fill text circle sep mm sep draw white color bic text bic minimum draw text black thick style sep draw thick fill bic black bic style circle cm thick white fill color bic style circle size draw white bic v bic bic bic bic bic bic n bic bic bic v bic bic v v v v v v v v v v frequency small include choose bic never select select frequently small select leave number leave independence depth always trivially bic small depth converse focus depth pick leaf note equal generate dataset fix summarize marginally real canonical threshold bayesian computed tree main tree propose refine usual alone consider exhaustive forest observe variable selection structural em consider heuristic put paper bound zero proceed step recall product move nonnegative zero indeed first neighborhood sufficiently away zero view function restrict bound away imply nonzero turn application line proposition note nonzero coordinate intersection minimum intersection lemma applicable b hc I nonempty asymptotic equivalence completeness coordinate substitution jacobian computation compact take restrict q taylor around since claim nonempty form interval minimum translation equal small sign coordinate gx fact support sign change say loss generality restrict asymptotically recall multiple nonnegative expand drop mix note nonnegative since nonnegative jensen equivalent amount case contain change sign coordinate form reflect version prove origin point origin origin small neighborhood point zero substitute h hx tree space correlation first applicable compact compact deduce theorem remain onto appear sum recall forest partition set leave set leave comprise belong inner leave forest leave disjoint sep circle draw sep label label label label remove square observe connect take component sum pair distinct node edge square eq also lie lie obtain node let three let square origin single suffice exponent incidence leave word leave vector terminal leaf newton satisfy define path include span hull hyperplane define inequality claim polytope vector combination incidence construct ii iii contraction tree node induction iii leave path leave base induction pick leave tree subtree leaf satisfie path extend leave leave path path path edge transform path give include leave lie tree degree generalization match necessarily tree origin connect neither incidence hull newton polytope incidence long hyperplane proceeding similarly ray newton polytope newton acknowledgment partially support union national foundation dms security university reproduce xy section thm thm construction thm thm di gaussian generally forest fisher matrix become along compute log canonical bayesian provide development treat laplace integral whose phase sum canonical apply laplace forest tree explore exploit recent latent forest mathematical information criterion widely guide forest bic generally long share asymptotic long asymptotically unbiased estimator kullback leibler behavior inference gaussian forest obtaining form canonical threshold expansion formally introduce object index paradigm induce contain unique node path conditional compare leave marginal parametrization latent moreover zero parametrize correlation correlation everywhere theory develop property parametrization map inner node connect leave label definite matrix three sign change node identify uniquely identity parametrization finite correlation identifiability matrix every lie algebraic intersect reader familiar root direct model specification markov structural seen apply independent write marginal hold rational dimension greater equal derive variance pair behavior laplace formulation adopt canonical threshold state rely present concern general asymptotic exhibit recall forest comprise leave forest model assign edge path exactly parametrization distribution namely extend forest arise data distribution lie forest paper begin review connection asymptotic likelihood introduce tree parametrization via technique geometry selection criterion simulation alone consider parametric inference determination derivation bayesian criterion behavior true marginal tie integral divergence distribution therefore integral neighborhood q analytic compact far sample infinity explain satisfie use nonempty analytically since g f take normal integration take concern fact hold paper concern loss generality center density positive q hence neighborhood positive q compare tree forest q obtain jj discussion parameter generate variance definite correlation matrix behaviour marginal tree square result context u u u case vector support right generality moreover proposition define zero part hull inclusion last empty compact bound turn linear integral clear newton canonical early nonzero part newton ray span zero size sep circle minimum b divide coordinate vanish eq gaussian latent leave dimension edge component number degree nod forest forest threshold model proof appendix leave space share node example canonical translate threshold tree behave model suppose leave label degree variable calculation example apply forest isolate edge edge repeat edge contraction tree least original node equal motivate wish forest forest forest forest analogy tree appear union theorem connect forest model parameter forest smooth positive share node two consider selection forest forest identically criterion implicitly forest degenerate whose give coefficient difference likelihood forest little consider criterion average possible bayesian short describe compute contain forest forest forest forest give q proxy side model system triangular solve recursively univariate possible bic give nonempty bic motivated dependent differ strict belong I correlation forest bic order lattice empty graph isolate maximal select forest optimize bic likelihood estimate em maximize forest model comprise latent inner introduction tree lie forest respect well choice repeat time black circle draw label label label b b b comprise lattice heat choose label independence rgb rgb rgb rgb rgb rgb rgb rgb sep mm thick color style circle inner mm sep draw fill style sep minimum draw thick black fill black circle sep draw thick white style cm draw color text inner thick fill text mm circle mm size inner sep mm fill black circle text rectangle sep thick black cm fill circle inner mm minimum thick color text black style inner sep minimum draw style circle size draw thick white text black sep minimum cm black inner thick color style minimum cm thick color minimum thick sep fill text inner mm minimum black inner sep minimum cm draw text color black fill n style inner text inner sep style mm white text mm thick white color black sep minimum fill sep black circle sep draw black inner sep mm thick fill color text black rgb rgb rgb rgb sep cm black fill color circle sep draw fill style cm bic text circle inner minimum draw thick black color text bic circle thick black color sep draw color text black circle bic minimum cm black fill color text style sep minimum color draw thick fill black bic style mm color bic rectangle inner sep draw thick fill color bic style mm thick text circle black style cm thick black black n mm fill text mm thick color n style circle inner style white fill bic bic style mm size thick style sep minimum thick text black bic style minimum draw thick color text style fill style sep mm draw bic thick black black bic style inner size thick fill color bic bic circle mm white bic circle inner thick white fill bic bic
diag operational composition entity bilinear diag element dot additive highlight bilinear diag provide insight common compositional multiplication entity category entity significantly category result entity predict entity relation category one learning improvement linear entity train phrase baseline compare entity entity parameter dimensional pre phrase vector representation technique introduce word word initialize art popular base completion evaluate structure relational deep hierarchical input datum figure embedding relation reflect release close country embedding learn hard ht table neighbor relation frobenius relation vector meaningful retrieve production production production award award organization role organization job company business organization university location people person location capital neighbor embedding lemma corollary definition ny usa li microsoft usa real entity people place often multi model multi relational biological multi relational category relational logic relational graph probabilistic path large relational embedding relational knowledge entity tensor bayesian study neural network powerful tool generalization recently learn entity entity represent representation entity entity neural relation operator represent relation entity report promise unseen basis compare result neither representation carefully entity vector pre corpus idea tend syntactic natural real world compositional phrase name movie name mean compose type entity include variant completion task dataset present several interesting finding tend scalability play entity operation superior modeling relation entity vector pre boost entity finding inspire embed predict vs triplet describe certain output scalar high vector project input low project specific formally network scoring relation triplet write scoring score unify transformation triplet reformulate denote r parameter transformation derive c r r relational scoring deep semantic learn pair neural project entity e cosine normalize network margin objective decomposition comparison among objective beyond score relationship triplet high negative triplet triplets triplet one objective minimize margin q triplet entity relation consist triplet subset frequent relation result triplet entity link task triplet entity corrupt reciprocal triplet accuracy evaluation gpu batch triplet negative triplet corrupt subject entity entity improve mini batch entity vector epoch k determine rate initially five slice
distinct none origin imply hull eq possible basis negative sign case hull suffice equality expand reduce coefficient coefficient coefficient lie hull otherwise equality compute coefficient expand automatically lie convex hull roughly lie convex hull rest separate member family let set hull none member real lie lie clarity positive vector lie thus one contradiction investigate analogue namely norm great question norm norm equivalence body interior respect standard euclidean symmetric convex symmetric convex vc statement prove preliminary lemma vc infinite consider interior ball norm closed interval terminology give subset pass note hyperplane pass operation set interior three implication close slight make section particular open ball bound send open interior close send convex convex collection dimension origin interior natural take union result element base choose interior appropriate interior interior bound lemma bound interior complete ensure valid assertion concern interaction hyperplane wu point form lie claim equality sequence subset choose real addition base subset n possible inequality q satisfied lie equivalence convex equivalent infinite vc symmetric vc nest convex vc convenient increment index become body contain regard sequence lie subspace lemma condition natural number hull closure set close additionally eq since empty interior remain verify body origin disjoint point lie observe cone segment exceed lie convexity lie convex symmetric body natural convex subset expand translate lemma subset important vc ball also prove vc collection ball dimension norm vc vc regard norm short mean close open ball regard dimension norm paper vc vc norm vc remain norm least precise vc let subset subset set definition finite vc vc accord vc dimension supremum dimension close form number convention diameter include affect dimension cube distinct positive diameter hand exclude vc collection proceed stage vc say prove cardinality agree claim dimension collection close close may closed bind exactly zero may proceed attain maximum integer point lie contradiction every point list leave least imply vc precisely assumption cube note lie di condition lie hence dimension three clarity recursion base even dimension exist higher odd exist satisfied origin lie subset suppose leave lie thus cube diameter cube imply cube lie still cardinality c dc fc c dc intervals df replace every odd exist may set accordance extend cube origin give diameter interval equal set form imply let cardinality
test criterion criterion considerably classification consist step encode extract whiten normalization learn finally classifier comprise pixel whiten finally pass patch sparse differ patch comprise apply thresholding extract cf fill text fill text width corner blue draw auto block image patch pre block pre unsupervised line dash training extract numerous insight principle whiten correspond trade look whiten recall whitening transform patch change precisely whitening transform patch whiten patch change entire affected indirect capture introduce high say round narrow define large eigenvalue width spectrum serve supplementary column gram notion understand whiten patch perfectly particular perfectly round whiten therefore remain randomization randomization random matrix matrix bound standard sharp purpose sufficiently particular round illustrate function entry gaussian random randomization link conduct toeplitz matrix local typical nearby natural entry toeplitz k ij toeplitz lead toeplitz matrix lead find feature matrix increase illustrate perfectly suggest increase matrix preserve global want understand split set unsupervised determine contrast statistical whitening dictionary randomize whitening create dictionary stack select whitening patch statistical whiten patch feature different number acc yes yes yes yes yes c acc yes yes filter dimension sparse filtering first result norm matrix transformation filtering form sometimes norm similar far obvious need htb apparent choice feature influence filter interested costly alternative cross filtering compute correspond basically indistinguishable three observation accuracy sparse monotonically highly peak coincide set test observation stop early optimize filter compute training intermediate peak version report claim confirm finally note version randomness involve spectral promise provide interpretation wide spread whiten patch quite considerably spectrum intermediate feature specific extended paper planning convolutional network cnns cnns deal recently many cnn substantial acknowledgment row proof entry invoke orthogonality obtain yield desire display corresponding row normalization multiplication prove normalize first write normalization multiplication eq matrix yield second claim claim claim normalize matrix display numerical cifar splitting result conclusion htb width corner text cm corner
improve case bias dominate estimation answer systematic improve estimation potential problem functional bias region additionally yield minimax overhead dimension large mle valuable going consider finitely among influence theory compress benefit carefully analyze practical size demonstrate efficacy entropy mutual discrete wide fundamental decision shannon two involve insight various paper I technique develop mutual employ estimator application tree empirical liu significant implement mle iii bayesian empirical mutual yield performance recently methodology functional show mle far minimax unbiased suppose functional differentiable smooth fall fall polynomial specify order polynomial smooth regime plug estimator reasonably hand parameter turn towards functional close sup functional polynomial order scenario unbiased power tool various experiment show methodology estimating entropy compare expense bias dominate methodology reduce expense slightly utilize polynomial estimate specifically show estimator present wu independently scheme achieve require mle improvement practice space horizontal vertical mle entropy demonstrate optimality therein essentially mle finding functional interested estimating estimate convenient impose joint liu dependence precise write liu solving solve span liu mle mutual connect word node span estimate liu assign tool design graphical research biology extensively reverse engineering expression dedicate theoretical property cl edge maximally extreme correspond reveal cl ratio cl begin reconstruct perfectly continue fail maximally require transition sample ic construct assign test important learn class class attribute conditional al assume dependence conditioning class probability conditioning attribute label precise factorize precede parent graphical light cl augment cl difference mutual mutual information posteriori map label attribute rule demonstrate section algorithm base reasonable mutual information specifically mutual way increase implementation computational improve classification list identical since alphabet theorem indicate clustered attribute cluster cluster original attribute bold implement cross error record separately error percentage bold reduction modify scatter modify relative none solid circle error top eight alphabet observation e light remarkably mutual expect mutual begin fail theoretical improve consistently require achieve acceptable classification conduct classifier size training testing implement time display error remarkable reduction scheme uniformly size modify one guess least bad adopt reduce modify outperform original method demonstrate justification apply mle take consideration might lead direct two stanford nsf discussion likelihood biology thank suggest wrong cl original mutual write entropy take also argue estimate denote estimator estimating consistently obviously suffice consistent sample theorem stanford edu widely technique recent introduce construction functional demonstrate improvement particularly scenario comparable result theory rate requirement message functional performance mle sample present highlight statistical achieve reduction require classical liu mle improve liu apply replacement network modern form year series remarkable paper since estimation article book indeed response popularity year accept parametric employ mle likelihood letter fisher possibility mle perform poorly example significantly improve upon cf le excellent overview create part see perhaps systematic
obtain minor since appendix simulation available upon request bandit randomization patient report simulation patient represent population arrival trial treatment pick patient decision arrival feed week patient process patient come assign patient random patient true preserve among take negative reward fed algorithm patient multi armed bernoulli admit greedy tuned patient patient patient set hard patient rate minimal obtain level treat display per average repetition instantaneous plot half cc also report patient treat duration trial randomization softmax trial treat patient must treatment obtain treatment cite success test test independence clinical weak advanced adaptive value contingency allocation contingency cccc failure greedy cccc softmax cccc success failure ucb cccc total ucb tune cccc success failure total patient clinical retain fraction patient treatment clinical accurately method treatment represent bandit indistinguishable patient look well patient treatment effect effect patient day treatment divide patient bandit indistinguishable randomization greedy softmax ucb answer question practical bandit algorithm delay feedback minimal randomize patient decision outcome negligible constraint arrival dropout pose context interpret failure require fill treatment outcome patient clinical equal effectiveness patient identify patient difficult treatment still return weak randomization return superiority treatment bandit bandit clinical patient patient successfully treat interesting patient still trial day effect occur patient suggest patient trial much easily guarantee usually strategy limit bandit notably address issue study goal bandit aspect affect reward relative cover type bandit surprisingly consistently outperform advanced softmax least find theoretical analysis significantly identify perform perform exploit design bandit identify bandit tune future arm improve reward certain algorithm ucb may setting half turn bandit problem trial clinical trial motivate armed bandit evaluate clinical real clinical use strategy simple randomization base patient trial treatment confidence offer reason clinical multi armed bandit efficacy medical far rich clinical setup unable difference among field network practical exist limited find broadly hope study encourage apply bandit preprocesse internet table detail clear result decision since treatment response determine entry indicate day determine patient period receive day last last day patient entry exposure treatment patient calculate taking divide patient present per day average measure test overall average average always patient patient treatment independence normally five meet number patient ucb tune ht ht c number treat randomization greedy softmax p like cccc total cccc softmax cccc success cccc success failure ucb tune cccc success failure treatment figure effect curve rating result patient randomization softmax tune rgb rgb rgb rgb proof stochastic armed reinforcement effectiveness present thorough popular bandit heuristic boltzmann sound secondly algorithm vary dramatically bandit setting perform perform theory even exploit practice heuristic relative affect reward find may subsequent evaluation turn attention trial clinical motivate multi bandit allocation study simulate outcome clinical trial adaptive trial successfully patient reduce patient trial treatment identify finding current allocation multi armed bandit trade automate gain explore bandit clean exploration exploitation comprehensive perspective simple generally bandit variance initially player slot player view many turn player select receive goal hand value hand playing specify alternatively express random denoting play arm classical suboptimal kullback divergence suboptimal solve armed match bound establish recent fisher analyse loose strategy algorithm limit large theoretical problem arm generalize setting evaluate extensive far compare include evaluation become comparison investigate paper optimally may detail criterion interpret extensive conduct thorough aspect bandit problem affect surprisingly characteristic remarkably outperform sound algorithm similar strongly practically varie exploit study identify aspect bandit number hope take account study viewpoint finding need formal sound simple clinical trial armed viewpoint turn bandit clinical trial design clinical trial practical motivate bandit describe refer clinical trial motivation bandit trial capture balance exploitation look treatment arm benefit clinical trial sized treatment level confidence trial dynamically reason decade theoretical modern adaptive clinical several design trial stop base sample estimation patient trial drop hand drop naturally trial promise adaptive design family thorough discussion literature clinical trial design adjust patient treatment assignment favor pursuit family adaptive randomization arguably trial randomization ad hoc heuristic patient winner though literature clinical evaluate treatment allocation particularly surprising many simulation aware clinical trial trial drug successfully bandit aim spirit determine bandit constitute effective trial strategy generally wish effectiveness question implement world patient arrival long time statistical base offer patient thorough overview literature simulate clinical find would clinical trial adaptive randomization effectiveness criterion patient treat patient treat significantly end trial algorithm attractive alternative strategy outline section setup present representative conclusion implication briefly clinical trial arm simulation section detail discuss obtain six four exploration pursuit heuristic capture handle exploration tradeoff play average maintain arm distribution handle exploitation tradeoff heuristic aware empirical systematically evaluate pursuit reinforcement latter ucb ucb sophisticated strong ucb solve armed bandit optimally factor make former heuristic know maintain empirical picking denote greedy obvious generalization select arm probability report accumulate summarize detail handle third relevant situation suboptimal arm trial every experiment repeat average arm evaluate algorithm arm equal behavior benchmark number arm respectively reward sample equal every arm deviation contain obviously separate bandit characteristic high moment distribution triangular inverse variance normal algorithm empirical always optimistic initialization initial find choice result criterion literature well possible third optimize armed characterize reward second half demonstrate aspect bandit tuning dramatically affect report every variance regret numerical achieve percentage greedy softmax mm reinforcement ucb tune greedy pursuit mm ucb mm ucb tune l greedy mm reinforcement comparison mm tune mm reinforcement pursuit ucb mm reinforcement greedy pursuit softmax mm ucb tune pursuit mm mm softmax reinforcement mm ucb greedy pursuit mm mm reinforcement comparison mm greedy mm pursuit mm ucb softmax reinforcement tune greedy mm pursuit mm softmax reinforcement comparison ucb ucb reinforcement comparison tune greedy mm pursuit softmax mm ucb tune boltzmann almost similarly softmax softmax outperform except medium setting ucb ucb little performance optimal slowly generate regret turn reward suggest advantage pursuit important algorithm affect differently variation handle number arm variance much find characteristic distribution surprising normally observe omit present pursuit softmax mm reinforcement quite surprisingly algorithm counter intuitive hard skewed tuning incorrectly although case regret large surprisingly bandit bad reward illustrate regret boltzmann tune every ccc initially tune study tune every strategy account three heuristic advanced suggest practically heuristic boltzmann close advantage appear substantial heuristic development sound boltzmann exploration open whether sense theoretical need balance exploitation throughout bandit strategy ucb theoretical ucb base softmax substantial improvement practice dramatically poorly possess
selector contain gaussian agree notation denote value counterpart arrange set cardinality p overcome elastic regularizer behave elastic sense within elastic paper report support correlate wherein statistical support besides current method norm formulation might fail rest sp dual present fast exploiting efficiently computable exist square directly c sr small root addition exist nonnegative unique op kp search motivated search part assertion accelerate search procedure execute algorithm complexity c sp u selector consist address width get upper quantity notation bound norm generalized selection prove technique group lasso k follow minimizer theorem choose natural interpretation special ds norm therefore admm concentrated efficiency hence side behavior matlab four operator sp ratio illustrate accelerate nonzero entry equally normal response roc h result correspond ds selector vary introduce generalize selector norm encode norm exploit flexible inexact conjugate proximal solve unified analysis utilize prove trivial show inexact admm support structured last sound support nsf yahoo constant proof q lie rewrite since distribute surface sphere gaussian lipschitz lipschitz choose old next gaussian lipschitz eq statement sr inequality pair pair follow without imply sign order focus violate similarly cauchy schwarz unchanged determine optimality quantity inside exist nonnegative root minimizer actually care lead give obtain theorem need theorem decrease r first simply monotonicity mention mm focus case satisfy one generality let know minimize construct choose decompose contradict k part part part violate second must violate uniqueness satisfy make follow contradict minimizer eq assertion note operator minimization constraint consider restrict unable situation violate indicate corollary increasing replace still modification decrease cauchy schwarz decrease modification monotonicity mention satisfie conclusion obtain first norm atomic prove satisfie consider every non vector element combination whose none convex norm thus norm atomic norm cone cone support cone define provide ease contrary e statement mm norm thought norm utilize technique specialized state enable henceforth understand overlap support go order prove gaussian cone nonempty convex cone cone lie normal construction proceed c appropriately let bind follow comparison q eq freedom complement bound therefore maximum use lemma pt mm ready follow mm inequality let eq bind note mm proposition corollary support upper need first confirm selector ds regularize establish ds generalize decomposable primarily focus sparse notable extension structured model suitable aspect norm induce atomic estimation constraint atomic aspect norm selector propose primal interior method generalization homotopy piecewise multiplier admm linearize motivated ds general inexact primal proximal suffice side bound width ball suitable support proximal support norm efficiently focus ball set need statistical guarantee rest establish section efficient error experimental section analyse selector ds approach approach similarity ds norm general primarily focus notable group structure paper consider organize present estimation support experimental conclude supplement suitable ensure empty section inexact present consistency subscript due applicable alternate direction method multiplier admm augment lagrange multipli control quadratic term admm q amount
use restaurant stochastic process exchangeable likewise prior directly random investigate prior gamma gamma binomial marginal suitably stochastic integer count binary restaurant count marginal sampling likewise process binary beta bernoulli gamma poisson marginal gamma beta organize preliminary bayesian random naive document categorization derive prior hierarchical product define continuous space scale evy measure gamma process sum base two continuous measure separable metric space concentration evy beta bc matrix sequentially new subsequently add introduce add row meaning indicate across convention name three stochastic define almost surely count name underlie hierarchical stochastic stand matrix process construct poisson binomial draw independently atomic j j atom nonzero matrix construct count count care specie mass pmf sum define let sum logarithmic pmf number kind pmf compound binomial pmf pmf wise count vector count matrix highlight difference result without deferred infinite dimensional unconditional pmf count separate absolutely component introduction count mass concentration prior arise conditionally nonzero pmf derivation count may verify random count pmf distribution count row column exchangeable column exchangeable recall new row construct n row direct calculation yield prediction express familiar add count crucially new count normalize play key role combinatorial appear gamma binomial process calculation interpret draw keep unchanged binomial introduce iii iv iii iv iii iv iv column original drawing identically add column arrive identically newly distribution different multinomial example new column random count poisson simpler fix binomial count count augmentation negative binomial draw gamma negative compound joint auxiliary count matrix scalar pmf distribution detailed derivation verify pmf compound count via permutation column differently row random maintain exchangeability construction count draw customer table new column add q row exist draw q j customer aggregate kt original mapping new follow logarithmic implication order count distribution sum integer gamma whose row sequentially construct row similar f random dispersion pmf pmf outcome beta conditionally process define jk jk derivation verify calculation pmf generate multinomial maintain exchangeability count analysis pmf thus row customer j r ice column ice thus analogy ice section provide description ice number ice implication infinite many ice exist customer similar random matrix exchangeable dispersion parameter relate marked dispersion atom dispersion however beta column I challenge sequentially construct introduce combinatorial appear independently negative binomial process share argument focus single dispersion dispersion large negative binomial process count submatrix doe submatrix maintain however indexing column permutation row bring column arise realization normalizing term performance construct order constant column choose k combinatorial analysis evident random count I column construction relationship insight pc column c construct negative ten via generate unbounded use one mode monotonically decrease addition count highly count decrease limit unique one identify count encourage th column distribution build use employ two negative employ parameter instead new column n p use expectation variance count allow fine count wise employ column logarithmic count column n j variance c count dispersion fine number random row exchangeable random row dispersion variance count simulate provide difference prior count count count matrix small range whereas count significantly count close gibbs exploit augmentation marginalization binomial distribution count distribution bring additional argument combinatorial random naive bayes classifier category summarize contain count exclude count count j com vocabulary detection tracking corpus appear category consist dataset use compare document process category I correspond infer affect neither long iteration collect mean row count collect calculate expense document term th c I row document term count example word count unique significantly vocabulary whole corpus could fast consider vocabulary corpus principled document contrast traditional discard tb binomial document count random count row right count document term posterior display clear restrictive observe limited heterogeneity adjust count closely expect prior tail distribution highly row wise heterogeneity indice training classifier row exist count training belong count treat feature iteratively row simply predictive use likelihood constrain vocabulary likelihood contrast truly produce fit produce multinomial classifier smoothing q predefine vocabulary discard document categorization unconstraine vocabulary grow vocabulary negative negative smoothing plot category categorization multinomial smoothing document fit matrix restrictive probability count moreover mixed nb tail help oppose multinomial classifier word vocabulary normalize count binomial provide length vocabulary section derive mass binomial matrix I infer construct add row predictive nonparametric bayes classifier classifier share vocabulary outperform multinomial classifier observable derive call process exchangeable count consider simplification represent row exchangeable exchangeable accord theorem gamma express k k k nonzero count pmf count row nonzero count across bring combinatorial question carefully arbitrary labeling index atom conditional likelihood q mass continuous formula directly ne n prior normalization arise column two count matrix although amenable complete update inference except share prior process define draw employ specific hence conditionally row differently focus marginalization conceptually simple compound jk pa jx express j mixed logarithmic lc binomial sum le sn mix
perform non convex cluster method moderately large impossible simple subsampling would efficient subsample new discriminant subsampling classifying belong elaborate subsample cluster discriminant modification goal subsequently towards computational efficiency small large primarily accelerate spc solution include assignment spc introduce concave function parameter choose center cluster cluster spc detect singleton due fact cluster solution relatively pairwise center spc enable spc combine spc subsample assignment remain iterate assignment subsampling spc order spc accuracy effectiveness spc separate need filter successful small cluster dataset new subsampling spc path cluster effect solution novel subsampling cluster estimate cluster provide user significance path step assignment design proportion iteration partition remain cluster identify contrary review subsample contain proportion include spc subsample subsample size big inherently operation would systematically loss lastly spc raise satisfactory simulated spc assignment section discuss consideration demonstrate brief represent spc penalize center concave penalty mcp controlling concavity merge spc utilize minimize cyclic initialized singleton gradually build center spc path decrease cluster include singleton path drive select base property mcp please spc singleton large cutoff cluster knowledge spc cut relatively cluster spc tuning determine neighbor initial spc couple path allow develop noisy produce single describe spc full likelihood ratio membership separate show grouping introduce distribution spc subsample parameter remain membership remain subset respectively suppose replacement subsample training remain index point cluster select index point gaussian denote kp proportion assignment proportion hand overall assign cluster rule take note identify assign estimate parameter update calculation next point discriminant analysis mixture generalization quadratic discriminant analysis generalize extended matrix mixture assignment spc prohibitive choose subsample subsample probably able cluster full recursion assignment step spc repeat random recursion repeat spc short essential select clustering rely cluster cluster characterize estimate include significantly check become assignment recursion increase recursion describe estimate assume test cluster sufficiently design negligible treat degree sort cutoff discovery fdr sufficiently many small corresponding discard otherwise control fdr subset cluster critical generally reject moderately small thus control finally cluster discard spc subsample recursion fdr mechanism automatic determination dataset sophisticated simply large size follow test loose give discover number assignment recursion consist find recursion note spc b b ba km b iy ic bb spc define cutoff determine keep big impact long value reasonably practice hypothesis testing mostly along recommend many occur chance spc rely violate especially subsample typical outlier spc incorporate overcome acceptable point remove discard perform unstable discover increase amount test cluster probably cluster subsample amount cluster possible tight cluster spc tuning determine however generally iteration cluster outlier terminate demonstrate simulate spc ability quality separate compare spc accuracy rand ari develop calculate different ari misclassifie identify merge evaluate cluster misclassifie right comparison proportion cluster total mixture generate radius center ari score use choose cluster spc dataset due operating limitation hold big time input include bic hyper please categorization appear force cluster volume package six dimension order spc scale figure considerable order magnitude fast even magnitude subsample find compare see depend noise shorter amount effectively identify reach spc show computational complexity much spc algorithm especially cluster amount subsample fact comparative original spc figure demonstrate middle estimate panel dataset spc amount indicate proportion spc report example small size though somewhat inferior spc small ari score cluster vary inferior spc risk create large mostly accept due substantial beneficial quality consistently good proportion suitable subsample satisfactory see ari gray subsample able fairly cluster scenario exception cluster subsample splitting cluster subsample size spc order fast mention estimate mean useful spherical shape complex situation certainly understand ari panel bottom inferior cluster tend remove clustered calculation center variance force assignment background sequential step particularly subsample generate necessarily small subsample considerably large subsample figure generally great produce simulate beneficial obtain compact finally demonstrate purpose correlate rest four six point proportion run ari spc result spherical expect tend split preserve albeit amount misclassifie also score dataset reflect mainly due cluster split see believe useful spherical datum dataset tractable two e across select profile perturb simplify grouping involve gene consist expression profile es cell across experimental behavior dataset second gene subsample section two large incorrectly include exclude apply cluster group run figure misclassifie gene misclassifie figure subtle difference big cluster distinct go cluster several gene involve biological b cluster fdr em central development pathway em figure term clearly cluster high level compare particularly pattern novel finding induction seven factor eight es clustered study cluster attempt identify potential cluster depict identify hereafter characterize cluster overall display distinct expression h approximately respectively separate go particularly gene go term cluster fdr em e surface pathway bind activity activity response dna stimulus complex organization activity activity act e go term indicate discover category bind large cluster p cluster go term contain third yet annotate yet six go mostly small biological significant go development transition material particularly interesting high expression control find sharp establish role es therein bind adjacent site uniquely gene responsible development significant classified relate heart development none large contain go low
satisfy due page limitation network page limitation substitute derivative therefore analyze change simulation iteration network steady evaluate run range steady reach minima reduce vice versa make department electrical communication institute signal theory communication mail com es investigate year network experimental apply rest optimum due incorporation subset computational need propose sparsity expression regularization system real data process stream simultaneously instantaneous estimate system impulse sparsity norm prominent amongst coefficient algorithm deviation steady achieved obtain fraction use rule aware employ update refer agnostic provide maintain uniformity computational burden complicated exploit adjustment aware claim validate via regularize trivially case norm scalar q noise require mutually node adaptive filter desire e refine two adapt combine receive b neighbor originally weight adaptation node estimation exploiting add scheme term node constant combination rule combination define par homogeneous sense aware major result individual index deviation steady neighbor exchange diffusion input assume spatially argument matrix top col w n w col carry stack matrix product notice term steady let term e take occur minimize highly sparse conversely zero node
achieve result qualitatively h al model realistic snps spike performance follow distribution simulate phenotype h figure phenotype choose considerable estimate considerably h h h resample genetic bootstrap auc phenotypes bootstrap auc deviation bootstrap deal fix effect account p select base risk denote lastly whether compare deviation set highly relative risk relative individual x risk prediction subtle aspect population control r case summarize phenotype ht identity variance section identity software fast invert run infeasible realistic mean individual focus vector scalar compute overall spirit formula eq plug eq use identity gibbs sampling control study sample obtain scheme mixed effect assumption normality environmental effect environmental effect genetic environmental genetic distribution variate naive gibbs reference drive phenotype I f
tool package project conclusion topic project document discrete treat word corpus words corpus word corpus vocabulary vector vocabulary probability corpus hierarchy generate word possibility probability model map topic sample distribution dirichlet word represent assign proportion sample corpus corpus topic represent sample multinomial distribution document multinomial sample multinomial distribution model parameterize probability topic topic symmetric document value begin sampling document network lda represent leave generate corpus repeatedly expectation algorithm gibbs consider posterior strategy chain sampling chain converge target selecting begin next newly probability stand document assignment word assignment document iteratively initial integer sized markov chain markov chain topic run reach current value gibbs derive unique document remove tag time assign assign word assign document assign discuss data preprocesse lda package project input lda project receive email server server email clean capture email server help far cloud computing process consist language fundamental scientific computing offer package sort code platform processing package remove package package package format unique current document index vocabulary computing tool environment operation specify use option parallel calculate count write program program receive generate implement option input path finish count ignore compare calculate speedup send account gibbs sampling efficiently learn result precision evaluate initialization convergent appropriate convergent gibbs tends reach independent initial topic topic winner enter party color email contact action medium email people receive health http entry email pay support country stay word reveal email health topic new contribution news information lda model word list part measure definition htbp word list top word word match capture correct stand choose tf tf represent major text classification relevance corpus raw document eq high term low word size tf case tf topic word free package project word obviously tf another likely document analysis network social certain people service topic analysis visualize word com dirichlet allocation gibbs train free server com word generate latent topic typical political news instance project contribute speedup project high tf
unfortunately pd pd property bx ib h bx functional lipschitz prove obviously since p pd pd p x n follow immediately similar prove respect versa fix r rx h nr r rd pd evaluate contour lipschitz less prove construct qx map lipschitz frobenius q identity x follow map identity claim lipschitz claim obtain obtain acknowledgement author support nsf dms also point reference point university mathematics center mathematical university college md usa mathematics scientific college md call problem specifically nonlinear map hilbert lipschitz term surprisingly independent frame span hilbert space apply iff product nonlinear phase retrieval reconstruction problem analyze frame map entire remain space continuous lipschitz literature e bi f bi real inverse restrict image surprisingly minimal lipschitz organization follow section nonlinear induce eigenvalue negative symmetric restrict necessary phase frame phase constant eq distance norm subscript choice compact particular nuclear frobenius expression distance condition state result distance induce additionally embed lipschitz topology however endow particular metric endow nuclear norm equal identity theorem together previous frame bi lipschitz metric particular clearly space lipschitz prove entire factor phase frame hilbert exist upper lipschitz explicitly mean lipschitz bound
irrelevant feature mf n class random train efficient mf online forest accuracy remarkably mf competitive batch forest fraction mf unable split choose split mf limitation forest influential machine extension mf hyperplane split instead axis align split acknowledgment e bl part hold fellowship college newton international fellowship european union framework fp agreement smooth label point fall leaf xt appendix detail training distribution exist posterior multinomial context special sketch picture refer reader explain inference predictive batch chinese restaurant wherein customer table leaf table customer parent restaurant class resort approximation particular smoothing smoothing interpret restaurant precisely customer restaurant table approximation follow leaf simply internal j k procedure serve fashion kk j add affect count leaf contain internal root summarize informally discount parent probability single pass jk predictive involve traversal starting contain leaf extend contain extension lead confident analytically along leaf gray leaf branch else point along root branch split equal lie probability branch j child contain location branch leaf node mean simply weighted fact discount distribute exponential forest initialize ty j x k k j p jk jx discuss associate trivially process binary processing leaf posterior count overall datum n factorial rf complexity asymptotic expansion hyper factorial discuss version nj depend nj j x nj nj j nj nj j nj nj label nj extent remove nj j x j j e u dx jj leaf else depth forest leaf fraction point leaf parameter experimental setup table report forest train c depth fashion tree true similar forest key mf forest ensemble whereas average limit whereas mf still multiple hence mf outperform scenario decision insufficient explain experimentally validate performance package tree leave constant leave multinomial leave correspond learner number mf pass support dynamic forest mf test accuracy achieve dynamic mf achieve accuracy dynamic dataset dynamic mf indicate usefulness guide split mf mf performance forest superior dynamic suggest explain fs fs fs fs mid fs unit college department page ensemble decision task forest test excellent candidate forest variant operate batch great demand forest require batch counterpart comparable work ensemble decision call incremental fashion remarkably forest achieve competitive forest forest fast computation vs tradeoff decade forest remain due robustness excellent survey novel mf present efficient agree online forest fast forest art dataset performance mf section conclude label focus class classification methodology supervise forest forest collection next point make standard expectation ty tree overfitte increase seminal introduce ensemble model average online example another incorporate label n n n tree incremental fashion irrespective order none random forest sample efficiently depth logarithmic sequence computationally efficiently forest define purpose rule predict root binary except distinguished parent zero leave child child leave tree internal parent location split tuple leave respectively rectangular along dimension simple partition decision tree split split root denote block associate red circle em introduce additional denote denote denote label upper training respectively b x small datum point node family refinement study care introduce algorithms nj add nj nj j nj nj n nj nj nj j start rectangle exponential split leaf node big leaf else internal split take probability u child tree cart split control total split maximum iv internal node illustrate consider family distribution range point distribution process possess self sample tree extend exploit datum upon observe tree conditional represent focus structure tree nj denote gray rectangle big rectangle extend outside split option distribution extent e r probability e abc consistent partition add data extent root split new contain I denote leaf prediction otherwise incorporate one leaf particular extension confident hence integration analytically depth forest vast attempt brief refer review forest batch setting tree split split optimize quality greedy manner random forest I bag random batch set rf bag node subset chooses subset choose split location bag tree split choose independent term unlike mf smooth test perfect totally randomize although difference forest grow tree every leaf every list split associate candidate split leaf node update sub base leaf minimum criterion good internal child candidate split process repeat memory deep cost maintain candidate quality incremental cart tree forest mf incremental decision tree single typically specify mf perform discussion comparison purpose fraction ii training time divide mini batch compare forest mf batch forest rf training mf author report
sensor time activity sensor aforementione fire change state pattern unchanged period unchanged presentation pattern sensor random begin say pattern presentation eventually form item item think present notice presentation presentation exist structure subsequent presentation pattern special high level pattern recognize item appropriately strength accord memory eventually presentation basis mean basis item see predictive contrast complete importantly without verify multiply pattern item none item allow delay sample depicted correctness item fire item shall accomplished initialization state repeat sensor yet establish review probability item fire item pattern repetition presentation repetition presentation step sample delay pattern item item present time essentially create share achievable tree another benefit decrease accelerate remark upon presentation item necessarily undesirable alternative parse turning pattern happen height claim chernoff bernoulli possibly response pattern since item activate input symbol fire recursively item iff item set item pattern next pattern encounter create whose item accord operational rule pattern must item item create item add item item pattern item whose high en none parent sensor round additional round sensor tree I round item parent item become step predict therefore item sensor sample exist item parent create presentation pattern item form input item create parent another level item round pattern create fact either lemma slightly weak number item fix pattern sensor fraction sensor involve long already common valid matter form since sensor n sensor pair valid already parent child pattern continue level round input arbitrary order time top stable sharp valid overlap distance pattern proof must draw fraction create come valid yet valid fraction sensor sensor sensor pick sensor level total expectation claim create create valid come presentation create number add create valid level claim quantity r high independence implement binary checking measure memory traffic input item presentation run every item presentation drop less traffic pattern percent pattern per traffic without restriction difference create early stable create predict show average create keep explore sharing pattern pattern obtain coordinate randomly ham consecutive pattern perturbation varied produces perhaps predict decrease pattern unsupervise currently one considerably pattern slow current intend certain aspect join ability briefly could graph basic direct suggest song et work chance far establish whole classical graph pair vertex case graph apparent connection elaborate introduce arc round ignore direction arc probability round specific fit rich reciprocal possibility little reciprocal connection back unnecessary carry three oppose small length item say probability hard see root discover process soon imagine create require algorithm see step synchronization simultaneous another whose implementation match furthermore join firing create assumes implicitly signal respective root early step fire fire become fire item symbol time availability sequence look form b form fine formation happen interference item star enjoy favorable establish introduce primitive intend capture implement item input later recognize activity pattern recognition predict part direct imply certainly lie point spirit work invariance recognition pattern challenge cognitive thing view object angle context high object reference language thing plausible analogous operation seem related machine besides primitive place huge sophisticated reason success elsewhere thing accomplish secondly sensor circuit among thing feedback loop eventually modification environment seem theory characterize kind environment evolution behavior acknowledgement grateful reference les early table see part signal stable approximately pattern size go pattern randomly varied pattern percent traffic third increase input pattern ht p pattern coordinate member visual possibility two sensor state sensor remain unchanged presentation next similarly state achieve appropriately strength zero activity aforementioned firing change present eventually current presentation mean difference item complete prediction importantly next verify factor state unchanged maximum sensor item random say eventually far think pattern follow presentation exist structure subsequent presentation pattern special level item pattern recognize example notation primitive predictive join extend join reasonably complex pattern involve phenomenon namely base traffic human thing computer intersection science task complex seem brain advance understand brain yet mind network solve nontrivial computational fashion reflect brain solve motivate present believe play gap inspire work les direct edge influence fact brain recent year visual example mt etc connection fire traffic hierarchy e g student brain prediction example vision visual also active input rapid field seem make layer visual control area visual third connection seem random graph reciprocal connection length direction would briefly brain connectivity way neuron call platform receive potential fire go world neuron vector formal brain read graph random know neuron fire cause neuron item element fire majority concept show join item exist establish via item stand conjunction main contribution implementation new item predictive join operation join predict whose prediction may specification discussion variety actually appropriately sensor word architecture happen adapt kind live operation nothing else create connect repetition share pattern recognize special item act already short item allow pattern section use item simulate algorithm create traffic generate presentation traffic consistent prediction particular cognitive quite bit issue control running algorithm appear agree well theory brain item consider disjoint shall disjoint set simultaneous firing majority define item create instant item henceforth item operation capability simple computation involve local computation strictly action argue capability neuron argue realistic need response operational ready henceforth item strength retain strength operational executed every employ intermediate neuron result fire neuron operational operational incoming memory operational strength small upon fire enter firing period fire model fire follow factor subsequently otherwise fire item enhance variation join join addition enter state ready firing predict whereby informally join intended item join elementary task create bc allow execute randomize something could perhaps component shall choose half expectation comprise strictly disjoint implement enter total operational ready well part context part reason discussion explain simplify considerably operation take fact memory parent identify come require double neuron
contextual track track prediction n truth track video specify flow extract infer truth label slack pay cost flow track optimize svm cut plane number constraint flow violate constraint solve iterate constraint video valid cut plane subroutine video subscript notation inference augment correspond negative inference behave somewhat differently constraint finding flow tracking generate relaxation constraint fractional rounding track incorrect tend tight relaxation flow track hamming ground label q indicate estimate truth critical aspect tracking criterion false false true true birth death consecutive link simultaneously routine consider pairwise propose decomposable link capture aspect account localization link rather constant loss empirically careful specification crucial order let four transition false detection detection true identity transition link virtual virtual virtual depending overlap transition virtual virtual virtual link virtual virtual ground flow practice specify ground need map onto video taking scoring window label assign track run simplify dynamic programming claim edge transition birth death false weight intra frame benchmark frame object object evaluate category comparative publicly evaluate percent label car trajectory trajectorie trajectory difficulty train special evaluation remove correspond training mining negative full sized frame frame overlap subsequence track link predict location frame via optical give good specifically optical candidate flow candidate candidate frame candidate repeat observe many raw post trajectory fit cubic compare publicly code first baseline scheme appearance publish successive default another baseline dp table various baseline baseline dp also remove overlap learn transition measure evaluation learn lp lp dp conduct leave validation lp dp outperform state baseline attribute appearance flow turns properly learn produce comparable dp flow much previous attractive feature shown learn lp mostly keep round approximation rounding inference via relaxation average time find lp round dp run cost forward relaxed produce often within relaxed global significantly round mt ml flow flow dp c c ml flow flow lp flow c dp flow dp flow c mt flow dp flow flow flow flow dp dp mt lp round ml relaxed preferable train conduct sequence cross validation dp lp relaxation inference inference metric stand inference slightly train lp competitive augment well tracking pairwise jointly optimize extensively evaluate traditional lp relaxation greedy dynamic programming performance support award programming successive show find find reader detailed sort short node short path dag frame last accordingly reconstruct variant minimize dp dp interaction apply find consist correspond subtract pairwise cost turn additionally pairwise node entire direct node end ji ji j ec track simplify construct instead edge cost original add pairwise cost pairwise cost backward turn backward turn often dp lp slow slow round video object pass dp use finish whereas dp lp round minute validation dp dp run backtrack cyclic note proper hash link list cache array second pass propagate label eventually might look overall pass still promise moderate optimization style default style style style default default style vertex default style default style default style vertex default style vertex default style vertex edge forward style edge forward forward forward style forward style forward style style edge style style style style style style edge style forward forward style style edge forward vertex default default style vertex default vertex default style vertex default style default style default style default style vertex default default default vertex style edge edge forward edge style forward style forward edge forward style edge forward forward style edge style style edge select style edge style forward style forward forward edge forward style default style vertex default style default style default fill style default vertex default style red style style default fill style vertex default vertex style vertex default vertex default vertex default red vertex default style style style style forward style edge style forward style style style style style forward style forward forward edge style backward bend bend bend backward bend backward bend bend bend style vertex default style default style vertex default vertex default vertex style default default style default vertex style vertex style edge edge style forward style style forward style edge forward forward style style style style forward style forward style edge forward edge style style edge style forward backward bend right bend edge backward bend edge bend style bend bend bend vertex default style default default style style default default blue default vertex default default vertex style default vertex vertex default vertex fill style fill style backward bend left style forward forward edge backward edge forward forward style forward style edge backward forward style edge forward style style style forward style edge edge backward edge style style forward style style forward edge style bend right edge backward bend forward style bend backward bend inner white thick draw thick draw frame video interaction track track contextual min cost approximate multi target track two track relaxation greedy pairwise algorithm across category enforce intra mutual co relationship within detection tracking topic advance build track avoid period since often find low often min matching min yield solution somewhat traditional generative formulation track draw estimate object discrete frame association observation track face joint inference track trajectory implicitly skip utilize successive could approximation associate integrating raise difficulty pairwise potential perform allow rich appearance maintain purely group combinatorial number rich undirected combinatorial often min cost pairwise track typical spatial object track object method order technique prediction affinity association task crf segmentation translation knowledge unique discriminative track death relation art category tracking begin track association min cost equivalent individual track markov whose video framework incorporate successive frame video discrete site site scale frame extract site frame collection track foreground generate background site track assume track behave expression map q appearance track site track find take log yield flow transition site know satisfy thus solve exactly specialized successive short path even approximation multiple short algorithm find apply original network edge track dynamic pass dp nearly variant aforementione track always key showing track integrate pairwise flow intuitive example would overlap boost occur object interaction site video integer quadratic addition discuss solution section describe conduct inference flow track min network explore build alternative global quadratic auxiliary integral lp round yield avoid expense relax cost objective wolfe algorithm relax keep video qp quite replace quadratic term new enforce program efficiently lp solver discrete track relaxed constraint round relaxation constrain optimize standard linear frank wolfe eq round heuristic subject integer flow execute round inspire
globally country country sum engine attention title past country country rt rt piece information rating ordinal nc plot distance plot cf capture solely historical cosine actor create share actor weight actor create actor counter membership large figure existence friend friend friend co target train rf predict rf normalize sale instance sale correspond predict decay demand release train week steady state volume predict release uncertainty great problem company term obtain late life cycle predict week uncertainty time next discuss construct sale interest forest construct drive go varied actual policy week prior week train week b issue historical demand transformation censor demand score reasonable value demand issue score censor affect compare one demand drive access product make fair comparison product demand market dataset past henceforth difference measure week live decision say objective sale volume impossible perfect average location city note insufficient neighborhood cost compute develop explicit impose function restrict expect transform term norm nuclear appropriately incorporate objective framework great potential problem quantile rearrange service level requirement auxiliary cf ms involve general expect distributional conclusion square quantile analysis predictive approach universal instead difficulty restrict portfolio allocation must apply equivalently constrain guarantee outside dataset run affine dimension onto intractable solve extend region maintain convexity planning quantity e positive end apply restriction norm show problem rule inefficient consider unconstrained develop condition proof convex every subgradient polynomial guarantee derive proof traditional framework derive rademacher c appropriately begin generalize multivariate give quantity also multivariate rademacher guarantee suppose confidence involve multivariate decision complexity linear conjugate consider sample relax restriction norm combine idea ms specific use optimal decision ms motivate tractable measurable together grow availability auxiliary potential impact asymptotic proof sequence random joint finitely consecutive stationary marginal distribution property sequence look ahead head nearly nearly metric mix stationary sigma sigma algebra subsequence total set square integrable value mixing establish mix mix satisfy mix cf cf thorough rate explicitly cf situation take example stock market dependent doubly stochastic arrival iid everywhere henceforth almost henceforth iid sampling hold n kernel iid mix process na I hold absolutely bound cost e kernel asymptotically preliminary stationary measurable mix sigma f measurable compact convergent nz nz restrict subsequence n weakly nz nz limit eventually k pg nz exist nonempty suppose loss c nz nz nz x x z exist apply consider least subsequence contradiction limit contradiction yield convexity compact restrict affine low semi theorem consider nz nz precisely e weakly f z nz constitute let hold desire statement iid mix consider mix hold therefore lemma turn yield desire I necessary choice I obtain hold yield use uniformity square choice iid boundedness consider combined boundedness equal hold apply manner converge yield assumption conditional support independence separable space bring inside establish suppose lipschitz involve lipschitz iterate expectation hand choice let inner choosing take left rademacher complexity class fix function note boundedness lipschitz hence complexitie z kx w f f complexity rademacher complexity get conjugate exponent term jensen norm follow result jensen expectation body via ellipsoid call oracle produce cut trivial membership weak constraint allocation construct portfolio observation factor eq generate security noise marginally minimize conditional risk b example serve space evenly circle space evenly figure advance per last simulate example portfolio allocation example operation management science three quantity simultaneous associate recent google search company news review traditionally ms focus period generalization priori cf presence identically distribute iid extensively foundation learn hand largely usually univariate base large address optimal decision uncertainty availability advance game query office sale manner ms good good decision account uncertainty absence mean aspect solution idea ml decision optimal joint decision future task full key contribution way construct paper motivate specific construction inspire great variety predictive ml forest rf summarize construction traditional erm ml construction erm multivariate value encounter ms construction limitation proposal study general cost full optimum limit decision surely observe demand via sale introduce term order content helpful analogue determination management international entity million unique world scan trading year internal company spirit public online source google combine approach improvement account toward counterpart construction predictive broadly idea approximate locally great cart imply forest rf imply xy mx identity leaf cm implicit construction focus period realize one make uncertain quantity multi extension uncertain realize subsequent decision period illustrate make auxiliary section real portfolio allocation decision portfolio consist uncertain return security interested risk loss negative quantile write extra decision return eq product location stage unit product store per satisfy location unit production know portfolio security analyst rating volume google searches company planning past weather forecast volume search us leverage marginal ignore datum q try predict observation guess approximate drive forest decision portfolio factor location predictive demand evolve process order simulate average solution distribution true drive decision performance quickly latter remain observe upon study general asymptotic proposal convergence observe guarantee mild ignore use appropriate outperform datum drive past construction effectively eventually motivate rf notable bad dimension relatively well e world uninformative normal performance dimension cart rf largely serve approach focus ms commonly active approximation drive optimization notable approach decision make robust cf variant literature inform far cf iid ml supervise wherein expectation regression mode interest cf cf function square specified minimax decision intersect erm criterion ml arise erm extensive cf ml ordinary ridge quantile regression decision erm policy equal cost management consider loss result square useful general ms decision erm show limited generalize class ms decision instead erm mode nn locally constant method notable connection asymptotic rigorously local pg even recursive method often form notably great former partition average cart popularity interest minimize observe true specifie one fail conditional require way generalize cost similar problem estimate predictive construction motivate construction take weight decision understand case motivate tie break index rule computation speed f variation neighbor regression neighbor neighborhood unitary nonnegative density ratio particular lead predictive square content measure fraction explain take average perfect know involve estimated cost deterministic counterpart third drive poor serve analogue prediction denote useful significant reducing leverage purpose predictive converge grow instead optimum figure allocation example knowledge correctly take denote successful property optimality proof optimization tractable give different complexity solve completeness separation oracle fix subgradient oracle call ix effective nonnegative nonnegative weight polynomially present optimization converge full sample guarantee mild condition asymptotic almost structure assumption constitute iid strong velocity variety collection mean historical world process chain evolve market daily google topic extend present rest iid mention follow existence continuity regularity either yy condition follow asymptotic nn optimal kernel hold kernel asymptotically I absolutely bound cost result cart observe distribution arm international medium lead ask identity figure location cd home scan capital hold cost medium mainly production medium secondary capacity location medium display back store storage sale studying restrict attention video particularly inherent whereas period demand determine demand release much trivial new demand sale formulate index product denote quantity uncertain demand capacity effect per add regression fixing capacity replace via optimal regression dual optimal conditional issue sale demand censor approach correct datum threshold censor appropriately I weight conditionally share support cost bound share event observable conditionally give past decision make realize cost drive internal company public source public datum company consist year aggregate sale week period sale week censor observation demand occur develop transform tackle sale cycle home sale sale week release pose great sale demand include chain location country google interface item item title title descriptor compose information france item title fr imply may title public box office review item desirable
produce would term tree multiclass five become fine fine grain previously autoencoder input different department ci decision every leaf allow every jointly backpropagation like neural even allow make soft root power perform classification internal operate build function leaf binary return node decision subtree depend multidimensional output regression many attribute threshold split relax linearity regardless leaf decision perfectly decision decision step grow search split reach improvement entropy child keep leaf accordingly prune tree replace leaf decision tree build multivariate decision sigmoid tree split stop reach descent tree soft tree leaf node degree traditional decision tree end encounter continuity induction exploit continuity backpropagation compute nonlinear train convert recursively leaf contribution internal toward leave train soft activation consider activation remainder mean response convex combination leave softmax power hard threshold soft threshold thus partition part node extend tree child node locality leave tree control child linear leave independent local path intuitively layer softmax threshold imply number leave traditional hierarchy tree essentially subtree activate activation node leaf subtree hierarchy representation feature must together mse com fm nh opt quantitative task ten binary uci task discriminant tree baseline separate final remain
strength smoothness parameter dependence describe simplicity carry water period bottom row fully bayesian collect hour hour standard intel core ghz gb become cost design work operation store base carry computationally massive traditional non depend allow communication low fix methodology use store extend spatio temporal demonstrate world sensible however get likely refined complicated full dependent explore work natural multiple measure principle stacking weight big require due size become prohibitive process acknowledgment material upon support national science foundation dms sciences institute support nsf sciences dms make aware problem spatial helpful anonymous environment comment discussion derive likelihood write determinant formula z j j follow desire sparse k diag j decentralize kalman filter j j j server n diag v rapid often carry distribute fashion store physical location relevant divide case inference move central low spatial scale parallel massive datum exactly cost communication size extend spatio temporal methodology distribute spatio particle filtering water sensor spatial random temporal datum capacity thousand past decade transfer therefore tool minimize movement computation situation arise massive relevant give costly avoid unnecessary goal analysis around relevant problem location parallel achieve work memory approach repeat substantial goal computer individual focus situation modification environmental science frequently usually contain environmental store center throughout aim measurement advance national resolution imaging conduct spatio temporal call total water measurement store consist spatially fine recent parametric covariance situation low model spatial scalability show full propose spatial present inference spatial low apply decentralized surveillance rao carry massive server operation depend suit distribute traditional computational ignore spatial purpose science massive describe evaluation considerable movement literature filter prediction sensor collect aware individual massive aware previous analyze distribute case article distribute context would correspond might ignore dependence substantial coverage block dependence work unobserve e g belong effort spatial dataset exploit split suitable organize brief review distribute describe discuss inference present important describe spatial extend spatio section total water measure sensor conclude interested spatial j give note order arbitrary way spatially measurement origin assume follow low scale trend vector assign form assume measurement massive widely class spatial spatial discretized mat ern parent covariance posterior numerical likelihood base inference normalization low rank j j us frequentist inference use bayesian server evaluate central updating sequential sampler various exact result evaluate suitably move create calculate transfer central calculate j take advantage parameter inference carry calculation evaluation assumption server unnecessary server constant likelihood update z j wishart distribution gamma spatial data amount carry inference final estimate frequentist particle nonzero weight determine none prediction location measurement support continuous spatial throughout manuscript diag p describe coincide desire spatio spatio temporal vector autoregressive jt z jt call spatio temporal effect temporal low first filtering mean interested obtain filtering collect time obtain decentralize nest filter outer filtering essentially initialize prior time server matrix server j central forecast collect might smooth p also actually information server filter smoothing time appendix small unknown spatio low rank approach resample natural inference use algorithm particle weight smoothing likelihood apply spatio filtering inference
sample spanning assume point restrictive lebesgue density combine proof link example visualize plot identity set dotted line represents represent estimate figure fig separable indeed suggest divergence section outline divergence distance exhibit make useful relatively straightforward fact attain value divergence express f dd pf pt requirement special use estimate deriving notation introduce slightly easy pf simplify notation estimate theorem problem binary formulate find distribution probability rate combine result eq tight completely go rate base affinity appendix affinity measure special bc widely use motivate develop new algorithm popularity bc mainly bind provide bc separability class result relate divergence f pf pf expansion chernoff local equivalence since induce manifold term tight since surprising since bc bc tight ever comparison sample come mean distribution overlap entirely separate integration empirically bind display two bivariate tight calculate never bc empirically estimate bc due variance estimator divergence consider distance rate two domain correspond source represent us distribution decision true similarly target identify error difference measure follow assume data distance minimize characterize labeling target us mean target target shift exist label g assume rule attain bayes distance matrice q label provide insight representation error source distance heavily word quantify versus feature separation class high third classification alternative consist distribution error analytically bc bc furthermore assume require c evaluate bind outline monte simulation assume parametric mahalanobi close regardless stress expression perfect knowledge empirically estimate machine learning reduce prevent performance dimensionality densely prevent task adaptation separation selection two seek bad minimize scenario da optimization forward search alg use parameter determine domain machine adaptation set minimize tune minimize domain j f empirically speech record patient classify speech speech speech laboratory consist include mixed disease pd patient speech phrase speech minute record speech split individual extract sentence envelope spectrum feature spectrum p slow detailed reader fs feature draw speech ensure separability selection maximize build training evaluate repeat ten result initial fast compare bc roughly restrict set classifier success stay bc method bias distance estimator seem separability bc converge bc bc efficacy adaptation problem evaluate order select different make training comparison selection algorithm bc assume come fs account domain bc domain compare attempt prediction performance source lagrangian classification use minimal contribution target reader involve source domain separation separation resemble fs present accuracy yield top fs train table high classification accuracy bc low trial utilize domain type scenario normalization yield accuracy low generate accuracy display accuracy feature propose improve additional criterion select minimize tendency safe e prefer informative helps build top return da fs feature return criterion similarly application top return da fs separation similarly feature source bc divergence bound error training first tight finding criterion speech rate alternative future around analyze understand convergence size fidelity furthermore bias estimator apply improve feature space text rewrite eq link manner begin bayes q next show tv q simplify begin relationship vs harmonic immediately chernoff upper distance tight clear theorem tight less use cauchy combine begin fashion identify target tv using express play statistic improve classification
method try partition ideally bs outperform setup well storage every contiguous bs contiguous location per bs mention mention bs propose exactly row comparative approach bs convergence note express common notation table method assume randomize partition block block row minimum eigenvalue denote method c method c x randomize op pd p able rich check optimization program subsection subsection show specialized instance generalization unconstraine suggest strict function framework notation space linearly generalization guess column affine concern easy see guarantee always use point alternatively fact fact connection framework suggest pick know guarantee descent sequence appear possible idea obtain minima special say alone strictly unique point fact easy see update framework block set precisely minima q contrast framework indeed numerous observe column dense store entry minor modification minima confirm method greedy unfortunately run briefly mention idea greedy pick block goal block follow immediate let partition strategy eq iteration immediate amongst choose block close norm amongst iteration well every th th serious computing gradient coordinate unlike run algorithm tell iterative block bottleneck remain fx time run operation run time per entry main requirement remain discussion put partition end subsection describe give explicitly deal secondary storage ht preprocessing store storage device p p fx section brief greedy strategy method let method block block k large amongst iterate requirement feature proving partition input effect possess iterate input make diagonal block diagonal well obtain convergence eigenvalue observe occur e k iteration index suppose operation detail reduce reduce node setup divide group associate storage device node storage device retain memory running modification preprocesse implement transfer hence operation second usual emphasize serial rest lead resource distribute advantageous delay incur storage device device device spread processor second dimensional setup copy section give matlab wish approach greedy conjugate gradient experiment realistic solve experiment highlight limitation strategy randomized scenario equivalently intuitive behaviour also use partition implementation moderately specification carry gb ram gb secondary space intel processor choose equal gb gb ram matrix break variable rule mas store file disk generate require approximately denote store individually entire operation result file mb entry define numerically entry dominant store file minute solve descent nesterov randomize define method implement block block partition maximum choose iteration require read submatrix plot sd bs identify individual randomized average method demonstrate intend superiority propose matrix highlight crucial randomize place marker trajectory descent bs respective conjugate take roughly bs take roughly period bs result subset column store finally arbitrary method matrix memory step involve secondary storage read preprocesse block randomize choose square origin comparative convergence key observe mainly matrix randomize particular almost large submatrix irrespective lie fashion lie experiment choose manner experiment specifically choose input block keep row index block remaining row arbitrarily partition block make loose use start origin performance paper resource desire row partition matrix amongst reveal block partition input almost simulation method find fast however remain open question arise web social problem characteristic consist million describe program bring challenge concern management setup adapt resource central aim simple descent one suit verification however method prove bring well amongst novel deterministic serial parallel distribute variant confirm conjugate paper essentially quadratic solving equivalent eq share aforementione nonzero involve resource available hardware technology main memory ram processor gb device observe store double I firstly storage device store necessity slice store secondary device third portion processor run memory reading processor time secondary storage location add overhead point iteratively improve processor hour million running discuss across processor practitioner usually processor limited resource per improve solution quality cut marginally usually mention decompose applicable block lie dimensional dense pointed serial leverage single processor availability setup henceforth limited sized secondary storage device number enough store dimensional solve develop far exceed main size serial processor convenience setup processor discuss distribute multiple processor practitioner prefer entry reason setup order run view say never entire course execution store move secondary hard consuming require load datum reason disk contiguous contiguous method addition prefer method possess run solution good scope distribute implementation parallel survey exist generalization one prove way forward discuss equivalent greedy establish bound way improve section give finally conclude direct classical broadly member require consequently unless solve high see survey discuss research gap art direct begin svd factor finite take dense give popular member elimination cholesky lack couple fashion setup method solve repeatedly improve moderately appropriate modification ahead method solve broadly cut plane direct every iteration current close descent gradient modification decide estimate gradient cut plane method feasible contain hyperplane centre contain new idea repeat difficulty efficient centre hard ellipsoid overcome alternative setup avoid quality member solve method begin simplex transformation sequentially function slow simplex run iteration include also iterative line search never end subsection reason fundamental use row matrix row fashion refer method traditional multiple per partition block round iteration improve estimate notation update clearly two partition use work choose emphasize choice guess pick k simple gauss see select iteration round unlike match index traditional arbitrary traditional method choose one round fashion coordinate
use package bm bm package http www cs bm ref available http es software c filter wavelet thresholding bm bm review rest terminology denoise method amp result amp bm bm call bm free amp efficiently minimum mse strategy denoise variety exist estimate standard image convenient amp package algorithms standard deviation package tune bm bm variant amp package skip without self tuning package tune challenge early effective effective small iteration table construct artificial additive white choose noise simply choose parameter maximize problem look optimize code recall evolution comparison bm increase attribute effective correct behavior bm optimize among value test contaminate notice level look control within amp amp typically stop less threshold demonstrate bm suggest variance remain reduce variation decide bm amp run implementation except amp calculating review insensitive exact wide pick effective amp calculation effective amp observe performance bm define value estimate intermediate denoise house measurement amp amp amp within predict respective online researcher evolution additive white reconstruction bm db amount measurement might expect denoise base amp robust outperform db c bm amp cs bm amp amp c amp amp amp bm bm amp bm amp dramatically cs amp presents signal balance extensive approximate amp sense variation denoise amp amp maintain performance amp evolution prove tuning amp use amp robust amp plug match amp since design denoise easy amp different area amount develop amp rely assumption signal follow support evolution validation future research likewise subgaussian matrix matrix fourier direction minimax amp state notational simplicity useful proof interpretation prove w exist last large evolution happen hence establish interpretation favorable amp e least favorable return suppose consider ax vector incorrectly part prove consider n way result low risk hoeffde hoeffding sample belong define condition derive risk bayes word risk step note define conclude dominate prove corollary remark conjecture denoise algorithm seek remove perturbation error devote amount compressive cs recover reconstruction answer natural effectively employ develop pass amp demonstrate choice amp offer state cs explain performance amp analyze critical appropriate fundamental compressive reconstruct dimensional signal measurement compressive measurement thought mapping model physical interpretation camera inverse transformation reference refer compressive sensing problem represent determined search recovery formally pursuit denoise first perfectly cs deal program extremely demand iterative develop pursuit iterative hard compressive soft linearity residual iterative thresholding extend extra eq average vector correction term illustrate figure amp effective straight line gaussian enable optimal lead employ amp work many signal majority imaging dct show classic signal coefficient far seek failure researcher elaborate recovery include variation sparsity hide non self representation image complementary non develop enhance simple denoise whether explicit employ denoise capture complicated recovery scheme inspire design approximate use denoise measurement advantage summarize analysis framework characterize limit suggest use amp assume belong signal class close employ treat receive employ achieve derivation applicable wide signal class amp employ correction algorithm many formulation require explicit exist denoise intuition amp eventually predict amp track deviation amp extend evolution mean square amp characterize amp amp also enable amp presence practical concern amp employ wavelet amp local denoise call wavelet thresholde amp original amp last pass message pass extensive compressed paper belong pass calculate pass simplified message employ state analyze amp signal introduce develop elsewhere amp whether approximate rely amp noise extensive writing paper aware pass algorithm employ pass adapt statistic major broad adapt recovery many noticed sparsity explore explicitly penalty functional encourage initially structure wavelet researcher dictionarie art penalty functional fit way relate et bm impose solve add noise miss spectra bm bad test finally emphasize approach application amp come behavior remainder devote amp connection evolution state evolution explain calculate correction summarize main validity set tuning performance amp goal estimate extension hyperplane point hyperplane move orthogonal hyperplane obtain repeat two step move project correct ease notation residual call replace assume figure apply unfortunately strong observe thresholding passing resemble gaussian rigorously prove class prove wise discuss subtle avoid non pass estimate similarly demand fortunately message pass algorithm amp correction derivation mp algorithm amp finding confirm display effective noise bm bm amp observation evidence simulation behave property leave empirical evidence theoretical implication conjecture main amp framework setting capital letter column transpose scalar element random event respectively expect two variable denote expectation value belong family index take require proper monotone level definition chapter class signal generic proper also eq every every proper since onto linear eq furthermore freedom cx complicated example signal sparse vector family short mention notational equal therefore straightforward see desire combine every figure signal instance image bm monotone monotonicity standard group monotone straightforward simple inequality independent monotone statement non improve monotone ingredient amp evolution measurement amp predict formally amp starts amp compare amp bm denoise amp figure amp check find bm mean amp wavelet simulation post validity explore conjecture range amp base require evolution call monotone noiseless amp monotone well amp claim monotonicity every induction every q hence converge conclude value combine amp value amp successfully successful goal measurement natural amp address evolution apply section transition threshold amp noiseless devoted presence measurement proper amp noise e amp recover noise sensitivity amp sensitivity variance level hence satisfy x calculation interesting emphasize noiseless setting amp theorem mse number measurement decrease certain certain variance variance perform amp rely effective parameter regard extensive diverse sure stein unbiased risk estimation scheme amp free produce performance amp jointly different notation set evolution change emphasize pick ask q note produce amp amp soft thresholding seem turn amp find greedy parameter tune call minimize denoise amp follow result prove optimality greedy strategy sense tuning induction hold monotonicity hence clear discussion amp difficult denoise amp researcher area denoise art bm employ optimally tune amp denoise employ scheme solve amp family recover outperform amp pose sense denoise recovery employ outperform amp use different amp framework amp section proposition amp recover signal measurement question ask recovery signal answer amp amp uniformly amp denote minimum accord dimensional integer fundamental recover amp class consider image amp unfortunately therefore evaluate amp optimality signal signal find amp employ denote evolution quantity amp minimize family note definition algorithm recover measurement amp mean employ amp unfortunately amp text book signal noise expect minimax family amp order recover amp require since proof appendix optimality amp less unfortunately answer example signal denote rest confirm standard normal amp involve amp ambient actual require case amp accord proposition class characterize signal amp sub optimality class natural image recovery amp eq proof state evolution low regularization recover cost regularizer consider return value many case propose solve accurately amp heuristic heuristic amp evolution theoretically sensitivity amp depend efficient amp optimally review amp optimization amp calculation correction employ passing study carefully direction researcher year area model approximate state analysis seem framework bayesian evolution framework evolution work observe goal calculate amp purpose amp state equation emphasize employ framework family definition set support eq side general hold state infimum information theorem set far key yet address provide calculate correction review correction relation give output popular signal literature group signal block soft calculate result mention elsewhere help soft denote x denote soft b word magnitude toward notation result signal diag employ rank divergence thresholding yield solution high dependent monte carlo work show calculate efficiently monte carlo obtain estimate signal efficient
replace shortest many coarse true distance sdp know fairly missing find location distance translate scalable sensor stress miss eliminate local approach exploit distance consider low classic integrate property reconstruction incorporate expect fidelity reconstruction section thorough evaluation array calibration simulate evaluate scenario vary number percentage distance present assumption pairwise hoc calibration local connectivity present hoc array vary uniformly diameter pairwise addition miss miss vary distances distance due limitation noise average quantify define evaluate illustrate deviation er rao quantify improve mc position cm reduce cm although mathematical calibration configuration pairwise figure improve express observation get term cubic room dimension simulate position configuration generate alternative implement sensor number distance source assume miss white require position ratio miss distance missing effectively handle require calibration miss e calibration present pilot synchronization model uniform model additional distance range cm diameter repeat configuration calibration quantify illustrate effect increase distance measure accurately small study room room acoustic room rigorously method approach set circular uniform array diameter cm array apart room dimension cm circular diameter channel array diameter cm deviation miss due distance miss scenario list repeat experiment reduce eight diameter cm c setup datum performance room cover big window rectangular room moderately time contain locate rectangular database sample pairwise distance two window parameter fourier compute coherence function frame state reasonable estimate confirm regard scenario nine estimation use rest demonstrate map stress completion quantify result belong stress cm comparative illustration mc propose well propose problem distance array method offer illustrate half alternative map poor entry stress sdp close mc mc property completion algorithm enable array partial pairwise distance demonstrate relation calibration channel circular also locate calibration list calibration eq position calibration line analysis propose calibration hoc partially recover missing entry euclidean distance project onto array calibration confirm exploit iterative reconstruction theoretical applicable framework ad hoc calibration find norm squared distance structure hence physical norm vector circle probability include structure miss necessary depict low probability location circular depict locate right upper bind figure circle eq constant union grow ratio increase grow positive great eq big extract use chernoff independent science foundation research interactive modal information management I acknowledge anonymous clarity manuscript ht symbol symbol circular square entry square cone radius define observe entry estimate calibration logarithmic distance error bar error bar correspond deviation mean calibration logarithmic quantify versus bar deviation ht position versus bar one standard error versus source total calibration standard pairwise effect mc quantify position error bar correspond miss ht evaluate number estimation trial cc c lr lr lr stress lr lr mc lr lr lr ht ht array c lr lr sdp lr stress lr lr mc lr lr lr array calibration c l lr sdp lr mc mc lr lr mc circle rectangle draw size hoc array consist propose miss entry alternative rank completion distance cone completion locally structure measurement know calibration level miss thorough theoretical achieve art hoc hoc array calibration cone completion hoc array sensor spatially acoustic ad hoc processing acquire sensor challenge attribute asynchronous position refer precise source localization separation study source specific distance source distance reconstruct array geometry self calibration source arrival source negative signal source signal arrival cross two source prior may et acoustic measure al exploit distribute platform arrival correlation use location since robust al auto exploit asynchronous spatially distribute acoustic impose sound align arrival measurement bilinear approach minimum sound chen likelihood distance extract array geometry joint source localization linear along arrival extract refined position delay source exploit structure rank enable position source minimal exploit calibration propagate equal direction coherence noise pairwise reconstruct line compact sub calibrate coherence field sound activate position response source use coherence field pairwise estimation practical assumption distant audio requirement ambient many provide enable calibration play increase goal pairwise miss arise measure activate device may acquire sensor fail capture leading distance ad hoc array distance base sound propose theoretical ad array impose distance euclidean square pairwise past year devise scheme recover know et reconstruct low et show optimality exploit nonzero regardless applicability array calibration proximity coherence signal approach imply connectivity pairwise far construct pairwise unknown enable pairwise approach goal combination measure array geometry pairwise distance cardinality final remove distance become effect entry ij ij ij ij sub setup assumption exploited state absolute scenario distance quantify position robust transformation denote distance summarize exploit rank recover recall distribute recover collect measurement rely carry ambient intuitively degree column recover define completion recover distance estimating row twice row row dominate characteristic thus entry decomposition singular guess provide minimization gradient last completion rank matrix distance modify composite property property modify step version transformation make output element characteristic error iteration algorithm cone figure serve dimension cone project follow inequality distinct projection thereby cost thus denote thus less summarize use search classic completion whereas yield reconstruction position completion well calibration provide hand singular decomposition loss incoherent great provide great explain great relation
nan orthogonal small close al consequently near suggest another author innovation science china foundation introduction china university mining meanwhile like molecular chen lin yu lda recognition small sample pattern recognition introduction usa g van computation implementation discriminant neural network al fold alignment dynamic journal theoretical biology perspective linear matrix multiplication al improve discriminant analysis vision application cm project wu com national foundation china grant natural foundation introduction china mining expensive recently discriminant multiplication scatter arbitrarily orientation may useful discriminant lose investigate nan guarantee orientation matrix discriminant analysis lda nan discriminant analysis nan objective remove irrelevant datum intensive processing mining powerful disadvantage scatter dimensional space usually centroid centroid scatter scatter matrix moreover generality assume realize scatter scatter scatter scatter singular sample overcome difficulty nan class scatter essence nan h satisfy theorem et nan nan lda refer et orientation matrix incomplete let suppose x rank pick matrix orientation lose discriminant choose criterion method satisfied sufficient orientation geometric since orientation eigenvalue correspond nan diagonal thus eigenvalue prove follow notice span span c therefore theorem give column position evaluate practice n upper
edge valid adjust begin node begin relate graph construct prove path path summation figure path score valid always label labeling produce specify latent come index label least different labeling clique pass position position except position form clique pass pass different clique difficult base world concentrate distribution mass rank reason since gradient objective analyze trend people maximize function portion regard higher dominate ascent latent label gradient degree rich trend rich trend also meaningful discover confidence concentrate task another control overfitte potential concentrated world find probable conditional viterbi definition search indicate search rank label mp p remain remain produce short probability map dynamic forward probability generate latent labeling leave labeling detail viterbi heuristic produce backward style compute corresponding try label remain label probable unnecessary remain p k label exact label large p condition k right n implement crucial design heuristic heuristic heuristic top probable probable easy way achieve find top score normally current current current heuristic algorithm never heuristic guarantee want try enough concern monotone heuristic heuristic informally formal monotone path generate monotone compute lattice search heuristic position set start heuristic viterbi score variable represent value step approximate try intuitively first intuitive alternative speed worse optimal approximated viterbi inference two step search labeling derive mean traditional estimate token compute way label marginal position naive exact rough estimation refer posteriori aim likely configuration complement researcher solve approximate exact search tractable require characteristic bayesian tight upper develop branch search map unclear crucial unclear concern make formal show np hard disjoint assumption importantly inference naive bound able fast latent latent field special popular language processing processing decode inference unclear try decode np furthermore able exact analysis decode hide capture conventional classification parse structure refine vision structure structure example area model advantageous representative widely syntactic parsing demonstrate latent recognition outperform widely svms hmms syntactic parsing without conditional programming viterbi nevertheless latent inference chain structure unclear limitation try simplify limited real distribute top try show reasonable field inference systematically combine top dynamic probable training class reduction hard basic notion value hard super lower know np clique clique labeling inspire consensus hide establish hardness analysis index define clique simplify divided score score
robustness statistic parameter shall type g follow composite hypothesis I variable proof independent statistic composite hypothesis theorem expression statistic normal easy verify property type test derive example justify level power test consider quality shape scale fact apply x nx nan hypothesis simplify similar testing q respect represent note asymptotically calculation asymptotic give asymptotically chi square freedom contiguous test asymptotic chi centrality contiguous centrality increase contiguous level r influence therefore influence become note always zero several boundedness second influence imply classical solid stability result derive function nan robustness asymptotic power contiguous alternative q q chi visible b statistic next level contamination proportion nan chi square power robustness different important applicability arise tune obvious control trade test pure contaminate robustness efficiency recommend enough robustness drive hypothesis testing asymptotic contiguous regarded efficiency example increase contamination increase approach construct minimum paper influence analysis carry justify influence remain contamination contiguous whenever influence influence justification behind power influence classical result exhibit chi establish test perform theoretically overlap one context model x great density derivative take function second order eq q frequently get get obtain note eq get follow axiom conclusion condition conjecture theorem exercise remark summary mm grant statistical university robust classical composite hypothesis base density power robustness property chi factor stable whereas confirm secondary density estimator influence chi testing hypothesis statistical procedure although misspecification small misspecification hard individually real life fold level test robustness robust stable neighborhood consistent arbitrary produce contamination nan contiguous robustness compare mostly develop examine robustness investigate global reliability although extended concept see besides consider statistic study reflect contamination nice review test however idea robustness study except chi overall contamination namely test popular develop empirically robustness fill develop robustness motivate researcher theoretical procedure also organize section section composite derive present justify result develop discussion choose conclude provide dominate lebesgue count two density derive limit case turn divergence density inference parametric density represent density divergence denote maximum estimating unbiased estimating equation nan equation behavior intuitively apparent n w np centrality central chi random centrality shall parameter define restriction rr type q type test statistic composite nan hypothesis chi degree possibility contiguous hypothesis consider increase move towards substitute equivalence nh power test contiguous alternative introduce crucial interpret influence influence function particularly robustness statistic observation underlie influence statistic view study influence influence establish divergence q function hellinger model know w simple hypothesis show adequate functional take test test statistic statistic al hold model power statistic statistic nan influence eq easy derivative contiguous important robustness look contaminate contrast contiguous contamination
cell alone find neighborhood connectivity converge quickly probable type chain detail similar recover know circuit generative broadly rest brain cell connect cell cell approach connectivity reconstruct cell hour divide cell type well approximated fit type distribute cell source closely reflect determine narrow medium wide cell fig reflect fig middle neuron author highly fine type even close agreement produce probable cell sort structure dark cell plot block roughly homogeneous collection along closely agree cell extent agree know red method outperform simple connectivity compare link fig far get particular distance depth lead ari human spatial region space essentially clique type extent layer variance continue substantially encode visualize cell type block subset cell cell group broadly coarse identify previous predictive curve inclusion source predictive accuracy additional conventional depth area type human recover connectivity repeat early unlike concentrated various project locate differ separate one chemical electrical initial cell connectivity electrical cell adjacency show identify cluster cluster position cell type coarse text label electrical determine inspection roughly neuron close agreement classification outline white cell reflect design neuron neuron vb mostly head together connectivity cluster pure correctly place single type reflect combination head combine mostly db neuron split thus reflect type entirely type artificial structure style spatial technology circuit identify processor complex bit homogeneity mirror original retain digital state contain additional identify spatial figure show horizontal vertical allow discover circuit like incorporation link accuracy discover base probabilistic probable parse cell connectivity carlo initializations converge similar obtain optimum broad offer adapt precise solution probabilistic become slow increase probabilistic area recent method work large expect close agreement exist become statistically cell know connectivity divide hand monotonicity connectivity class cell never less know insufficient neural wide functional discovery manual cell day type change understand modern allow discovery circuit development identification quantification phenotype quantify neural activity neuron across connect neuron one way already molecular marker could important cell appear molecular selective visual allow comparative quantitative aid combine connectivity way towards rich connectivity relation type one experiment brain rich model become human huge similar molecular biology gene find evolutionary biology ultimately resolve link connectivity matrix define connection cell material extension connectivity exist number latent cell assignment connectivity cell latent hyper jointly vector indicate cell step per parameter preprocesse runtime thank read manuscript discussion manuscript review amazon web services education grant derive test run kk manuscript text feedback interest material email berkeley edu probabilistic model incorporate entity extend take define cell material assume cell belong connectivity base well function jointly posteriori map assignment global bernoulli spatial cell type place graph process assignment determine automatically cell belong class number new global grid learn depth thus indicate conjugacy allow depth depth contact contact mixture mixture representation distribution contact study posterior monte anneal burn transition construct ergodic ergodic sampling lack conjugacy explicit assignment motivate auxiliary explicitly represent duration scan employ transition hyperparameter independent slice component slice hyperparameter global parameter discrete tuple transition chain kernel together otherwise chain random visualization full proxy prediction accuracy compute fully potentially overfitte bias favor compete collection computing cluster latent one vary dimension reasonable accuracy clustering dramatically generative setting collection markov chain pool probability profile connectivity actually inference mix chain evaluate sample distribution area proxy truth appropriate figure ari total runtime change initially temperature runtime characteristic jump nearby run ari importantly regardless exact variable parsimonious largely connection serial yield place contact cell originally select provide ultimately contact cell area thresholde entry input center logistic publish isolated chemical originally cell position distance axis normalize direct electrical undirected distance poisson extract gate source unable consistently source ambiguity effort create six terminal example connect likelihood probability poisson explicitly count neuron distribute learn rate cell close cell per parameter source code source code run along figure available please contact publication access multiple graph handle simultaneously extend product likelihood grid space poisson space lambda set adjust rand ari clustering identical ari become negative
denominator resp bind upper take numerator term experimentally cc x whereas context variable absence direct pathway exposure lead improve lower certainly case relationship consumption relationship plausible analysis allow direct unobserved keyword analysis explanation study conduct cause law want causality discuss topic claim ann drug death cause observe hand medical create address happen ann take drug question ann drug assess cause prediction formalize denote ann drug code resp address query decide I prefer associated decision situation query drug would try imagine would ann take drug ann actually drug likely drug address long purely condition know nevertheless answer introduce response exposure regard exist choice response observable observable fact contrary address name triple define ann q ann knowing actual response probability response ann drug quantity suppose experimental tested take ann live variation proportion compute table make causal x experimental drug ann drug death ann exchangeable individual subject exchangeability identify datum never observe never dependence inequality experimental risk death case trivial exceed often assess converse false find likely causality ann refine pre condition original fail adjust ann ann conditioning ann even ann sometimes refine thus know deduce ann refine bound wider far perhaps conditionally assumption ann x able observational ann joint observational bound experimental table observational live thus find deduce ann take drug involve pathway exposure causal split effect experimental ann additional evidence refine formalize
approximation semidefinite fast importance sample hadamard paper extend show perspective improve notion bound convergence approximation understand kk call original nystr om low within matrix use score column kk interpret implicit set form kernel accomplish variant inversion algorithmic statistical adopt exploit generally compare nystr sampling answer open zhang et divide parallel prove evaluation require evaluation notice comparable om leave open fundamentally well indeed nystr improve obtain statistical combine approximation let cast minimization reproduce kx reduce optimization solve matrix datum sequence square estimate point shorthand g risk estimator decompose decrease matrix proof need cubic running approximate low operate small matrix contain sample nystr om matrix multiplied produce preserve desirable om trial also random perhaps satisfy structural behavior concentration one statement section nystr om begin upper estimator construct place nystr om process uniformly zero equal sample column find b easily change notation arbitrary deterministic statement property randomness via highlight adopt offer clear result satisfy dependent algorithmic improvement theorem k monotonicity replace suffice second random concentration notion statistical leverage use one art bound concentration among constitute tailor yield still result integer replacement accord hold conjunction prove result eigenvector square matrix projection soft truncation smoothly smoothing close analog expect ss exhibit sampling robust quantification tail much effect optimality distribution reflect particular tight prove leverage ridge nu formulae na compute ridge quantity statistical literature classical hold I base sampling ridge leverage information column uniformly small accuracy high leverage become essentially equivalent away performance point still ridge na compute matrix well roughly take approximation description construct construct th time cholesky multiplication approximate leverage thus run note involve everything compute dimension correct inversion compute formulate memory construction involve additive multiplicative quality nystr om approximation probability remark column sampling square extensively randomized gram multiplicative scale instead virtue multiplicative weak minimum eigenvalue denominator concern think like sufficient column sample need easy see ridge score suffice albeit number particularly subsequent completion preliminary eqn leverage whether iterative technique score rank regression real dataset consist database provide method goal fold simplify setting ridge h nb linear fm nh fm nh former goal dataset freedom result small degree freedom work synthetic sequence uniformly score nearly optimal fact prove distribute ridge score importance sampling beneficial see density unit center unit observation figure leverage point low approximation leverage provide nystr om ridge effective improve upon attempt achieve combine improve rank involve analog notion score sampling score run depend algorithm formation full dimensionality addition unclear obtain logistic finally well recent eqn acknowledgement thank zhang point bias inequality exist sequence iid p ik kx px give fact estimator eqn given hold standard bind approximation appendix get bind derive low approximation appendix finish low fourth multiplicative bind l ik
define characterize function commonly kernel radial rbf rbf exploit sensor representation dictionary space sensor sensor support mapping feature kernel formulate eq associate sensor incorporate sensor share pattern sensor integrate collaborative share sensor ms feature kernel need express multiply fidelity reformulate mi dot dictionary atom dot product atom coefficient class kernel represent dot sample c ms experimentally method signal significantly improve effective interference unfortunately noise interference time kernel integrating learn incorporate improve analytically show enforce noise interference structure benefit assumption describe interference transformation sensor test still datum type dictionary description adapt corruption similarly trick mm ms classification involve kernel datum conduct nine sensor consist four sensor fig type sensor human lead human person people people run whereas one people people group test vary addition ideally would discriminate human datum difficult researcher laboratory run two nine starting sensor backward separate run collect accurately actual raw short period test subject sensor addition arbitrary useful detect location physical occur identify maximum detection second physical sensor run nine sensor divide overlap visually demonstrate signal sensor test visualize sensor sensors segmentation extract segment first accuracy fig visualize observe distinct framework ms exhibit performance sensor utilize ms classification sensor interference achieve examine sensor close take close look understand obvious fig boost sensor multiple demonstrating improvement alone average improvement sensor range among demonstrate even sensor type different acoustic event always achieve collective fuse always close cm sensor multiple sensor combine ms ms data cm sensor multiple sensor ms ms svm ms ms l cm cm h ms ms ms discuss unknown component sensor optimize effectively dataset validate model ms noise somewhat ms truly discussion interference likely co turn problem throughout broad enforce complementary information homogeneous heterogeneous sensor enhance moreover ms well beneficial counterpart sensor broad incorporate carefully classify induce domain significant improvement ms notably well sensor utilize ms l always single sensor ms averaging examine yet interference structural sparsity ms multi signal effect interference propose various source different interference linearity practical collect laboratory reveal complementary multiple significantly improve sensor appropriate joint bring classification interference sensor induce tool understand adaptive efficient although verify specifically rather locate present brief firstly close take f derivation sketch optimal derivation skip include limitation point yet converge paper collaborative sparse take heterogeneous sensor interference incorporate interference low rank multi sensor co locate simultaneously record physical event test training efficient convergence guarantee collect laboratory effectiveness propose automatic sensor discriminate classification multi sensor numerous detection sampling simultaneously locate source physical event exploitation complementary feature multi fusion classification information source sensor improvement classification mostly two category decision decision di di collect sensor decision incorporate method visual acoustic sensor furthermore compare di di counterpart paper sensor datum category appeal sparse interest inherently sparse dictionary approximately significant signal bandwidth storage efficiency also separation classification recognition furthermore allow simplified structure thus effect noise sensor novel sensor effectively scenario interference scenario collection external inherently stationary collection normally appear interference manner interference sensor spatially locate thus interference external effect extension collaborative induce feature method structure sparse ms impose within sensor sparse ms interference ms sparse regularization integrate enforce row sensor trick ms ms ms multi detect organize introduce sparse interference efficient direction multiplier solve optimization problem representation experiment vi conclusion recent year signal recently belong cp sparse simplicity presence discard description though fidelity assume coefficient particularly sample vector classifier problem employ coefficient drive extensively investigate sparse provide approximate collect variety precise classify test vote answer nearby spatio compactly p sparsity support row sparse row extension aforementione observation show application e sparse regularization sparsity common representation fit system capture simultaneously seek exploit homogeneous heterogeneous handle sensor employ di sensor aforementione describe perform selecting source decision propose ms classification sensor collaborative illustrate notation dictionary contain call modality sample feature modality pc cc belong segment signal segment simultaneously partition sensor segment sensor reconstruct atom dictionary support correspond measure physical observed sensor approximated matrix row pattern atom index solve generalization representation use ms joint representation present simplify lasso decide residual associate interpret assigning collect environmental environment source arbitrarily magnitude dominate accuracy obvious remove clutter fortunately certain band environment coefficient experiment respect account entry arbitrarily magnitude nonzero represent clutter sensor since type scenario contaminate dictionary develop al recognition remove clutter take advantage retrieve form matrix encouraging row entry wise compute slightly account presence noise capable large term interference often external source sensor sensor platform interference pick corruption interference car pass nearby interference frequency many signal alone record contain also intrinsic underlie train portion record span especially locate hence interference multi sensor representation rank interference l expect tackle problem extract rank correlate information sensor ms component wise low jointly balancing note dictionary interference look practical pca separate sparse rank hence structure address ensure bring flexibility side restrict accurate segmentation key source separation heavily incoherence atom base active area year explore large dense interference sense recovery representation assume interference change slowly rely project interference onto current anomaly constructive able deal collective structure across capability extract inherently exist outlier depict cover flexible model ms collect band nonzero sensor failure rank corrupt certain zero column interference component location distribute consider rank versa presence low interference eq ms capability exploit multiple sensor coefficient enforce level boost even incorporate coefficient study theoretically prove well beneficial task rather come group coefficient enforce active inside force active two level sparse search among sensor interference term ms concatenation matrix label encourage wise sparsity group regularizer minimize account interference group sparsity parallel extracting interference appear coefficient term label decide form become furthermore I ms interference constraint case sparse sensor propose
marginal confirm relevance covariate confirm explanation intercept fit capture addition fitting show three burn profile proportion fall profile likelihood interval limit compute indicate prior little impact respective inferential purpose mcmc width cccc sensitivity distribution distance standard hellinger distance default prior hellinger distance hellinger posterior show prior hellinger distance hellinger distance prior posterior plot cccc priori result parameter substantially parameter parameter posterior prior default prior hellinger posterior change difference difference conclusion unchanged since change small random default std report analysis beta one emphasis place sensitivity prior index worker indicate company add intercept fitting criterion one gain burden attractive dispersion beta mixed hellinger measure distribution show beta dispersion insensitive slightly unchanged social da ep model vast area linear mix response poisson despite proportion situation usual adequate likelihood inference mix demand call integrated nested laplace allow inference discuss model compare obtain mcmc life collect beta produce easy handling discuss mixed model reason popularity generalize glm effect flexibility hierarchical repeat extra variability view extension probability variable poisson binomial majority find despite flexibility response index traditional base adequate bounding ignore display skewness regression model identically distribute beta propose follow link glm extend precision covariate explore beta regression correction develop genetic recently variable unity interval overall prominent beta implement package r statistical add correction mixture mixed model analysis beta series rate beta bayesian gaussian specification prior distribution parameter inclusion inference two solve effect presence approximate adopt carlo come overhead upon attempt informative prior inference mix attractive specification prior novel numerical inference integrate nested computation enable prior accurate guide describe bayesian discuss measure model flat aid issue idea default choice assess model straightforward analytically available carlo mcmc technique fit computational moreover implementation problematic user program software user nest laplace model focus marginal replace deterministic marginal implementation call perform serve interface usage density several goodness procedure specify sensitivity include goodness fit predictive detail develop general hellinger assess change adopt assess choice shift change hellinger correspond hellinger density happen whenever assign density assign vice see eight area health education health development attribute stimulus united unit close life quality conduct social service da life survey eight unit company first divide worker individual health education capability provide life relevant question
show correct solid peak wide peak narrow likely end intensity original curve correct solid basic seem bias interval attain frequentist spectrum section note peak I bayes invariant wide unclear high energy unfold empirical regularization quantification hyperparameter wide appeal classical strength validation discrepancy unfolding unfold yield happen smoothness value appropriate true intensity peak rapid potentially biased situation case suitable highlight interpret mind main frequentist quantification solution bootstrap resample interval serve estimate moderate attain nominal take confidence effective bootstrap part presence intensity bootstrap unable probe away bootstrap blind unable account confidence bias problem elaborate scheme one regularize quite surprisingly I bayes even former take likely bias bayes coverage serve useful especially interval nevertheless clear frequentist interpretation interval unclear interestingly alternatively perhaps link would significantly help bayes least asymptotic acknowledgement wish discussion de ed call unfold spectrum elementary particle measurement due resolution detector arise formalize intensity unfold principled quantification uncertainty inherent argue deal satisfactory attack unfold bayes coefficient expansion unknown employ marginal maximum hyperparameter regularization drive hyperparameter credible empirical bayesian interpretation understand use confidence intensity methodology simulation real large inverse physics uncertainty quantification monte carlo em study unfold produce european organization nuclear powerful particle order interaction elementary particle produce trajectory energy use vast experiment analyze conclusion law complex quantity challenge unfold particle detector detector could g production particle due induce stochastically version distribution ill inversion map unstable perturbation trivial exhibit spurious take additional plausible non realization unfold intensity represent detector hand inference nature approximately unfold furthermore account observation rarely use datum early certain energy physics value heuristic provably account effect incorrectly recently problem practice difficult physical interpretation impose early stop iteration unfold positivity deal significant strength quantification solution strength analyse scenario quantify uncertainty spectrum analysis rarely account relate work unfold characteristic momentum top production propose unfold aim principled main bayesian expansion selection regularization monte carlo maximization frequentist quantification previously unfold account poisson positivity impose curvature solution unfold emission discretized difference unfold unfold interested scale enable intensive one quantification physic rarely naturally discretized expansion appropriate well contrary parameter one unfold receive attention recent good choosing level noise approach use regularization become computer frequentist uncertainty quantification credible confidence statement hyperparameter interpretation standard moreover quantification explore albeit computationally construct intensity ignore sensible use place regularization parameter enable automatic bayesian quantification dependent methodology carry bayes frequentist need background produce role formulate detail form real invariant mass experiment conclude remark km circular locate unit physics powerful accelerate particle move direction lead detector experimental every ns detector per detector detector capable variety range discovery study detector source unfold principle experiment compact france diameter operation international matter new particle interest physics particle decay familiar particle energy trajectory particle record create mcmc produce estimate histogram particularly attractive strength comprehensive relate bayes g chapter idea regard denominator use technique maximum maximizer non evaluate monte carlo integration approximate question space region rough maximization em algorithm find approach originally later little apply unfold log compute hyperparameter constant maximize hyperparameter maximizer incomplete coincide enable find hyperparameter involve integral monte replace approximation metropolis hasting sampler monte involve eventually arbitrarily maximizer find hyperparameter compute intuitive interpretation produce summarize understand tune match well match become iterate monte expectation equation least correspond reasonable intensity prior behave mostly intensity plain equation considerably take depend hyperparameter constant plug available combination agree ability partially estimate rise bias discussion resample conclude section note bootstrap intensive procedure outline since bootstrap replication matlab parallel toolbox computation generally roughly fold setup demonstrate unfold simulated data mixture true process fs e ef noise variance boundary discard setup setup deconvolution problem discretized histogram bin discretize spline knot unknown indicate ill pose set paper ghz intel relatively sample start iteration start spline significantly ill pose unfold number iteration metropolis post burn convergence size whole repeat replication obtained resample minute run whole min bootstrap core core bayes unfold pointwise percentile ill intensity obtain estimation pointwise percentile peak despite tail moreover percentile cover interval true intensity plot na I bayes confidence long percentile interval bayes unfold sample size pointwise percentile I bayes hyperparameter converge easy sample regularize rate make component diagnostic component chain converge close boundary convergence bar interval straight add appear slow plot histogram sample lag corresponding effective regularization ill pose parameter burn autocorrelation effective slow mixing apparent trace mean base exhibit also depict negative mcmc behave one try fall experiment converge variation hence increase sample enable probe bias point take illustrate iteration increase posterior diagnostic plot final indicate mix intensity represent near peak bias correction correct capture true intensity always come cost visible boundary basic albeit price slight conclusion coverage simply repeat several observation amount suggest subtract intensity normalize na I confidence interval unclear length move intensity bayes bootstrap blue band consist pointwise basic bootstrap show na I curve curve unfold size independent integrate integrated fs scale straight indicate slow illustrate unfold invariant spectrum publish produce decay almost particle decay decay resolution serve unfold intensity remarkable precision detect energy particle two particle rest reconstruct angle track mass preserve particle invariant mass spectrum enable uncertainty mass often simply call width half near peak dominant source measure energy resolution energy principle mass peak ignore resolution decay
topology cluster decision merge big therefore topology entire step nearly time decision invariant high become nearly invariant gain enhance clustering choose proper combination policy recursion know policy exploit benefit include metropolis coefficient determine neighborhood tend determined recursion large block block recursion involve cluster minimizer big group section well simulate agent agent observe ki identity also cluster belong belong load two namely step underlying connect fig agent blue simulate grouping begin non topology plot fig metropolis steady decision time cluster merge big link active link cluster topology active link fig topology imply interference cluster network connect steady fig topology fig fig separate big collaborative involve obtain trial first red obviously steady form cluster simulate node initial five topology five cluster fig respectively curve average theory fig learn detailed conduct segment sub enhance interference prevent node normal furthermore adaptive objective step size technique establish sequel introduce complex multiply side vector independent hold I recognize similar recursion drive immediately follow examine covariance next recursion evolve recursion eq jensen jensen get property norm converge satisfie lyapunov identify jensen q rhs substitute I first rh substitute extend denote arrive establish stochastic recursion I condition expand gx gx around denote gx jacobian e real parts martingale difference moment namely asymptotically weakly mean unique lyapunov recursion positive obvious unique condition easy recognize assumption eq jensen eq weakly mean follow convergence region constant rhs b fact marginal rhs chebyshev likewise substitute since drop subscript mean c moment verify substitute yield process neighboring share common many agent belong objective learn neighbor ignore result enable cluster attain accuracy carry probability ii false alarm mis detection establish correct arbitrarily diffusion consensus adaptation learn unsupervised distribute agent applicable wide gradient incremental determination np even topology consensus focus size adaptation learn response problematic size consensus diffusion modify step would decay active implementation gradient consensus size grow unbounded network suffer problem regardless especially context change motivate diffusion proper consensus exist algorithm agent common correspond minimizer interested different work investigation separate cluster considered collect arise location separate need agreement important appear multi problem introduce different different vector adjacent assume formulation many scenario problem involve multi problem appear assume fully network square minimizer mse moreover know belong estimate adaptive fundamentally study mean square risk agent handle broad situation adaptation objective word study component truly objective avoid belong agent interested application sensor move direction share beneficial amount agent belong link neighbor agent make cluster learn well highlight quite label neighboring agent exchange may certain accordingly devise strategy allow agent result correctly attain performance intra letter letter letter inversion trace besides use semi associate assume minimize accord minimizer agent mutually exclusive consist cost minimizer I different cluster share common aim q j agent network cluster topology minimizer employ strategy collaborative cluster mean every consist neighbor belong cluster circle agent well cluster neighborhood neighborhood cluster split sub fig challenge cluster completely cluster already cluster introduce group denote agent know topology information five two group five merge group partially neighbor fall group fig trivial cluster member cluster lead agent access information neighbor leave cluster devise enable agent automatically time turn solve evaluate sufficiently enhance network adaptive summarize main consist cluster denote link total denote cluster one agent indexing rule generality accord cluster consecutive index likewise index agent index index accord indexing belong belong either agent belong formulation section aware cluster agent initial aware cluster information learn group share cluster group group start neighbor procedure grow enhance collect individual minimizer equality stochastic guide relate ensure minima pose problem relate gradient process approximate true vector gradient approximation unbiased moment regularity assumption group strongly low hessian j assumption relate condition purpose function denote j denote random aggregate noise across I write noise martingale difference moment lipschitz function easy minimize cost cluster available partition group one group information gradient minimize cluster formation include shall argue able gradually leave stochastic minimizer w indexing definition I subtract side recursion nm recursion indexing eq express isolate automatically network stability agent loop primitive condition appeal sufficiently error recursion sense network square recursion strongly primitive left square stable long dynamic recursion dependent random recursion lemma square steady approximated long nm albeit continues drive similarly recursion term size term mse recursion immediate dynamic term variable two detail reader detail motivate low turn e correspond centroids evolution evolution couple arrive need equivalently indicate prior primitive frobenius inside unit eigenvector associate eigenvalue normalize add p entry ahead denote block unit rank rhs represent contribution eigenvalue ahead centroid eigenvector stacking indexing rule dimensional compare p g matrix hessian collect group recursion stack recursion describe centroid expand manner accuracy sufficiently give via error positive lyapunov know within steady metric asymptotically representative steady sum give examine normalize error steady lemma mean original recursion low centroid steady norm covariance bound jensen normalize finite steady moreover apply positive enough let taking gm semi nonnegative ii large desire equation verify positive accord gm equation reduce rhs continuous lyapunov network steady error satisfie assess hypothesis recursion use weak normalize solution lyapunov sequel distribution variable f fact lemma weak follow asymptotic normality converge triangle dimensional sequence sense rhs vanish likewise lemma verify variance lemma term vanish vanish vanishe allow enough sufficiently small individual say I individual minimizer th block let r possess obvious k pair agent distribution joint need decide pair difference k I serve sufficient true otherwise hypothesis difference know available pearson criterion ratio unbiased covariance predefine mn mn q central f stochastic sampling steady state sample carry replace identity square available quadratic vector k appendix dominate step chebyshev
use assume usual provide base question reduction sum contribution use policy q h n h three term suffice use come entropy entropy completely prove prove relation unconditional expectation note iterate conditioning apply conclude apply information theoretic moreover reduce entropy must q outcome true indicate inequality strict object question pp f hx achieve low example notation example strictly dyadic distribution next answer answer collection support question provide among binary set indicate etc consider vector matrix code object location characterize vector must exactly locate thus observe answer question describe lem seed fix event rewrite two nc j jx imply explicit characterization immediate follow du kf du previous analysis greedy dyadic setting question suppose object locate uniform dyadic possibility suppose otherwise dark subset mark object predictive I distribution history external source useful expect dyadic demonstrate proof lemma respective kf verify equal complement use conditionally provide explicit probability let random distribution poisson denote probability poisson binomial put together characterization probability mass mixture poisson binomial mass non allow environment description concern policy equality asymptotic dyadic policy definition dyadic introduce useful partition ask time ni du martingale martingale converge variable nk direct precede assume almost prove q lemma surely imply nz dyadic ask object line single case dyadic actually greedy deterministic respectively process deterministic law large illustrate normality entropy dyadic nk present greedy despite dyadic greedy value provide outperform dyadic section inequality dyadic policy already question expect reduction might greedy dyadic although deriving seem impossible following greedy fix history recall borel subset distribution policy conditioning rewrite question dyadic question theorem f kf rewrite borel borel continuity construct union element construction corresponding mass exist sr attain define class greedy policy low take previous argument lemma dyadic policy circumstance first question dyadic dyadic question consequence interestingly simplify specifically dyadic answer aim multiple testing vision bioinformatic entropy two policy dyadic employ pre set thus dyadic policy relatively certain assumption answer answer paper know square entropy measure differently question acknowledgment like thank chen eventually lead nsf institute grant spatial nsf nsf fa adapt time maintain interval obtain choose interval portion create version sequential differ policy design rather design case object question noiseless aim object set characterize dynamic programming equation curse dimensionality prevent tractable computation first explicit optimal greedy maximize dyadic split fine dyadic dyadic easy relative greedy intensive robust possible numerical outperform divide benchmark dyadic showing distribution asymptotically sequentially subset noiseless answer study devise method question find finite question accuracy dynamic tractable bind minimal analyze method similar object game game person million query find yes allow lie time game continuous problem less probability least lie analyze among work consider probabilistic originally dyadic generalize policy object work previous multiple object subset game game tell reveal thus answer either otherwise consider problem answer count localization construction code search auto collection failure electrical computer screen state problem value represent location interest assume ask series take answer answer previous question randomization previous call choose question policy sequence subset ba ba highlight expectation simply distribution equivalently distribution compute proportional learn final denote policy question characterize optimization attain infimum partially principle via dynamic prevent force easily policie greedy dyadic step forward borel subset dyadic cumulative density dyadic I value dyadic policy one sized illustration dyadic set dyadic generalize object ready main dyadic binomial theoretic inequality second inequality trivial computation observe answer question present performance strictly dyadic last equality special dyadic policy illustrate question entropy posterior figure dyadic right policy benchmark benchmark dot line lower dash question dyadic policy solid greedy identify single question location question bit object set somewhat problem screen
feature method redundancy maximal relevance q measure pair already satisfy greedy feature redundancy double relevance modification information effective mi feature selection pairwise redundancy mutual greedy heuristic independently criterion share algorithm know search weight instance evaluate relevance redundancy dependence etc arithmetic operation apply evaluation iterate conduct compare construct top bind information may result select achieve conduct result four cs implement implement sense typical incremental matlab slow cs time consuming admit drawback try lp execution conduct ghz gb fig accuracie four ten x consecutive depict ht ht ht ht ht ht ht ht ht r cm c p avg fig verify superiority perform well particularly five vs kp synthetic control accord six select good feature illustrate superiority subset pairwise approximation apply begin come clearly synthetic dna fig select independence rather measure redundancy avoid optima perform bad redundancy perform superior consider feature kp seem effective reason pairwise redundancy analysis effectiveness strategy performs control inferior possibly many e suggest relationship dependence obstacle determined break cs dependence select feature consist inferior eight nine synthetic end select estimating g result inaccurate obstacle feature illustrate cs feature grow stage addition cs evaluate radial may output end radial technical performance dataset fig thus fit cs determine term super efficiency score candidate feature currently obvious super large mi based mention rf selection select feature take index cs super salient strong conduct relevance redundancy analysis selection individual cs explicitly candidate take output dependence constant super evaluate efficiency subset increase validate effectiveness four widely classification ten uci difficulty sample since conditioning dependence drawback whole cause world improve criterion take future tackle super apply evaluate input orient thus input evaluate consider trade output pareto measure enhance ram consider future make output thank anonymous constructive comment helpful foundation china fundamental central fellowship foundation science technology author acknowledge financial china gray gray gray gray gray gray paper novel separability cs strategy relationship handle class super rank via feature super conditioning iteratively eventually viewpoint empirical verify feasibility superiority propose classification pose challenge recognition get large type become computer internet amount type rate realize important frequently reduce dimensionality remove irrelevant redundant application acquisition result speak filter select feature classifier agnostic criterion g score mi uncertainty schmidt mi efficient although information search combinatorial investigate construct former discriminate np usually conduct hand accord criterion find optimal contrary latter perspective many belong kind manner heuristic intra inter class distance etc however sort relevance especially high never redundancy lead redundancy relevant redundant high relevance redundancy apply selection relevance criterion relevance redundancy denote select perspective manner representative make trade simultaneously relevance redundancy one comprehensive conditional original set typical relevance select effective mi mi feature selection criterion note belong magnitude dependency rather efficiency estimation hand mi research indicate pairwise etc mi eq parameter correspond example correspond joint may detail item multi index employ programming unit production structure recognition mining attract increase attention effectiveness order recently zhang nature give overall among rather arithmetic operation overall sum relevance redundant drawback redundancy every consider thus feature take index propose selection separability label apply handle super efficiency via conditional score remainder paper organize metric section super implementation measure experimental evaluate compare representative discussion present summarize conclude remark mutual mi quantifying dependence mi variable form measure sense random give variable interpret nonnegative super method concept use represent cs mi evaluation super search identify salient evaluation currently select greedy follow end newly select currently subset th cd measure concept carry irrelevant advantage attention redundancy conditional dependence frequently mention dependence metric classification class assignment predict class label accuracy reflect include exclude feature redundancy e may redundant class example mutual mutual candidate guarantee dependence distribution feature redundancy dependence e histogram label evaluation evaluate feature cc label label cc ct cc capture label word mi artificial feature illustrate new place class hence mi capture absence partially sample label measure sample keep nonnegative whereas g candidate magnitude I select candidate scale system constant candidate feature take evaluation execute show greedy search ht cc cc model simplification standard focus hence avoid focus gap make sometimes class often influence select separability feature ff rf p pseudo code loop step stop make mutual medium scale loop totally bad line super efficiency candidate solve complexity analyze estimation mutual process
competition aic cv criterion poorly contain frequency weakly identifiable stand setting model small correctly select covariate table perform identifiable bootstrap aic outperform base size criterion employ likelihood identifiable nonetheless size conclusion selection criterion generally lead criterion well among aic criterion accurate display good performance finite criterion considerably criteria select regressor mean relative dispersion table employ regression competition table major city unite propose scheme take covariate show selection regressor mean selection scheme contain code beta regression people logit link dispersion include covariate quadratic transformation assume regressor include dispersion std constant people people ht correctly residual plot see detail diagnostic tool refer standardized th figure residual slightly still fit similar envelope indicate envelope point lie envelope band relation mean number people interaction relationship number people response dispersion beta fit dispersion two criterion vary dispersion criterion aic expect likelihood criterion bootstrap cv quasi cv typically lead alternative mean dispersion application discuss acknowledgement acknowledge financial information aic widely aic measure sample aic bias tends select criterion likelihood cv performance aic variation dispersion regression discuss aic bootstrap validation dispersion practitioner usually interest select yield broad criterion information commonly aic maximum maximize log likelihood bias bias aic asymptotically correction expected likelihood aic correction expand cover regression autoregressive show aic analytical correction difficult model analytical well certain restrictive analytical obtain correction aic explore class criterion bootstrap outperform aic sample additionally expect maximize paper approach expect adjustment nonparametric bootstrap cv cv parametric bootstrap cv quasi cv modification autoregressive model cox beta model tailor modeling proportion dispersion regression generalize beta dispersion goal model selection beta performance regression extension new regression monte empirical density measure discrepancy kl minimize kl follows formalize candidate sample denote possible family class small contain family use fit collection fy fy fy k fy fy times kullback notice evaluate require asymptotically biased minus adjustment aic sample aic develop develop class extend regression autoregressive asymptotically aic aic estimate discrepancy aic applicable regardless derivation bootstrap estimator bias small lead reliable bootstrap recommend algorithm quasi cross treat cv validation obtain cv another variant give study field proportion beta beta distribution quite since shape value index function variance respectively vector dispersion dispersion beta observation regressor likewise regressor dispersion include additionally strictly monotonic parameterization probit log complementary cauchy discussion beta identity perform beta likelihood respect information respectively maximum numerically likelihood goodness transform ratio maximize maximized function measure correlation propose regression investigate variation simulation carry analytic derivative beta replication large value notice yield bootstrap aic criterion covariate logit evaluation criterion process dispersion parameter identifiable identifiability covariate terminology identify differ relate uniqueness small present result dispersion cv aic cv c aic l cv parameter dispersion size sequentially nest dispersion nest candidate select three
minimize aic criterion bic know towards modify cross aim aic correspond sure bic potentially consider practice lasso approach select coefficient exception cross validation rule formula remain determined well known formula word condition rarely application choose property important selector proxy tend bias true bias estimate bias shrink residual unnecessary less optimize aic poor study result simulation jointly asymptotic normality cross validation also lasso thresholde selector base resample like exist calculation criterion achieved calculate single appealing new rule aim true predictive wavelet smoothing localization massive carlo simulation inspire compressed phase selection phase estimator operator employ sparsity wavelet conclusion regression fully selector instance property indeed nan one threshold quantile connected power entry good motivate extension rarely except total former control I gaussian control discretized bridge thresholding asymptotic counterpart threshold specific structure universal quantile must single cross involve establish orthonormal parameter thresholding selection letting tend conservative nan concentrate compress rich interesting vary role carlo range nine region time identifiable lasso experiment massive study calculate characteristic coefficient thresholde oracle estimator include correct property oracle oracle include small model practice oracle fdr stability sl serve benchmark discovery number cv ss method offer fdr know whether impossible predictive performance test choice performance preferable performance certainly many gene repeat one set factor record coefficient select predictive square respective coefficient risk tend select pointing capture right consider conservative bic take set leave observation partial least cv ht explore rule perform simulation fix sample screen nonzero randomly index select finally conduct let vary sparsity ratio fdr cv sure sl median fdr change sl poor fdr concern ss lowest closely follow linear wavelet regularize show employ ill pose radial block bottom cluster radial profile consider encounter galaxy emission function see end intensity measurement line intensity radial profile tend zero infinity hence simplification q profile expansion ray eq figure center function bottom radial employ sparse wavelet rescale sake identify go along haar wavelet factor wavelet noise ratio vary bic sure method replication fdr mse mse select sure fdr fdr fdr mse expect show fdr conservative snr lead fdr selector selection seek controlling behavior nan recover like stem smooth thresholding universal good low fdr single technique cross support finding extension generalize author science estimate noise sense selector identify covariate thank yet selection propose quantile extensive high positive low discovery achieve predictive keyword discovery lasso universal threshold relate covariate level consume technique microarray identifying significantly reveal disease modern device covariate therefore analyze exceed concentrate relate model error estimate also performance overcome paper concentrate last estimation decrease bias prominent ridge principal square three estimator assume make reasonable regularization technique govern control trade
h operator concave e g easily lemma assumption indicate table usual satisfy penalty affect analysis penalty nonsmooth general nonconvex nonsmooth base motivate right side solve follow update dc weight nonconvex nonconvex norm easy lipschitz update still nonconvex fortunately w globally ii iteratively lipschitz computable backtracking estimate section decrease monotonically continuously lipschitz sum inequality equivalently accumulation sequence algorithm subsequence k g k j semi subdifferential exist w j exist jj algorithm follow satisfie hold loop generally solve follow divide singular experiment effectiveness repeat frequency success plot legend function performance alm nonconvex accelerate lie relative plotted logarithm mcp since globally gaussian real singular dominate recovered image channel apply matrix ratio task nuclear solver admm solve code admm try tune report real image replace pixel add image scad plot evaluate nonconvex situation truncate nuclear nonconvex surrogate rank nonconvex concave monotonically increase problem able general nonconvex rank convergence nonconvex nonsmooth limit local real demonstrate outperform interesting future nonconvex problem combine alternate multipli admm acknowledgement research foundation international centre office lin support china computer national technology laboratory school university com mail edu cn edu cn surrogate functions nonconvex penalty enhance vector nonconvex singular value enhance recovery solve nonconvex low nonconvex exist concave monotonically iteratively norm nonsmooth setting closed real monotonically solver observation aim nonconvex nonsmooth minimization problem singular nm ng monotonically increase nonsmooth eq lipschitz nonconvex iff vision fall square loss adjoint know rank many segmentation work prove incoherence nuclear near rank violate solution nuclear may approximation nonconvex norm norm absolute deviation scad logarithm mcp property function rank another norm nonconvex sparse dc difference convex function program nonconvex dc function dc program programming reason nonconvex proximal proximal nonconvex solver even q low nonconvex minimization relate square relax iteration general nuclear loop compute efficient solve exist nonconvex surrogate norm extend penalty table observe nonconvex
value response consider follow smooth integrable space real within value variable unknown order turn affect estimation note estimate density term flexibility stem unknown form drive grow literature nonparametric functional estimator functional distance local mathematical dt generalise product admits value real value predictor distance associate infinite dimensional predictor second kernel predictor theoretical semi metric increase explanatory non smoothed principal smoothed semi use derivative reader real predictor express dimensional value order gaussian binary consider discrete notice x gender case natural ordering include rating discrete express determined function unknown know smoothing always nonparametric estimation prominent divide part square increase decrease select balance square bias regression value regressor functional rv estimator functional design predictive certain measure curve integrated mean integrate optimality cross addition validation appeal affect optimal inferior accuracy instability error regressor residual obtain functional approximated leave q represent residual residual parameter approximate leave nonparametric estimator residual square since gamma ig density hyperparameter assign small squared keeping result hyperparameter along possibility sensitivity table kernel bandwidth uniform bayes express parameter estimate h independent correlation error metropolis burn period first iteration record burn mean observation kernel heavily affect residual may cause use study observation low region express b goal way compare estimated error estimation replication briefly describe simulated build draw take compute also replication x function generalise value real predictor binary categorical error average function nonparametric regressor regressor latter regressor accuracy improve discrete ht cross continuous value residual estimator bandwidth term integrate discrepancy replication discrepancy form bandwidth discrete continuous model un panel panel path reasonably mix ergodic credible se se density se prior ig l cauchy em language check diagnostic diagnostic pass replication I regressor regressor observe two value two discrete point regressor smoothed bandwidth irrelevant regressor smooth exceed deviation prove phenomenon functional focus spectra obtain pure protein piece grid e observe protein obtain chemical note split group display ht give member member protein help forecast nonparametric estimator original allow learn sample allow testing forecast correspond cross bandwidth estimation table improvement accuracy log coverage functional validation ig ig cauchy ig ig cauchy measure paradigm attention generic method output generic good accuracy speed three likelihood prior empirical probability iid replacement replication remain large functional collect penalty functional functional optimal regression estimating estimating regression nonparametric regressor investigate forecast predictor functional propose nonparametric functional type regressor give forecast error among investigate forecast functional good regressor affect forecast accuracy remain transformation forecast compute compute grid point forecast solid dot pointwise vertical bar ht estimate admit mixed regressor unknown expression exist establish mathematical bandwidth difficult bandwidth marginal use study simultaneously estimate density prediction attempt type regressor extend regression local improve take local well forecast concentrate extend regressor covariate dependent model kernel nonparametric admit mixed regressor partial admit type heterogeneous optimal crucial adapt type choose among semi curve derive detail course compute outperform estimation fix thus practitioner good semi curse way bandwidth consist introduce quantity show minimizer quadratic procedure h produce spurious aspect bandwidth good phenomenon unbiased contrast autoregressive
ignore science datum work medium text help propagate offline general rather online contribution seek close cm methodology google site collect extend upon early research level compare find hold significant management message set additional practitioner power propagation help value comprehensive use root stream cm keyword word google effect type aspect text help propagate message understand offer limited view gap investigate medium google site computational classifier sentiment seek replicate early effect notion significant social medium propagation google factor message contribution co media site broad management precise word propagation insight practitioner word help need number medium comprehensive google sentiment advance linguistic start poisson seek sentiment message digital expect management towards word result analysis finding offer conclude remark take production phenomenon capture production logic especially create attribute active active message propagate essence much great challenge find connection link effect role study identify narrow definition influence analyse influence interpretation text reader wide message message increase propagation conclusion message affect interest community around necessarily share nothing common community important co social medium understanding create simply news select propagate make like medium elaborate text mechanic social message form stream user connect form facebook twitter google prominent allow content add possibly news stream one company meet message propagate possibly community fan page readily social medium past find medium traditional mass medium consideration channel challenge hard seed plant medium reach take medium state word social medium establish current research draw inconsistent picture fuzzy potentially differently difference key message sentiment message message offer hypothesis message medium count receive answer question complete social medium foundation repository base contain name list contain localize exclude demonstrate website exclude profile check assign manual check google selection somewhat nine randomly google page retrieve extension package table page naive package ratio measure keyword sentiment state concept seek network analogue scenario higher able google google receive attention simple glm glm message represent model poisson summarize q side represents receive message frequency message day among identify medium day contain age post reliably number take include relationship extend allow comparison mix conceptually intercept slope graphic produce section describe regression sentiment message incorporate company difference mixed analysis message initially compute poisson start character hour step control receive test successively ratio yield significant appropriately baseline day analyze compare propagation nest compare sentiment type evidence random require contain incidence deviation value heterogeneous sentiment ht establish word could validate
indicate algorithm alg tracking improve additionally exclude loop far penalty cardinality figure agree come price experiment take around algorithm introduce model choice track line alternative would mcmc move refine show mixing normally enough joint tracking performance particle association result show upper half sample tracking burn mle histogram uninformative maximum mle mle available intractable obtain propose poisson derive survival beneficial figure mle data time mle properly step min target true obtain biased show black sample solid line show show bottom axis normalised axis black estimate mle vertical axis show horizontal normalise horizontal iteration linear develop case da run update mle filter proposal move transformation random root accord nonlinear approximate transform rd input nd include extend hide sigma obtain sigma description find approximation produce smooth q suggest sample accord description mcmc explore approximate sequence sample particle denote dirac locate time consist independence sample propagate condition sample resample multinomial resample w forward density gibbs valid hmm well mixing follow simulator accord w lx ba ac uk school mathematics university propose bayesian multiple tracking monte mcmc posterior target birth constitute problem comparison compete significant improvement mcmc tracking parameter sample target reduction continuous new approach performance da linear gaussian compete demonstrate target infer accurately track object fact number birth new target death false clutter may record observation record say jointly infer track chain continuous discrete comprise target death time exclude parameter small discrete refer da target approach da demonstrate algorithm linear exclude da technique state metropolis hasting indeed unbiased fashion tracking combine integrated known marginal metropolis context appeal inefficient likelihood take become product form simultaneously estimate acceptance elegant particle state tracking essential tracking single incorporate e track see ignore maximum method propose contribution several interesting comparison da reduce da iii track online tracking incorporate iv obtained build agreement mode remainder describe model target static propose novel da linear track static estimation tracking mapping capital case letter random lebesgue write explicit transpose denote commonly use q observation paper r capital letter small respectively resp value set letter set element target number surveillance death target birth time survival evolve transition addition target process target target target addition observation target clutter appear superposition clutter measurement precise target birth time newly target certain specify target specifically time evolve time target measurement amount identically parameter detect index non association detect target permutation decide zero birth death target detect measurement target collection unknown time assume survival probability mass lebesgue target state satisfy likelihood time miss relevant terminology hmm correspond time hmms correspond birth death irrelevant correspond mis clutter birth birth birth death trajectory otherwise take life transition convention contain irrelevant k ki otherwise birth time target contain clutter appearance measurement index want description illustrate introduce correspondence description evolve introduce name name name style name name b description unique description main serve depend novel posterior distribution interested static regard general extend assume density concentrate np association sample nz essentially mcmc however paper mcmc da place obtain run particle target problem slowly estimate association design sampler old avoid encounter apply particle accelerate birth death clutter set linear hmms irrelevant apply calculate mcmc da applicable unbiased place obtain target slowly estimate particle change order rule designing small particle accelerate mix use deriving include mcmc go notice association hasting algorithm applicable reversible dimension consider index dimension finally reverse propose jacobian reverse target random termination death I decide detect whole procedure grouping measurement contain kalman filter proposal step matching finally nz death reverse birth track acceptance birth move whose reciprocal corresponding death p thus move exploit distribution observation compare birth move birth move consecutive mis note assignment also birth choose target track either backward execute decide decide assign clutter probability posterior forward extension repeat reach extension e death observation information extend part proposal denote n forward approximation posterior propose obtain birth move base jacobian reduction pair among reduction discard discard probability reciprocal acceptance move probability propose resp extension extension move merely forward extension move make hidden add instead continuous dedicated change successive time measurement move modify rate move combination merge switch move measurement update move modification assignment diversity choice enhance introduce locally move change move old clutter sub move differ old merge new move sub remain link move become observation time describe noise mostly modify move reason proposal resp w propose one w nz acceptance state q locally unlike move modify move first propose mainly measurement propose state modification proposal density change proposal first move modify move n acceptance ratio move state move joint state move explore hmms independently constraint move first ignore directly live long mixing prevent fortunately mcmc particle efficient trajectory leave k invariant perform whose admit marginal follow backward idea second loop nz k k sampling order rule see conditional smc backward b obtain posterior sample parameter prior execute algorithm algorithm nn nj mcmc move algorithm explore move explore prior implement run invariant z pp z gibbs hmm posterior posterior hmm conjugate represent gamma commonly resp trial posterior beta birth rate posterior conjugate hmm state comprise plane target move constant velocity
process latter sample particle degeneracy whereby implement particle output particle filter idea filter within two metropolis gibbs marginal appropriate state value extend implement also common possibility use particle calculate accept depend new input value particle conditioning obtain run I I intuitive use within algorithm ignore approximation eq proposal acceptance unknown particle proposal likelihood correct particle particle sampler target involve iterate implement path model observation calculate analytically filter informally particle condition time condition particle filter simulation call conditional sampler stationary regardless filter mix autocorrelation act act blue red act particle mcmc pseudo variance respectively act trade include information carlo error become value alternative work correlation ignore simple adaptation particularly update ensure diversity particle maintain depend smc sampler particle twice ease keeping standard strong dependency slowly reduce particularly due uninformative chose informative particle pseudo particle row middle row bottom correspond approach sampler choose implementation inform pilot run particle particle particle conditioning ny ny trace improvement run long period reject output output able mix calculation effective roughly fold effective cpu conditional plot run bottom column column consider mixture population genetic possible present individual genome individual come unknown vary wish individual come example assume allele population x I unobserve variable conditional lx conjugate dirichlet population prior mdp number population present assign population individual population belong population particle particle particle subset plot posterior population analyse people cause filter likely plot overcome propose pseudo contain population individual give subset individual label actual arbitrary population individual uniformly change recursive proposal easily adapt particle filter fixing update purely run storing implement algorithm particle filter store information implement plot trace bottom leave hand run particle remove burn keep every substantially many substantial period run filter variance hence avoid estimate respectively particle suggest augmentation new model move rest careful substantial situation perform poorly volatility help break make slowly particle filter early likely inconsistent similarity marginal augmentation gibbs latent implementing expand mix correlation variable sampler augmentation flexibility particle mcmc key variable suggest variance posterior acknowledgement author engineering sciences ep k consider calculation distribution gamma pz x x n I individual write sampling give process prior individual take distribution poisson truncate value varied em augmentation mcmc involve process move parameter generalise beyond idea introduce move latent variable generic way latent choosing amount particle trading particle mcmc observation improve scenario enable keyword gibbs particle b carlo call filter efficient unobserved process deal within unobserved value move use widely area biology implementation move particle unobserved alternative inefficient particle slow move paper mcmc augmentation variable particle mcmc algorithms particle px
unclear train deep autoencoder fair previous train svm layer significantly regularizer autoencoder table noise dataset performance beneficial powerful lrr rand cccc cccc j u mnist rand compare training unsupervise affect unsupervised autoencoder train epoch validation training example similar joint since supervised case train true pre suggest joint beneficial supervise unsupervised joint deep appropriate joint initialization htb rand conclusion unsupervise autoencoder circumstance autoencoder could train jointly could view generalization train multi layer autoencoder stack autoencoder well compare highlight potential unsupervised success autoencoder superior deep jointly autoencoder platform investigate usage volume unlabele consecutive bottom long show mnist rotation mnist random mnist image rectangle convex generate denoise autoencoder scheme generate autoencoder training train diverse traditionally greedy wise employ prior training suboptimal investigate autoencoder view stack two layer autoencoder single global jointly optimize autoencoder layer act layer empirically joint training scheme learn learn high layer representation find usage achieve deep framework platform efficient usage type grow volume introduction learn various application recognition face recognition exception initialize unsupervised follow appear ingredient supervised give grow remain technique amount method latent prior deep network high variable local important architecture deep performance layer wise disadvantage distribution bottom representation layer something furthermore layer summarize wise focus learn auto layer disadvantage fail effective multi auto various setting joint objective input cope wise allow powerful view layer global reconstruction attribute make confirm representation learn approach consistently outperform pre algorithm demonstrate superior deep amount datum label consume remain continue improved volume procedure engineering challenge difficult apart layer value mean joint measure input layer readily respect post network decompose cover distribution tends assume learn occur hidden layer recursively trick delay motivation simple easy greedy wise optimum bottom learn preserve likely learn lead optimum reason therefore autoencoder make consequently possible burden prior background briefly review autoencoder variant expand primary autoencoder basic autoencoder layer reconstruct activation take put encoding decode input decode layer function choice want reconstruct actual denote encoding include input input however training fail around set meaningful denoise denoise autoencoder idea pass reconstruct force trivially objective equation formally illustration employ autoencoder layer try reconstruct back final autoencoder stack deep jointly final well represent simplicity exclude illustration autoencoder propose autoencoder achieve robustness small perturbation frobenius jacobian activation respect thus would prefer activation stay input vary frobenius hide activation change representation represent manifold prefer would cost deep try deep I eq architecture decode sequence follow stack layer deep autoencoder autoencoder stack autoencoder autoencoder encoding layer reconstruction interpret modular train autoencoder joint exist relate technique perspective regularizer dirac delta equation recover perceptron mlp decay exception mlp commonly unsupervise replace identical ordinary mlp straight forward see training construct modify train slightly rate follow greedy tune however domain behave apparent represent decode word note modify somewhat surprising behave special practical equivalent differently data interpretation model decompose decompose bottom respect regard tune together likely take fact capacity add later hide unit architecture tune maximum generative stochastic modal try approximate intuition mostly unimodal easy autoencoder corrupt one follow imply distribution true like denoise deep denoise autoencoder autoencoder framework empirically analyse train helpful recent regularizer autoencoder joint mnist dataset split mnist variation dataset validation shape employ shape classification classify classify example dataset validation also foreground visual tie autoencoder unit activation optimize prop factor sample per mini good validation deep denoise autoencoder contraction level deep autoencoder gaussian mention section goodness denoise autoencoder chain train joint training estimate measure window converge true number window generate dataset window likelihood test table method rr dataset l mnist rand sample mnist mnist consecutive autoencoder notice fourth consecutive show training illustrate long qualitative purpose show however train spurious log test whereas drop illustrate prior training model f objective focus reconstruction reconstruction testing achieve less case previous advantage
algorithm since estimation compare shrinkage optimal singular infinity signal remain asymptotic remarkable square rule shrinkage matrix connection suggest iterative statistically remain reflect deep phenomenon datum correspondence rather direct correspondence discussion serve principal correlation decomposition transform total count rank estimate formula use original probability q logistic desirable regularize parametric jj efficiently solve iterative interestingly contingency table obtain shrinkage correspondence provide principled comparative reproduce competitive benchmark adapt experiment outperform correspondence different exist reproduce simulation experiment accord four snr repeat autoencoder sa define iterate apply run appear stable truncate truncate asymptotically threshold shrinkage asymptotic soft soft stein sure suggest assume noise addition sa tuning parameter r r snr bold make strength namely mse snr high low snr conversely snr surprising estimate happen lasso select meanwhile shrinkage function sa flexible scenario accurately nearly indistinguishable move vector term mse illustrate phenomenon expectation represent three component concentration third concentrated corner adapt poisson vary describe baseline motivate value r r c r rv rv mse also table measure coefficient correlation iterate rank ability appear stable much align one whereas shrinkage though motivate regularization generic finally use analysis collect organize associated available correspondence analysis visualize association highlight profile well method ran build original perform classical correspondence rank well regularize report rv respectively coordinate population one rank parameter course work however sa set section make good rv rv sa compare ca representation figure bottom obtained contribute represented use package emphasize correspondence analysis use visualization appropriate affect look like seem know table thus regularize transform regularizer adapt enable noise create pseudo dataset want induce estimate stable autoencoder intuition concrete bootstrap remain whether bootstrappe extend discuss grateful helpful david stanford fellowship people w support stanford fellowship begin establish bias two q conclude separate rank constraint meanwhile add plug unconstrained q solution next get verify expansion meanwhile everything define fix general monotone monotone positive cone definite follow proof fix symmetric decompose eq monotonicity matrix conjecture prop prop prop example prop comment prop stanford stanford develop transform seek basis respect noise simple isotropic non method estimator iterate scheme tuning datum analysis parametric bootstrap low play role scientific include collaborative filter genome wide imaging motivated suppose noisy admit parsimonious statistical try recover observe regime accurately classical center svd form believe close approximation noisy improve usually make close accord several sure provable identically distribute gaussian singular order new start shrinkage motivate classical allow encode want recover perspective oracle approximation solve know guess result choice parsimonious isotropic reduce singular shrinkage always poisson singular around iterate fix iterative job underlie isotropic singular shrinkage strong property fact resemble one propose discrepancy perturb conversely signal induce want vice experiment autoencoder attractive sense close view useful outside induce estimator reduce shrinkage non stable autoencoder efficiently view easily solve
evolve covariance smoothness subsection need notation convex density parametrize first decay covariance decay couple instant constant q assumption hold infinity denote go converge topology useful introduce eq select step repetition course g q description subscript replace convention function key role q differently limit depend q start analyze first rhs combine bind expectation rhs despite nature apply combined definition insight expression eigenfunction orthonormal kx correspondence calculation jx rkhs norm rewrite rhs take derive compact bind permit virtue class operator jensen lead proof assumption partition compute reconstruct measurement need cast field input discuss numerical progress allow appearance relatively embed sensor system environmental powerful resource application environment even become reality although single trend adopt area attract considerable relevance try function service monitoring center originally review environmental control compute coverage treatment begin similar environment dynamic partitioning limited coverage communication constraint cite cost coverage problem assume function area high advance hard distribution area refer strategy propose assume noiseless moreover parametric approach assumption recently parametric approximation gaussian unknown robot guarantee function simultaneously coverage robot centralize work design find visit cost collect draw pdf extension replace convergent sequence coverage mechanism location agent move domain interest pdf allow current markovian happen often pdf ensure case reproduce technical detail give propose discuss section simulation let paper partition centroid partition partition centroid function introduce define fix local point centroid partition consider problem classic center initial cycle iteratively step namely increase converge generate pair radial kx move agent unit agent basic computation capability denote position q capability store take perform computation partition robot agent base explore environment estimate partitioning goal hereafter ec base collection agent take compute trajectory agent dynamic base centroid position receive agent assign reach central base new additionally position varie agent simplify way every suitable establish centroid reduce goal density uniformly irrespective instant measure concrete specific rule report random gaussian constrain support determine follow estimate use define determine direction variance location agent instant trade exploration exploitation tune level heuristic function maximum report begin domain posterior consequently certain threshold e uniformly update agent centroid phase algorithm perform unknown function estimate reduce much switch radial consider kernel exists give kx jt pr collect input location location thank eq statement exist agent switch agent centroid trajectory generate generate partition implement coverage team combination four bi posterior grid switch phase allow variance favor agent movement centroid example display contour plot agent clear contour profile maximum number iteration average dx vx minimum posterior time figure circle ideal computed circle estimation reconstruct establish agent movement trying see input markovian allow happen infinitely discuss also appendix good
first term component integrate evaluate conjugacy crp hdp hdp auxiliary alternatively kind dim view analogy drive customer group act restaurant cluster crp probability restaurant restaurant customer customer accord restaurant specific crp table assign represent atom crp crf restaurant draw customer restaurant crp crp form restaurant crp aggregate study marginalization result establish link nest suitably endow dp distribution formally collection measurable q marginalization dp base let formal ji marginalization sketch step either side equality marginalization marginalization demonstrate application model three world state model datum unknown unknown cc consistency context topic datum unique letter string international conference letter letter generate document document variable drawing word iteration content ground successfully identify string demonstrate observation observation recover overlap character provide appendix c c word word title author word word dataset dataset vocabulary exclude stop contain publication comprise testing dim bag sift dim bag tag text level use hold hdp require document hold score comparable use univariate tag author collapse gibbs iteration burn context informative aspect level context achieve information explain author additional hold suggest induce simultaneously beneficial induce value show example discover author supervise incomplete via fast likelihood dependency discrimination derive understanding evaluate work content information yield fall illustrate distribution discover context topic appear research year top search histogram close google discover time use google wide bag sift image tag exploit observation bag outperform sift tag sift evaluate class truth report cluster metric mutual rand baseline min standard propagation ap ap document euclidean hierarchical hdp ap run hdp content affinity similarity consistently metric wide cluster content onto clarity display reasonably separate missing encounter context document observation c report observation utilize big top bottom proportion demonstrate approach single utilize level discover collapse experiment world domain content level encounter real model applicable domain ingredient bayesian form prior thus establish interesting bridge product yield new present deep main marginalization property derivation inference marginalization measure move measure measurable endow measurable set measurable disjoint set let ns finite collection property union write easy furthermore dirichlet ga ga accord measure every n space space marginal measurable set rhs let form note h h n general recall dp dp mixture realization generate proceed nest dp measurable set ji marginalization whereas result proposition still vice versa immediate swap result proposition still prior I h arbitrary measurable integrate stick break atomic placing sequentially conditionally k stick break lastly stick break put thing display integrate exclude document document context index recognize restaurant crp count conjugacy leave conditional exclude finally last context ji v conjugacy j j l therefore evaluate q sample give make marginal conjugacy exclude ji argument restaurant ji content context j j ji use q jointly auxiliary hyperparameter hyper similar previous q q use al assume concentration hdp previously count z last variable kt eq auxiliary accordance cluster statistic come content topic word mean soon improve contribute rich confirm document affect repeat al evaluate utilize dirichlet type wishart binomial word assign context set count suppose gibbs proof theorem utilize context group dirichlet block construct product accommodate observation model possess nest integrate variable collapse extensive world utilize text content e g student group group group information grouped set diverse model public health consider analysis cluster group source cluster document information example sift tag reduction content form document word problematic due vocabulary sparsity occurrence typical hdp document utilize limitation suggest alternative document joint expect improved document predictive recent jointly cluster specify advance hdp adjust cluster none context content utilize product accommodate possesse nest interesting context mixture collapse automatically utilize work subsequently exchangeable group model reflect formalism dependent dirichlet level another dirichlet
dataset pca method size ari determine make noisy introduce spectra datum see pca although robustness speed experiment processing face crowd outline pose fall dataset image extend outli scale center point project sphere report center geometric median denote remark dimensional dimension run projection distance degree subspace figure subspace point face face evident version comparison performance order pca subspace close robust close subspace face approximation subspace slight low face estimator fact robust outlier aim non recent successful relaxation aim ability obtain truly complexity empirically minimize convex minimizer robust outlier relaxation energy even strong power success verify iterate synthetic project observation theory case nevertheless experience seem potential reduce dimension massive make randomized implementation recommendation dataset ari david help zhang comment manuscript helpful encourage define energy satisfie pca scale eq monotonically decrease easily establish let arbitrarily desire inequality different pf l gd assume consider prove equivalent versus try initialization initialize whenever initialize subspace write run face crowd outlier find display test follow describe onto subspace calculate truth ambient htb htb htb add add face perform dimensional proposition hypothesis mn fast consider around dimensional dimensional possibly portion lie nearby subspace fast median accuracy modern collect increasingly dimension massive analyze high subspace subspace capturing find moderate singular decomposition svd svd stable fast progress pca follow corruption sample underlie presence outlier devote develop numerically formulation good compute ambient least approximated address linearly high underlying minimization difficult nevertheless support prove energy minimizer relaxation rich may theoretical model careful indicate competitive particular method classic enjoy efficient last decade fast singular review pca et fast however practice corruption clear emphasis quantify study success instance show achieve achieve completely corruption wise corruption within datum algorithm adversary former often initial datum sphere dependent energy relaxation target relaxation accuracy possibly however non estimator minimize competitive empirically accurate obtain eigenvalue initial involve subspace sufficiently competitive implementation online algorithm suffer denote vector seek subspace manifold subspace denote gd dl angle motivating discuss minimization propose heuristic establishe convergence summarize usefulness approach lastly robustness next motivate last approach minimization problem among subspace regularization relaxation energy remark constant continuity minimization experience center geometric successful center geometric surface natural geometric median subspace domain relaxation sufficiently iterative procedure summarize notation center desire regularization default l uk find affected replace exact fast advantageous pca although subspace seem convergence see without stop iterate difficult guarantee desire geodesic guarantee future create operation find top scale empirically notice treat copy basis subspace various real compare relaxation notice difference lower report case set median r mirror descent md try algorithm outli competitive report though md important algorithm also aim directly step run use md pass size iteration ht accuracy versus bind subspace superior synthetic run compare art distribution within random subspace experiment restrict reasonable estimation percentage runtime error recovery subspace identity comparable magnitude truth subspace ambient dimension fix also perturb add runtime percentage generate dataset average demonstrate problem high end fast exclude percentage display
solution sm implement runtime eq log proportional theorem assume trivial random interpret burn distance analysis leave future period independent even initialize burn period epoch variance optimum different point take iteration starting converge bad heuristic datum stochastic motivated regime readily knowledge far ht hybrid plain line see legend f iterate generalize see legend comparison implement use several step random ball hard require perform burn display pca version iteration though tune logarithmic rate choose sub exponential leverage finite exponentially behavior occur perform similar mnist size pre process divide deviation square root hybrid run decay step initial hybrid perform mnist singular vector generalization display qualitatively simplify presentation use remain divide multiply step ds orthonormal follow place establish recurrence relation begin epoch evolve key suppose since epoch subscript denote orthonormal denote condition quantity whereas simplify focus epoch drop simply constant tell sufficiently small numerical depend fix evolve recursion certain mc ct stochastic assumption fact verify cc armed fact version simultaneously long expression statement remain prove cb inequality confidence c part analyze epoch ready lemma pick take upper eq apply epoch therefore accuracy size constant recall whose discuss apply correspond possibly numerical svd convergence assumption scale factor require iterative leave open dominate inferior second convex compare draw parallel strong pca problem runtime however factor analyze initialize sufficiently optimum experimentally satisfactory might optimally choose finally formal square relaxed dependence acknowledgment fp intel ci institute science foundation thank lemma early sm runtime code runtime sparsity theoretical exposition reproduce epoch unit n assume zero value dimension iteratively add n store scalar line store follow implement ensures implement accordingly principal component fast intensive runtime scale reduce gradient analyze apply inherently analysis fundamental wish singular identity prominent application principal pca consist specify possible numerous let write simple find later extended solve exactly decomposition runtime prohibitive common alternative power norm show involve I pass pass prohibitive dataset alternative deterministic algorithm much update runtime flip stochastic incremental rate know slow medium prohibitive stochastic pca provable advantage avoid hand scale involve applicable runtime well mention logarithmic factor perform single scan build technique somewhat different crucially strong unconstrained attempt alone pseudo code appear execution inner execution outer step epoch mi n helpful pca rule repeatedly perform rewrite thus zero project sphere give show convergence relatively slow variance stochastic inspire reduce change encourage variance stochastic rewrite begin compare eq type add pick step decay long rather control lead decaying variance compare remark
issue impose nonnegative constraint solve capture low representation relationship basis e nuclear e noisy corrupted add corrupt call balance encourage corrupted relax norm focus popular alternate however auxiliary linearize alternate direction method lagrangian function problem update iterate add algebra scheme partial differential respect complete tb z update lagrange solution subsection associate vertex edge set give graph reconstruction recover derive naturally rank guarantee come fall capture structure note feasible solution insufficient sr practice noise small interested structure normalize summarize normalize x I obtain solve column make affect robust robustness improve improve make show strategy datum sparse subsequent separately simultaneously learn want preserve much minimize plug construction arrive reconstruction simplification embed refer ef ef commonly optimize follow augment lagrangian fix reduce alm lagrange code find http www di fr tb parameter lagrange multipli update follow alternatively ef summarize get difference great problem evaluate currently popular semi implement matlab run state server intel core ghz processor public database line lrr empirically different set generality dimensionality experiment subsection carry classification database select method label sample otherwise propagation utilize combine harmonic optimize function laplacian function elastic fitness term laplacian combine graph framework select range labeling report result case extension ef consistently achieve error labeling cutting error extension compare ef improvement representation robust lrr dense lrr base inferior contrary graph lrr graph graph case property construction effectiveness semi take semi supervise discriminant recognition database aim intrinsic infer set run image image test recognition rate graph consistently lrr ef tb ef nn nn graph lrr subsection examine sensitivity include sparsity deal corruption emphasize property base percentage corruption empirically vary setting average rate decrease ignore sparsity sparsity low graph whole tb subsection joint propose embed datum apply fair keep ef baseline ef learn keep ef raw demonstrate necessity discovery ef well prove proper discovery ef htbp novel rank supervise low derive informative suitable graph help reveal embed support science science foundation national foundation china aim discover intrinsic structure semi build non obtain structure discriminative extremely good jointly within framework term embed ef extensive publicly demonstrate construction semi supervise embed category application recognition label prohibitive unlabele utilize rich considerable vision appeal success represent unlabeled affinity pair sample sample formalize regularize despite many mainly try accommodate likely normally care manifold effort way good informative characteristic high neighborhood rule construction sr usually characterize global datum overcome liu lrr weight hereafter lrr result dense construct informative dimensional informative representation point hull ensure enforce coefficient low embed facilitate subsequent improve separately learn optimize representation learn graph improve supervise contribution graph ensure low capture global structure sr base enhance embed learn datum suitable consequently semi extensive demonstrate improve multiple evaluation framework robustness conduct propose influence review work graph section detail embed experiment section reveal intrinsic local capture structure whole manifold neighbor distance capture capture
france superior de e high education hyperspectral sense resolution conversely spatial infer fusion availability retrieve formulate minimization preserve account different noise regularizer across hyperspectral accounting nature regularizer live subspace lagrangian shrinkage multiplier convenient spatial linear link outperform simulated life hyperspectral imaging fusion total variation method multiplier way world spectral image term represent scene find sense focus field air sensor context common hyperspectral difference application dependent high resolution near resolution correspond somewhat narrow em spectrum cover spatial resolution offer cover large result small band cover band red spatial resolution high extensively address fusion latter band image range resolution significantly factor ill pose fusion hyperspectral typically act computationally asset often cover large cover band dedicate trend underlie relatively number signature corresponding material scene scene underlie material call spectral explain resolution use resolution regression exploit fuse hyperspectral via gauss jointly rgb reconstruct rgb impose sparsity hyperspectral match pursuit induce song dictionary pair test band old signature framework bayesian prior distribution work foundation work publish expectation monte deal chen treat fusion joint hyperspectral simple around inverse used fusion optimization regularization spectral order method multiplier admm augment lagrangian explore inherent redundancy technique efficient hyperspectral allow hyperspectral literature fusion blind sense inaccurate blind assume unknown make support response estimate response correspondence band sensor work extend process clearly establish present remainder organize describe present deal sensor present conclude hyperspectral think three array tensor however notational convenience band contain pixel band bold denote e g observe hyperspectral band spatial denote hyperspectral measurement spatial hyperspectral sensor spread assume band circular assumption deal blind allow vary band blind relatively regard advantage fast transform inversion costly operation periodic totally experimentally find lead fuse reduction amount base admm correspond complexity image work column subset account subsample spatial resolution identically band band band straightforward model response hyperspectral large usually live I translate description dimensionality dimensional space original normally work hyperspectral band typical infer briefly physical give pixel linear pure signature spectral signature numerous algorithm address vertex linear singular decomposition h rectangular diagonal contain singular increasing truncate low hyperspectral discard singular subspace complex hyperspectral reduction incorporate truncate svd denoise try solve ill pose therefore need adequate th compute vertical discrete periodic total purpose mean transition edge extensively formulation isotropic isotropic hyperspectral band band band one normally band work align band denoise hyperspectral reduce dimensionality subspace span svd def formulate regularizer eq first term impose explain shall discuss selection section quadratic difficulty direct use transform involve deal consist employ unlike primal dual require equation primal method admm auxiliary call four notational define eq augment penalty ready admm complex much simple augment relative auxiliary datum regularization initialization k minimization quadratic block cyclic efficiently fourier three respect solve matrix advance optimization scheme kronecker product simple computational gain splitting seem describe condition column presence close knowledge alternate optimization simple without know much strong minimize response set hyperspectral image delta impulse summarize present square correspondingly important associated response channel band band band contiguous somewhat follow reasoning l h def b couple quadratic represent intensity ms patch imply leave except definite optimization uniqueness subproblem matrix expression parameter correspondingly scale normalize dc spatial hyperspectral describe index quality experimental comparison fusion truth simple shape compose material digital library measure analyze national visible red imaging capable contiguous spectral signature randomly build image create hyperspectral image direction color representation hyperspectral fig color material capture four band unless noise hyperspectral image snr db synthetic hyperspectral university image image span spatial truth hyperspectral dataset image paris observe resolution provide hyperspectral hyperspectral fusion hyperspectral image resolution hyperspectral image hyperspectral ground evaluate fusion take ground image ms se truth ratio l angle angle estimate denote index image paper report degree quality window denote segment segment band truth deviation index image simple computed code wang fusion hyperspectral access ground inspection experimental perform preprocesse step hyperspectral first band remove information band band manually truncate subspace preserve strongly varied normalize band quantile hyperspectral band normalize b spatial share since ten result topic adapt two stein verify experimentally parameter yield image influence quality regularization stop work always yield run experiment aim spectral spatial response life check b yield quality dataset method spectral response take overlap hyperspectral band since original apply hyperspectral fusion image mind image number hyperspectral straightforward hyperspectral image restriction number band quick comparison restriction resolution impose restriction divided substitution family band transformation well may hyperspectral gram schmidt adaptation schmidt gram schmidt resolution band band modify band band expand spatial way difference gs gs band result hyperspectral improvement gs fusion include gs space intensity principal component image replace method contribution bt pixel combination spatially expand band extraction information spatial produce pass operation fig various evolution fig root square rmse truth band c life datum snr db db r gs bt outperform except index bt find publish hyperspectral channel hyperspectral consequence publish access implementation zhang henceforth comparison implementation resolution therefore interpolation input method work method author decomposition transform table method restriction available work image pixel
probable process reconstruct space source auto feature extraction calculate analogously ar ar calculation classic dissimilarity retrieve shape database align signature task addition exist call elastic dissimilarity optimally fig center form accumulate dynamic programming recursively respectively except initialized deal series call dissimilarity usually take difference deal multidimensional choose euclidean dissimilarity e perform series backtrack variant equivalent unnormalized apply computation common parameter formula deal series length round two general carry constraint prevent estimate go beyond computational stand main benchmark systematically source require extensive rigorous assess could say superiority unclear pay aspect assess consider statistically significant sec turning observe challenge chen real value recent outperform formalize specifically function otherwise suitably two initially improvement absolute difference relate sec perhaps extension algorithm penalty dynamic time control sample eq well choose final dissimilarity difference cope include aspect metric triangular inequality metric characteristic dissimilarity retrieval main behind jump cost dissimilarity resemble bottom jump visit jump cost increment want jump jump cost magnitude consider fx similarly introduce series parameter control advance dissimilarity measure efficacy similarity commonly ratio understand classify item total measure use nn implement relate error suggest simple near neighbor publicly set time repository series set comprise shape range length detail refer properly assess classifier need follow scheme balanced item balance balanced error estimation regard repeat fold precise estimation avoid bias ratio split provide allow ratio implementation agree error ratio one modify algorithm interestingly rl fc last rank across data position sort order accuracy datum indicate characterize extract fc feature bad elastic intuition sample important order magnitude euclidean one several set match set superior rest fig next measure statistically difference separate apart comparison make aforementioned global choosing solely hence otherwise could measure utility ahead look ratio gain couple set kind contingency reason contingency table euclidean gain euclidean mostly sec look perform measure stage ranking stand construction rl euclidean majority give across peak perhaps present fairly accuracy robustness quality incomplete plot ar parameter range indicate reasonable choice measure potentially seen seem consistently range good combination datum lie potentially accuracy set comment large tracking path advantageous account small measure track may advantageous set conclusion derive ar coefficient choose ar low classify group statistically suggest sub sequence variant could potentially join group however seems consistently consider distance often measure take time competitive generally euclidean statistically significantly find fc measure course exclude measure variant suited rest sec compare mostly train test error unseen list notable assess vs assess regard regard step pre step well discuss therein book wang prevent low dissimilarity series inclusion assess depend consider sensible approach implicitly consider pre strategy z invariance phase invariance ar invariance correction instance become usually consider usually accuracy improve set far potentially reduce use unsupervised cluster candidate quantitative strategy series emphasis impact scheme empirical multiple important necessary get picture approach unified analysis tool interested series similarity cluster acknowledgement make repository time series similarity core many particular similarity ingredient lack comparative study rigorous quantitative strategy extensive evaluation series principle family available datum come wide variety accuracy accuracy meaningful conclusion equivalence accuracy measure finding follow methodology researcher consistent criterion inform baseline time series scientific indexing stock fluctuation medical g motion location shape effectively transform determine give resemble retrieval clustering exploit outperform elaborate alternative tree perceptron logic boost multiple classifier correctly deal calculation measure continue future generic purpose readily task last aspect highlight desirable measure type wang year pr fu nevertheless make efficacy apart interesting oppose straightforwardly time similarity control quantitative framework come various scientific newly theoretically attractive efficacy measure vast case difficult consistently remain quite accurate measure look similarity measure efficacy chen competitive none three initially generic behind relate specific dissimilarity string comparative deal classification study introduce usually corpus lack besides usually significance formal difference rarely impact field sure baseline sensible choice perform pool time series additionally storage issue approach restrict stage aforementione sufficiently cover decide accuracy contribution previous formulation evaluation sample rigorous statistical assess superiority accuracy assessment rest paper firstly outline application sec explain sec end sec deal similarity vast comprehensive enumeration scope present book wang book fu measure auto selecting avoid measure small set lead alternative measure consistently aforementione measure include way dissimilarity measure group measure compare temporal approach alignment dissimilarity measure indexing
mm fig image pattern neuron axis linearly scale template history issue template variability technique template consideration closeness method template light representation calculate dot give mathematically cross correlation think traditionally template feature thus equivalent kind template template audio broadly audio short duration short song type song entirely test notable tendency improve basis learn fig believe performance far rather unit co variety order combinatorial layer abstraction temporal energy pattern perhaps advantage large dataset explicitly template fig field broadly sensitivity combination harmonic strong spherical k method design involve secondly show partial consideration may influence mechanism measure song unit biological paradigm worth current automatic motivated scientific volume come source simple effective boost training within impose extra effort learn similar report principal volume effective confirm large volume increasingly become dataset feature whereas raw spectra much recommend species recognition dramatically across together achieve peak classification quality make demonstrate data volume exploit trivial domain demonstrate community publication collection least label publish open acknowledgment project research website sound source fellowship ep g early fellowship ep availability compose sound file file request sound specie list record website website automatic sound computational monitoring communication classification useful crucial ensure big acoustic summary information learn automatically often outperform manually transform manual yet introduce feature volume sound inspire technique prove domain experimentally representation diverse database forest classifier demonstrate limited bad conversely substantial boost spectra computational particularly notable scale activation substantial audio annotation interaction choice representation automatic specie classification useful number big crucial huge volume audio audio volume much sound hold sound digital scale without manual intervention manual segmentation song section lack segmentation since audio classify specie study least far survey often manually specie application unclear study limitation remain question intensity large modification robustness distinguish ten hundred typical advantageous label specie present project name provide stimulus research classification challenge report benefit make available standard focus choice outperform spectrum evaluate role aspect four enable perform dataset specie boost little cost dataset demonstrate boost label task amount audio explore reason follow describe experiment learn classification raw audio generally input even input duration audio would audio magnitude time transform audio duration sound indicate energy frequency frequency carry information content transform originally reduce dimensionality traditionally dimensional automatic speech spectrum keep coefficient approximately reduction advantageous manual inspection cope high see modern algorithm cope well available discard little capture specie design represent speech yet human differ production spectra classifier could matter rather design representation manually automatic feature topic aim signal compression procedure unsupervise operate unsupervise component pca pca projection create operate inherent machine feature despite method semantic relevance already make feature surprising without feature add however transformation reveal machine expansion one layer neuron connect intend instead simple scalable modification classic audio identify specie specific aspect patch representation delta sometimes cf study variant frame time variant automatic separately four sound forest systematically feature spectra coefficient pre processing decision treat full audio purpose divide short duration window produce overall configuration random forest classifier test binary parameter combination rigorous hundred explore issue recognition separate aspect character dimensionality test item total duration france frames uk uk frames frames represent large classify two consist publicly project per challenge private evaluation final partition half addition sound environmental sound duration minute annotate list median approach adapt website many specie cover specie specie include retrieve retrieve allow system song call sample range widely vary characteristic example typical sound file class distinguish label specie use separate label song call strong specie list audio file region implicit collection audio noise audio monitoring manual audio tp width mm mm discuss feature transformation drive characteristic inherent study first relate mean cluster centroid unit direction angular modify update input centroid euclidean move centroid cosine angle update spherical mean spherical find overcomplete approximated multiple discover simple representation dimension normalise pca configuration one spectral frame sequence spectral frame short temporal spectral number frame index frame offset fig basis alternatively think stack stacking frame give find mean require consider volume author minibatch update stream stream online mean apply centroid amount centroid adapt case spherical optimisation presentation learn true single pass pass reservoir apply pca whitening start window decision pool overall window second audio decision purpose training decision aggregate audio mean specie across window reasonable default motivation audio strong factor window audio cost evaluate feature combination stage auc auc systems property unlike unbalanced true example probabilistic auc tell negative always good probabilistic rank list lead evaluation relate rank difference auc apply mis position statistic rank l label dimension spectra spectra spectra mean frame frame two test statistic general glm use package test glm interpretation auc glm odd ratio experimental fold repeat version glm fold grouping interaction mode decision pooling duration whole audio testing configuration result pool glm effect exclude case estimate odd dataset annotation therefore explore source recognition quality strong overlap mean testing systematic example generally short mean pool feature training specie wider expect strong train possible set intrinsic dimensionality difference potentially give degree freedom classifier capturing run create projection feature simple form learn feature degree classifier run plot fold mm plot fold mm mm plot mm mm mm width mm mm plot width plot bl fold bl fold dataset mm mm mm mm bl fold mm mm bl fold factor vs ms ms ms ms ms pl none relevance distinguish auc measure ranking tendency outperform outperform feature learn spectra except compare switch learn effect strong feature conversely performance large fact deep aside feature dataset switch raw spectral feature boost though feature across give classifier reach map chance configuration reflect observation score annotation insufficient full mode dataset auc dataset small boost auc switching combine outcome effect aggregate find effect reduction inconsistent extent improve performance tp mm mm mm bl mm mm bl mm bl width plot bl mm mm bl mm bl cross scenario dataset middle datum boost tell firstly contain audio secondly include setup perform expansion audio annotation contain lead accommodate well attain training poor tp mm mm except train turn broadly preserve dimensionality improvement change ordering though part feature overall effect estimate random small validate map red binary model relevance layer first list attain two second list evaluate classifier entire comparison decision system audio one outperform variant audio mean notably attain notable subset substantially work volume peak reach auc actual develop winning attain report peak auc full private
optimal passive learn datum w w ks k w x w b k w w inequality tell complete sample k first w w due give list elementary inequality let first check k corollary ready prove iteration argument prove three technical get complete proof lemma want prove apply take union classifier respect particular isotropic say furthermore mean zero identity isotropic constant sample iteration also data I isotropic f k k w add inequality surrogate linear inequality theorem require lemma present large hamming bind separated set make separate bind reduce switching subsequently fact obtain follow fact taylor bayes classifier satisfies condition condition point satisfie last lastly represent label decompose synthetic propose request unknown subsequently active learning define divergence parameter depend furthermore sufficiently small follow set semi mp hold prove corollary easily depend jensen prove use concern separable theorem fix denote code length I hamming ready prove first comment always even always remainder shall bound repeatedly pick request case excess w corollary definition robust base active homogeneous pass origin analyze corrupted impose noise low achieve margin condition membership synthesis scenario stream selective surprisingly show separate provide insight increasingly make unlabele sample algorithm access capacity request specific hope label informative achieve improvement passive noisy polynomial two access unlabele decide point point stream margin show condition statistical make active usually also low one stream base set surprisingly algorithms distribution margin active detail homogeneous unit algorithm later base query budget bound characterize cf eq key parameter available robust disagreement bad discuss development convex variant near amount exponential risk optimal present developed concave change shrink like surrogate function hinge logistic algorithm bind factor stream algorithm exist satisfy excess bound adversarial manner unclear low apply low log separable show exponential contrast polynomial membership query synthesis section analysis datum distribution cf focus low low bayes optimal classification hyperplane stream algorithm active inspire stochastic connection active optimization analysis build base active necessary keep track classifier optimizer bayes shrink analysis construct adversarial fail synthesis active responsible dimensional inspire convex assume joint instance space draw goal loss indicator paper consider df linear w consist angle characterize hyperplane increasingly w risk base selective request manner formally stream operate point sample distribution unlabele accept request conditional finite selective operate selective setting access entire pool make query request margin active learning stream query set active learning introduce query synthesis query label request shrinkage iteration allow adapt value excess basic idea request passive reduce scope classifier htbp failure rate xx w final key depend query divide evenly difference sample budget previous tuning free active learn query optimize algorithm surrogate sketch defer appendix defining notation f b k respect acceptance technique probability good carefully iteration long possible exist phase phase everything behave excess upper bound classifier speak apply risk constrain appendix appendix density slight modification defer appendix fix suppose distribution unchanged detail active stream imply low stream selective synthesis margin base unit facilitate synthesis condition distribution speak constant eq classifier use synthesis remark eq deferred density imply establish angle output bayes membership query label satisfy excess bind omit dependency excess corollary stream fix denote excess omit section defer assume remain construction tt rigorous intuitively want distinguish budget imply kl condition lemma constant deferred fix design distribution construct th hypothesis illustration indicate actual solid dash green respect distribution point classification hyperplane hand must bayes classifier must hold data
obtain ise irreducible ise configuration zero boundary prove minimal ise configuration configuration singleton configuration technical follow form subgraph short vertex long htb rectangular rectangular singleton give singleton ise unique correspond rectangular configuration configuration singleton configuration prescribe lemma configuration one prove rectangular max singleton connect rectangular prove two configuration mean irreducible technical rectangular configuration max result lattice e exist q tb consequence connect configuration denote rectangle simplify connect illustrate large connected simple component rectangle lie outside depict component repeatedly scenario move indicate box position either singleton move outside vertex start scenario swap remain site swap swap instead singleton configuration remain swap site situation swap tb pt c swap swap swap swap swap swap swap rectangular otherwise reach configuration lie latter rectangular configuration let proof show dimensional dimensional lattice small expand space dimensional show section prove since lattice hold version example hyper rectangular configuration swap prove sufficient lattice sa analogous omit detail proof connect rectangular configuration connect component rectangular rectangular lastly rectangular analogue vertex adjacent neighbor indicate forward ise suppose show irreducible various nan either presence long interaction homogeneity alternative test statistic define normalize constant minimal sufficient ising define could correspond take lattice otherwise pdf analog sided indicator high adequate goodness fit ise two statistic use sided pattern form vertical consecutive ise choice indicator range interaction alternative q normalize test homogeneity square configuration statistic degree homogeneity expect statistic small homogeneity homogeneity dt nn normalization statistic describe ise result lattice analyze ii generated ising ise model interaction ise periodic simulation compare ise ise compare ise interaction three ise value way equilibrium chain configuration metropolis remain available website author chain iteration generate six statistic homogeneity test sample chain carlo models statistic depict statistic observe depict standard distribution none reject data level hypothesis homogeneity ise model homogeneity homogeneity test test reject test homogeneity show difficult interestingly count adjacent pair recognize ise ex pair adjacent pair b experiment chains monte carlo configurations plot negative value model depict quantile hypothesis ise ii rejection moderate interaction large ise count adjacent pair generate ise homogeneity hypothesis reject test b weakly homogeneity recognize overall decide organization count adjacent pair homogeneity test seem line pair adjacent pair ex application spatial super resolution highlight component density picture lattice circular ise configuration adjacent homogeneity adjacent remove step analyze study statistic chain illustrate figure homogeneity sample pair size near neighbor discard homogeneity configuration max singleton component swap size respectively size configuration singleton contradict write adjacent connect large decrease increase singleton become result configuration max singleton maximal interest q minimize equivalent maximum attain rectangular configuration bound since obtain singleton configuration rectangular give component box statistic thus let complete reduce sufficient rectangular noting tb write rectangular move figure move indicate htb need move type rectangular move result configuration rectangular side length b swap join lie repeat subtle lie type configuration rectangular lie move swap b swap configuration lastly configuration swap depict theorem thm conjecture thm thm example thm ise simple interact system originally interaction physics widely process usually fit lattice ise basis develop statistic goodness fit intractable thus develop monte fit avoid spatial organization ise mechanic appear ise thesis usually arrange interact play mechanic see interaction area e social ise single paper goodness model apply contribution series paper asymptotic approximation irreducible markov starting build spatial test difficulty guarantee markov discuss testing specifically ise model simple state preserve sufficient irreducible introduce conditional statistic set move pass configuration preserve statistic markov basis building irreducible markov perform find basis polynomial principle gr basis technique lot exploit computing basis intractable toy section lattice consist compute ise lattice infeasible gr basis technology observe network node collection compute method computing easily chain connect simple move goodness fit irreducible inspired perform relevant overcome computing ise contingency network previously algebraic notation lattice basis ising prove construct chain change ambient construct irreducible markov chain hasting move uniformly move ise acceptance markov ise algebraic consider coefficient index configuration ij set give subgraph restrict vertex addition let polynomial definition directly ideal model induce color diagram composition lattice algebraic software markov degree instance generator configuration configuration sufficient move three side configuration degree move gr basis grow lattice large
unsupervise information overcomplete latter section semi unsupervise supervised tensor mixture let error include tensor iteration initialization rd empirical uniformly sub satisfie factor vector therefore reasonable regime I q give unlabeled sample rip random remark rip eq supervise mixture column true w satisfy error arise empirical moment orthogonality latter large minimax label much number unlabele overcomplete semi furthermore sample complexity unlabele minimax regime brevity semi supervise high regime notice regime noise magnitude general tensor concentration unit dimensional sphere reasonable appropriate negativity assumption require assumption high norm assumption weak incoherence furthermore discuss uniform draw column sphere hold spherical discuss reduce symmetric unlabele semi overcomplete build initialization initialization slice tensor condition follow sample condition rd condition sphere output estimate column bound whiten base huge complexity mixture incoherence mixture employ whitening set comparison sample well regime sample incoherent factor spherical mixture mention result gaussians spherical gaussian whiten complexity analysis regime unsupervise ica noise ica view already semi ica initialization give let eq recovery give symmetric high initialization column uniformly draw sphere subgaussian nonzero ica output h addition estimate ica mixture weight ica theorem assumption weight ica see assume observe overcomplete ica efficiently learn fourth moment unsupervise previous explicitly characterize initialization svd slice propose setting unsupervise state semi initialization rank given consider estimate symmetric ti initialization rank uniformly draw sphere subgaussian variable setting permutation addition appendix ica suppose subgaussian probability nonzero precisely suppose subgaussian theorem provide unsupervise ica change supervise unsupervised guarantee sample mix rip remark detail condition ica sparsity ica one nonzero entry ica dense guarantee also learn incoherent alternate obtain handle arbitrary enough expense sparsity small high expense reduce incoherent impossible identifiability method incoherent dictionary independent arbitrary expense extend analysis bad section consider noiseless code condition assumption universal represent normalization factor limit sparse code dependent sparsity dictionary random let propose dimensional appendix dictionary atom level quasi regime sparsity time noiseless noisy sample section learn draw sphere impose component effect component notice tensor compute multilinear section discussion option fix criterion experiment subsequent stopping error constant random initialization initialization depict ratio recover vs observe overcomplete work theoretical room improve guarantee initialization l algorithm recover estimate error average initialization perform stop overcomplete model last number normalize theoretical claim square recovery ccc c avg avg avg weight e acknowledgement part microsoft nsf award award support award award award nf notation operator define tensor th hadamard entry guarantee provide prove semi convergence provide iteration good behavior asymmetric inner analyze rank generic generate rank perturbation weight initialization eq formula asymmetric loop satisfy error approximation tensor local identifiability result tensor local good guarantee present exploit r run perform mm perturbation condition generate guarantee condition follow define initialization semi supervise variable tensor guarantee tensor appendix result apply algorithm regime dominant noise regime observed complexity describe theorem global bound concentration case apply concentration theorem requirement change sparse prove ica difference theorem exploit moment expand multilinear exploit expand eq assumption involve odd norm impose apply prove I latent linear ica ica unlabele semi supervise enough matrix bernstein concentration norm tensor concentration mixture theorem expand difference claim combine claim term ne similarly set column partition set belong value apply argument restrict set inequality contradiction assumption contradict rip partition care product fix however fine partitioning kt furthermore constrain net define possibility location bound give net net rip satisfy property unit l lemma section separately like intuitively like adapt simple idea bound norm rewrite symmetry imply vector b random variable norm construct net triple sum introduce definition satisfy term bound bernstein exploit summation exploit bound bernstein bound rip property bound show compare prove norm bound triangle step exploit argument union triple net union triple argument net triple probability claim random satisfy rip vector partition consist rip noise e b rip let rip cauchy inequality inequality sample bind however bernstein utilize small suboptimal additional get bind small column bind consider net partition product definition let sum fall begin term form bernstein exploit bernstein constant net union net hold net ready last q variable remain addition provide net matrix product respectively inner empty construction expand lemma lemma summation weight sum bernstein summation bound apply bernstein take triple net bind triple bind prove ica th moment tensor eq eq proof claim term perturbation bound term claim bound subgaussian subgaussian later median distribution negligible get desire order prove bound median take order rewrite partition summation accord fall summation term show subgaussian use subgaussian bernstein probability term bind imply summation ready h nx h spectral net construct unit ball net subgaussian entry subgaussian subgaussian claim least constant bind net close net nd outer good apply concentration suppose subgaussian equivalently term xx w ab bernstein apply summation inequality indicator subgaussian indicator variable suffice summation q bound bernstein imply least set hold theorem similar perturbation version sparse ica eq ideas claim loss subgaussian dense argument desire net unit ball net uv would claim standard trick two h h difference negligible polynomial inequality bounding triangle vector claim product addition product empty restriction exploit exploit partition summation value exploit last like directly need analyze tail phenomenon give tail subgaussian recall specifie bound chernoff bind subgaussian subsequently tail case summation least range summation q subgaussian least summation use union separate nonzero summation subgaussian summation subgaussian bound term range second union perturbation nd sparse similarly q construct eq variable subgaussian indicator bound summation variance eq lemma conjecture remark bold bold guarantee model overcomplete regime dimensionality spherical sparse bound empirical analyze recovery set exploit label rough refine establish spherical initialization svd slice tensor overcomplete efficiently tensor unsupervise semi overcomplete representation tensor hide identify latent disease community observe latent lead gain speech vision largely attribute moreover overcomplete achieve overcomplete observe overcomplete know provide model overcomplete representation gain mostly domain obtain label typically large develop novel guarantee overcomplete gap wide tensor decomposition employ mixture markov guarantee requirement dimensionality drawback behind work mostly tensor overcomplete mixture code overcomplete pose latent exceed incoherence redundancy establish make enable overcomplete regime compress sense sparse paper guarantee code exploit tensor asymmetric update tensor perform symmetric power update highly overcomplete efficiently tensor decomposition moment tensor unlabele initialization setting require guarantee concentration tensor argument learn tensor method summarize incoherent component basically impose soft orthogonality constraint draw dimensionality supervised tensor prove extremely unlabeled label sample note minimax bind rank svd slice moment initializations ica ica constant component ica fourth moment number unsupervise ica provide learn sparse setting give initialization special ica mixture ica extend dependent sparsity bad guarantee overcomplete recover decay order decay point bias update alternate objective fit learn leave suited non negativity topic incoherence sparsity method believe formulation well suited establish concentration draw mixture ica concentration norm rely net vector sphere net loose fine grain distinction employ concentration propose vector classify sparse dense correlation large refined impose factor impose isometry rip rip gaussian mixture noise constraint vector rip constraint bernstein separately final logarithmic level geometrically therefore overall additional combine analysis somewhat mix one establish bound fourth moment assume involve hide activate case hide correspond sparse ica establish bind depend partition bind tight empirical mixture ica coding novel involve tensor concentration conjunction alternate guarantee recent establish local global incoherent combine concentration range variable learn complexity topic analyze iteration extend nonparametric work require overcomplete require whiten input condition tensor provide improved sample incoherent learn overcomplete challenge ica fourth third overcomplete slice provide careful perturbation handle two fourth slice ica mixture roughly low every dimension call style h name x covariance proportion mixture component spherical spherical generalization overcomplete parameter problem special empirical empirical assume variance overcomplete scalar hand setting also matrix ica random signal perturb noise latent independent gaussian noise depict representation ica circle draw style minimum size inner black hide x observe x estimate ica formulate j ica model ica constraint sparse dictionary ica study code briefly work concentration high tensor concentration norm estimate sample mixture concentration rd moment mixture describe respectively recover norm difference condition randomness hidden notice hide sample correspond hide matrix satisfy see remark introduce noise regime theorem regime dominant concentration mixture theorem whiten whiten step orthogonal eigen whitening lead result apply concentration norm analysis eq estimate singular factor bernstein result scale rip property sample propose rip adapt rip rip spectral model entry prove concentration net argument construct vector net good result small argument net incur factor complexity get bad key usual
successfully employ impact medium patient medical text particularly appeal label approach base transform dimensional dense expensive special heuristic novel learn adaptation structure learn dense feature reconstruct language directly low outperform adapt pos web occurrence source driving method domain adaptation correspondence al autoencoder well suit heuristic furthermore correspond subspace face amount occurrence information avoid tradeoff induce directly tendency nlp per template rather treat induce fill template embedding give dense representation embedding skip gram negative yet method useful template skip gram induce embedding sigmoid embedding embedding target dense feature vector template nonlinearity representation apply embedding concatenation vector pos pos several web review answer well use development sentence representation web target unlabele sentence pos treat svm add dense feature basic feature lexical embedding correspondence et autoencoder target domain aside distributional stanford pos development word template distribution word embedding set table slightly use principle embedding compare
overall cloud conclusion solver apply alternate admm noise convergent handle long exactly decomposition iteration count require critical iteration rough start solver solve hour switch formulation accelerate proximal improve solve size f alternate augment exploit separability splitting system minute partially approach apply fista solve dominant cost example video formulation half variant admm insight interesting split problem depend low maintaining context encounter factorize thresholding approach potentially present leave future work restriction necessary particular huber smooth arbitrary operator operator embed index include transform g main formulation differ functional fall problem study find datum case norm write set origin define interior example trivially non negativity exposition implement must solve simplifie denote simple square main computational solver key functional decompose denote canonical give ex dual explicit formulae ex tx also asymmetric negativity model formulation challenge fast formulation focus challenge advantage issue project onto define product ball efficient onto project median ex ex ex long straightforward efficiently onto form scalar sort conjecture median reduce ball arm state may projection onto mn alternatively inner onto well depend singular minimization projection onto ball reference intersection note negativity theorem efficient accelerate ex deal quadratic whereas hessian solve descent objective operator rx il hessian remove separability instead cross coupling term bold prevent nuclear norm hessian coupling potentially use solve non smooth take fashion expect trick algorithm code software need testing purpose variant solve program cost vary solver reference l benchmark design required pick stop plot rather table dominant multiplication core randomize since number value order calculation without convergence unfortunately involve incorporate set challenging routine test create rank singular haar measure uniform add noise equal entry exponential long tail noise capture partly partly give reference solve find advantage solve different formulation non norm normalize residual show extremely simple formulation quasi variational section sequence show jump start solver accord proximal make slowly accelerate coupling depend converge slowly perform reasonably bad solver wrong likely bad smoothing several test show test accuracy knowledge solver sample uniformly distribute white test vary range error term nonzero feature converge accuracy need try ht show achieve test poorly competitive use impose time two quasi initially long long svd interestingly explanation subproblem increasingly hard warm start easier consistent regard ht conclusion largely similar ht turn band point major frame camera issue together scene frame hundred application clean remove error great model approach remove far full span iteration randomize anomaly original pick remove camera camera scene cloud cover review formulation new denoise denoise formulation show propose newton state innovation fast method synthetic removal publicly principal pursuit source separation corollary remark ny research ny ny introduce principal component
incomplete unless properly property carefully testing establish result complete family necessary condition subset contain density differentiable logarithmic satisfy positive finite function establish distribution many situation open differentiable continuous definite exist density degenerate two relate routine derivation asymptotic model like poisson partially acknowledge lead manuscript conjecture remark hypothesis robust robustness intuitively extensive property test form describe derive divergence context class test illustrate robustness result hypothesis component statistical help systematically claim basis testing mainly decade theory direction researcher yet formalize later hypothesis tool widely practitioner criteria serious robustness misspecification develop property know involve density continuous motivated minimum divergence density divergence test nan true point simple powerful utilizing although concrete robustness property alone general test recently family pd assume asymptotic condition simply similarly condition al condition description overlap condition take keep relate nan end extended rest test theoretical derive remark theoretical numerical finding choice appropriate propose briefly generalization conclude remark recently propose popular divergence power divergence family read family case divergence respective expression equation coincide pd family read family therefore natural reconstruct statistic divergence nan equation parameter behind correctly nan hypothesis generate density consider employ ideal choice put coincide statistic rest statistic although vary asymptotic first prove asymptotic divergence require minimum test derive nan al condition I routine minimum sense replace minimum divergence asymptotic nan independent observe face value equation parametric offset asymptotic parametric study theoretical test also power require achieve test suppose al normal quantile tm robustness property statistic robustness statistic robustness statistic result cover develop absence robustness ignore test minimum contaminate contamination proportion degenerate contamination order influence evaluate nan function second influence influence robustness independent density power however unbounded mle mle contamination test robustness contiguous parameter order size also contamination contiguous one tend confusion nan alternative neighborhood huber contaminate influence begin contaminated density variate matrix chi centrality propose contaminate square freedom random prove let taylor taylor series expansion around simplify get probability ta employ theorem diagonal ii part series central central chi function derive contiguous contiguous contamination robustness independent important follow corollary coincide asymptotic previous give alternative truncation contiguous case power contiguous asymptotic hence asymptotic contamination form series expansion linear independent chi central expansion approximation many usage finite ga g u f g hypothesis asymptotically normal zero cg chi square exactly otherwise independent corresponding interval misspecification solely illustration study cg extent interpretation influence whenever value bound extent follow routine omit scalar u u contamination hypothesis univariate use limit thus ordinary consideration nan density assumption recall normal asymptotic compute contiguous alternative alternative plot almost whenever second divergence power simulated power interestingly power power case illustrate give picture power base approximation work produce value close simulate surprising approximation nature simulate power much close contiguous present table contiguous hypothesis freedom centrality contiguous turn upper chi square distribution freedom independent loss efficiency pure offset robustness extend present influence zero influence unbounded always use robust statistic value influence seen decrease influence power numerically figure significance robustness perspective power influence nature derive replication contaminate type contamination scenario empirical sample three contamination give fully size combination table nominal compare like size somewhat value close consideration h r power contiguous hypothesis contamination figure size respectively power size power stable power contamination scenario although present calculation representation power size contamination contamination almost power decrease contamination away true combination contamination distribution contiguous satisfactory alternative contamination finding result parameter practical usage th illustration implication chi subsection mean chi contaminated define subsection independent show present contamination proportion various early correct highly produce confidence interval even contamination remain chi theory discuss effect contamination present examine contamination proportion generate quite contamination slope slope chi factor bound contamination stability unbounde far robust illustrate divergence within
carry primitive neuron template store dimensional transform template module template illustrate simple signal product cell inner pool nonlinear function cell bin histogram could moment moment mild moment could invariant signature complete characterization moment suffice notable complex smooth histogram transformation histogram complex cell group observable within partially observable group pool signature compute normalization constant transformation template neighborhood signature imply main module recursive architecture build invariance factorization stack paper latter representation extraction layer invariance shift property signature compare audio track sec evenly level majority voting frame classify global label voting track discriminative strength vs rest multiclass classifier result window achieve long stationarity add modulus lose invariant stable art combine classifier spectra coding achieve combine multiscale base fourier ms audio alone attribute instability frequency instead instability mid invariance add pool template template audio template k collect moment layer concatenation moment template base notably nd layer local translation explicitly neighboring frame pool subsample pooling window frames operation cnns field cover impulse template reduce pool shift template layer randomly set although drop method question architecture speech relevant music transform template manner c build invariant module stack provable invariance usually strong assumption insight weak deep stability lc rd translation invariant translation binary theory stack invariant hierarchical network currently rather invariant representation stack question transform moreover systematic evaluation music audio representation towards limit audio signal capacity invariance end theory pathway unsupervised di technology representation stream module invariance propose module extract build hierarchical mid level representation signal projection template template induce transformation result signature guarantee unique invariant transformation stable constitute network aspect audio representation music convolutional music music annotation detection rely automatic speech recognition transform stationarity window ms music signal apart acoustic music shown require content identify scalability specific approximation analysis scale music variance leverage projection propose architecture learn invariant analysis invariant improve music classification unsupervise feasible store transformation deep network cnn cascade wavelet transform frequency cnns speech tune neuron representation primitive dimensional computational principle audio network many form transform convenient mathematical formalism difficult normalize haar discriminative high within distribution sphere follow er
chain compute execution speed observation delay methodology intend prior bring big term e compute last accept ultimately inefficient unless consider together address remark stress analogy delay slice decomposition proceed simulate constraint delay sampling conversely scheme slice slice prove delay appear poor additional processor dynamic processor mcmc make ahead walk metropolis say reach figure subsequent future convention odd share parent master evaluate na collect master serial core core serial run update fundamental requirement underlie driving reject simulated leave start acceptance trivially satisfy actually metropolis proposal elaborate riemannian far ahead time scheme limited processor worth branch quite substantial rao could improve efficient towards exploration exploration acceptance branch reach static define thus acceptance sequence store average iii towards exploration branch increase per advance reach thing determine candidate assign child add illustration chain reach processor force next line second static possibility symmetric say strategy almost processor iterate examine candidate take processor b notation eq candidate next processor add candidate rejection candidate thus three step last core exploit correspond next candidate branch top useful involve future computation step select tree complete core return reader describe methodology straightforward major improve acceptance delay instead reach extra algorithmic reference give depth add assign k assign candidate k computationally individual split acceptance bias directly u decomposition set strategy show large involve approximation basic acceptance product straightforward remark delay remark reason nonetheless actual poor actual example code cluster ghz core use communication core combination delay acceptance upon delay soon likelihood start bring unit number delay efficient mcmc represent logistic box quite hasting delay concentrated parameter particle detector experiment large search particle decay particle background team provide public reproduce behaviour make adequate combine delay acceptance regular burn iteration sample first logit covariance mle obtain burn core delay run counterpart average draw iteration obtained delay acceptance algorithm repetition namely relative ess huge concentrated distribution posterior clarity similarity mixture offer challenge reference fisher jeffreys readily available jeffreys past drawback posterior hoc correction rely improper posterior distribution jeffreys derive matrix whole establish beyond goal progress sufficient allocate dominate event remain improper associated exhibit implementing delay costly integral form integral analytically involve integration ratio costly metropolis acceptance jeffreys determinant time delay apply accord improper pick second valid therefore opt small choose second multiplication translate n I repeat value compare metropolis implementation metropolis hasting version rely delay without implement maximum processor availability reason graph report result gaussian hasting algorithm delay second conjunction sequence histogram remarkable particular high variability case aside switching occur mh sample delay acceptance minor balanced delay even reduce number draws delay reduction overall acceptance size time less hasting delay acceptance ultimately respective say reduce acceptance since require little since version delay acceptance broadly time reduce acceptance nevertheless suggest time merge overall computational advantage mostly prior costly gain term mainly exploit thank helpful acceptance massive help cluster fundamental partly paris bs paris paris paris paris mcmc hasting distribution huge cost strategy idea generic divide acceptance division variate consider part computer hand example keyword big mcmc acceptance jeffreys running mcmc algorithms execution algorithm direct illustration difficulty solution issue literature likelihood handle unit computer consensus prior evaluate approach acceptance rather rejection sequentially computation propose acceleration modification algorithm present gain computer realistic environment regression benchmark mixture jeffreys conclude hasting acceptance decide value decompose ratio accept successively uniform result markov sequentially stop mean costly final hasting algorithm metropolis acceptance value target test preliminary step detailed balance argument take arbitrary decomposition associate q balance purpose metropolis particle mcmc decompose metropolis modification hasting likelihood delay acceptance good probability original hasting
edge appear array importantly exchangeability invariance property place counterpart provide exchangeable exchangeable analogy uniform representation consider positive equivalently jump carefully evy characterize able evy activity sparse alternatively compound family yield infinite evy measure building framework able considerable efficient statistical procedure graph evy utilize hamiltonian dense infer graph thousand million enjoy former model whereas connection nonparametric interpretability desirable node tune law straightforward interpretability hamiltonian monte carlo efficiently rapid power propose bipartite undirected importantly prove formulation graph cast exchangeability non explore section sampler simply apply efficient computation large network organize background exchangeability array measure important foundation propose present background form building undirecte bipartite graph present exchangeability present section specific case dense empirical carlo computations extensive analysis variety structure network build construct brief exchangeability discrete array thorough exchangeability present survey abstract notion place table l structure arise exchangeability discrete exchangeable law exchangeable measure mix examine time associate exchangeable evy process focus recall arrays discrete consider special row node scenario distinct identity bipartite array likewise array jointly exchangeable matrix undirecte exchangeability fundamentally important concept model exchangeability network triangle star separate invariant recommender derive crucially assume adjacency exchangeability consider throughout point examine notion derive de style theorem jointly exchangeable definition exchangeability measure lebesgue also array jointly exchangeable surely poisson place within yield graph exchangeability adjacency flexible class functional refer exhaustive countable disjoint e increment poisson evy laplace transform evy characterize increment note jump evy infinite jump jump surely model finally throughout tail evy intensity q poisson implicit weight often person message associate count message tag could undirecte direct direct node due relationship much gain carry atomic illustration restriction ccc auto grid circle color color draw blue cm leave loop bend cm edge bend left node bend auto cm every style edge node atomic simply give measure informally individual construction imply finite primary similarly atomic indicate edge arise undirected edge self could person page equivalently specify undirected condition random ij ij ii yield respectively mass drawn measure model simulating graph direct depend total value form undirected consider correspond cox attractive practically theoretically use power law exact sampler dimensional practice activity normalize background random probability variable distribution partition exchangeable partition symmetric argument rewrite though bipartite let set allow atomic direct similarly atomic graph formulation introduce whose jump correspond bipartite slightly formulation general exchangeability enable insight depend intensity provide refined choice jointly permutation da follow directly exchangeability rate l evy tail evy evy bi inverse evy intensity undirected evy intensity analog formulation extensive illustration arbitrarily yield enable interpretability ji indicate expression application l evy tail evy q slowly vary function satisfy constant equivalence notation degenerate evy intensity follow asymptotic derivative edge restriction obtain poisson infinity activity sub infinity link evy intensity evy intensity vary slowly xx direct consequence appendix theorem scale node activity evy intensity disjoint group simulate undirected direct transform undirected one imagine simulate direct infinite jump approximate possible evy intensity resort applicable evy intensity inverse define sample weight method direct problematic must one poisson bernoulli draw instead cox direct edge simulate inverse evy iw z show scheme examine various link evy generalized process graph hyperparameter undirected graph case direct bipartite poisson increment dirac delta recalling follow equivalent os enyi lead dense graph edge grow poisson representation poisson either trivially empty interpretable remarkable know l evy intensity jump jump jump include special process tail evy intensity gamma display enyi b gamma c stable exact sampler mass stable plugging process variable give direct undirected challenging scope power law undirecte direct incoming twice almost surely behavior corresponding proof sparsity dense whereas nod undirecte almost infinite activity evy intensity proof technique analysis simulate undirected various os r enyi graph nonparametric model explore exhibit power heavy degree show cut tail plot number number growth os enyi cc er ba distribution various node lead dense graph versus note growth os base explore empirically follow interpretation figure relate slope distribution overall network law overall direct interaction large determine decay law degree pure small infer hyperparameter hyperparameter conditional restrict decompose corresponding measure correspond total remain point poisson distribution evy probability laplace poisson random identifiable bring information homogeneous derive also want assume improper prior evy pdf hyperparameter interest w simple miss poisson convention efficient propose hamiltonian hmc within log posterior case q total mass admit analytical base need metropolis ratio summarize sampler rest hmc metropolis hasting graph count hasting computational linearly step hmc mcmc iteration hundred thousand efficiently hmc collection bipartite posterior exponentially intensity particular gamma total stable additionally describe appendix l evy intensity preserve identifiability update metropolis calculate latent symmetric repeat detail model regime run mcmc chain use matlab computer successively indicate rather remarkable node degree degree displayed show method model os enyi run chain specification informative expect section weakly factor convergence trace plot converge node os star sparse base graph relate measure costly implement formulation describe graph prior aim report connection mcmc output social circle facebook political connection network student california protein power united act bipartite mat method network email connectivity link www pages nd range edge empirical l l name nb node nb ci facebook mat www run chain specification posterior credible trace respectively parameter fail provide small circle likely dataset connect three infer note top network dense evidence subgraph spatially may highly though community sparse capture future generative note analysis remarkably leverage otherwise discussion reasonably network l see tail behavior cutoff explicitly cutoff dense cutoff tail bipartite article projection create dense contrary construct undirecte count two count great edge dense issue overall appear well homogeneous law cutoff tail perhaps dense cutoff extensive work dimension overview produce draw edge product parameter similar rescale projective another edge node interaction belong dense latent node embed latent factor case edge probability possible extend approach highlight connection generate configuration proceed follow odd node connect edge obtain discard self loop repeat work place generative projective modeling exchangeability network tool theoretical herein represent important building development incorporate attribute etc thank derive feedback value stochastic relation slow grow grow activity activity equivalently follow homogeneous poisson law yield theorem combine almost almost consider infinite activity jointly exchangeable w imply conclude finally q moreover surely thus eq imply graph construct exchangeable symmetric array surely law large almost thus almost surely combine dominate statistic value symmetric hx x x consider martingale
situation minimax separation characterize mean enable detection leave dependency sequel contribution know unknown combine canonical distinguish symmetric tailor dependency propose minimax covariance normalize competitive relatively test base top eigen propose test moment achieve rate detection obtain projection skewness statistic suboptimal signal term mahalanobis mahalanobis minimax minimax detection express regime minimax detection rate top ep n top minimax rate r unknown low bound extremely test nd moment mixture support mean estimate responsible parameterize certain estimator consistent dependency left selection work dynamic constant dynamic set give positive integer large ps analogously replace contribution support maximize consistent minimax estimator nontrivial asymmetric show support consistent imagine cluster motivation selection meaningful accomplish indeed mixture methodology test nontrivial suboptimal variant bring nontrivial skewness normality motivation prove rate moment hard control nan issue emphasize table moderate testing share pca detection gap amenable procedure contribution propose test wise relaxation sparse table wise precise statement rate regime test maximal top eigenvalue precise slow optimal symmetric nd sign moment note mathematical technical argument derivation reduce standard distribution put chi detail degree control bound approximation gaussian random already cite dimensional none offer real theoretical difficulty mathematical area gaussian focus design polynomial identifiability optimize exception analyze canonical mixture gaussian center note sparsity spirit propose appear initial present publicly feature wise goodness slightly suppose nevertheless rate relate cluster mixture two gaussians dimension identity exhibit method propose identical method relate obtain specialized author discriminant comparable instead close literature component expression work iid center normal know closely work lead sparse relaxation see closely tackle selection context propose correspond step study method strong lead concern reference sparse diagonal important general covariance indeed eigenvalue covariance unknown special diagonal coordinate wise discuss issue sparsity covariance mixture proof defer low notation matrix principal euclidean denote inner etc appearance covariance minimax q fix test use knowledge covariance fix minimax usual testing reduce design low proposition apply know display thus respectively lead maximizer principal standardized variance competitive moderately set vector guide top ss standardized observation direction roughly reliably detect versus fix powerful nc universal simple inclusion argument test remark notable difficult compute reason leave implicit test proof proposition say reliably detect consistent let sequence asymptotically mean nc constant strong assume dynamic range small isotropic boundary roughly statistic show top principal eigenvalue instead powerful satisfying however proposition mahalanobis distance schwarz eq symmetric methodology seminal applicable assumption test variant case mean asymmetric set minimax minimax detection distance degenerate sparse dimensional setting deduce become minimax phenomenon occur problem remark reduce reduce note testing center covariance bring sensible hypothesis high set test normality normality reject general direction calibrate test substantial discrepancy proposition come moment heavy tail concentrate enough fourth central absolute moment aforementione large minimax boundary variable unable original situation mixture symmetric argument maximizer normalize denominator maximizer align motivate consider estimator consistent effective seem fail simple condition otherwise omit simple low bound term mahalanobis mahalanobis relevant dimensional reduction simpler sparse asymmetric meaning shall asymmetric symmetric due ability symmetric cover pseudo test fix setting sparse note interest normality make sparsity assumption along necessarily direction versus substantial obtain issue enough third leading eq critical c universal achieve note minimax substantially unable statistic analogy consider estimator despite statistic satisfactory strong refer reader coordinate wise support covariance unknown lead linear analysis testing matrix eq eq estimate covariance estimator place yield convention square diagonal matrix ij depend long diagonal bound detection denote sequence critical asymptotically powerful constant maximizer phenomenon suppose qualitative difference statistic covariance proposition involve mean away mean result grow situation value result size subset precise compute practically central concern consider section eigenvalue tailor sparse nature motivate method arguably test implement inspire arrive statistic correspond statistic n testing asymptotic eq universal universal adaptation stronger somewhat prove asymptotically c large spread method even achieve rate multiplicative factor incur analogy sparse recent work apply polynomial plant clique time definition covariance constant adapt work calibrate set significant testing level detection eq small range bound condition fix spread reduce performance multiplicative factor rate extent intrinsic polynomial come instead principal need relaxation eigenvalue learn apply detect covariance simplicity eigenvalue semidefinite semidefinite relaxation perturbation soft jk jk mdp relaxations time semidefinite program computationally require eigenvalue grid statistic universal level go ingredient valid find mdp tend nan hand follow tend factor sdp worth clearly algebra binomial random chernoff binomial chernoff generality step begin brevity place resp resp prove asymptotically vector l sr sr entry sum turning moment rely distribute size integration follow denote derive follow enough go zero corresponding expectation small expectation zero occur k last kp first compare expansion ad hoeffding u dx nr nr dx nr k nr iv leave hand x zero derivation hand side supremum negative away side occur imply sr ep sr iv simultaneously positive sr know test equal turn independent variable bound sum consequently w q q since decompose iii k rely depend ii ss fourth need apply iii entail line numerical correspond nr us fact cauchy schwarz line inequality either rademacher hoeffde eventually iv stochastically binomial chernoff start eq v si iv get sum variable proof fix proposition prior sr op sr x turn approach computation follow formula let n assumption taylor rely conclude comparison parameter rademacher decompose separately closely argument without proof reduce use observe tv variation due contraction space triangle translation tv r tv distribution mean calculation application schwarz go tv nh mahalanobis schwarz tv l distribute sum exist go infinity expectation p k thus line hoeffding assume standardized mean bound degree centrality e x wishart apply lemma turn row similarly case performance need control theorem iid normal thus conditionally central freedom non centrality get generality define event bernstein tend q independent conditionally equal satisfying comes chi plug max p large rhs go conclusion small simply tend constant condition rhs rhs w standardize nan n wishart matrix although hence probability go universal appear degree centrality q note work standardize go ss subset union eq tend chebyshev hence tend eventually eventually let maximizer j j standardized assume write control numerator denominator separately euclidean unit bind denominator dimension maximum possibly wishart union derive numerator possibly apply derive variable control proposition universal constant integer cardinality continuous deduce simultaneously together diameter opposite cardinality lead tend note chebyshev inequality ps powerful suffice numerator control chernoff since generate standardized assume simple define bind nan proposition numerator distribute z euclidean norm triangle schwarz cauchy inequality bind subgaussian metric proof section constant appearance since subgaussian packing semi diameter pack coming add nx come deviation tend one calculation chebyshev chebyshev inequality denote extraction us quantity soon development give thus need amount study sign elementary calculation function decrease symmetry powerful section introduce q differ definition show tend shall uniform control absolute center namely appearance control use bind constant hence nn u deviation follow subgaussian subgaussian apply maximal cn get tend last control second moment u explain c n sc control statistic obtain u development conclude proof proposition without work statistic concentrate rely universal cardinality surely decomposition deduce simultaneously control combine together range cardinality moreover v large term p use powerful converge towards asymptotically extract generality translation standardize observation except x iv control denominator probability tend numerator define sign I sign second sep going go remain control deviation combine argument proof sign ss valid go loss v proceeding early chebyshev inequality n elementary increase derivative indeed powerful focus subtle tend chebyshev working conclude chebyshev strictly q variable diagonal find tend moreover fact use tend tend denominator eq conditional variance j holds conclude consistency one detailed omit proposition selection prove argument omit positive imply concentration denominator universal since bound j consequence chebyshev w w controlling follow decompose apply conclude application chernoff standard define get k x back moment x sparse prove subgaussian deviation transform v u taylor get u v k k already lemma taylor true application chernoff yield desire result rely concentration bound rademacher rademacher develop n inequality converge consequently fix superior r hence euclidean w n triangle schwarz inequality n v schwarz iy n apply
node group section dedicated selection generalization sbm briefly review challenge heterogeneity latent model weight self loop however generalization often graph without loop deal characterize individual characterize may either admit loop model specific random characterize behavior would behavior characterize latent exist property study extensively probabilistic dependency distinguish occur literature may continuous network characteristic summarize dimensional block sbm review section explain computed size set configuration huge computational model independent corresponding form nice counterpart maximization latent raise latent optimization combinatorial complexity besides result observation chen yu perform variable finite mixture respectively latent general variable index panel observe distribution depend parent pz pz z pz py factorize get panel give clique dependency structure prevent oppose hmm graphical shape space probabilistic legend observe variable line inference chain former aim rely suffer limited network hundred algorithm latent handle size random appear surprisingly sbm sbm model position define graph two parametrization direct graph connection node parametrize latent vector kind direct distance replace normalize become q case recover translation whether restriction ensure total include social putting rely author observation argue w multidimensional approximate position fit form second step author acceptance rejection conditional automatically position stage stage simple second procedure cluster sampling besides may determine rely conditional latent define connection node sbm induce science network neighbor protein introduce dot respective propose power diameter independently identically uniform fix moreover connection node dot product latent interestingly dot infer estimate rely also provide position label namely label concern consider apply web page wikipedia mention much already quite people compare take impact already investigate describe space latent space suppose simplex namely use conversely penalize dimension view model extensively mention use interaction popular simple prescribed degree uniformly among degree satisfy ensure know view precisely apply world wide graph company limit graph independently discuss start node I unobserve view sum multinomial consist characterize relation undirected loop direct loop easy generalization loop distribution note matrix model social network biology protein protein world etc weight early version appear conditional useful gaussian etc distribution induce thus mixture dirac strength distribution restrict cumulative cdf continuous zero coordinate connection cdf dirac mass sbm truncate poisson multivariate simplify assume varie consider structure connectivity take depend whether assume induce exactly clique already unconstraine sbm cluster detection sbm subset behave existence start discuss identifiability parameter sbm know undirected sbm solve sbm non valid sbm sbm estimate sbm less sbm rest current estimation sbm binary sbm first limit binary sbm group sbm namely nz formula handle early relation bayesian attempt develop heuristic given replace simple decompose follow eq kullback leibler iterative starting maximize quantity expectation automatically factorize instead go back kullback divergence variational instead optimal factorize unchanged current distribution graph result sometimes respective mcmc sbm gibbs gibbs iteratively factorize distribution right mentioned approximation convergent regular precisely prevent sbm ensure procedure accurate see weighted network version variational approximation appear graph detail thesis one approximate posterior realize factorize approach sbm node note implementation weight graph package weight binary sbm model consider proportion drawback moment general consider propose suit binary optimizing whose parameter look pseudo simulation approach graph group connection within likelihood result node procedure iterate perform theoretical parameter estimation method explore binary precisely estimate proposal reference associate discuss rely degree fast concern estimation sbm concern sense iteration local empirical exhibit reference empirical procedure variational estimate variational nice variational consequence approximate increase maximum convergent parameter graph increase kind dirac factorize distribution possibly maximum variational proportion assumption unknown require fundamentally different sbm see sbm composite surprisingly limiting ensure amount proportion limit conclude regime go infinity next recover automatically theorem formal start recall community additional intra group large outer connectivity cluster posteriori result rely behavior variational binary sbm establish state parameter group dirac actual value configuration also result converge rate convergence sbm yet dedicate posterior latent neighborhood estimator sufficient dirac locate actual condition highlight case sbm recovery converge product configuration explain tend sbm related use different consistent sense tend sbm modularity community difficult consequence remain unclear reference establish procedure rely modularity satisfy sbm modularity issue recover tend separability author spectral cluster sbm group binary group group refinement sbm receiver motivated study graph except allow group grow size nearly restrictive note provide cluster setup degree increase node sbm computationally demand network even space unknown question introduce posterior evaluate undirected graph integrate criterion entropy z proxy term refer proportion whereas whereas observe study approximation criterion formal exist bic correspond around recently gap order aim default often use assign sbm belong multinomial assumption analyze network play overcome limitation node possess choose membership link sample fairly strategy simple network propose overlap membership vector respective probability class binary overlapping sbm present logit z much membership relationship chemical specie etc network publish contribute contaminate column asymmetric membership membership draw first context simultaneously variational inference recently propose toward sbm assume indeed situation connect extend value sbm take play role sbm control degree orient version asymmetric generalizing likelihood regular detection sbm partially explain accounting desirable well understand structure important species covariate edge take ip presence edge logistic regression covariate raise focus covariate propose probit depend covariate act membership author
maximize regularity establish point observe far different allow exist matrix uniformly leverage property establish propose measure time empirical straightforward section evaluate influence synthetic generation building employ generate uniform maximum large inside generate sequence baseline hazard algorithm topic standard hazard indicator mention tweet tweet survival hazard short update record indicator illustrate newton baseline hazard week worth interval size show mean vector underlie ht see quality quality give influence interest exhibit relative next insight measure generation htbp rank one approximately realization user action generation rely picture twitter link account draw account u financial post york increase activity acquisition twitter tweet extremely trace twitter active tweet back tweet table tend mention mean self activity pay medium account political media influence twitter utilize build public pass date account tweet nj avg http rt mass http rt huge http http co http stand worker w keep track pay rt thank I community http take hope nj http thank rigorously apply constitute many algorithm account publish create adjacency data score denote great r rank propose financial york journal cycle cycle next estimate analysis max china hold death frank run use raw consistently influence include explanatory table coefficient mean influential twitter successfully tend influential life pass instance influence r std intercept age df std error intercept propose influence age characterize influence scale platform information user comprehensive demonstrate direct correlated influence broadly direct closely community modeling inference detection goal adjacency media platform incorporate fundamental action social outperform topology perhaps massive volume estimate relatively straightforward identifying involve analysis message relate appendix algebra element necessary hessian partial respect expression cross baseline hazard partial likelihood parameter ignore straightforward go concave expand definition rate define replace hazard integrable integrable converge number converge converge notation employ condition theorem simplify want establish convexity dimensional zero expect entry derivative derivative semidefinite definite convex complete ten influential account period financial post prominent several rank financial table period table score significant influence regression transform score moderately std influence df std intercept r estimate std intercept df variable estimate std value intercept age theorem theorem introduction remark measure twitter counting extract medium twitter text interaction influential sense generate platform service web apply interact capture influence maximum covering year twitter member news business volume increasingly complex datum create insight area growth social public easy exchange news idea area business network analysis vast amount careful content propagation product platform importance business twitter twitter million twitter lag behind facebook nevertheless presence mechanic twitter basic communication account platform allow short mid message daily twitter follow receive whenever follow serve primary spread platform account tend interact channel direct way copy another tweet tweet name symbol mechanism mechanic user rise message together use constitute corpus enable search query build capability create flow interaction account influence capable drive discussion topic valuable twitter user constitute active business service scoring employ good message reason sophisticated widely use ranking result network popularity necessarily influence follow account million propose account twitter social media platform ability message action interaction account count action intensity basic underlying activity account reveal world member united house united states two year prominent post cnn e g white house influential interaction direct give tend e edge tweet influence twitter indicate closely importance purely solution organize introduce modeling propose influence synthetic concluding remark section presentation future development correspond node twitter whether symmetric vice versa principle dynamically static explain twitter platform account message account mention vast majority let total generate topic message capture response account mention counting process hazard use cox hazard specifically hazard process account positively
neighborhood regularization belief propagation square euclidean distance originally maintain next convnet sift descriptor sift bivariate alignment seed sift convnet feature nearest sift flow sift row version align sift flow second image target image cat convnet case sift instance alignment measure predict target small great point wise align correctness truth lie bound box height pick compute visible target show per category indeed convnet capable sift field table tv sift flow sift transfer convnet semantic parts image train classifier datum extract feature convnet rf lie place vs sift five activation pooling layer convnet layer sift come specifically testing convnet feature expect high train car cat tv sift precise understanding response cat histogram location include maximum lie sift seem sensitive location convnet fine field motivate final experiment base convnet localization beyond sift feature histogram response pixel take cross cat mean convnet plot feature despite large work sift sift annotation ground bounding inspire slide window part detector predict location cnn demonstrate deep investigate cnns scale window descriptor descriptor region cell give field hard mining ten close truth example dense descriptor sift eight consider within bin ground time bin negative detector nearest neighbor gaussian take neighbor find cosine fix standard output detector combine tradeoff validation high location define width prior detector train outperform sift margin knowledge dataset five set rescaled box annotation predict sift outperform sift satisfactory despite offset noticed window regression fine prediction criterion car cat sift sift sift five cat sift conv prior view alignment intermediate implicitly convnet classifier understand correspondence field convnet ones visual support program reference california berkeley cs berkeley edu convolutional neural net improve detection understanding establish fine pooling whole label success correspondence precise localization effectiveness correspondence evidence convnet feature fine field size perform alignment conventional object advance convolutional neural net dramatically improve specificity rely large pooling region job coarse localization extend fine modern able field correspondence pool task well suited hand provide convnet conventional considerable image alignment task face motion object alignment correspondence across variability alignment supervision require class joint model unsupervise method hand unsupervise optical match densely sample sift correspondence motion sift recognition pose fine grain categorization depend variation category challenge localization human pose pool input convnet convnet spirit pair dictionary average convnet feature vector associate patch field table number replace patch near neighbor database densely database one million every feature match rf sized neighborhood throughout region notable e cat replacement visually specific cat get replace differently color shape
mx unknown use sample n pointwise alternatively build set quantile confidence set include smooth ordinary smoothed regression mode issue tool htb q process limit assume smoothed discrepancy supremum gaussian process couple behavior bootstrap consistency f theorem limit confidence modal set level sample kde conditional ny estimate pointwise pointwise ny ny correct coverage nx ii estimate prediction coverage prediction select kde prediction subscript denote smoothing lebesgue uniform define roughly speak estimate manifold manifold bias variance manifold proposal select e minimize trade versus mark line display modal estimate uniform set display uniform local modal comparison smoother also illustrate modal regression method conditional capture main next population modal base underlie pointwise prediction lebesgue modal somewhat abuse lebesgue length prediction consider denote gaussian variance important make population reflect state define several quantity minimal center moreover q show signal modal small way modal know expert vast function parametrize variance usage assumption consider simple tool use inherently base joint mode without component mixture model role modal regression specify instead flexibility regression kde tune figure give linear modal package specify run eventually high bandwidth reveal regression trend domain specification carry trend across assumption independent volume cover extension would address modal conduct cluster lead proportion modal roughly analogous cluster mode apply modal path start cluster shift accord iterate update point arrive mode jx determination modal jj immediate running shift point datum example run mode estimate modal place modal regression modal modal seek modal former mode second investigate nonparametric modal kde points modal regression confidence prediction kde compare relevant message offer method develop construct usefulness practical treat predictor finish example normally distribute surface come modal green modal identify local surface sufficiently large assertion kernel theorem write mode unique empirical point eq divide side away inverse mode prove assertion focus follow repeat argument mode local mode local assertion bx nx yx imply big involve thus supremum maximum vc envelope give proof theorem theorem local argument integrate still omit technique define empirical process proof couple anti convert coupling constant constant recall see constant envelope let center ba verify assumption vc type envelope inverse closed mode thus pick fact coupling pick triangular constant nh nh eq nh essentially theorem current basic detail depend probability index density approximate measure completely take estimate derivative kde determine maximal space putting result prove prove mode w another necessarily distinct local mode mode eq since mixture component away bad scenario define since term mode obtain local mode definition attempt solve hold see sufficient combine conclude theorem three step consider pointwise prediction set extend prediction summarize four mode apply first uniform gp note prediction pick regression mixture center kx mx thus reference z require involve inequality apply equation hessian explicit determinant namely hold need eigenvector therefore former case correspond modal regression usual simple kde techniques latter kde behind modal tie class alternate regression response variable unlike base modal regression would modal favor conventional answer level mode reveal illustration example fail capture trend produce band improvement well rigorously modal economic formally mode simplification joint consist general smoothly change local behave surface call focus derive kde nx plug author thorough modal prove modal regression derive error metric set base plug regression select bandwidth kde draw comparison suggest modal ridge begin basic recalling previous describe item end simulation response set classic modal denote modal value smooth local set rely yx kde joint brevity modal eq subscript efficient computing shift algorithm gaussian mean describe kernel mesh convergence iteration straightforward update indeed ascent nx yx implicit step actually attain critical size property modal union implicit dimension illustration univariate htb modal factorize eq connect parametrization open call convention write nonempty mode twice modal manifold yx jx joint modal function manifold guarantee mode smooth modal smooth smoothness modal condition hausdorff omit classic notion smoothness hausdorff think theorem interpret statement continuity hausdorff modal manifold merge vary though occur contact manifold figure modal manifold leave curve close look slice unimodal saddle htbp
sampler draw center order avoid vb mh assumption derivation appear covariance matrix mh ground model reasonable mh return true see mh consistently variance ht retain datum actually deterministic retain prior unknown posterior look spirit classical leverage statistic literature covariance classical score impossible naive distinct covariance draw evaluate leverage effect manual leverage score plot manual great one assign retain leverage indeed assign component affect affect affect way complex chain correlation acknowledgment suggest response helpful comment berkeley suppose x linear collect proposition proposition approximation interior apply particular track write eq part remain natural eq zero dimension finally eq lemma target use covariance particular return block matter form next order consider characterize posterior posterior derivation highlight column invertible since stack follow point exactly log interior derivative follow matrix multivariate simplify sufficient vector denote kronecker calculate derivative sufficient statistic statistic interest effectively ignore submatrix variational variational property cc simply multiply v apply normal draw correction transform fisher information use correction formally argue correction variational require correction coincide approximately proportional might affect whereas us analogy parameter asymptotic multivariate take find mode normal em equation transpose complete correspond variance fix variational leverage datum first leverage leverage model score x formula denote scalar form stack recover vary variable analogous fit improper light statistic term correlate respect quadratic multivariate apply consider cc x tx calculated tx tx cc tx since diagonal new perturb observation plus z px p x I imagine unobserved statistic point since matter property expectation convenient stack statistic single covariance sub covariance derive mind notation shorthand body block ccc interested eliminate immediately complement matrix two group refer everything else give cc r r x r I first r r x quantity aid write z r r r x q use version term exhibit power perform analysis substitute thing plug r r final rgb california berkeley increasingly collection interested analyze ever fit old paradigm capture practitioner uncertainty across method fast major model uncertainty interact develop family model particular
pass boundedness finitely accumulation generality limit side boundedness kkt solve subproblem solve quadratic solve suitable write perform processor ghz ram unconstrained subproblem inner condition approximately eq penalty update discuss minimizer closely relate residual thus typically terminate approximately describe place exactly ease reference approach inexact exact randomly generate generate row space standard gaussian test report table nonzero use matlab cpu second termination penalty termination usually recovery slow method inexact objective phenomenon intrinsic c c inexact cpu cpu e e c c inexact cpu e inexact exact cpu cpu cpu e e e e e e e sparse processing constrain widely use minimizer always nonconvex nonsmooth study existence penalty regard local minimizer minimizer penalty solve sequence via gradient prove method kkt preliminary solution system appendix solve optimization continuously moreover bound uniformly method pt integer arbitrarily globally however globally convergent th termination criterion convenience one statement induction use hard use relation relation finitely iteration hold statement induction statement ii hold hence induction finally statement total inner execute one follow ii em solution iteration hold algorithm corollary support partly research grant author partly grant possibly intersection ellipsoid induce tolerance incorporate fitting constrain exact objective existence regard local minimizer minimizer subproblem solve prove kkt preliminary demonstrate sparse penalty proximal nonconvex optimization follow continuous q nh necessarily convex locally avoid suppose region nonempty flexible accommodate wide important imaging science processing variable study refer reader emphasize function nonsmooth nonconvex non enable various induce concern one popular bridge fraction penalty incorporate must application gray incorporate lead substantial flexible range solve extensively decade study scenario nonconvex minimizer construct optimization minimizer closely resort important exact constrain development nonconvex non objective bridge use hard must satisfied soft advantageous knowledge paper various penalization follow problem minimizer global minimizer minimizer projection minimizer onto feasible produce minimizer solution exact smoothed suitable scheme penalty minimize possibly nonsmooth smooth globally nevertheless globally include subproblem addition accumulation approach solve approach sparse solve suitably result rest organize present notation preliminary material study regard minimizer optimality propose update scheme penalty experiment conclude number entry sup norm quasi norm pi ii n r function infinity elsewhere center ba recall subdifferential horizon subdifferential mean eq subdifferential coincide subdifferential subdifferential finally separable reader explicitly throughout nonnegative rank well concern corollary explicitly case start q indeed pseudo last follow follow simple next reader finite upper exist bx bx assumption desire lemma auxiliary concern generality maxima attain optimization complicate next consider increase twice continuously differentiable check optimality stationary monotone consequently minimizer nonconvex regularization bridge equal happen equal negativity building regard minimizer find minimizer nonempty special possibly penalty emphasize little everywhere lipschitz modulus local local subsection locally assume except local minimizer local whenever I since minimizer apply b conclude exist whenever assume loss generality minimizer globally lipschitz continuity modulus assumption hold modulus inequality minimizer minimizer concern minimizer say globally modulus concrete maximum value let discussion fix continuously differentiable attain approximation show conversely minimizer minimizer minimizer minimizer admit minimizer whenever minimizer modulus eq combine hard e conversely study necessarily feasible minimizer old take optimality old concavity follow immediately h follow immediate suppose minimizer penalty optimality condition look model motivate check whenever subdifferential explicit satisfying use hard kkt solution assume motivate kkt kkt exist case kkt equivalent check sx constraint stationary point conversely kkt point comment stationary facilitate exist focus local inequality follow inequality follow minimizer low nonzero derive magnitude lower low except ax b lower bound minimizer finding globally computationally inefficient form nonsmooth solve sequence smooth counterpart see hard show form solve consider adaptation differentiable globally objective observe locally globally appear fortunately capable verify applicable problem hence continuous
carry assign eliminate average testing number node test spend cpu whereas cpu fig result zero also consumption interval satisfactory section conduct diabetes concrete repository one report speak rate rank output calculate generalize inverse rapidly possess acceptable robustness sometimes point sort transformation come symbol become treat resolve operation method sort effectively transformation keep activation function satisfy radial belong function include pose use effective department china china extreme promising learn single hide layer feedforward nevertheless random layer rank effectiveness effectiveness propose improve make weight rate prediction accuracy experimental problem classification regression performance feedforward machine feedforward neural extensively mapping natural artificial cope technique address approximation ability compact make neural field prove continuous feedforward activation furthermore systematic condition activation al bound strictly construct show almost activation method advantage tune network architecture algorithm accomplish minute large might learn possess generalization svm activation among one fast superior overcome input weight bias column sometimes train predict point feedforward network method overcome machine properly input matrix extreme learning machine use bias sort due constructive algorithm activation use radial basis sort radial basis activation diagonal bias output rank simple operation give weight bias spend short fast possesse implement make fast generalize inverse follow bias strictly correct theoretically constructive actually complexity give discussion arbitrary ii hide name sometimes cause overcome extra bias acceptable keep column add definition inverse di j jj da confusion concept follow constructive sort transformation dx dd w x em w w ik kn ik kn p w kn ik k affine sort sample order sort correspondingly weight ensure non network activation gx dx dd square actually gx gx gx ax select obtain b b therefore complete w n em b w ii calculate forward effective algorithm training x cm plus minus ex ex ex plus ex weight calculate output detail practice fast choose bias instead selection big less sometimes random bias singular difficulty accuracy section orthogonal instead singular one summarize extreme call new basis ib I correspondingly sort ij I x gx
fall value polynomially make logarithmic hard conclusion know parameter depend matrix section bind clearly condition suffice importantly estimate parameter actually precisely source literature acknowledge difficult logarithmic perform nontrivial operation truncation concatenation require operation require operation run require include multiplication follow take theorem proceed maintain inductive hypothesis epoch begin step index value vector goal next estimate let algorithm decompose help keep matrix set track angle matrix orthogonal maintain inductive epoch incoherent eq define everything hypothesis satisfied assume base handle indeed probability incoherence satisfied suppose break loop algorithm goal gap analyze statement hold terminate terminate line reach obeys orthonormal satisfie define line return accuracy may iteration conclusion step initial inductive hold round hypothesis hold lemma statement imply requirement favorable bind epoch inductive round occur lemma base summing show precision produce round approximation tt return subsample tt node tt tw tu indicate argue noisy constant statement lemma thing top singular wish show invoke theorem long union condition close return find sufficiently case long lemma establishe conclusion imply inequality algorithm identify gap suppose hypothesis value q let read large q first conclusion analysis gap particular must verify inductive follow imply eq know reflect top vector denote actual use theorem perturbation close eq similar computation q favorable imply appendix unitary eq truncation probability incoherent precisely q ball orthonormal orthonormal projection close eq furth gram schmidt eq thus eq incoherence computation line conclusion incoherent step depend condition hold final holds follow fast completion frank wolfe fw analyze also naive algorithm code fw aim diagonal spectrum sample low fixing per smoothing median implement frank wolfe parameter improve cost run change qualitative dependence eigenvector large wolfe completion measure frobenius error recover outperform metric one reliably converge find eigenvalue measurement basically progress happen svd converge small quite find fw svd begin still outperform like frank wolfe cite fw depict reason reason note fw converge sample converge matrix error illustrate frank wolfe pre specify spectrum run frank wolfe algorithm observe entry predict fw much fw fw fw never near entire acknowledgement institute berkeley take probability subset thing convenient begin choose observe let split l l u rt divide suppose return include independently let let definition show generation treat notice independent consider collect upon record q similarly perturbation similarly singular perturb recall refer angle principal decomposition perturb q q angle span matrix unitary two close orthonormal unitary v v row ir let orthonormal suitably eq index indeed restrict angle fix denote suppose gap collect repeatedly chernoff version matrix concentration subset entry conclude chernoff sequence adjoint median proof prove noise proof index coordinate expression drop identity suppose q probability apply chernoff proposition constant I markov proof choice appropriately inequality complete constant claim compute expectation markov inequality attention choice conclude least claim together small indeed union least show term favorable q polynomial rank matrix number well alternate minimization recover subsample amount processing partly due applicability recovery guarantee feasibility semidefinite program dimension immediately dimension research effort large scale nuclear solver preserve nuclear alternate solve solver alternate scalability minimization less despite progress number th target condition serious completion unknown approximately singular decay typical alternate decomposition alternate truncate crucially sub routine solver appear kind dimension running emphasize main sub time alternate work resolve question variant achieve standard alternating framework improvement completion noisy completion first incoherent certain compare sake begin completion result consequence subsample include nonzero result straightforwardly rectangular matrix state intuitively coherence standard vector formally orthonormal vary state formal factorization expect exponent minimize imply small several logarithmic near step run nearly reveal except overhead discuss apply matrix close typically capture arbitrary assumption incoherence euclidean state entry correspond coherence norm argument frobenius well show generalization recover statement compare parameter enter suppose formally assuming discussion relate overview overview understand understand basic matrix start iteratively fix objective typical repeat generality square exploit square update interpret noisy rough spectral initially ignore since like discover value order achieve number arise alternate truncate svd onto subsample behave run fix call run alternate minimization instead suggest et number dependence unfortunately see run serious noisy completion hope black imagine day spectrum matter rather matrix run error prevent converge far might problematic singular arise residual arise multiplicative difficult ensure intuition argument seem single alternating however dynamically maintain proceed epoch proceed begin epoch converge singular prevent say remain vector correspond perturbation ignore polynomial importantly motivate epoch approximation singular point identify block singular principal mean alternate factorization epoch crucial always minimization subsample original purpose prevent accumulation basic say gap care gap might small additive super pay able must make sure singular issue face coherence incoherence completion outline sure incoherent rough estimation preserve incur alternate handled build extra involve take alternate incoherent related number polynomial crucially build minimization black box note work sufficiently particular polynomially al nuclear guarantee however dependence aware fast polynomially another nuclear
model call suggest global graph previous separate package execute complexity exponentially variable np network problematic high dimensional biology protein scalability impractical restrictive either dag address reduce independence score computation core processor make core hardware algorithm decomposable meta heuristic say si efficient basic backtracking optimisation examine implementation backtrack gain variability dag alternative software framework implementation set suffer section among share ic pc gs illustrate first dag bn originally suggest consistency symmetry correct treat false positive arc dag parent child illustrate step separate separate set increase keep computation greatly know I exception skeleton enforce third arc direction equivalent identify decomposition arc undirecte identify contain multiple dag uniquely skeleton step step dag complete direct acyclic variable symmetric symmetry drop parent child equivalently give place undirected limited symmetric pair direction arc I direction arc still undirecte recursively adjacent direct lead undirected arc symmetry make translate enforce construction g x jx reduce test introduce positive false type limitation focus model overall acceptable causal model large set many describe version step inclusion inclusion case backtrack undesirable structure store backtracking compare implementation compare define give correspond backtrack node si code learn already bn contrast node si b equivalent learn would si pc information omit brevity package implement node pc implement backtrack instance backtrack vast investigate various alarm arc expert alarm message unit monitor device arc bn diagnosis clinical condition marker arc base physics united assessment plan recognition action arc develop context linkage large linkage genetic marker arc expert interpret disease physical dual core gb ram size fact backtrack variability display three step mean step si test si pc step convenient show perform test intensive part amount implementation pc different pair arc merge step backtrack require constraint user use master process generate alarm alarm cl cl counter generate pass output affect structure raw run backtracking perform cl cl counter cl r inter cl counter average result link size run parallel si follow add small never fewer complete instance parallel overhead range attribute cost well competitive bad performance scale biological si pc control count former predict patient diagnosis gene explore presence system biology bn genome quantitative trait use human disease dependence nucleotide publicly available miss number biology genome wide association si pc average consider test student correlation see observe overhead normalise running consideration make across overhead surprisingly overhead seem absolute overhead comparable suggest strongly depend seem little effect run implementation comparable framework create implementation root ic particular part execute firstly limit overhead keep part see backtrack improve backtrack different motivated suggest increase dags speed gain competitive parallel computer come least core outperform backtrack even implementation reference overhead introduce high scale efficiently finally important consideration gaussian variety structure several implementation improve overhead might dynamically become bn likely little benefit overhead parallelism window avoid directly bn partial update would also overhead scale bn average variable latter learn copy could avoid share memory suggest overhead never modify hand bn overhead various bn likely leave even allocation scheme bn dramatically introduce variable rough proxy bn depend impose sparsity tool overhead keep constraint use four biology modern core hardware preferable backtracking develop processor dag distribution refer arcs connect represent
class maxout percent classify clean training adversarial model appear deep build part likely process model behave reasonably thin pt network adversarial form bad perturbation incorrect answer phenomenon focus nonlinearity argue primary cause explanation quantitative set yield simple generate adversarial maxout make discovery neural adversarial correctly classified wide architecture train adversarial blind nonlinearity deep combine insufficient average insufficient regularization purely unnecessary cause adversarial fast make adversarial practical adversarial training dropout generic dropout model average significant reduction adversarial change nonlinear family rbf fundamental designing linearity design adversarial possible tradeoff designing optimization nonlinear demonstrate variety include bfgs reliably adversarial imagenet example example indistinguishable adversarial misclassifie training softmax however optimization modern machine determine naturally datum use euclidean regard indeed already design adversarial perturbation though yet maintain clean adversarial many individual digital pixel discard dynamic rational perturbation feature formally assign discard storage problem adversarial adversarial perturbation maximize subject max assign magnitude weight dimensionality activation perturbation output sort force closely signal signal amplitude dimensionality previous example suppose linearity simple explain softmax adversarial suggest perturbation maxout behave way easier sigmoid tune perturbation neural obtain refer adversarial computed backpropagation reliably cause variety imagenet softmax classifier set adversarial maxout cifar standard deviation adversarial example example angle reliably adversarial misclassifie favor interpretation way confidence imagenet whose gradient small bit image number consider logistic gain intuition adversarial train recognize sigmoid function derive adversarial perturbation note sign gradient regression somewhat activation training eventually confident happen adversarial simply adversarial good logistic multiclass softmax treat softmax single weight decay deep multiple unit decay necessary decay maxout coefficient cause get decay coefficient cccc logistic model adversarial somewhat unlike deep adversarial universal layer unit assign discover desire obviously specify adversarial adversarial somewhat augmentation usually actually augmentation occur way beyond dropout benchmark partially adversarial bfgs may work guess work use error reach adversarial large original maxout cause model slightly adversarial make original maxout stopping terminate decrease flat adversarial therefore early adversarial five generator train weight trial rate result mnist though indistinguishable fine tune adversarial adversarial error adversarial training show robustness error train adversarial still highly confident misclassifie learn change train adversarial perturb interpret play adversarial request case human replace copy nearby insensitive change small training box correspond norm zero inefficient adversarial dot case difficult fact case set noisy noisy maxout noise base pixel mnist adversarial fast sign clean point space considerably rbf invariant generalize view rbf unit different tradeoff curve unit direction rbf point decide quadratic rbf adversarial obtain error train aspect set adversarial often agree non readily account behavior capacity consistently label adversarial rational adversarial example subspace product different see contiguous sign explain adversarial misclassifie fairly probability misclassifie another explain adversarial neural network learn reference approximately stability adversarial deep maxout softmax rbf misclassified maxout rbf maxout class maxout correctly number drive rate exclude mistake predict maxout rbf component behavior maxout generalize significant behavior cause adversarial example reliably large classification occur thin manifold occur plot make train maxout show softmax example see unnormalized linear wrong classification region move input input curve positive box indicate classified input hypothesis adversarial example hypothesis generative training constraint cause model confident test get good classification mnist differentiable differentiable mnist mp sure adversarial rather non top model find rate remains alone cause adversarial maxout mnist network use seed initialize generate dropout select error adversarial example design entire adversarial design member fall adversarial perturbation summary follow observation property dot product generalization adversarial across adversarial perturbation highly align function train perturbation adversarial example like rational direction adversarial clean example adversarial regularization control experiment fail reproduce simple regularizer decay model optimize easy capacity adversarial adversarial perturbation observation concern class easily class rbf modern ai design relu maxout lstm sigmoid carefully fit adversarial correctly imply truly ask confident point confident often incorrect work identify problematic point ease motivate development procedure locally acknowledgment thank helpful thank article concept degenerate input human classify belong want positive input classify degenerate input something separate binary classifier want near would
artificial forest decrease hand ib average noise build mechanism beneficial next diversity instance weighting scheme compare use achieve compare method weighting outperform scheme case biased weighting weight weighting technique forest count average bold ccccc ccccc dot count count count count f filter validate filter ensemble filter choose algorithm achieve filter yet filter classification bold value table average set bold significantly accuracy gray cell l ccccc noise ensemble compare compose ensemble compose mlp three section high contain low nine forest high correlation significantly beneficial compare ensemble weight filter base filter ensemble bias filter handle significantly high surprising handling decrease see noise filter ensemble class class noise approach consider ccccc ccccc count count count completeness compare representative gain accuracy single despite accuracy higher consider surprising perform focus mlp high accuracy consider vote significant within mlp weight filter handle classification safe compare mlp base weight mlp mlp level noise mlp achieve represent weight important weighting investigation topic examine diverse examine voting find outperform less diverse handle knowing filtering achieve significantly low classification accuracy able outperform consider handle bad filtering statistically technique despite handle voting ensemble exhibit possess base classifier gray widely induce bias less effective broad bias diverse prediction instance instance noise handle technique algorithm across keyword filter weighting learn voting machine learn accurate generalizing instance world generally attribute attribute noise consequence summarize fr frequency know case task instance relate work examine handling approach theoretic remove misclassifie towards especially artificial efficacy handle technique give upon datum handling technique artificial ensemble base classifier ensemble class accurate diverse hypothesis however none explicitly focus classifier inspire principle ensemble select diverse algorithm prediction dependence hypothesis could two classify way set algorithms base classifier vote set filter technique weight voting base explicitly diversity account find diversity significantly improve filter demonstrating noise set vote diverse classifier achieve high classification handling technique organize section label noise diverse inherently examine class find class generally thorough handle fr learn algorithm design error prevent pruning completely change handle noisy instance problematic boost instance place single svms hinge misclassifie instance instance misclassifie wrong modify possibility maximization remove noisy weighting instance criteria filtering receive generally result especially artificial broad handling always classification noise add et al also filter examine predict efficacy near remove misclassified learn van idea bag theoretic learn noisy instance remove instance wrong low label filter potential discard significantly filtering consider special instance assign instances pair influence discard filter clean automatic datum correctly label train decision include correct value instance increase instead single exception set diverse learn algorithm explicitly handle label bias examine diversity effort diversity measure accuracy ensemble study diversity classified misclassifie distinguished class effect ensemble boost bagging create selection knowledge diversity presence voting handling approach instance heuristic bias classify determine noisy estimate e specific multiplying infeasible though sum hypothesis non trivial attractive discriminant classification discriminative lower examine diversity refer meta classifier distance make hierarchical agglomerative dendrogram connect representative conjunction color package terminal classifier output training mlp backpropagation score node reach leaf node near label forest leaf return cover probability however induce class misclassifie filter find generally produce biased misclassifie algorithm c rand forest ib count examine backpropagation forest tree rather count sum meet compare bias seven weight biased technique list brief cited repeat near remove misclassified near neighbor base instance least correct instance removal filter training continue long remove filter remove fold default remove instance misclassifie base distinguished filter base classifier choose filter three ib misclassifie majority remove misclassifie training correct correct filter validate partition equal time subset instance misclassifie iterative filter partition induce misclassifie induce filter original tree em belong class criterion clustering form default finish since filter removes finish large set set finish table compare count represent time accuracy bold achieve contrast represent accuracy significantly many handle measure previous artificial improvement bias nature handle broad handling datum noise briefly application handling set compare technique add handling significantly investigate important example highlight handle beneficial handling accuracy gray significantly l ccccc mlp rip count count filter count conjunction terminal option explanation either package package graphic terminal graphic macro ltb lt lt lt lt lt lt ltb lt lt lt lt bp r r ib conjunction see explanation use package graphic terminal need graphic macro ltb lt lt lt lt lt lt ltb lt lt
generally kronecker nmf kronecker basis nz equivalently change rest proposition stem enhance ghz run window code observation additive independent gaussian nonnegative multilinear rank sparse tensor entry tensor meet specify noiseless version world observation signal interference recover normalize zero solver version nmf subproblem subproblem run directly procedure update fit involve computation probably ill issue algorithm although seem speed consistently level base average algorithm robust without sensitive iteration quite helpful noise consequently improve robustness investigation essential uniqueness tensor datum core tensor generally db component factor failed also guess cause convergence simulation factor substantially improve essential local minima largely th object experiment pose simplicity category randomly decompose r denote k time local superiority technique pca extract randomly list table outperform part moreover core generally impose whole allow adopt computation propose analyze iv iteration fit algorithm acceleration c mi fit th c c face recognition face database gray randomly decompose training unfold feature matrix knn classifier distance measure run factorization relatively also discriminate analysis tensor considerably problem seen accelerate marginal version affect recognition database basically accuracy computational load guess unique originally physical basis know give show c face without need impose sparsity nonnegative tucker powerful multi nonnegative give localize representation multilinear order free learn procedure reduce subsequent substantially flexibility indeed well establish contaminate various discuss uniqueness nmf uniqueness decomposition justify propose promise nonnegative tucker tool nonnegative latent high often high major low multilinear tensor significantly cost function besides dramatically reduce run quite flexible well establish incorporate substantially improve uniqueness curse dimensionality tucker world justify validity propose tucker decomposition alternate rich behind task signal field component highly gain decade rapid development observation data component specific temporal smoothness properly exploit base successfully color often cause perspective model widely apply deal high widely tucker decomposition pattern cluster denoise etc great success nonnegative graph datum preserve regard nonnegative prove powerful analyze nmf interpretable nmf extensive area code matrix respectively element integer large set tensor element division tensor define kronecker rao product column wise kronecker zero uniform nonnegative tucker decomposition gain recent advantage structure illustrate give part representation face represent possess multilinear tucker decomposition lack uniqueness curse indicate fact core exponentially unconstraine tucker provide meaningful core sparse discover significant curse exist perform update rule exploit special multilinear tucker suffer term especially large quite develop yield take tucker decomposition paper unconstraine tucker tensor access big thereby considerably proceed overview present together investigation important section iii review flexible simulation section vi high propose notation mode fixing index matlab unfold column th I follow concern mode frequently mode due property mode product tucker core connection column rank factor nonnegative factor bring key purely additive often may effect factorization ability localize th core broad application processing understand th matrix tensor core tensor equivalently column sample combination basis vector extract feature regard eq tensor involve multiplication base counter multiplication version ir nr mu update choice term descent multiplicative multiplicative see rule manner update update improve total idea nmf practice unnecessary subproblem execute converge tensor tr multipli method th respectively mu core tensor analysis lipschitz provide initialize converge nmf solve roughly speak inverse fast stability sometimes component necessarily nonnegative nmf factor algorithm q similar generally guarantee many accuracy approximate tensor respectively exact practice entry hence intrinsic often incomplete weight miss decomposition miss straightforwardly prefer step weight tucker decomposition complete tucker approach allow entry dimensional deal miss randomly approximation datum partial subtle difference whereas select satisfactory framework scale govern tucker decomposition curse dimensionality lack uniqueness former increase latter due tucker limitation tucker although far analysis uniqueness still miss uniqueness nonnegative multilinear multilinear multilinear nonnegative multilinear uniqueness uniqueness nmf nonnegative multilinear rank simply r ns trivial nonnegative obvious nmf exist trivial n contradict essentially
recurrent input current vocabulary element follow rnn length recurrent connect backpropagation propagate multimodal neural rnn deeply simple rnn word layer word embed recurrent multimodal softmax input firstly network dense word dimension vector secondly dense encode word find calculate vector sentence use vector initialization initialization sufficient treat activation embed input embed layer dimension calculation recurrent denote activation vector unit relu training deep vision differ rnn sigmoid relu hard sigmoid backpropagation conduct rnn temporal heuristic stop step hyperparameter good property relu stop early multimodal connect model b layer recurrent image part extraction connect multimodal please add together multimodal multimodal scale force non process rnn model softmax generate layer vocabulary rnn adopt sentence set length denote generate softmax context calculate equivalent set backpropagation learn part embed layer recurrent e mention layer part convolutional feature widely previous multimodal use improve gradient multimodal layer image deep train rnn sentence retrieval sentence retrieval image generation straightforward start word word calculate next next selecting perform pick sign generate image treat measurement top sentence might probability sentence frequently appear look probability generate sentence query normalize ignore sentence annotation consist around location action annotation sentence annotation one adopt available separation work image extract five sentence annotation annotation adopt provide image annotation describe image annotation image sentence retrieval task automatic translate give sentence treat sentence translation sentence reference description correctly sentence reference sentence reference sentence stop generation sign adopt sentence retrieval retrieval measurement recall retrieve sentence task retrieval rank retrieve result well tc exactly evaluation metric plot match retrieve sentence retrieve sentence sentence description often find subtle sentence publish score htb b f base rnn generation table serve model architecture rnn conduct fair include context necessarily correlate section sentence word content might although rnn fail rnn perform well term outperform comparable publication section curve percentage retrieve sentence sentence image word third multimodal rnn outperform publicly report retrieve rank retrieved sentence htb cccc rnn state devise feature method sophisticated avg method cnn detection confidence strategy object performance even well htb cccc cccc sentence text r random avg devise avg publicly rnn base rnn well metric table show htb cccc r devise avg rnn rnn network retrieval query retrieval query rnn sophisticated explain multimodal university california com multimodal generate description content description sub sentence deep convolutional interact multimodal whole rnn validate benchmark dataset tc rnn retrieval art objective retrieval description become education retrieval blind thank rapid computer brief review treat retrieval sentence address query sentence annotation exist lack ability contain unseen propose multimodal recurrent network task novel sentence sentence retrieval formulate learning
checked still check computational pre gap determine simulation effort considerable simulation consider implement broken batch equal algorithm bm w batch batch condition require geometrically common choice application bm usual batch storage entire store clearly memory soon serious check updating prefer simplify end suggest plan provide strongly consistent bm specifically could nn bm record merge storage batch already illustrate plan interested direct technique checked examine criterion pre specify user previously batch accordance simulation effort regard perform modification implementation deviation yield measure time procedure new plan applicable storing entire iteration ess sample way define ess implement package alternative approach ess strongly ess calculation produce study relatively systematically correlate ess one termination combine equivalent ess specify set ess practitioner particularly diagnostic gd gd gd rule statistic spectral frequency markov chain gd diagnostic gd markov apply deviation bayesian application weather dataset implemented terminate comparative study several advantage consider temperature collect weather start r package illustrative record nearby weather model limit continuous temperature specification uncorrelated measurement transition temporal cs spatial specification scheme package use interested reader particularly interested ts upon simulation conduct relative standard plan set batch batch number rule coverage gd check iteration effort confirm cccc ess cpu gd table statistic criterion equivalent sampler available ess cpu effort term variance illustrate tradeoff estimate bias work relative standard high setting default plan total gd compare gd ratio ratio quality computational posterior deviation comparison ratio significantly concentrated one gd agree gd application imaging fmri study change activation bold contrast course single patient stimulus brain acquire activate region intensity imagine patient brain divide voxel lattice time bold voxel spatio structure analysis fmri interest introduce activity relate level eight four four attention subject picture four second second later stimulus star sign plus sign present four second subject picture period second sentence take second standard standardize snapshot voxel voxel bold intensity alternative conventional feasibility baseline stimulus activation amplitude transform external stimulus proceed brain stimulus preprocesse response characterize formulation turn consist gamma amplitude convolution measurement appropriate distributional temporal spatial article single summarize ti tv tp formulate notice voxel activation identification nonzero end indicator rewrite selection averaging square voxel effective inferential specify mass prior incorporate spatial mrf interaction voxel regressor two voxel article six immediate neighbor reciprocal euclidean voxel external incorporate reflect knowledge eq address place I density two wise design update baseline activation amplitude symbol selection problem figure visualize linear previously deviation memory issue add batch nominal coverage symbol jt summarize interaction small eight slice task symbol voxel activate compare gd confirm chain voxel sample stay inactive although comparison termination compare two ratio estimate deviation clear ratio gd require gd expensive adjust reach truly high automate fashion tuning parameter attain confidence practitioner routine modify relative control approximately deviation ess ess desire affect simulation result balance depend estimation interest single sampling summarize simulation requirement storing solve issue limit estimation quantile arise dimensionality multivariate adjust interval nominal require adjust one volume width multiple interval separately explore state mcmc dimensional setting date proportion type stay fix reader acknowledgement author paper grateful recognize winner competition second work dms remark pt university challenge terminate chain automated stopping terminate uncertainty moderate illustrate high simulation current bayesian spatially correlate involve weather fmri image sequential uncertainty well dimensional simulation fundamental determining terminate simulation often encounter inspection mean extremely challenging moderate practitioner resort terminate terminate stop applicable mcmc collect scientific temporal association dependency involves mcmc considerable time study modeling spatio temporal fmri develop selection brain economic public analysis assessment inferential uncertainty involve thousand carefully describe game report lee employ fixed choosing lead practitioner utilize visual truly randomly width rule implement theoretically justified terminate accurate scientific unknown markov high dimensionality associate create additional extract marginal width report terminate fall
simplify adaptive deconvolution seem treat deconvolution counting intensity bayesian method prior intensity aim show satisfactory first frequentist asymptotic behavior posterior computation obtain still theorem process adapt count multiplicative intensity abuse sequel surely every integer depend asymptotic follow illustrate poisson intensity analysis censor popular patient patient represent patient censor hazard iy finite markov path integrable transition intensity state independent intensity counting process respectively aggregate reader p aggregation intensity true assumption satisfy illustrative introduce right censoring model support concentration inequality imply process empty surely consider endowed f f f ft concentrate monotone intensity consider parameterization increase density mixture uniform prior increase intensity dependent study distribution belong families lebesgue eq cumulative go denote lebesgue measure transform neighbourhood define define q consist general posterior interest positive sequence integer x stand sup mass lebesgue satisfied assumption dealing consider process quite neighborhood sup neighborhood hellinger propose derive intensity nh nj replace nh right hand example poisson censor n n kn old mild derive finite empirical prior distribution let observe jump recall intensity function done provide parametric estimation involve satisfy advantage hence involve mass dirichlet strongly influence propose additional information correspond weakly apply three determination dt simulation true consider use appendix dataset correspond dataset intensity plot intensity second column strategy compare strategy criterion quality computational empirical fix comparable simpler hierarchical moreover thus even hyperparameter suppose influential fix turn conditionally variable accept observe ar dramatically automatically phenomenon randomly number approach correspond concentrated around credible posterior simulation intensity prior strategy lead generating phenomenon execution empty last case prior narrow prior lead phenomenon large impact empirical hierarchical cc cc fix block ccc empirical prior hierarchical empirical plain dot hierarchical line concentrate hierarchical line true estimation dot third strategy line plain line line confidence dotted throughout may line bayes dirichlet mixture ordinary measure nu mn u n l sn sn l j k accord inequality repeatedly sequel abuse l verify assumption f proof partition interval diameter complete k u n c l jx proposition f l verify verify difficulty n k jj u nu j proportional super case ordinary since rate suitable need eq f f k nu cover rd know lemma inspection reveal b piece c u u j nb np c reasoning proof u l p jx proof need indeed na exponentially bayes contraction wasserstein metric remove particular every exist path prove appeal property deconvolution mix approximate mix mixed density use os ss short super characterize regular kernel fourier vanish p k recall super case take constant super x x intensity still follow subsequently kullback leibler evaluate second q exist loss generality dirichlet uniform distribution index dirichlet mixture distribution base index base type e inside therefore decrease increase second control control poisson intensity enough n nh use combine nh assumption compact enough nh together soon homogeneous proposition constant proposition similar notation note markov inequality imply use test prove sequel empty almost use nh tt tt x remain enough n enough deal denote may change k n tt tm k ce cv k q exist n iterate set k j j b kb j tt j tm cm j use j tm n radius center lemma constant q conversely ns q choose result tool convenient function recall non pseudo c since implie non negative n tt u nd u positive enough eq u c end lemma argument assume ty n ty ty ty n ty assumption imply since soon v tt v q inequality true enough u nd lemma depend ty tt nd depend use since nd depend end lemma detail posterior process decompose jump jump artificial truncation slice sampler breaking introduce w I term rigorously dependent take account detail initialize initialize c w sample identity iw u nt k slice sampler slice follow prior stick breaking k know component component exceed I component parameter class follow p independent h class appearance translate perform directly resort simulate accept detailed consequence easily could theoretical number x value p get cm section posterior prior provide seminal estimation intensity poisson former also deconvolution latter provide posterior concentration counting prior depend posterior dirichlet process mixture process title approach principle independently task prior hyperparameter dependent fall bayes statistical sample family integrate specification say throughout drive hyperparameter sometimes explicitly case regard nonparametric case mixture variable another empirical bayes mixture procedure adaptive empirical literature method claim fully way frequentist asymptotic bayes give little know general bayes asymptotically close framework noise wavelet selection investigate asymptotic posterior condition frequentist strong merging maximum empirical take frequentist behavior contraction rate pseudo empirical concentrate converge probability probability fully far consistency seminal article identically observation iid observation elegant powerful methodology rate kullback type however take prior develop drive contraction distribution spirit apply mixture application dirichlet mixture section estimating process mixture tool posterior distribution dirichlet study minimax collection exist due deconvolution error model base drive choose deconvolution process density model finance social huge practical establish behavior intensity process extend result intensity monotone b e e b j e nu nu control iid lebesgue prior class density denote typical tail n mixture investigate prove mixture lead contraction h super key ordinary gaussian eq article treat rhs extension regularity consider instance satisfy old case follow hellinger tail rhs empirical bayes satisfy k
daily stress include size result stress distinction non data split testing test accelerate dimension cross strategy hence trait train subset evaluate significantly vs rest vs visualize understand space find mutual uncorrelated room exclude produce make interpretation complex turn feature decrease outperform chi average approximately range perfect maximal inequality maximum maximum bag common metric accuracy reduce feature pool make efficient mobile classifier algorithm support radial basis iv neural network find classifier rest forest tree predictor tree forest form accord margin h h training empirical vote vote error characteristic forest tree number tree generalization validation train number select validation pairwise among correction chance agreement chance simple percent agreement take account agreement occur conservative selection replacement fold fold cross scheme order prevent overfitte call software empirical risk solution overfitte work penalty balance distribution probable model estimation iteratively basis exploratory analysis metric dataset test metric sensitivity random forest show model performance test l ci acc specificity metric validation provide substantially heterogeneous fold resample ht min st mean rd indicator weather iii mobile phone trait weather condition combination trait mobile phone vi weather infer mobile phone report specificity accuracy run class label neutral label score include score size neutral outperform return accuracy specificity weather weather weather table weather activity alone endow pairwise combination feature set neither weather weather activity simultaneous usage classifier feature mobile accuracy period non period outperform recently report combination mobile however make video audio stress stress environment pose concern people reliable investigation predictor stress association trait contribute predict daily stress focused analysis mainly association important played also experience daily stress create stochastic daily stress recognition use report limitation user study trait recognize mobile phone regard weather wind predict regard mobile phone proximity select proximity interesting play interval seem finding relevant face tie stress sure investigation capturing confirm available association interaction conclusion message income stress increase pressure stress people severe stress may detect consider demand increase severe stress stress financial health year american low technology reliable tool recognition life trait call weather reliable stress essential scope area applicability people capable rich life transform stress target assessment treatment stress record patient stress access longitudinal identify recurrent significant environment stress become serious health employ early stress stress motivate people behaviour inform toward strategy mobile develop increase stress suggest stress management relaxation technique also student availability concern people collection fact study variability background subject homogeneity investigate people daily stress people activity individual trait individual suggest type source drop performance moreover robustness generalization together provide daily stress reliably necessity activity environment stable mobile usage pattern advantage privacy moreover stress mobile phone device apply several real world situation application device clinical application daily situation stress management semantic innovation grant stress life cause disease researcher stress require sensor continuously propose approach daily stress reliably recognize behavioral derive mobile phone weather trait concern individual person obtain class daily highly power recognition methodology behavioral sciences mobile million mobile allow access huge stream daily device behavioral location device physical communication phone call message correspondingly availability continuously grow huge stream interaction problem finance work problem stress know life cumulative stress play broad physical cognitive disease measure daily life availability early help prevent life feasibility sensor sensor heart rate setting exhibit person argue research opinion tool field behavioral science make sophisticated world work daily detect stress well activity include execute phone people associate level weather environmental argue act sensitivity finally impact factor activitie weather stress proximity interaction person turn might automatic recognition daily stress vs type activity weather people activitie proximity diversity proximity behavior weather environmental along trait internal stable people daily compare weather condition mobile phone simple family weather condition mobile phone weather mobile phone find obtain evidence reliably individual trait drop drastically factorial stress seven comprehensive approach feature weather mobile phone weather phone feature weather mobile phone body stress detection focus measurement infer stress see heart variability provide reliable stress level comprise sensor carry continuous monitoring situation speech production study acoustic research analysis speech variation example stress detection life environment sound quality public place reliable stress study analysis despite provide monitoring employ survey daily stress stress participant seven item neutral subject consecutive daily stress fig high stress stress score neutral variance high rl decrease accuracy decrease index weather weather weather weather call social trait stress negative examine name trait researcher daily daily daily people event tend trait student show less affected stress five trait measure subject answer version question mean scale trait raw trait relationship health weather weather show weather et al six weather wind power air pressure affect negative affect reveal affect effect air pressure weather temperature pressure iii vi wind metric source weather alpha weather daily source weather previous characterize mobile phone predict people behavior trait table proximity feature max deviation chebyshev basic day backward move window possibility past event current subsection describe detail feature fall phone usage behavior
start vote ensemble determine simple majority agree upon majority choose say result chart figure effectiveness frequently retrieval priori extract combine consist scientific different science word tend run slowly raw nmf singular svd principal component analysis normalization two decomposition quite reduce create dataset mean proportion document h input clustering ranges misclassifie misclassifie reasonable ask solution low curse distance metric dimensional clustering surprisingly internal like coefficient one must careful high clustering metric clustering agree without dimension consensus matrix simplicity without consensus cluster well suit representation provide consensus iteration common choose consensus single consensus much great average table typical across algorithm consensus consensus consensus cluster methodology consensus similarity chain consensus sum accord section row consensus mean permutation diagonal block diagonal block diagonal degree nearly dependent deviation partition represent coupling without merely perturbation dominant stochastic block dominant continuous block cluster examine consecutive gap indicate nearly structure consensus cluster suggest might consensus clustering matrix count number sometimes helpful consensus traditional plot fail picture may discuss look similarity collection display eigenvalue information look consensus consensus build ensemble pair prefer dimension dimension create method initialize initialize dimension reduction cluster cluster determine clustering although original available ng collection result consensus form mean dimension reduction convention observe plot singular ng datum indicate reduce preferred reduction principal mean proceed fast clustering step associated initial figure might cluster distinguish iterate consensus use consensus input figure either scenario cluster eigenvalue clear repeat change present document reveal single h superiority certain consensus compare accuracy algorithm dramatically interesting contain consensus matrix nature poor author come ensemble mean c c consensus nmf cosine consensus consensus consensus average cluster iterate consensus number agreement cluster column table run round voting accuracy table consensus nmf analysis collection tool analyst tool chance solution pca particular internal validation metric mean solution difficult compare tool consensus framework work difference accuracy achieve algorithm herein present combine multiple input exploratory cluster discover matrix build value often appropriate determining cluster algorithm agree example succeed cluster dataset refined iteration picture might agree consensus practice multiple together explore varying purpose approximate improve cluster consider consensus discuss individual consensus aim solution another emphasize agreement ensemble solution cluster equally relationship draw whether favor iteratively encourage agree upon similarity reflect agreement iteration cluster agree favor agreement definition framework consensus present similarity high datum refine algorithm agree nearly determine random determine succeed number cluster method consensus provide average hundred text mining biological datum without hundred thousand guarantee analysis aid tend separation cluster tool make informed tool dimensional hoc science hope quick develop individual task become many mining cluster stem vast number information fact question analyst answer determine consider separate answer group point agree cluster essence suggest herein problem determine determine final solution algorithm consensus form voting ensemble proceeding majority user year consensus researcher challenge ensemble generally wide variety result produce ensemble consensus minimize distance metric clustering define clustering clustering metric information maximize median partition heuristic propose believe bind importance clustering inaccurate share cluster away optimization solution act move consensus introduce impose reach accept object introduce notation since algorithm ensemble n jk data individual simple clustering ensemble record prefer think ensemble consensus cluster ensemble consensus clustering value reasonable colored circle clustering result consensus isolated expect truly give cluster break total clear far break break way block diagonal structure offer traditional cosine traditional curse space mean address entry consensus compare cosine cosine similarity range document among benefit consensus depth adjacency output available something know object cluster previously
quantify se rs ensure comparison tie differently function loss tie outcome summary difference rs instance rs capture variability value average value rs provide comparison rs comparison final extract interval gain generalise additive interval specification function default thin regression relate expect fit seem roughly level dot rs outperform primary quantify experiment al early thus single experiment overall address complexity assessment provide find e generalise pointwise confidence include fit output al analysis generalise reasonably q dispersion h name negative binomial thing method mis match conjecture work classifier mis match quality pool high mis classifier mis metric sub mis match informally mi ill suited match improve mis match great quality mis match bad fourth task complex decision match bad fourth conjecture al budget word increase label gain select pool range would practical early al belief mean length notable significant confirm exist analyse detailed disagreement se algorithmic compare result disagreement disagreement vote l classifier logistic machine setup way binomial fit reasonably se confirm al mixed confirm hard may behaviour se stage classifier se result significantly well svm good classifier help examine variety range factor conclusion overall fail often consistent largely confirm belief literature se show e enable recommendation relate determine al study complexity assessment methodology assess award learning improve two central al broad quantify need al detailed complexity assess performance experimental learning field need motivated concern label consider question help question answer enable researcher tackle difficult al benchmark method variability difficult example notably show negative valuable overview al suggest still thing understand broad include classification simulation systematically careful raise contribute assessment issue application assessed structure present classification feature denote label consist classifier unseen objective label image diagnosis valuable label image obtain systematic guide al set typical would pool budget request al receive oracle se informally se metric decision al label provide newly label initially train improve classifier variation effect study factor analyse nature classification difficulty boundary express decision smoothness since affect pool intuitively task hard pool decrease label determine performance open experimental evaluate combination performance answer question attention create mixture gaussian curve involve way varied b modify cluster smoothness varied transform another input label high require classify quadratic forest rf vector machine label pool figure trajectory al set label happen repeatedly create trajectory pool overall trajectory denote point increase amount label several trajectory score interval score process budget score outperform rs whole budget begin benchmark score terminate score detailed section al instance benchmark rs rs substantial trajectory figure score se comparison shannon vs difficulty contribute novel assessment complexity quantify al gain scalar summary address preliminary issue benchmark al outperform decide benchmark option large improve benchmark al rs variability evaluate rs instance interval thus benchmark trajectory context understand al benchmark budget budget iterate entire pool amount label grow minimum
proximal prox inf eq prox operator would restrict function solution indeed proximity restriction g require satisfy summary toolbox ball lambda parameter reason ball compatibility solve image goal would closest firstly know position miss pixel result assumption secondly image patch color variance simulate operation want recover mask relaxed rewrite measurement patch q mask play fidelity signal tradeoff fidelity measurement vice play note trivial affect convergence proximity toolbox object compute non provide operator prox structure function role operator tv norm implement tune setting select log summary display step indicator proximity infinite previous prox x eps projection suppose indicator implementation note lead evolution since select different solver observe present lead compute solver backward gradient consequence matlab gamma select define iteration define display reconstruct done follow prox prox lambda lambda lambda gradient well define domain allow solver forward backward solver solver converge might problem parameter regularization ball choose allow compute I original sigma I noisy I sigma prox tv original prox x eps lambda f prox prox lambda lambda gamma lambda backward gamma lambda ii close toolbox proximal splitting use specific general try stay mathematical precisely matlab toolbox design problem split compose solver operator file user implement toolbox convex lower assume nx problem smooth method generally forward algorithm fall term generalization operator unique well proximity useful tool directly try find sequence proximity survey excellent splitting relate toolbox core toolbox proximity quick common toolbox largely inspire toolbox develop file backward continuously lipschitz without use default assumption constant exist q stepsize also allow smoothness step proximal backward algorithm f start two matlab structure function take latter matlab vector matlab act stepsize control convergence default classical augment lagrangian technique
sparse noisy real success utility quality source rest overview detail framework year formalism transform machine area node edge characterize transformation heterogeneous node type detection within fall interpretation weighting algorithm set type multi work leverage share information across source overlap iteratively select edge approach aggregation ground know subset require ground validate section list intuitive need similarity computable rough overview collection learner whose performance well seminal form learner think weak present boost make arbitrarily good allow pure learner equally good graph graph representation quality application standard bandit receive explore minimize many suggest take adversary know everything make advance expert similarity learn potentially bad artificial reward care end seek care final knowledge immediately largely heuristic even bandit optimize guarantee useful nevertheless successful bandit hope graph technique bandit list maintain element distribution reward receive multiply control representation edge update next section sketch implement unweighted undirected expert vertex combine produce global give round four part produce aggregate measure quality quality edge round end round maintain non edge weight graph run cluster produce observe effectiveness application update weight define two use present use first call intuitive notion superior across tie idea consistent simply evaluate algorithm algorithmic agnostic measure cluster idea many neighbor quality cardinality neighborhood neighborhood vertex conventional mechanism neighborhood metric due stronger adjacent brevity index experimental use consistent utility edge plug guide key every inefficient give improve fix weight extreme pick total runtime time balance parallelization addition alternative edge roughly round relational stay inside boundary consider suggest reject suggest discuss respect edge union initialize copy p ip u vi mu h iw pp ip algorithm synthetic structure list vertex block occur block simulate global stochastic block draw block model within cluster block represents represent refer representation graph community well naturally extend formulation community er enyi example note er instance noise combination er primary comprehensive database research extract subset publish science topic field author graph title least three word common exclude paper summary dataset experimental couple community structure evaluate notion reflect quality source quality assignment capture quality representation representation community structure disjoint clique clique modularity capture modularity compare nan produce also remove cross perfectly modular trivial display graph fraction total quality capture weighting source community weight contribute equally weight quality metric graph converge round far exceed modularity tell able discard graph round type weighting requirement total edge enyi appropriately preserve amenable select title recover community share topic sense title well proxy capture division modularity induce snr appropriately usefulness outperform overlap ground clustering produce modular enyi converge weight r ec modularity sparsity weight modularity na na na graph aggregation graph respect demonstrate application community detection
signature large additional implied exist place sum number bound idea contribution weight bound hand formally ta line cauchy schwarz g normalization assumption signature assumption size definition let edge k apply set definition signature except negative completeness version great set negative correlate either choice signature expand similar definition expand signature signature expand expand signature set straightforwardly general signature note expand lemma adapt bias expand large sample analog hold formal statement step purely magnitude edge help contribute two sign argument use overlap magnitude recover signature almost iff deal magnitude recover case state formally expand output use refinement conclude number furth closeness order want analog large constant equal absolute nonzero equivalent I triangle omit prove bernstein standard deviation iff probability tx j tx would variation otherwise sum variance probability te j te te weight know focus jx j signature signature tc secondly property signature expand set expand signature analog claim w similarly proof use claim bernstein know expand signature remove element contradict indeed expand signature fs thm thm thm claim exercise pt coding sample unknown overcomplete heuristic provable hard find provable dictionary et al algorithm incoherent et handle guarantee design provable algorithm work matrix individually notion motivate limited enumeration try combination unknown overcomplete dictionary fit used machine feature recently code influence processing dictionary super resolution provable program nonconvex unknown know general hard combination decode regression compress incoherent matrix even sparsity satisfy isometry matrix recover hard heuristic algorithm widely design direction mod reference however provably recently al give practice overcomplete dictionary prefer provable overcomplete dictionary polynomial incoherent weaker incoherent fundamental limitation intersect dictionary regime seem deep regime dictionary matrix weight bipartite allow albeit slight real life dictionary probably raise dictionary current time enumeration similarly learn discuss dictionary nonnegative partly analogy algorithm traditional discuss exposition purpose refer coordinate though apply vision definition effect add could object one specific version restriction dictionary make nontrivial formalize property life instance reasonable life dictionary feature involve imply dense match speak observer know produce assumption one presence pixel affect interest et al incoherent pairwise product sense count fairly secondly one incoherent one rip matrix dictionary check weight think constant large practical purpose unclear ng neighborhood intersection pairwise neighborhood matrix dictionary random entry close comment check need claim return dictionary equivalent true large constant assumption optimal sense significantly violate intersect feature characterization nice long valid expectation equal normalization magnitude weight lot also variance intersection neighborhood two feature entry large contain entry constant simplify notation g say nonnegative pixel small mn recall dictionary seem try extract assignment assume nice property recover become easy recover via overlap incoherent intersect fail slightly intersection among tend mean value simultaneous unknown therefore look subset pixel aggregate pixel consistently predict value signature identify signature good signature size similar correlation signature separate signature set correlate size idea try expand column pick column result expand expand
copulas student direction initial log increase new evaluate correlation algorithm terminate yield reasonably good choice present maximum likelihood elliptical copula task fast consume advantage suitable copula elliptical copula copula tool dependence allow model flexible describe marginal family copula strength use decade mathematic distribution unique theorem encode dependence structure vector margin distribution u copula refer example elliptical copula copula multivariate elliptical elliptical elliptical copula dispersion depend location copula elliptical especially copulas normal student gaussian student degree freedom univariate student copula stress copulas correlation general margin elliptical copula correlation description reader close elliptical square practically problem elliptical copula student conjunction efficient student copula focus elliptical copula space symmetric estimator correlation elliptical copula copula exist method student copula appendix elliptical copula efficiently student essence less project kronecker delta maximization student true maximizer maximizer ascent would seem move hereafter fact directional directional trace diagonal positive define symmetric iteration project log method reference small although perform numerical also outperform simple size scheme describe estimation algorithm exact widely software package calculate solution inverse hereafter exact estimation package copula slightly execution estimation fix gradient execution time case dimension converge second always second matlab hour every copula generate test description generate sample copula estimate parameter original value obtain gradient positive h dimension fail converge provide plot deviation generate eigenvalue imply method especially correlation extend copula estimation procedure extensively fast technique latter moderate numerical student
use iteration arbitrary substitution variation would gradient whereas variation variation stochastic bfgs introduce subsequent instantaneous implie show variable iterate optimal argument proving result instantaneous twice instantaneous hessian exist q eigenvalue hold follow linearity observation assumption typical inner proposition gradient variation recall small eigenvalue instantaneous constant eigenvalue hessian instantaneous hessian fundamental instantaneous definition variation eq e inner product bind write curvature compute solution proceed recursively matrix positive upper descent general f whose average specify assumption eigenvalue state taylor upper bind hessian substitution side observe third term side norm eigenvalue upper yield upper use lead low second positive semidefinite eigenvalue small far note aside term sake define relationship imply sequence almost surely per infimum norm surely observation probability relatively minor nuisance care assumption true infimum distance optimality realization statement stochastic substituting yield since nonnegative satisfy converge almost vanish infimum sequence nan square low eigenvalue apply expansion optimal simplify cauchy schwarz substitution simplification infimum nan consider establishes convergence low variation gradient section instrumental proper effect curvature curvature estimate eigenvalue term dominate ensure progress towards argument complement characterization introduce define sequence step give parameter sufficiently parameter objective objective satisfie expect value imply convergence sense sgd convergence improvement mark experiment sgd problem small particular definite matrix instantaneous minima surely minimum condition control variability instantaneous large note optimum comparison iterate study represent optimality stochastic process number require certain text instance discrete run gradient realization distance much iterate iteration since correspond evaluation conversely upon function modify j realization sgd order text realization convergence time text advantage fig keep yield family well function ill condition condition sgd respectively cf function fig ill condition condition family average average family sgd spread small go moderate start moderate value decrease text stochastic compute section problem instance cf interpret distribution trend apparent decrease ii decrease go start go moderate deviation monotonically decrease stay increase thereby yield payoff pay evaluation good half turn actual time moderate suffice provide curvature interval moderate number comparative dimension method define value determine interpret failure sgd respectively fail rare realization realization spread eventually well median well fail exceed fail furth smoothly evaluation need stable implementation hyperplane separate set contain feature vector find hyperplane support separate point separate vector deal introduction measure hyperplane support constant minimization sum hyperplane measure norm desirable common square g give uniform upon rewrite substitute general explicitly attempt select instantaneous take loss training half belong class likewise uniformly random overlap range know less parameter progress gradient large sgd stepsize use performance process sample processing feature objective sgd process vector conversely method dimension process fig acceptable reduce progress difference time translate show build vector record percentage correctly classify repeat histogram sgd uniform test exceed classification far performance comparison suggest investigate difference regularize version bfgs vector bfgs regularization induce oppose proximity requirement stepsize value vector process value reach regularize stochastic bfgs jump permit consistently occurrence always recover regularization amount curvature occurrence stochastic behave sgd stochastic objective introduce corresponding argument sure behave expectation far prove show particular significance dimensionality exhibit target develop advantage improvement cardinality spread true lagrangian duality lagrangian eq combine lagrangian optimal minimizer must eq multiply rearrange inverse log multiply rearrange consider trace cyclic argument latter scalar observe log determinant opposite determinant inverse substitute determinant rearrange term expression order dual eq gradient plug lagrangian compute hand verify direct must argue directly term observe hypothesis must write conclude belong semidefinite sequence rate constant satisfy sequence bound time prove rearrange compare inequality true substitute substitution simplify equivalent conclude substitution conclusion combine lemma show stepsize satisfy optimality gap side gap recall standard completeness g assumption eigenvalue hessian take taylor expansion bind side argument zero minimize imply norm substitute side double f conclude form hypothesis rewrite identify substitute remark mm mm edu appear bfgs newton method problem objective require argument prohibitive dimensional order objective hessian incur computational descent utilize deterministic gradient determination direction gradient rate advantage counterpart upper argument bfgs develop use function value determine optimization determination instantaneous function average problem common resource wireless convex conventional determination f intractable unbiased sgd limit dimension easy newton achieve rely gradient estimate unbiased generalization devise objective quasi retain advantage deterministic counterpart quasi newton near see bfgs quasi method structure regularization avoid function brief sgd bfgs continuously curvature previous curvature bfgs retain modify stay bfgs bfgs make large completely specify resolve bfgs require close possible constraint matrix
different prox sdca starting find separable time indeed enough say influence tight describe use arbitrary update paper sampling unconstraine minimization strongly assumption special method besides close coordinate analyze primal dual step sdca sdca analyze establish gap exception novel primal specific obtain variant pick similar prox always pick complexity n match term choose nice mini variant sdca specialized svms loss besides accelerate mini batch method zhang detailed comparison well dual however propose coordinate dual sampling coordinate accelerate efficiently implement distribute environment variable describe non serial first analyze datum partition speedup serial sampling paper however reader uniform find variant exist mini batch stochastic dual certain mini batch drive use inform speedup datum illustrate consider datum appear factor excellent predictor behavior special uniform sag gd gd serial primal mini ms gd analyze specialized dual coordinate serial generality far arbitrary uniform accelerate arbitrary purely accelerate variant simultaneously primal algorithm interpretation outline update sdca compute several select proceed result deal specialize nice relate stochastic speedup main loss describe random value chance proper assumption positive always stepsize shall formalize notion smoothness strong constant brevity subgradient dual maintain maintain proper vector n p iw let describe convex subsequently way numerous variant allow option actual update entire process repeat interpretation fix iw decompose w pair optimality current primal prox sdca adjust choose option example special serial sampling dual prox prox prox sdca primal prox perform convenient notation nh nh iw nh elsewhere positive shall formalize compact establish average hadamard equal merely diagonal matrix equal especially set computing eigenvalue impossible albeit perhaps suboptimal identify identification read special ji inequality x h function dual possibly knowledge method strongly besides study stochastic single pointing serial oppose parallel update dual serial uniquely characterized turn serial give large small loading select cardinality terminology nice suited indeed processor available assign dedicated processor processor compute access assign major depend influence processing choose processor available processor lemma nice nonzero block theorem extension case straightforward matrix form rank formation time format loading phase problem work compute easy sampling direction reader non serial example group share two associate random stepsize parameter serial sampling separability assumption assumption satisfy ji separability form partition fix form index belong ji ji later variant analyze huge tb computer simplicity block partition set assign dual partition iteration parallel pick locally node compute update dual lc consideration sample distribute primal distribute number active equivalent case improve constant instead lemma partitioning instance without positive scalar assumption sequence dual q analyze coordinate descent specialize nice specialized serial sampling iteration dual random cover general rate dominant two term simple serial sampling seek quantity give improvement uniform probability dual variable use serial uniform dominant separability let fix lemma sampling cardinality corollary serial composite deal albeit separability serial specialized sampling specialized serial uniform sampling vary degree partition balance speedup perfectly balanced uniformity si perfectly perfect speedup factor nice nice combine table second line fully dense speedup obtain get fully ht fully fully nice assume study speedup nice nice speedup factor depend level problem quantity provide low speedup last involve speedup modulus achieve course regardless matrix frequently regularizer mini inequality give phenomenon plot increase speedup data speedup sparsity beyond ht exist mini mini analyze stepsize special author mini sdca specialized square complexity descent vary mini size extension mini equal consider table assume table complexity sdca regime simplify complexity sdca ht sdca n nn n showing x third follow monotonicity claim fact regime match linear sdca roughly big outperform accelerate condition sufficiently order already distribute involve depend partition variable lemma say partition negligible fact vanish pick average similar nice interpret analogous note first perfect mini condition receive nearly mini batch sdca name propose variant analyze much bad ignore expression dominant term strict lower bind clear gap sparse perfectly well measuring amount specialized specialized serial speedup mini nice sample particularly suitable unless big implementation frequently increase matrix way necessary understand machine ignore cost nice sampling contour speedup axis contour nearly straight mean speedup factor approximately well plot hence contour average sparsity analysis first strongly g write separable satisfy lemma bind term bound side eq therefore write q option case g ia inequality list problem formulation application include regression multiclass focus square hinge loss main message dataset serial sdca practice theoretical speedup term perform several sparsity dataset table option c size sparsity ph support svm regularizer example smooth strongly option specify linear support svm hinge define convex primal update hinge section uniform serial prox sdca sampling sdca three w describe set number
three signal directly generic open claim retrieval core emphasize potential question symbolic computation approximate algebra retrieval lead guarantee like question use principle article contain result independently question section describe phase algebraic retrieval usual variant retrieval major algebraic algebraic complex ground close include make algebraic treat separately algebraic obstacle regard field restrict derive bound accurate version nz usual mathematical generality amenable algebraic know take observer backward original reconstruct rank hermitian mathematically observer equivalent apparent assume hermitian knowing say algebraic closed include algebraic reason let assume symmetric rank thing image complex want symmetric hermitian algebraic problem priori image restrict variant fundamentally much easier amenable algebraic rise question explain treat algebraic namely write write reconstruct assume elementary computation equivalent original phase retrieval know rule field analogue treat allow range though specific final retrieval retrieval set map notational clarity analogy matrix determine possible signal less whereas set symmetric almost forward mapping projection pair phase notation follow n nz write signal uniquely measurement factor ambiguity avoid identifiability signal cf yield set namely back real instead consider order algebraic short technical technical map algebraic book logic know definition irreducible element map perturbation sense identifiability perturbation paragraph identifiability algebraic concept valid model model tuple kk measurement measurement formal forward fulfil section namely irreducible prove statement formal slightly technical identifiable statement identifiability remove remain identifiable borel open identifiable identifiable closed hausdorff condition open imply condition slightly technical fulfil case crucial translate signal remain whole signal important note iii priori introduce terminology brevity identifiable proposition principle exclude middle state identifiable perturbation identifiable hausdorff continuous positive hausdorff perturbation identifiable signal perturbation direct consequence zero identifiable different perturbation identifiable situation signal perturbation identifiable signal signal identifiable axis corollary signal perturbation signal regard property make statement measurement three generic identifiable identifiable three case mutually exclusive exhaustive generic perturbation perturbation differ set identifiable differ signal usual signal signal identifiable perturbation identifiable simple non matrix origin identifiable restrict origin moreover exactly open identifiable signal example neither regard identifiability notation call measurement tuple signal identifiable identifiable perturbation keep mind terminology identify matter whether show completely identify identifiability irreducible process model irreducible fulfilled case irreducible analogue characterization measurement identifiable identifiable open identifiable proper closed hausdorff subset open property open variety generalize property perturbation occur two condition measurement regime identify borel neighborhood identify identify borel map x describe prove keep complement condition call regime perturbation identify identify perturbation remain proposition completely identify context omit mind terminology regime identifying completely identify definition identify mutually exclusive exclusive identifying reformulate identify regime completely regime identify regime measurement measurement completely ia ib proposition allow analogue case identify generic regime identify three mutually exclusive exhaustive measurement regime identify generic identify analogy define terminology case keep call measurement identify generic differently happen theorem identifiability I n na nz yield contradiction identify summarize infer identifiability variety signal closure statement virtue projection identifiability signal complex ir restrict imply imply tool statement let nb nb generic measurement generic imply identifiability identifiability recognition one technical due involved complex xx yy yx xy non one imply therefore summarize infer identifiability yx identifiability xx yy yx xy establish statement closure signal xx yy yx xy imply combine unitary connection deduce original complex let nz nb unitary equivalent generic theorem identifiability n projection extend irreducible succeed complex retrieval bind threshold generic dedicated verify recovery generic signal uniquely determine orthonormal basis vector must recover may assume measurement determine generic derive j loss generality choose allow measurement generic global one pure measurement signal measurement linearly apply measurement identify signal highly crucial measurement phase algebraic regression mean principle tool algebra formulae identifiability retrieval algebraic similarity consist assume identify formal variable give polynomial ix x ib ideal fitting performing symbolic object work slightly formal projection rise polynomial estimation x reconstruct retrieval fundamentally real complex whereas part relate consider certain retrieval focus explicit inversion formulae compute linear system equation write invert singular explicit answer numerically stable main idea ideal ideal contain part use polynomial generator orthogonal scalar set z I z refer experiment detail notational would reader instead stress numerically explicit inversion formula recognition experiment magnitude measurement inversion formula ideal outline comparison alternative classical retrieval alternative fouri set deal semidefinite ideal cause limit number shall performance ideal measurement uniformly measurement also standard haar set outcome performance comparison depend exact range poorly inexact noise number threshold hence measurement ideal perform whereas accurate acknowledgement science algebraic algebraic complement topology variety write union algebraic algebraic variable let algebraic borel open irreducible call point contain irreducible proper subset restrict algebraic state algebraic algebraic field irreducible converse proper algebraic algebraic continuous union irreducible prove algebraic irreducible follow condition set iii iv apply irreducible closed proper hausdorff imply zero variety open complement irreducible equivalent borel open open neighborhood imply irreducible elementary complex appear lemma properly literature intersection complete zero nf h variety converse irreducible intersection intersection analogous complete ii closure preserve therefore closure variety irreducible variety part algebraic property contain put obtain check start map map relate projection matrix generic resp hermitian generic open dense onto dense notation irreducible observable complex similarly statement follow ii notation assume n image choice suffice proof main case generic identifiable identifiable three signal perturbation identifiable identifiable triple formulation
decay determined circuit iteration scale implication choice minimize setting clearly statistically keep primarily reduction need execute small occur acceleration virtue computation stochastic hardware technique neural back network demand necessity hardware technique enable architecture big execution mini dense feed propagation weight mini inherently sequential achieve level parallelism dense result mini batch well hardware perform fast classification deep digit mnist comprise image pixel digit scale hardware precise proportional bit backpropagation generative pre architecture convolutional conjunction hardware conjecture deep initialize layer perform feature stack autoencoder train fine batch qualitatively initialize randomly pre error drop notice achieve error control clear trend preference bit neural classification range extend include dataset neural perhaps acceleration machine purpose deterministic hardware circuit hardware descent run stochastic hardware noise procedure variation somewhat bit demonstrate deep accelerate propagation hardware framework inexact error stack level hardware com highlight design boundary practitioner stay hardware hardware software co methodology intensive trade digital circuit machine theoretical impact simulator recognition extract sensor dominant algorithm enable apart dynamic binding well presence unnecessary consequently technique randomize linear become software stack interface yet application continue purpose traditional unnecessary degradation overall system robustness noise approximate introduce reasonable expect corresponding energy per computation fast often less hardware hardware degradation term metric software prove conventional processor success dominate time execution kernel dedicate computation interact processor hardware truly benefit entail careful closely couple include optimize interface trivial design consumption cost also equally model hardware readily software cost typically substantial feasibility target common circuit usefulness hardware adequate development address observation computation datum machine use digital stochastic circuit approximate abstraction introduce abstraction analyze computation gradient execution handwritten digit problem train error rate circuit day period decade method processing implement multiplication main application perspective multiplication computationally implementation multiplier resource area hardware bit multipli stochastic circuit hardware arithmetic parallelism computational computation bit variable stochastic generate compare draw encode estimate occurrence arithmetic operation multiplication logic deterministic computation scale appropriately bit perform logical implement express bernoulli view large central multiplication use mean proportional bit variance number multiply describe matrix nd counting bit vector bite possible vector processing parallel cycle number uniform fed gate generate bit produce counter refine inner computation normalization multiply tune suitably adjust improvement interestingly stochastic circuit compute low circuit analog circuit circuit compatibility logic rapid verification low circuit circuit extremely bit sequence include generators parallelism address limitation circuit diverse come device propose device bit sequence architecture speed overall propose discrepancy quasi generate speed circuit effort circuit need bit hardware feed component build give research question regard discussion learn investigation compatibility stochastic valuable insight hardware design formulate well typically
objective design capability correct model quickly also world repository california position ct slice human measure position none case integration keep another would involve process gps differentiable might partly ct slice database learn repository learn learner goal slightly property parameter little goal clearly uncertainty suboptimal gain expect posterior latent notably bayesian design unlike extensive evaluation active well candidate selection try find apply like validation batch wide range scenario choose query success active successful reduce datum acquire active selection choose datum paper learn selection discriminate class generally early uncertainty primary relevant provide evidence performance work minimize discuss minima exhibit belief wrong significant leibler model capture evaluation relate refer generate remainder active discuss detail world random finally use successfully use field range show active however mainly minimization class aim select model set selection mainly criteria bic need approximation parametric validation statistically subset class among selection sample choose must try try version competing find sample within concern prediction disagreement multi scenario hard define kullback member focus prediction approach optimally shannon information expect expect measure gain traditional may design recently bayesian disagreement exploit equivalence order tractable mutual adjustment measure variant algorithm arguably classified finding amount information great change process introduction pt pt circle plain shape dash circle dot shape dash dash edge random variable indicate hyperparameter observe far input model arcs dependence model depend eliminate dependency iterative candidate generate next add approach predictive class predictive introduce criterion criterion log point expectation augment gps entropy point maximize predictive entropy readily shannon expect entropy way subtract shannon maximize entropy eq kl q insight mutual transform space surely empirically local sample minimize suboptimal confirm current model augment quantifie capture relative knowledge decrease pm pm model expect quantitative experiment design optimize whole iterative augment explicitly hypothesis exponential hyperparameter correct compete hypothesis correct wrong curve objective choose objective kl kl divergence eqs low divergence objective first wrong hypothesis happen ground noisy actually occur wrong already compute already support increase decrease actually correct maximize hand direction lower recover fast objective measure cross hold compete hypothesis lead minimal prediction discriminate help additionally uncertainty different
fix proportion scenario analysis subsection decrease go recover theorem conclude whiten uniformly spread close pure one one exhibit heavily intuition behind design good efficiently practice original noiseless noisy approximately pixel could plus arguably sdp solution see let easily posteriori sdp method implicitly add violate observe extract intuitively extract well assess spread alg alg noiseless ht robust kind surprising make robustness let base identify extract prove generality column affect singular g ai factor base allow robustness process post process orthogonal moreover process outperform recursively solution significant would w algorithms alg hull matlab code test matlab cpu gb subsection effectiveness take wrong show happen fraction properly ht cc right explain perfectly perform potentially entry middle toward hull noise detail report ht ht c sdp improve noise correctly identify slightly oppose fast example perform sdp hull hence fact simulate noisy hyperspectral table report remove angle clean spectral signature figure display abundance correspond ht cccc variant material properly oppose slightly pre condition analyze make pure robust whitening yet effective analysis aim sdp provably error pure hyperspectral preliminary hyperspectral plausible large large derive theorem image make pre process pixel algorithm sensitive outlier enhance hyperspectral practically influence pure pixel result paper successive applie need increase decrease infeasible feasible domain compact objective attain product generate contradiction optimality relaxation solution relaxed optimality relax multiplier sum equality give combine satisfie mm theorem design enhance pure search blind hyperspectral analysis focus successive pure recently robust resolution ellipsoid sdp high consume generalize sdp multiplicative fast contribution allow robustness pre whiten interpret solution robustness whiten perform optimal relaxation sdp extremely compete sdp several set hyperspectral whiten algorithm nmf hyperspectral blind aim signature material call signature pixel combination signature let hyperspectral pixel noiseless case signature signature abundance pixel abundance sum constraint factorization nonnegative blind play exist pixel pure permutation combination blind pure assumption therein difficult provably successive describe note nmf pure assumption refer separability separability presence near nmf image noiseless permutation illumination pixel successive robust pure alg ht project onto extremely implement operation closely generation successive simplex see provably q www satisfy pure algorithm sensitive conditioning beneficial robustness column apply result corollary simply noise view condition corollary follow column improvement especially denominator hyperspectral signature similar essentially course otherwise solve approximately orthogonal conditioning volume origin column ellipsoid via ellipsoid noiseless optimal recover transformation precisely prove satisfie full rank w w projection svd dimensionality see see truncate svd cholesky truncate operation sdp however use constraint sdp namely remove volume ellipsoid base sense build approximation whose error provably expensive sdp consider computationally alternative focus contribution analyze robustness prove satisfie let away give eq w solution third combine yield obtain fact near allow nmf rank satisfie apply identify column q theorem need high solution increase fast approximately research understand whitening noise whiten blind analyze pre whitening case pixel generative whiten r rr rr frobenius assume noiseless gaussian whitening keep recall orthonormal alg ht truncate svd alg alg simplicity equivalently assume already whiten square q observe objective constraint active multiply optimal satisfy optimality multiplier together svd obtain robustness pre whiten volume ellipsoid moreover factor solution feasible eq q identify w hyperspectral subsection whiten pre whiten robustness analysis precisely tight bound denote whitening condition yy sufficient subsection use q h h singular value see robustness whitening satisfie whiten plug tight fact pixel th column q whiten pixel match upper relatively spread whitening perform wherein whiten generative
inverse parameter write set regression express b inclusion regressor prohibitive entail resort employ fully factorize distribution approximate inclusion regressor form encourage dpp elegant convention specification distribution factorize guarantee dpp investigate effective versus variational posterior random vector ensemble e also indicator dpp field propose variational variational entropy dpp entail form show effectively learn approximate namely tu base dpp summarize u unnormalized linear variational posterior advantage estimate adjust sure initial sample dpp set inclusion element cardinality tr cardinality require close solution logistic computation compute linear process encode long approximated supplementary material benefit algorithm automate include extend learn relevant diverse easily credible predicting bernoulli dpp posterior adapt change stay algorithm conjugate warm greatly experience cg converge quickly dpp per iteration computing likelihood regression matrix element converge experiment cover section supplementary section diversity dpp map strategy failed map six baseline orthogonal pursuit generalize glm glm elastic net forward spike dpp factorize mean standard convex elastic orthogonality induce dpp approximate posterior parameter match select average stage different diverse also diverse e although auc method diversity basically show gene assess pathway involve breast cancer use five community within preferred gene cycle dna breast module cancer activate cell role cell tumor understand breast survival may investigate anti circle grid point suffer g california pairwise distance mse measurement method diversity relate well outperform close dpp method seem well grid particularly measurement construct non construct gp determine implicitly spatial site space grid observe also ensure broad grid cover domain center basis vector different scale spatially spread overlap diversity measure month locate united sensor method report mse report select perform outperform method good balance reconstruction area measurement trait interesting prior similarly diverse propose elegant encourage select item encode omp information model similarity experiment section dpp active computationally variational far variational rely dpp svd number hence parametrization condition number matrix conclusion quantification modal omp demonstrate dpp robust omp supplement dpp factorize posterior modify adjust solve ii ii draw current approximation py tp w diverse enable diversity covariance pursuit information employ spike variational dpp generalize extend field learn efficiently fast property motivate come bioinformatics tumor gene gene explore application spatial statistic diverse feature classic enable create compact interpretable feature promise interpretability prevent regression focus selection diversity set compact easy play fundamental application cancer increasingly recognize present heterogeneous mechanism task tumor select interaction exist explicitly diversity subset pre balance model complexity combination sparsity fidelity typically encourage diversity show unstable issue successively bad orthogonal pursuit omp proceed way wise orthogonal previously orthogonality implicitly maximize diversity establish omp flexibility define diversity product view assign approximation impose particular measure feature regression classification variational dpp appeal encourage define alternatively offer appeal map sampling interval unlike approximate dependency dpp approach conditional al learn variational require fortunately operation efficient regression marginal framework close work al suggest variation call uniformly propose determinant dpp prior posteriori select although prior choice make contribution propose use dpp decomposition develop family available include fully bring advantage I relevant view ii feature sampling iii define rather iv compute marginal inclusion condition feature set computational review diverse identify diverse gene tumor diversity respect gene finally optimal point process
truth structure researcher utilize hierarchical learn occurrence hierarchical structure organize multi introduce learn space finding multi label refer label training graph structure node acyclic dag graph two node correspond parent label large scale label space make propose tackle occurrence arise dense way maximize word rely fix length context interesting word representation reasoning introduce adapt log scale efficiently state make hierarchical multi maintain human source reduce space occurrence formally basic predict co occur label maximize introduce I correspond vector output label respectively label share occur softmax activation connect activation label neural network vector whose element hide label together label parameterize well update gradient code give variable length hierarchy short code output decision tree label label bit unlike tree denote take otherwise node label computation substantial carry experiment index child control library database child real reasonable cycle wrong annotation different abstract level introduce difficulty visit traversal never end unless stop follow pick depth detect pointing probability rarely see co objective limited right occurrence learn structure co occurrence inter relationship locate opposite group part health term leaf manually representation capture label occur close see figure illustrate together likewise effective treatment co occurrence show strong relation pointing leave learn representation representation occurrence hierarchy well occurrence representation example analogy qualitative upon reasoning hierarchy specifically regard label representation train hierarchy advantageous train hierarchy kind representation probable analogy hierarchy poorly type answer question something conjecture method one hierarchical term predict predict analogy answer learn disease post stress behavior cognitive rational brief disease post stress external capture structure occurrence expect hierarchy change firstly hierarchy originally child contrary child keep new type develop environment group figure modify previous learn co previous result cluster since parent observe around please co occurrence original explanation hierarchy learn representation learn demonstrate hierarchical structure occurrence identify occurrence qualitatively observe though intra still analyze limited example choose arbitrarily currently check bring classification pre interested extend reasoning relationship engineering department computer universit discovery institute research tu multi label underlie pattern paper
gaussians covariance component publication theoretically additionally interesting optimality support datum much distinguished dimension approach mixture discriminant addition assignment return complexity cluster ambient require assumption high dimension relevant extensive research supervised feature typically largely extract arise many cluster patient base characteristic drug etc cluster select assume even employ projection onto individual coordinate pre processing suffice relevant feature clear step suffice cluster mixture spherical relevant unimodal motivated spherical gaussian mixture high computationally efficient simultaneous primarily number feature feature feature objective induce penalty penalization learn thus solve np papers iterative paper exception latter consistency np paper relevant may another cluster pre screening feature marginally unimodal learning history particularly computer community emphasis assumption paper spherical ambient rely isotropic whiten multiplying covariance feature covariance hence line question optimal norm build apart computer community propose spherical however either approximate provide statistical separation minimax bound spherical marginal thresholding feature restrictive selection separation minimax perspective combinatorial search tractable base component assignment identical discriminant leverage estimation optimal hence correspond discriminant lda label plug sample classification rule mixture plug cluster covariance assumption cluster natural drug responsible expression capture cluster equivalently dimension work relevant coordinate coordinate occur matrix consider special case zero optimal additional assumption guarantee feasible use restrict property satisfie similar penalty assumption performance seek identify need relevant formally relevant unknown price direction liu linear classification step availability mixture mixture combine moment mixture learn dimensional exposition choice cluster tie estimate recovery n ji parameter univariate datum mean discard variance put skip small apply datum terminate failure invoke discard invoke return state rate normal regard identity permutation standard cdf appearance account permutation eq feasible optimization problem eq q I parameter least define section permutation
figure recovery respective run principle recovery roughly orthogonal tight stay next design increase canonical basis hadamard parameter noiseless iteration show predict theorem decay however suggest indicate dependence atom cc training illustrate noise dictionary hadamard coherence stability coherence training signal criterion generating generating dictionary ratio create decrease increase parameter show average trial reflect theoretical whole range stay quite gap hand determine conversely recovery mainly level noise noise go step run oracle iteration trial curve prediction theoretical stay level enough sized coefficient signal almost half hadamard slight perturbation therefore show theoretical translate algorithmic exist point direction future response dictionary learning generate coherent signal precision present show identification locally possible derive level roughly somewhat algorithm complexity compare sparsity project local iterative sign successful step iteration multiplication serious drawback criterion believe local maxima confirm preliminary local seem global near generating dictionary behaviour strong guarantee important want radius radius generating alternatively could extend important extend convergence algorithm arrive level version instead pure exhibit research gap interpret relaxed frequently sensitivity algorithm extend guarantee exactly reflect practical equally weight analyse symmetric coefficient integration extend k c soon original dictionary large assume perturbation dictionary perturbation I q expand try perturbation maximal inner attain typical p mean long c q absolute collecting objective compare need remain suffice say hand simplify symmetry coefficient assign increase component permutation x kx expectation calculate familiar calculate decaying satisfy satisfied almost surely expectation get almost surely almost surely I soon case symmetric integration argument constant permutation subgaussian noise generate signal local maximum conventional scheme calculate perturb attain case response inside outside fact sequence soon maximal response perturbation define fix maxima attain hoeffding z union eq subgaussian v e split expectation sign get symmetric either term last bind q taking see imply proceed soon analogue employ low conceptually combine split constant outline idea concentration cover admissible dictionary note noise replace averaging lipschitz expectation I n n free need perturbation perturbation recall perturbation ball know radius perturbation net argument signal substitute make sure expression split soon second choose ks large probability tried therefore follow sure right condition eq four choose c ks usual except probability inequality present stable identification overcomplete coherent possible training signal criterion criterion well signal ratio translate scale recovery achievable thresholding sign identification k finite criterion facebook gb gb estimate reach actually concept decade high every linear overcomplete size ambient component representation efficient processing scheme compress analysis learning address fundamental question eq unit column development well experiment start aspect theoretical insight separation predict expect approximate tool compression efficient dictionary justify dictionary tool source algorithm source identification er basis interesting overcomplete alternate locally correct dictionary aspect common successful identification order incoherent dictionary give sparsity usually signal sufficient global simple thresholding sign locally introduce present analysis give identification stable recovery sample size find identification implication identification collect letter deal abuse notation collect maximal inner restriction dictionary transpose transpose collection constant follow singular element unit frame g symbol growth aim codebook vector see extreme case dictionary sparse allow atom solve algorithm assign turn ask getting learn q therefore sign k assign atom updating atom normalise sign detail go formulation sparse formulation common mod except however onto atom eq maximum use partitioning large singular sign training oppose problem effective learn bring underlie give hold simplify random perturbation behave dictionary behave optimisation simply absolute response q local optimum signal perfectly quite local randomly sparse page foundation provide suitable identifiability principle asymptotic signal expectation coefficient follow also suffer first coherence respect decay unfortunately size incoherent dictionary sign indicate guarantee equally sized therefore identify dictionary incoherent dictionary ambient therefore extend stable task noise unbounded white noise want identification finite convenient consideration model subgaussian parameter employ typical gap generate support norm frame frame symmetric sphere let subgaussian x ix satisfy outline ingredient add substitute condition generating sign sequence actually accommodate relatively level perturbation find sub gaussian quantity iid variable q signal balance white possible expect eq noiseless identify generating local close small coefficient quality say thresholde correct even ambient fraction decay turn size either noise noisy normalise version response coherence unit assume c kx sx ix satisfying unit frame frame coefficient distribution unit accord kx non component almost surely coherence r ix respectively proof find lipschitz property mapping sum expectation cover admissible close
necessarily treat stability stand eigenvalue condition equilibrium stable conclusion hold replace example process stable recurrent unstable upper situation node diffusion obvious reality reflect topology recurrent impose difficult propose stable sparse stability impose transpose raw coefficient implementation penalty extremely focus introduce matrix helpful one want knowledge preliminary allow turn negativity directly step replace programming negativity variety package much start define modification refer decrease satisfy observe practically converge step value inner loop apply enyi like generate sparse follow number choose randomly rest node exclude node ad independently I first recurrent popular dimensionality screen induce ad hoc roc false simulate network accord early third system matlab sde sde stochastic ex cardinality collect compute averaged margin roc curve necessity penalization use network stability network ex parameter forecast time concern snapshot obtain one process error long forecasting extremely excellent term forecasting study cycle publicly expression periodic dataset expression record point cell identify factor connection factor target gene evidence comparison connection detail stand confirm connection evidence interface observation pattern stability correction average subject much hessian note definite w k p easy equivalent eq apply require solution position q sl property thresholding variant ij ij globally solution minimizer remain line theorem separability prove existence equilibrium show existence equilibrium determine verify well uniquely lyapunov dynamical eq lyapunov chapter stable equilibrium exponential shown omit modify global modify improve reduce obey vanish handle maintain decrease f result desire sequence follow operator globally lemma apply convergence p soft rule l evaluated give u ds von lemma corollary draw capability model phenomenon physical mechanism identify parameter extremely modern observation strongly rigorous variety sparsity recurrent direct control estimation recurrent network stability lyapunov integrate sparse excellent forecast learning estimation dynamical lyapunov identify topology scientific dynamical evolution level gene influence follow connected topology detect behavior dynamical great stock brain social topology underlie network devote develop appropriate model linear commonly human network equation change activation nevertheless lot instance strength unbounded combination widely system gene stock matter strong go beyond capture nonlinearity must activation proper world mechanism behavior exist feedback loop dynamic capture kind say effect recurrent successfully bioinformatic financial forecast circuit computer vision often I brownian direct collecting connection direct describe fundamental extremely modern big available observation available face addition analytical difficulty hoc aim identify complete dynamical well automatic topology recurrent interest many dynamical structure gene number point consistent compressive rely alone limit address network propose system real interest reason practitioner limit shrinkage algorithm rigorous guarantee form quantile recurrent screening control topology moreover recurrent stability follow introduce recurrent regularize regression recurrent screening topology identification dimension condition recurrent system stable detail penalize among penalty perhaps enforce relaxation take interested enhance nonconvex nonlinear penalty may nonconvex observation learn great meet modern response sl recurrent subject penalization notational w subsection unless otherwise propose problem matrix version estimate thresholde shrink fit denote jk update thresholding proceeding rescale associated j penalty penalty elastic penalty instances suppose nonnegative iterate l prototype proof usually span coefficient intercept e penalty intercept set prototype update column convenient integrate cf formulate rewrite minimize proper penalty network popular penalty scad penalty parsimonious matter ignore validation time consume nonzero meaningful prior availability resource control contamination add regularize sparsity shrinkage tune indeed usually sensitive many researcher fix similarly recommend add mild optimization fall penalize thresholding operator
large matrix therefore datum outli analysis allow large could rank outli rank use objective formulation transform derive operator proximal proximal low outli k costly implementation build neural represent proximal splitting train network fine tune proximal splitting either constrain dictionary possibility first dictionary network term outlier result decrease increase bad reconstruction output outli change result non good lie act outli regularizer applying would autoencoder sparse select array apply element define norm sparse sparse parameter apply norm allow store prevent outlier leave sparse code far sparse function new setting k process use framework something happen shrinkage element large instead sparse operator k sparse autoencoder kind manner proximal operator code proximal operator operator u u function need apply function soft sparse shrinkage derive look original shrinkage complete look descent match sparse proximal change also include come norm non since part mnist contain interested sparse compare error regressor classify digit classified percent code error produce representation suit classification change fine store value similar error code select learn dictionary picture one typical pca dictionary sparse code network case digits segment produce present usage completely prior due objective automatically
motivate hashing algorithm core use batch stochastic context neighbor provide scan modified locality lsh space hash standard hashing product typical variant equivalent analyze always note computing collection project hash similarity coordinate example consider convenience introduce estimate permutation combine projection hash unbiased inner assume projection definite definite constructive basically expand datum recall nonzero hash basically length except note package format adopt trick bit hashing example understand express inner product easy also inner consider difference put require permutation projection expect would heavy tail normalize actually quite small web th th algorithms corresponding repetition var theoretical estimator unbiased formula typical tailed datum text use learn weight tf simply well formula extreme variance raw become essentially theoretical dash count panel panel input solver core kernel section estimator product expand permutation location linear mean projection e inner achieve choose dataset combine achieve suppose apply loss th entry project obtain svm expand form exactly always experimental result panel expand th entry original middle panel kernel expand datum hashing accuracy reach mention hashing svm basically idea keep develop essentially equivalent sample applicable binary suitable metric appropriately unlike kernel often help justify type core projection permutation many extension currently rbf allow flexibility improve performance type panel core right bit hashing projection time feed expand linear another line extension apply hashing view inner product hash inner product potential compression context sublinear approximate search layer projection top store sign project sign bit provide indexing capability allow sublinear neighbor lsh popular hashing variant inner product scale linear e outperform sparse core accordingly kernel fed classifier confirm line hashing scale expectation estimator c moment regularize report accuracy drop uci repository view comparison test example need l test lin linear core kernel
bayesian update output calibrate predictive unlike hypothesis whether plausible possibility quantify plausibility use highest credible finally whether phase challenge reliability answer test prediction assessment along embed formulation statistical e discrepancy physical knowledge predictive reliable model organize procedure assess capability introduce idea predictive illustrate mass remark challenge propose real world process describe constitute prediction simulate observational available scenario necessary available system credible fact observational uncertainty prediction endow characterized uncertainty need raise concern ask prediction part solely represent highly reliable applicability however highly reliable augment embed reliable reliable reliable foundation validity idea abstract prediction discuss validation process uncertainty background discrepancy discrepancy model validation generalization use reliable make mathematically eq example mechanic momentum energy scenario scenario include parameter problem quantity mechanic tensor system implicitly map require relationship unknown fully define case indicate embed scenario parameter scenario embed particular setting entirely calibration embed enable prediction calibration observable measure experimentally observable embed simplicity exposition embed either incorrect perfectly error seminal true output statistical directly observable original treatment incomplete analogous calibration datum pose require would way validation making model representation capability accuracy challenge observational datum imply reliability infer extremely direct mapping construct alone modeling alternatively mathematical relationship construct provide test influence prediction recognize error embed physical represent uncertainty additional uncertainty even inherently additive choice must stochastic drive physical well consideration computation tractable principle develop physics uncertainty specification dependent discuss observe embed composite model appear structural uncertainty furthermore transfer validation process calibration embed embed assess assessment gain reliably prediction design illustrative broad validate unobserved need prediction interact physical phenomenon generalization embed model phenomenon model molecular chemical need embed mass momentum quantity close calibration scenario parameter hyperparameter generalization situation scenario depend new quantity characteristic experimental describe experiment model quantity provide direct regard embed experimental calibration involve additional embed ideally none formalize introduce description element additional embed denote observation express prediction model model model quantity must pose model generally general determined validity assess prediction associate prediction state generally model stress determine embed apply prediction embed express embed consistent need argument abstract appear composite q experiment form provide meaningful error uncertainty embed small preferable model quantity composite experiment avoid uncertainty model exercise embed embed pyramid experimental input low exercise generally control numerous ideal embed base pyramid hierarchy pyramid experiment exercise increasingly expensive commonly limit higher higher embed critical pyramid experiment system complexity provide therefore generalization predictive validation process simulation complex system idea example characterize abstract generalization composite likely g must calibration model calibration assess accuracy involve activity assessment briefly specify uncertainty calibrate e g speed acceleration least well consistent exist phenomenon inverse determine require output impose knowledge furthermore uncertainty uncertainty calibrate parameter approach uncertainty parameter serve predictive approach discuss however representation powerful rely bayes knowledge observe give uncertainty parameter condition uncertainty guarantee datum less calibration indeed ensure match may output experimental check consistent notion however much acceptable reality metric approach determination tolerance opinion consideration uncertainty acceptable discrepancy outcome uncertainty include cause available intend plausible uncertainty insufficient outcome uncertainty mathematically uncertainty use tolerance obtain process yield observable helpful acknowledge credible interval define particularly interval belong outside invariant undesirable mean conclusion validity consider observable period specify density define small observation tolerance less outcome set observation skewed multi modal multi modal region consist disjoint peak credible valid far validity confidence embed regime assess prediction fundamentally available justified agreement alone uncertainty potential important determining prediction primary need address sensitive aspect embed effectively inform calibration domain applicability answer central assess prediction rely characteristic instead subsection consideration prediction credible sufficiently small purpose important nature inform prediction discussion far assess embed inform credible reliable aspect assess prediction make consider example inform validation fact current context extent little unless reliable e speed understand chemical reaction alternatively could highly quantity validation sensitive cause embed validate example validity calibrate pyramid situation would validation pyramid sensitive composite prediction depend state recognize include argument pyramid test pyramid insensitive failure assessment scenario sensitive determination close constitute sufficient sensitivity knowledge approximation composite scenario suppose largely due embed discuss valuable assess whether represent quantity insensitive reliable prediction independent involve inform calibration outside clearly regime expect scenario composite embed structural response however scenario linear embed magnitude well model model range calibration matter check composite applicability way straightforward predictive assessment ensure sufficiently rigorous uncertainty perform uncertainty requirement decision determination requirement scope know uncertainty check requirement second know simulate involve composite sensitive situation able one identify insufficient could case issue reliable knowledge system model represent incorrect detect illustrate aspect predictive apply simple involve position mass act mass reliable must specify potentially embed force truth system otherwise exercise take velocity execution information system information exercise conclusion describe available several high fidelity model embed model force observational thus constant embed composite truth appendix exercise independently observational validation regard si si si confident physical highly accurate physics si forces coefficient available fail reality si linear adequate problematic si cause notice warm move energy assume temperature would temperature build position mass table position actual truth use actual variable different si si si si confident adequate thus standard calibrate use specify take denote composite perfect parameter alone deviation si si way represent reproduce variable randomness intend variability need lack knowledge value coefficient positive variability normal uncertain must however fundamentally si determined characterize uncertain alternatively si goal cover infinite characterize pose well dependent develop complex far eq model coefficient model predictive necessarily different separately action validation describe model inference solid pdfs narrow highly posteriori map true approximately value change calibrate figure htp figure comparison interval plus make difficult base give uniform distribution show tail alternatively distribution existence nearly si lead plausible prediction uncertainty contradict assertion important uncertainty error characterize thus prediction combination obtain equip well description valid prediction allow enable concrete hypothesis mechanism output confidence observe situation si assess scenario si specifically range variation coefficient energy study temperature govern competition quickly add quickly energy one temperature slowly conservative necessary truth conservative prediction qualitative phenomenon present validate calibration assessment assess marginal pdfs bayesian htp solid post pdf result dashed label prior set broad si value validation unlikely si reproduce show comparison quantitie si much uncertain agree less show statement calibration cc compare calibrate prediction available phase complete dependence mass assessment extent separate set uncertainty fit result vary large mass datum htp pdfs somewhat well deviation decrease demonstrate htp show distribution shift variability consistent move check informative si indicate domain applicability check calibration conclude trust calibrate si model credible need make htp general unknown assign validate ask uncertainty specify however simple uncertainty pose well validation process predictive building enable reliable reliable augment less reliable composite dependence embed composite model allow process require specification fidelity embed connect lack calibration use uncertain aspect model validation plausible uncertainty validation uncertainty finally satisfactory predictive view maker mass understand lack confidence modeling confidence know reality nothing generally present research development issue address application challenge outline model propose respect know quantity broadly applicable uncertainty datum critical concern dependency arise datum qualitative information important make inference tool construct qualitative difficult express tool need kind qualitative commonly regard physical reliable discuss qualitative also characterize applicability predictive validate scenario model dependent model applicable technique develop calibration datum measurement quantitie uncertainty need large domain applicability embed model allow well automatically execute associate curse dimensionality expensive naturally introduce knowledge physical phenomena address acknowledgment material work security award fc na grateful david many discussion discuss situation coefficient vary temperature determine heat support decision make need generally experimentally observable assess inform challenging validation observation determine consistency ensure predict quantity limitation dramatically reduce effort decision imply observation agreement validation prediction
situation subsample rather bootstrap amenable bootstrapping subsampling although originally forest bag forest analysis work subsample forest prediction much sampling forest one paper adaptive neighbor use slowly tree small recent move beyond black box forest prediction give propose rigorous asymptotic prediction forest amenable make forest grow forest number action rigorous overview study bag compute particularly formula time subsample covariance respect formula low status school education classify select test replicate otherwise line smoothing spline context forest empirically work bias correction analysis provide show forest prediction motivate classical justification solid spline freedom dot connect setup inferential framework visit classic feature plot forest median house measure status prediction bar distance spline note nearby bar relationship forest error end back term coefficient throughout build tree bootstrap enough carlo matter carlo detail identically distribute admit infinity regular definition subsample subsample exist subsample forest govern reduce decrease conversely large tree grow deep get result describe course put used start state condition recursive leave fraction split sense tree form device sure random forest become local meanwhile theoretical simplify exposition sometimes get effect leaf ignore without spirit tree adaptive neighbor tree use split prediction paper forest fully conditionally know arbitrarily bias align rectangle infinity find tree use training split cart tree consistent even everywhere cart however rather cart separate rest neighborhood towards say cart subtle enough affect similarly estimate forest cart learner understand cart forest library promise proof back develop normality briefly capture effect projection projection expect asymptotically tend projection automatically abstract chapter projection directly around definition argument incremental predictor argument proceed weakly predictor subsample thus back forest motivate potential near random meanwhile device cart predictor nearest predictor operate neighbor consider near align contain ty decision axis learner tree predictor always value often good get formally show quantity establish show incremental suppose feature infinity thank ready show incremental proof follow lemma regular function uniformly bound incremental constant show subsampling proceed classical analysis flow motivate let variance base pair forest base accord restrict moreover abstract forest estimate level forest theory inference forest theorem prediction subsample setting considerable monte subsample bootstrap correction ccc ccc mse cosine cosine cosine e cosine e e describe accuracy bootstrap replicate synthetic distribution construct rx k average analogously absolute divide metric average test rather get cosine relative mse decay predict appear error yet regime decay high bootstrap classic uci repository set due predict log divide set parametric synthetic perform despite ccc ccc auto e e metric accuracy subsample replicate rule noise lead box validation measure like study subsample subsample satisfy establish normality show generalized density involve showing hold see pair equivalent independently distribute try continuous tie standard get gamma incomplete eq letting expression thus quantity around index sum loss regularity probability converge obtain meanwhile stein suppose identically show expression also projection write decomposition quantity interest individual tree forest equivalently
consist estimating minimize infinity appendix inner ridge optimize ridge scenario j ignore gradient given attribute namely behind lasso algorithm let b directly recall always case attribute sparse improvement harmonic dd require may available consider moment exactly run modified use upper count run starting ignore analysis formulate output dm value become achieve improvement moment attribute prefer improvement regime regime reason easy infinity norm remain upper hold sufficient ridge join regime line plug k plug k eq use q plug section test analytical claim conduct set experiment control mnist digit design ridge regression consist moment attribute moment tries require ridge offline attribute utilize page attribute efficient use represent attribute quantify avoid normalize ratio prefer upon definition exact quantity analysis scenario define fold increasingly average zero error bar algorithm result start phase upper confidence conservative find split attribute evenly inner improved linear easily scenario algorithm define exponent decay ball attribute namely independent corresponding expectation entire analogous stochastic addition connection adaptive additional improvement future learner prove complement room improvement partially grant grant direction use full idea beneficial coordinate rather single popular along instead size uses th attribute learn gradient across arise paper discuss run simulation decay exponent datum experiment htb offline erm erm htb adaptive attribute bad version improve understand risk algorithm actually perform gradient enough estimator state lemma randomization eq convexity lemma first see tr calculation give recall state probabilistic moment nz prefer bernstein fast pay factor attribute first realization sample phase hold trivially actually phase part conclude similarly bernstein arithmetic conclude plug dm plug lemma constant assume therefore prove directly completeness second vector examine expect respect lemma us value c next relate linear lemma proceeding lemma fact induction hand q combine rearrange complete proof proof eq lemma bind randomization hence bt triangle q note dc ic equal yet di I assignment probability minimal value attain equation hold institute science analyze learner subset attribute training ridge regression geometry sampling learner probability calculate data improvement excess state large main knowledge knowledge simple amount achieve improvement complement analysis claim effective whereas life learner per medical diagnosis patient learner perform diagnostic cause physical likely diagnosis attribute whether site email address pay cost known attribute observation formally attribute reveal select attribute attribute reveal feature include subset predictor minimize target discrepancy generally loss expect vector unlabele particular problem ridge scenario regression one online gradient descent behind scan calculate unbiased attribute algorithm sample excess online attribute additional factor interpretation provide low establish fact improve develop manner attribute moment advantage utilize examine reach optimize principal able budget begin moment namely risk scenario summarize notation old ridge km km easily prove bound always previous previous dependent moment sufficient rate distributional algorithmic elaborate attribute coincide online full scenario limitation approach second moment advance computable address phase phase simple phase always sufficient prior knowledge attribute rest organize follow describe develop scheme case knowledge moment variant prior factor ridge improve experimental simulate connection scalar letter font indicate index indicate pp proper triangle expectation randomness attribute respect respect randomness framework learner represent goal learner weight minimize entire follow standard training minimize expect fit require norm schwarz old assume loss generality attribute scenario limited attribute popular budget budget total total exceed bind budget efficient scenario exist full attribute scenario imply full thus trade linear regressor constant regression gradient scenario attribute expectation perform ridge prove correspond regressor regression ridge scenario call ridge gradient descent build current step direction result project ball square attribute attribute sample sample attribute build calculation obtain unbiased reduce estimator minus building unbiased set bt mr ki slightly notation show analyze bind therefore develop sampling estimate bound every need solve multiplier yield inner minus use follow strategy probability prove superiority generate lemma moment attribute idea attribute formulate output run calculate equation recalling algorithm always well coincide however dd exact knowledge knowledge task case moment initially depend next unknown prior equation address split phase run moment train slight modification upper order return output order apparent use moment method never get estimation formulate value perform phase second turn km bind assume guess proof bind squared bound factor analyze proof treat join single estimate constant factor
suited exploit short spatio unable investigate novel cache block trace trace vector explore count address gap propose dp mixture capture covariance however particularly dp multivariate generalize lead cache step historical capture trace cache experimentally trace storage mining modeling categorical count block trace exist policy cache least recently use ahead strategy sequential pattern nearby correlate extremely short interval typically find correlated interval predict alternative spatio novel count exploit cache capture trace trace sequence request often span million interested trace spatio arise long access certain read capture aggregated view partition histogram slice span get aggregate slice count count instance request adjacent together rich temporal dependency aggregate vector aggregate portion small count dimension count understand spatio temporal common correlation hide extension temporal inherent storage trace suffice since mixture type model kind adjusting datum parametric variant study decade correlate count parametric multivariate gaussian expensive sparsity datum extension sparse modeling select aware world trace technique outside apply often count mixture extension exploit propose novel technique dp poisson dp methodology tractable discuss extension hmm cache aware address take first cache long spatio dependency memory experiment world trace show improvement particular baseline improvement mt multivariate complicated designing mixture unknown reference aware unknown truncate dp study amenable extension model immediate good another sparsity emission component parameter aware work mixture density specialize investigate prior count cache cache storage cache warm cache trace different long temporal cache serve study trace trace grain short correlation specialized type large cache performance predict event operate contain amenable trace cache application persistent medium disk cache medium request cache cache constitute cache application else cache retrieve disk cache cache high improvement measure cache cache observe part derive place cache operate trace term hour operational run much restrict phase repeat even exploit trace partitioning phase slice second let access request sequence count bin trace arise range access repeat albeit hence dependency vector correlation markov choice follow chain induce cluster count denote variability trace k suitable scenario motivate non technique dp cluster correlate capturing hmm exploit count hmm predictive ability hide map raw request various slice latent aggregated operating aggregated like choice block load cache happen hmm dp algorithm deviation usual viterbi possible much slice viterbi technique fraction prediction would consist loading cache correlate count model count temporal dependency parallel along mixture propose dp dp base hdp hmm mixture challenge design correlate exploit dp mixture challenge follow hierarchical hmms fix dp hdp data hdp discuss dp brief overview collapse hmm dp detail supplementary throughout notation subscript sparse latent aid hdp detailed collapse variable likelihood py z old hmm dp know arise know require sum possibility exponential supplementary summarize standard dp also computationally motivate modeling dp suppose derive j variable integrate alg eq l z z l kb hmm dp exclude u supplementary hmm dp active sample variable benchmark trace likelihood evaluate effectiveness available block trace commonly storage microsoft choice trace trace aggregated aggregation number bin dim slice later trace two experiment understand well next cache dp detail compute dp dp dp margin correlation aspect necessity model independence dp hmm inherent datum r r trace hmm dp dp mt mt mt mt mt mt mt trace simulator augment simulator block simulator describe supplementary appendix prediction improve simulator trace plain capture portion trace dependency ideally data train see explanation trace pick trace repeat fine bin memory aggregation count find baseline attribute sensitivity trace superior bring focus hmm dp term train hour run algorithm sparse dp finish trace sparsity latent improve efficiency time c hmm hmm name dp dp mt mt mt cluster avg outperform hmm sparse hmm perform hmm dp trace spatial model handle fit good bin dp without trace sparse trace trace baseline parametric baseline suitable application trace vary infeasible exist dp model focus correlation type capture range ahead hope show model trace trace long present trace predict understand read schedule cache size prediction play important role incorporate capture disk scheduling issue hmm dp structure far explore datum hmm dp lead efficient leverage block trace cache improvement world trace outperform trace perform experiment publicly block trace microsoft represent diverse trace comprise week worth allow study long range temporal eliminate trace write percentage focus read cache present remain trace validate result comprise collect hour divide trace vector phase operation r rd mt mt
unit diagonal diagonal learn inference hide convolutional inference generative model three convolutional layer filter activation fully activation generative replace varied final step detail ex step step hyper epoch importance monte carlo numerical report test likelihood gap mcmc convolutional inference step report mutually exclusive combine specification variational specification practical practical transition satisfy eq x optimally choose reverse iterate converge running follow tr allow way detailed balance improve posterior bind make practical balance ensure transition balance hasting rejection transition first generate finally analytically metropolis interpret reversible jacobian evaluating target carlo estimator variational posterior estimator rao carlo calculate respect short attractive computationally demanding path accept reject alternative omit acceptance transition operator reduce respect gradually reverse importance sampling transition p normalize density reverse look like old log ratio variational strategy specification base guarantee inference satisfy impractical far variational last consist multiple different effectively become take distribution put iterate chain offset add iterate inverse set take bound effective reduce potentially input suggest optimize mcmc step optimize mcmc sequentially maximize variational improve optimization boost iteratively unnormalized posterior new maximize local simple exponential form improve variational approximation accuracy advance perform variational approximation synthesis variational monte incorporate rich inference variational fast objective option trading computation parameter quantify observe datum relate specify rule quantify simple conceptually imply computation intractable resort method inference monte carlo explicit case latter nonparametric get parameterized approximation maximize bind eq maximize minimize perfectly variational start distribution rather mcmc draw choose outcome variable converge exact advantage mcmc approximate time practice getting long interpret chain expand set auxiliary free posterior marginal approximation distribution since close fit choice optimal reasonable specify optimizing special auxiliary like z rewrite subscript highlight possibility different chain flexible parametric approximation choice operator however transition inverse variational without operator variable initialize transition ratio low insight behind parameter estimate low obtain gradient application chain transition operator case backpropagation obtain bind solution gradient h initialization hmc low tv z tv x l l omit monte discuss approximation respect stepsize mass hamiltonian algorithm local improve add hmc reduce thereby use mainly calculate additional derivative rule hamiltonian variational step expensive reduce mcmc computational dimensionality variational hamiltonian hamiltonian tune hasting step reject transition optimize marginal assess technique find may posterior short rely theory hamiltonian consider estimate death large contain cancer city occur city count compare expect assume low dimensionality contain integrate numerically calculate approximation number step hamiltonian see hamiltonian benefit realize iteration
exponential bias bayesian nonparametric model conjugate beta odd representation gamma couple general conjugate family representation likelihood conjugacy conjugacy continue place broad class mixture limit exponential however whether similar notion conjugacy bayesian literature family include familiar name poisson name refer aspect underlie bayesian l evy construct property conjugacy parallel known conjugate multinomial bernoulli break biased obtain separately significant formalism posterior conjugacy nonparametric analog provide constructive conjugate prior specification size representation broad process trait us trait trait likelihood point subset trait trait make trait trait allocate new trait yet allocate trait grow challenge countable trait frequency prior calculate posterior trait frequency trait nonparametric integration three principal conjugacy marginalization conjugacy dimensional exponential conjugacy turn beta hyperparameter iid bernoulli conjugacy certainly popular cardinality arguably parameter prior pair generally exist though process include classical bayesian dimensional family construct nonparametric biased trait play role year allow exact slice particularly useful representation general show family directly integrate trait trait beta bernoulli show generate bayesian exponential constructive exist marginal build nonparametric bayesian nonparametric calculate posterior development introduce generate automatic conjugate exponential likelihood size derive conjugacy discuss view trait trait express recall measure trait trait index many tuple consist trait trait descriptor trait weight q th point trait allocate degree individual allocate trait trait degree point belong trait belong treat observe bayesian topic model vocabulary occur document might document document topic concern posterior conjugacy exponential fact especially trait order trait specify measure well algebra subset consider measure random almost surely particularly form measure completely random property without treatment follow part component component construct e location location atom finite infinity deterministic random since measure generality distinct independence random variable ignore conjugacy representation cardinality countable deterministic borel product countable subset generate ordinary component start poisson countable ordinary yield place component incorporate atom atom infinity ordinary ordinary typically part hierarchy extensively elsewhere affect point trait henceforth measure proper ordinary point attention trait trait atom distinct location distinct discrete infinity helpful component point specify assume iid locate atom location take locate atom may state form impose likelihood formalize restriction recall trait trait unbounde collect countable infinity trait require infinity location represent trait sense know location atom advance trait discover priori countable trait countable location atom ref require countable trait location ordinary component must countable atom trait frequency infinite mass ref implicit part allocate trait finitely thus number atom every correspond fix finite restriction atom restriction finitely nonzero particular atom countable atom consequence purely mixed discrete henceforth discrete write requirement allocate finite trait translate requirement construction form mark ref thus capture call ordinary measure beta component feature multiply real factor contain weight hyperparameter achieve mass beta improper pair process specify finite mean integral proper e restriction imply range like posterior fix atom distribution ordinary proper jointly weight dimensional prior proportional q theorem ordinary atom location well draw fix atom locate weight atom deterministic purely without knowledge atom generate note know ordinary atom return atom atom form mark consider iterate restriction satisfied calculate location atom put normalizing atom density third component posterior exponential rate update hyperparameter fix likewise show conjugacy hold desire next conjugacy conjugacy establish conditional location rewrite poisson bayesian nonparametric first atom weight rate ensure ordinary component characterize proper weight rate distributional location atom finally range ensure improper gamma either integral finally hyperparameter discover poisson highlight poisson process atom weight gamma atom gamma conjugate likelihood process odd bernoulli exponential weight atom support odd parameter probability successful odd success failure write emphasize location atom ensure ordinary hyperparameter range ensure improper beta require beta proper hyperparameter restriction summarize component previously beta atom conjugacy odd highlight result corollary odd bernoulli process atom weight atom process conjugate prior odd find bayesian prior build representation despite biased random hyperparameter satisfy stick break stick describe proportion remain stick break describe stick break stick call representation reason draw thought draw limit proportion atom atom choose representation useful familiar atom inference truncation constrain sum explore past notably beta though term stick mass sometimes refer stick popularity stick beta case slice general discover previously unknown representation general random q location atom simulate location weight come come cf ordinary generation countable demonstrate weight finite familiar case biased representation atom ordinary discrete atom atom atom atom atom atom atom moreover enumeration break enumeration observation atom location atom atom atom atom atom atom atom atom process first poisson eq take component posterior equal poisson finite total q atom find atom identically across atom independently summarize give detailed calculation thereby trivially conditional location component write biased representation beta derivation bias gamma poisson process summarize let fix weight ordinary conceptually focus cf integrate canonical comes distribute iid marginal form let atom location example marginal chinese restaurant prove since integrate dimensional generally marginal representation atom write ordinary jointly union weight new atom location moreover express atom location induction assumption weight atom weight mass line let agree development cover present atom atom location new atom location eq eq finite let repeat measure inductive hypothesis hold biased find prior exponential atom trivially satisfy generate fix ordinary suppose conjugate provide atom location distribution atom corollary beta discover new iid condition q eq summarize location distribution ordinary iid across accord location atom distribution construction atom atom calculate posterior general prior draw bayesian nonparametric model notion family allow specify automatic likelihood
second robot system action angle robot currently receive simulator controller current angle angular velocity angle velocity physical randomly choose action simulate noisy draw contaminate deviation probability dynamic density pair transition useful model three scenario right roll thus case joint summarize redundant aim draw video identify action vector shape ratio offset axis near reverse driving transition dimensional dimensional output collect transition summarize bottom method successfully high dimensionality reduction conditional estimate least dimensionality density mutual denominator ratio effectiveness extensive computer q let express proof derivative approximation eq substitute partial gaussian derivative cs ac regression informative multimodal preferable challenging dimensionality first dr execute dr propose novel single shot perform key formulate need dr method extensive various computer reduction however analyze informative possesse appropriate naive approach density kde kde nearby problem nearby separately estimate density decompose form p thus problem method mini solution efficiently compute cope aim input reproduce hilbert possess systematic model available overcome alternative sufficient theoretically optimal rate dr dr promise accuracy preferable regard paper propose dr dr include execute therefore density usefulness name robot method conditional dimensionality let dimensionality identically dimensionality expand form theorem conditional independence therefore reduction minimize denote represent relation span equivalent member class loss conditional negative kullback leibler member loss ce pearson divergence sharp include easily critical develop method denote coincide search geodesic gradient estimate entire estimate computationally expensive execute randomly achieve small derivation conditional density minimize even maintain th gaussian locate may subset center increase advance notable appear compute analytically similarly normalization analytically essential include denominator uniform reduction accurately well exist experimentally experimentally investigate usefulness dimensionality reduction maximize gradient manifold artificial dimension execute conditional density neighbor square cross validation least square inside behavior plain method normal plain well due dimension artificial right leave measured norm clearly profile much discuss ratio smooth density p p loss q kde
per person illumination individual image correspond illumination change correspond pose illumination expression database consist intensity algorithm subspace subspace generate normalize pair basis perform svd subspace vector next generate projection apply point separately green either claim see experimental choose different label pair project evident figure project along analyze separation give dataset dimensionality project separately matrix dot visualization face element dot white represent inter dot consistently dark separability project reduction quantitative result discriminant regularize random preserve embed dimensionality dataset perform yielded compare method make dataset class evaluation test compute projection accuracy deviation similarly rp run c c method pca rp dim acc see section rp dim acc rp dim acc lda dim acc pca dim acc result table report find degenerate hence dimensionality first class class dataset show table performance multiple subspace sufficient independence disjoint iterative algorithm learn projection reduction three world reduction example proposition datum union application preserve independence subspace trivial design dataset dimensionality compression theory face texture segmentation reduction nuclear simply image image datum think traditional subspace principal application datum preserve reduction although try preserve geometry datum try preserve dimensionality independence subspace disjoint subspace subspace line idea vector sufficient find projection aforementioned handle corrupt say subspace dimension subspace margin separate margin margin maximum dot either vector angle definition subspace specifically dataset goal reduce continue lie formally let propose subspace number require label vector require motivated disjoint unit orthonormal tv jj tv show symmetry lie need respectively thus say projection plane line lie along along line angle separate angle principal principal dimensional subspace disjoint subspace plane project subspace two line argue add dimension plane project subspace one project concern notice least linearly forward already would vector would computationally handle though label circumstance specify try label dataset attempt sample heavily underlie principal two subspace projection subspace pair separate margin subspace submatrix repeat state idea span close repeat cosine opposite subspace opposite handle margin subspace equivalently express identity disjoint local minima gradient w thus lagrangian principal principal pair local minima w r minima setting
r poor posterior show bootstrap obtain member return sample construct finite subset agnostic bayes bootstrap possibly infinite model hyperparameter build agnostic e reflect repeat hyperparameter ensemble expensive would train predictor always matter trick exploit observation accelerate construction fashion maintain contain risk update cost single multiple share detailed v h run updating behind run gps deal construct powerful mcmc accommodate probabilistic order predictor ignore nature determine comparative agnostic predictor probabilistic traditional predefine adapt ensemble three building select search rs practice often superior ensemble construct use hyperparameter mat ern kernel scale gp evaluate hyperparameter configuration performance test several collection perceptron win frequency converse sure outcome chance use sign derive posterior significant significant substantial come datum convert multiclass merging class benchmark collect collection cccc cccc rs r rs rs present win represent redundant complement add method conclusion sign colored dot dot report obtain observe outperform method outperform rank look outperform concern forest yield generalization performance elaborate method look challenging space vector agnostic alternating analysis column clearly significant degradation term speed opposite left reach significantly exception well gets quickly outperform cccc rs cccc rs cccc rs cccc rs rs automatically construct ensemble hand adapt hyperparameter produce ensemble method attempt ensemble uncertainty risk extra generalization dominant task hyperparameter fortunately progress sequential optimization method validation ensemble properly select extension automatically ensemble recently paradigm agnostic bayesian confirm selection important make machine expert science recently report success successful hyperparameter configuration learn increase improved configuration converge find good configuration well instead combine good win entry netflix competition variety different helpful comparable performance likely differently input produce error average hope majority well globally dominate average much however ensemble system automatic construction method thus method error exhaustive explore space recent oppose agnostic weighting effectively generalize space model hyperparameter efficient set confirm regular hyperparameter follow agnostic bootstrap construct ensemble discuss notation setup task refer member predictor hyperparameter training hyperparameter let obtain set assess quantify incur target task minimize hyperparameter selection hyperparameter list select good example grow hyperparameter yield outside well replacement search inefficient inform test address limitation automatic hyperparameter consist treat learnable learn hyperparameter must gaussian representation promising configuration gp assumption conditional pf nf r k ij acquisition success q equal cumulative function normal acquisition maximize equation perform ascent initialize chance global optima expect offer exploitation face fit gp test initially empty acquisition hyperparameter procedure mean function either marginal hyperparameter sample detail hyperparameter good predictor suffer hyperparameter preferable properly extend ensemble
evaluate color green green blue missing evaluate effectiveness compare admm try recovery completion demonstrate final admm admm try recovery numerical lr admm try try lr almost quality admm computation well reasonably extra find admm notation name admm adjust admm admm adjust reference deal matrix lr show dct operator stage two note second singular estimate stable merely large lr admm important use fidelity admm recover sharp play rule synthetic certainly sensitive course jump way true try jump singular low limited good develop foundation china k recover large arise image medical imaging kind formulate problem operator recent use norm minimization nuclear major limitation nuclear singular correspondingly paper besides completion method extend validate superiority etc approximately naturally decision nature firstly approximate rank norm I bind unknown low recovered optimization nuclear reduce sampling projection incomplete problem result nevertheless suboptimal application operator together overcome nuclear nuclear minimize small singular truncation correspondingly q way aim extend completion algorithm euclidean space frobenius variable projection operator denote transpose low rank optimization attract interest algorithm briefly review influential reformulate sdp interior scale project subgradient computation concentrate decomposition solve scale slow uv parametrization factorization matrix decompose rank priori dynamically adjust alm multiplier programming arise admm widely et solve apply value though nuclear replace nuclear convex solve critical note rx xu u r rewrite get minima propose later similar sparse learn specifically present support short number detection identify reconstruction support detection reconstruction advantage prior information true signal decay heavily seek try al propose special detection completion contribution particular new call solve introduce solve mention subsection conclusion give algorithm elaborate approximation key define variant recovery beyond problem model discrete transformation dft framework procedure start e recover estimate recover idea iterative initial fix svd update author study completion algorithm plain pure nuclear via initialization x return explain nuclear regularize base process large feasible try aim well approximately often decaying extend previous detect vector value nothing specific implementation support show repeat estimate singular thresholding singular spirit significant jump singular minimize false sort straightforward last jump prescribed value decrease last jump unlike propose apply absolute value value look example cardinality computed jump neighboring reflect stability large cut threshold heuristic admm originally norm form extend original admm nuclear deduce subproblem enough convenient follow conclusion satisfy max particular matrix decomposition conclusion adjoint reformulate constrain problem lagrangian lagrange multiplier admm decompose task small subproblem involve admm ignore derive subproblem easily scheme admm approach update eq iteration iteration subproblem elaborate subproblem explicitly solution equip form express admm solve subproblem form remark eq scheme study omit analysis attract lot attention task prefer noiseless norm accelerated propose among accelerate proximal line al extend completion paper solve general completeness short overview meet possibly differentiable continuously lipschitz add construct q conclude iteration subproblem q iterate framework subproblem close closed well also omit efficient solve convergence alternate multiplier condition result name whose subproblem kind mention could form stack contrary process put correspondingly function n reflects equivalently transform lagrangian lagrangian linearize admm prefer handle easily update proximal iteration update update calculate update computation solve elaborate solve form ignore concentrate obeys rule solve adjoint operator show side accord achieve eq remark begin omit admm subproblem solution efficient problem validate part effectiveness hand real admm refer extensive accuracy
extract balance explore consistency trial trial provide plot misclassifie trial use large trial suffice theorems markov discrete rely non ground truth bias toward cut circumstance evidence weakly connect weakly decay word regime certainly perhaps lie component geometric log together acknowledgement grateful part research ds grateful nsf grant dms support support nsf dms nsf author counter counter counter definition counter counter corollary counter remark pt pt pt von establishe algorithms sample ground truth minimize functional cut minimizer cut minimizer sample hold cut result scale connect nearby leverage cluster optimize objective quality partition separate introduction functional cut meaningful introduction balance term functional closely cut functional multiclass cut theoretically computationally utilize cut algorithms expansion approach cut truth sample converge precise manner towards partition discrete subsection informally consideration investigate consistency graph vertex weight point connect scale geometric average desirable resolution take increase represent consequently discrete work precisely take consistency hold cut notion tool modern study minimization random graph provide study limit minimization consistency consider linkage maximum unfold consistency spectral rigorously von work eigenfunction limit sequence recover go zero normalize cut minimizer discrete functional minimize specific set discrete functional prove well one von von normalize knn graph graph hold quite minimizer functional close functional functional balanced graph cut take numerator balance c introduce simplify remark balance term appear multiclass pair note cut multiclass balance partition functional domain way term weight boundary boundary smooth surface cut regular boundary notion geometric measure theory mathematically subsection area tendency balance term refer pair equivalent cut read eq consist point set boundary extract connect nearby precisely decay zero basically describe point increase investigate balanced cut minimizer balanced partitioning uniform present cut unique balanced cut rescaling surely converge towards minimizer notion partition precise let let n c definition convergence discuss conceptually rate scale lebesgue compactly associate optimal cut cut example optimality relevant machine minimizer involve hold dd graph cut determine still valid parameter connectivity balance graph cut remark despite graph minimization effectively fact choose appropriate initialization use bridge rely notion optimal min proof convergence carefully subsection property desirable minimizer statement cut prove cut balanced notion recall subsection total variation illustrate also investigate related main n expand introduce partition turn useful characteristic function ix characteristic need borel interest coupling measure second distance understand focus absolutely continuous respect lebesgue pass discrete formulate way borel borel plan formula exist little match make lebesgue statement convergence proposition enough find sequence map important map convergence occur consideration control norm ic eq convergence partition ambiguity arise partition previous give subsection discuss let represent graph f convergence make remark consistency consider minimizer sequence minimizer need enough compact enough conclude cut energy approach extension viewpoint namely absolutely continuous lebesgue end respect see representative discrete think flexible discrete restrict correspond notion analysis geometric measure domain need extension whole extension give function variation total characteristic hausdorff measure variation finite derivative chapter simple precisely area relate weighted weighted formula rigorously formulate precise definition formulate either relation measurable q deduce ratio cut cut indeed minimizer lemma close balance implie imply cut imply every fact state beginning minimize follow calculus variation minimize continuity converge semi continuity total complete precisely subset similarity context similarity proximity kernel otherwise main cut domain satisfy converge zero point weight optimal balanced cut converge one cut subsequence cut surface scaling establish cut utilize problem particular minimizer minimizer limit remark hypothesis scale must distance measurable sequence eq theorem prove outline rather work indicator eq denote suitable nx rescaled coefficient show indicator subsection show functional establish converge toward indicator toward convergence follow functional discrete function subset start nu u iy nb nb nu nu nu analogue prove suitably indicator function set minimize balanced cut proper subset satisfy sequence exist converge prove claim first balance n nu fact every change obtain nu deduce proof arbitrary arbitrary show subsequence assume limit trivially enough us map inequality hold want start know since imply n nu subsequence subsequence sense convergent subsequence suffice hold sequences nu balanced balance cut sequence follow q nu subsequence convergent subsequence subsequence minimize particular characteristic c imply bound cut nu n nu variation invariant translation assume either continuity balance measure region region strictly zero bound away zero consequence turn sense consequence subsequence u instead moreover subsequence subsequence converge theorem balanced cut analogous space set collection comprise indicator cover multi class r collection disjoint lebesgue additionally imply set lebesgue definition may equivalent balanced minimization balance cut particular want sense define want follow kernel sample satisfy functional converge topology way proposition way omit analogous argument subsequence minimizer finally argument subsection adapt proposition argument constraint perspective due smooth partition meet next lemma multiclass combine nu ny n nu statement iii statement dl proposition deduce orthogonality subsequence every almost k k side equal conversely contribute inferior involve due orthogonality convergence recovery remark due belong let c fact define finite empty intersection lebesgue assume without mutually disjoint finitely manifold embed start construct recovery piecewise dense set variable claim recovery partition define partition consequence functional proceed particular assume otherwise nu nn inequalities tv rr balance term deduce establish r piecewise existence immediately balance denote smooth exist induction hypothesis enough simplify denote hypothesis equality r contradiction bound disjoint subset satisfy mutually q subsection belong consider converge correspond symmetric let x b let smoothed eq symmetry produce q word prove open rotation every constant summation chebyshev eq claim lebesgue property smooth lemma example lebesgue measure measure countable nan partition hypothese lemma combine co formula imply everywhere particular subsequence lebesgue almost subsequence continuity formula subsequence along subsequence analogous hold well relation subsequence extract subsequence satisfy previous lemma complete cover exist imply contradict denote boundary combine convergence show combine continuity total variation disjoint complete
budget pac completely subset happen arm low bad bandit yield budget set arm still introduce successive bind error bandit permutation either propose good confidence inspection knowledge drawback share ucb bound identify complexity multiplicative analogy three low bandit theoretic permit quantity subgaussian propose tight bind complexity uniform sample close contribution fix bandit lower bind budget setting bandit bandit complexity first towards new two confidence budget setting mathematical result permit line sub increment iterate logarithm permit efficient matching kullback make relate probability model lemma technical aspect generalization minimization framework draw draw let model stop relative low fix confidence straightforwardly bind identifiable satisfie make family continuously parametrize mean let pac without arm order arm arm arm long pac apply total number monotonicity yield obtain arm lead low relaxation paper every denote mean prove perspective algorithm design exist twice order natural gaussian common exponential follow kullback leibler family sample kl complexity eq bandit popular website version present user page model probability user feedback user two equally whether armed match different algorithm match bind provide sample pac identifiable let uniform particular obviously consider direct c variance imply strategy consider quantity analogous chernoff explicitly property illustrate two exponential bandit indeed tight change use hand inequality change involve use cumulative modify order alternative low bandit good pac property obtain use satisfying similarly bandit performance guarantee closely match possibly term elimination bandit section armed gaussian bandit reach rule expect elimination two subgaussian cover bound subgaussian enjoy bound subgaussian bandit case introduce coincide pair sample normally pac algorithm versus type type recommendation rule choose empirically arm sample match rule satisfie lower prove among elimination pac elimination threshold asymptotic low point exhibit preserve sum I variable obtain non choose iterated surely achieve goal prove elimination pac illustrate significant average reach conservative allow rely arm base governed ensure end round schedule deterministic match low elimination reduce effect elimination suggest feasibility bernoulli observe particular little gain together stop rule quantity introduce section aim bind provide bind arm sample uniformly determine algorithm arm subgaussian precisely theorem stop respect exponent significant exploration ti ta drawback propose sequential stop stop consequence ratio two proportion pair likelihood denote display consequence I interpret relate kl close armed threshold use guarantee stop eq result analogy conjecture conservative exploration lead conjecture armed bandit obtain theorem confidence bernoulli comparison set bandit present obtaining bandit low yield failure consistent algorithm note previously confidence comparison family distribution fix budget consistent satisfy kullback leibler symmetric hold hold family arm arm upper side arm complexity equal model q recall always define theorem precisely check strategy budget bernoulli exponential static arm appendix every observation arm interest bernoulli model unknown exist bandit comparison quantity bandit strategy arm satisfie see describe approximate universal arm varie observe indistinguishable provide allow budget armed bandit bind armed bandit budget lemma state prove nonetheless able low bind spirit simple leave room improvement bandit model exist bandit bandit q gap modify statement blue focus arm experimental design budget fix setting bernoulli illustrate bandit difficult right budget star set circle probability logarithmic probability carlo purpose plain line straight log set report elimination form three exploration provably pac rate almost pac bold green symbol specify symbol function slope complexity conservative time comparable rate significantly run maintain failure empirical symbol symbol testing seem error sample introduction number line stop matches slope huge gain use use prohibitive bandit algorithm exploration rate elimination stop rule stop empirical exceed exploration plain rate rate elimination bandit rate mostly coincide elimination sophisticated strategy experiment pac relatively easy compare budget pac algorithm pac case green observe probability algorithm usually sample budget pac design requirement budget draw arm preferable counterpart much bad exploration whereas predict budget difficulty provide principled way budget stochastic bandit complete testing sampling match certainly generalize gaussian show stop criterion comparison complexity behavior specify alternative confidence ie sub dependent arm confidence set algorithm perform arm notably gap investigate analysis improve budget understand complexity arm greatly identifiability common alternative bandit introduce element relate bandit differ expression simple change distribution provide proof let show conditional jensen successively arm ratio rewrite combine inequality extra ingredient provide low sum error probability absolutely measurable optimal arm order problem resp algorithm absolutely respect write show apply inequality q conclude exist show induction statement eq gx initial assume let measurable nz hence statement let briefly rule recommendation arm sequentially number arm give generalized distribution identifiable bandit eq unique let time whereas likely happen thus little precisely inequality tt ta define display construction picture convexity family kullback divergence may bregman twice argument admit relate uniform confirm indexing natural bb equality achievable elimination chernoff tt show pac upper last inequality go go one dt upper exist one get follow whose help bind quantity follow implication true q proof suffice map ss ss ss conclude optimal find region independent random ts suffice hand side infinity small proof omit every subgaussian super martingale sd dx stop confidence separate mention note involve exploration rate bound conclude use stop q large event show prove conclude bandit event give note algorithm every let optimize possible satisfying bound static arm rely family g interestingly proposition apply strategy arm theorem multiply chernoff random direct computation therefore one show b expression x ng arm contradiction exist h b propose change arm assume bandit bad easily armed
small high normalize document target choose hold subject loading automatically test geometrically spaced loop sequential due fix resource could screen hence main sequentially read disk ram hold feature hold termination exhibit use automatically select step performance open one average automatically range plot screen key test structure allow quantitative student fellowship innovation fellowship fellowship university award city distinguish gold wang receive electrical high university receive ph university focus signal distinguished award city fellowship engineering receive sc b university ph electrical university wu engineering paper seminal paper system com university award interest process learning prediction medical fmri definition wang lasso problem combination vector variety dictionary screening quickly identify subset receive solution resource speed solution intuitive understanding screen illustrative dictionary screening respect dictionary heart many column field term lasso seek serve lagrangian constrain formulation extensively signal vision literature introduction prove application range recognition recognition speech classification recognition classification text document dictionary iteration bottleneck address context classification zhang et collaborative scheme improve speed face application xu collaborative superior representation solution considerable recently target screening quickly identify guarantee remove reject solution appropriately zero solution screen run mode significantly dictionary solve lasso second reduce often gain lasso moreover since solver conjunction solver heuristic select voxel statistic fmri fan excellent review base feature formalize approach algorithm similar screen elastic net logistic sis false seek column false spirit remove non bind screening test call examine focus close problem screen remove approach result intersection elliptical dual execute require memory seek strongly moderately significantly reduce lasso focused concentrate screen efficiently full moreover foundation apply regularization exposition within development feature exposition geometric developing emphasize architecture examine intersection half consume execute test space examine region test describe carefully perform study scheme screen line successfully size allow screening begin review basic tool especially interpretation screening detail region form region plus hyperplane sphere plus hyperplane show spherical iteratively refined basic test screen screen screen eq analysis minor throughout instance nonzero say feature target objective problem result without accounting define purpose screen use particular enhanced parameterization solution dual via appendix contain origin problem nonempty form half space fig feature eps eps dual feature unit two feature unit function seek set inequality contrast may unique call lie ss dictionary column primal solution satisfy primal partition feature select index reject vector solution screen virtue memory hence without screen normally metric select solve suitable denote solution reduce problem assume b inequality simple worth state active solution resp solve resp conceptually obviously solve screen unnecessary hold resp resp create partition dictionary logic iw region potentially implement construction partition compact convenient encode partition region rejected select denote reject special follow region reject feature particular form region bound region spherical arise half space define onto ball c know give screen simplify depend subspace span problem constrain increase test reject also consume execute subsection simple sphere test insight test bound closed block close expression tt r screen lasso also theorem parametric n st rr sphere well screening want computation performance hence answer outline dual feasible point radius bind fig solid require specification call example feasible homotopy algorithm solution path actual instance sphere sphere st spherical variety test quick place test exposition safe sphere assume improve default center spherical comment far test core center projection onto strong sphere notational simplicity r sr sphere false rule fraction advanced version rule rule assume solution form residual also rr radius yield false sis intend nevertheless translate sis sis dictionary marginal b sis criterion lasso decide default spherical sphere algebra sis sphere particular region test spherical illustrate brevity form hyperplane hyperplane sign hyperplane convention indicate simple geometry relationship ensure nonempty sphere hence require subset sphere half intersection nonempty lagrange primal solve screen consist half sphere area nm screen u lt dt cv uv theorem continuous check calculus test insight test situation two function ds boundary test reject extra rejection bar rejection discuss nonempty sphere half contain ensure proper test reject disk maximize select spherical simplify boundary hence hand feasible base intersection bounding sphere check angle fig safe specifically screen triple exploit employ provide entail specify sphere solve evaluate scale close safe favorable circumstance refine bound obtain dr radius smaller tight spherical summarize cd provide suitable half space tight bounding reduce select maximize current since proportional residual make one space rejection sphere examine examine intersection bound two half examine allow stand trade rejection efficiency h ic half space sphere form parameter ir ir I r half intersect nonempty sphere half solve yield correspond na q region q theorem correlation I assume provide half bound c seek half select maximize radius feature sphere simply alternatively solve yield must algebra inequality f side compare examine rate increasing impose test unit proof generalization result general sphere constraint bound special strong claim test complex alternative nonnegative composite similarly q call implement arise also include illustrate calculate product iterative execute product use complexity feature test test composite mathematically region test weak demonstrate intersection intersect compact trading ease despite limitation sphere st st construct spherical default spherical ball green region circle test implement sequentially key innovation base ellipsoid ellipsoid ellipsoid fashion intersection half ellipsoid volume except refinement rather tight spherical encode test dictionary refer shot screening test equivalent rejection primarily obtain bind alternative screening examine idea safe previously solve instance dual instance help form sequential value specific et performance sequential screening geometrically outperform uniform shot test rejection power propose homotopy solution solve homotopy potential use homotopy variant homotopy feature loop via solve sequentially instance help continue solution merely help sphere center fact bound open feedback adaptively sequence instance scheme robustness feedback diameter rule ensure stop decide feedback scheme dual regularization effectiveness screen v f ft ft nonnegative logical evaluate h I ht j j r lt k r ti k screening test describe require logical indicating reject algorithm fashion product hardware computation running dictionary unnormalize pass normalized recommend simplify set unnecessary point operation f select feasible half selection evaluation basic implementation keep
localization art inference widely quickly intractable truncation meta model latent accord function property interest result specific crucial underlying volume state visual general idea function property least connection idea neural furthermore context researcher inference efficiently implement model notably design suitable challenging require knowledge considerable implement particular major contribution paper propose approach completely specific adaptively simultaneously em emphasize underlie accurate importance preserve selection gps analytic shot base adaptive target efficient et meta inference generative gp number benchmark code code spike equal pick spike case linear regression posterior shape mixture regression expectation reveal map publish translation match straightforward implement truncation extension maximization em optimize step adjust maximize g become variable mean ps n exceed turn subset relevant truncate otherwise construct rank relevance try marginal probability correctly exist predict prediction ni probability state word relevant state sort relevant latent per mass latent space hand posterior summarize occur use compute posterior calculate wish generalize em graphical want free way datum flexible exact concept marginal use gp iteration em mean contain information receive simply one loo compute gps chapter namely relationship early posterior apply probabilistic hand sample inference approach sparse generative dictionary point multiplication g consider latent maximum center posterior yield g latent n h singleton bar repetition recover basis converge truth frequency furthermore flexible relationship reasonable apply cluster target n em previous rbf covariance visualization first gp selection cluster select latent show rbf selection assign rbf initialize randomly gp datum approach successfully easily find assignment quickly away identical converge result figure rbf assignment solution relationship input priori strongly role success likely potentially suggest flexible diverse column mask third mask predict location grind show model function figure verify graphical challenge computer cope object object location latent object rely invariance apply construction mainly location appear predict next reduce construct accord object limitation tune scan costly function predict possible component could maximally regression image example pixel initially seem model dimensionality perform fast original selection scan make distribute scene rough entirely idea go convert information object selection exclusive component vision background randomly appear image generate optimize em use truncate successfully object mask quantify learning location distance hand accurately location selection avoid explicitly whole selection gp enhance speed infer cpu core inference cpu core gpu parallelization propose achieve fast approximate relevance
proved prediction define minimal borel estimation another way hope sequence theorem individual predict borel suppose lie unbounded compact considerably argue future convex ergodic surely loss algorithm asymptotically extended version article sketch ingredient mainly state lipschitz perform well ergodic infimum borel borel make thank let variable defer extend article space complexity involve design priori consideration expert inefficient practical restrict expert cost loss convex lipschitz result long material tree omit main body suffice index range rr calculation eq proof substitute induction bin besides decomposition bin child conclude tree depth e one node leaf select leaf replace subtree root inner node therefore binary tree conclude simple forecaster hoeffding jensen substitution sum first entail continuous uniform lipschitz see borel exist since inequality conclude true subset equip every every proof ergodic begin martingale resort eq integrable entail right term infimum set borel measurable super martingale sigma jensen inequality implies surely state france online prediction ergodic process lipschitz bound ergodic time learner ask prediction next bound stationary ergodic process knowledge function consider limit strategy surely loss estimation try design review forecast vast majority like year neighbor estimation window expert adopt sequence main individual sequence observation ask step predict knowledge past past forecaster cart finite forecaster observe forecaster predict environment cm forecaster suffer clean layer advantage computational efficiency discuss remark square build lipschitz function occurrence part implement maintain associate deep simplicity set precisely lemma good x argument get sum conclude pt forecaster two time respect argument imply constant bad small estimate know individual sequence see expert respectively vanish average respect case bind instance loss upper em one consider bound absolute define calibrate online rectangle draw fill white fill r label rectangle rectangle rectangle rectangle maintain region space pair integer refer node convention child denote associate terminal thus step associate predict local observation observation receive leaf tree update two cutting give root first coordinate split go split third histogram near neighbor ergodic process unable deterministic neither neighbor manner histogram histogram divide strategy predict partition optimize number bin trick regret bind function computationally inefficient happen height distribute datum allocate space observation un yield improve need online substitute diameter splitting proof control locate index diameter associate em basically proof induction defer recall inner node splitting step h em total appendix article tree exactly nod depth node em bind dt th start exist inner therefore get thus conclude cumulative regret sum cumulative incur lemma loss incur satisfie jensen inequality conclude section later forecaster sequentially step ask forecast strategy form vanish increase regret htbp forecaster feed feed straightforward corollary integer combine basically form
excellent quantitative software obtain piece literature raw text source make list character number character one process string ease software preprocesse handle text text scalability classical text employ alphabet author also class standard subsequently implement represent believe document create choice text gram simply object respective feature specify gram construct functional gram frequency store dictionary enable text information character information must create add text frequency call elsewhere software package major lie automatic identification wherein author influence simplify problem approximately detect give capable representation expressive practice establish equal capable produce document correspond software seek subsequent accord stress allow specify two computational many relationship previously calculation relationship module allow upon object metric query operate search character document length document document document decide query part feasible within time formulate connection document obtain document document sample define size ease create document return one determine similar text similarity misclassification interest similar influence question put text rate use combination two text tendency text l l c misclassification classify text many true text close hyper text usually within hyper consistency hypothesis identify interesting result intuition training text differ variability author text due suggest vary greatly notion loose word word author write intractable effort analyze one gram coarse tool face classify broad grouping grouping categorization book remove direct variant classify classify preliminary hypothesis fact heavily rewrite variant sound explore fact identify direct direct variant variant negative broad analyze text gram magnitude small gram sized true style however believe quite due short test probabilistic uncertain viewpoint case preserve heavily method preliminary finding true origin text software perform demonstrate efficacy open field also recommendation several arise result develop hypothesis develop rely author analysis hope confirm efficacy text several progress exploitation yield upon train avoid knowledge support area classifier author highly area see author focus existence extent author influence many style establish piece text behind belong text claim may characterize text write author form paper instead propose text computer science result side computer insight figure classical represent subject apply quantitative brief historical seek address quantitative bc parent day early evidence suggest choose regard major play death interest play great work either lose corrupt play effort original copy continue much year play heavy ht claim capture attention induce gram representation idea sound word allow additional produce author pose capable heavily modify proximity death fourth death study reflect power closely write death european numerous particular consensus despite closely play leave mark history fall city great great span bc historical writing little actually life significance body record require historical author time rather art bring remain extent fact underlie attention believe interest ability fundamentally ultimately character capture text sound encourage importance build character character text letter alternative gram implement inverse frequency purpose identify document importance word receive high text corpus term corpus belong adopt probabilistic achieve representation make appropriate probabilistic able robust quantify useful author entire space definition discrete follow study mahalanobis metric standard average common text svm anomaly text author algorithm whether piece text write solve point offset hyper slack kernel length vector take lagrange multiplier use lagrange multiplier utilize rbf kernel degree small moreover rbf project infinite expressive separability experimentally derive determine text author author metric document computationally expensive match possible length character length document comparison expect time consume string query wish follow character demonstrate begin
tail satisfy tend word particular embedding mention recover embed multidimensional follow classical scaling scale estimate decomposition diag dd specifically tend pt hard see define formulated cone program solve practical scale instead show euclidean matrix take devise compute shall alternate algorithm refinement von alternate projection design intersection close sequence htbp xx q k projection intersection pos node pos pt node consider evaluate projection observe close alternate projection readily apply input evaluate specifically matrix th lead submatrix clear replace zeros efficacy numerical experiment motivated protein relationship counter activity mutation understand interest study derive infect follow five long sequence study alignment sequence substitution substitution fairly report analysis similarity convert dissimilarity distance apply left figure derive stress c noise classical medium shrinkage difference one run shrinkage multidimensional repeat bank symbol consist atom stress value report compare noise standard shrinkage medium shrinkage classical distance score atom distribution mean shrinkage distance reconstruct figure htbp proof equality follow together next trace trace n ensure imply minimum also exist follow basis word obviously contradict second argument point argument x follow f light imply statement pt nx nd r n characterize kolmogorov geometry conditionally semi orthonormal span form matrix v last clear recall last negative definite taking yield p fact imply na f r f nr n nr pt cm corollary definition recovering fill simple kernel apply shrinkage pairwise distance allow imply consistently number increase programming application embed euclidean multidimensional scaling regularization trace norm euclidean noisy arise context object amenable provide dissimilarity molecular standard measuring use al encoding insight metric respective employ method numerous dissimilarity successfully convert semi definite kernel play canonical multidimensional aim object dimensional euclidean object distance preserve al al chen others popularity extent embed reflect largely exploratory tool another interest reconstruct distance determination molecular nuclear short demonstrate chemical shift need three euclidean distance result observe translate location become euclidean distance occur goal graph euclidean embed embedding object domain molecular determination case realization score measurement call eq g stand convention identify suggest embedding obviously embed particular high refer embed dimension euclidean embed dimension molecular determination dimension similarly multidimensional realization q correspondence euclidean column regularize estimate al tradeoff goodness fit hereafter semi definite encourage low al many goal operate characteristic statistical define difficulty understand identifiable distance latter preserve translation estimate subsequently challenge notion kernel resolve associated kernel characterize amount shrinkage project pair onto subspace offer ability induce low characterization suggest use version thank structure et expense principal alternate explicit characterization establish discrepancy consistently approximate embedding rest section exploit duality matrix explicit characterization geometry cone efficient proof correspondence euclidean minimum despite identifiable resolve ambiguity minimum associated euclidean identify associated euclidean obviously q positive semi kernel matrix translate uniquely semi hereafter write embed result different unchanged reconstruction score avoid require embedding center embedding unique distance uniquely characterize among correspond trace embed pt restrict minimum trace map minimum trace viewpoint instead clear addition center kernel figure htbp column sep em rd rd similarly distance actually explicit distance whose diagonal projection soon onto
select user rate analogous establish item recommendation cluster hold constant completion accurately condition far hold completion eq imply recover rating hold show exist majority user name recommendation summarize necessary notion co co differently r rating similarity two u un u r clustered entity recover select user select preference vote e name high preference determine vote cluster co pair item majority vote recover step decide majority vote practice verify hold hold rich contain rich likely pick rich user select ii rich select cluster rich heuristic early combine three use similarity rich find directly user iii similarity user cluster identify user user compute similarity super super high work original detailed hybrid cluster recommendation hybrid user set select accord similarity user similarity set define super rating user eq select high similarity generality super user rating emphasize dataset phase transition user rich call ii denote select item modify select item accord define super majority item e select super loss generality super rating iv high similarity denote iv item rate mu un r voting reason rating item region rating region rating entry entry vote htb test netflix recommend user movie user word movie believe rate highly user movie rating movie receive comparison binary rating rating predict rating testing metric accuracy top movie model accuracy error correctly recover recommend continue recover preference view metric among metric restrict recommend whose rating give hide accuracy movie recommend error compare case user rate compare rate htb low significantly rate free un rating netflix noisy perform recommend error error error htb sparse rate error since summarize c c item item index cluster rate user user rich negative function constant positive rich rating rating rich user bound k theorem user cluster preference except rating item word otherwise separable condition change r equality hold equation calculate follow case rich p rating user preference rating case rich sparse rich pg define rich sparse sparse user cluster z chernoff consider user independently variable apply chernoff fact e p obtain e preference bernoulli apply bind cluster consider rating scenario rich scenario chernoff sufficiently large choose assume rich similarity z rich user pick user rich pick rich normalize similarity z normalize end rating eq abuse true preference item km km voting user q cluster occur voting produce preference theorem policy rating item cluster cluster preference agree entry except rating item verify user long change rating item cluster contradict hold argument prove correctly give cluster rating item abuse far rich otherwise define majority voting give chernoff verify hold co cluster collaborative filter rich user develop similarity entity item cluster recover rating matrix large netflix experiment majority user rate item basic remark furthermore give rating size correct thm lemma thm proposition wu collaborative filter website recover user user item cluster user item cluster algorithm noisy cluster recover algorithm well popularity friend netflix overall error recover importantly co cluster scenario recommender user rate majority noise add recommender recommend user example amazon netflix suggest like paper call netflix user large item discrete item rating view row item rating user goal recommender recommend may sparse mathematically pose unknown entry user preference filter recommender multiple predict user practically rating exploit solve item rating typically justify state write view user feature mathematical abstraction situation look movie actor assumption item group cluster provide item strong rating matrix column rank strong predictive power well emphasize actually verify justify study result assumption whose matrix agree observe know result rank problem popular heuristic minimization objective nuclear show condition nuclear reality operation fast minimization know portion mention know entry select repeat improve name know guarantee recently establish obtain assumption nuclear require purpose one view good date make apply rate one vote user cluster user cluster subset similarity user identify top user rate user vote optimal model think well behind rank simplify presence rich user item cluster applicable case user cluster cluster completeness believe dataset significantly item call presence sparse user well know rate movie rate movie good presence rich exploit completion contribution cluster rating information entity entity user comment apply item devise similarity dramatically improve find number rating user easily provable logarithmic cluster co achieve logarithmic mention verify rating even assumption clear contain user use exploit require user recommend combine similarity popularity friend netflix later propose low item fraction entry matrix denote user computation score computation computation total computation voting dominate regard since please section actual easily implement similarity majority rating notable algorithm rating matrix add score computationally rating basic find order preference user item assume level preference recommendation recover matrix cluster user u b separable user fractional separability preference preference cluster index assume receive preference hold fractional separability user receive preference observe assume inconsistent rating ask time generate illustrate pass channel create pass channel reveal true preference bias rating preference contain let user user information user item rich user
power weak setting get consideration power test outperform alternative test dense substantially test reflect discussion preferable capability maintain line case power test increase numerical result alternative strength gaussian long range strength nominal significance h alternative sparsity nominal significance distribute numerical test screen outperform test alternative spread structure maintain nominal good power recommend sample powerful application sample research large setting test generality nan hypothesis whereas let entry uniformly draw integer setting sparse magnitude entry diagonal pool weak follow use generate I v identically independent draw long dependence impose non generation mechanism study autoregressive ar identically j nj mt gamma z I impose structure simulation propose significance ex ex hc ex hc test hc nominal gaussian block covariance autoregressive process distribute model strength sparsity nominal block diagonal f level signal along test significance level strength nominal matrix result test summarize propose reasonably close relatively improve fails maintain dependency section hc maintain nominal small dependency maintain nominal significance reasonably control fdr identify set ns ns ns independently value procedure level type originally insight genetic also analyze illustrate focus patient group fusion employ approach scenario exclude set small number retain gene mf gene display biological insight development type clinical aim level sample test employ control significant matrix generate may error number set identify identify find gene propose testing discuss overlap within explain relatively large identify ccc ex mf screening stand investigate gene set test set activity disease category recognize development find connection type supplementary material list disease test screen fdr control association set may biological reaction test enforce structural assumption alternative extreme self whose extreme marginal statistic target guarantee correlation max moment unknown invertible covariance study may difficulty complex strong experiment perfectly correlate include power test develop screen principle pre ratio numerical superior sample screening yet maintain satisfactory power datum disease gene appeal preliminary classification setting feature broad kolmogorov discriminant powers supplementary supplementary material online technical proof result data pt section population testing procedure employ critical heavily rely structural therefore scope applicability enhance power test preliminary screening test gene practice screen gaussian test equality vector field particularly modern quantitative finance subject small furthermore measurement possess issue multivariate mean paper dimensional sample hypothesis control scientific low traditional extensively examine normality deviation asymptotic statistic statistic use test statistic aim norm detect relatively dense type preferable detecting signal medical problem anomaly testing derivation limit critical structural unknown impose guarantee w whose validity moment satisfy pc restrictive verify applicability asymptotic rely heavily structural expression pathway associate covariance real concern point extreme value maximum usually convergence although suitable concern genomic image extent least setting validity assumption approximation gaussian utilize however vector account dependence organize hypothesis test report assess datum supplementary material discuss preliminary screening propose throughout w nc nc n k diag diag n sample consist identically distribute observation set type form q refer statistic intuitively ns cv ns cv cv ns cv cv cv properly expect size cv alternative traditional calibration setting unknown nan motivate limit critical ns cv w ns critical cv cv ns w w cv f ns w ns w reject detail wide explore drive characterize closeness process test indeed test naturally sample statistic n nx k ns ns cv ns cv ns section follow quantile compute w two cv cv ns f test consistent require covariance develop test restriction grant wide applicability validity procedure entail estimator proposition include k supplementary material natural employ low coordinate uniformly r pn logarithmic allow depend heavy enforce wide scope extensive form long test operator still matrix empirical eigenvalue employ enforce valid much large value enhance power propose procedure feature expect irrelevant provide substantially power alternative coordinate construct statistic index exclude upon analogue ns k ns ns original statistic suitably select coincide probability power comparison test procedure maintain significance preliminary aim original statistic discussion advantage problem focus reduce h result ns k p f pd problem respectively asymptotic power thousand device efficiency monte mathematically ideal show nominal propose alternative mean marginally u unknown impose follow mild order th uniformly sub tail one assume establish validity test nominal significance asymptotically numerical base nominal sample small augmentation bind power test without screen eq agreement statistic region complement sample test screen test ns condition screen nominal either sample size distance test let hold sequence satisfy alternative asymptotic type propose base control significance asymptotically part iii imply two counterpart test covariance either condition nan part specify theorem analogously property pre screen identical omit supplementary material asymptotic sample screening test covariance assume nan h h simulation several evaluate performance screening ease exposition one problem test hereafter hc hereafter three hereafter higher
consider tail master tree worker compute master evaluate improve estimate report back master meanwhile master constant master assign execute path estimate worker proposal become likely worker unlikely likely pick worker posterior subsample depend track difference value master difference empirical approximate correlation high actual decision implementation require master worker master basically bayesian framework target master core evaluate core core core intel processor serial worker problem target eight gaussians model molecular real real functional experiment spherical distribution provably convergent adaptive study expect perform point evaluation phase early proposal reject approach target outcome inherently predict estimate uncertain incorrect divide batch subsample min max result mixture run sequence vary worker produce chain cumulative speedup iteration burn compute two chain table phase obtain speedup speedup achieve burn efficiency drop achieve logarithmic speedup sub logarithmic number round explain whole range initial worker burn proceed cumulative fall logarithmic speedup difference scheme burn overall speedup burn necessarily decrease monotonically initial region truly region speedup maintain system mm burn predict predictor burn burn predictor quickly correct predictor vary almost evaluate typically wrong eventually opposite incorrect speedup figure achievable speedup molecular gaussians evaluation accord convergence condition step drop logarithmic speedup improve switch resource leave present inherently serial often transition countable predictive use parallel require focus mh predictive predictor accept effective predictive estimate prediction respect evaluate correlated variance difference state evaluate higher prediction justify resource great execution I achievable sublinear core logarithmic achieve liu helpful award health lm google e large carlo exploit approximation chain parallel accelerate without subset available exactly equivalent serial initial burn serial core model modern learning appealing represent latent real rarely amenable require approximate form target may challenge new inferential target density examine exploit taylor process regression randomize development stationary low factor evaluate arrive attack parallelism difficult mcmc hasting inherently modal chain decrease achieve sometimes chain attack use execution approach sometimes attention past decade seem iteration stochastically reject randomness initial future chain think single tree immediate effective challenge correctness exactly serial treatment pseudo randomness serial treatment risk introduce scheme require core speedup improvement speedup acceptance rate reject improve speedup heavily extremely speedup something still scheduling use speedup relative adaptively adjust acceptance available fast approximation though learnable schedule increasingly far improve approximation schedule approximation insight error small evaluation evaluate large expensive current incremental show synthetic system parallelism speed serial unlike system achieve near speedup core speedup eventually logarithmic core hard chain evaluation incur evaluate determine acceptance move slice expansion focus case expensive sometimes many expensive arise easily decompose item e achievable cost aggregate partial accelerate source parallelism class sampling ensemble accelerate share chain parallel chain parallel implementation generalize elliptical slice mcmc algorithm use parallelism execution accelerate idea literature core evaluate slow result I scheme core tree respect core node maximize depth summarize idea static fix acceptance version context anneal acceptance level tree computing estimate mh alternatively identify perform combine source parallelism obtain mh core usually improve scheduling exact unlike stream mcmc incorporate estimate acceptance tree proceed use external number generality uniformly hypercube back operator hasting slice etc countable qr disjoint setup metropolis hasting tuple try mh delay mh create large variant slice sequence converge elaborate usual intend purpose separation generate highlight point randomness view separately generally case evaluate candidate burden mcmc observe pseudo function evaluate tree yet reach node eventually remainder straightforward point uniform proposal correspond highlight leave evaluation core speedup iteration proportional fall scale perfectly evaluate proportional making turn whether indicate
loss embedding sa word loss sa rd say th pair token call word frame word train model variant adaptive magnitude use accumulation past slide window hyper momentum precision small use setup though update thereby implement train dnn stack unit output evolution cosine show cosine train overfitte unsupervised system necessarily like unit phone appropriate hmms make phone linearly etc discrimination pairwise token token linguistic belong b category minimal shape keep center phone vary cosine embedding discrimination p share performance ignore remove layer vice discrimination base order identity fully dnn phone label task show correct understand nature encode detail unit within corpus compute take across phone phone category intuitively large encoding median kind phenomenon regard code layer phone unit layer doubly code third ie task reveal code inspection phone reveal doubly rather discrimination suggest layer example would inspection code red phone relatively localize localize network discrimination moderate information different type need complement phone representation european pour france de de paris de france dim et train task speech possible share discriminate second discriminate two theoretically linguistic plausible acquisition put language recognize discriminate construct representation propose neural word help acoustic phone embedding
estimate show excellent term demonstrate gene set counterpart rna seq form q library genomic region genomic henceforth gene many poisson datum choice marginal gamma poisson rate estimate model gamma conclusion allow see sample appendix group crucial component prior differential de distribution determine jeffreys seq primary trivial de great distribution also g gene close joint resort carlo cycle call distribution set conditional dispersion parameter metropolis hasting posterior fortunately choice integrate define tv far discard step step setup sample iteration ghz bit processor software require assess publish take rna seq count drop assume calculated coverage posterior derive sub coverage indicate accurate dispersion h red star posterior de show posterior absolute de run drop count parameter star gene heterogeneous gene know ground improve upon exist method detection underlie truth generate consist divide gene choose small gene gene de individual expression count expression set individual level consider simulated dataset compare package receiver operate versus threshold take approach express perform well de much rich thing distribution parameter gene addition output analysis next distribution set gene difficulty posterior indicator input group competitive differentially express gene problem gene count rarely determined attempt accounting difference gene cause uncertainty describe posterior henceforth ff testing significance follow fisher de gene set roc considerably ff g develop bayesian rna seq count proper inference uncertainty sample change mass detection analysis conceptually suffer bias frequentist demonstrate show detect thank several anonymous research center molecular joint integrate update measure distribution follow bernoulli distribution contain article closely stage variation article binomial also poisson type describe fairly typically conclusion gamma pp update heavily implement binomial possible resort update account fact follow burn phase propose iteration previous iteration compare section main indicating hence leave simulation setup model th unit effective ess ess sample amount mcmc autocorrelation measure effective binomial even run account high accuracy two ess min try negative binomial hold severe mix dispersion identify sequence differential expression sequence posterior inference efficiently appropriately uncertainty account differentially gene posterior excellent detect package interface rna become
refer cluster similarity shape hard specify past cluster able shape initialization lead decide specify initialization clustering remain challenge many well clustering exist mainly limitation firstly access feature low probable significantly clustering even ill clustering secondly cluster unified limitation ensemble crowd explore base validity measure crowd agreement clustering unsupervise manner treat aware triple similarity regard common neighbor reliability clustering linkage unify term gp comprehensive literature motivation provide propose term accumulation capable ill incorporate conduct several follow review work technique ensemble crowd source aware triple propose consensus term link cluster ensemble combination aggregation aim clustering base cluster member ensemble step clustering give step clustering consensus base clustering cluster different initialization via repeatedly via project generate base clustering I consensus important past consensus category co iii base partition maximize partition clustering median problem complete finding huge partition genetic cast find algorithm median median clustering embed perform convert take consensus clustering time base clustering evidence accumulation co association agglomerative sl co association li analyze co novel utilize normalize wang accumulation take category partition clustering hypergraph structure cluster partition similarity partition hypergraph meta cluster formulate clustering ensemble bipartite node exist cluster partition disjoint set node ensemble approach implicitly assume clustering contribute low clustering ill clustering weight base clustering validity et exploit validity index vi connectivity ci si index di assign weight ensemble extended weighting partition need vector suppose many framework li clusterings optimization process deal lin cluster library partition selection weight preserve partition ensemble weight accordance cluster cluster fuzzy ensemble aim partition hard specific different cluster partition intersect cover denote cluster refer base initialization th denote look provide partition final partition solution generally refer consensus ensemble base diversity dataset construct ill clustering may significantly poor clustering one clustering regard evaluate quality know truth develop wu validity inter cluster intra cluster al deviation connectivity quality instance cluster connectivity often neighbor utilize coefficient evaluate coefficient measure instance cluster whereas average instance inside applicable suppose formulation clustering utilize distribution ensemble crowd quality individual science crowd consideration collective opinion crowd expert ground truth truth suppose crowd clustering compare opinion crowd individual base clustering crowd agreement clustering denote base agreement crowd member compare crowd member crowd agreement crowd agreement basic quality collect opinion crowd hold mutual base suppose ensemble connect triple cluster reliability neighbor share common cardinality coefficient consideration sharing measure cluster base cluster intersect intersect common al utilize justify reliability view source cluster quality cluster quality base cluster consider correspond base propose aware connect triple measure regard reliability cluster accord greater big influence influence clustering I l coefficient cluster thus coefficient cluster common compute maximum cluster similarity define coefficient eq two consensus utilize deal ill describe accumulation link assign specific cluster cluster original affinity assess co occurrence ensemble clustering ensemble instance otherwise similarity pair occurrence evidence accumulation similarity matrix idea matrix reliability clustering assign quality co follow eq cluster definition thus clustering map utilize pair wise occurrence reliability member co construct agglomerative consensus clarity htb accord build agglomerative obtain consensus cluster clustering lack ability treat partitioning formulate bipartite compare ensemble base distinguished aspect firstly utilize crowd agreement see exploit among base quality clustering unsupervise secondly integrate similarity bipartite instance treat link common bipartite twice two instance link link node link regard link via section instance cluster use incorporate clustering exploit reliability via crowd estimation regard graph utilize node disjoint treat possibility disjoint lead come probably force link contain together clarity method initialization evaluate eq bipartite construct partition cluster cluster output eight baseline approach criterion parameter discuss section base section bit windows server intel processors gb ram experiment eight world uci repository use namely benchmark evaluate quality consensus information share clustering truth instance instance share scale link cluster influence big gp r suggest cm clustering base clustering construct base pool outli pool base clustering repeatedly parameter initialization cluster hierarchy clustering width trade paper cluster thus cluster pool baseline approach apply base clustering pool experiment choose base clustering clustering ensemble ensemble ensemble evaluate performance combination clustering propose gp partitioning whereas pair wise co agglomerative cl sl cluster sl drawing base pool ensemble ensemble repeatedly average cluster bad base cluster ensemble clustering ensemble fig consensus clustering clustering clustering dataset clustering win ensemble clustering totally clustering clustering run tie count win associated base consensus clustering number clustering cluster benchmark c gp cm cm true true gp cl sl cl sl sl cl sl cl best cl sl sl cl sl sl cl sl cm true gp cl sl cl sl cl sl
xy test derivation rotation axis value rotation align estimate dependence three degree dependence corpus relative language block combine expression comparative genomic qualitative phenotype illustrate relative b large dependence varied median pairwise sufficient although dependent great demonstrate due tight concentrated becomes predict powerful ccc e plot repeat draw dependent size powerful demonstrate world parallel european corpus documents es english da broadly language variable first statistical language dependence language group remove stop word apply tf feature representation kernel bandwidth per distance dependence find language cccc fr da language high group language ccc source da en fr en fr es es solid rate despite advance child survival year genetic depend location biological process tumor location order hypothesis treatment tumor obtain child newly paris organize block block category third contain segment characteristic variable median pairwise distance tumor expression empirical dependency finding literature support tumor location determine source dependent build hilbert schmidt strictly powerful order demonstrate test performance identify language determinant dependence wide dependency currently framework construct author fellowship european fp describe novel two determine variable second measure schmidt criterion measure powerful independent unbiased statistic favorable property quadratic time matching effectiveness real identify language corpus tumor dependent dependence statistical many context dependency research non string diverse covariance correlation covariance instance test ranking partitioning approach problem multiple dependency dependence present visual brain automate source language match one language respective learn basic statistical determine two variable dependence third hilbert schmidt independence covariance relative dependence take measure derive joint test utilize result hoeffding determine statistic variance statistical construct uncorrelated subsampling synthetic language identify relative compete determine whether statistically question statistically nonlinear relation however address influence closely detect factorization notion schmidt work independence associate covariance uniquely express determine respective like separable xy unbiased q ij sample write u tuple draw u eq uniquely reproduce hilbert source target xy xy r xy xy xy statistic asymptotic consistent computing simple attractive effort implement correlation result respective kernel associate uniquely reproduce space estimator u statistics respective r xy variable follow joint base test p xy conservative estimate quantile achieving counter integrate result first axis rotation conservative xy also measure implement variance consistent converge calibrate even compute form collect combine unbiased empirical bound everywhere negative sample give statistic prove population eq test relative two equal sized set denote drop pair first pair x determine space
time despite near dynamic computationally allow gp accelerate dramatically simulator posterior population individual conditionally give summary collection invariant simulator design sample accelerate first wave find good polynomial wave exploratory determine rough log set threshold truncation error gp take replicate diagnostic guide selection accuracy gps improve successive wave reflect decrease validation report predict region accuracy application wave model increase wave wave wave space figure approach gp approach require evaluation chain probably accelerate similar ridge difference two scale usually accelerate examine evolutionary biology specie divergence intractable demonstrate various methodology branching unobserve randomly consist acceptance cutoff make acceptance simulator estimate scheme due successful wave simulator replicate solid line figure could run abc differ red expectation plot various flat ever mass near dot posterior obtain local adjustment estimate substantially improve abc trend decrease gp accelerate knowledge impossible expensive perform simulator evaluation monte gp allow universal degree great many model supervision build diagnostic wave gp building poor poor model design gps raise gp number calculate produce time simulator gps implementation upon reduce carefully example number simulator replicate location detail carlo exchange enable bayesian distribution calculation process computational determinant accelerate continuity reduce require computation approximation evaluation population computation collection complex simulator model phenomena parameter return output simulator abc enable simulator require range primarily biological science nearly abc complex abc prior realization simulator accept tolerance return conversely tolerance resource key simulator simulator rarely simulator run simulator computationally abc require hour computation posterior moderate extensive explore mcmc sequential monte smc previous simulation know function guarantee learn gp accelerate abc method resource gps accelerate idea successively rule build log find simulator abc abc exact one intend replace step proportional acceptance kernel get uniform abc interpret believe represent relate simulator measurement observation simulator discrepancy mass believe simulator distribution approximate monte repeat begin build simulator output instead project e base summary statistic relate function estimate simulator evaluation indirect indirect auxiliary approach mapping accelerate abc majority smooth informative return mcmc simulator test likelihood vary positive instead estimate ia priori mean accurate inclusion prediction design take mat ern length scale likelihood variance variance estimate help avoid identifiability improper inverse prior integrate analytically maximum ensemble multivariate update train value simulator carefully order minimize point simulator need quasi discrepancy advantage extend number monte carlo smc support manner place region space translate prior priori design complex certain maximum order posterior accurately capable predict sequential iteratively region build threshold discount side leave gp likelihood still gp currently degree multipli trade accuracy use cause wave extend determine simulator gp draw additional new simulator new together build gp model prediction wave far whether fit
sampling condition number search convert correlation synthetic experimentally validate network distribution count topology strength quantify assess method pearson benchmark model american module come round round datum contain filter remove sequence total small zero require round histogram justification parametrize topology representative successfully association well whose architecture respective range control association relationship condition type size distinct instance generate maximum fidelity method range covariance selection selection refer design compositional baseline reference pearson neither robust estimating determine however interaction correlation include code case ability precision curve rank accord confidence pearson prediction rank infer star selection summarize condition sample topology dimension blue pearson random area precision recall vs different bar one sided test line trend certain perfect follow degree reduce infer scale free high maximum band scenario significantly pearson correlation outperform compositional limit recover portion test well synthetic hundred curve rank final date nonetheless high confidence interaction edge precision representative suggest outperform art network test scenario superior accurate connect component length abundance help cluster versus topology incorporate recover network regime prediction edge rank edge stability pearson synthetic edge true topology color correspond kl bar side degree define number edge figure show degree type scale characterize exponential degree interaction cluster relatively reflected measure predict size topological centrality degree centrality centrality short path unique four method agree core network term salient feature unity figure comprise edge distinct negative association dense correlation negative method comparison total unique edge scale edge prediction respectively network eight respectively predict edge property distinct centrality scheme interaction observation explain edge indirect due alone indirect edge explanation hamming suggest american attribute instead band free cluster type component infer interaction specie understand ever number sequence study strong environment diverse study alone develop interact specie environment throughput compositional interaction construct challenge estimation inference interaction dataset generator underlie engine know sparse covariance robust transformation context abundance realistic look dataset benchmark inference two benchmark demonstrate addition sample agreement band number also demonstrate direct correlation community rough inference assumption underlie experimental design statistical synthetic datum term water american inference network reveal observation network appear composite network evidence scale band like important advantage neighborhood ability knowledge scientific principled manner grouping relationship improve verify specie interaction context covariance neighborhood similarly network agreement empirical confirm free structure network globally scheme scale network include although interaction covariance key address question example design gene area reference therein interact sequence could incorporation understand evolve association structure development association develop might perturbation art inference rigorous addition flexible principled mathematical incorporate association improve prediction serve sophisticated modeling hypothesis relevance environmental acknowledgment thank alm discussion manuscript presentation work cm abstract rna environmental sequencing population diverse environmental identification require tool challenge unit dataset compositional count detection relationship spurious secondly sequence hundred hundred association additional sequencing address combine develop compositional inference assume reconstruct synthetic benchmark validate tool generate datum state synthetic scenario predict american project sequence interaction routine component experimental biology collection american project bring recent research biology aim objective community unobserved community model appear lead concept steady study environmental covariate relate context observation disease status new connection infection diversity goal interaction detect typically association measure measure sequence result read common operational quantify proxy population environment population successful pearson correlation sufficiently survey spurious count total community term classical fail method compositional correct permutation design compositional bias yet correlation association correlation arise connect expand point pose hundred whereas hundred assumption development great sequencing challenge increasingly dependent simulate influence via datum draw negative accord pearson correlation green negative red thresholde relevance edge threshold notably importantly underlie inverse sample symmetric approximately correspond non entry color identify true correlation induce strong e correlation node although metric inverse depend help potentially introduce generation realistic comprise compositional apply transform unlike seek concept independence informally e abundance graphical conditionally independent relationship explain alternate avoid detection correlate ensure detail undirected link node represent gain considerable popularity network biology recently biology art synthetic generate realistic synthetic network diverse topology date verify gold exist synthetic reflect actual strongly impact network recovery performance scalable engine I feature realistic benchmark network apply real iii invertible agreement theory underlying method american project likely membership ii topology section present statistical current available material synthetic datum module summarize key introduce describe generate realistic dataset h consist synthetic datum count topology statistical suggest fit marginal generate data proceed synthetic count pre ratio compositional select graphical mb glasso assume sparse correct subsampling dataset low variability select edge output invertible input discussion typical sequence w nm j raw j pm composition unconstraine simplex lie simplex restriction simplex application covariance covariance exhibit closure advance achieve simplex ratio study compositional statistical term statistically equivalent transformation unit compositional gx mean composition vector transform component covariance transform absolute j j j dimensional sample serve serve basis abundance add pseudo avoid association abundance dataset network association undirecte represent association formal unknown encode variable undirected field family entry term precision adjacency factorization conditionally conversely entry conditionally inverse thereby association fundamentally distinct estimate correlation though highlight biological b two provable dimensionality neighborhood solve maximum reconstruct optimization key tractable provide reasonably association comprise inference scheme selection introduce mb inference independence denote column follow tuning aim necessary local edge entry view choice consistent neighborhood present selection read element scalar tuning expression distribution graphical encourage sparsity diagonal mb originally normality distributional estimator large problem include inference count nonparametric approach additive model use association transform inverse pd matrix ensure entry estimate infer value sign diagonal inverse wise covariance approach advantage obtain associate subsequent edge discriminant parameter control final rather empty criterion criterion scheme ability star repeatedly subsample incidence retain overall stability star empty edge accord inverse practical advantage theoretical available characterize asymptotic infer topology neighborhood precision underlie recovery interaction network highly nod evenly neighborhood addition theoretical implementation infer practice scale grow advance increase engine rely star abundance absolute comparative scheme remain biology reverse considerably advanced understanding applicability gold experimental realistic context gold realistic synthetic generator outline generation fit count specify e topology combine normal generate user topology approximate structure univariate function normal correlation cdf
boundary element equation dense recursively divide base structure low matrix depend tree instead article restrict efficiently arise fast applicable characteristic distant though dense employ store vector summation oppose factorization represent interaction several arise green compute expansion chebyshev interpolation entry low chebyshev available rank find algorithm solution software package relate fast article black box online matlab com reader reader box note exploit sparsity operator run box compatible wide eq chart demonstrate scalability usually highly obtain chebyshev improved chebyshev product one illustrate compare kf enkf assimilation implement continuously track co result model pilot reservoir simulator pilot co depth core assume predict co pressure wave build flow use baseline wave co vary co velocity delay conduct survey every apart acquisition geometry integrate propagate reality ray assume line connect source receiver limit assume due co express vary location invariant represent cell co induce velocity delay measurement background step perturbation observation delay contaminate ratio snr eq noise realization equation major source walk dynamic set assimilation monte enkf parameterization noise structure uncorrelated kf enkf assimilation enkf give reliable quantification spectrum eigenvalue step kf enkf eigenvalue eigenvector insufficient rest tail yet kf effective covariance assimilation rank number need explain total enkf suffer insufficient ensemble information embed kf enkf kf expense cost assimilation method computation pc ghz kf hour minute tool fast kf storing overall operate comprise offline online part offline measure form compute monitoring normally form associate cross cost storage enkf linearly enkf produce gray propagation walk gray kf enkf kf enkf gray risk ensemble linear forecast forecast kalman filter quasi assimilation datum rapid adopt random forecast gaussian error generalize covariance algebra propagate comprehensive kf enkf select chebyshev interpolation kf within less computational effort enkf realization ensemble carlo spurious estimate cross variance drive quasi monitor acquisition network forecast advantage full forward rely less dynamical walk kalman solution enkf well enkf linear pde forecasting rely increasingly collect desirable kalman continuously quality monitoring material work support national technology award advanced inversion modeling set national mathematical award dr berkeley national physics global energy stanford quasi datum assimilation kalman quasi assimilation continuously movement challenge tracking collect advantage provide continuous high flow aid analyze impose monitoring assimilation computational requirement paper filter kf dramatically reduce storage kf produce practically take dynamical tailor assimilation problem apply co enkf numerical enkf demonstrate usefulness walk monitor progress field operation co track enhanced controlling monitoring provide sample monitoring use large temporal vast monitoring collect quasi continuously continuously temporal resolution co important exploit temporal result challenge analyze arrive high resolution flow time monitor use space process algorithm kalman kalman kf powerful processing arrive continuously improve dynamical kf give good solution kf represent quantification characterize maximum posteriori extreme quantification crucial inform decision data kf kalman filter size meet computational requirement high operate matrix computational limit kf coarse heterogeneity kf kf still reduce reduction particular work type kf project reduce dimension resolve approximate rank reduce singular seek matrix correction direction ensemble kalman filter enkf find low construct root matrix method gain popularity efficiency storing covariance reduce kf dramatically approximation carlo method approximation slowly size enkf statistical enkf computationally sample size ensemble although like version enkf reduce enkf fast fourier fast method accurate alternative reduce structure associate allow reduce rank near inversion krige covariance generalize matrix solve approach spaced grid realistic hierarchical incorporate develop computationally filter present employ walk widely medical dynamic monitor rapid accelerate kalman computationally enable cost kf accurately reproduce minimum mean kf processing cpu minute feasible implementation approximation filter tendency filter error covariance rest arrange follow kalman quasi assimilation introduce representation physical enkf respectively subsection propagation follow matrix subsection synthetic demonstrate kf enkf monitoring examine light govern value measurement state assimilation recover observation system govern behavior measurement relate matrix vector evolution simplification practical rapid assume subsequent equation forecast adopt specify state jointly kalman filter compatible evolution kalman implement ii obtain step measurement refine state measurement kf walk state give time operation gray gain nm k nm require step assimilation number major kf kalman kf prohibitive kalman filter originally carlo kf enkf
prior enter way explore metropolis suitable proposal yield hasting transition translation metropolis adjust langevin evaluation require differential pde draw density likelihood computational dominate forward replace typical scheme evaluation example metropolis hasting advance allow refine evaluation add grow outline spirit previous sketch previous effort evolution suggest connection argue lk acceptance repeat else previous rather approximation construct local nearby subset evaluation effort polynomial expansion advance refinement allow infinite number proceed depict set evolve become region approximation ever small increasingly change allow asymptotically posterior convergent metropolis kernel behave obviously sufficiently approximation local center might radius generally increase ball early sample sparse approximation relatively ball imply become approximation refinement refine approximation make useful explain refinement cross indicator explain refine approximation evaluate change substitute linear construct draw ball contain may square operator approximation empty omit option indicator assume gradient function lipschitz constant hessian constant parameter separate fill lie near quadratic long show compute geometric poor geometry consider refinement need rigorous bound approximation reasonable converge fall square unless geometry sample carefully design sample inner unity ensure rank subsequently decrease put less emphasis distant sample derivative process subroutine produce sample represent appendix numerical approach multiple output construct separate fortunately construct scale select separate portion section discuss need explain perform candidate choose symmetric behave identically treat avoid coupling whether move refinement refine fit naturally essential establish criterion cross error intend proposal true forward error leave strategy computing sensitivity produce whenever indicator exceed threshold indicator inside probability full variation leave computed acceptance reverse acceptance capture forward computable variety interpretable make user exercise either forward feasible mh criterion purpose ensure quick run refine criterion efficient may asymptotically position combination refinement increase cross choose practice require perform nearby computing evaluation radius improve geometry sample ensure maintain quality location ill ball point obvious simply cluster induce problem type design near new maximizer optimization initialize constraint ensure ball thus operator find separate quality reveal inner minimization simplify outside optimization likely global meaning build although produce set limitation might problematic easily add process surrogate natural application gaussian section process polynomial describe adaptation indicator compute jj compute distribution produce regression gps use exponential kernel hyperparameter endow gamma find correlation endow likelihood maximize construct gp neighbor use mostly include combination pure unconstraine later quadratic handle separate predictive summarize algorithm simple else start often find proceed chain construct state draw indicator accept reject compute repeat else target posterior asymptotically algorithm approximation via interpolation point replace fix modification essentially direction seem affect representative choice substantially concept require check one drift true hold empirically refinement sensible point check prove approximation notation throughout new let target write rx rx rx px lx lx collection time time distance satisfy proposal x p x briefly interpolation stability correspond weak widely condition geometrically ergodic density assumption lyapunov distribution inequality vx define useful hasting markov chain r yx important piece markovian process state say generally couple stochastic aa z evolve markovian process algorithm denote identity tx tx observation allow fairly naturally stochastic kernel note define tx extend tx tx fy despite hasting appendix suppose compact away envelope mention degree use proof difficulty far happen algorithm globally establish mcmc remain perform section describe three local posterior dramatically compute absence analytical estimate compute chain compose posterior produce model thorough standard sampler focus evaluation algorithm representative use performance type approximation infer parameter ode genetic circuit field pde conclude illustrate figure perform course must walk several nonzero approximation discard combine initialize point run contain discard burn evolution chain measure consist divide frobenius accuracy comparison figure mcmc cost show model mcmc baseline show reflect reference variance length small acceptance error respectively low high value increase reduce surprising chain eventually improvement accuracy chain consider predict show efficacy chain validation refinement accuracy reduce seem relatively insensitive criterion jointly decay decay measure theoretically sound increase improve robustness summarize accuracy impact fast quickly chain validation value decay indicator denote circle exceed compare figure observe change refinement setting percentage balance apparent interestingly refinement approximation become progress htb plot truncate clarity give previous approximation ode compact wish switch inference differential algebraic switch six observation endowed hypercube gaussian broadly highly inform largely gaussian adapt proposal use metropolis size algorithm mcmc figure reduce cost quadratic approximation proposal fall outside prior without run htb involve diffusion pde leave detail pde suffice purpose pde solve resolution coefficient define endow take field pde posterior shift significantly pde forward gaussian strategy parameterization relatively adaptive local previous switch accuracy indistinguishable demonstrate true model approximation regressor suggest regularity domain approximation htb dramatically reduce yet address time although store perform near search might challenge find problematic storing trivial modern finding set neither implement outperform run measure genetic switch pde hour spend qr surrogate spend near run run competitive expensive forward fix offset evaluation run demonstrate metric class construct local surrogate expensive introduce approximation metropolis hasting refine approximation result markov employ variation employ process regressor thus span use class significant forward pde problem local regularity log small although quantitative manner capture great decay almost evaluation grow process show bias decay quickly discrepancy primarily cross evaluate primarily advantage construction remain significant room local approximation exploit surrogate share forward surrogate correction use metropolis langevin mala hybrid hmc availability approximation finally far mcmc acknowledge scientific discovery advanced office science advanced award support thank additional scheme restriction quadratic regressor entry scale original local regression sample ball interest correspond scale magnitude define rescale rescale compute sample column desire numerically stable qr factorization may remove qr qr tx x lyapunov constant non vx satisfied tx kx vx vx unconditional x vx markov start inequality inequality vx combine vx vx conclude adaptive correspond condition requirement hold follow short fix compact support p cover compact within collection clearly clear exist base combine occur set nx lx remainder claim put together proof fix satisfie exist let x lemma draw bernoulli variable success r success sequence couple write I choose complete show theorem drift time away trivially event let complete lemma place long whenever set sharp care take mode recall briefly almost surely either drift chain exist p px p constant lagrange polynomial associate ix p I definition also denote acceptance kernel target show drift infinity tx satisfie proposal satisfie sense proof tx v dy dy x dy z since assumption claim x lemma eventually satisfied also large ignore ray also ray one element cover either great random add cover particular success affect take borel I part almost sure infinitely contribute surely put return infinitely compact recurrent satisfy lemma drift compact recurrent satisfie finish proof analogously follow item assumption claim follow envelope lemma envelope sentence compact lemma origin ready proof satisfied function lemma great surely inequality constant depend assumption show w decay uniform sample denote except initial choice fail validation event pass walk sample point converse compact proposal ergodicity whenever decay rate example justified certainly find
misclassification associate misclassifie bound maximal point svm likely classified bound identify act model bag influence influence leverage contamination original define base hyperparameter ensemble hyperparameter stability ensemble hyperparameter determine estimate bag account contamination hyperparameter resample potential contamination instance design contamination separately increase model experiment tune also obtain misclassification class base enable imbalance bagging hyperparameter parameter learn classifier unlabele label ensemble bootstrap increase bag provide intuitive mechanic semi art benchmark comprise label fully false positive improvement exist unlabeled instance inaccurate problematic negative acquire amount iii label include web page bioinformatics variant gene virtual screening drug share common final fundamentally instance precision since target biological recall anomaly go refer instance give unlabeled respectively contamination contamination true instance g contamination positive contaminate unlabeled difficult supervise approach estimate contamination distinguishing proxy distinguish positive negative assumption learn violate application various outlier performance contamination ensemble machine several split conceptual category label distinguish cluster weight individual change penalty misclassification training bag rt try negative inferential classifier supervise step convergence mc approach relate bagging svm bag supervise penalty unlabele positive penalize misclassification emphasize optimization kernel slack variable misclassification tackle technique svms bagging base training bootstrap separately contaminate bag svm additionally relative positive unlabeled bag svm significant bagging table illustrate resample contaminate subsequently bag advantageous learn approach computed decision approach potentially contaminate resample contaminate contamination use increase increase original contamination half instance due contamination increase converge contamination equal contamination decrease empirically repeat measurement expect equal contamination decrease introduce bag strong essential grow diverse ensemble bag ensemble model voting decision view bag interpret approximated instability success bag lead tree bag bagging explain instability relate intrinsic variability predictor influential bag explain influential effect resample contaminated insight mechanic bagging
disagreement majority label control minimize easy equal high target indeed risk could imply performance self optimize note connection da minimize keep every fold learn vote label fold risk correspond mean fold k ib semi svm learn domain self da algorithm divergence pac da labeling use nn nn adaptation da target weight real base minimize vote control disagreement da transfer label unlabele self justify self advantage label da generally consequence necessity lead instance description learn learn vote thus adapt corpus ii da direction self accurate distance imply closeness lastly usefulness g relevant machine generalization bound cm test datum key information specific weighted vote present justified target risk vote perturb region marginal appear study influence labeling deduce hyperparameter promise result bayes expansion supervise machine transfer survey strong learning task spam filtering adapt one receive different scenario call domain adaptation da arise model label da latter situation address da us deal covariate self labeling learn classifier auto label intuition measure easy divergence discrepancy disagreement classifier enhance disagreement control divergence much differ da scenario perturb pay attention designing label close special da majority classifier call disagreement risk vote advantage theoretical derive restrict classifier framework vote pac bayesian scenario mind supervise da labeling every self labeling help self source marginal close unlabele region self deduce original name well nearest label pac da theoretical basis synthetic review pac usual introduce bound majority vote set value call input space output stand ss sm sample value belief aim majority vote empirical risk real sometimes classifier domain empirical ss usual pac bind risk predict first drawing correspond risk accord eq pac bayesian deterministic pac set different respective marginal sp sm td majority low recall risk real every disagreement target reflect usual favorable da divergence small achieve promise disagreement usefulness pac bayes da remain vote regard state tackle drawback novel majority vote equation indeed source majority elegant non real extend da classical relation loose tackle tight relation notion define margin b positive convention h sp sx know express distribution marginal p numerator correspond moment b risk denominator moment margin disagreement relate disagreement counterpart majority justify elegant principle program learn weight measure minimize denominator disagreement fix classifier regularization performance j view suggest domain disagreement equation relate target risk gibbs risk deviation source disagreement tend majority vote da vote algorithm label rewrite label come tb b tb recognize true divergence true since label labeling labeling tight still valid da point one bind target tackle define labeling label close thus investigate rise follow label justified b ed st labeling pair marginal resp target source counterpart match self goal use maximum matching step unlabele thank belong affect true else construct actually region coincide algorithm indicate st good good da
c ccc train gold element target map vs correct map increase map correction similar test language maximum map adjust decrease trend brevity setup shoot least across space affect strong simple traditional query adjust availability employ incorporate different learn objective show correction setup simplicity future plan extent different kind representation objective affect work pose understand mapping start grant extract linguistic label empirically propose proximity map lead improvement realistic cross labeling image retrieval extensive co occurrence word corpus learn manner paradigm manual annotation bottleneck domain image signal must associate available mapping domain apply induced entity originally test decode function fmri activation vector representation training read mind shot label outside vision exploit translate word learn promising technique manual supervision encourage term return correct top less shot chance specific qualitatively map contain item universal map intrinsic reducing shot relevance classification setup severe map elsewhere leave affect problem setup get adjust bring post process attractive least square train mapping stand retrieve shot map retrieve often whole phrase near query happen take account invert convert score retrieve base score empirically rank keep shot domain translation pair language label contain training test ts ny ts tr vx stand regression source straightforward least label estimate vector source retrieve return accord similarity map common use cosine query precisely position similarity integer cosine q stand near brevity item search counting list omit subscript occur space near query return know similarity converge dimensionality cosine linguistic converge know increase qualitatively observed tendency become bad target map different source space compare map element vector english word english translation pair english item consider simply instead return solution target high alternative present formulation many tie want word rank cosine break tie translate cosine map query rank follow equation implement cosine break test method work corpora seed vector co occurrence context rely neural representation shot induce seed irrespective report seed word set evaluation word word representation try word representation word sub estimate draw token wikipedia tokens en bins frequency sort literature generally word medium frequency useful also use translation dictionary english test query entire report translation accuracy english occur one translation entire predict test standard method well map test instance use english improvement use solely map simply need supervision well improvement standard decrease add low measure effect add actually whereas affect improvement important medium although number frequency bin observe similarly many gold c ccc size regularization nn ccc train nn c cosine test vector left tend word realistic low might point cosine mean l nn translation correction case wrong
I impose nuclear norm perform svd follow project singular onto ball norm computed constraint put sparfa guarantee sparfa tag question set tag tag question define complete partially tag concept learner large tag tag pls perform compare sparfa unobserve real efficacy average monte carlo sparfa predict unobserved algorithm five course electrical consist answer answer question introduction probability answer answer university consist learner answer question consist answer collect see university dataset value model refer dataset dataset compare sparfa predict unobserved learner computationally efficient sparfa convex cf sparfa tuning run conduct carlo trial sparfa sparfa require min intel core processor reduce sparfa nuclear experiment collect school conduct amazon dataset value answer question response fully tag manually assign question sparfa dataset simplify geometry average bad profile learner tag simplify geometry tag learner tag percent leverage tag profile pls provide feedback learner strength resource pls tag recommend tag tag moreover pls average entire class plan sparfa incomplete learner question sparfa performance learner response significantly reduce b k q c x x recently analysis sparfa learn ordinal e correct learner response underlie term concept use learner concept profile association question difficulty sparfa powerful include difficult optimization la build algorithm automatically use sparfa unobserved method theory sparfa computationally content completion convex advance system learner provide learner automate education experience large sparfa introduce la learner stand analysis resource e video sparfa ordinal learner course sparfa learner response govern sparfa joint ii learner profile solely response provide analysis enable pls automate organization analyze course suffer lack principled concept reason affect learner determine interpretability concept pls learner expert manually approach intensive massive online sparfa utilize cross extensive sparfa run value sparfa automatically select analyze response course assessment sparfa recent account ordinal sparfa compare conventional sparfa sparfa learner perform la variety real factor analyze response factor value response achieve superior predict learner priori collaborative sparfa parameter identify extensive scale require author learner examine decay automate mc recover value extensively recently mc rank binary ordinal learner scenario typically binary ordinal next investigate applicability mc sparfa aim unknown ordinal learner answer let underlie rank denote model logistic unit contain response represent quantization boundary quantization boundary bin boundary unknown directly detail equivalently logit
synthetic set benefit stepsize rule big strongly perform dual formulated finding computation uk lead convergence problem solve dual detail equal accord admissible stepsize number number nonzero element e tb access ghz processor core support hardware choose coordinate iteration scale convergence iteration fast per average time expensive q case coordinate minimize strongly optimal convergence counter theoretical stepsize uk synthetic distribute regularize loss nonsmooth regularizer coordinate technical exist matrix regularizer relevant increasingly modern describing encode ram hand read among manner method exist strongly type arise frequently strongly encode encode box e strongly convex propose squared convergence method accelerate efficiently computable cd big computer partition partition computer parameter union sequence iterate accelerate output iterate store update way store computer computer pick compute scalar parallel use capability z k k z scalar step proximal backward fista compute one indeed computation processor element step deterministic scalar step reduce algorithm sum execution want directly influence u diag propose generic coordinate descent make extend accelerate coordinate descent complexity iterate satisfy lx k lx suggest small satisfying propose new stepsize matrix row partition empty characterization convenience denote compose schwarz equality reach feasible disjoint prove differentiable section fix df identity view apply previously quantity define although definition follow stepsize satisfied computation however operation power twice nonzero element quite pass instead easily computable run immediately much still satisfied computable improve inequality need check trivially plug side deduce submatrix corresponding term negligible vanishe increase complexity computable appear hence partition near know run enough bind partition apply reasoning need approximated computable tight upper view easily stepsize eq q ease reference call see pass compute discussion cm yes yes next small e quantity small let prove let lemma small h j tu generalise element vector idea sum think preliminary accelerated also twice expensive problem formulate correspond convergence influence compare different stepsize influence derive replace lem lem replace note problem big follow solve order benefit new dual htp duality htp evolution epoch epoch
plan two concept regression kernel give source space reduce mapping weight flexible radial rbf knn kernel audio audio video common cca form mapping v v nh correspond eigenvalue done retrieve video audio versa representation h update layer update minimize learn merge originally learn merged compute layer update embed method evaluate another perform read video label range embedding rbf report oracle train set simulate soft globally like learn transfer neural multimodal learn feature representation plan map maintain get regularization room audio recognition representation video challenge compactly represent modality address create short report compactly aggregate information variable plan reconstruct generate motion audio input audio improve would acknowledge contribution valuable constructive suggestion development like thank student course university fall deep framework transfer neural network fine tune initial semantic learn analogy preserve abstract learn semantic modality modality modality audio task multimodal progress report dataset multimodal deep modality pattern modality main focus modality multimodal multimodal extremely resource thus imbalance datum modality example label readily read video learn imbalance modality learn transfer imbalance selective transformation transfer well task moderately transfer modality intractable due drastically semantic transfer modality fully exploit neural specifically layer leverage embedding multimodal modality audio letter map read albeit space level knowledge within tractable transfer transfer speech read flexible modality parallel corpus formal report leverage improve multimodal dataset show language addition improve machine lda text svms often multimodal multimodal modality recently deep neural learn multiple modality single modality share representation audio video final multimodal infer method modality work availability corpus modality therefore address allow modality transfer leverage task target resource notable study dataset exhibit completely transfer apply transfer semantic level neural knowledge entail network top counterpart comprehensive topic modality learn make abstract audio video audio tune audio perform reconstruct new audio reading multimodal illustrate modality truth label input audio video lie concept build output h x h v layer net abstract representation fine tune input audio lastly reconstruct previously perform audio reading letter contain region represent audio contiguous example contiguous video frame vector
link linear demonstrate variety text superiority patch filter external database non mean goal clean decade problem remain fundamental one variety highly regard date noisy find reference apply unknown q example local weighted patch denoise denoise performance reference patch noisy patch former know practically external less expensive internal training patch image orientation search patch image plausible often fail patch patch patch regard internal work rare patch extent external show theoretical large developed sampling large external denoise noisy helpful database database database database contain external database obtain practical face camera scenario image ct concept external database tailor denoise bridge address suppose algorithm utilize emphasize early etc less reference patch may look extend internal external database patch likewise treat external video feed image problem force straight forward method solve utilize database theoretical denoise formulate drawback easy yet challenging database external image exist iii propose group minimization fix basis mse spectral operation improvement strategy improvement optimization patch present thresholding method improve detailed proof proof paper rest iv conclude remark vi foundation propose linear brief review highlight limitation denoise patch linear minimum square mse truth assume symmetric basis diagonal contain become orthonormal note let give patch follow column see give truth span wiener shrinkage achievable oracle filter question surrogate answer case achieve minimum minimum eq identify part problem choose dct basis pca however basis fully understand depend ground truth stack dct apply wiener pass estimate unclear relationship determine denoise discuss patch patch patch nn drawback patch may truly denoise task discuss improve without loss assume return first distance project zero energy denoise norm row zero similarly illustration show go back norm ensure orthonormal interestingly surprisingly classical component pca summarize observation practice possible user define sparsity minimization perhaps underlie enforce tensor bm slice see dimensional transform bm dct haar default set sufficiently similar dct location dct coefficient flat final haar transform sparsity stationarity dct axis essence sparsity true utilize singular recently stack dimensional array seek orthonormal array denote mode phenomenon tend bm patch mask adaptive subscript add consequently pca sa emphasis pca bm component arrive group notice play basis word noisy share dictionary patch train basis latter expensive adaptive cccc b patch patch formulate patch emphasis overall insight penalty claim equivalent important building formulation systematic nn equivalence nn understand closed line segment must either check versa correspondingly clearly claim know nn possible choice penalize reference problematic due patch similar patch similarity share concept order short path try patch regularize matrix way relax geometrically use patch one notational simplicity diagonal shape adaptive bm bm denoising pre learn apply project filter image desire component framework role delta bm assume measure uncertainty result become sensitive suggest additionally incorporate covariance provide estimate denoise pca assumption perturbation pca implicitly dirac denoise use generic database usage concept generic global cover concentrate mean local reference prior propose sample thorough justification denoise datum see subsection compute truncation mse variance trade reformulate penalize norm introduce penalty define section optimal solution ideal zero desire require similar component demonstrate effectiveness propose example refine result new solution consistently th ccccc f h db db db stand denoise stand denoise database bm bm pca four denoise method modify search external database iterate specific internal bm default window include influence window denoise denoise external comparison window identical external bm good patch method compute bm pca first threshold weight function patch reference patch default patch fair mention train database train database patch dictionary implementation external denote image denoise internal corresponding new database external mean deviation patch slide quality structural consider denoise purpose identical text handwritten signature bar code capture add external different font show denoise noise yield db benchmark insufficient database denoise method size conduct window bm patch redundancy exploited extend external cc window external test image level pca variety make bad level increase informative average yield use generic usefulness database learn training build train contrast propose fully th noisy db db database affect offer insight clean like database compute patch indicate noise level decrease linearly database distance moreover slow significant low condition consider camera captured suppose properly corrupt goal demonstrate help clean view noisy could simulate computer vision consist view add view visually compete area indicate method remove noise fine consistently db superior confirm database denoise denoise denoise capture corrupted facilitate recognition tracking denoise use simulate randomly image face database denoise row one face still generate plot average b database image denoise ccccc runtime sec sec implementation matlab runtime database image similar code intel cpu table runtime indeed significantly external runtime magnitude pca patch svd speed patch discuss particular perturbation answer font text view
factor ard percentage denote take argument frequency correspond energy scale control rule physical ii natural iii estimator fraction ard mask narrow predict therefore test examine detail prediction ml state analytical characterize low paper hamiltonian spin degeneracy embed localize spin continuous site interaction term positive wave choose width far choose center notation site descriptor machine predict function tr well prediction lead exact particular include approximate one numerically exact carlo ed weight size body wave note retain axis axis fourier detail ed interaction vary half bandwidth finally varied interval also include ed temperature lead maximal solution randomly test divide ard representation fraction green frequency green polynomial hence learn fraction write accurate coefficient available ed calculate code give q evaluating transform real smooth learn learn derivative evaluation transform function polynomial polynomial polynomial offer act statistical come direct well green time give fourier transform calculation large noise calculate devise fast chebyshev polynomial interpolation chebyshev expansions chebyshev free toolbox odd rapidly around different database sign security use easily replace odd sign odd however difficult thus use odd great representation either obtain small fine learn reconstruct therefore coefficient present full infinite aim band real frequency choose frequency ed rely temperature axis take unit twice combination set predict ml fraction since mention contribute use calculate ard show dot ard example dot dot result learn green frequency particle though see low predict good frequency qualitatively fairly predict green fraction able frequency physics blue dot dash circle learn length line learn prediction attain equivalent physical act prediction predict learn fraction inaccurate even perfectly relative visual clearly small number systematically correct see b converge length totally random introduce dense member combination descriptor database minimal prediction none component difference descriptor descriptor ever database half scheme result predict quite small enable closely behave really green try polynomial parameter predict polynomial derivation effective result show fig example discrete frequency also curve fig curve learn function correlation number slice coefficient green model test representation expansion operation long expansion superior learn improve promising material representation problematic thus serve intermediate interaction frequency qualitatively create even less drastically therefore ml really solve ed representation handling material model logical logical learn number highly nontrivial inherent example concern representation office science energy f l numerous implement thank critical reading laboratory office u contract de ac l respect cost set definition kernel fix example kernel matrix ed new hamiltonian effect approximate weight hamiltonian site reproduce hamiltonian reproduce match site equal infinity could representation require green half energy correspond approximately green hamiltonian specify set minimize several wave calculate eq procedure ground recursion orthogonal precision force orthogonality lose complicated numerical calculation eqs modify state normalize define interval write multiply side polynomial l ref also eq polynomial expansion green ml therefore put k ml two highly nonlinear von j laboratory il usa computing national laboratory basic quantum body matter physics polynomial number size machine dynamical theory full extremely demanding survey material provide complicated situation simplify perturbation expansion theory interpolation development ml explore complementary ml calculation interpolation analysis method solution dft molecular force molecular dynamic body arise application dynamical theory physics material science information material interact quantum define zero dimension solution effort algorithm accurate provide rapid preliminary range material refinement conventional formulation relevant quantum material density would body correlation strength must specify green state effect body implement determining correspond need whereas approach dft output total energy potential key devise term sized material tool optimize paper address know priori namely work ml consistent green follow ridge sec test sec calculation present type green sec polynomial shown prediction look summary conclusion regression exact detail polynomial present polynomial approach learn represent green self energy descriptor describe appropriate infer integrable nonzero technical often integer sometimes study axis frequency important hamiltonian correlate different coefficient expansion respect constraint predict spectral issue ml invert variable inversion principle many eigenvalue ill condition consider pose proceed posteriori approach involve made discretization define cutoff discrete orthogonal representation spectral set seem
typically computation require easily large setting rest organize include sure false surprising connection exist penalize random propose neighbourhood thresholding refer refer neighbourhood lead decrease certain theoretical property appendix introduce connect edge constant sure screen set false negative raise neighbourhood answer must first assumption eigenvalue assumption covariance quickly naturally appear let constant propose expected value define would decrease positive rate false asymptotic pp furthermore screen property theorem hold sophisticated obtaining propose perhaps recently result connected thresholding connect pattern lasso think stage set model component word consistently consider generate set partition sized ji precision via create eigenvalue identity rescale finally perform control investigate extent control practice diagonal completely successfully however satisfied reveal control large value investigate diagonal simulation figure vast majority furthermore simulation column block consequently identical monotone two h simulation large red vast element correspond sparsity publicly available data microarray patient high standardized control biological dependence give gold take equally graphical lasso refer quantify accuracy treat gold detail calculate calculate graphical agree uninformative result split summarize regardless whether gold graphical great gold size accurate obtain sure procedure recover sure framework set theoretical present particular ensure dramatically still contain tend unlike eigenvalue advantage approach sparse graphical operation require operation practice element tend range graphical expression datum acknowledgment nsf grant dms dms grant dp research fellowship first reproduce sake distribute constant constant p ia ib ia ib bound ia ib ia ib I variable together constant imply imply establish omit uncorrelated eq constant satisfie side eq uncorrelated conclude q b ap argument finally conjunction assumption sure screening show pp show q p c next pp false furthermore bn bn pp consequently expectation desire c f c f last fact pt obtain possesse illustrate graphical modeling interest graphical vision processing particular extensively compose thousand infer hundred gene expression consequently dimensional feature graph edge form conditionally equal model recover attention recover brief consider penalize likelihood incur non scad entail precision type aforementioned dimensional efficient high precision
stage sf discount discount parameter formulation easy projection project onto lagrange multipli ghz intel core cc post style domain xlabel ylabel post style mark ylabel gauss gauss plot mark none xlabel ylabel gauss gauss post upper xlabel ylabel probability gauss gauss xlabel ylabel smooth legend col index space average rs ex xlabel ylabel legend pos south col sep index col rs sf xlabel ylabel smooth legend pos south table sep average index col rs xlabel ylabel legend south east col sep sf col average rs sf xlabel ylabel legend east col space sf col sf axis none xlabel ylabel gauss gauss xlabel ylabel legend south table delay col point rs xlabel time ylabel smooth legend pos east rs alpha txt index col rs grid alpha delta txt ylabel east col grid delta txt col rs n x alpha txt figure discount cumulative reward discount total road user discount set sf rs sf throughput measure road metric delay road show reward present average sensitive long neutral variant discount average perspective sensitive outperform risk neutral amongst discount rs sf though computational inverting traffic throughput sensitive neutral setting observe policy parameter illustrate algorithm early confirm plot converge similar observation indicate rapid observation return novel actor sensitive discount reward actor ascent lagrange discount point incorporated sf gradient hand actor compatible feature proof traffic result neutral counterpart future would risk sensitive trajectory discount bound portfolio application solution good knowledge rate approximation actor corollary proposition pt sequential problem may minimize measure variability addition maximize among common finance discount decision first variability give criterion formula devise algorithm fast policy ascent multiplier difficulty gradient incorporate perturbation average actor algorithm usefulness rl actor multi simultaneous smoothed functional sf criterion infinite markov process mdp discount reward develop plan dynamic value reward gradient performance measure case refer representation gradient actor rl maintain algorithmic actor whose action whose actor address whereas policy difference prefer minimize risk usual optimization criterion criterion incorporate induced variability uncertainty uncertainty mdps inherent stochastic sensitive mdps g make maximize variance percentile unfortunately markovian stationary computing tractable although risk sensitive history operation finance attention machine mdps work reinforcement result utility framework base transform occur et measure et short actor risk criterion return obtain episode discount set algorithm additional actor risk measure discount summarize contribution discount reward variability return maximize return policy return see section definition lagrangian relaxation unconstraine simple show state operate underlie td purpose latter lagrangian simultaneous perturbation smooth functional sf discount sf simultaneous perturbation refer introduction function require evaluation parameter evaluation irrespective useful setting algorithm prefer also original perturbation certain hadamard sf vector use perturbation originally sf enhance propose sf scheme variability policy follow identify definition solve discount derive lagrangian discount require sophisticated lemma suggest simple alternative employ compatible feature action function parameter show advantage compatible develop actor neutral serve calculate discussion usage obtain square actor employ bias ordinary differential equation locally policy stochastic essence slow view fast principle td fast algorithm converge bellman operator multipli policy update track asymptotic converge equilibria ode saddle lagrangian moreover feasible I policy upper demonstrate usefulness actor formulation minimize behind control reduce variation road sensitive long discount high neutral cost neutral variant discount easily financial remark risk taylor expansion much easy actor limit require tradeoff take consider lagrange multipli tradeoff variance formulation know ideal expect formulation despite replacement formulation point author gradient stochastic path devise actor discount discount function every state mdp compatible neutral set short setting discount employ simultaneous perturbation estimate hessian propose unlike dual ascent optimize multiplier rigorous describe rl set mdp discount actor present actor optimize section algorithm discount experimental present result discount cost setting conclude remark future reinforcement rl agent environment goal long term interaction process mdp tuple action space reward denote probability state space act markovian action condition rl problem find optimize long maximize discount policy actor define policy adjust policy place make markov irreducible actor finally denote action stationary finite similarly discount pair define return sum discount encounter start discount variability reward introduce action bellman straightforward bellman bellman unfortunately monotonicity dynamic dp measure policy actor candidate discount mdps parameterized satisfie infer use randomization paper sr popular risk extension discount optimize lagrangian convert unconstrained lagrange saddle saddle achieved operate lagrangian objective unique saddle dual ascent tuple minima maxima r set gradient lagrangian x u gradient constitute discount fact derivative q last policy gradient provide reward help actor discount motivate stochastic sf far function initial simultaneous actor follow actor optimize sensitive perturbation simultaneous perturbation stochastic smoothed sf purpose optimal procedure parameter nest inner loop stochastic loop run parallel identity project onto compact keeps lagrange multipli interval analogous slow fast use td approximation equation ensure scale lagrange multiplier slow operate mdp particularly suggest usage ascent lagrange multiplier complicated simulation employ simultaneous gradient policy parameter classify include rs former use sf rademacher sf correspond rs sf employ sf perturbation rs n sf policy function initial observe next state reward draw specific lagrange return policy function actor illustrate operation involve loop instant simulation take simulation reward state temporal td value function value function hessian lagrangian sf update descent direction gradient multipli outer constant trajectory discount ensure enough describe td section present first actor respectively actor respectively denote th subspace approximate linearly project consequence diagonal bellman square functions govern reward transition weight project bellman contraction cf first aforementione see lagrange multipli rs hessian estimate algorithm estimate involve sf perform part twice sf perturb respectively recall simulation update rs devise rs j j actor gradient estimate n update last lagrange sf second work along outline reward state neutral reward differential satisfy mdps criteria variance occurrence action pair sensitive mdps sr discount convert discount l differential action square respectively satisfy equation derivative side replace thus rhs without change integral replace reward rhs advantage difference td differential n statement feature actor average mdps algorithm rule actor operator discount proof satisfy plus fast update intermediate input parameterize observe state reward although unbiased use bias bias actor algorithm upon td estimating consist lemma show rx claim put l ratio sr actor recursion sr variant sensitive actor td actor present discount multipli sr actor algorithm approximation variability discount define close variability measure discount average average reward average compare necessity trajectory actor risk actor use ordinary ode order rs modification sf sf second recall rs loop inner loop td evaluate square outer stochastic update policy descent lagrangian slow ascent multipli recursion slow static slow recursion rs g saddle objective step give td fix bellman utilize lyapunov recursion track ode asymptotic limit early constrain mdps recursion overall saddle td estimate lagrange parameter govern converge eq td policy parameter perturb surely fix perturb realization perturbation outer perform td recursion infer establish argument q far sigma field ode globally stable negative observe infer aforementioned sketch imply final theorem assumption latter continuous uniformly stable martingale integrable verify ode martingale td recursion td loop converge purpose recursion update rewrite due track ode equivalent descent limiting depend ode q operator ensure evolution ode stay point limit interior boundary boundary rs rs sf lagrange lemma sake completeness ode ode let perturbation solution rewrite td inner loop recursion recall converge parameter policy taylor expansion use equality see line discretization ode lyapunov trajectory piecewise interpolation cf ode step first converge evolution limit recursion define function q proof recursion vanish recursion claim l envelope economics ode interpret equation generalize envelope rhs ode time differentiable point maxima next evident actor recursion tuple local convergence saddle far also gradient differ sf theorem establish manner ode rest td involve recursion similar early whereas recursion rs rs sf descent newton limit ode see stable analogue rs sf lagrange exist almost estimate almost claim l j claim lemma first method separation converge td equivalent rule recursion recursion discretization ode rest claim sf establish employ rs sf claim proof claim rest identical ode good approximation actor algorithm true even actor incorporate risk linear scheme rigorous could follow argument quickly analyse recursion td converge loop trajectory iterate error td one fix asymptotic normality asymptotically limit variant aforementione asymptotic scheme score counterpart normality rs rs eigenvalue refer detailed unstable equilibrium possibly equilibrium situation include randomize recursion place
rbm family explore author mix propose realization simulate set regularize easy sample construction keep allow replica certainly computational simulate address observation achieve stack training jointly replica neighboring exploit sample long move applicable paper boltzmann machine deep key parametrization visible learn jointly accurately reflect posterior define distribution v simultaneously equation proposal mean analytically neighbor replica consider swap figure f rand iv ii v deep implement gradient state gibbs layer greedy layer share similarity consequence rbms parameter temperature e smoothed temperature simulating correspond rbm reflect despite difference accept reject still rbm rbm stack whole deep network propose common rbms ensure rbms unit adequate move learn rapid ensure rbms ask style traditional potentially important consequence rbms simultaneous standard rbm notable exception upper layer rbms reach move ask change rbms rbm phase use mcmc traditional reflect rbm sample distance jump correspond particle neighboring rbms rbms share maintain model acceptable swap ratio rbm energy begin reflect distribution rbms provide diverse negative rbms rbms share parametrization layer rbm replica share mcmc move dt bias well likelihood rbm configuration single figure show rbm couple ratio likelihood good layer curve train bottom field ensemble mix property chain facilitate get high world deep hierarchy mix idea individual hierarchy layer concentrate rbms interesting boltzmann negative draw diverse simply even layer model propose state describe auxiliary carlo sampling rbm wang random add simulated system achieve differ relate auxiliary layer rbm wang think dependency latent variable rbm op universit boltzmann rbms approximate rbms typically many computationally gibbs poor mix novel machine belief deep level hierarchy dramatically increase ergodicity auxiliary hierarchical conjunction asymptotically guarantee simulate rbm experimental confirm gradient boltzmann requires draw sampling preserve lead popular practice rely markov training allow configuration statistic bias decrease gibbs step offset ergodicity incur boltzmann temperature temperature space become particle quickly energy landscape configuration local minima original nominal distribution thus explore serial despite benefit remain computationally mini pt require expense efficiency jump high rejection possibly couple adaptive simulate cast relies mostly occur back success factorial enable inference gibbs sampling analytically un configuration derive qx mcmc expensive burn update maintain method modal mode anneal reflect reason author use phase pt work instead simulate difference parametrization temperature act scale low close uniform facilitate leverage fast mix replica neighbor move numerator denominator swap
influence reflect linguistic supplementary material group start n unobserved quantify term product dirichlet multinomial conjugacy remain q however sample collapse slice material foundation infer central use process temporal reveal social readily model interaction analyze discussion argument power take language identify influential content significantly comparative approach characterize influence compares demonstrate linguistic infer turn move beyond dynamic word bayesian ability argument open market demonstrate model pattern also influence network turn infer et linguistic investigate latent influence parameter determine discover associate nine represent format party represent ask sometimes united intend seem movie limited cast people entirely room combine movie consensus set ideal explore strength generate movie market open seven member bank must meet four time vote year financial person discard contribution post concatenation frequent remove stop relationship salient characteristic provide supplementary parameter experiment generative contain length generate time round duration token infer blue circle infer average bar indicate accurately value hold sometimes express standard evaluating higher comparable occur split predictive analytically logarithm via draw bayesian language provide probability supplementary set influence involve equally sized slice slice take probability use perform variational bound available set language lr lrr lr model synthetic dc united movie infer infer describe report supplementary argument speak minute infer linguistic reveal rest infer reveal illustrative case unite remarkably network influence infer network illustrate quantile material figure bar posterior ultimately side side ten infer argument first support pattern infer al pattern status comment ask question unlike focus movie discussion consensus reflect influence infer linguistic turn influence infer influence receive show top significant influence extent initially vote ultimately vote vote vote content similarly vote discuss suppose last three confirm influence et show influence vote influence position much htb exploratory range available length token divide correspond second neutral outcome result depict aggregated averaging subset influence arguably fisher notable policy result correspond pre continue sparse influence result neither finally relationship play role role much relationship result policy strategy oppose economic latent linguistic constitute research political influence take explore combine bayesian al model scale person et capture influence linguistic tie likelihood likelihood hold type supplementary network tie extremely infer suggest reflect informative investigate turn promise direction future exploration discover latent via linguistic demonstrate synthetic compare influence variant linguistic model meaningful potential social latent influence member market separately content word influence acknowledgement early part information nsf finding recommendation necessarily bayesian generative evolution language unlike infer focus via linguistic permit validate use capability influence market demonstrate social dynamic group social datum find use datum social group people interact another achieve goal extremely complex social take g content study social question influence political researcher infer traditionally analyze structural link network facebook however state link exist observe proxy infer social relationship concentrate explicitly turn move beyond behavior present dynamic capture influence upon substantial within indicate interact person increase person extent increase depend power influence drift closely accommodate language linguistic linguistic reveal reveal reciprocal language model idea language self mutually doubly point form mathematical foundation et al depend take interaction multivariate person return make interval extent event increase time decay person couple people via
key backward algorithm recursively forward pass conditioning force transition great draw state sample forward become full state backward synthetic hmm utilize true rate express temperature sampler instance zero intractable correspond ignore complexity tend towards sampler cause transition duration temperature beneficial term sampling observe respective prior sample transition sequentially chinese restaurant crf dependent table sample chinese restaurant customer keep track table sample customer customer simply care exist customer rao posterior hmm restrict represent cf dot index make tractable use gibb accord note concrete hmm map state correctly three dot white even though dot distinguished duration hmms hdp hmm able stick weight hasting row sample hyperparameter depend mcmc technique explicitly hmms specific state inference take place describe principle apply possible transition merge merge state forward must ensure merge weight state gamma stick break weight remain must update update accomplish beta stay break accord ensure incremental infinite delay duration rd distinguish short assign unique emission explicit duration hmms infer existence duration name instantaneous temporal driving due quantum macro system pixel complete entire quantify effect characteristic must understand hmms duration temporal change datum expert four characteristic correspondence surprisingly hmm duration treat proxy map mean hyperparameter give duration share give hdp variety state believe hdp hmm bias accordingly could specific right hmm mean mean top duration emission transition state initialize hmm score histogram distribution duration region consistent observe duration influence hmm therefore wide test hyperparameter find similar value number major mining analysis either two gamma mining make change interpretation one show mining set posterior location well concentrated year finding black dot infer state infer histogram horizontal show occur structured parametric hmms hdp hdp direct hmms structure hmm construction follow exist infinite encouraging construction minor avoid state problem prior encourage parametric hmms hierarchical factorial review area research practical persistent state hmms demand efficient review advance nonparametric construct perform infinite variant enhance posteriori generate right explicit duration parametric hmms e domain recognition natural language write biological recognition parametric hmms long recognize identify fundamentally combinatorial determine highlight seminal exhibit kind nonparametric replace introduce mathematical overhead comparison bayesian approach learn inference appear directly address state cardinality hmm name specifically usage hmm hmm confusion extensive hmm characteristic duration duration characteristic segment rapid short segment desire segmentation steady towards infinite hmms rapid state flexibility derive infinite hmms transition duration hmms thing bayesian leave hmms visit practice particularly tree unknown cardinality largely parametric hmms duration hmms hmms hdp introduce generative explore em integer indicator measure th x x discount prior consist state discrete time endowed emission distribution follow usually state distribution element latent state learn hmms machine learn describe collection expectation likely latent explain max viterbi state infer must hmms number maximum combination penalization criterion alternatively cross procedure single model goal reason alternative model placing encourage small imply hmm small subsequent total characteristic sparsity difficult intuitively might way interpret graphical consequence beyond bayesian estimate include transition hmms let hmm carlo monte use compute next single hmm hmm every might segmentation data step insight hierarchical canonical transition hyperparameter control specific transition vary many diagnostic try observable clinical signal want restrict latent reach hmms restrict topology restrict hmm visit leave right hmms kind topology restriction encode instance use hmm transition hmm may visit restrict topology encode explicit duration hmms suggest hmms tuple long order time impose integer latent state consist tuple remain duration duration transition duration observable endow placing perform inference hmms hmm large state conceptually consider hmms possess state transition bin specified additionally encourage dirichlet process goal infinite allow refer review construction addition hdp generalization extension hdp hmm hdp hmm offer issue dirichlet hmm hdp hierarchical bayesian review hdp infinite tie hdp link top ensure countable tend concentrate mass dp allow different dp stick break meaning namely state specific emission distribution state popularity similarity construction finite base sequence draw hdp encourage state persistence hdp extra self large probability hmm similar chinese hdp hmm hdp generate mechanism otherwise hdp hyperparameter count state et theory parameter heterogeneity persistence markov hdp hdp modify hdp state duration require transition hdp zero generation hdp offer hdp distribute hmm duration poisson negative binomial even phenomena interest novel nonparametric generate hmms explicit duration distribution transition conceptually closely hdp hdp claim duration heterogeneity issue partially add hmm hdp rise different future build unclear would model like hdp hdp hdp except construct structural zero use process construction structured state define slice snp gamma totally measure define base draw measure possible totally almost similar sum pg p choose one choose set number transition example restrict would zero normalize distribution zero snps transition unnormalize normalize produce base hdp hmm restrict formal procedure base concentration way let realization concentration discount h certainly case draw parameter stick prior distribution introduction measure critical snp state collection disjoint whose track uniquely satisfies gamma restrict pg restriction projection ps interested draw role serve draw transition distribution zero drawing allow normalize distribution yet choose encode hmms lead infinite hmm difference partition necessity construct atomic understood portion product space act subset namely eliminate space control manner restrict atom transition structure note random requirement duration transition duration nearly hdp rise maintain range transition simplify discussion give notation state denote draw transition perform sec explain grow method structure specify infinite enforce define restrict increase model region let plug hmm hdp hmm hmm process construction hdp set discount letting variance simplification hdp mathematically hdp hmm recover
non uniqueness prior knowledge section upon kernel introduce advantage little impulse system realization whose give smoothness impulse impulse apply estimator parameter characterize variance paper integrating highly large novel expectation iteration sequence computational effort notably tune kind parameter recently retrieve user parameter order follow use identification organize background base present conclusion end linear transfer I drive output corrupt assume sample know restrict write u px pp vector characterize evolution input input switch constant input switching collect vector entry need piecewise input room load monitor amplitude frequency full amplitude problem obtain impulse response sufficiently sample arbitrary continuous setting focus discrete know problem follow toeplitz input output available follow gaussian impulse response covariance scale amplitude identifiability issue describe usually draw kernel give call scalar interval decay velocity impulse recall provide matrix hyperparameter hyperparameter em obtain noise compute toeplitz close admit retrieve solve scalar solve domain initial randomly keep provide bayesian kernel em blind identification initialization repeat update experiment specifically pick phase equal large piecewise experiment generate experiment noise time variance output estimate impulse response compare estimator criterion ls impulse response base least quantity system kb estimator nb input correspond kb known mean fitting monte carlo toeplitz need result six group nb access estimator know carlo one degradation increase blind median approximately trend ht identification impulse response process stable assume unknown perform maximization elegant permit computation wide class shall attempt belong adopt note random variable function two address carry optimize write recall correspond maximizer plug back respect conclude se propose blind system impulse unknown realization introduce
discretized enable valid physical experiment illumination situation paper object simplicity discretization axis heavy denote use view around obtain express mathematical embed corner locate mf ff choose point kk fix compact format object care vector convert index convert define illumination abuse otherwise translate notation collect compactly dft everything stack matrix lk alternate ap general object nm nm note information phase retrieval amplitude solution find vector p project correction entry replace entry th preserve information recall commonly ap ap solution proper ap well easy verify sense nature constrained search close characterize behave like might different entry furthermore mention matter locate p length dash decrease dash decrease curve emphasize nature ap find locate theorem nonlinear subscript take amplitude lose follow frame subspace manifold frame choose due proposition range main count frame theorem lead unclear us analyze ap signal high frame fourier transform report unique zero operator inverse distinguish global phase theorem proceed immediate consequence note also n mr dimension know claim prove conclude unique phase emphasize p quantification p p p prove contradiction k contradiction emphasize imply convergence algorithm lemma equality equality r ie ie mb ie ie ie p ie I direct view dimension real positive ray le mb inequality mention imply convergence decrease relate monotonic lemma inequality hold eq hold since inequality eq evaluating hence indeed note please numerical case imply similarly continuous p I ip finally hold locate inside set converge nontrivial convergent since converge convergent converge show locate claim condition ap algorithm situation ap generic unless ap finite ap globally indeed note force suppose still clearly infinite would expect ap series imply simplicity ib simplify imply well clear ap converge author notice frame generic hold theorem mention metric projection successively project say initial intersection point nontrivial ap setup know generic frame intersection claim ap existence initial point ap understand ap eq z mr q transpose real restrict solution locate evaluate calculation first fact derivative evaluate fact c ie curvature direct expansion derivative gradient hessian ap I set theorem locate mp illumination window pixel geometrically describe illumination window via image require illumination window match overlap assumption phase illumination illumination maximize phase interact phase illumination window phase forget amplitude indeed show synchronization function relate pixel relationship synchronization study hermitian matrix illumination expansion diagonal entry overlap th illumination window preserve matrix overlap function I ambiguity difficult distinguish view affinity illumination q show illumination scheme intuitively illumination window overlap fourier determine overlap edge I u please pixel illumination amplitude want pay reconstruct idea maximize regard amplitude section relaxation discuss lead initial ap take affinity vertex relationship follow phase unitary transform indicate affinity recover relaxation relationship connection laplacian define affinity affinity purely encode graph denote next graph positivity phase mention generalization laplacian affinity precise take view generalize random status vertex modify encode vertex relationship I I jj ji ii contain evaluate synchronization framework setup converge heat associated laplacian eigenvector parallel manifold literature mathematical propose consider amplitude amplitude consideration truncation amplitude eq evaluate functional equivalent hermitian phase eigenvector call synchronization ps performance optimization play essential beyond describe illumination form dark sort harmonic form represent circular denote illustrated top denote connect together produce illumination illumination connect frame intuition behind synchronization data experimental wide second amplitude iteratively adjust circular limited experimental pixel technique row size pixel figure frame pixel cover first odd random fractional shift interpolation illumination experiment eigenvalue synchronization p p nm gold complex gray scale onto circle complex notice decrease monotonically good convergence ps produce start also typically overlap compare illumination two produce illumination start yet algorithm large increase field illumination among frame apart weakly resource result iteration ap start lead notice experiment new algorithm acceleration enforce improved frame wise technique adjust every exist frame wide combine frame synchronization start kernel synchronization large eigenvector q replace wise repeat step ps initialize synchronization conjugate maximum synchronization ps ps frame phase lead range synchronization across phase write z find large eigenvector expand frame kernel yield synchronization justify understand build illumination frame phase accord synchronization cg cg frame frame wise ps ap convergence figure ap ap noise simulate use proxy distribute simulate variance iteration ap linear reconstruction limit ap global phase check ap inverse transform retrieval synchronization ps construct guess ap problem synchronization ap leave mention least illumination accurate detector range response information per detector channel second need investigation since uncertainty count illumination position incoherent detector discretization etc algorithm ps synchronization architecture base memory fourth ap image uniqueness relaxation last cg iterative study finding phase synchronization scheme support basic energy science advance scientific sm wu thank discussion acknowledge gpu test band limited phase complex illumination illumination content ps truth ap start ps start illumination c ps convergence ps htbp illumination ps select value row convergence ap ap algorithm right ap ps truth ap start illumination describe select top ap ps ap convergence ap ps bottom ap ps illumination scheme ps select high ground convergence ap start ap ps ap lead wrong illumination describe ps set high ps start ps change ap randomly line low numerical empty lemma corollary advanced light berkeley national laboratory berkeley mathematics nj mathematics stanford stanford solution synchronization construct initial accelerate speed light far apply varied ray particle short
ie ie u u inequality uniformly pr py ct pr pr ar remain theorem sufficient unique solution show dual condition svd obey uv f optimal solution unique next shall feasible perturbation objective subgradient convexity norm long unless lrr u inverse u operator u follow hold I I I u uv right high individually distribution nonzero sign matrix lemma prove remain prove f seem widely adopt easy fortunately devise approach q ii relate relaxation consistency coherence third coherence define leave singular u demonstrate coherence approximately constant dictionary condition uv f aa uv tu u aa uv nc simplicity require supervise environment demonstrate effectiveness generate accord l create randomly value bernoulli dimension subspace vary fraction trial successful successful weak produce solution able meet learn specific exclusive nsf dms nsf fa li also support iii svd ground truth corruption rank second coherence third onto resp identity subgradient vector large singular nuclear frobenius sup expect variance subspace prove namely characteristic enough interpret phenomenon coherence increase necessary accurate characterize parameter extensive uniformly logarithm coherence proportional logarithm constant law law induce approximately law coherence sphere notation see large divided component uncertainty vanish similarly coherence clarity million simulation calculate ideally happen provable datum subspace equivalent block analysis subspace law underlie denote sign sign sign h e u tb exact alm alternate minimization update update update lagrange alm solve convert equivalent alm minimize augment lagrange f fix update lagrange multiplier convenient corrupt elegant reality strictly incoherent inconsistent natural grow keep accordingly lrr overcome lrr namely mathematically dictionary lrr coherence potential deal coherent obtain dictionary environment promise result often high massive missing method probably arbitrarily estimate consequence develop explore several decade robust principal build upon exploration rank rank low except restriction location cardinality nonzero either magnitude give scalable fashion theory tell recover follow besides theory vision imaging imaging theory powerful reality matrix perfect latent require incoherent condition hold reality sense reason low beyond widely quite may demonstrate whenever parameter cluster go coherence keep drop well handle nevertheless coherence condition shall parameter discard interestingly impose environment approximately environment lrr advance identity far lrr lrr presence extra sufficient advance low rank e dictionary coherent elementary dictionary subsequently effective algorithm utilize construct lrr demonstrate motion promising include paper problem recover coherent version regime widely typical example coherent coherent practical understand standard coherence assumption excellent relate nature insight regard lrr special dictionary understand lrr well coherent spirit long understand factorization useful remainder organize summarize notation coherent corrupted algorithm complete demonstrate capital accordingly etc abuse denote space column onto space abuse notation projection onto matrix singular nu six norm singular large denote frobenius singular denote nuclear sup ij low letter list notation reader physical raise coherent observation basic notice characteristic indeed structure excellent quantity property characterize coherence first standard basis coherence call coherence introduce r notice calculation analysis work see behavior different thus adequate consider individually prove regard successful considerably go column zero else widely coherence accordingly unnecessary indeed realistic interpretation subspace subspace domain texture rank behavior coherence cluster verify ease citation phenomenon phenomenon coherence please affect nevertheless could accurately corrupt high however usually kind extra appropriate modality structure mixture subspace face sharp contrast much devise identify success condition nature lrr show lrr main proof defer noiseless column satisfy numerical sense z column aforementioned need coherence worth note restriction probably requirement purely subspace ask necessary indeed elementary criterion dictionary lrr confirm lrr avoid rank unnecessary lrr exist everything sparse set observation noiseless reality dense lrr need modify consistently near property solution please refer proof noisy svd dictionary numerical lrr handle coherent ideally environment construct environment dictionary also satisfy kind supervision long rank form contain interestingly even given introduce heuristic coherent except extreme could achieve improvement straightforward firstly estimate utilize construct dictionary post modify design encourage condition indicate construct n lrr already exact fail recover produce weak great equal sense double although program much lrr assume fairly I notice use far dictionary iterative converge simplicity assume entry
issue generic gs summarize technical need characterize solution proof function scalar define technical slightly convenient sequence sublinear define suppose ready u vx xu inequality apply convexity combine divide side rearrange eq definition eq apply relation view rearrange term side rhs establish recursion gs easily procedure k convexity last convexity add definition subtract inequality convexity eq combine view definition establish gs assume satisfy q vx n vx u fact clearly inequalitie fact observation various option specify gs feasible limit compact q identity imply use hold implie observe simplified view complexity gs algorithm find bound eq bound view outer gs moreover iteration subgradient argument hold gs worth requirement use much easier happen nonsmooth present slide still iteration gs nonsmooth component oracle jensen sliding replace exact except modify remark noise note specification generic schwarz apply moreover last previous divide side immediately note outer algorithm ambiguity search generate iteration accordingly origin eq ready q bi saddle nonsmooth assume prox satisfied solve aforementione accelerate solution reduce properly phase sliding follow h establish consequence require stochastic corollary definition follow convexity take side induction follow phase perform observation subgradient previous stochastic subgradient firstly possess subgradient evaluation exhibit secondly establish light tail assumption shrink slide situation nonsmooth approximate linear semi problem easy convex write subsection application loss group regularization nonsmooth slide approximated class us dy c definition nesterov show differentiable gradient ready present gradient slide study property search smoothing slide replace outer iteration respectively view plug easily relation outer observe observation conclude inner reduce outer iteration access maintain subgradient evaluation note use b aforementioned access operator associate present reduce gradient evaluation show evaluation significantly reduce accelerate especially exist compute gradient many application bound solve composite slide generalization nonsmooth bilinear saddle point point theoretical property associate gradient slide practical certainly estimation slide bound gradient subgradient expect proper incorporation search interesting future composite optimization summation nonsmooth relatively nonsmooth order slide skip gradient component require maintain total subgradient similar composite smooth component strongly develop slide smooth nonsmooth component nonsmooth bi structure keyword complexity slide nesterov c program form smooth nonsmooth function satisfy composite appear many correspond certain fidelity regularization enforce property solution paper information subgradient situation subgradient many gradient need order find exist order computation require evaluation recent effort direct lipschitz bound show variant prox eq develop enhance accelerate gradient also observe summation together would expect appear unclear problem significantly bind subgradient evaluation bottleneck first motivate mention example many enforce variation n ax mb mr relatively nonsmooth sparse case arithmetic operation b regularize lx stochastic subgradient arithmetic operation need arithmetic operation computation box simulation evaluation efficiency solve composite briefly summarize firstly method namely slide solution significantly reduce total skip computation require idea iterative solve accelerate proximal slide secondly consider case nonsmooth oracle search refer subgradient gradient slide develop stochastic number evaluation stochastic stochastic deviation light return generalize slide class composite convex respectively nonsmooth approximated smoothing slide subgradient retain prox exist slide establish devoted stochastic slide composite slide situation convex nonsmooth conclude remark make notation terminology necessarily subdifferential nonsmooth differentiable constant denote integer denote denote natural provide review gradient apply subsection prox control development prox place geometry say modulus differentiable respect prox prox initially bregman reference therein prox prox prox constant prox grow multiply distance prox subsection briefly work nonsmooth e search iteration proximity imply move proximal method find method multi sequence build proximity sequence specify accelerate find iteration evaluation also aforementioned proximal type subproblem difficult nonsmooth address
sparsity formulation group sub overcomplete sparsity achieve regularizer write similarly collaborative highly reasonable rank flexible compare joint sparsity row sparsity neighbor non joint rank state nuclear ht f h sum able summation nuclear norm propose group pattern rank prior within across l gs lr train c c gs lr hyperspectral toy example hyperspectral assess imaging generate table label imaging contain band band ground interest label table gs l sparsity toy atom pixel green location atom coefficient belong blue dot clear row many row activate atom demonstrate clearly image region worst result prior due area joint give laplacian via outperform admm low group yield show time lr l significantly computational window size many narrow joint contain small region laplacian give prior review five structure sparse propose classification confirm prior flexible compare latter work well impose high prior hyperspectral pixel wise pixel assign predefine one hyperspectral represent small rather plausible compare incorporate structured appear far exploit spatial neighboring dictionary prior sparse classification also consider prior classification image ne procedure pixel label numerous surveillance develop rapidly technique find effective efficiency wide variety svm modification classifier rule solve task often art also rely hyperspectral belong low subspace homotopy solve insufficient employ contextual neighbor spectral test dictionary eq wise sub total class scalar determine sub dictionary operation belong suffer coefficient fortunately reconstruct either dependency neighbor inherent incorporate classification sort three neighboring pixel lasso rank lasso c enforce structural collaborative collaborative lasso structure incorporate logistic contribution assess conceptually call mixed pixel combination low group prior take advantage group prior encourage use one regularizer section investigate role structure impose norm structure discuss laplacian sparsity group sparsity small consist highly correlate sparsity assume vector sparsity pixel pixel whose neighborhood hyperspectral image atom neighbor pixel recover follow ii indicator operation belong atom coefficient neighboring pixel different even neighboring mention section joint enforce neighbor pixel fall homogeneous region use dictionary atom laplacian sparsity difference pixel belong introduce weighting characterize similarity neighborhood laplacian characterize similarity spectra possess enforce coefficient allow vector pixel become sparsity
optimize costly allow rely efficient recommendation bias world social setting recommend build analysis contain item per roughly uniform recommend estimated sampling order repetition analysis period day date note recommendation first effect recommendation recommendation sense sense agree recommendation introduce weight strategy shown extract reduce constant recommendation focus simple situation constant recommendation item recommend item reduction elaborate possibly complex trade solution hence item eq implicitly independent draw pi pi coordinate pi pi gradient compute coordinate integrate majority system adaptation process influence interact system difficulty algorithm historical analysis propose weight impact extract network recommender rank supposed interest moment daily experience movie book music match ad display website history aspect aside recommender system sort consequence system obviously order ensure recommendation monitor ad article record monitor one strategy offline click item would profile numerous variation range account rating factor influence historical recommendation time database quality associate offline production start user good generate action attribute model user especially recommendation end state influence recommendation production want offline item online cause offline item winner take probably unbiased strategy address literature knowledge modification offline evaluation reduce impact general weighting shift propose optimize discard reference rest organize describe weight reduce section practical world social information item historical recommendation instant user list item decrease ranking present product recommend time possibility item recommendation instant user decrease finally offline scheme reflect business user item exhaustive probability pair quite probability favor profile online evolve uniform soon modify procedure calculating take joint large system accord small probability weight point two moment directly fall recommendation give guarantee influence induce evolve influence responsible snapshot database online system discard importantly profile online platform profile example user via practice obviously illustrate htb implement roughly static evolve various influence recommendation recommendation recommendation modification quickly probability recommend decrease phenomenon recommendation recommend curve recommend second always historical particularly easy illustrate external way record probability subsequently probability offline user overall need keep offline computing select item reduce bias contribute reduce offline modify offline business without business propose weight covariate via weight selecting
mind eigenvalue exist straightforward gram distant completeness gram follow I equivalently optimum get theorem follow eigenvalue coherence measure lower investigate deal unit norm gram coherent norm dictionary since deal measure measure gram eigenvalue gram atom eigenvalue linearly atom allow atom independent weighting coefficient dictionary nonzero zero sparsity sufficient duality gram atom minimax consequence different sparsity atom distant dictionary bound norm atom coherence norm sensitivity resolution inaccurate perturbation oppose large ill condition pose instant reduction upper instant aforementioned upper gram dictionary dictionary coherent dictionary pose topology span distance preserve associate dictionary namely distance coherence isometry issue preserve space establish connect note atom uniquely represent dictionary provide atom theorem dictionary isometry property quasi isometry inner worth isometry isometry product extend exist equivalence less deal quasi isometry quasi isometry aim bridge gap isometry product quasi respect inner product exist isometry pair isometry expression satisfied inequality become tackle issue show eigenvalue eigenvalue take bound isometry constant w r inner product measure criterion unify condition establish quasi isometry induce connect functional framework impact topology future extend new projection kernel span function equivalently eq substitute yield degree university receive ph security technology france associate systems laboratory university france interest representation interest application wireless signal hyperspectral paper award machine signal past review lemma share approximation challenge end sparsity quantify sparse construct one one distance eigenvalue analysis share independence pose prove quasi isometry induce filter gram essential datum big reference therein process essentially model include support machine gaussian radial network neural seminal learning rely sample interesting enforce interpretation tractable within last development sense advance online bring sparsity process number need growth formulation literature collect call measure investigate criterion couple radial resource construct pairwise another approximation criterion explore approximate atom process kernel development compress atom kernel extensively dictionary comprehensive structure limit cumulative knowledge criterion introduce filter square affine projection ap recursive filter develop dual framework space update widely instance second framework estimate extensively algorithm overview point relationship connect feature space aforementioned criterion independence atom number associate bound sparsity conditioning provide quasi deal dictionary bridge gap framework section picture illustrate find banach compact consider error output fitness regularity solution hinge vector machine formalism hilbert candidate base incorporate use rkh reproducing call commonly optimization show functional estimation expression duality regression recent eigenvalue bound isometry distance isometry expression matrix vector relation equivalence unfortunately versus problem essentially identity preference small norm normal equation therefore minimax eigenvalue consequence norm functional dictionary tight bound detail constitute datum processing sensor recursively instant instant add drawback control growth instant expansion order instant fix investigate instant prediction paper restrictive unit instant select study detail former latter next throughout quantity dictionary stress th gram denote study section term construct since formulation denote dual explore impulse quadratic instantaneous deal approximated yield reduce insensitive ap propose present comprehensive filter dual framework framework eq available analogy ap formulation report drawback feed span dictionary current subspace lead implement formula independently online couple instant unchanged diversity exist several diversity measure arise unchanged contribute significantly therefore discard dictionary arise atom removal provide discard least diversity characterize dictionary distance distant correspond project substituting construct measure parameter criterion follow atom comprehensive dictionary composition dictionary approximate follow satisfied project derivation remove become approximation constructing investigate coherence characterize dictionary correspond correlation atom two analysis quality
variance scale dissimilarity representation instance define linear primal common weight determine dissimilarity individual classifier trend choice linear majority vote well comparison measure p bag total alternative subspace simplicity subspace default validation performance popular cover range recent paper dd instance dd dd point ratio bag near instance step maximize machine attempt bag label likewise extension boost instance boost round convert bag svm apply gaussian similarity instance supervise toolbox default unless ensemble base svm dd method base dd mi svm minimax dissimilarity derive dissimilarity used ensemble classifier subspace sort dissimilarity dissimilarity dissimilarity dissimilaritie informative dissimilarity bag support concept instance ht medium engineering technology thesis detect outlier automatic sort towards ph laboratory interest david physics thesis learn structure receive ph thesis university supervision work two year recognition laboratory development classifier alternative performance criterion graph problem elegant classifier university ph institute development recognition work technology laboratory pattern university multiscale multiple object bag feature instance bag represent use dissimilarity bag prototype bag bag prototype representation number bag approach combine strength ensemble consider subspace use state multiple many face weakly training example overall locate formulate case vector bag label apply image patch could present real successfully activity document categorization computer diagnosis category rely recover bag classify output bag example bag instance category often collective contribute bag bag classify bag instance supervise learner bag performance one object bag set reference dissimilarity dimension prototype th successful study training instance alternative demonstrate wide however dimensionality content bag preserve dramatically redundant feature prototype bag analogous general third ensemble train decision test phase dissimilarity translate subspace correspond bag ensemble dissimilarity preserve therefore potential dissimilarity ensemble preliminary meet expectation ensemble dissimilarity ensemble method depth furthermore result insight success dissimilarity ik n bag extension bag positive instance formulation fraction consider bag dissimilarity prototype take bag db b dissimilaritie bag instance bag informative access dissimilarity relevant dissimilarity might strategy instance dissimilarity informative case correspond bag minimize form classifier classifier weight norm typically non therefore influence large discriminative classifier discover informative redundancy find feature redundancy feature reduce individually dependency cross problem feature choose could argue redundant dissimilarity minimum distance exclude instance part neighbor bag dissimilarity instance average third mind particularly classifier strategy address aspect feature reduce classifier resample introduce I overall ensemble ensemble subspace base classifier vector redundancy influence diabetes irrelevant diabetes responsible diabetes redundancy select relevant decrease simplify possibly relevant classifier example successful image microarray hyperspectral furthermore many sample
result ratio optimization decay geometrically error decay geometrically study focus technique could understand problem paper extend direction three treat assume draw condition violate mis develop em concrete analysis em algorithm initialization pilot initialization gaussian plug particular recent initialization interesting acknowledgment grant nsf grant dms center us nsf science technology center grant agreement like helpful section relate analogue wide range ascent size smoothness gradient em step omit theorem begin proof vector shorthand iterate piece yield claim base em mixture previously equation take auxiliary central condition constant verify condition symmetry fact immediately bind contraction fact make elementary fact function place begin define expectation value devoted bounding fix let orthonormal diagonal diagonal entry define event condition condition note bound consequently standard gaussian combine piece eq recall yield apply term term return whenever function previously sphere denote sphere put piece suffice rademacher process triplet consequently contraction q satisfie operator discretization put together piece conclude generate rademacher sign show parameter vector sufficiently combine sufficiently small combine combine chernoff imply sufficiently state algorithm split initialization bind result splitting need iteration optimally dependence establish turn hoeffding inequality eq piece yield claim triangle interval hoeffding inequality probability put together piece complete provide proof corollary level follow corollary splitting update begin prove maximize verify condition use q note write z notation establish upper eq claim smoothness since need show suffice immediate lemma rescale weight vector orthonormal transform also rv rv place begin scalar guarantee reference note separate case namely eq taylor series integral see auxiliary claim since cauchy second bind next combine cauchy use piece complete turn argument event reference event constant event stage control event section goal measurable conditioning turn cauchy observe need x yy combine step return conditional cauchy schwarz effect schwarz cauchy schwarz vi five combine select sufficient select claim section treat taylor function schwarz step event measurable note event put sufficiently consists accordingly let yield turn recall lemma bind introduce shorthand long equivalently substitute upper three component constant measurable let z successive eq simple upper piece conclude decomposition equation claim probability state give vector singular hold consequently apply claim iv follow vi tail ii parts iii combine matrix write ii appropriate choice noise note elementary scalar particular norm claim eq independence note complete proof consequently sufficiently signal condition upper yield bind bind sphere q event gaussian tail remain apply contraction expectation moment sphere result imply contraction piece decomposition need uniform x consequently apply cauchy gaussian put together piece cauchy put piece appendix result present corollary splitting verify smooth strongly equation hessian fix show smoothness concavity hold scalar pattern claim scalar eq need coefficient bound corollary completing remain assumption need indicator one position ease understand matrix sub exponential condition ensure rescale sum matrix identity correspond tail thus I sub sub remain reference lemma random sub constant introduce shorthand show variable variable q use since argument sub gaussian since complete note obtain variable gaussian sub tail q follow bounding variance equation implicitly understand let one consequently cauchy schwarz since sub argument expectation schwarz inequality eq bound find c cm cm ccccc bin yu electrical sciences california berkeley prove algorithm em analysis divide part treatment infinite result update sample global maximizer likelihood characterization em likelihood ascent em perturb ascent leverage develop canonical incomplete mixture covariate high mle theoretically miss practice likelihood complex extent concern maximization growth incomplete model return optimum goal gap guarantee rich em work g introduce modern among establish work establish paper show unimodal certain regularity optimum behavior algorithm despite popularity em sensible little interesting em initialization statistically instance mixture regression empirically good performance initialization refine encouraging type behavior understand related goal address tool suitably sample em estimator analyze alternate case directly em mixture problem mixture regression follow exact natural alternative generalize perform em analyze case em concern population mle ball completely population version certain around concern gradient subset around remainder follow regression miss em introduce concrete corollary concrete give characterization initialization em complement theoretical confirm theoretical prediction em along suppose joint belong parameterize rather datum observe component component structure goal via namely observe variable population maximizer violate identifiable non identifiability difficult computationally expensive observed algorithm suit bounded holding successively maximize notation easy specify consist maximizer requirement relax instead find optimum closely variant ease compactly extension constraint arise project problem straightforward additional projection condition algorithm namely form number population level statistical observe eq equation expectation analog population em fashion analog em popular variety literature review specific base balanced denote density assume equally variable component draw sample variable form example operator close population analogously replace expectation mixture population em operator step analyze update mixture regression recent sample pair equation observation assume design regression underlie regression observe symmetric regression closely retrieval albeit weight update maximization operator form calculation em expectation analyze mixture regression canonical use algorithm covariate introduce covariate directly corrupt component involve fashion em joint gaussians conditional give assume em operator population counterpart em form counterpart eq return case maximizer converge sample converge result concern operator develop operator operator relate population em oracle verify section concern em sample base operator population result bound base sample addition per update stochastic begin analysis version turn vector classical must condition play concrete three section relate population update mapping mapping fix update virtue self satisfie maximize set close inequality condition g van de leverage consistency regularity relate condition condition involve fix exhibit triangle ii section parallel early repeatedly second sample begin quantity operator fix analogue population operator ball vector large enough ensure bind least large ensure sample splitting round identical omit since follow obtain establish population function variant inspire stochastic decay section recursion gradient compute denote onto center iterate radius iterate remain satisfie gs initialization update instance result stochastic order expect relate operator ascent operator algebra iterate choice q integral dx claim model prove population reader summarize theorems c thm strong concavity thm splitting concavity gs r thm splitting thm em well population level develop concrete class previously update previously provide bound difficulty mixture ratio form necessity ml quite slow researcher justification separate snr condition universal em corollary mle fraction rate involve large snr concavity apply conceptually detail quite technical guarantee corollary standard previously guarantee involve proof em split achieve dependence pruning reach snr weaker require oppose fig eps fig eps em optimization decay geometrically decay geometrically provide rough guide integer choice bind perform minimax error computable since quantity contraction iteration qualitative prediction test predict geometrically trial standard panel curve versus curve versus geometrically optimization decrease geometrically tolerance qualitatively appear fig snr scale ep value snr predict figure apply vary snr iteration expect geometric scale snr geometric error analysis geometric analyze em regression model apply condition suitable guarantee population operator locally snr operator contraction decrease function however functional
n lemma tell feasible proper give ls first ls denote operation follow l l l lp theorem notational lp present recovery recover lp go singular satisfies constant l nn n monotonically e notice right tend tend eq q lemma one see x function happen illustrative comparison property normal th row method recover experiment comparison include experiment time mse estimate tag lp tag give mse oracle tag aic give tag tag become curve lp oracle htbp picture exactly efficacy support choice portion successful trial conclude empirical small successful choose small successful exist technique exploit tuning cross method part parameter choose cross test dimension evaluation realization experiment perform measurement know set element period system row noise tn tw assume occur period recover display fig theory obey assumption proposition exist increase system parameter vector sparse method explicitly require open quantify sufficient guarantee recovery resort borel exist explicit characterization another question suitable proposition remark unknown efficiently traditional estimate recover lp thresholde de support formal derivation associate property true go consider formal definition obey consider unknown nonzero assume appear place material demonstrate find matrix sense different compressive sense sense compressive theory value satisfie state notational assumption covariance persistent pe wider require gauss markov unbiased noise gauss please raise perform property term scad smoothly deviation optimization problem later adaptive recently two method bic concern apply could happen please discussion possesse consist lp linear whose finally detail vector capital bold second though formulation selector point selector lie identity formulation point selector behave respect however behave due operation decrease zero advance try illustration path equal increase solution computational scad needs non suffer discussion compare step soft detect second precise description step need step propose solve soft thresholding also computational burden aic
likely generation number consequently likely optimizer dataset optimizer select access l wise qp validate dominate plan achieve figure gap dominate dataset low efficiency large machine cache dominate across fast amazon least fast higher slow epoch interesting lp amazon decrease algorithm try validate architecture report performance improve attribute validate dataset row b show bottleneck increase become converge fast fast b gibbs validate fixing good plan tradeoff strategy two assignment replica accurate region find converge epoch seem preferable run access neural speed classical detailed popular inference row bipartite gibbs sampling calculate row access gibbs establish achieve neural contains contain neuron connect across consecutive layer neuron function label goal deep maximize label descent de network al sgd seven million benchmark call mnist process use classical throughput baseline quality report mining memory optimization database include extensive related trend statistical processing database put system challenge processing develop statistical point tradeoff system goal study body aware range mining memory improve locality decrease cache mining cache temporal locality association mining net work consider hardware free execution aspect machine seminal use machine group discussion decomposition locality year language help extract parallelism two goal trade hardware recognize memory change landscape improvement et et li tradeoff advantage bandwidth new affect hardware aware machine tradeoff benefit prototype demonstrate tradeoff interesting current thank team team sharing acknowledge research project fa fa national foundation award office fellowship google finding conclusion recommendation nsf implement scientific describe worker different strategy worker affect worker worker need node try protocol rely operating system worker second evenly worker node worker replicate svm throughput second operate worker dense extraction text protocol storage advantage store require sparse allow parallel storage scatter improve cache throughput overhead operation synthetic sparsity sparsity dense vs tradeoff intrinsic design current optimize dense major column storage study data storage strategy conduct experiment major intel boundary unable pick access cache gets therefore always access store ccc target hardware et al et et al hardware efficiency relate consider increase hardware mining k mining neural cache frequent pattern mining cache include spatial locality improve mining structure temporal locality careful study type task system almost core study parallelism task parallelism frequent mining number pass al implement load memory usage locality tradeoff pre et al implement parallel memory optimize reference memory locality cpu none optimize affect computation consider worker technique least system design role dense sparse computation computation kernel dense dense mapping model beyond consider vs storage study community traditional database technique sure hardware modern hardware hope future language language pattern effective trade insight mathematical optimization task community look asynchronous recently establish ji detail fair tune run throughput use combination batch type statistical efficiency hardware storage compression na size batch size together experiment try size parameter contribute converge dataset two report overhead overhead scheduling tolerance make fair conduct scheduling tolerance impact claim gradient strictly epoch epoch batch slow cross different architecture epoch scheduling computation particular loss epoch second epoch second scheduling use calculate cause implement tradeoff hardware impact throughput totally seven parameter relate hardware parallel measure throughput find throughput music set speed surprising gb high throughput trend language help user parallel program experiment tradeoff help high performance quality implementation logistic regression music try locality try curve figure different surprisingly hardware e core apply strategy hope illustrate scalability follow et al create million page al page validate scalability randomly example create finish grows cause sub dataset whole model fit cache sample tuple equally datum tuple result linear leverage however call leverage specify tolerance acceptable epoch example score datum experimental set time compare music use tolerance music importance slow tolerance increase music tuple detail represent access neural illustrate graph run factor calculate connected factor assignment factor gibbs conditional proceed prove protocol theory aggregate factor figure b gibbs row non correspond get column sample throughput e generate general code modeling topic implementation implementation application illustrate stochastic discuss contain sgd fashion inside one invoke path different layer stanford edu tradeoff access first execute memory differ incoherence share tradeoff discover tradeoff study valuable prototype engine least sgd patterns architecture management machine via amazon ec result support google pick tradeoff goal paper system utilize modern hardware sometimes hope identify next system solve core system study configuration study preprocesse class machine traditional study statistical traditional efficiently execute fundamental tradeoff need step describe precisely minimize loss several complete call pass may explore current pick discover explore several store row wise brain access statistical different hardware tradeoff systematically tradeoff storage access prototype include supervise least different find converge differ access method access support develop select nearly study mechanism share explore sharing informally share processor replica three treat e event approach share architecture part visible core end epoch scalable hardware perspective grain communication core processor uniform memory google care hardware may communication dramatically find beneficial natural hardware responsible cache coherence processor coherence worth method technique batch dramatically reduce processor runtime improvement technique fact maintain effectively update total across processor partition aggregate memory partition may replicate replicate datum list conceptual treat machine system exploit unify implement modern architecture learning row call distinction datum read perspective paper distinction analytic method pass epoch sgd google oracle epoch read single read computation scan capture trend statistical memory coherence storage memory read execute read atomic access critical incoherent memory converge rely atomic modern processor empirically costly protocol popular include sampling solver al element memory distinct classical coherent prototype allow region share processor share per simulate nothing access distinct system column wise row prototype several epoch pass wise row row take applie update model use access include gradient descent order bfgs set typically use row iterate conceptually iterate read row method de approach order popular access briefly illustrate multiple node multiple cache quick gb data architecture cache coherent use machine name local amazon ec configuration sp tradeoff optimizer consider access versus hardware experimental paragraph execution provide initial model list solve method first capture argument index access argument pair index zero entry receive study modify single variable specification specification contain execution execution plan execution plan thing core operate model locality describe locality locality engine explore pt read write write replica assign tuple core replica one key synchronization frequent synchronization need converge find possible asynchronous version model separate together core reduce take execute epoch use observation strategy loss epoch intuitively replica information redundant phenomenon execute time finish show strategy svm locality incur request rule pattern suffer hardware replica worker strategy system partition replica avoid computation epoch partitioning row resp column wise access method column replicate replicate dataset copy redundant statistically benefit average low hardware read frequent dominate read point surprisingly epoch epoch illustrate show run use tolerance figure strategy give within epoch observe region cause execution loss hardware efficiency epoch choice epoch number surprising epoch process c music music forest ls music forest amazon amazon report deviation tradeoff enable speedup validate affect experimental diverse set vector lr ls programming programming qp include determination choose music forest forest benchmark analysis amazon customer google google art function take loss hour low measure measurement tradeoff request request request unit manual conduct second secondary storage memory four descent lr wise access implementation possible improve model variety differ cache machine tune buffer disk os tuning version try locality machine implement validate system extra tolerance task scheduling comparison protocol search parameter mini size report good result number local logical core machine logical tradeoff local always give time fast difference great lp qp order fast order choice tradeoff
dc express say dc dc clearly h ff dc function proposition dc dc dc explicit problem proof proposition propose performance reference therein popular dc program cut branch inefficient scale difference suit solve dc program convex obtain solution algorithm reach minima improve linearize convexity solve convex project stochastic exposition solve subgradient ht label choose update project subgradient solve linearize problem perform stepsize stepsize moreover accelerate minibatch instead minibatch traditional feasible solve iteratively constrain eq solution propose algorithm recall correspond dc p around current many criterion terminate terminate maximum terminate satisfied concatenation convergence verify change q last thresholding section evaluate digits image dataset section focus methodology break mean dictionary ii sparse soft compare last technique subsection encode able model unless state empirically worth validation soft approximation sparsity manual procedure choose equal subsample whose entry change set quite algorithm set denote iteration several value test small minibatch gradient iteration gradient first soft comparative study image learn atom randomly proportion randomly atom merge successively code solve optimization package dictionary optimize correspond function set generic neural use mini stepsize cross choose make three mapping cross linear train feature approach set task finally dictionary simultaneously encoder texture build texture take patch texture patch vs vs outperform size agreement author empirically learn dictionary classification task vs unlike crucial test dictionary dictionaries price ht classification thresholding fig evolution objective sgd solution sgd reach ht cifar dataset class rgb comparison restrict scenario later section illustrate classification report vs task thresholding soft classifier dictionary outperform size small dictionary reach reach atom illustrate show class separate reasonable fig b observe minimize whose class summary encoder traditional unsupervised way sparse code classifier dictionary fashion table largely near moreover mnist bad highlight benefit technique compare train optimize unsupervised pre architecture outperform outperform classifier dataset interestingly tune classification fast technique mention aware achieve mnist incorporate translation training shift version digit go class cifar pixel advantage deal sometimes invariant result high classifier feedforward rbf svm sgd last entry sign intra atom encode set cifar linear last relu net layer relu error relu report confirm superiority svm outperform sgd challenge surprising stochastic one adequate training give difficult report classifier use atom dictionary atom discuss solution generic algorithm complexity classifier algorithms complexity classifying classify mnist respectively require product nonlinear svms linear vector linearly practical cifar extraction involve multiplication control complexity order computation slightly scheme sparsity highly beneficial help data level mnist cifar dataset mnist cifar penalization mnist cifar exhibit predictive complexity precision code homotopy last computational classify test need test mnist ghz core gb ram early thresholding scheme commonly optimize descent oppose compare testing confirm whereby good minima unlike descent critical stochastic descent descent sensible stepsize choose stepsize sgd beneficial prevent interestingly stepsize solver solve solve intermediate optimization stepsize stepsize rule unlike sgd heuristic slow learn hyperplane dc dc significantly outperform experiment result consistently lead classifier compare near show handwritten digit mention encoder competition encoder act behavior competitive parameter network gain bad work reveal layer reach find network insight soft thresholde coarse negative mapping problem gradient proceed iterate recursive proximal operator stepsize nonnegative otherwise impose condition stepsize first eq precisely correspond soft soft coding go proposition dc dc dc recall dc p derive dc acknowledgment like thank anonymous quality proposition representation show provide many recognition major limit scale scenario consider yet alternative code extraction nonlinear tailor linear cast dc program solve dc conduct generic classifier appropriately classifier scheme train soft thresholding development efficient method vision decade volume produce instance internet technique large focus one task vision goal permit predict computationally focus research decade separate classifier feature choose convert heart popular popularity classification complexity scale recent nonlinear compact overcomplete dictionary beneficial processing denoise nonlinear conjunction architecture code work task drawback prohibitive vision power requirement map unlike vs vector scalar map give represent simple procedure nh illustrate classification implement vector multiplication linearity soft successfully architecture feature remarkable encoder couple result provide proper objective soft classification soft thresholding pose learn comprise control regularizer prevent overfitte dc efficiently iterative solver extensive image exhibit remarkable comparable sparse rest paper organize highlight section dictionary classifier algorithm extensive digits scheme highlight difference technique draw connection aspect share similarity architecture code extraction apply dictionary known obtain particular gain dictionary additional dictionary specific knowledge especially optimize code variant still trivial discriminative soft thresholding view coarse motivate soft extraction code efficient predictor parameter learn code approach soft close require solely accuracy moreover approach purely supervise often tangent thresholde single easy implement tie code appendix consider quite architecture layer neuron combination activation neuron activation choice logistic sigmoid tangent neural hide unit define layer represent connect classification architecture recent activation result classical tangent top nonlinearity plausible representation architecture differ architecture unclear appendix impose restriction neuron negativity necessity regularizer enforce sparsity scheme work choose thresholde nonlinearity coding provide independent motivation turn motivation deep simple particular architecture network generally stochastic descent directly exploit structure estimate classifier classification scheme incorrect regularization prevent thresholding hinge
cut label vertex common unlabele suboptimal bagging prediction bag ensure receive label extremely subset without replacement describe obtain average average probability predict protein genome level five describe interaction row column protein come absence structural dot vector combine genome database matrix indicate protein protein protein indicate know gene profile number isolate impossible sparse examine q connect time receiver roc vertex vertex edge label svm sdp result roc normalization sdp column show column field protein understanding biological huge class protein roc improve slightly deviation graph similar versus sdp simple topology describe uniquely suited reason operate protein biology describe thank thm topological functional category yield exist understanding protein slowly build sequencing characterize ease dna sequencing protein database would third unknown number difficulty shift biology necessity large identification protein characterization hundred major shift determine gene numerous throughput produce protein protein interaction protein etc picture vast amount protein protein protein represent natural vertex group graph simple contribution algorithm effective predict protein perform chen code locate detail boundary define sum sequence create small removing create informally n call order show order order apply arrive irreducible ordering irreducible minimum complement
different mmd test base probability let performance sequence distribution replace mmd error mmd base test rate case finite demonstrate mmd comparable divergence test test traditional test fr test mmd five test laplacian distribution two distribution laplacian laplacian mean variance test see mmd among mmd test well suggest mmd advantageous two sometimes mmd much mmd mmd use three comparison figure probability case mmd perform test mmd daily maximum usa anomalous datum day year temperature error average anomalous detect two place may perform apply mmd base set mmd plot error see mmd test fr test test error converge fr mmd mmd see increase mmd problem anomalous sample mmd free detect anomalous scenario reference sequence scaling go infinity develop scaling scenario performance appealing test demonstrate useful mmd solve nonparametric measure variable recall th remove test analyze anomalous anomalous follow quantity affect change affect change imply satisfied zero fast anomalous analyze obtain q clear satisfied exponentially computation multiplication analyze generality anomalous sequence np apply kernel multiplication analyze without anomalous anomalous notational stack define independent distribution independent satisfie divide component affect affect hence combine hence pick satisfied converge respect complete generate bound hence enough eq conclude clear respect include computation multiplication test generality anomalous constant sn q large enough therefore inequality enough np test include multiplication hence cm ex cm anomalous kernel embed h poor edu department electrical university nj usa department bioinformatics nc email ex anomaly detection totally anomalous sequence identically draw whereas distribution distinct scenario mmd mean reproduce hilbert rkhs develop test consistently develop show numerical demonstrate test perform competitive traditional anomaly detection consistency test maximum discrepancy reproduce hilbert study anomaly detection totally sequence anomalous sequence detect anomalous sample distinct assume test anomalous datum cognitive wireless either channel issue channel utilize channel improve spectral study whereas paper detect anomalous dna detect computer computer detect modify study assume priori detection nonparametric explore distribution arbitrary li decay case discrete utilize major challenge limited anomaly difficult building decay challenge approach tool solve g accurately propagate anomaly detection traditional distribution estimation intermediate fr well arbitrary discriminant use probability approach sake implement test base anomaly demonstrate test test specifically distribution hilbert idea distinguish distinguish embedding justify certain kernel laplace kernel rkhs naturally carry embedding easily rkh mmd mmd complexity test mmd approach density estimate distribution build test avoid propagation mmd sequence generate anomalous interested large regime sequence go motivated dna detect datum clear become anomalous large increasingly challenging detect anomalous correspondingly anomalous sequence regime go mmd computational complexity increase analyze consistency assume sequence increase characterize I study without scenario free anomalous suggest reference advantage reduce sample need help nevertheless lack reference exploit example anomalous contain anomalous characterize impact anomalous sample behavior order guarantee consistency theoretical traditional statistical approach mmd compare result mmd perform among well perform real mmd base section theoretical guarantee test sequence remark embed mmd anomaly detection sequence I arbitrary priori test sequence anomalous generate anomalous case priori respectively interested go infinity sequence fix applicable anomalous comment denote converge assume available reasonable collect exploit anomalous probability performance let anomalous index anomalous claim sequence say consistent large become increasingly e case anomalous consider priori test large anomalous sequence characterize anomaly reference sequence anomalous sequence far apply exponentially consistent see appendix equal sample sequence far threshold increase somewhat surprising threshold decrease intuitive detect anomalous sequence increase thus applicable value far away close anomalous base understand test asymptotically positive characterize anomaly anomalous unknown priori test applicable hypothesis anomalous imply value scale level lack extreme case occur order anomalous sequence dominate dominate also exponentially nan compute average complexity desirable anomalous introduce capture anomalous sample result anomalous constant unknown scaling substitute consistent anomalous sequence reasonable impact increasingly sparse anomalous anomalous explicitly tradeoff anomalous number anomalous scenario case study example anomalous anomalous fully compose sequence compose generate impact anomalous negligible test anomalous characterize condition consistent anomalous suppose bound constant appendix sufficient detection general similar role except pick consistent anomaly reference anomalous sequence consistent condition test consistent computational reference sequence guarantee consistency applicable role enough exploit reference consistent seem counter fact sample hence small become contaminate sequence eventually gets suggest test although reference help although without complexity sequence build satisfy e replace substantially consider less base understanding scenario build requirement pick anomalous domain knowledge characterize condition anomaly detection
receive marginal decompose aside generally likelihood importance alternatively within mcmc parameter conditional observe apply state consider perform parameter distribution estimate sample metropolis hasting accept sampler proposal generate mean covariance covariance pilot simple identity user simple case alternatively mala view define time particle particle probability value qx px sample adapt filter prior calculate calculate normalise assume user propagate particle k normalise filter particle weight detail filter place mcmc parameter use within accept metropolis hasting see focus likelihood iteration compute p stationary particle score outline idea particle back slight abuse denote path particle px step index particle particle time increase particle filter algorithm increase variance expense quadratic instead use rao substantially maintain particle follow replace discrete distribution obtain shrink user idea however actual affect rao calculate depend summary add ii f store vector shrinkage degeneracy significantly reduce rule reliable estimate equivalent auto considering use density unlikely reference point proposal bias estimate assumption resemble control degenerate behaviour efficiency analysis limit acceptance unbiased estimate article eq proposal q early mala assumption independent source variation log target independent interested acceptance jump n distribution define q density next consider overall efficiency term computational cpu additive justified number particle act noise variance limit acceptance optimisation mala optimal scale mala variance differ slightly particle rwm however important rwm increase scale mala rather mala sampler opt contour efficiency function leave panel leave panel variance scale jump variance figure rate considerably sensible variance sensible achieve acceptance rate inefficient choose noise acceptance estimate tune particle exact gradient would mala particle mala sources eq keep particle mala proposal view mala mala retrieve noise degenerate limit must mala limiting impose term mala proposal necessary mala limit behaviour notation hold non acceptance rate ii degenerate acceptance non degenerate relaxed expense variance lead three distinct describe limit overall scaling limit value approach iii corollary rate part I rwm study article iii efficiency scenario proportional possess therefore plot reveal mala firstly evident estimate without exhibit rwm behind bias term dominate rwm decrease proposal exhibit mala allow proposal decrease acceptance monotonic vary regime resemble rwm resemble regime mala fix let fix nevertheless acceptance change slowly value acceptance lie use scenario outline expect iii estimate particle linearly would estimate estimate gradient procedure unbiased particle eliminate still raise question trade see support tend infinity seem applicable finite filter fully adapt filter meaning attain mala discard burn estimate calculate particle alg update mala proposal increase mcmc sampler gamma distribution parameter sampler transformation jacobian term mala assess option ess give approximate h scale option detail left panel vary particle initially increase increase computational setting support acceptance panel regardless centre regardless scale predict panel rate reasonably predict consider mala compare rwm moreover method linearly linearly alg however particle observe expert probabilistic mixture autoregressive expert economic cycle nonlinear non noise record data g release significantly ensure expert expert expert growth assume measurement could sample vice versa see cause sampler mix slowly would efficient implementation whereby integrate mala rwm implement particle run alg discard use mala take pilot prior constrain transform hyper comparison mala minimum per simulation algorithm rwm min mala max table improvement term effective particle mala algorithm effective approximately mala however take cost proposal perform walk account order proposal cost present empirical particle mala particle rwm apply pseudo filter compare rwm particle mala establish optimal scaling practitioner mala proposal particle mala significant particle execute large reduce variance look particle mcmc markov nc x eq moreover sum central limit see second n n combine prove proof converge zero schwarz zero theorem separately therefore denote statement consider hasting ratio mala let term mala mala rd taylor mala double taylor expansions usual mala simplify examine refer multiply mala proposition lead term degenerate mala proposal condition mala scaling list satisfied alternatively rate proposition possess ix x nz nz u nz nz bx u I mala derivative derivative let also q form produce taylor expand term odd power integration respect target assumption provide form straightforward imply part analogous use bias present
latent entry major direct graphical bayesian belief undirecte refer markov network represent conditionally independent parameter consist graphical model convenient main reason encode distribution addressing need test speech bioinformatics rna analysis homology detection alignment genome identification nlp pos due structure store thousand maximize length problem handle traditional table direct acyclic dag represent conditional table show quantify relationship node completeness guarantee since constraint guarantee unique variable parent parent know probability l affect markovian bayesian causality make mrfs vertex dependency clear causal influence one node link dependency neither cause undirected undirected graph conditionally whereas mrfs assign positive value clique subset potential function function potential clique graph calculate sum integrate potential mrfs common language processing time model dynamic system whose output speech synthesis hmms whose satisfie long need factored hide notation mean take specify distribution evolve work annotation ms propose apply keyword purpose build semantic system limitation hierarchical categorization large interaction another hierarchical gene author allow aggregation simple complex small observed immediate immediate allow although bayesian network massive automate ms third macro protein scientific attempt integrated relationship clear accumulate growth term chemical difficult identification prove protein structure diversity along absence standard representation result database format use format format use major analytical automate ms analysis mostly complete abundance spectra mass abundance peak make distinguish database exist suffer produce nature irrelevant attempt process program produce incorrect thousand literature datum mining technique resolve great probability iv x l semantic allow learn gradually integrate attractive big age need include result use new used variable annotation ms predict evaluate annotated peak level ms probability well manually annotation suit ms automate integrated ms call ms annotated peak annotation ms layer node assign ms profile table ms ms parent direct parent layer child ms layer frequently child layer occurrence parent identity parent massive ms ms node ms parent annotation run ms peak ms technique new time record demonstrate precision train peak annotation tool dataset peak ms latent term provide com operate job extensive million job million search hour recommendation team want discover build engine query order relevant traditional engine tackle search represent job engine language search represent place class root place child node form term back user term store belong frequency connect term net vb se se health care table probabilistic similarity share parent filter technique search term distinct final graph node perform among ghz processor core ram term discover evaluate send review discover search return pair discover ratio discover relationship search use relate big business analyst software software big science sale analyst mining datum sale project master modern mining major issue probabilistic scalability set focus datum modern computer system sensor attempt scalability scenario hierarchical datum design regardless automate mass bioinformatic semantic discovery search large job test entry computer david helpful suggestion improve university valuable discussion suggestion thank university valuable time share annotation york scalability become crucial requirement graphical suitable represent massive arrange structure level expect million kind bayesian network represent level single hundred value ii level usually also network predefine top parent
nonconvex formulation local optima tune carefully remain quite large combine go around limitation matrix combination expensive regularization dimension attempt frank wolfe similarity formulation find eigenvector scale consider dimensional efficient solve lie word nonzero typically much small goal sparse scale pair besides instrumental additional motivation learn thus psd learn onto psd cone furthermore psd prevent allow project basis rank basis jx allow easily learn notice text represent bag count basis natural thought encoding term term topic parameter consist triplet build implicit click notational convenience degree hinge similarity aim minimize average margin triplet involve next f k frank fw compact iteration move towards minimize linearization minimizer domain fw enhance call describe basis basis move towards possibly new basis reduce active away determined line add convenient way compact memory away step provide completely algorithm f fw observe gap able find approximate appealing high algorithm update careful storing allow well identify ignore find time line search available bottleneck find forward indeed consider element take memory iteration variant mini heuristic expensive forward direction mini draw replacement mild assumption replacement probability deviation mini eq decrease mini find forward detailed avoid follow find basis result shall next dimensionality compete predefine validation contain proportion irrelevant word representation binary split split detailed split dataset train validation weighting minimize optimization gradient similarity random project entry draw bilinear triplet similarity learning except machine dimensional also svm nd text paradigm multiclass constraint neighbor nearest due instance per label tune heuristic tune tune c bold dimension reduce accuracy learn similarity early stopping learn nn notice perform bad projection generally outperform space generalization information pairwise consider weighting table svms variant early linear svm although outperform function negative entry good ability selection similarity iteration show incorporate may overfitte data characteristic ability diagonal nonzero pairwise score co occurrence fast computation superior make nn ranking etc also worth attribute extra capability draw svm investigate recall learn psd equal run space dimensionality assess nn performance iteration give compare projection similarity rp note tune separately size show early feature eventually heavily dataset sign sparse achieve form similarity combination operate frank wolfe world confirm robustness noisy rely similarity partially contract w nf ap purpose annotation view conclusion herein either imply proposition edu carry california many machine mining learn powerful scale dimensionality efficiently parameter decomposable one specific sparsity wolfe learn incorporate provide control overfitte enjoy strong memory depend high dimensionality many application processing vision biology dimensional ability score crucial ranking similarity measure datum
matrix mean random support allow grow grow associate square loss include x x mainly study compute specify estimate sparse observed tuning parameter regression appropriate omp algorithm parameter write mapping tuning refer discuss select let sample plot sparsity level sparsity level forward algorithm refer sequence estimate consider output dimensional level appropriately sparse however visible unknown turn sparsity reliable measurement matrix c present besides additional appropriate reliably identify clear evaluate compute stop fall threshold quantity decrease support compute sparsity line algebra write proposition say could furthermore ensure algorithm box lasso blue score two plot show estimate loss stop see performance remark select fortunately independent popular regression performance insensitive choice involve residual compute additional complexity assume stop change modify computation nearly lasso lar level tune one increase solution finally minimal loss sparsity implementation alternatively depend path solve increase compose stock return let zero use compute score want measure red horizontal plot clearly mainly vary narrow manner identify tune reliably cc say property various state follow entry eigenvalue block open analyze incorporate scale improved setup highlight propose output property thus reach equilibrium decrease tune omp greedy interestingly lasso start select parameter select lasso theoretical appropriate constant accurate estimation motivation scale furthermore lasso root lasso norm entry implicitly threshold empirically sl pl want sl pl show box pl sl range sl pl sl nearly score pl pl relatively insensitive choice compare sl advantage lasso scale lasso apply specify vertical specifie coefficient apply total solution may true significantly reduce real data attribute community normalize contain gene patient value present detailed reduce set tuning subsequently process stage require cross propose computationally select tuning parameter contribution agnostic sparse appropriate select grow thus asymptotically regression drastically solution problem significantly stability motivate future example interesting sparse vector thank cox paper grant nsf w nf unknown vector seek stop stop rest step use use follow equation side bind right rhs ii write fact next upper rhs upper rhs rearrange definition write tail follow combine use state definition remark call thresholding tuning prove specify finite setting reduce parameter domain measurement imaging furthermore basis several solve omp analyze require reliable vector comprehensive
neighbor converge relax keep close neighbor centroid classic relax mean soon prefer prescribe convert assign neighbor assign centroid let mean convert experimentally convert regular surprisingly score often mean smooth mean landscape minima q eq dropping mean get right present comparison value initialization fair comparison extend heuristic summarize contribution empty event heuristic cluster event happen heuristic well merge method bring expense third objective convert relax mean potentially many local optima exploratory minimum definition fact close cluster many heuristic minima heavily depend mean method heuristic take account empty cluster event tend increasingly occur mean round lot show seed center objective merge splitting heuristic converge minimum finally generalize center convert iteratively relax minima grouping intra cluster inter cluster point let cluster one yet cluster minimize minimize globally hard programming exponential yield separate copy equivalent intra distance sum inter cluster square distance heuristic propose hardness search heuristic heuristic heuristic partition good discrete yield contradiction p ie initialization singleton say add time close centroid online single mean initialization cluster close center convergence single cluster convergence close center mean say heuristic improve partition partition pn empty number partition heuristic cluster converse heuristic exponential open initial crucial clustering several initialization replace require initialization reach mean build cluster seed minimize point global problem euclidean apply bregman organize follow empty heuristic heuristic mean performance objective point associate close cluster convert relax experimentally contribution discuss mean start seed center iterate square centroid assignment repeat monotonically decrease guarantee denote mean perform polynomial point report optima mean may minima pdf centroid pdf second exception h nk number get center cluster random empty uci repository consist classify random count phenomenon notice dimension note tendency vary heuristic cc compute million empty empty cluster produce empirical empty toy heuristic mean demonstrate empirically notice rise avoid surprisingly show empirically give meet cluster exception current center usual method table allow minima table compare heuristic without dataset initialization observe minima mean build tend random minima provide single increase intra variance statistic mean point cluster heuristic min avg decide merge accept iff merge operation mean example cluster close obtain keep short since improve detailed proceed center primitive j center deterministic implement operation center find force hyperplane predicate compute determinant sum among point mean heuristic pick pick distance keep accept respectively stop iterate mean macro heuristic assignment move centroid correspondingly heuristic last stage merge merge merge split monotonically function converge finite number since decrease
concept return concept exist agnostic sample case oppose learn private goal private privacy pac hypothesis differentially pure privacy parameter note learner require requirement requirement neighboring matter consistent mechanism preserve sense similar return database concept every return capable approximate database close term close class concept database output description improper predicate class coin database algorithm pure set output computational improper transform except probability complexity et predicate provide theorem state database require big element bound predicate away recall operate database output state necessarily concept database particular imply always database always fix na close database private mechanism laplace sensitivity neighboring laplacian generate noise output preserve differential privacy goal choose maximize input database sensitivity exponential differentially private mechanism output show exponential sf generic private release solution much input parameter sensitivity two gap differentially private database sensitivity hold database differentially private query unknown trivial release round answer operate database composition elegant scenario care meaningful answer care receive privacy sensitivity query sa proceed round unbounde pure privacy differentially private execute algorithm output least pn chernoff concentrate learner concept class demonstrate learn jx jx learner prove exist improper class complexity alternative simple complexity proper learner consider execute quality return else random intuition whenever copy differentially fix concept chernoff least label appear quality big enough therefore hand fail whenever good need e number requirement however zero concept dc ax ax empirical typical straight forward application fail observe give concept unique every label every high execute privacy answer execute solution da contain subset cardinality learner proper private class proper learner stability label sample p concept hypothesis error change significantly hypothesis learner motivate construction simplify aim hypothesis approximately error second diverse two assumption make hereafter remove choose differentially private refer point zero differentially lot lot every many interval refer interval interval line dot correspond inner thick interval define five divide switch locate contain zero assume generality one close concept contain zero close concept argument parameter h frame sep min min plot thick thick define find privacy sufficient hypothesis explain interval differentially private return length length interval shift h thick thick thick thick interval zero lot one switch inside contain zero zero large suffice lot recall therefore interval preserve exist complete attempt one laplace specifically interval contain zero interval contain contain label sample diverse label hence one many utility noisy reduce noisy big indeed big length zero moreover contain one simultaneously operate differential set search summarize length good length choose start threshold reduce tool formalize analyze tool later construction learner notion enable concave concave quasi consist order database sensitivity parameter call quasi exist otherwise label goal view concave define correctly classify concept satisfie find inner frame max max target dot sample point quality quasi solve problem privacy preserve figure range quality database recursive call parameter choose return otherwise l ls label exist l l jt pt see label recursive range return successful call approximation one k pt partition interval let property w mechanism label fraction mechanism good utility rectangle sep dot thick identify upper decide whether good check define simply high start call let I iterative monotonically decrease n proceeding privacy observation sensitivity proceed execute sensitivity bind recursion preserve differential note sensitivity recursive call mechanism twice recursive call execution mechanism differentially private differentially correctness number call recursive call claim quasi concave non point exist nn ns recursive call ensure output perform recursive satisfy lemma call execution denote power inductive need recursive claim function step quasi next recursive appropriate lemma plugging get index quality bind recursion h two return recursive inductive ls r lemma interval might interval therefore contain therefore concavity quasi concavity quality quality rp left interval quality ap proceed exist contain point concavity quality sub length exponential ensure probability step mechanism output hence probability output sensitivity must exist gs gs establish prove output obey k fact lemma utility choose mechanism straight function database mf fail assume solution mechanism define choose I km km show must operate private separate database size private string every query fraction algorithm mechanism point define restriction otherwise input initialize let mechanism quality bb c mr b utility element mechanism denote begin event label happen succeed step choose mechanism case event assume contrary exist iteration mean appear contradict proceed simple input mechanism mechanism exactly interaction preserve differential interaction applying preserve differential concept query execute denote every privacy immediate ki c jx x j al get pure next approximated initialize empty privacy call convenient maintain decrease call database subset call execute either recursively small appear next property initially empty contain point quality j pt execute algorithm quality quality privacy parameter label successful return denote otherwise divide might z union quality recursive call database twice laplacian mechanism mechanism interaction differential call interaction mechanism preserve differential last interaction preserve entire mechanism differential c c define function step draw proceed analysis good occur coin fix execute initialize database label step define interval suffice probability mc pass occur recall plug inequality execute valid define define shift must b iteration define interval part give future recursive none recursive call interval range yet event whenever execute event happen recursive call none define point event happen continue assume occur begin show event denote step occur define intersect future total thus least none occur ensure add draw event consider iteration iteration define step empty consider iteration define occur particular contain interval mean execution whenever execute initialize ne event occur throughout denote occur argument triangle therefore q pure private al pure operate database slight different differently concept label differently concept construct I f fx q fs exist database proper otherwise solve let bind possible else second kind concept define construction separately database query sample private relationship privacy reduction private lower bind bind term use learner query complement necessary show arbitrary restrict via element show produce w r different hypothesis imply hypothesis small must start technical database exist sized hence chernoff hold case q good matching trivially fs show mention first step private show imply modify define dc label notice class connection operate database predicate utility c error happen return event existence obey every happen mc ensure proper learning able use task concept concept treat randomly exist divide laplacian set output step first output big differential privacy next analysis step execution denote neighboring let output neighbor identical change database mechanism f overall two private algorithm private private utility execution h note moreover I close assume case label c hence get c label tc eq predicate proper note assume efficient pure class contain concept prove low reduction similar appear show require proper learner learner ignore part maximal cardinality every moreover cc acc c number remove hypothesis ga sm least database chernoff every appearance good database contain randomness distance ensure big moreover whenever construction yield learner learner derive necessary require database exist cause twice every operate database lemma increase database lemma equation exist proper learner privacy reference necessarily identity reasonable privacy individual consider label denote private database label pac concept differentially private definition algorithm see correct learner constant learner learner private ns mb b concept add quality describe construct unlabele every realize use mechanism labeling property mechanism private utility event set hypothesis choose event existence thus ensure choose mechanism ensure event happen algorithm nc chernoff bind hypothesis happen least happen least see privacy model recall label private require privacy label scenario privacy publicly preserve privacy scenario preserve differential private semi class non learner specific privacy complexity see label ignore privacy task know semi private learner guarantee privacy case privacy guarantee private learner complexity et helpful discussion idea gray theorem corollary conjecture complexity private pure differential task differential call consider task observe quasi concave instance allow construct align private learner relaxation private label vc completely characterize learner label privacy constant privacy collection individual privacy private task preserve privacy task also differential privacy differential privacy privacy privacy individual require individual affect formally pure differential privacy one output whether database significant private privacy private private operate classify private focused pure et show construction hand traditional learner term learner exactly domain picture change private learner come must evaluate many complete complexity learner recently give show dimension randomize communication separate show improper private dimension class sample differential privacy significantly satisfy pure privacy observation privacy complexity separation pure learning give task computationally pure computable notion predicate private agree fraction give pure differentially bound partially support complexity difference simple differential simplify exposition omit variable length domain release immediately learner approximate differential suffice tool proper private axis point function threshold maximize sensitivity call growth problem define concave solution order concavity quality observe solution recursive solve quasi iteratively define
convex objective objective monotonicity curvature reasoning prove rich mathematical language repository solver use solver include solver source interior solver solver representative inspection concentrate involve affine language convert time produce framework model language solve modeling framework times l l package notably fast science fellowship stanford atom atom monotonicity curvature affine affine affine positive atom atom cone atom concave sdp convex solver l language sdp exp simplex interior interior c primal x begin minimize solve begin I subject e x p subject solve subject x usa david describe translate language tree representation global infer problem rule programming pass dramatically reduce verify choose programming software verification checking modeling translate user solver language gap mathematical form composition computation convert solver description dynamic familiar technical language matlab match orient implement depend inside project technical compute high abstraction abstraction generality toward abstract formulation mathematic code language separation operating benefit operate parse solving familiar specify mathematical constraint concern structural form example affine program lp affine call quickly convex concern devise division solver structural problem appropriate purpose associate language make formulate solver jump design significantly outperform purpose solver target program exponential frequently automatically solver automatically embed matlab use concern idea language notably specialized parse requirement include optimization convex problem solver programming describe represent constant another kind simple expression matrix z variable positive symmetric nonnegative eigenvalue semidefinite treat positive variable z automatically underlie derive change expression word expression parametrize compose norm expression hence call addition atom argument useful feature arithmetic array indexing transpose valid indexing multiplication expression atom involve atom call expression type atom sign top atom argument curvature curvature atom affine monotonicity atom monotonic method wrong sometimes example return define use suffice five implement variable result apply atom represent leaf node atom refer leave structure closure name acyclic dag dag evaluated already assign evaluate evaluate aspect head together child expression automatically unique identification place problem reduce pass dot rectangle fill minimum n n n standard strict strict convex consist minimize maximize constraint x object appear example construct property objective inf x notation display figure p program value annotate solver dual checking approach implement model range microsoft convexity determination hence whose convexity kind affine expression affine constant constant affine affine curvature use argument concave convex curvature need level atom atom affine curvature expression convex curvature affine concave infer say order curvature argument atom appear expression atom example convex convexity derive easy add atom allow user expand function recognize expression monotonicity quadratic observation observation implement sign monotonicity rule monotonicity atom expression multiple monotonicity type multiplication enforce rule add multiply apply child end return else statement enforce appeal code mathematic since time since multiple rather constraint expression expression affine affine expression convex concave objective concave sense satisfy form recursively problem iff call cone convex n x k specify cone extend problem affine constraint constraint trivial rewrite solver problem language list rewrite
see interested science sequence semidefinite base e setting guarantee thus meta guarantee comparison local descent g optimum reach search latter succeed highly optima local minima must minima ball likely suboptimal minima square minimum achieve good x mx capture guarantee correspond expand need way elaborate explain behind prove analysis polynomial x stem surprising extent notation terminology operator notation make low degree moment necessarily formal operator order emphasize often subscript degree system polynomial polynomial hard see definition imply point support polynomial write proof say polynomial equation satisfy equation bound degree degree polynomial cone sum square follow assume linear boundary convexity square combination polynomial contain contain cone algorithm mx theorem variant proof dual instead try existence sense author summarize follow solution degree seem come trivial good access wrong next match find plant let characteristic es minimize satisfying degree describe estimate constant let discuss language algorithm base eigenvalue proof true predict proof degree able prediction nature expansion rao however approximate set tend yield proof closely moment largely discrete variant language help match theorem characteristic set proven distribution quadratic moment iy ix remark produce large take constructive moment first assume eigenvalue positive define equal second shift carry least positive test satisfy ii expectation case capture heart claim imply scale scale choose random therefore need polynomial expectation spectral concentration ingredient linear constraint u p lemma sampled satisfie event establish behave polynomial result old sum satisfy b inductive iv triangle norm degree invariant used norm iv iv b together follow establish u md j moment vanish I big magnitude sample md probability hausdorff vector upper maximum problem replace hausdorff ax show relation u factor depend hard unit pa allow determine polynomial constraint u p linearity transformation conclusion lemma idea argument one column different latter degree even merely sum connection mention predict particular give perhaps candidate heart may seem priori completely subset expansion geometric notion expansion equal otherwise enough mass sx see moreover project allow sense dominate heavy x vector capture expansion span theorem know hard omit reduce question maximum polynomial design could resolve could question interesting imply I w norm simply dimension unit subspace contain subspace frobenius define equal trace equal use q hand inequality exist eq hence I x depend bound dimensional sphere require see reference imply tend zero want distinguish graph achieve game improve exponent whether could namely two degree even vector give every evaluation seem evidence consist several natural come instance need parameter value turn proof fact heart proof show constant reasonably degree mean yet question provide reference diverse method theorem theorem traditionally tailor development surprisingly necessary great problem predict class computational polynomial algebraic geometry control theory program diverse quantum verification recently game square particular tool bind obtain solution optimization square new guarantee interest possibly primary semidefinite programming theoretical understand solve efficiently one regular regular set let set large vertex leave start multiplicative hence computationally purpose often purpose notion graph people system understand linear vertex hard compute assume fact quantitative infeasible maximum independent assume polynomial tend hard arbitrarily approximation trivial discrete yield computable efficiently graph bound away isolated keep sophisticated come hardness efficient rarely match tight hardness computing already know give non trivial formulate body conjecture defer hardness challenge trivial beyond reach complementary mean result efficient one broad sense polynomial perform game find going isolate unified complexity meta predict technique quite setting time view common setting give hardness existence conjecture imply vast show class include meta constraint closely relate well correlation precise yield efficient though actually factor summarize hard capture concrete meta already improvement conjecture attractive question conjecture discuss promise approach potentially game latter beyond problem tool context yield meta notion summarize could understand instead game focus implication direction suggest probably survey least somewhat conjecture turn eigenvalue approximate expansion exist mention absolute hard give approximate reader also imply optimality merely become become quantitative relation surprising imply tight connection hardness problem constraint problem priori nothing stick survey unique conjecture proceeding skip thought satisfy property restrict two author size game hard structured structured conjecture round game name survey square application expansion relate hilbert yield yet false plant combination evidence evidence subspace much paper issue game excellent survey survey survey focus around problem explain semidefinite survey entirely survey mostly implication hardness meta hardness manuscript actually understand go beyond basic lp sdp description topic volume topic develop researcher nesterov meta programming
consider generate swap neighbor inclusion metropolis transition inclusion vector allow adaptive comprise real let tx produce draw calculate weight sample reverse proposal unique predictor swap initialize predictor treat priori example framework scheme useful sampling inclusion enhance stand add remove neighbor subject variate discrete draw independently define forward mixed remove define reverse accept detailed posterior lemma version move function algorithm follow draw define move consider ar swap construct forward r sm use define reverse remove neighbor move j use function define reverse predictor k r r r define reverse swap neighbor correspond reverse pair forward reverse neighborhood proposal pair move scalable linear g smoothness adaptive mcmc within section predictor rapidly move complementary update inclusion pair scheme pair divide explore predictor move n transition predictor nn cd n index predictor define pd pair swap empty p jk ps move facilitate rapid exchange active component hold component e transition proposal pair swap proposal maximize optimal empty pd ps index randomly select update move particularly predictor across replicate run real predictive consistently move table provide section appendix transition kernel preserve k define fix upper number allow component predictor little variation ratio remain active additional subsequent let per per budget neighborhood individual update initialize one inactive component vector fix budget active neighborhood budget k il propose version chance add move chain burn period convention descent score increase include converge equilibrium see stationarity maintain score pair select predictor form neighborhood function pair crucially predictor predictor move allow neighborhood subset neighborhood particular neighborhood initialization expect neighborhood mp vast predictor retain neighborhood maintain paired swap letting maintain importance score predictor neighborhood function reverse neighborhood recommend term square rmse initializations independent replicate complete minute sample thin subsequent every draw data dimension ard exponential predictor suffer accuracy function final outcome concern simulate fix x degree nonlinearity vary consider process component display empty burn explain active sort importance observe evenly example graph predictor active thresholded quantile relationship important marginal importance th jt across namely fraction predictor recovery identify important predictor inclusion addition dramatically see section active utilize appear inclusion graph important predictor pair proportion co inclusion well show two effect additive predictor bilinear effect regression interaction successfully recover interaction way predictor configuration explore persistent predictor component replicate mode difficult effect neighborhood subsequently discussion predictor identify mcmc coefficient produce component model histogram empty roughly univariate least half edge graph extreme sparsity nonlinearity additive structure inducing adapt underlie smoothness scaling interaction fail component model configuration median dash line figure rmse report replication standard interaction quantification modeling confirm mcmc sampler pair inter move provide reliable plot test consistent excellent value line band rmse appear average rmse lasso ard covariance regression important primary gp mcmc estimation ard predictor dimension development interactive reason two generate predictive summarize ard iteratively improvement ard enable superior even small rf remain remarkably increase predictor replicate rmse ard map rf plot predict enable band four statistic offer effect mostly high count vary moderate example validate satisfy budget move adaptively rf compete dramatically lasso clearly ensemble nonparametric good good ability accommodate degree interaction rf rule leave fraction compute predictive hold rmse average split appear pt crf across component sort sized model dash datum primarily capture component inclusion identify predictor effect mostly interactive effect compare variance partition dataset move real display bar b algorithm trace variance serve measure dash hold test mcmc move var active empty component size appear vertical line middle marginal right predictor edge co across gp marginal split predictor accommodate probit future package accommodate predictor e development approximation enable mcmc enhance score finally mcmc move inclusion tackle local share predictor section component move subsequently interaction drive force pair move sampler obstacle additive fairly e move follow component subsequent pair move enable second pair move separate enable move toward grants es institute environmental health st recommendation author reflect section pair move neighborhood efficiently introduce neighborhood preserve stationarity pair move sampler predictor adaptation draw approximate posterior draw update beta l ap cd pd ps cd aggregate scale covariance let stand vector aggregate realization lx kolmogorov consistency gaussian response sampling proceed draw draw posterior quantile wise credible addition inclusion configuration thresholding probability run configuration evaluate invert neighborhood sampler update inclusion vector require iteration grow pair control exceed iteration cholesky aggregated enable involve overview inversion cubic order select control stationary pair move proposal balance verify pair remove add symmetry check swap swap proposal choose probability proposal reverse remove neighborhood contain q swap restrict pair reverse case inclusion vector remove swap probability pair reverse section construction predictor denote reverse pair likewise reverse paired expression move proceed pair pair swap probability move neighborhood construct section empty km pd ps pd j cd pd ps hence comprise pair swap preserve stationarity pair sampler predictor importance consider plot datum plot comparison section plot predictor importance bar trace additive interactive offer minimax wide case predictor large sample predictor effect response bayesian implementation interactive efficient markov hyper specification light computational consideration explore inclusion offer improve diverse real platform regression keyword model multiple try metropolis selection weak focus parametric linear regularize shrinkage assumption often model naturally occur predictor response relation globally remain quantify effect parametrization adjust non linearity several smoothing regression predictor relation mathematical performance various setting computational poorly evaluation importantly nonparametric assume predictor greatly curse set variable small e ensemble learner additive dimension address curse dimensionality may actual behave black box forecasting additive structure univariate ignore predictor unknown predictor additive interactive resemble distinct add learner boost efficiency avoid overfitte contrast interactive high divide piece add together enable learner seminal work gp indicate specification could suit motivate recent attractive interactive set additive interactive away extreme single bound smooth include minimax correspond restrict component one predictor increase develop additive interactive abstraction offer match rate significance present detail hyper allow pattern maintain adapt stochastic try metropolis additional strategy well state art interaction diverse section provide evidence interactive attractive platform high propose chain sampler effective extension work cx definite mean refer continuous supremum pair additive interactive perform joint proceed sequentially parameter regression conjugate analytically marginalization obtain marginal multivariate back fitting proceed enumeration intractable large sampling metropolis draw diagnostic stability sampling scheme posterior inclusion update inclusion random propose flip quickly grow move look assume inclusion vector walk toward add predictor rather remove search search quickly demonstrate develop explore straightforward adapt search additive add neighbor remove swap swap
conjugate analytical insight base include cg cg distinguishing include conjugacy general numerical principle conjugacy instance generalization primarily inspire cg contrary preferred justify proposal light algorithm issue parameter automatically note direction bfgs cg see think extension framework regard see role conjugate chance bfgs update quasi newton anonymous eps fill receive paper dependent namely generate conjugate preserve conjugacy value obtain cg provide cg solution wide real year reliable solver framework either consider considerable specifically aim pde constrain framework often specialized reliable iterative symmetric numerical analysis context detail linear generality implicitly assessment system people work great software development literature analysis become address equivalently reduce cg suitable choice primarily intend efficient mainly inspire quadratic generation parameter scheme conjugacy cause precision computation intend numerical experience clear proposal assess similarly currently proposal outperform cg carry selective extension cg symbol indicate definite symbol sect review cg subspace promise detail relevant conjugate direction motivate member property class sect conclusion include table process conjugate generalize share say cg iteratively cg step stop else k tp r ap symmetric often application cg residual search direction impose conjugacy condition prove implicitly satisfy practical computation fail property lose consequence sect detail purpose table cg eq expression generalize exact fulfilled process method cg cg cg generate sequence satisfie yield inspire order cg recurrence correspondence may cg extent resemble recurrence direction cg cg recurrence direction possibly conjugacy cg offer cg preferable solver statement briefly conjugate truncate latter rely direction direction outer perform step cg approximate table form framework suitably call curvature direction optimization see conjugacy introduce cg conjugacy loss great importance hand rule assess parameter satisfy thus conjugacy solve accurate introduce cg additional cg latter cg sect aspect work effort since independent necessary address current direction impose conjugacy previous automatically matter framework essential control computational item iterative might generalization indeed proposal item respect cg latter user conjugacy direction precision sketch table cg cm else compute else compute cm direction reveal difference cg compute eq coordinate conjugacy specify detailed sect double recover sequence compute impose orthogonality cg result cg additional inner scalar table evident provide term recurrence conjugate condition involve computation hereafter positive cg relation directly conjugacy direction e ap inductive yield conjugacy property trivial lemma theorem simplify available relation remarkable avoid storage step require storage cg stop condition result class cg orthogonality hold ap ap k coefficient k r tr inductive q along r tr hand inductive lemma prove likewise cg iteration assumption solution reader may integer chance table cg stop else else analogous cg prove see suitable inverse far minimize ap span indicate ip ap bp cp bp assumption table ia ia I latter yield tr ap p tr I tr yield functional inverse recall direction observe geometry might substantially contrary kp k r consequence directly similar cg cg might possibly observe cg step storage additional idea store base example approach may cg may equivalently explicitly impose conjugacy pair implicitly impose cg cg recall worth hold modification tp kp cg indeed cg k cg necessarily item cg relation satisfy cg possible latter conclusion order cg condition latter table relation red else cm cm else r cg satisfie position lead scheme red table recall substantially impose unique alternate cg analyze combine preserve cg
send describe algorithm order get start expert knowledge quantify transform level dimensionality iterative quantify fuzzy rule linguistic test realistic environment different robot environment result different application play central tracking mobile quantify fuzzy genetic fuzzy mobile behavior motion behavior whose robot environment evolve range etc order environment operate environment internal properly convenient cope interpretability rule fuzzy logic represent design mobile preprocesse raw sensor obtain sensor usually high mapping preprocessing automatically level describe embed avoid expert input variable controller sensor robot mobile e g capable dimensionality meaningful description proposition kind expression set level variable expression low fuzzy proposition formal evolutionary fuzzy rule combination fuzzy logic genetic system aim balance interpretability rule conventional refer low tree describe learn preprocesse mobile quantify fuzzy learn unconstrained e proposal design mobile internal robot sensor structure learn robot sensor preprocessing embed able linguistic interpretability use validate statistically combination preprocesse tracking tracking move obstacle structure present advantage mobile learn show point relevant conclusion machine evolutionary neural widely genetic fuzzy even combination evolutionary mobile get type logic knowledge evaluation function evolutionary function genetic fuzzy alternative membership membership distribute expert knowledge rule reducing learn main behavior curse among evolutionary position learn sensor competitive proposal learn involve label unconstraine multiple approach adjust balance competition rule behavior comparable proposal output control category establish level modeling hand provide sensor relevance group significant decide analyze individual range since gap frequent environment usually consist measure measure right mobile low stage traditionally knowledge preprocesse expressive meaningful within quantify fuzzy proposition useful belong clearly set proposition conventional preprocessing stage high variable group fuzzy high fuzzy apply reasoning tb evaluation evaluation equal equal min check ex robot two individual sec approach produce individual context structure define compact v terminal symbol separate leave two consecutive group order level symbol linguistic prop ii linguistic fig prop prop measure linear linguistic linguistic label linguistic approach universe space uniformly var five linguistic individual example label linguistic membership mask label limit apply linguistic use measure finite receive triangular search mask tb velocity initialize two linguistic velocity fitness calculate th define meaningful output example maximum minimum regression several desire interpret individual code individual fitness population ability support cover final line admissible cover covered support calculate cover total combination strength generalization matching go define objective information proposition rule important individual initialization population operator generate individual proposition proposition proposition one select individual accord follow criterion linguistic high r b individual copy yes yes proposition combination take similarity fig take partial merge could rule individual individual proposition eliminate partial merge combine proposition done minimum performed mutation two generalize rule high value mutation high confidence rule cover discard select min example give similarity individual cover therefore proposition mutation proposition modification among possibility generalize adjacent repeat decrease proposition adjacent process proportional example low probability modify proposition modify modification among possibility velocity proposition strategy mutation finally mutation mutation part select fig membership close label close one tb mutation selection replacement steady state new population epoch rule criterion fig limit vary stop consecutive iteration maximum stop regardless ends add moreover mark cover algorithm line fig part select subset rule follow basis rule rule base code indicate th good rule rank fitness execute set last implement search threshold well objective controller suitable robot move high control player robot software environment also real robot range amplitude scan loss generality robot sized distance value input principal component extract variance cover total use preprocessing change percentage input configuration configuration preprocesse preprocesse tb statistical significance table fold algorithm mean deviation eq tb train c straight concave convex concave convex min sample fold cross example preprocesse validation low configuration show include able adequate algorithm mobile controller good validate controller environment different difficulty assess velocity path execution simulated environment fig indicator linear velocity linear consecutive cycle reflect along robot robot parallel five calculate environment present five symbol table path period time controller velocity form purpose reliable robot alg preprocesse min min tb hc alg cm home home office home home office home office min home home office home home office min home home office home home office environment hc alg min min environment c except distance perfect situation result obtain low like curve get adequate velocity smoothness robustness fail environment recommend compare result hoc detecting table significant environment velocity corner velocity change velocity robot value preprocesse c hc alg cm office office office office finally proposal rule behavior purpose comparison expert sensor weight linguistic local evolutionary orientation table common environment environment learn embed method rule linguistic denote robot describe continue zero velocity get robot cm typical number base multiple show term standard deviation input tb alg output min c c proposition straight min straight concave concave knowledge rule per thus demonstrate general basis mobile tracking application behavior describe literature guide robot predefine guide person show goal service service path team build surveillance environment operate area environment numerous author able path uncertainty control measurement robot static difficulty combination perform people track environment task safe must endowed implement allow order mobile must application guide robot predefine initial new people make necessary avoid return predefine quickly reference guide come close obstacle maintain addition robot track object move mobile behavior fusion track controller controller robust neither avoid obstacle obstacle describe obstacle side solve depend obstacle detect robot value robot establish distance safe tracking behavior angular robot place move object different robot coordinate robot coordinate robot robot negative indicate robot move point move robot move angle object robot angle move object track perfect robot keep htb colors code medium path grey move obstacle path dark grey f place move behavior validate different environment try reproduce fig track medium grey path follow include robot mark velocity moreover without environment place track grey path indicate trajectory robot avoid successfully robot predefine obstacle generate robot obstacle return predefine path quickly possible move fig light grey represent robot also medium grey trajectory object go situation controller execute execute corner close robot behavior move obstacle fig light grey follow robot track grey avoid follow moving avoid show
break tie combine enhance accuracy execute time integrate draw kernel execute membership partition ensemble obtain fig illustrate cluster randomly datum cluster consensus membership sample matrix similarity complete weight meta meta vector update accordance less mean algorithm kernel portion need store classical expensive avoid eigenvalue large solve optimization rate accuracy computational cost ensemble additional empirically sample reduce considerably binary random membership express give matrix u minimize n n complete add kk n identity special matrix first twice q mean approximation introduce trade speedup denote coherence adapt gap refer experiment examine different sample size range satisfactory memory vary cover table small medium imagenet mnist processor performance demonstrate scalability improve implement matlab matlab toolbox available house ghz processor limit imagenet pyramid cover imagenet mnist kernel algorithm imagenet consist million concept know sift descriptor extract handwritten available image represent dimensional training test compare approximate performance compare step algorithm find representative mean imagenet employ pyramid pyramid effective take histogram pyramid mnist neural range directly demonstrate naive class time matrix cluster initial final clustering calculate ari adjust rand lie matching partition algorithms table table list run kernel algorithm speedup mean take need cluster later execute pyramid pyramid play mean column ari algorithm mean partition partition show achieve kernel achieve lower sample significantly achieve indicate insufficient center randomly kernel mnist spend calculation simplicity size fast ari column inferior partition see amount except algorithm well comparable mean cluster c c time cluster imagenet strategy table fig table pre sort index execute non great high complexity sampling fig inferior compare also produce scheme uniform strategy strategy spend cccc imagenet comparison eps large type scalable use dramatically reduce requirement survey us forest forest attribute qualitative like slope distance use group type region resource information network datum dimensional seven class represent traffic represent traffic set currently infeasible store reduction take hour large demonstrate non scalability eliminate algorithm evaluate range increase cover employ pairwise similarity employ set tune optimal number cccc run mnist calculation compare cluster less algorithm effectiveness run algorithm fast achieve error set imagenet forest network cccc c take averaged run set show ensemble especially cover significant improvement cluster small thereby efficiency avoid kernel restrict center small algorithm yield well popular integrate propose algorithm enhance enhance scalability plan acknowledgement office grant foundation research association ci use matrix partition full extension include definite coherence coherence column contain ss provide positive constant value row j ss combine equation pt extend enhance tight kernel approximation ability various gain popularity iterative ease run term size large cluster mean cluster center well demonstrate requirement quality employ cluster meta advance storage year massive amount generate service audio one tool amount web medical set group hand kernel cluster employ distance object cluster ability capture linear perform distance cluster som kernel neural propose mean simplicity efficiency addition several equivalence replace euclidean function storage kernel cluster scalable thousand learn cluster adapt use low kernel name follow idea avoid center vector span subset portion kernel lead speedup approximate explicitly yield efficacy relate scale kernel algorithm incremental cluster implementation restrict requirement datum cluster center center though complexity memory match unless large superior kernel mean fact mean center combination word span x n center small small ii simple denote give similarity point cluster membership
text collection practical multiple leveraging share potentially enhance multiple news ad etc explicit social make available interaction user complete observed leverage correlated component affinity among type wherein entity low dimensional represent entity leverage share attractive often view interest capture recommendation collective completion additional collective statistically ill pose decay observe observation localize individual entry oppose measurement completion development statistically optimal collective previously analyze collective completion trivial recovery collective collective completion completion challenge collective extension complexity exist completion leverage low key collective normalization observe general collective structure joint may behaved enforce avoid case assumption relaxed contribution algebra collective identify assumption feasible tractable collective consistent collective subset collective exactly collective logarithmic program adapt collective matrix scalable algorithm significantly use tradeoff accuracy paper simulated besides relate probabilistic seminal rank collective al wherein parameterize share factor collective author collective factorization completion guarantee scalable algorithm probabilistic collective develop algebra analyze collective denote letter etc matrix denote singular norm etc affinity primarily represent list affinity entity type entity affinity relation wherein pair type affinity relation entity relationship collective denote entity imply either collective graph connect handle entity instance collective common entity view convenience introduce alternate equivalent collective matrix paper collective represent form entity statement concatenation list collective represented identify block wherein block representation collective collective provide j collective possess factor dimension denote value joint factorization exist collective convex extreme symmetric sec collective hull interest also iff atomic program collective collective denote basis far without noise e pose dimensional etc low impose ground truth collective collective entity see rest projection onto iff matrix entry significant analogue assume incoherence basis c onto factor dominant atom need pose subtle challenge collective undirected assume equivalently odd cycle induction verify note collective completion necessary j n si k cardinality scheme expectation learn give entirely consistent depend convenient derive collective rank collective use atomic suitably modify sense practice program collective algorithmic consideration state recover truth collective cardinality cardinality type collective incoherence requirement meet kn cn kn kn free collective scalable adapting solver collective atomic state convex program collective cast solve loss z z u propose estimate primal error curvature function iteration involve large eigen non converge eigen accuracy collective rank entity sample present paper factor collective low also impose component feasible component standard matrix estimate standard complete entity independently recovery complexity sub completely share jointly collective collective also cast standard complete block collective partially coherent fail strict exist task collective optimally share dimensional sample exploit narrow subspace analogous show condition exist adapt introduce al c vs supplementary sm km lemma follow assumption sec exist ty f proof supplementary material complete construct partition r sign satisfie follow analogous proof fp use inequality use section intend ground collective entity entity collective
extensive accuracy suggest greatly partially explain regret near neighbor knn latter simplicity category conceptually organize instability minimax section propose classifier achieves present comparison exist neighbor stability devote couple take regard label object probability borel classification define rule x py define distribution classification instability x distribution define classifier measure instability measure classifier denote distribute copy training instability classification instability obtain classification ease procedure knn knn gaussian setup vector identity versus knn calculate classifier aggregation decrease sum view balance knn classifier minimal mark slight lead improvement stability confirm theorems extensive experiment sequel study function nx first show derive two condition satisfie contain q integer say support denote lebesgue center radius lebesgue deduce function distribution respect margin condition worth note type addition hold newly minimax lower seen bind old marginal optimal set requirement large stay long take lastly slow increase introduce attain review weighted neighbor explicit propose novel call moreover theoretically sense fix distance weight ni ni reveal compact nonempty open ii conditional absolutely continuous lebesgue differentiable exist dx sx x ax volume asymptotic knn pointing condition see assumption ensure vector near section classifier special subsection classifier sequel instability asymptotically appendix sketch detailed include nx ni e dx nx last area proportional asymptotically large knn meanwhile instability serve develop term expansion call variance instability ready procedure regret determine minimize acceptable lagrangian minimize ni depend multipli lead knn proposition accuracy classification instability lead respect minimizer weight classifier rate define assumption procedure corollary stay away approach theoretically comparison exist knn procedure significantly improve regret classifier knn neighbor apply near neighbor subsample majority vote sufficiently particular replacement approximately resample lastly achieve minimal denote classifier knn difficulty weight classifier step knn knn ratios notable ratio merely dimension corollary plotted figure stable knn procedure large ratio knn equal less bag variability phenomenon ratio bag knn accuracy furthermore quickly stable vanish regret note great compare ratio reflect characterize corollary constant improvement ratio figure show ratio function fix dimension increase increase ratio get grow htb investigate improvement large regret relative improvement percentage expression show logarithm large confirm introduce simulation base validation tuning subset proportion misclassification parameter value lead weight calculate summarize subset train label classifier j step repeat training search tuning minimize cross validation preference weight tune select whose risk pre choose set minimal instability subsection form derive ratio least sampling region replication calculation theorem risk see minimize lead figure along asymptotic asymptotic one carlo although slow phenomenon theoretical corollary accord ratio classifier ratio indicate increase however appear value cause classification error estimate issue classifier classifier subsection neighbor tune equally space comparison equally spaced fall simulation underlie probability choose df toeplitz entry classification empirically verify comparison combine latter test datum replication figure classification procedure case regret even explain dramatically small particular improvement procedure procedure knn procedure phenomenon advantage prediction big summarize obtain minimal achieve minimal scenario large see appendix slight procedure minimax knn bayes bayes sim subsection investigate knn machine survival diabetes heart heart information dataset randomly procedure error specifically procedure improvement addition knn procedure knn agree slightly illustrate accuracy significant improvement knn error error acknowledgement like thank mathematical sciences massive part thank communication em independently bayes n n n ease nx x p last prove call hypercube mapping denote positive define partition satisfy collection hypercube absolutely continuous show variational specifically eq hypercube hypercube supremum rademacher corresponding p n lemma nx part separately show nx appendix nx nx nx x c properly ni n ni nk ni ni ni ni detailed discussion constant substitute du b du plug du iii plugging iv lead desirable conclude e valid independently identically sample without give norm nx ni boundary x nx x n ni n show analyze complement combine apply normal yield derivative respect lebesgue measure tx theory integration manifold since r similarly argument imply contribution ps nx mm w apply let ni imply term due modify expansion accord sr rs r r density normal tx x ax ns dr argument substitute difference du tx dt du iv
use collaborative item easily optimize factorization nice mainly representation new ml yahoo regard ask rating use mainly quality protocol simulate realistic process income proceed user user denote user dataset model remain validation apply subset ii answer rating answer question rating rating rating predict rating system predict regard miss explore quality approach use equation collaborative mf present item knn pearson cf inductive model train rating phase testing tuned gradient size latent thus parameter grid randomly initialize figure table evaluate split evaluate predict provide result less information obtain ability additive user four obtain mf score yahoo knn representation belong moreover knn slow unable deal beyond able predict user time mf ability realistic mf item benchmark select item item rating popularity entropy g baseline rating inductive coefficient cs user pca depend positive rating concern modification consequence concern norm smoothly start interact provide easily integrate translation representation recommendation I warm learn regularization explain learn choose incoming result set protocol new set illustrate add warm extension link warm promising percentage add american iii star episode iv star episode vi matrix private lose selection recommendation past user mix rating informative technique distinguished neighbor item similarity item infer factorization technique collaborative major limitation process ask paper comparative question criterion like popularity greedy minimize seed also consider choose fit user branch correspond answer present node allow contain regressor model one tree warm interesting usually adaptive user usually one rate collaborative filter representation user latent translation depend allow user rating recommendation direction certainly incoming process consist review item user build ask new investigate I acknowledgement article rapid approach collaborative filter rating rating approach handle user start context ask initialization rating present good ask efficient representation context start show rely extract useful highlight seem crucial go intelligence gain parallel recommender system field research variety application aim product facilitate experience recommend recommender implicit movie star item song post movie actor common recommendation additional context method problem representation compute user us user user rating proper rating item latent representation user representation denote classical rating dot denote predict learn sparse rating objective observe rating user representation item loss propose since item nature well adapt practical application item require unstable limitation factorization approach rely representation e must indeed mf cf base interact rating make recommendation method suit focus method rating ask recommendation approach context simultaneously learn rating building representation inductive rating latent allow contribution formalism user formalism build simple nature user dataset start quantitative effectiveness qualitative learn organize generic representation present discuss related collaborative filtering propose rewrite detailed integrate start consider item building collect set rating compose item rating opinion typically movie provide rating movie select relevant collect item user thus rating loss cf free balance simultaneously build formulation easily mf base obtain computation complexity idea concern build recommendation rating representation suitable cf
finite increment eq collection q prediction depend segment roc confusion roc define interpretation follow plot receiver operate roc curve join roc integral curve roc curve curve roc sample observation parameter roc voting threshold word parametrize forest parameterized nn well prediction true roc would generate since evaluate prediction area allow true disadvantage course classifier divide make evaluate discusse divide classifier denote simply evaluation set accord depend nn trees forest divide run testing usually set often repetition consider train explain follow allow classifier labels label assign evaluation testing iterate q particular fold one set figure illustration process disjoint cross validation run multiple parameter determine fold commonly use evaluate simple suffer possibly train often furthermore conversely disadvantage repeat consume chapter evaluate accuracy f divide training evaluate chapter explain methodology forest genetic dataset nucleotide snp thesis reduction come heart study detail algorithm cross heart file thesis patient control patient file storing simplicity two count list snps dna microarray patient determine three snp minor probability near directly disease transform transformation let minor equip domain nn goal section explain methodology classification genetic cross project genetic projection nn forest random genetic virtual laboratory increase increase snps classification merely threshold forest result obtain technique method forest able predictive accuracy area roc considerably higher nearest chapter conclude thesis limitation short thesis two nn forest dimensionality random term thesis nn highly desirable learning algorithm finite thesis justify distance supervise closely thesis compare approach genetic study reduction comparative selection random considerably nucleotide snps important selection classifier receiver operate curve previous area thesis science perspective although evaluate random correctly snps entire genetic important community validate predictive genetic trust ideally snps genetic snps label snps concern certainly improve thesis concept control whether included ignore thesis recall thesis snp assign find raw dataset normally snp high snp check condition snp satisfactory snps control thesis control remove snps start snps considerable remove question remove related disease experiment control remove demonstrate forest exactly snps score yet high previous evidence result dimensionality reduction quality decrease snps efficiently case quality reduction handle three snp thesis classifier give predictive classifier machine neural advantage snp discrete distance simplify pair snps common simplification assign snps theorem provide margin would continue develop reference study classifier margin limitation thesis new predict snp information hope address limitation thesis future far classify disease day accurate whether patient disease discussion science parent heart science subject intersection mathematic valuable large technology large magnitude information big practitioner computer often cloud compute big science notion detect email detect email spam amount past spam spam email spam give equip spam spam algorithm classifier information observation mathematically present observe denote know major challenge big dimension computational map label would simplify commonly know learn near machine dimensionality include component pca discriminant lda main goal thesis dimensional dataset disease disease heart severe lead heart attack variation repeatedly variation account prediction disease individual variation dna base call nucleotide snps successful diagnosis understanding behind variation snp dimensionality thesis explain random novel theory mass introduce thesis recognize science predict majority vote important theoretical consistent rough arbitrarily predictive ability become partition hyper belong forest generalize trees bootstrap label vote unlike forest consistent excellent practice dimensionality euclidean lemma sufficiently absolute multiplication variable training via nn project pairwise preserve guarantee second reduction mass problem natural marginal separation infimum integral metric simplify consequently dimensionality take value coordinate finite correspond two eq dirac occurrence observation mass separation dimension coordinate distance detailed comparative label contain nucleotide snps heart study nn genetic space prediction forest select equivalently snps dataset apply prediction predictive f area receiver roc curve result predict disease subset genetic area snps snps snps area roc high whose snps classifier contributions thesis theoretical contribution thesis consistent although result direct consistency nn euclidean three universal lemma sufficient nn thesis see thesis introduce completely mass distance thesis practical thesis nn heart study disease nucleotide apply practical contribution thesis approach apply achieve curve ever obtain thesis structure foundation explain concept chapter discuss genetic dataset thesis disease genetic single nucleotide snps genome association chapter survey generally application chapter near finite forest include classifier chapter discuss feature selection thesis outline motivate justify explanation distance three area receiver operate characteristic chapter method divide evaluation estimation classification dataset chapter genetic consider thesis include snps methodology projection nn random genetic list reduction projection chapter summarize first projection approach brief comparative finally chapter conclude thesis address limitation list direction throughout theory thesis mathematic science thesis orient intend reader thesis explain reduction genetic result biological biology genetic keep master thesis biology question provide thesis define predict label dataset notion dna level genome study genetic formalize algorithmic sample applying theory base coordinate label sample label pair pair merely classifier define satisfy false classifier classifier since proved know universal consistency important supervise prediction infinity define learn call size far define universal become terminology learn family classifier take consistency rule take sample thesis simply always rule stage train predict explain know provide understand biology behind thesis health brief dna nucleotide explain genome wide association collect introduce work literature involve united number death disease disease occur build heart lead heart attack death understand important care population increase show diabetes stress prevent accept biology heavily regard whether disease predict high purely individual thesis attempt answer term information nucleotide explain biological study pass trait parent trait molecular dna encodes trait pass human dna contain repeat together complementary commonly dna string omit determine individual double paired structure dna base base pair million organization person genetic genetic dna genetic pass parent variation sometimes dna precisely trait variation pair dna nucleotide base nucleotide snp normally variation position nucleotide snp allele less term allele snp dna possible major allele minor copy minor allele copy allele dna variation snp minor million genetic among human dna level extremely certain disease explain genome snp associations physical trait nucleotide association trait consider trait dna microarray individual snp trait correction study snp mostly disease coordinate dataset individual microarray due microarray possible explain study detail include precise master thesis consider genetic predicting disease snp genetic thesis science study predict trait result publish survey past logistic regression receiver operate roc explanation show associate publish supervise compare support predict diabetes curve snps multiple snps observation forest ms snps roc curve ability machine dataset disease genetic snps disease modify classifier accuracy considerably also study disease two area roc curve snps forest study fairly argue trait investigate disease focus certain supervise reduction explain chapter explain science near consistent introduce section universal follow space equip distance nearest classifier rule search near classifier pair regression x distance x normally odd tie value life application determine figure illustration nn generally accord define negative add classifier sort observation equivalent nn nn introduce cover develop publish euclidean list together universal consistency use euclidean section explain particular classical generalize find nn theorem generalize consequently theorem universal require proof classifier condition non finite q notion convexity jensen along lemma entirely base pair classifier accord estimate quick scalar function jensen proceed show indeed estimate regression classifier q measure theoretic jensen jensen hold lemma right arbitrarily uniformly continuous middle term consequently eq since x prove satisfie generalize lemma prove consistency version whose order induce depend covering radius respectively ball sphere open radius dimensional finite sphere finitely require figure construction note sphere ball need circular prove imply since since span figure triangular generalize first depend unit still near inequality mark x less universal nn classifier nn eq assumption chapter demonstrate metric consistent classifier explain another tree chapter introduce science decision train predict forest decision construct decision tree build majority classifier build binary predictor explain classifier implementation classification cart label divide disjoint splitting feature homogeneity two recursion terminate divide contain class long improve label decision force geometrically correspond possibly axis consider observation provide formal explanation decision notion homogeneity term along tree coordinate assume ij discrete label homogeneity maximal contain classifier subset weight entropy entropy entropy label coordinate decision training entropy training split coordinate termination meet class termination partition associate observation observation would accord calculate recursive tree model parent divide node sample split recursively termination upon termination become leaf correspond dominate predict observation pass tree predict leaf fall package programming life predict representation example build new sound feature great observation classifier generalize first forest bootstrap take bootstrap replacement construct q second selection find build tree new decision prediction forest prediction classifier training predict total forest depend split usually validation explain excellent effectiveness easily generalization forest justify show classifier consistent fact common distinction theoretical produce accurate life absolutely support excellent predictive run chapter supervise forest generalize former classifier detail chapter introduce field extremely discuss concept section introduce explain section popular selection novel mass big genetic thesis include coordinate nucleotide computationally expensive may constraint technique often dimensional simplify e space training feature extraction reduction project threshold map property hand feature reduce transform simple projection define expansion method borel reduction introduce far easily feature extraction projection function introduce domain assign mass method integer exist map visualization distance theory nn distance map project pairwise guarantee simple constructive finding extraction work pick training x x tx include thesis justify via random matrix multiplication supervise section probabilistic provide proof constructive finding multiplication tail sub tail sub sub tail variable imply generate constant consist sub tail entry satisfy hold cn theorem generate pairwise enough great usually practice determine optimal purpose relate sub constant sub tail conversely suppose sub tail combination sub gaussian suppose real uniform prove tail euclidean prove transform distance classifier reduce save analogous also either discrete explain new distance method distance wasserstein explain infimum measure think minimal moving probability distance infimum extremely difficult distance else discrete metric space eq mass simplify distance simplification mass suppose hamming probability measure observation respectively theory exactly x consequently sample induce probability probability thought estimation
month date place much scenario unlikely person c give name absolute month difference agree disagreement since modification compare string simply name may piece token name token token token transform mean total token disagreement reader detail construct disagreement level except disagreement disagreement moderate disagreement take nominal truncation classified inaccurate nearly truncation parameter inaccurate fix year collection parameter field amount prior priori extreme scenario month name inaccurate versa context point posterior eight concentrate gibbs sampler supplementary discard burn eight partition although way eight concentrate day inaccurate posterior concentrate record entity record result coherent indicate field inaccurate record family name think inaccurate month fairly quite get equal day month think inaccurate name concentrate name become distinguish record therefore record probably finally partition posteriori record quite record assign partition properly uncertainty record record emphasize example determine posterior evolution membership gibbs sampler supplementary material five contain large file depend file heavily influence explore sophisticated datum generation corruption tool contain tool field permit generation adapt default describe characteristic file simulation file either seven involve include gender file seven phone number gender name jointly table frequency name name set name source phone number eight digit two field include default generator contingency serve contain category eight interval create allocate randomly select assign accord poisson interval allocate contain uniformly possibility miss error string error optical recognition use finally possibly name family name age gender phone pt name family name age interval agree gender agree agree phone code performance amount file field synthetic per file create indicate file disagreement name constitute prior truncation carefully truncation scenario believe file optimistic run iteration discard runtime implementation part language second file include comparison ghz processor start long chain report wrong another source gets code name although collect region indicate occur potentially give six token example record overlap token token la pair meet constitute remain use illustrative name standardize compare supplementary material record level either year month pair introduce record present f indicate belief likely exact still expect field agreement year death ij death high still go truncation interpretation record c truncation remain field believe error become unlikely magnitude indicate probability observe disagreement disagreement example p ij priori year death death expect year two year disagreement specification probability disagreement year table finally day death believe expect report node width pair never appear together group obtain sampler present sake record preprocessing pair graph package subgraph clique illustrate trivially pair preprocesse step partition get color width pair appear group together chain never appear appear group together method ensure output partition partition unique record minus cell file contain unique interval greatly vary file summarize different region leave percentage correlate report relation panel show percentage year ground truth important whether take region death record identify treat label record ground decision check idea partition partition file also like result change choose alternative prior one optimistic sense one truncation table subtract truncation additional keep fix table point indicate bold recall sensitive truncation recall robust optimistic agree finding file number application balance optimistic issue pairwise component classify pair pi estimation posterior triplet vary gibbs surprising treat model hoc methodology decision prior show illustrative realistic methodology time indicate report methodology usage distinguish point partition population important file account future multiple file incorporate record linkage procedure acknowledgment document discussion comment suggestion ball green file help generation synthetic brief standardize name implementation application node nsf census research nsf keep accurate account record detection independent status pair record ad hoc fashion file pose file group record present file interest ensure decision implementation incorporate file available decision multiple united truth record refer entity file entity file miss existence file need wide health census quality improvement arm receive report report come degree detail file step keep accurate form united occur report friend multiply leave national front sign agreement united henceforth report occur focus individual country information publish family member friend addition name occur friend detail provide nontrivial file variability record miss datum challenge difficult reliable record miss process field file linkage multiple file usually collection process assume file despite principle article approach record linkage model approach train know record pair type decision status record pair neither decision record truly record record ad hoc recently detection linkage decision distribution file file therefore currently categorical continuous model field name address phone detect field often advantage base pairwise comparison record meaningful record linkage basis decision currently take account decision modification detection problem propose build literature decision partition file closely relate record datum idea disagreement field introduction file natural similarly organize propose methodology deal illustrative address detect times united truth conclude file record record file grouping record entity entity represent think record file call representation convenient nonempty subset article entity record representation computation consider eq contiguous file record represent inefficient record alternative arbitrary cell record file safe assume entity file assign label potential entity entity labeling entity lead partition indicator label relationship obtain specify notice element possible fix get record rapidly record grow practice file fortunately early stage inference file entity record compare agree field completely agreement field normalize distance see divide set approach disagreement field appropriately fashion field divide interval disagreement agreement include high complete disagreement record comparison record record linkage construct inference require range functional field build disagreement variable generic long question threshold build level specific disagreement disagreement extreme disagreement practice simple number obvious early detecting reduce inferential record translate fix matrix turn assign record group detect refer survey divide file record categorical field reliable unlikely pair record field gender code record appeal divide type expect would unlikely record predefine ideally check record event date date death date record true containing field fashion context unlikely among record distant naturally assess ideally comparison string metric complete comparison comprise file still obvious combination different disagreement inferential disagreement field age record meet behind approach distinguish record probably record pair unknown candidate comparison present record partition constrain record already practice much partition file heavily rely able candidate medium file comparison ij comparison though realization comparison array compose observe among intuition formalize pair entity regardless record pair entity assumption widely employ file record linkage intuitive formalize comparison leave observe comparison fix depend depend formulation candidate factor ss partition take I proportion pair assumption decision undesirable denote z z n measure label label notice structure prior appropriately investigation commonly encourage formation cell dirichlet multinomial partition compose cell describe criterion record common record field miss comparison pair record incomplete field record assume miss base inference observe decompose record ij ij sum probability obtain arise miss comparison partition scenario except th record model membership arbitrary labeling use present use refer entity ratio square hand represent testing refer entity accord record pz I take proportional label exhaustive state cell record partition ratio likely get cell material gibbs parametrization assume field take distribution record model multinomial disagreement rewrite conditional specification parametrization pair f binary model record linkage
perform quantization train classifier joint manner optimize simultaneously column encodes pair verify obtain also iteration procedure radius classify three classification rate classify divide result bar generate classifier close superior bar output sketch cc black curve naive sketch number use mean recover rank track essentially constant bar correspond deviation face sketch classical sketch focus broad constrained showing theoretic view novel iterative scheme know iterative hessian derive showing grow dimension gaussian cone take addition also evaluation reveal optimality classical nuclear program naive sketch behind iterative minimizing square subject problem norm especially data sketch base cauchy paper technique obtain solution acknowledgement support office grant national dms addition microsoft fellowship appendix verification first let singular invariance showing case pick matrix row use row balanced let jj j mp show claim packing semi conditionally x denote result observe statistical estimator bad square infimum suffice first k kullback kullback leibler kl divergence eq q setting sketch feasible program piece suffice claim bind successively iterate optimization update original adding shorthand vector belong yield claim error estimate star notation complexity set star shape integer small refer localize measure bound square constrained constrain square together illustrative convenient shorthand claim rip property imply use width since see hull low property rip property vector norm consequently conclude claim upper straightforward e constant theorem width recall minimum singular g put throughout adopt shorthand claim expect respectively shorthand u event return prove bind combined bind great claim lemma u inequality g final view vector lipschitz constant concentration q final definition put piece claim cm section definition lemma berkeley edu california berkeley electrical department approximately solve square constraint assess way minimizer approximation focus randomized square surprising sketch sub present original least square unconstrained constrain constraint approach real experiment past decade datum procedure interesting arise frequently vector constrain simple case unconstrained class include constrain program nuclear ball enforce line book random solve possibility study vector approximate version dimensional square involve new substantial substantially solution assess term solution example unconstrained least square random size paper well reference similar base statistical leverage analogous sketch whereas past sufficient cost approximation notion minimizer oppose prediction course cost bind derive solution sketch instance use al unconstrained ensure cost bound satisfactory normalize quantity grow sketch observation expect provide population illustrative order suggest order require undesirable regime sketch precede rough least sub poor behavior least sketch unconstraine red correspond curves correspond sketch apply projection correspond algorithm mean optimality problem standard square regime sketch nonetheless main square use size underlie hessian sketch iteratively refine chosen logarithmic background turn statement theorem least hessian section consequence defer summarize equivalently background class randomize include well orthonormal low solution observe motivate investigation investigation sketch serve iterative hessian construct optimal type randomize restrict attention matrix sketch I particular instance vector rademacher refer straightforward point view however disadvantage gaussian multiplication random operation multiplication randomize sketch define sketch hadamard fast hadamard transform random uniformly vector give multiplication instance give sketch sample row canonical different weight I lower balanced section apply kind begin consider ensemble square namely constrain maximum characterize necessarily large refer apply eigenvalue symmetric discuss lower also involve measure norm pack optimality appendix combine theoretic understand sub ordinary simplest imply low sketch undesirable proportional reveal surprising sketch accuracy match optimal sketch round see precise figure panel curve previous curve round use fair mse sketch relatively flat bound good drop square involve variant linear ball entry fix square solve randomize early unconstraine nuclear sketch apply return bind analogue hessian suffer namely require hessian building novel iterative hessian sketch match square reasonable sketch begin underlie summarize event hessian sketch long dimension large ensure hold give hessian problem optimum sketch optimization approximation yield sequence whose decay geometrically iterative sketch take return summarize follow sketch combine compound event tn iterate lower base implement sketch datum show proportional choose panel illustrate result convergence increase geometric sampling assume sketch choose least corollary immediate consequence omit illustrate understand improvement sketch sketch sketch solution guarantee sketch sufficient scaling size note square use randomize sketch conjugate gradient lead reduction however type specific least square least implement use hadamard sketch iteration total total width pair unconstraine width bound solve case say involve constraint interior solve g guarantee equal consequence particular square prediction minimal square random dimension iterate obtain accurate original square approximate goal quality show expect measure ordinary least square error consequently tolerance perform roughly summarize run algorithm bound great confirm ensemble behavior confirm bar figure bar height average error run use sample confirm green bar dimension finally bar run sketch total large bar run bar dimension red bar correspond number twice constrain relaxed basis pursuit user radius zero entry illustration constrain keep
surface range agent starting collect surface spaced entropy use entropy position continue overall maximize initialize random partition continue tend enter py py intelligence angle cognitive side great area many computational intelligence act intelligence multiple characteristic aggregate especially relevant recognition effort broad theory goal effort inspire physics agent act maximize propose path derive entropy way walk global work paper arrive perspective concept intelligence computational sense entropy facilitate intelligence prediction reality training set show meaningful simplify include apply presence sparse paper discuss underlie collect discuss implement discussion discuss abstract like aggregate complicated role play present theoretic intelligence entropy drive paper attempt intelligence accumulate take intelligence randomness environment definition intelligence environment follow draw utilize discussion implementation artificial intelligence computer even discussion intelligence disagreement typically school thought act think entail problem solve direct behavior intelligence processing event event prediction optimize ci entity agent together event provide intelligence discuss research key theory amount uncertainty random among interpretation much deep nature core physical reality central physics theory review formulation denote denote content expand rooted physical reality central concept everything chemical entropy map straightforwardly system give serve summation physical reality relate exact state specifically shannon information remain boltzmann serve convention concept important limit although wish much set intelligence member input map map converge intend reflect fitness eq definition minimize involve norm shannon take log minimize minimize repeat logic sign section intelligence entropy concept element find minimize satisfy whenever take state always energy transition increase entropy environment equilibrium due suppose decrease entropy rest entropy amount great equal area physic extend conclude intelligence intelligence process discussion first unsupervised algorithm shannon minimization behavior act consist element group like neighborhood entropy use genetic member reach organization avoid case one operating system source enter py please prototype concept optimize test comparative
eq negative pointwise light make appropriately outli phenomenon order phase transition phase plausible robust correspond two component show outli outlier remove negative margin real occur indicator tend remove case margin concentrate margin indicator open interval relaxation non margin take variant classifier svm dc us validity numerical use outlier contaminate original contaminate decision svm outli parameter maximum function plot panel leave show respectively panel denote percent value inequality agree violate accordingly unbounded kernel robustness confirm region right panel get setup contaminate kernel work poorly bad case beyond violate thus learn provide useful mm ccc plot cb compare generalization robust robust dataset present library language negative running learn standardize zero deviation randomly split robustness chose label change add label robust svm accuracy learn five cross select often contaminate outlier express I kx iteration element hull decision present decompose lemma ii inequality violate index assume assumption ii lemma let contaminate dataset index outli way define note cr c imply ji k hence boundedness gram contaminate contaminate contaminate make possible outlier q therefore primal problem contaminate rational data sample negative outli obtain decision svm base contaminate define contaminate bf bx margin drop dependency permutation estimate unchanged contaminate non contaminate datum guarantee q contaminate non lead addition statement hold assume mean hence equal great negative number due ranked lead ii resp margin express long result sufficient theorem reproduce rkhs inner boundedness kernel become empty empty therefore index assume prove dominate eventually slope argument proof em vector successful popularity serious cause outlier deal robust outli bound investigate point contamination still give contaminate show regularization algorithm formula guarantee grid validation work candidate experiment explain svm world data misclassification balance maximum problem separate plane reproduce space variant svm generalization study svm drawback remarkable separate mainly misclassifie sample misclassifie significantly affect svm outlier svm penalty hinge loss misclassification convexity cause unstable outlier unbounde put one way instability replace simple hinge loss statistic statistical long kind mathematical analysis influence measure robustness variant robustness convex rkhs influence rkhs estimator yu et function convex study learn provide deal standard regularization provide tune introduce variant learning remove relate robust svm main contamination information contaminate inequality conversely prove inequality violate partly boundedness one desire help grid conduct study assess kernel consider property outlier paper setup review devote svm dual intuitive great robustness evaluate investigate statistical order generalization numerical conclusion appendix let notation natural finite set express reproduce hilbert space rkhs denote rkh resp resp I produce output test sample accurate training take rkhs endow kernel misclassification penalty hinge loss precisely svm decision threshold interval range avoid preferable input coefficient support provide support infinite space reduce quadratic rkh non parametric statistical pointed speak average permutation decision express minimize margin svm obtain hinge svm regularization instead svm svm make provide appropriately variant svm parameter svm replace outlier outlier variant influence outli indicator intend ratio outlier formalize rkh margin zero difficulty robust robust detection linear classifier solve semidefinite problem level margin include middle difference learn method refer robust emphasize robust rkhs kkt difference dc use algorithm programming efficiently obtain base derivation dc dc objective value finite objective argument robust svm loss programming terminate consecutive phenomenon convergence define compute sort component multiplication unchanged decision outli indicator picture geometrically dual lagrangian slack negative multiplier geometric expression training bound non rkhs reduce domain label dual distance estimate scaling rkh evaluate robustness measure evaluate influence estimator training sensitivity case refer quantify outlier contamination necessarily large amount contamination contaminate take family contaminate dependency drop space let express boundedness rkh decision boundedness contaminate condition contaminate dataset retain indeed unbounded regardless since prove rigorous proof omit rkhs great conversely trade ratio result correspond outlier reduce svm necessary greater rational svm strictly robust svm formula label negative integer note guarantee point decision less boundedness bias positive ratio rkhs
assume variety far objective machine generally rigorous guarantee investigate issue human take axiom mild axiom unfortunately interestingly reasonable crowdsource computation crowdsourcing involve perform generally computer human machine learning inference typically employ process worker infer algorithm infer solution design dependence response objective likelihood negative worker em optimization minimize problem obtain optimum extremely successful function meaningful computation appeal convex study extensively performance rigorous setting thus reasonable human convexity provide vast body two natural crowdsource convexity inference crowdsource mild axiom computation ensure interestingly one modelling indeed human select ability ability higher represent line number use ability worker regard ask crowdsource worker either notation choice say look procedure answer worker receive optimization base relax associate question inference x subsequently every ask worker worker ask question represent worker matrix element value ask answer worker worker response sequel optimization program axiom answer completion round step execute round infer leave soft answer jx towards natural axiom recall function axiom manner think maximum worker scalar distinguish lx lx lx w lx w informally axiom say report worker report less model highly crowdsource pose major axiom incorporation axiom suffice solely make continuity objective inference exist crowdsourcing aware fall subsequently constructive absence requirement model three axiom list translate worker observe ask scalar upon axiom list worker identical axiom axiom present example crowdsource show list two axiom denote receive crowdsourcing assume worker answer question worker incorrect answer parameter response else identifiable simply scale define likelihood consider subspace worker ask reduce list mean hence obey thus p coin worker model answer correctly true question true worker answer incorrectly else simply attention obtain coin additive assume worker interest worker f perform early restrict attention asked verify increase thereby property answer else entropy principle therein impose must form j minimization subset verify attention present property respect satisfy role axiom axiom crowdsource axiom strong property property hence axiom jointly multiple ensure satisfie axiom construct plot claim good use crowdsource objective crowdsourcing model reasonable permit framework throughout complexity alternatively problem bayesian non impose convex weight make objective albeit perhaps expense capture objective satisfy axiom present continue apply certainly complete absence indeed model theoretical instance model logarithmic appropriately may scenario minimax well result importantly date crowdsource alone convenient although theory extensive unfortunately exploit human computation paper incorporation
focus lead interested additionally convert free bind comparison interested quantity relate ex state theorem ex proof appendix implicit completeness make include proof ex plug bind reduce splitting reformulate fix c free style implication bind follow immediately completeness term low base argument obtain use appendix dependent star also survey variant robust noise additionally complexitie splitting case vc reader quantity analogous notion characterize complexity membership name characterize choose label point extend function classifier minimum sufficient quantity characterize learn return objective excess rate target define quantity prove low purpose star hx h exactly classification equivalently respect td prove label active source precise basic observation additionally refine thesis state relate quantity star obvious reading find extended way star versa learn familiar extended dimension equivalence formally td td previously instance become another remarkable set state lemma proof include appendix specify star center simplicity discuss replace though directly term bind equivalent logarithmic however imply extend classifier q u prove relate proof combine immediately imply lower involve appendix bind recently space compression q see u uv u uv v u v u h u specify vice immediately imply xy active budget return arbitrary bind analogous aforementioned bound purpose free independent take possible combine find match factor h follow quantity xy complexity passive show passive algorithm achieve xy though go idea xy though typically loose question case hypothesis though bad star range specifically proof h show gap quantity specifically p low sometimes within universal discuss appendix show sometimes tight aside divide interestingly process establish relate disagreement coefficient hold measure inequality quantitie passive aside minimax active model offer well discover unknown minimax label active vc class passive express combinatorial dependent equal maximized choice derive sense express bad sense express distribution label label complexity marginal achievable complexity exploration aside factor important minimax fix remain challenge open deriving consider kind guide restrict focus dependent bad bad free present result begin proof tn ta h gx follow collection vc union collection measurable vc integer ax algebra universal let vc xx x likewise interested theorem useful vc collection universal log xx hx hx x least h g imply g x h note log straightforward net relatively universal n mi x mi denote vc lemma let take g ie eq definition equivalently complete union variant lemma specifically exist universal xx variable x I result xx mi ie ny chernoff give least eq number sequence I q probability define n ny I I nx ie element assume x x kt py rr verify lemma continue improve follow immediate implication k ti tx trivially hold modification execution request return x k k xy tx h py p behave er er th er er achieve latter guarantee probability achieve er er probability prove exist hx implication behavior vc establish law number van result effectively guarantee high random make address fact strong obvious data turn relax classifier instead partition measure least partition property every constant final claim markov article proportional term cover bad value case let imply intend nontrivial simplify log proposition thus final follow xy xy xy xt choice instead least er complete bound part straightforward establish log technique finally third part technique analyze disagreement active technique modify technique place factor label eq give return xy h xy xy er xy xy er xy xy upper attention prove bind net u technique learning query treat instance specify label budget mm td j g gx ref jj particular let see return satisfie nx f agree return u vc u f nx un log least xy xy er xy xy well reasoning finally establish xy h input j label gx v eq chernoff integrate satisfie execute step particular denote define event conditionally k gx fx I gx since hand k satisfie event probability j k xy xy k k kk k define event e kk k kk induction imply upon reach setting technique three part also simplify proof let applied sequence simplify notation budget argument return else let else return subroutine label datum counter request let partition second commonly significant directly partition query repeatedly majority original discard point identify end high rate effectively distribution favor remainder xy xy xx x xy appendix range q xy xy return subroutine denote event least imply eq proceed characterize behavior subroutine via sequence exist j p law law mx mx p p measurable union exist I xy j f xy xy first statement jensen second inequality every j x inequality assumption inequality must establishes establish second monotonicity let hx hx union right hand side side side eq mm lemma agree subroutine lemma hoeffding inequality total value imply k recalling xy imply hold xy likewise one two clearly subroutine reach condition occur value q xy xy xy xy sign xy j f k measurable chernoff distribution probability law hold apply behavior algorithm begin budget retain run event xy proceed induction clearly xy v mf xy x f imply hand xy inductive hx principle er xy xy apply law union event v next k imply fact note budget question request algorithm produce budget budget question address lemma budget chernoff imply consider budget trivially remainder note term right chernoff imply lemma term note li j f section imply well li k g k martingale theorem apply probability k j k hand chernoff conditional distribution least event least hold let g furthermore ii last inequality ii plug iii iv combined bounding iii iv bit reveal q plug hold event sufficient reach return therefore guarantee run budget request xy er request execution subroutine represent running request budget subroutine ensure step ever condition algorithm request number budget take suffice reproduce budget value obtain implie summation therefore fix combined imply plug appropriate already appropriately final budget budget return return budget execution exceed infinite execution take infinite budget execution er xy xy er xy theorem note noise parameter dependence selection discussion far leave issue theorem know x least produce er xy passive n size universal xy therefore xy two bn simply correspond give statement theorem slightly factor log turn lower take technique contain lower bind bn bn prove bn bn bn bn bn case begin fix budget size produce er xy er xy note k kk therefore either particular furthermore dd xy note purely statement note log range recent base establish term remainder xy x xy xy xy xy verify redundant therefore plug complete aside logarithmic refinement bound away prove apply insight improve rather merely known star instead bound work modify variant exist constant size xy see thesis survey relate therefore prove sample passive match bind simply prove upper bind x x x markov x imply xy er q lemma budget produce xy er imply log log bind turn establish recent survey contain prove theorem imply otherwise q imply log term bind simply examine argument also strong expression state large state upper leading bind fix follow b variant universal least produce er er upper relax log also helpful measure z ex h xy ex ex xy xy h ex ex ex xy xy xy xy xy xy exist fact hx r lemma distribute random every I gx r gx r hoeffding union particular let xy xy xy xy therefore since arbitrary xy unnecessary purpose ready ex ex mh mi h b ref xy ex ex directly ex furthermore suffice support base fix furthermore z rr z r hypothesis q fix inductive handle nontrivial distinct z x hz gx r inductive fact imply j b hz hz z z k hz hx hx hz gx hz hz gx h h r b z r consider complement j hz redundant include clarity z hz hz ji hz I g hz ji z j z h h h j characterize since g b z r h z well case b ex ex ex due supremum specifically nonempty therefore take eq q usual xy xy ex ex present ex ex ex lemma focus finitely probability distribute imply union event occur let fx gx q qx gx gx hold ready ex implicitly original proof q b ex ex ex g fx gx q ex lemma repeat finite f gx inequality side divide immediately follow imply would thereby complete end denote g gx gx value entirely x fix finite mx gx complete proof inductive nested base nest inductive nonempty q x qx yy argument follow mi x ji x j ji define k kk ix k ji mi k mi j mi ks I k I kx kx I k z inductive r argue prove star k k z I kx z k z jx f h f z jj kk kx h x h argue imply induction give minimal specify fix specify respect star center u x j j v j u ts v j jx jx kk match star star lemma immediately ready fix let td monotonicity maximize monotonicity td remainder establish proceed sequence mh inductive hypothesis u u u u u specify star center since star centered specify inductive follow induction next subsequence distinct u last mh u r ks element h u g g ir u h u v v h h v maximize proof fix x h mi ix ix ix h ii x e modify proposition log set let denote chernoff hx gx I integer hx gx hx I establish g gx hx gx g kolmogorov gx h fix recall maximal rr see kolmogorov kolmogorov I gx g gx pe x b classification realize furthermore classifier r upper classification disagreement see divide maximize universal logarithmic factor bound theorems x sx vc star argue increase match logarithmic let ip ip furthermore define ig g p ig break tie also ig ig therefore reduction learn distribute j j j p algorithm execute request simply purpose request request label attempt request termination valid active learn budget request internal randomness return ann p iy p inequality linearity least plug strategy produce valid budget choice choose specifically j internal random conditionally distribution variable sequence ij er total von von probability independent plug bound noise construction proof star low already upper bound theorem bn ip bind prove maximal star appropriate therefore dp ip x ip ip px er pf establish proportional lemma apply hypothesis x h recall x ip x k x p dx upper sometimes logarithmic factor upper sometimes near logarithmic already consider proof last low p imply marginal k px px additionally since support distribution disjoint thus I lemma logarithmic establish upper sometimes sometimes tight choice furthermore theorem sometimes tight logarithmic factor tight first lemma x h p ii active xy er xy guarantee xy index subsequence hoeffde inequality bind subsequence run subsequence return subsequence empty return classifier note request label label request er er er evaluate complexity imply er er xy er strong sequence disjoint may j complexity law event er h er xy xy choice space prove sx star star sx fact factor specifically set define pa pa pa p ip ip together since distinct classification without request return note request every either thus take x ii equal negative denote return union x xy xy xy xy else xy xy xy regardless request xy er suffice q constant dominate existence tight measurable p pa pa px ip ip p bn every every contain star imply similarly respect low sometimes within logarithmic within bn bn tight slightly involved fix match logarithmic case x f yx trivially ix ip I hypothesis class imply star eq claim always proceed include bind p p x p yx x yx yx sign p yx p trivially ix ip I ip therefore dimension star vc imply plug factor sometimes tight within logarithmic factor since bind conclude sometimes within logarithmic factor specifically liu establish minimax label vc always passive good regime label low regime measure call interestingly previously learn star active strategy nearly complexity design learn condition sample machine primary annotation protocol sequentially point initially access unlabeled consider able pool label datum process continue must hope sequentially select effort already thereby require produce predict learn often significant observation result survey capability request sufficient desire substantial question gap quite case question specifically interested case request low model reveal establish propose active significantly smaller active get key study set applicable arbitrary range minimax complexity depend structure hypothesis roughly complexity minimax complexity find g classifier minimax passive nontrivial interest fall category minimax complexity essentially fortunately improvement passive datum focus various well learn place unlabeled minimax label thesis thesis upper minimax complexity achievable complexity achievable effectively measure capable behavior complexity active result reveal improvement minimax passive value bound beyond initial study recent literature develop provably advance seminal agnostic active thesis thesis label intuition inherently hard passive significantly passive pattern scenario passive learning scenario improvement unlike show accurately capabilitie nontrivial class vc reflect survey passive although minimax passive learning unable confirm truly really gap close work reason range noise regime reduce match basic surprising introduction particular dimension minimax complexity sometimes basic argue characterize combinatorial reveal label complexity active fact minimax interestingly dependent star maximized distribution include star summarize main active select estimate imply prior upper bound reflect minimax passive learning non possible imply hypothesis vc roughly minimax logarithmic factor imply literature upper bound complexity measure refer complexity star star study exhibit hypothesis gap upper bound demonstrate gap aside factor result vs respective minimax complexity passive bound base analysis query determine highly modification innovation net cover low largely directly combination survey incorporate note focus result reveal low aforementioned strong low assumption restrict leave work characterize marginal article introduce follow model work define combinatorial star lower bound complexity include discussion among scenario complexity correspond star maximize relate star concept star write contain section section follow section appendix rest make formal simplicity x b arbitrary statement similarly condition classifier independent active individually require scenario leave question number unlabele label particular indeed xy x jj iy proceeding initially access select index request index request continue round label formally active conditionally independent ready specification learn respect denote produce base distribute define classifier vc classifier h hx gx px proceed additional notational help simplify statement proof value function write equivalently express define define remark van van mention issue implicitly event study set corresponding specification collection xy f bn xy xy xy xy xy xy h xy xy xy er bc h study correspond optimistic case pac study name noise slightly discrimination form state passive however weak assumption condition equivalent imply thesis contain within related study admit distribution widely literature complexitie learn bound express slightly prove typically factor proof refine aside believe optimize bound additionally relation model since low theorem contain study log likewise case bind theorem simplicity explicitly include theorem variant basic point determined sequential test course present repeatedly able distribution give resolve argue cell vast majority knowledge datum discover vc give point partition label point majority cell effectively majority note simply apply repeat strategy distribute would determine label partition number sample become note less agree noise classifying excess er xy learn effectively favor discard become gradually agree every pass combine provide fraction classifier bind decrease decrease try classification able reduce request component essential case introduce namely value active disagreement request classification disagreement recent separately section relate star dependence state comparison protocol label passive minimax learning passive classifier er xy request run determine return one
subproblem principal seek outli anomaly seek detect outli magnitude seek recover nonzero coefficient outli variation problem deal relax contain properly hold table lead outli numerically efficient far unconstrained call merge discuss signal outli signal ad backtracking analysis try ii measurement follow satisfy feasibility belong optimum theorem detection graph perfect achieve b perfectly note theorem factor smoothness upper signal trust prevent influence outlier similarly deal relax merge combine anomaly provide provide clean use clean admm implementation graph outlier stop several bridge indirect bridge monitor opinion dataset dataset classify political either label correspond bridge feasibility indirect bridge structural health monitoring bridge system build acceleration collect bridge put bridge collect acceleration mass gram gram simulate acceleration near neighbor node represent eight represent acceleration shift symmetric undirected directed allow direct graph weather united record weather per day weather geodesic pair weather represent weather represent eight close weather graph weather graph shift weather normalize direct contain measure norm rating represent near node eight user signal I acceleration algorithm acc mse rmse absolute mae acc mse mae ground indicator split training validation part choose validation appear include completeness describe laplacian restrict undirected shift symmetric large classify adopt describe label thresholde classification labeling remain underlie filter also require sophisticated labeling unlabeled acceleration acceleration assign acceleration algorithm remain show average test ratio adopt mse obtain use collect comparison identification labeling matrix apply propose expert opinion classification propose include factorization describe minimize similar different factorize nonnegative low internal contrast hide fair non convex miss predict temperature matrix temperature weather adopt dataset temperature describe pick day case signal temperature pure well label increase algorithms graph mae outperform completion cc rmse b mae completion recommender dataset task predict rating incomplete rating purpose signal temperature average test advantage exploit rmse rating completion mae rating many opinion expert image opinion combine formulate solve classification ground simulate expert label opinion labeling mistake opinion matrix expert classify content split easy hard expert chance expert correctly easy make avg sign avg method opinion part entry tuning parameter report ground improve result scheme completion provide application robust apply robust signal identification contrast manually part algorithm graph signal validate detect label feed accuracy accurate classification cc semi supervise describe robustness randomly label feed algorithm together correctly acceleration compare signal provide labeling ratio general multiplier show signal matrix subproblem solution validate identification opinion acknowledge support fa well institute technology award lead improvement follow principle page reader decompose introduce residual capture rewrite equivalent form f operator note move function put equivalent augment lagrangian thresholding aim solution aim solve solve thresholding setting satisfy update lagrange multiplier mm electrical engineering pa usa pa usa com graph signal recover corrupted formulate multiplier principal analysis relate theoretical bridge recommender recovery completion semi growth generate source include social citation unlike image novel lead extend signal underlie graph signal generalize signal transform classification detection processing generalize series concept tool denoise classification root definite real edge weight approach root build operator elementary generate shift shift graph arbitrary real nonnegative within problem time deal undirected graph graph assume smooth corrupted assume neighboring cast signal optimization solution alternate direction multiplier recovery theoretical new graph completion validate bridge temperature recommender expert opinion work review exist recover observation technique wiener filter wavelet thresholding lose part image video representation take number recover low originally noisy decentralized recover rank corrupt measurement separate image two background foreground corrupt relate signal recovery include interpolation signal uniqueness graph recover random contribution novel analysis novel graph nuclear novel anomaly signal review graph recovery section subproblem completion section recovery recommender opinion briefly domain line commonly graph signal supervise node connection node quantitative underlie dependency pattern representation assign vector graph fourier transform expansion invariant simplicity complete expansion eigenvector form signal content weight coefficient graph qualitative underlie signal quantify use magnitude normalize guarantee call symbol shrinkage define signal multiple completion denoise graph section propose appropriate assume outlier outlier distant variability measurement magnitude small magnitude furthermore certain large graph denote component graph section counterpart completion minimize variation anomaly robust recover sample total access task signal true signal variation signal variation quadratic graph cyclic matrix combine recover subset subset typically rank recover rank model assume subset index recover condition also graph represent structure matrix difference corrupt recover low index associate identity detail part appear completeness signal seek entry incomplete noisy signal smooth formulate case exist iterative measurement derivative close invertible merge objective trade solution set zero invertible denote try recover error part condition smaller tight smoothness technique smooth bind smaller close estimation graph subproblem general seek graph column signal case formulate addition completion next call constrain problem solved project split component formulate proximity iteratively iteration feasible differentiable component convex proximity
measure infer year connect graph appropriate temperature trend correlate regardless reliable information observe experiment give observed temperature signal learn quantitative build graph reflect term connect two small compare learn comparison visualization focus top comparison edge graph consistent confirm score quantitative comparison infer topology graph learn score value trace recall verify result disjoint learn measure obtain laplacian spectral clustering learn disjoint cluster fig dot mainly blue flat region especially centre lie mean algorithm record information cluster together capture information measuring confirm record california per month learning would like infer similarity measure variation cluster compare result overall metric though challenging indeed accord description cluster obtained learn base measure c cm move national vote support lead like infer capture vote neither obvious relationship preference partition spectral blue cluster speak speak five red primitive consider among conservative cluster membership fact agree close european cluster political voting behavior cc confirm b national mass year cluster seven percentage support great largely speak margin confirm demonstrate topology smooth enforce smoothness signal gaussian impose factor numerous appropriate capture entity furthermore focus present paper impose analysis leave comment dr ed de mail meaningful crucial success handle especially process meaningful readily particular desirable graph processing application admit certain signal variation topology adopt graph impose lead signal propose enforce property minimizing learn demonstrate graph topology laplacian processing datum signal vertex set weight undirected signal vertex represent entity weight reflect pairwise vertex carry observation measurement numerous world structured currently graph domain e graph intrinsic entity desirable topology entity present ill pose associate topology topology pre define meaningful structure graph edge capture value temperature significant variation represent interest friend represent vertex interest datum graph representation graph multi view scalar potentially define fig signal however obviously objective topology speak would edge signal bar point represent negative respectively bar reflect graph smoothness unobserved term latent traditional transformation topology joint signal consistent latent specifically impose prior generalize factor obtain principal signal smooth signal base uniquely signal smoothness datum laplacian central process new operator latent iterate graph variation minimize upon graph world infer topology art closely idea estimation importantly framework processing perspective approach rigorous framework signal property classical insight signal numerous entity amount signal effort process generalization wavelet dictionary processing inference domain capture whose importance processing represent topology e loop usually consider enable generalization notion frequency fouri signal develop matrix signal laplacian operator central kernel graph via regularization process laplacian permit real graph signal represent behavior condition converge intrinsic laplace riemannian manifold may manifold benefit application view zhang et eigenvector optimally build priori domain devoted graph topology metric evaluate smoothness signal fitness learn valid topology fitness property statistical manner help understand adopt graph linear brain principal correlation distinct consider region behavior perspective link explicitly particular amount community graphical gaussian graphical small singular another consist know graphical exact zero entry partial log therefore correlation emphasize mention precision usually zero row however result precision matrix global property graph signal rather correlation straightforward convex order rather work learn topology adjacency classical regularize rank laplacian infinite basis natural impose together fourier assumption lie use degenerate see precision laplacian generic much graph laplacian commonly precision assume degenerate precision recover noise free classical lead principal component probabilistic interpretation highly successful analogy signal propose interested specifically map scenario quantity laplacian quadratic confirm gaussian similar scenario component signal graph ready introduce framework impose come joint change accord noiseless version observation recall quadratic eq usually smoothness laplacian signal propose follow objective zero frobenius respectively acts permit trivial valid laplacian furthermore trace function diagonal entry similarity elastic impose adopt alternate scheme solve solve cast convex minimizer symmetric triangular main therefore low triangular convert rewrite problem subject point computational graph instead split alternate method multiplier form second hermitian cholesky factorization compute alternate summarize htb input output laplacian update section graph present compare learn visual quantitative comparison existence graph evaluation criterion commonly namely performance partition pair vertex class graph package stop reach absolute experiment namely finally weight framework propose adjacency precision determinant regularize determinant precision regularize laplacian diagonal loading interpret priori consequence problem case one experiment similarly carry synthetic vertex base euclidean distance follow generate vertex square radial basis rbf width r enyi edge ba graph ba experiment add exist degree exist er graph science former network ba er graph unitary laplacian normalize generate show signal visual comparison laplacian random lead quantitative paragraph choice
require experimental excellent dataset convolutional network demonstrate key convolution layer serve seminal contribution improve deep convolutional deep convolutional machine model pool contiguous pooling robustness variation shift reduce move maximum map block pooling average max stochastic probabilistic max generative deep highlight deep start map multinomial distribution feature analogous impose pooling demonstrate yield generative statistical readily implement jointly use bottom bottom layer refinement phase jointly may readily goal obtain maximum find unnecessary expensive attempt parameter view alternative convenient learning dictionary deconvolution layer convolutional simultaneously nonlinear nonlinear testing network approach inversion test still aspect deconvolution operation hierarchy dictionary joint leverage generative model top operation hierarchy map imply contribution employ beta separately via layer proper top allowing mean feature map deconvolution experiment convolutional representation gray analyze jointly dictionary hadamard element shift w ki z ga b b may look complicated conjugacy admit gibbs bayes order pool use employ feature move layer learn stack upon refinement pooling proper never tackle yield discuss present pool closely model start sequentially stack parameter layer serve sec parameter layer jointly close stage input dictionary element view entity discuss spatial dependent perform layer contiguous part block location block one pixel pixel stochastically large block hence process proceed impose question bernoulli give multinomial model bernoulli follow multinomial statistical latent denote multinomial imply equal entry I element block zero first position zero phase datum learn use activation sampling multinomial correspond gibbs stack pool continue learn element via continue top layer top pool generative constitute refinement learn excellent initialization subsequent generative process l l multinomial map corresponding block block convolution w equal block multinomial multinomial unchanged discuss refinement via multinomial size refinement constitute initialization refinement understand element visualization layer associate multinomial show upper image capability deep convolutional well activation expensive issue filter explicit deconvolution follow though step must learn dictionary framework model accelerate project datum plane test perform top element map plane strength subsequent classifier dictionary map datum plane multinomial top dictionary plane element dictionary convolutional shift pool shift pool pixel layer deterministic retain deconvolution deconvolution must infer element plane detail aspect material conjugacy component close efficient gibbs supplementary detail convolutional accelerate operation update pre discard burn sample burn refinement ml collection sample spirit yield posterior select sample discard burn dictionary view plane refinement trial use hyperparameter hyperparameter perform matlab execute cpu memory refinement minute deconvolution acceleration via realize recently convolution mcmc batch vb layer train widely mnist testing digit layer dictionary layer initial dictionary large value pre discard indicator element summarie classification compare second top send support vector kernel multi classifier via fold cross layer dictionary element computation close simple rate learn similar refinement step testing report examine learn visualize dictionary map observe qualitatively refinement average
generalization rademacher finally analyze representative affect behavior comparison domain keyword adaptation deviation wu david research types adaptation learner receive source domain know adaptation zhang without concerned representative adaptation datum one adaptation cover multiple domain combine paper previous exist result regard case generalization representative factor affect discuss choice meanwhile representative cover target include additionally quantity follow integral quantity entropy rademacher generalization representative domain adaptation asymptotic representative support finding brief conclude appendix inequality proof jk domain respectively let stand respectively differ differ occur n kn k denote combination empirical empirical empirical approximate precisely adaptation david zhang david david david integral exist recently give investigation integral sign trivial domain rewrite form quantity discrepancy formal briefly quantity introduce exist one summarize follow upper david quantity recall condition place restriction contain function task discrepancy quantity mention quantity match instead author upper resp minimize upper summation note address paper condition next aforementioned discrepancy definition integral relationship integral probability three possibility differ occur two difference difference measure distribution moreover another quantity trivial labeling integral discrepancy distance show simultaneously specific labeling meanwhile though set definition rademacher generalization achieve incorporate complexity covering refer entropy class covering cover metric norm derive bind situation adaptation know source domain long applicable adaptation free clarity presentation notation sample z k z n k norm easily omit z kn frequently class random either rademacher complexity q version take base entropy bound domain adaptation hoeffding derive hoeffding deviation deviation inequality base present hoeffding type representative domain adaptation bound bind tf least risk respect three coincide hoeffding deviation incorporate expectation variance type result type type domain tf bound hoeffding two limitation affect satisfactory analytical inverse trivial since tf use lead compare strong bound affect representative second type alternative hoeffding type next provide detailed bernstein type refer hold kn convergence moreover observe rate varie especially imply become become contrast hoeffding bound affect adaptation detailed representative rademacher generalization representative domain class function give domain rademacher source domain derive adopt domain adaptation coincide assumption domain match replace derive type inequality bound follow notation take define match k omit hoeffding tradeoff rigorous tradeoff discrepancy measure achieve discrepancy w entropy tn part infinity accordance process event representative domain risk hold process distribution coincide match kn kn theorem generalization decrease small cauchy set minimize side hoeffding result imply fast domain process choice essential tradeoff number kn fast convergence relatively large representative domain adaptation kn kn lead type accordance analysis well experiment verify generality domain target tn n tn tn sample source n regression combination coefficient repeat time average increment choice big fail become big recall big means datum situation adaptation accordance present analyze representative support finding fact discrepancy fast convergence slow far away fast rate rate slow accordance finding domain source david uniform source condition classical multiple source know domain domain domain extend target source combine meanwhile analyze classification task introduce condition vc capture extend rademacher property learn source target particular metric domain provide mechanism domain theoretical paper study adaptation setting previous work also representative term apply type generalization representative domain adaptation uniform entropy respectively point process uniform convergence rate finding discuss hoeffding type hoeffding type result representative type complement cover adaptation include zhang generalization bound result theorem base obtain result adopt martingale hoeffding type concentration obtain generalization certain specific hoeffding source k hoeffding result result suitable coincide domain hoeffde generalize domain expectation compare inequality suitable inequality coincide one domain match explicitly reflect right completely satisfy hard analytical incorporate parameter multipli type deviation cauchy affect result
extension pr pr polynomial establish pac number target unlike standard vanish treat define notion hardness hardness result furthermore ergodic system termination trace construction structural output playing strongly consistent ergodic map generator structural construction lemma synchronization role synchronization specifically transition synchronization necessary infer cross distribution distribution first process synchronization ergodic minimal recursively b b symbol specific use definition cross give stationary ergodic generator minimal yx assume equivalence note yx b yx yx note synchronization stationary ergodic string yx string projective stationary encode sense definition yx complete proof string solve lemma suitably note establish string string ergodic exist inference derivative symbolic string occur follow immediately symbol string string number occurrence string symbol imply derivative derivative string respective cross negative summing unity bx stationary respective string x analogy describe seek derivative string stream history explain symbol assume history former stationary next cross occurrence row vector yx induce illustrate admit furth average predict weight choose strategy nan statistical causal strongly positively keyword education positively causal directional causal infer education search frequency education reduce immediate full search correspond keyword trend node degree arc google trends http www trend search normalize entry indicate sum google website trend long strongly political keyword table google trends education freedom ex music ex keyword easily expand theory new topic note interesting search datum correlate illustrate datum series keyword education keyword correlation positive education causal environment lower seem plot causality carry unique interesting causality series neither full keyword symbol stream symbol colored new existence dependence stationary source cross sufficient broad causal causal flow development open mechanism diverse intensive scientific pt north yshift north east cc use statistical relationship branch really causal causality hard construct difficulty causality practical operational restrictive dynamical trivial nevertheless computationally evidence dependence yield tool calculation causality symbolic stream ergodic precise compute stream linearity specific dynamical structure explicit sufficient fairly propose pac probability search google trend choose keyword causality keyword illustrate fail insight correlation causal correlate dependence consider past value maximally apply contrast future would reveal carry unique prediction causality early preliminary text causality statistic expert largely sound operational causality notion lack consensus causal relationship perhaps mathematically causality concept know know attempt influence statistical universe knowledge universe denote contain forecast however expectation simply causality future future past contain redundant exclude within definition note applicable causality say cause set intuitively cause unique immediate notion causal address primarily interested obtain mathematically lead algorithmic encoding universe series available consist fx j extra affected necessary j universal cause n say identically mean far structure causality discuss commonly employ causality find series self function cause implication dynamical specify ordinary differential able perfectly construction imply may concern causality require cause additionally cause causality necessarily require cause imply sense induce causality particularly white process cause general fix spurious two common causality mean easier satisfied incremental one ahead operational variance forecast cause bivariate causality cl dl side lag lag operator root mutually individually constant use determine significant predictive power nan strictly illustration notion generate ergodic alphabet b possible produce respectively class string class mapping symbol alphabet edge conclude end linear restrictive structure bivariate analytically limitation nonlinear autoregressive wavelet transform heuristic allow quite pre suppose causality show sensitive non attempt completely causality series integral similar nevertheless absolutely bind integral additional stationarity separate variant quite nonlinear stock factor return stock price trading volume despite application parametric test limit beyond financial interest detect causality specify obvious hand box advantage influence variable parametric system dynamic leave parametric although nonlinear causality detect nonlinear parameterize behind residual remove additional influence origin causal priori dynamical objective indeed infer heuristic influence linear appearance dynamical well absence assumption source sequential variation ergodic stream explicit early generator stream probabilistic stream model causal represent generalize refer infer machine structure nature causality logical influence stream identical additionally show causal existence trivial direction independence ability find dependence carry causality require ability base prediction impose produce generative suppose structure therefore carry addition stream identify stream stream symbol stream symbol stream value indicate strong thus causality quantify immediate future stream stream importantly directional see merely infer existence establish significance passed fail relate build state strong connection indicate past completely test inference impose latter perhaps approach ergodicity test process absolutely regular mix certain minimum regularity way one essentially stream nearly ergodicity figure class prediction pac infer asymptotically well investigation parametric test go beyond binary testing quantify causal influence observe pre particular influence show impose stationarity pac rest notion elsewhere completeness difference exposition section presents framework cross introduce causal directional infer self stream sake pair data stream causality multiple future section causality source trend list keyword upon literature formalism brief overview completeness alphabet possibly unbounde denote identity infinite denote stre length denote also value moment calculate strictly generator ergodicity able sufficiently algebra infinite induce ergodicity extend countable notational brevity class string equivalent extension string relation induce equivalence string right invariant equivalence induce equivalence construction final mark marked alphabet initial recursively extend impose probability unity symbol probabilistic however lack state additionally strongly remove dependence ergodicity formalize generator mark unique probability countable immediate imply mark corresponding initial extend recursively nan finite imply mark imply generate yield initial generate initial mark probability whereas mark canonical representation remove state dependence representation I non entry row mark associate stationary string lead begin unique canonical mark uniquely induce set construct stationary use induce include representation mark representation contain exist begin stay copy mark strong initial state minimal initial mark strongly exist map l permutation transform encode realization correspond represent index generator space represent ergodic state connect component correspond strongly node label iff initial us equivalence strongly right terminate immediate map contradict ergodic hence generator possible distinct exist string contradict conclude connect argument valid label unique state encode associate relation z initial state equivalence refinement identical realization synchronization synchronization state analogous context identify stochastic generator translate synchronization machine history determine top finite history machine bottom string always synchronization remove arc trivially state string hence generality x induction construct finite x ij tx contribution arise simultaneously state nevertheless see qx n satisfie arbitrary string search string order string alphabet find entry follow string scaling imply string computation symbolic symbol symbolic derivative specify alphabet symbolic string count occurrence imply symbolic symbolic thus probability refer symbolic recall satisfying reading unknown string imply class single state true state loss strongly q complete long extension specifically long non establish state class describe identification string observe corollary establishes string arise inspection geometric structure construct different derivative sl geometry sl hull combination string hull consider string probability derivative hull corollary number drop factor case kullback kl probability pr pr pr ergodic evolve assumption require dynamical specifie string process specific alphabet strictly algebra denote space map ergodic algebra map consistency additionally xy b effect segment vanish cross induce cross cross stationary cross tp ix derivative stationary ergodic assume recall equivalence derivative generating em b capturing dependency capture dependency evolve letter alphabet evolve letter alphabet capture dependency differ former specification could alphabet probability symbol generation symbol empty see pr dependency need formalize define appropriate call cross probabilistic equivalence relation ergodic clearly forget actual notion map string output alphabet identical respect formally finite state alphabet alphabet alphabet possibly parameterize mark marked cross finite noting extend recursively denote argument ergodicity drop without unique minimal figure transition generation transition alphabet alphabet alphabet different input alphabet next investigate may ergodic ergodic equivalent essentially evolves independently string affect symbol distribution string simple state copy ergodic step shift case figure calculate follow stochastic process ergodic single k kn nn assume canonical ergodic dependence ergodic respectively symbol b b b immediate minimal process alphabet represent possibly large graph specification minimal realization force projective vice involve remain transition state conclude complete proof b initial distribution causal definition claim one string string string lemma conclude next symbol claim symbol compute string imply state respectively b sequence induction hypothesis vector pr pr pr pr b w pr b w directional stationary ergodic establish capture well suited directional flow quantification directional introduce representation represent direct direct label composition strongly encode get merge equivalence let encode denote connect establishe full q equivalence imply map imply complete projective version projective strongly projective corresponding note projective operate operator alphabet second second operator indeed additionally projective preserve project choice equivalence choice state distribution invariance encoding alphabet state composition stationary denote encode state j states matrix establish stationary ergodicity stationary b symbol well process look coefficient entropy bit letter observation symbol ready define causal definition give causal ergodic finite causal dependence ratio due absence b entropy discrete process produce alphabet inner stationary lemma noting equivalence class correspond class string q coefficient x entropy establish statement x x b converse independent minimal x complete computation
generalization value let overlap meaningful sum average fully average gain map binary transition probability less refer factorization unbiased realization process generally markovian proceed parametrization minimize hamiltonian vanish find transformation play correspond read shall hamiltonian ise field map reduce limit variable overlap eq clear bs hmms infinite operational regime order transition transition alternatively intensitie intensity simultaneously objective straightforward model factorize overlap separate domain admit factorization uniquely proceed regime weak irrelevant regime maximize gain spin focus two verify refer regime domain end correspondence configuration generally domain support uniqueness reflect entropy consist inherently relate exponential degeneracy intensity indeed observation linearly thus indicate degeneracy regime three look positive old stay end positive form pair rule super spin play role regime odd build positive super regime small pattern implement separate domain spin advantage implementation already spin incorrect estimate one iii calculate approximately spin opposite map remove uniqueness overlap calculate spin determine close analytic form calculate fluctuation separate domain write normalization checked even spin likewise originally spin need seem surprising symmetry expect overlap gain define differently agree gain active large thick opposite recognize correctly change decrease recognize domain scenario one fourth spin case mirror symmetry respect overlap gain supervision lead overlap gain meet previous gain correctly flip overlap fourth spin spin see possibility possibility exist possibility gain fourth spin reach remain inferior one describe domain case active realize relate mirror symmetry overlap gain furthermore straightforward calculation gain scheme domain supervision one domain spin flip semi recover one perform simple suggest spin negative supervised overlap gain confirm thesis gain even domain still budget spin positive spin employ cf small efficient spin inside spin map configuration active estimation regime separate spin converge picture estimation unsupervised map odd gain spin yield maximum thereby gain supervise remain aim analytical prediction focus trivial regime high noise intensity exponentially many map observation recall domain run find sequence apply finding focus domain see domain figure show belong domain fraction inside rather instance belong domain solid error plot figure use infer align overlap infer hide gain gain noise simulation near perfect intensity switch eqs interestingly branch agreement remain near perfect intensity however agreement branch give assume start gradually break carry correspondingly way employ write error original active yield low whole intensity intensity slight curve analytical active inference symmetric hmms expression relate within approximation specify prediction active bs hmm observation correspond map find domain odd inside domain domain one focus filter separate weakly domain extent section always assume apply heuristic suitable measure maximize domain reduction mirror symmetry fourth scheme spin domain fourth spin relate unique principle domain since spin spin find wrong spin sequence however spin configuration beneficial another discussion uniqueness wrong appear configuration extend task assume joint problem might bs hmm extend robustness strategy extent optimal hmms answer examine imply viterbi return several candidate likelihood energy intuitively analogy physics domain generalize apply instead ise random ise system situation dimensional type recognition vision human heuristic regime relate tend confirm rather
person tv fc ap fc fc fc yes yes cnn acc baseline fc activation improvement visual third object map supervision localization comprehensive recognition task representation visual recognition convolutional cnns rich straightforward approach method fair activation aggregate activation scale wise essential replace activation significant mit use task task introduce scan data outperform method descriptor devote representation bag design descriptor representation kernel order major descriptor invariance property advance visual convolutional cnns jointly whole class stack processing million power training recent imagenet contribute scale cnn extract independent successfully apply generic combine activation show task detection fine grain attribute recognition image activation generic way response second response geometric variation common random augmentation though use prevent average multiple activation help geometric invariance cnns activation activation achieve invariance characteristic recognition fed patch pre cnn activation aggregate fine scale introduce activation paper discriminative robust geometric figure utilize cnn activation activation state fisher scale wise demonstrate scene classification pooling activation demonstrate object confidence map localization label object bounding box meaningful mechanism pooling scale performance representation fisher neural activation review add kernel visual extend bag model descriptor respect although across possible classifier linear dimensional descriptor aggregate descriptor intuitively direction descriptor kernel improve additional follow activation cnn fed network patch extract scale densely inefficient redundant perform extract dense activation replace fully connect image large feed modify output cnn thousand dense level extract extraction second per c scale activation naive sec fc activation cpu gpu image generate pyramid minimum size feed scale activation activation merge pyramid descriptor aggregate explain cnn activation descriptor adopt kernel cnn introduce modified pyramid contain scale local activation extract apply aggregate activation fisher merge pooling cardinality since concatenation improve fisher framework finally overall illustrate scale characteristic traditional activation cnn activation represent aggregate perform scale wise fisher important fisher label horizontal denote scale number obtain fisher give activation pool fisher combine activation accord patch take traditional kernel visual descriptor sift densely descriptor encode gradient detail mid cnns fc fc represent level posterior visualization region activate cnn different property fisher sift sift cnn wise dense descriptor framework encode performance fisher accord demonstrate clear sift activation come fisher perform sift properly low level aggregate cnn activation poorly activation ccc pool label horizontal pyramid possible aggregating scale cnn activation however dataset activation contribute balance examine pooling pool perform experiment pool five number pyramid superior rapidly finer involve activation fine scale level dominant form fisher vector exhibit increase cnn activation evaluate scene type rather use quite class precision grain number cnn compose convolutional layer three perform top validation evaluation henceforth since nearly cnn henceforth simplify five convolutional layer connect convolutional mostly cnn compare demonstrate cnns seven scale default seven scale cover procedure representation pyramid scale pyramid resolution define feed reduce pca activation consequently versus rest mostly implement library framework cnns perform comprehensive method compare state descriptor pool summarize cnn activation cnn dataset fc perform ap ap improve improvement regardless augmentation baseline sn activation scale representation ap gain representation activation utilize activation baseline exploiting scale baseline verify multi far naive fisher kernel pool significant multi activation encode option concatenation without raise proportional outperform pool far representation recognition pyramid representation construct pyramid region middle time difference rich sp redundant various activation perform compare outperform possibly superiority fisher seven quite way suitable aggregate neural record dataset fc complementary discriminative stack stack performance complementary summarize augmentation multi perceptron mlp ground box representation use pre augmentation use box annotation gain adopt well cnns source imagenet task pyramid pooling cnns fc slightly compare fine fine tune nearly stack low augmentation fine tuning believe augmentation truth bounding box major pre cnn performance low art perform among class class object demonstrate benefit activation fine handle pt description fc cnn ap fc pool ap average fc naive multi scale pyramid fc concatenation wise pyramid fc
compare discrimination large eigenvector u u k maximize sum unweighted variance without difference pca direction second p distance direction affine span anchor remove effect locally anchor show visualize maps equation anchor metric furthermore manifold anchor straight experimentally approach fisher outperform manner mahalanobis space instance differential fisher exactly cosine finite discrete transformation intuition anchor similarity transform thus distance anchor point speedup learn low linear ensure finite discrete large importance unlike margin triplet constraint triplet near neighbor instance neighbor neighbor matrix loss cosine problem use project sub triplet simplex study symmetric margin cosine cosine fisher metric problem manifold differential instance need mass discrete distribution interpret define similarity base geodesic geodesic form straight local metric would metric allow geodesic distance follow parameter similarity fold inner angular pairwise distance point separately similarity svm inner angular select inner cv use triplet triplet constraint three class learning evaluate statistical student ranking schema find point difference b manner eight mahalanobis rkhs significantly four bad eight statistically significant well score good predictive follow svm explanation score understanding performance speedup anchor select low reduce anchor fold inner train split cross significance accuracy achieve accuracy map instance similarity induce metric induce robust discrimination anchor distance unlike psd interpret experimental svm acknowledgment wang partially support education research innovation number award ap award coordinate smooth coordinate approach gd r gd gd term accord definition r gd similarity anchor fisher information semi definite significantly crucial role learn task metric euclidean address satisfactory manner lead last global call single computation instance follow project vary locally flexible local address limitation metric smoothly learn metric riemannian metric geodesic computationally expensive approximated geodesic form straight line along unfortunately distance flexibility first hilbert rkhs global mahalanobis space define mahalanobis rkh space induce semi transform psd kernel psd keep similarity similarity anchor riemannian density bias region low dominate learn riemannian direction orthogonal effect locally irrelevant remove knowledge first algorithm flexible various similarity moreover local metric algorithm distance form evaluate dataset metric instance dimensional vector type manifold manifold metric probability learn different type simplex etc manifold similarity differential map compute similarity define induce riemannian riemannian space intrinsic instance category density metric large discrimination anchor orthogonal manifold new distance remove locally irrelevant dimension anchor remainder terminology denote exist define coordinate contain smooth np interested variable otherwise fisher metric manifold distribution replace give explicit statistical otherwise leibler kl divergence fisher probability fisher approximate hellinger cosine importantly manifold g equivalent fisher distance smooth tangent riemannian induce pf f nd pf p smooth coordinate jacobian function g psd metric follow lemma relation metric map geodesic endow geodesic endowed fp appendix assume geodesic form line approximation learn anchor k differentiable non similarity distribution outcome I map similarity define discrete learn instance intuitively instance onto base lie ignore reader anchor give anchor empirically cluster norm control
smooth experiment evaluate precision baseline evaluate standard instance learn preprocessing normalize bag average bias implement suitable computing near neighbor object detector recently window annotation fine tuning annotation detection method detector note besides bit annotation bound annotation meta annotation annotation annotation detection efficacy recently construct window neighbor instance proposal feature mode account windows method intra background sign onto metric set contrast protocol scoring detection table show detection dataset method per object detector optimize develop object set window initial model refinement detection supervision object source code website thm proposition supervision vision since costly image submodular automatically discover set positive window formulation leverage quasi provide improvement art classical paradigm object instance bound exhaustive labeling costly dataset massive annotate visual different weakly without box object detector goal supervision learner label object access annotation box start object million selective formulate discover contain detector smooth recently proposal detector prior weakly supervised detector improvement achieve relative weakly weakly discovery mid visual number object formulation level presence absence present challenge image implicit correspond initialization early effort focus center simplify clarity focus design initialization helpful bias work detector generate box annotation challenge focus design shrink mid use discover visual occur discover element provide discriminative mode draw connection shift challenge segmentation object address pixel co submodular share submodular idea rectangular window detection however level label classic multiple think bag rectangular specifie contain specifie category instance label typically find convex mi practice heavily initialization focus extensively initialization initialization method approximately greedy initialization refinement produce detector however far improve alternative mi optimize auxiliary objective bound novel objective solve unconstrained bfgs experimental improvement bound selective search proposal box ultimately box box neighbor box neighbor optimize occur iii multiple graph ii kb image connect implement occur positively image equally close negative image consequently box neighborhood neighbor box equally box close picking box green highlight neighbor box box neighbor cover cover redundant relevance maximize box many neighborhood complementary large additionally cover gain close neighbor fs fs ft ft thus thus finally sum submodular also submodular obvious algorithm factor say intuition special covering filtering minimum merely single smoothly sub relevance visualize class experiment might mode shift htbp htbp review enable unconstrained smooth optimization analogy object binary want learn typically bound box amount find bound box contain image result exponential choice solve scalar svm formulation eq
datum show bss leverage competitive unsupervised feature bss unsupervise well bss leverage score run leverage score primarily feature heuristic svm support bss large dataset support bss select closely approximate singular perform namely contain task regularize svm formulation since pre multiply repeat experiment five around randomness bss experiment bss bss increase right projection extend provable guarantee empirically bss score comparable method guarantee full datum construct appear progress make full provable advance approximate dataset direction see svms sp support nsf theorem corollary science institute ny usa computer department institute ny accurate svm deterministic respectively supervise supervised prove feature bad case set worst thereby ensure pose world often well state art provable linear support svm theoretical result svms numerous technique work feature selection supervised preserve minimum relative case thus open supervised setting select preserve margin support vector error data label separable primal constructs maximize geometric distance hyperplane separate separable soft norm lagrangian formulation soft quadratic regularizer hyperplane construct relate result lie hyperplane width bound sample monotonic provably selection unsupervised guarantee margin run deterministic algorithm svms deterministic logarithmic margin sufficient linear margin margin margin solve suitably support prove margin whereas get strong deterministic select select optimal optimization datum prove within within effective dimension combination pure preserve non trivial svm practical heuristic bss allow main unsupervised score unsupervised elimination sampling qr method leverage come provable supervised art heuristic empirically provable empirical survey feature weight formulate perform step method formulate sparse svm selection rank radius bind work include doubly machine penalty involve bottleneck formulate fisher fix show al projection margin preserve bss leverage select regularize learn identity matrix decomposition svd contain contain singular matrix singular vector spectral consist replace rd whose lagrange multiplier determine dual imply entry datum ir cm relatively simple margin obtain full score say comparable feature bss rescale margin matrix optimal eqn feasible solution eqn combine towards difference rewrite get z opt opt combine geometric margin supervise leverage margin svm say feature ensure comparable bss feature sampling rescale part replace result follow result leverage sampling leverage score rescale margin radius ball radius sample subspace bss consider minimum equal b n n b b ball n center minimal b point radius clearly bss leverage svm output leverage fold scale like bss approximate bss scale time offline matlab r intel processor gb ram bss comparable suggest pick column satisfy choose high among column euclidean never compute quadratic program dataset feature point point relevant varie construction run select fold synthetic pick supervise bss feature repeat five set select ht supervise unsupervised l music education read iii uk bss read education
generalization locally optimize factored distribution essential step factor factor iterate place jensen logarithm q furthermore since equality contribute additive though entropy inside maintain factor form invoke optimize entropy rather conditional seek marginal begin modify multiply repeat iteration section compute efficiently multiplicative lee must compute multiply divide multiply multiply avoid storing array implementation total arithmetic source mathematically make relate correspond typical alone nmf learn represent dictionary discard st evolution activation learn fix divergence nmf measure contribution bin common magnitude fourier typical approximately reconstruct separated multiplying consider output mask audio take arithmetic operation per array sensor audio signal value enough wave array enough issue wave linear square bin fix geometry problem bin take single design matrix bin parallel array interpret frequency treat marginal source allow tie together source require direction account sound source choose appropriate multimodal come true desire projection begin factored model force finally factor number time mask source symmetric source priori environment source multiplicative update reduce resource requirement example eq multiplication compute multiplication divide multiplication multiply similar intermediate memory input output never array even mass f f td df simplify since nonzero denominator sum define define get arithmetic multiply take operation sum multiplication multiply resource requirement arithmetic factor supervise nmf use use clean audio still resource cost traditional separate db mask mask nmf instance confidence instance available version paper ghz intel core gb ram demonstrate random sentence construct two different array mix file delay relative separation directional less directional nmf consist f nmf two receive channel mix audio audio clean algorithm source directional nmf source directional nmf two directional close direction fit speech background directional speech center distribution location source applicable well array rigorous geometry work particular separate closely spaced acknowledgement thank paris david suggestion regard theorem assumption method audio sound propagation separation greatly remove nonnegative factorization method audio source sound provide potentially arrival bin form frequency direction tensor source advantage much traditional supervision clean source resource traditional array extend technique audio literature apply nonnegative stack drawback decomposition gain post cluster
dynamic equations follow eq let predict diagonal error enough make linearization accurate eq compute density substitute substitute membership along switch likelihood maximize posterior label arrive posteriori map current drop thus substitute f apply linearize temporal logarithm ignore diagonal large row sbm local initialize membership node change posteriori procedure employ initial membership spectral initialization prevent getting begin priori time step calculate multiplication inversion size dominate apply neighboring assignment visit substitute invert log inversion matrix assignment local search reduce multiplication search note search algorithm specifically visit neighboring execute separate core reduce inference four covariance noise relate prior form edge initial x diagonal state mean observation observation state assume invariant dynamic sbm relate advantage plug hyperparameter denote time furthermore diagonal affect entry structure estimate assume neighboring index exploit function propose hyperparameter non survey hyperparameter linear em maximize note propose procedure make involve density gaussian binomial rule binomial reasonable sbm correspond recall small approximation linearization approximately linearization approximation kalman filter filter filter often well computational argue pose linearity observation taylor negligible matrix entry denote bar generally suggest simulate dynamic sbm investigate synthetic generator class state evolve construct snapshot sbm since proportional variance term suggest sufficient square filter use pf g actual pf tracking confirm sufficient pf pf limit approximation network initially split class time randomly assign simulated network baseline static stochastic time spectral baseline propose gibbs anneal applicable posteriori tracking outperform slightly track slightly less second fig method achieve low mse priori posteriori bad priori observation proportional perform extremely poorly set evaluate adjust rand index adjust perfect expect accuracy adjust posteriori offer class achieve accuracy estimate true utilize estimate expense computation core ghz intel processor able outperform computation sensitive sensitive hyperparameter probability conjugate distribute rand hyperparameter posteriori ab note choice fig extremely choice certain rand index close assigning recommend maximize modularity strength partition ground modularity class correspond community dominant modularity extremely mse apply set suffer significantly evaluate baseline number node priori require second posteriori number search v denote increase hold utilize temporal suffer poor recover state show time class number notice magnitude expect al variation membership unlike space sbm result high tracking fig near require inversion covariance could achieve significant noise space rand tp mit reality phone activity student mit year construct dynamic physical measure nearby device exclude begin experiment week participant serve excellent network aggregated network physical first year business school student work compare accuracy posteriori show membership posteriori agree membership heterogeneity within community heterogeneity participant spend proximity time posteriori fit actually demand edge sbm adapt change edge accuracy compare email email week step correspond send cc addition role within company available use class place remove send unlike truth experiment task comparison link link link new edge current remove latter address static sbm alone operate individual predictor move combination link individual evaluate receiver characteristic metric undirected ccc sec posteriori dynamic auc alone priori add membership advance method method roughly auc magnitude obtain diagonal logistic interval tp examine reveal trend increase week inspection content send week confirm cause normal week week correspond event role another highlight fig select confidence show edge frequent discussion six know role begin increase fall notice peak align three week reveal peak email activity event increase volume identify edge across internal dynamic evolve furthermore temporal estimate would fitting characterize dynamic stochastic static either priori posteriori utilize inference procedure comparable accuracy apply base email trend trend steady edge financial investigation examine temporal class reveal examine send predict email propose evolve would provide source code anneal kalman toolbox matlab grateful particle thank comment paper iii effort development analyze network represent either snapshot observe time rich phenomenon dynamic static dynamic manner extend kalman demonstrate monte demand network estimation kalman interest complex biological phenomenon range protein interaction formation naturally network research represent snapshot interest aggregate literature complex phenomenon social researcher examine aspect shrink structural include dynamic social node correspond people edge correspond presence indicate occur characterize network utilize dynamic first propose combine type state commonly static social evolution model become sbm block increase employ present kalman augment monte yet accuracy demand mcmc true state analyze dynamic email interesting trend identify aggregate total send invariant time model applicable datum array social fit lee proposed attribute multi
user easy security access special htb familiar oriented analyst capability part worker difficult adjustment factor multiply produce technical factor get call formula environmental ef multiply f product ef final adjust calculate take calculate support machine implement inductive obtain pattern suffer drawback neural minima rather minima fit mean part suffer either drawback goal map space dot term cost point measure target deviation call basically I kx tx j goal target ignore implement support regression em string regression look em penalty value range default regression range different value rbf sigmoid default calculate actual effort train lastly default parameter predict eight four article software development improvement various technique scale e b f effort software em collect project use input scale element value calculate scale x predict selecting divide learn select optimal step fold generate validation criterion operation find validation select response check error rmse magnitude prediction accuracy test indicate visual comparison step effort various result use effort present test purpose validation partitioning training validation validation remain cm cm remain learn partitioning fold varied range generate operation ht model cp polynomial validation validation ht c model sigmoid rbf choose error validation sigmoid choose base validation c finally train test testing effort use error calculate cm effort test observation q square calculate divide rmse deviation actual effort datum implement software effort follow generate square squared squared mse cm coefficient cm square mse cm regression cm mse cm evaluating strength relationship actual effort kernel correlation point predict minor result predict effort sigmoid plot variation effort correspond show little hence dispersion predict data model exhibit various method effort rbf base htb actual effort estimate data htb comparative accuracy accuracy relate c table display section I less value use effort develop orient estimate effort develop optimize use study comparative assess compare result obtain outperform similarly obtain rbf outperform computation membership available soft particle optimization genetic ga science currently science technology interest software department since interest software engineering engineering management international member usa job software estimation early software detail improve vector getting map nonlinear kernel transform output software approach diagram effort project optimize keyword orient software development several concept abstraction play development effort model project effort effort line paradigm human effort early stage software feasible benefit use point diagram effort product help accurate software measure number actor multiply factor case actor determination simple complex number transaction widely decade limitation limitation effort software effort weight outperform base ba propose machine model public software organization usage produce software effort effort software extension point uk effort
software computational block sect write matlab net represent cnn potentially operation please detail basic neural block serve complex implementation train imagenet stock matlab http www imagenet imagenet mat net load imagenet mat I I I net net normalization describe preprocesse take intensity range layer language I image name convenience matlab score class extension feed average network encoding compute derivative propagation basic implementation gradient example example cnn cifar imagenet probably scale imagenet gpu adequate cpu disk highly recommend imagenet suggest imagenet imagenet convert image height imagenet manner every forget ram disk copy setup ready cnn enable multiple go able describe interface matlab take input return array pack map image arbitrary shape implement work backward direction well order pass third return derivative block parameter x take specified property w multiple matlab rest describe focus analytical refer matlab help implement compute map formally bias filter oppose subsampling array implicitly zero convolution fully array width use filter index various array usually field affect output later connect instead former output handle additional flexibility channel bank w filter group grouping use stream filter slow dedicated block deconvolution implement transpose filter output tensor imagine reverse use bank obtain convolution transpose transpose softmax channel convolutional location softmax exponential normalization operator ground apply across summing combine numerical stability compute vector location option eq q cnn mapping entirely enough hence output filter subsampling sample fall window stay since wide input generate sequence convolutional start signal width sequence signal recursive seem operation obtain approximate exact without filter input signal call determine affect level filter quantity width input layer discrete usually case operator odd delta continuous discrete sample matlab convention extent signal centre support operator coordinate hence offset application centre calculation convenient express convolution operation form I extract storing matrix eq I express expression eq formula derivative array likewise formula use implement convolutional inefficient fast approach allow leverage implementation understand convolution transpose convolution rewrite matrix happen index derive convolution transpose input may outside range convolution expand formula infinity recover involve filter likewise fairly tight depend element possible uniquely instead tight summation refined finite pool output element usually pose max relation exist binary order normalization eq q relu vector sigmoid output compute channel process derivative respect indicator bottom derivative follow note take evaluate simple divide numerator denominator eq simplify obtain array eq softmax array derivative little rectangle em width thin black gray true ex false convolutional toolbox simplicity flexibility cnns new cpu gpu imagenet document provide cnn implement give toolbox toolbox implement cnn document start cnns list build block combine cnns technical one discuss view direct document translation local toolbox contain thank modular create combine new one usually parameter result output learn suitable cnn architecture train thousand million conceptually train point result vector update minima derivative rule use capability default solver top library require iterate vast important large particular gpu reasonable integrated gpu capability design cnn layer software relu operator building cnn sophisticated several world cnn back matlab code architecture computer vision cnns contain fundamental building cnn convolution b filter bank bias derivative cnn suitably implement topology chain block current classification look start point implement cnns descent cpu gpu mnist cifar imagenet state cnn obtain connect image input real array dimension index dimension last dimension represent auto node f east west north south formally stack height width operation identically dimension ability operate batch dag network output output l x f block right node right block right west node west f east dot dot east west east south north south south simple l cnn interested effectively auto node distance datum dot dot w w z right east east f west east west dots east west loss west east south south dag work chain derivative work pass symbol derivative derivative block f composition simplicity drop subscript compute derivative auto distance f block w west east south east west derivative fact first derivative element shape beyond storage derivative storage fact compute latter apply recursively block chain section suggest modular programming interface cnn parameter message cnn derivative block
different unlike rnn require high rnn unable hence current slow fast rnn consist connection hide unlike partition module module module module module period sort module period module propagate leave slow module module standard output q activation time step activation simplicity bias rnn mod execute period period module period recurrent partition block module time period evaluate part highlight period part contiguous matrix triangular forward pass execute evaluation retain output step calculation illustrate ht retain speed module focus provide speed module error module execute activate module activate add module speedup rnn neuron exponential setup detailed derivation rnn lstm activation approximately rnn period initial weight deviation value descent sgd nesterov style momentum ht approximate much accurately task train target whole create ms sequence point interval input linear network summary epoch square decrease set separately keep find rnn rnn crucial forget high encourage nine rnn fail seem improve show big rnn far par rnn get output network five average generation lstm rnn second audio speech dataset arrange order make technical critical recognition competition example partition ms ms window emphasis channel normalize mean architecture softmax layer hide layer whole momentum input stop training epoch lstm forget gate neuron divide evenly exponentially follow give substantially lstm irrespective rnn rnn generation well lr lstm c rnn rnn learn inherent module period period intuitive option period back propagation would alternatively evolutionary closed low period adjust frequency set module size grouping module hard provide superior speech standard approach speech first translate rnn recognize module detailed internal taking place understand class reinforcement rnn assume rnn total neuron connect recurrent rnn half module operation step exponentially period per less recurrent typical evaluation rnn conservative acknowledgment research foundation grant reinforcement learn fp challenging identify distant recurrent ability theory cope virtue short connection long modification standard architecture rnn rnn partition processing input computation prescribe rnn rnn improve preliminary audio lstm rnns recurrent feed connection classification prediction rnn train difficulty dependencie sequence vanish specialized neuron backward order optimization preserve inform random allow training momentum gradient modification rnn error back performance sequence contain term dependency dependency solve different module rnn hide different discrete rnn rnn train module number slow module rnn test supervise using word preliminary outperform lstm provide simultaneous sequence deep variant state result neuron order principle time grow network add recurrent connection connection attempt enable rnn handle dependencies neuron activation bit technique technique use serial rnn neuron decay
psd soft element solve trace psd solve sdp need tune way single switch st projection tuning sparsity penalty solve algorithm trace relative run outperform spectral baseline superior report whereas sequential convex norm tailor rank motivation formulation inference first well understand formulate super using allow notably trace norm observe use support limitation work investigate support future nuclear optimize prove claim nan term large coefficient decomposition permutation decomposition obviously systematic enumeration possible right orthogonal show singular purpose express svd write primal check subgradient equal must satisfy ij z admit decomposition z equal claim decomposition svd disjoint support decomposition attain convexity decomposition contain prove claim consider positive semidefinite prove optimal lemma norm ab q equality maximization convex variational formulation infimum jointly convex elementary analysis also symmetric positively homogeneous j ij prove uniquely use characterization subdifferential characterization subgradient atom ba g norm z reason orthogonal vector nuclear norm ab inclusion middle equality therefore atomic induced atom rearrange eq k ik conclude start ij ij I I b g ia g one similarly op attain right give operator span show take u ia op g op I inequalitie g j I op j op lemma third j dt dt dt disjoint random chi square moment take intersection I jt let ab k q norm fail universal trace notation working norm notation decomposable norm norm point define subgradient norm j jt b lead hence lie cone follow enough ensure n let start th common cdf let denote pdf fu fu fu du v assume jensen cdf error inequality due deduce vector standard check euclidean cone cone index large absolute denote I otherwise ks subdifferential subdifferential let w rewrite coefficient expression subdifferential show mean statistical dimension use normal obtain take plugging show well hand strength lead eq statistical axiom rgb rgb pt electrical engineering stanford universit e paris est imagine des france centre computational france paris france paris france atomic number nonzero factor bilinear slow bind statistical formulation algorithmic scheme propose leverage promising range machine prediction phase dictionary sparse rank factorize sparse factorization allow storage interpretable accurate situation interaction highly overlap admit generally matrix explain superposition principal component sparse high genomic view convex instance note solve plant clique heuristic solve problem lead procedure right factorization optimization hardness generalization mild semidefinite sdp relaxation principal successive investigate coarse investigate convex nuclear investigate investigate naturally relaxation element basic however norm norm sdp find principal favorable thresholding work formulation guarantee new regularizer low multiple provably norm pay np resort procedure solve theoretical gain contribution norm factorization involved rank nonzero right surrogate build upon characterization nuclear support problem bilinear pca formulate norm however compare first slow upper insight trace cone dimension norm superior task factor gain vanish norm vector norm scheme approximately regularizer bilinear quadratic consist provide solution principle numerically focus one simulation linearly decay overlap integer index number support e set index vector entry elsewhere inner matrix notation stand number norm standard frobenius norm trace nuclear singular dealing form j outside allow formulate start define rank quantify introduce atomic tight relaxation operator norm construction constraint wise factor section relate norm norm establish define component concept solve problem notion incorporate j recover share rank follow proposition might collection sum inequality problem sparse consist symmetric approximation want relaxation aim instance atomic norm introduce atomic definition rewrite plug usual singular usual relaxation simply deduce expression follow trace singular generalize call sparse share number usual strictly large leave singular next differential subdifferential b restrict motivated matrix define define atomic atom atom whose element polytope coincide polytope q norm cut atomic norm atom alternatively instance recall norm characterization norm formulate nuclear infimum norm show nuclear induced atomic norm induce atom nuclear induced atom norm nuclear induced support completeness norm vector sort unique theorem find hull scale unit constitute see nuclear norm interesting factorization know nuclear norm nuclear elastic net briefly norm involve noisy low noiseless simply generally matrix priori input observe mean small involve feature convex instance combine retrieval note rewrite form transformation view parameter well feature map assume cluster point cluster low space design mean form row mean exist block sparse matrix dimension try matrix low although wish psd plausible suggest formulate component variance natural relaxation although psd ia follow proposition psd rank psd psd matrix write matrix may interpret successive explain less replace impose replace atom definition consider atomic precise formulation psd expand formulation proposition psd matrix psd approximate although norm formulate guarantee solve let optimization span symmetric np convex involve third heuristic approximate involve commonly recursive manner lead sample possibility force orthogonal component orthogonality motivation dense consequence thresholded pca clear motivation relaxation pca name direct sparse aim solve eq smoothing regularizer trace norm matrix community norm construction norm suffer conditioning atom build tighter rigorously interested depend bring support assertion compare enable avoid aim find sparse component optimization portion unit ball inside psd cone compute proximal map theoretically benefit new penalty low factor discussion building technique recently penaltie norm derive deduce square denoise interest rate norm easier rely denoise wish observation corrupt additive penalty norm control set study random noise order derive estimation provide expectation norm entry expect norm consider oracle estimate immediately follow control estimation error oracle eq derive upper error different call single ab ab ga ab ab ab immediately plug upper bound estimator respectively respectively straightforward matrix atom upper suggest denoise comparison table trace column magnitude instead penalize reach change norm enough conclude superiority trivially incoherent obtain decomposable trace rate still incoherent weakly valid rank low upper statistical norm closely relate powerful asymptotic geometry quantify nonsmooth regularizer penalty essentially point denoise sense quantify measure width intersection concept cone statistical induce theoretical exact recovery norm convenience linearly norm scale norm constant briefly related matrix tangent cone closure cone I standard normal result iid probability soon addition phase situation large situation corrupted assume noisy satisfy z least subsequent section compare technical let simple number actually regularizer statement hold prove satisfy follow informally trace scale nest meet matrix property norm meet nest tangent consequence satisfy hull decomposition c follow hull fact red belong plug show atom good good use norm upper norm probabilistic expectation fact inclusion tangent inclusion ball statement recovery consider norm realization exact recovery support cone cone vector support easy show computation substitute go appendix er add reference ok tangent suggest norm provide improvement recovery performance present specific characterize performance norm turn explicit estimation vertex immediately plant clique estimate coefficient recovery support depend signal atom ab ba compare note match logarithmic dimension trace combination table result vector trace kk p k km mp norm counterpart element plant notation mean norm atom bad dimension atom statistical decrease alone bad trace rate equality ab bring improvement bad statistical dimension statistical trace norm theoretically regularizer lose support specific follow upper dimension sparse coefficient entry sort absolute atom strength upper bind reach case hand atom bad standard never raise utility lasso complexity different regularizer elastic net tangent point half cone elastic net always dimension proposition improvement note degree freedom match logarithmic term aim improve proposition situation generally proposition match low statistical note cone equal tangent cone tangent exact statistical similar vector sort equation active say element surprisingly dimension proposition number degree element match consider aim perform unclear sparse matrix norm symmetry yield ab map fail recover universal appendix equality ab must grow ab dimension support small decrease upper norm bound tight sense suffer ambiguity become sensitive reader many involve sparse rank work column subset let optimality often component use working solve grow sequence zero throughout typically useful regularizer notably group lasso also optimality subdifferential writing ij j current approximately subsequently violate add initialize previous solve descent iterate proximal modification solve minor amount replace
large dimensionality play modeling typically produce candidate involve covariate compare researcher selection former kl deal devote extend chen chen liu al aic frequently use tuning mode penalize wang wang et zhang al fan fan inconsistent grow size implicit fix dimension case recently liu selection misspecification principle generalize linear lead aic bic prior probability motivate principle generalize bic liu dimensional counterpart misspecification misspecification high answer question gain motivate response functional size dimensionality regression criterion oracle working consist aic bic ignore misspecification reasonably select work fail model selection model newly suggest work significant establish misspecification high expansion different principle challenge technical justification incorporate misspecification set prior connection chen chen fan organized introduce misspecification present key quasi provide selection main technical supplementary material entail large denote deterministic practice work data misspecification generally occur true choose generalized link work db z contain value nf n observation vector define kl work close true two play role selection misspecification f asymptotic expansion kl divergence principle list prove property constant assumption establish standardized response liu major setting converge neighborhood wide dimensionality allowed impose normality condition normality dimensional glm liu next introduce additional principle hc n n n n norm naturally accommodate mild ensure restrict expansion kl divergence grow shrink except require lipschitz entry wise mild sensible bounding prove set principle mle expand lead aic compete drop quasi hereafter tending generalize liu high substantially due dimensionality correctly asymptotically demonstrate study aic substantially misspecification expansion latter introduce contrast characterize impact misspecification provide accurate criterion f assume hold liu aspects contrast previously justify liu simple plug enjoy consistency misspecification crucial practical implementation reveal work misspecification setting compete nonzero vector correspond bounded md locally zero posteriori model ease quantity quasi likelihood condition tend nc replace model reflect effect misspecification correctly specify reduce bic probability subscript candidate motivation far away glm larger sensible complexity motivate exploit extend bic chen chen show additive hold asymptotic new term non polynomially grow tend n np theorem consistent penalty fan side view sum misspecification counterpart dimensionality whole asymptotic expansion divergence principle introduce high dimension misspecification investigate selection bic error multiply effect misspecification involve covariate five function true regression result table latter supplementary term two confirm necessity fan specify multiply selection inclusion probability oracle prediction regression logistic pn rest response success section choose regression logistic interaction model argue oracle correspond regression five since replace table latter available supplementary show phenomenon section model dimensional multiply oracle error rate gene expression set positive negative nb set control project consist positive trial set fan exploit screen apply retain choose time median median table table good l median classification worth parsimonious expense effect model misspecification generally real suggest involve misspecification expansion selection percentage nb despite misspecification misspecification newly factor dimensionality complexity establish consistency contrast capture misspecification general adaptive correctly fan consistency criterion sample misspecification principle additive problem beyond current topic present theorem save technical material notational throughout proof specify order state constant notation response convenient euclidean main event calculate continuous full column definition n continuous concave concavity log positive entail event global maximizer must belong interior neighborhood hereafter condition herein due grow taylor expansion q line taylor expansion rewrite take derive eq negative product n entry respectively next obtain bound sub tail condition exist see notation let tail derive last th obtain norm dr thereby q choose hc choose stand omit subscript sequel establish require possibly n specify grow intuitively understand try put restriction nm large ensure mle coincide shrink hereafter mle unless recall e ne complement equality follow definition term expansion around evaluate tr n b regression last inequality verify order I n n n n n es n cp ensure show establish supplementary e cauchy schwarz yield entail yield similarly q c asymptotic use derivation restrict conclude proof view expansion square arrange increase large eigenvalue side n p result norm
reader development contain trait may importantly handle different govern unobserved p depend arise multivariate brownian realize follow probit formulation outcome underlie continuous threshold threshold alternatively assume order state map relative threshold identifiability non order trait value adopt observe dimension determine finally monotonic transform example distribute brownian diffusion give rise element node multivariate distribute unobserved trait variance manner characterize trait trait share descent trait integrate trait recall density function one equal weight short path node I nod augment trait latent convenient factorization factorization augment truncate normal illustrate four include tree annotate trait realization along code trait realization trait state figure modified package freedom specify aim learn mcmc development computationally kernel exploit scan metropolis scheme employ metropolis proposal full enable problematic tie also attack evaluate metropolis acceptance appear high form nf repeat limited algorithmic idea computationally sampling illustrate pre propose order traversal post traversal proceed imply internal root visit compute conditional proportional normal traversal precision characterize reader f ff distribution hasting approximate full conditional remainder distributional far collect wu conditional quantity normalization u u traversal integral solve n multivariate identify u pre partial vector precision scalar f toward must latent trait trait ic ic ic cc partition correspondence trait matrix generate possibly augmentation example manuscript range explore involve try metropolis simulate f hasting acceptance proposal start occur valid become mcmc mcmc chain towards probably employ proposal center assess relationship trait look pair wise correlation non fall great strictly less scientific lie comparison involve pair possible identify diagonal structure trait evolution demonstrate example order trait factor likelihood straightforward high adopt integral estimate possibly present estimate marginal likelihood efficiently comparison different path q parameter sampling employ numerically path natural path since require lead guarantee normal independent corresponding univariate whose match large trait cdf trait univariate analysis threshold map assume open interval dimension simplicity multivariate present wish assess type evolution report wise correlation reveal trait trait analysis additionally second trait highlight change feed trait order state model outcome position instrumental determine infer state bb regard adaptation examine compare trait bb bb bb formulation order latent bb bb inverting order sign trait marginal indicate bb factor bb bb bb bb bb share evolutionary estimate latent distance present compare account share history notice correlation pairwise orientation orientation contain evolutionary weak orientation account history surface provide rapid drift challenge drift site grant insight analyse site b protein site vary period major allele frequency suggest limited contain variant trait latent without generality site assess structure latent pairwise zero estimate site site contiguous suggest include drift present credible interval include positive range site trait see association evidence correlation coefficient site drive site latent assess use discrete biological problem show structure tool general threshold latent markovian argue trait vary spend state univariate order reconstruction simulate perform trait consider already comparative biology correlation requirement account lack sized interval would discrete correlation credible interval constrain prevent recover example two trait continuous trait root internal motion lead significant improvement compute successive post traversal effectiveness multivariate motion evy traversal improve regression gaussian latent perform integration build dynamic programming truncate conditional though truncate accept highly find rate become reference state model dimensionality mainly improve identifiability symmetry interpretability trait represent entry trait despite different choice change link briefly determine outcome common simplex make trait interpretable alternative evolution model tendency investigate whether identifiability relaxed explore trait branch comprehensive analysis acknowledgement lead result receive european fp grant agreement agreement trust health grant ai author acknowledge reference service dt provide constructive manuscript orientation length length control orientation orientation site page en sn ns dy dy dy sn ns vi ns sn ns sf site site align code correspond latent trait site belong department mail school public health human david university california ca usa center usa trust trust genome united understanding trait modern evolutionary biology assess type simultaneously control evolutionary molecular us trait trait trait state single along history trait history framework finally em evolution phenotype interested assess among trait genetic trait determine link alternatively selective environmental pressure act trait outcome trait affect pressure aim comparative purpose trait simultaneously combination type discrete outcome discrete outcome also tool hypothesis regard comparative trait trait control share evolutionary history dataset markov trait allow include trait assess transition
impose construct unique diagonal incoherent note fraction perturbation suffice make impossible condition l converge approximate guarantee exactly rank special guarantee noise n q present key standard argument detail iterate incoherent perturb appendix symmetric iterate sparsity ii iii repeat fold recover significant b demonstrate conv art solver experiment experiment real foreground result average pseudo code knowledge instead th tune conv incoherence cccc see interestingly remove step arguably dynamic foreground keep step take fast restaurant frame resolution moreover visually extraction well corner counter similar background video require non pca projection method match method experimental interesting match model recovery result improve noise need investigate decomposition beyond structured sparsity pt acknowledgement aa would like acknowledge nsf grant microsoft fellowship acknowledge grant grant acknowledge fact question recover rank unknown support project matrix projection establish require input run need run require per contrast complexity exponentially iteration synthetic establishe improve exist convex alternate projection principal pca preprocessing denoise carry pca implement sensitive attempt force outlier overcome pca reconstruction topic community detection input seminal work solve relaxation elegant expensive run large poor require carry complexity drastically pca exponentially accuracy gap singular rate global convergence prove minimization recently completion work method grow match subject sparse perturbation reveal gain relaxation run low rank thus rank nearly match pca sparse technique constant zero theoretical enjoy art inexact lagrange multipli time level accuracy real foreground separation visually separation establish contraction set sparse projection perturbation vector suffice establish correctness next hard thresholding inspire similar eigenvector enyi exploit characterization taylor reveal perturbation eigenvector adjacency subgraph thresholde contraction argument contraction case alternate stage alternate value argument perform rank hard procedure need hope perturbation convex pca robust past seminal work incoherent relaxation eq nuclear nuclear typical solver involve convex set soft spectral domain non incoherence upon match requirement recovery sparsity incoherence yield additional exact recovery entry rank robust would plant problem additional work weak assumption incur xu et specialized exact provide tuning relate multi alternate multiplier consider multi step multi block random work non method pca hold still intuitively project onto appropriately section robust formulate find lie project onto one ht sparse rank initial remove perform matrix initial hard thresholding progress subject large perturbation alternate computing perform thresholding entry certain gradually decrease proceed reconstruction naive extension condition matrix singular singular perturbation progress propose proceed stage projection thresholde run stage lower singular
continuous regret hand another relevant rl action mdps reward sub policy finite whereas policy htp iid primary simulation result demonstrate correlated bandit feedback mdp optima around optima possible equal step maximize experiment optimistic bandit alternative space predict bound per big identical empirically though complexity empirically fig requirement scale predict thm near optimality brief requirement grow polynomially usage iid regret whereas require order node iid create mdp function upon take value environment agent receive state priori reward rl optima rl policy mdp optimize optima algorithm design set mdp succeed find evident converge stochastic approach computational optima requirement suggest benefit approach online mdp global knowledge policy search current version learner unknown iid example require simple dissimilarity cumulative smoothness notice dependent reward policy partially mdps introduce new regret bound simulation feedback broad report iid introduce notation indicator create depth internal e leave nod n h need introduce last time time step expand node expansion expand coincide phase bound depth construct depth tree iid expand expand summation obtain solve high expand within empirical estimate depth application inequality upper depth e term choice eq recall definition sect term confidence interval confidence confidence interval bound probability first regret trivially suffice never imply remain term sum union imply combine ready bind tc step far instantaneous regret bound martingale difference probability proceed measure start characterize actually event node node immediately select value iterate parent node since cover space least include maximizer hold expand side high inequality definition p tf regularity node always exist leaf see parent select far simplify provide preliminary instantaneous select refine parent event rely rhs simplify need depth binary node depth twice node expand also recall parent select parent cover combine second term cauchy schwarz total focus summation child sequence rely time rely cover internal cover sum invert deduce notably term choose lead regret bind eq lem union final analysis assumption inequality iid random episode consecutive episode first select episode horizon arm episode arm objective build q notation episode condition time see definition residual follow bound martingale proceed arm episode total episode far rewrite sum lem grouping divide need need previous number episode step start episode notice unchanged begin apply result lem lem episode previous episode except episode large termination episode become large time episode horizon episode thus invert previous obtain episode probability lem statement simplify side objective achieve homogeneous h q lem previous case bind hoeffding high probability confidence estimate interval event concentrate arm could argument reward therefore inequality concentration eq arm event hold complementary step lem term definition step bind definition sect decompose depend event interval instantaneous rewrite interval confidence regret iid hold bound expect ready regret step difference step decompose instantaneous regret lead regret unlike sequence difference extra need derive follow hold definition nn every last episode coincide episode event result parent proof arm entire episode sequence simply episode instantaneous statistic episode immediate put bound lead final combine lem prove final nc hold reward generate store correspond branching time regret decompose depend since expression easily bound lemma rewrite total generate bind inequality expand fact large twice invert bind need probability event leads I c plug together depth lemma optimize statement online confidence algorithm bandit regret bound dependency challenge reward whereas reward generate iid process art well weak smoothness previous reinforcement sum reward sequentially optimization arm objective cumulative relative global focus immediately condition identically contrast bandit armed bandit relevant internet online game policy mdp mdps paper sect introduce first policy mdps build advance iid regularity e smoothness guarantee linearly number rely heavily iid feedback introduce sect explore arm space tree optimistic part arm insight achieve necessary expand optimistic sufficiently accurate even iid ergodicity mix sect correlate match sect iid dependency define require require though supplement development iid structure complexity runtime make scaling meet improve space iid sect benefit sect mdps arm formalize possibly relate arm reward context time arm dependent reward since infinite refine setting generate ergodicity ergodicity regardless arm time average follow mix finite exist mix stochastic reward trivially iid maximizer exist denote maximum learner observe differ contextual contextual bandit arm immediate reward function input contextual context next may reward current reward problem maximize reward sect reinforcement reward differ regret prove discount reward seek minimize binary cover detailed covering root cover index convention area region partition overlap arm algorithm whenever assumption dissimilarity equip diameter open ball bx x exist constant bx b coincide lipschitz arm require characterize optimality dimension set arm sake clarity dimension near optimality tree confidence iid arm tt tt tt tt tt tt tu correspond select framework discuss variant iid design reward arm iid reward arm alg ht tree ip ht tt hierarchical reward algorithm keep track optimistic begin episode episode episode valid I reason accurately bandit feedback assumption reward correlate reward actually lem mechanism expand node obtain mean feedback general iid variant reward arm full episode arm iid use complexity proof supplement reporting depth generate threshold guarantee fact estimate expand number grows depth grow report regret iid condition event immediate mean reward supplement perfectly match check show structure expand result require iid discuss sect although proof mostly literature move iid call technique main issue average different episode concentration inequality lem supplement episode bound technical derivation hold generate accord iid aspect iid iid major w arm iid mdps boundedness phase lem depth also coincide case node episode reduce selection cost cost node boundedness depth node still cost time unlike extra due truncation provide space space scale observation increase factor
coherence depend acceleration result noisy corpus stochastic mix channel set noisy clean impulse noise partially consist room mc corpus source near circular array diameter task error gmm enhance order noisy feature compute dimension show development test dnn acoustic achieve gmm negligible lead acoustic train clean noisy multi yield confirm coherence exploit dnn frequency resolution reduce improvement show necessarily exploit signal front input propose cloud speech recognition geometry device adaptation oppose sense arrival recognition achieve spatial dnn speech recognition environment dnn real multiple knowledge direction arrival bin feature acoustic rate challenge extract enhance spectral recognition automatic markov gmm hmm wide extraction employ extract contain signal efficiently acoustic neural network neural acoustic learn manually transformation outperform amount structure trend replace stage implicit array spatial channel feature gmm exploit signal aware noisy may principle noise estimate dnn acoustic exploit inspire towards spatial information field acoustic speech noisy environment treat coherence surrogate temporal variation aim dnn acoustic describe instantaneous coherence speech integrate dnn task outperform consider speech record th component short letter frequency axis auto f ratio coherence desire characteristic complex mixed sound bin convenient computation cdr diagram extraction signal domain correspond extraction term log combine power compute triangular filter apply show extraction enhance multiplication gain describe estimate weighting extraction apply sound field amount noise dependent expect estimate neural network enhance trend acoustic replace implicit use coherence coherence characteristic sound array may therefore array spherical array require acoustic model coherence system reaction change coherence change ms window frame shift ms transform dft factor triangular weighting cover code feature speech training corpus speech highlight
diameter bx side pick one optimal slowly bound see condition boundedness extension meet oracle action chosen long grow problematic controller automatically turned keep design robust back state safe system consideration controller rely controller replace available use controller come leave safe input safe initialize controller controller pick controller reasonable controller prevent exist happen theorem safe q consider family deterministic markovian smooth parameterization regularity running concentration trajectory addition optimal controller immediate r action step illustrate mdps mdp feature dimensional zero indicate mapping state transition probability state prior dirichlet nd tp ts ts action pair ts nd dimensional show frequency ts ts action dirichlet time choose next parametrize gaussian share similarity generality subgaussian gaussian without generality compact assume column corollary performance parametrize problem controller available parametrize suppose time satisfy resource management problem control resource chain describe evolution find class admissible action compact leave conjugate imply get propose algorithm thompson thompson distribution thompson compute propose choose time unlike thompson implementation thompson computationally may subject largely mdps sublinear consider finite horizon policy issue arise policy set significantly set approximate computation policy discount let result limitation trajectory follow policy policy general action obtain mdps space linearly still computationally even planning purpose simple server control control follow next web book book http server incoming queue connection assign process drop request second denote control long service usage server bound determined load usage operating point operating sequence gaussian diagonal deviation operating http server measure provide control purpose cost algorithm optimistic maintain find attain loss solve policy play objective solve consume regret avoid repeat time deviation horizontal axis amount process round change frequent bad prior regret right bottom prior horizontal mc vs chapter fx non inf admissible state action pair assume mapping variation norm sign measure admissible pair contraction banach substitute transition set negative inf admissible action transition discount discount policy exist satisfy bound additional assumption gx define one thank hold eigenvalue one large eigenvalue inequality definite matrix plug bind part semi nonnegative minimizer loss hx bx trivially fail proof decompose regret bind map deterministic let control oracle second follow change number last change multiplicative use old last cauchy schwarz second collecting inequality change together continue assumption replace along trajectory reason get cost thank check define x mx ax mx thank apply corollary satisfied na dirichlet p e thus theorem thm remark university general smoothly markov problem design posterior maintain unknown reduce importantly analyze show performance computation tradeoff method web design control randomly controller mass produce production pattern maintain good control rather appropriate knowledge transition dynamic cost would history apart policy resort suboptimal suboptimal compare optimal question computationally add reward estimate horizon discount sampling first reinforcement mdps failure regret factor course apart interested excellent unlike allow infinite subject regularity cardinality secondary compact phase begin compute draw algorithm keep uncertainty important element allow nonlinear allow go scope quadratic linear linearization policy achieve long run average loss resort measure dependent slow policy sublinear converge get
position contribution assumption make lipschitz logistic error convex hinge loss nevertheless smoothing algorithm framework still continuous ill condition usually larger add purely regularization purpose well complexity loose useful algorithm e find gradient require g pass form dense contrast operate complexity far take iteration fair incremental term pass expect precision divide complexity example batch batch complexity gradient complexity exploit average coordinate batch weak dependence method present batch saddle saddle function I presentation convex saddle assumption strongly saddle point primal accelerate coordinate complexity also mini suit present first apply saddle still accelerate uniform sampling batch complexity define much discuss method coordinate update extension primal recent batch complexity feature sparse norm penalty computational depend compare art optimization include batch sag coordinate comparable tx uniformly execute I randomly pick execute update analyze dual method idea quite saddle alternatively maximize minimize since dual maximize expensive reduce computational picking maximize computational iteration update primal give instead directly quadratic strength specify theorem auxiliary rule step acceleration fast present introduce mini natural mini coordinate mini mini pick index achieve first disjoint assume select add mini processor update coordinate compute accelerate batch operation ignore delay take basic surprisingly mini single since basic mini convergence choose mini establish iteration complexity mini batch assumption obtain e ensure equivalent inequality denominator recall corollary iteration complexity mini batch achieve batch less iteration extreme batch primal see discussion relate iteration pass mini batch lead efficient prefer choice mini batch batch approximate minimax specifically meet requirement complexity need extra exist either lipschitz smoothness guarantee px run minimize hence imply substitute old suffice hand q x denominator proof first hold second employ unnormalized bound establish addition fail saddle function method case formally continuous saddle scalar modify saddle employ mini add become saddle perturb effectively minimize assume convex lipschitz continuous mini px shorthand inequalities function vi saddle lipschitz lipschitz continuous px establish inequality smooth strongly smooth strongly handle omit obtain sublinear use perturbation complexity mini drawback convergence specific unnormalized norm batch r propose achieve accelerate sublinear rate extend complexity conjugate coordinate ascent efficient method coordinate ascent sdca coordinate pick update increase objective zhang sdca batch complexity ill vast coordinate nesterov randomize lot activity analysis composite variant study batch sdca sdca zhang propose accelerate mini batch sdca primal sdca mini show sdca vary size sdca ill condition problem zhang develop accelerate proximal sdca achieve outer outer loop primal loop regularization contrast straightforward coordinate recently lin accelerate coordinate method convex enjoy extra primal pass splitting equivalent condition formulation whole propose admm sublinear rate complex regularization mapping update sdca efficiently combine method dimensional operation computational per iteration exploit structure dimensional case penalty cost per iteration depend case update ta ta k p jj denote pick value update update delay imply update value iteration compute subsequently compute j x simplify similar assume calculate combination consequence basic algorithm solve problem update accelerate quasi bfg adopt adaptive scheme improve bfgs suggest gradient sag sdca conduct three compare simple quadratic synthetic generate ill condition standard consequence lipschitz cc horizontal pass vertical algorithm output pass entire global regularization coefficient relatively condition sag substantially fast bfgs fast sag sdca notably batch sag sdca name news classification obtain reflect news form hinge e bfgs result substantially stochastic batch decrease l become compare sag sdca opposite stochastic relatively fast get close comparable px fast method relatively ccc news focus maximized strongly minimize property inequality accord randomly specific event happen old sigma field generate define expectation q representation sum index divide I define ta I ta update characterize step derive strong convexity function inequality last definition relation ta term eq absolute eq inequality recall assignment define recursive implie eliminate hand side second equality bind prove relation establish sigma variable substitute average relation inequality q row expand satisfie plug assignment notice q inequality recursive calculate examine equation close equation q
prohibitive alternative efficient wishart p convergence element location conceptually draw wishart scale dependence reach within moderate novel posterior baseline reversible sampler show may build jump sampler way doubly intractable wishart calculation acceptance ratio newly graph reversible jump provide substantial avoids ratio invoke wishart nonetheless edge ratio cholesky double reversible auxiliary convenient acceptance remove flip edge consideration algorithm employ sampler wishart use sampler follow graphical accordingly usage direct double computationally scheme decrease proposal essentially accept mix chain poorly introduce birth death removal birth death change death independent death birth death poisson death analogous birth birth stationary birth death observation birth death use double factor exchange auxiliary novel double continuous dct current g create permutation variable accordingly compute accordingly draw time event accord probability death event birth event validity propose subsequently brain double reversible novel conditional independence follow scatter construct enumeration show expectation three matlab p execute double jump double expectation calculate discrete quantify leibl consider performance true contrary good find fast apparent finally efficiency substantial increase dct whereas time slow kullback expect precision visit double reversible jump mse kl model e dct bayesian assumption underlie structural simultaneously estimate connectivity functional collect subject reader acquisition preprocesse step volume correction derivative filter hz result region signal voxel standardize brain dynamical change produce direct region express correlation brain suffer drawback indirect alternatively partial correlation capture correlation matrix coupling must reveal connect word connectivity dct execute discard burn algorithm identical edge kullback leibl probability connectivity majority high probability show functional right correlation indicate functional salient expect correlation direct pathway well unobserved high partial interestingly edge associate weakly couple away algorithm direct result birth death continuous accurate estimate substantially fast functional connectivity simultaneously work improve sampler introduce move single edge correspond contribute efficient graphical acknowledge economic education innovation van david acquisition fmri macro bayes factor center prove prior matrix doubly partition development direct wishart estimate infeasible propose direct efficiently approximate graphical metropolis algorithm substantially art structural connectivity use fmri area amongst example gene amongst dna segment customer connection population link cognitive gaussian zero precision fully gaussian prove conjugate restrict decomposable wishart monte carlo wishart resource due wishart bottleneck wishart scale wishart fit dependency graphical wishart goal cognitive understand population couple pathway population connectivity correlate pattern population connectivity connectivity
article multiple sound source denote source vertical concatenation let band time please implementation two q originally human study prove part ratio space equivalently due phase nearby cost section model affect experimentally validate frequency refer acoustic wave signal express respective interestingly sound complex relative relationship may self none source correspond contain source common speech binary threshold value average activity sound source frequency sound frequency sound direction mind central entry sound variation variation mixture sound source aggregate information hundred speech white spectrum theory white noise power spectral density nan entrie sound acoustic vector sound mapping feature sound technique feature direction mean white principle possible direction main apply estimate direction firstly input space high training ill secondly nevertheless sound white predict accurate sound direction dimensional role corrupt hence parameterized regressor estimate estimate low sound advantage regression instance matlab implementation available generalization gmm analytic sound direction condition expression mapping train one transformation plus locally transformation reconstruction source spectra consequently eq assume mixture view low dimensional provide affine lie space summarize em evaluate posterior respect step optimally partition minimize affine thereby capture acoustic manifold justification dimensional single source localization affine covariance instead dimensional regression one full covariance datum localization speech already describe value activity seek condition assumption direction namely whose respect normalize predict general sound localization sbm sbm number source code available include sum variable give covariance p pz notation lead inversion conditionally denominator depend neither develop side formulae bayes inversion mixture proportional kp tp numerator simplify term expression localization evaluate source setup acoustic head leave resolution vertical one pixel horizontal vertical horizontal pixel relationship convert degree head camera place middle room computer fan well training room room room room room train people front device associate ground visual detect fig allow face localization method localization plane error localization manually correct pixel position sound evaluate expect sound convert degree accuracy vector fourier ms window ms yield window hz feature typical two head although head room robustness room validate different green testing training manually place position lie plane parallel front long white corresponding position record training refer straightforwardly importantly localization source dataset generate select mixing position source mixture ground truth refer live head camera distance vary scenario scenario narrow field camera person count english person speech whereas people count overlap language consecutive narrow view camera people count language english remain quasi position paragraph live source train live particularly variability emission distance head etc people head carefully noise segment along align video frame segment generally supervise mapping sbm sbm apply segment acoustic sound video mapping sbm training per white camera take pc sound histogram histogram pseudo probability sound length obtain horizontal image sound literature localization dataset result method correspond outperform matlab comparable binary single error localization standard fourth time propose number component choose degree decrease angular decrease localization seem significantly room decrease outlier none comparison baseline algorithm dependency image dependency model white right number maximum approach perform baseline histogram key propose sound rely sound solely base correspond base take minute standard compare sbm histogram peak mask initialize previous regressor horizontal pixel coordinate sbm sound localization variational estimate binary mask train histogram strongly source source dominate frequency bin train white noise mixture pair sbm show calculate outlier cc ccc ccc c sbm use pair type white speech speech speech mixture source db distance horizontal localization degree localization source speech sbm outperform term source sbm expect yield ratio db mixture though aggregate frequency plane high introduce activity reduce average white speech sbm yield demonstrate prominent sound source localization respect scenario critical correctly people party poorly well white sound sparsity white accurately map mixture error slightly sbm possibly use component sbm use affine angular cover transformation sbm times sbm mixture matlab pc suitable application iterative sbm em fast expression component sbm choose bring localization second localization increase localization suggest dense record minute source pair supervise count language circle find row successful localization typical localization full fr scenario square detect available team fr examine test source completely mixture localization sbm yield speech fail intuitive source locate however fact unlikely frequency match localization heavily second speech segment source experiment type overlap mixture plane perturbation vary source spectra section ms slide analysis position segment necessity sbm frames sound positions sbm frame participant sbm localize two source correspond number localize last column row localize face yield fig sbm single number return source position actual abc source include number algorithm position source another fig source supervise simultaneous sound localization require segregation point view addition camera base implicitly start train use white localization noise source single relatively frequency inherently scenario test make sound direction correspond pixel location numerous use mix sound corpora third audio alignment jointly face localization light experiment reliably sound source sound advantage explicit transfer parameter turn room position room cope position room alternatively additional factor variation investigate devise map scale parallelization reduce live experiment plan use detector automatically adjust window take markov grateful anonymous serious highly comment suggestion receive sc sc mathematics engineering france specialized research computer graphic vision universit france ph mathematics post communication interest learn sc electrical engineering sc engineering ph computer de position national en head team interest audio processing area member associate international conference computer project receive ba degree physics electrical engineering technology member department electrical imaging laboratory computer physics imaging vision modal foundation award student special distinction fellowship fellowship fellowship sc ph signal national et physics materials audio laboratory deal aspect model code synthesis audio visual speech separation computer science associate member team france france electrical localization linear address audio multiple location efficient prior neither segregation start gaussian directional source extract measurement length white reliably enable realistic audio fusion namely speech signal onto align audio modality thus enable discriminate face release novel corpus room quantitative evaluation localization sound accuracy art method source supervise regression visual fusion address sound acoustic head robot analyze interact environment shape setup phase band signal spatially narrow relative directional form sound sound direction source mix direction spectra assume tf acoustic power dominate source simplify tf relate direction valid extent mixture speech party state assign grouping select peak accumulate channel iteratively localization expectation intensive segregation estimate dominant account vast localization along simplified sound information must identify head either several source encode use train single source multiple source compete robustness map ccc sound head camera place head head isotropic filter responsible sound localization sound audio compose marker front head device location marker record red circle location train sound circle square face detector view need sound approach recently artificial network feature infer unknown direction advantage stage acoustic spectrum accuracy condition setup room position room etc rather simplified method segregation devise directly simultaneously source strongly scene although inspire mathematical
follow present large knowledge standard split compare split reproduce publicly available wide book collect used purpose seed sentiment explore effectiveness handle either hybrid annotate dictionary annotation strength amazon dictionaries corpora review sentiment context sentiment customer review amazon bias part speech build sentiment classifier field annotate twitter target review syntactic feature build svm classification build boost corpus twitter hash back review customer review concern sentiment problem sentiment english sentiment propose expand english translation scheme build business review build seed sentiment effect preprocesse normalization removal sentiment simultaneous system independently summarize contain movie review collect divide division neutral star rating consider positive multi modern sentiment review collect wikipedia page system sentiment considerable example publicly train research publicly tweet wikipedia opinion review wikipedia book review review categorization largely category sentiment neutral three category reason ambiguity token internet tend positive rating entity opposite language complexity language language different standardized challenge language name entity sentiment compound phrase work provide baseline sentiment analysis set sentiment per user number review book review token review review token user user review avg review book book median token token review avg token token sentence review dataset review rating negative neutral english show star notice review review rate positive neutral review book month review book book non book book perform processing step tag multiple dot dot character heart symbol character character character compose filter review release format review book review user median review book book review book token token per token number review much large review believe review book book rate book review positive review book datum include review color red represent review example review sentiment rating sentiment rating ambiguity review neutral review review book rating mean rating rating mean review rating book user review vice versa figure review review notice book user review negative review review book review per statistic review rough sentence count token sentence review token work dataset sentiment survey classifier sentiment classification add sentiment classification neutral moreover effectiveness number category unbalanced class category datum mini compare three work neutral review neutral positive rating map neutral neutral important reader review neutral set review category size class unbalanced equal proportion collect data review balance unbalanced count unbalanced notice unbalanced setting exceed pose challenge try feature c c balance unbalanced neutral review part test set balance unbalanced show feature explore two review neutral rating two wide balanced unbalanced gram range gram range gram contiguous degree show number review unbalanced table gram range bi gram range token tf token document gram tf normalize exist remaining define document frequency word frequency word use area sentiment benchmark library use default classifier nlp bag bayes binary term occurrence bayes describe linear select maximize margin online hinge order positive margin alternative improve cope advantage use machine optimize multiclass versus cost cost pattern feed forward linear neural account simple distance majority neighbor c precision recall neutral neutral unbalanced recall svm unbalanced tf feature show training number evaluation perform task sentiment inclusion hard confusion neutral class label write human get mark weighted accuracy class review positive review class q q positive negative c passive logistic knn indicate tf weighting bayes naive gradient knn near evaluation perform weighted accuracy c c passive perceptron knn table task five set unbalanced contain much compare unbalanced reliable precision unbalanced test tf despite unbalanced well dataset unbalanced evaluation proportional good overall svm consistent passive perceptron automatically difficult compound training compound english c passive perceptron knn sentiment classification compare table indicate manually compound phrase permutation combination extract seed utilize useful svm inherently sort gram gram negligible end zero automatic ordering weight classifier select weight positive sentiment low sentiment remove gram gram n gram operator sentiment idea use table sentiment compound phrase effectiveness domain specific sentiment experiment goal stand alone previous negative stand several million lead
reason claim pool vast bagging stack generalization technique propose stack generalization context subject stack divide collect trial trial come train cross dataset classifier train portion learn second level classifier create ensure diversity success approach test combination pattern observe subject specific combine stack way dataset covariate shift weight logistic regression great plain stack generalization pool sg inferential purpose within basic simple shift trial trial draw analogy decode across stack stack sg simply sg cs extract difference baseline sg reach sg show decode subject decode single subject reach motivate brain decode experiment predict group train trial unseen extreme difficulty across address across subject formally show belong sub account datum ensemble stack generalization variability across aim vs compare across subject propose consistently predict state stimulus brain record category stimulus denote decode build predict activity evidence information light neural decode subject frequently accuracie discussion ideally trial meaningful group difficult structural subject inherent environmental variability trial generative practical common classifier subject provide empirical propose solution problem across subject purpose early create across extent subject divide three main deal training feature space paradigm present brain computer interface eeg data device stimulus subject stimulus subject completely image fmri example multi identical task test propose formal definition decode subject instance transfer assume differ aspect transfer acquire label contribution solution enhance dataset motivate ensemble learn decode efficacy stack generalization covariate article decode across propose learn covariate shift stack experimental section across transfer necessary stack briefly standard basic application across subject category subject channel record moreover binary stimulus marginal record predictive domain task face task house trial record target record definition assume task example face face house transfer aim help target transfer difference domain decode available subject domain case decode face available set transfer call accord transfer learning aim train identical availability divide category share probability differ category name solution brief review record target convenient happen also convenient importance sampling minimization dataset test penalize
b stock variance helpful stock detail quantile stock stock predict particularly stock return quantile general significantly zero large quantile relationship return additionally lags researcher capacity entire financial consequence approach tail claim cross tail financial prominent include quantile daily market index return financial put gs belong investigate cross stock index show bootstrap confidence quantile replicate lag gs reach market trend risk take reach market exposure stress test reach peak market gs peak meanwhile market reach peak influence impact way figure either exposure wide manner change impulse function require partial economic index economic state variable highly persistent integrate quantile quantile generally remain individual economic cross interest management table size rejection frequency box test second lag box column rejection tuning rejection frequency box statistic rejection column lag critical value rejection self table material include theorem figure axiom theorem conclusion theorem example exercise notation propose measure apply directional limiting nuisance employ consistency bootstrap normalize use detect stock excess use stock return provide predictor stock return supplementary material quantile stationary hypothesis set series prediction whether unconditional quantile compare pointwise band literature statistic paper et several advantage directional conceptually appeal simple base heavy tail consideration allow long lag apply quantile approach stock return exchange rate issue limit nan limiting allow nan absence even structure look useful interested measure degree across quantile strong limit long run variance quantile conduct propose bootstrap valid investigate carry efficiently normalize statistic whose bootstrap methodology explicitly mention version result fact cross cross autocorrelation lag stock stock study apply cross risk paragraph derive let series density quantile iv consider serial dependence arbitrary quantile quantile serial dependency quantile single time become process moment invariant monotonic construct sample analogue unconditional quantile consider deviation quantile directional directional dimensional entry possess usual interested testing absence directional x k location normal accommodate lag test lag use confidence interval special sup p small improvement cross contain use among mix coefficient satisfy function kx derivative interest rate chapter ensure uniquely quantile ensure differentiable describe weak nuisance estimate may slow convergence address bootstrap block strictly sequence geometric scalar positive denote growth original I pair observation procedure take use estimate solve asymptotically negligible finite lag confidence interval maintain use length vector cross procedure sample nan eq repeat b bb b jointly directional percentile following provide statistic fix directional alternative vector directional range quantile lags test quantile follow alternative normalize necessarily idea lee call chen improve divide normalization self asymptotic framework et al construct replace population quantile lead analogue eq element follow asymptotic nuisance nuisance employ bootstrap normalization technique bootstrap x self subsample recursively normalize impose follow density control strictly strong mixing assume assumption hold v continuously differentiable weak z z z v partial k z suppose finite partial alternative argument theorem self power performance process identity kx commonly model volatility economic literature reference therein median quantile finite bootstrap save case table rejection box statistic critical replication bootstrap critical replicate adapt white later size property rejection frequency nominal case median median close table however rejection examine performance self normalize setup repetition report nominal quantile little size quantile show moderate size nominal size power period self size lag low bootstrap apply directional economic variable stock extensively consider stock return return forecast economic predict stock return relationship whether economic quantile stock represent tail return ols return certain quantile return specifically stock return conditional stock return information et al inferences quantile regression regressor unit analyze show predictor stock return point lag application lag stock return predictor however predictor highly persistent price autoregressive root motivate work establish bootstrap strictly persistent leave future stock predictor autoregressive coefficient variance unit stock daily
steady theorem da da c approximately virtue conditional eq also update sample become multivariate distribution limit equivalent asymptotic gaussian learning problem problem accuracy adaptation concentration estimation class choose clustering fully approach show cluster apply five lead alternative computational adaptively asymptotic severe overfitte tend ht communication message receiver bit transform symbol receiver perform decode probability error quadrature amplitude alphabet db measure channel quality successful receiver know recognition detect data point choose corrupt snr db cluster show fig detect lr imply new characterize db use reach decoder grow dirichlet gaussians motivate propose complexity drive concentration parameter number digital communication observation assume multivariate precision parameter wishart joint definite cone conjugacy us class posterior conditional class expression probabilistic conditional obtain posterior hyperparameter recursively would greatly simplify rule multivariate obtain update inside complete square integrate obtain recognize wishart th become eq ease wishart interpret update equivalently integral within expression inner determinant I h sufficient eq separate term euler divide limit q result term bound divide take limit lem law p base case trivial give q particular hold desire corollary remark proposition assumption sequential low mixture easily computable streaming assume asymptotic dirichlet grow rate limit digital communication show optimality bit error datum dimensional process make however variational optimization approach effort fast require pass adapt sample arrive author class label algorithm greedy selection fast impose heavily model incorporate account discretization initial analytically stability adapt adaptive adapt asymptotic call basic idea greedy novel parameter adaptively greatly logarithmic asymptotically cluster asymptotically behave detect digital communication alone error rate number organize review sequential upon sequential growth class adaptively experimental nonparametric component denote th latent search summarize completeness distribution eq observation consider calculation iteration within assign count parameter growth number experiment sequential even critical fully specify conjugacy recursively compute hyperparameter I inverse wishart interpretation positive definite likelihood iterate detailed remark allow concentration parameter show number class grow step follow update innovation toward thus innovation manner assess limit see appendix theorem suitably gamma form class alg asymptotic concentration need discretization tracking extremely innovation previous innovation n k mixture model choose innovation use kn initial choice sec q conceptually currently mode reasonable good model
precisely one type forest maximal vc cube latter forest complement complete cube possibility argument follow vc class binary maximum vc remove vertex binary cube vc let case vc cube claim vc class necessary increase vc set anchor six edge type sharing complementary vc coordinate cube close section associate example vc class yield scheme function vc boolean form sum maximum dimension function value coordinate cube symmetric associate mapping binary symmetric vector notation discussion coordinate class match value prove novel argument basis dimension choose function class hence coordinate class distinguish vc exactly next collection complement contain novel boolean exist vc ordering order complete anchor single coordinate equal anchor anchor coordinate put give iterate number iterate reduction contain coordinate anchor cube iterate form leave possibility coordinate overlap pair face happen coordinate order coordinate iterate reduction coordinate obtain leave coordinate face meet cube maximum dimension collection boolean maximum start cube form cube order sum distinct vc cardinality sum generating vc projection maximum vc cube hence consist element cube cube collection sum clearly element cube onto maximum claim chapter main vc vc dimension therefore collection vc compression scheme embed class satisfy conjecture attention vc place vc exhibit embed vc maximum develop generalise bounding class bounding face also believe bad offer reduction union develop three may boolean vc university electrical sciences berkeley usa theory compression class equivalently date statement possess cardinality vc positively class super vc dimension embedding compression scheme complex maximum vc class vc class vc possible embed maximum investigate recursive procedure vc vc class binary class recursive embed vc lemma discover system vc diverse computational empirical road automatic verification former bound cube meet equality cardinality vc view collection unique coordinate form tree important increase complete vc study class conjecture compression immediately conjecture determine converse finite sized compression scheme beyond provide deep notion vc maximum class conjecture bound practice date towards later david existence maximum follow recursive dimension class coincide recently form compression scheme scheme sufficient expand maximum conjecture technique embed vc dimension relate count vc low complement dimensional edge face show uniquely meet first present consider incidence dimension provide closeness vc maximal maximal class cardinality class compression scheme come establish vc vc project cube cube class secondly cube application vc class produce collection vc vc embed class vc improves embed compression compression via embedding recursive vc class resolve must demonstrate possible compression cube class class binary cube classify boolean vc classes cube sect contain complement vc sect develop new class sect sect vc demonstrate maximal vc sect sect cube sect conclude chapter consider cube call terminology derive evaluation interest concept classifier class concept point support outline number family concept vc exhibit combinatorial vc concept word vc number form binary vc extensively process equality concept without increase vc call trivially maximal definition maximal class canonical convenient type nc ci ic ic vc cube iff iff complete complete vc maximal iff properly contain union maximally equivalently iff maximally convenient complementary concept projection cube show maximum vc vc vc proceeding invert reconstruct cube place cube splitting along produce maximum obtain series predict concept admit compression mapping k unlabeled sample vc little rich class unlabele compression class embed positively conjecture without vc maximal maximum vc focus argument prove maximum section integer bound meet equality prove class cube maximum word string cube must count partition layer contain vertex layer bottom vertex zero correspond bound graph use prove lemma connect vertex edge consider edge orient norm class tree necessity converse euler characteristic euler characteristic number forest iterate euler iterate since way choose euler define iterate cube coordinate vertex conclude iterate consequently rewrite maximum vc cube theorem conclude iterate tree structure iterate follow minor iterate color color color anchor differ class collection iterate integrate geometry bipartite cube edge cube cube whenever former subgraph contain iterated direction embed immediately preliminary interest projection maximum first complementary complementary vc procedure find vc class embedding class vc class contain complete argument correct vc cube composition projection induction vc vc complete complementary maximum cube complement cube contain vc concept dimension strictly correspond cube map onto direction reduction cube claim claim relate binomial vc binary unless vc follow direction vc iterate iterate come reduction class long iterate iterate long complete since union multiplying cover contain direction direction maximum assume binary cube repeat application reduce hence projection imply vc example vc moreover exhibit contain class negative show dimension vc vc cube pair cube contain vc origin cube proceed coordinate respectively roughly complete string zero one majority coordinate immediate vc less vc complete collection cube anchor exactly anchor element value anchor consist coordinate majority one give contradiction conclude contain vc dimension tb abuse pair intersect dimension zero vertex one form zero class maximum vc take original maximum class well contain small vc cube deduce iterate meet moreover consequently structure iterate cube anchor anchor anchor majority see require majority anchor contradiction coordinate entry clearly belong must show
describe processing describe calibration result discuss advantage calibration conclusion future bayesian parametric binary generalize histogram calibration possible propose challenge score programming classifier th index induce calibration model specify motivated variable discretization calibration marginalization close assumption class equal uniform closed solution total locate bin class instance bin interpret define prior partition boundary bin contain boundary boundaries eq training equation bayesian mention call average calibrate total number instance prediction exponential tractable apply dynamic describe section dynamic discretization classifier output define subsequence high model correspond respective score bin compute bin score composite decomposition give decomposable choose composite subset repeat process derive well possible complexity programming procedure programming programming particular method use assume one correspond denote score optimal model cache analogous backward highest low correspond respective cache property equation use bin prediction remarkably since specific equally space map store calibrate retrieve section evaluate run experiment logistic lr whose calibrate make tailor lr outcome linearly separable test well instance show three real problem predict whether person dataset real categorical feature remove instance instance calibration model test uci well calibration application diagnosis single emission patient classify instance equal positive test instance instance calibration dataset lr na I allow comparison tailor na I classifier achieve discrimination usually well classifier frequently real contain finding g sign laboratory outcome community acquire examine patient predict patient outcome medical total patient divide classifier calibration testing I bayes discrete transformation dimensionality exist unstable previous performance calibration discrimination due lack linearly separable method excellent measure acc area roc discrimination calibration error statistic diagram partition ten fall expect calibration bin fraction bin mean post bin empirical fraction instance bin low calibration model comparison evaluation show bold see superior superior reason section perform real real generally retain acc calibrate important calibration sigmoid recall function restrictive produce calibrate fashion near separate plane limitation include bin calibration another calibrate calibrate one choose monotonicity increase restrictive boundary limitation however limitation monotonicity see violate relatively poorly discrimination capability secondary classifier recently introduce application tie ci bin dataset calibration performance histogram post achieve among discrimination accord limitation first single datum calibrate bin select around train prediction table base simulation appear promise outperform make restrict unlike disadvantage algorithm search histogram version use remain algorithm perform thousand show complexity method train respectively method bin binary calibration complexity calibration nonetheless efficient training thousand particularly calibrate decision analysis plan explore average extend finally calibration margin auc acc rmse lr auc acc bayes nb acc auc acc auc
want trace synthesis program would return would consequence language language synthesis engine start initial guess next guess return stop guess element synthesis guarantee terminate iteration language monotonic fact correctly thus identify program put restriction bound example far increase synthesis result synthesis program similar argument show inductive synthesis dominate formal program restriction type countable set minimal terminate terminate decrease synthesis program correct program program program synthesis dominate theoretical inductive synthesis automate synthesis speed identify class technique synthesis similarly two variant history whether variant enable synthesis beyond minimal interesting kind synthesis technique first towards understand synthesis inductive synthesis technique guarantee terminate program perform analysis inductive technique investigate whether space correct synthesis mistake synthesis power investigate whether use program inductive kind history inductive relative synthesis technique synthesis power technique history bound dominate science find optimize critical loop purpose program specification give specification advantage specification specification automated verification proposal specification automate synthesis iterative inductive synthesis technique kind validate program produce intermediate subsequently inductive iteration refer synthesis synthesis conduct examine impact nature successfully use synthesis controller set synthesis raise consider predefine example engine synthesis minimal inductive synthesis return significant produce aid conceptually localization synthesis minimal specifically second produce engine see example define technique history synthesis validation engine program localization accurate notion force produce engine previously whether increase candidate space program terminate terminate correct successfully terminate increase decrease power program program successfully power none none strictly good mistake bound enable program synthesis synthesis input integer output specific program space denote integer discover synthesis engine consider radial ordering order synthesis counter start initial always produce discover boundary one terminate arbitrary rectangle still paper question even termination infinite question synthesis synthesis widely study extensively receive limited good knowledge theoretical nature inductive inductive generalization previously learn string formal task formal language iterative inductive string string language procedure propose formal learn language formal language algorithmic classify include learner memory grow infinitely bound communication learner arbitrarily response query membership algorithmic paper gold inductive elsewhere infinite learner set theoretical inductive gold language identifiable limit identify language use stream learn example call learnable language negative example term learnable learnable learnable language none language identifiable text include regular language context language also class infinite learn example simple example vocabulary string form vocabulary string language guess index see use identify correct example language vocabulary language fail fact positive language currently merely presence negative learning begin guess next guess language survey classical example input noise string target language might detail survey present inductive generalization synthesis engine step synthesis design response store stream inductive generalization arbitrary intermediate synthesis engine algorithmic restrict learner synthesis engine rely availability explicitly memory intermediate respect differ learner synthesis engine engine teacher teacher analogous synthesis engine technique use kind query verification contrast restrict verification query investigate produce bound verification meaningful powerful verification provide simple help design trace differ trace source enable synthesis analysis verification use aid section preliminary notation natural minimal natural range argument assume tuple tuple computable recursive language total complement language convenience mapping language distinguish different program language number index element trace order language string operate numerical tuple order order element language denote minimal order language say language correspond brevity intuitively define component encoding program inductive synthesis consist synthesis identify correct language index language overall synthesis follow let candidate program correspond language target synthesis engine set synthesis correspond candidate program inductive produce iterative produce intermediate conjecture program develop useful section section definition target formal trace language denote synthesis technique employ formally formal intuitively return language way presentation non otherwise engine language predefine represent guess correspond intuitively along form language converge finitely p il language synthesis inductive arbitrary engine n language represent synthesis language say p kt il next synthesis history arbitrary generating case generate synthesis engine generate order element language element empty mapping l engine engine trace could synthesis language similar follow second verification engine inductive vary rest investigate prove replace verification engine arbitrary verification engine minimal inductive synthesis system non intuitive synthesis minimal summarize theorem synthesis arbitrary trivially intuitively simulate converge simulate two phase language return phase need trace cache language formal maintained store simulate iteratively multiple micro simulate micro storage component minimal map map language simulate minimal program program map know map mapping intermediate program know intermediate counter record part already variable initialize map know make one synthesis candidate program need find minimal
case negligible case tv considerably always interaction contribute way prominent initially optimize confirm find interaction separately interaction dependent tv tv novel novel interaction interaction good several outperform traditional model model specifically ht far early dimensional improve factor entity figure recall interaction model clearly pairwise slightly within traditional way model lag member rapidly model improve fast outperform importantly practical limited stable context independent model practically exclude pairwise pairwise recommendation perspective interact influence user inactive status affect consumption pattern argue context similar focus recommend direct interaction argue independent recommendation hard slow aspect dimension fairly kullback know state execute totally context different band little dimension explain perform pairwise ht cp sc c sc epoch cpu easily time practice scale linearly feature practically useful depend operation preference complexity accordance pairwise model method qualitative pointing factorization although advantage outperform flexibility regard also quantitative comparison algorithm machine fm machines fm factorization rating prediction explicit subsampling rate rate preference pair dimension determine two corresponding build sa basic therefore composite drawback training lot unnecessary partitioning basically result exclude certain pairwise learn sgd monte carlo mcmc four fm key fm fm pairwise dimension preference fm feedback computation implicit feedback either fm build sa extensive et fm basically either user item sum vector aggregate feature word pairwise keep drastically fm interaction drop experiment leave accurate incorporate explicit implicit use loss model model although pairwise rank strategy differ basic context comparison pairwise respectively appropriate preference special case factorization aware problem way much importance allow novel without quantitative factorization machine context aware factorization implicit require item occur fm fm factor converge fairly optimize fm optimize h way traditional way interaction include outperform fm case fm include context advantage measure similar test exclude depict twice fast interaction subsample feedback train twice time core cpu time multiple core lift impose sa attribute fully multidimensional attribute entity g tag include dimension tag apply entity transaction accordance attribute analogously interaction property irrelevant variable represent strength attribute assign compute entity transaction feature entity multiplication w property property feature vector property update fast assign high phase normal compute remain phase therefore stick direct optimization combine outperform g show include extend outline create attribute indicate item item token item create feature token compute treat recommender user preference change great help currently refine recommendation broad interest news transaction visit actual transaction exclude actual item context target outline entity thus dimension consist transaction associate attribute actual omit attribute attribute item occurrence item item event assign feature note different matrix incorporate gap less minute stre filter rare token normalize entity experiment run usage justify simplify compare classic interaction basically actual actual replace cf aspect item user interaction item improvement summarize need predict suggest information improve perform place basic start incorporate recommendation importantly preference implement separately optimize implicit feedback feedback datum demonstrate usefulness aware certain preference user well traditional composite refined well never model interaction recommendation novel model generally able multiple incorporation additional recommendation several path work well ignore great connect nonetheless characterization context usefulness help easy current maintain scale another could meta learner context acknowledgement receive european union fp eps fill rgb com economic aware recommendation algorithm focus recommendation topic gain lot feedback optimization context lack tool allow dimension space preference importance largely propose factorization dimension easily experiment model aware recommendation scale life circumstance framework explore preference aware real dataset implicit dataset increase model outperform one novel outperform art factorization multidimensional incorporation framework information great capability propose factorization framework feedback recommendation flexible dimension allow point important recommender system filtering tool relevant content factor gain popularity user preference practical content item explicitly user retrieve instance use type implicit feedback unary preference feedback explicit familiar negative feedback inaccurate item click direct preference miss interaction typically user aware item unary infer negative preference consider feedback spend distinguished feedback highlight importance recommendation aware system refine extend additional may user recommendation briefly latent factor work feedback strongly applicability factorization minimize preference usually necessarily rating iteratively optimize preference sgd optimizer dot optimize rmse user item dot user item state optimize monte e dot product every e context optimization strategy however preference explore method interaction former preference dot interact entity dot interact entity item proper argue proper due flexible experiment create factorization preference feature matrix allow recommendation task new research preference property work aware recommendation restriction preference restriction applicability real world besides implicit address weighting weighting enable dependent weighting miss hypothesis scalability interaction number make life recommender follow build basic introduce usefulness context aware clearly art incorporation like item etc potential briefly review representation aware rating straightforward attribute similar relational database usually atomic nominal attribute discretize attribute transaction combination attribute dimension item dimension location item location classical recommendation scenario individual present distinguish item property omit item item attribute attribute value attribute contain simplify major factorization fm attribute dimension limit dimension refer single sa ignore grouping attribute conceptual attribute dimension information grouping assume extra interaction interaction complexity simplify limit prominent restriction mean attribute svd require single user additional attribute binary item dimension attribute binary entity descriptor descriptor token rate basic sa extend multiple inspire latent cf approach preference estimation framework rely sa main user dedicate dedicate contain help preference context location interaction device preference etc epoch sized rank sized km ei yet bias add weight model implement cg small dominate transaction run operation compute member transaction co equation need change ds show factorization mf feedback predict user item aware factorization dimension predict preference tm tm tm tm w tm classic mf explicit weighting preference item rate include svd however recommend increase training demonstrate usefulness novel without feedback set evaluate tv user pair set period least depend mml mml month tv week music day day focus recommendation configuration rank predict primary recommend recall live recall well important metric recommendation proxy recommendation accuracy offline world available rank base metric map ndcg test comparison query metric optimize measure test epoch case epoch time epoch feature practice effect context recommendation find generally aware consider sa dimension context item transaction item improve implicit context state context transaction consist transaction tuple contain recommender exhibit first expect repetition aggregate offset aggregated need bin possible band length event band week week people differ hour beyond scope music item item complementary sequential type introduce use item information pattern sequential consumption sequential transaction set information test context test event would result preference accurately dimension put dimension usually interaction mf item tm interaction u dimension preference remove contain potential
blind extraction propose employ predictor blind source blind source generalise mixture blind separation bss extensively past base order class canonical cca matrix signal sum autocorrelation signal maximization autocorrelation generalise free mixture free mixture always traditional cca noise noisy successful accordingly cca blind source structure effectiveness generalise cca bss sec simulation show section instantaneous mix bss source employ spatially source delay solve cca vector maximize e proceed maximize correlation combination shown apply cca cca become bss simplify q multiply side follow generalise eigenvector cca recover proof denominator cca mixture white uncorrelated source signal give identity cca problem add white white e variable normalise noise correlate correlate component depend estimate generalised bss extract simplified next brief maximization source presence ss ss normalised eigenvalue draw maximize successful extraction normalise extract remove next procedure proof matrix different lag robustness denominator maximize function eq reality likely positively recursively technique online update updating case normalize yield predictor extract closely approach predictor implementation source show extract instantaneous predictor function give respect successfully numerator denominator propose noise predictor similarly note follow predictor blind ahead correlation source lag signal delay lag reality property meet requirement cost indirect assume ss matrix diagonal numerator autocorrelation clearly provide early draw successful extraction minimum normalise signal normalise estimate preliminary
cc demand history add file cost unit consumption cc optimize cache traffic take account associate place file cache request file cache instantaneous file denote cache content period choose cache initially instantaneous file store cache instantaneous q expectation file associate storing file cache total amount traffic minus cache focus find cache horizon factor bandwidth depend popularity solve initial cache content follow period horizon switch ignore expect immediate reward cache np branching exponential bad solution relaxation cache add file sequentially start file popularity cache full file cache discard partially file cache existence popularity output main instantaneous reward file obtain information cc want file discover storage capacity file static cache mab feasible cache feasible combination cache content cache instantaneous file cache instantaneous demand iid reward know popularity divide part due know due policy period reward optimal file arm reward probability find reward least iid reward policy period notice cache empty incur sum switch linearly mab play depend arm ensure sample reward arm positive less period regret bound time play switch arm play computation cost period period switching switching period period play switch th arm arm play period play h initialize cache file bt f f bf ft additive square grow decrease play reward file file popularity profile algorithm arm switch notice algorithm avoid popularity bound definition good opt opt opt say occur counter update bad draw bad switching n behind apart period bad period increase guarantee high reward include play zero cost switch two arm combination l l switch count switch eq growth rate study additional depend switch regret bound function number switch bad combination rapidly grow slowly sub linearly logarithmic imply switching imply switch cost period turn imply regret tradeoff switching grow regret cost fm complex cost logarithmic order knowledge prove bound switch g notice every period every cost remove logarithmic uniformly arm period period arm extend consider algorithm period growth iteration cache content iteration demand file period cache efficiency horizon cache traffic popularity cache efficiency popularity profile low cache cache close cache demand cache due cache replacement cache cache cost file user request popularity skew cache efficiency reach notice due term follow trend albeit cache efficiency profile cache cache study cache cache behaviour slightly cache efficiency cache cache size cache cache algorithm due profile file popularity rapidly file cache file cache file cache size file cache ht instantaneous slowly depict algorithm negative cache efficiency switch high compare traffic small negative bind confirm content population area hand cache small request period replace file finally file impose cache always available popularity skew wide peak cache memory store file popular file cache cache efficiency contrary fact cache user request constant occur file replace ht content cache content pressure popularity unknown storing cache cache file cache model combinatorial mab switch trivial cache content file popularity profile well know account switching network bring benefit cache efficiency skewness cache chernoff inequality hoeffding bound support realization ignore clearly consider period bad period counter update play event solver output combination time period bound j l n j j b n b j j n j arm play period update period fact counter play arm update monotonically period due f j solver fail period theorem n notice mean bad period l j bad period f prove plug opt opt cost relate switching switching switch constant switching bound r I j n f n u f f f tm b l switch maximum switch switch bad arm play sum play fact zero obtain period consecutive play finally b j state theorem j respectively moreover monotonically I b j jj b since second increase increase apply line separately sum j prove bn b j rate property fm fm fm lt use use find plug power side rhs subtract four great prof complete prove induction induction bb b j nd uk uk traffic wireless terminal cache store cache controller store popular content cache traffic practice popularity profile file observe instantaneous store cache demand associate place cache cc gradually learn popularity profile cache capacity cache measure amount impact file cache skewness popularity show popularity profile quickly parameter represent portion internet traffic delay video streaming growth wireless continue fraction traffic locate traditional traffic become grow pressure bring content end user part content receive great bss form edge content news cache locate wireless bss bss reliably user without business edge mobile operator raise exchange traditional third party store g video rate netflix file popularity advance cache even scenario popularity locality bs require significant take relevant popularity profile advance popularity good strategy instantaneous cache file internet request user cache request overhead cache file file cache request request call cache cache content cache popularity storing cache instantaneous cache capacity good file cache traffic popularity profile advance observe request file multi cache management contribution paper address content place new cache formulate mab performance propose popularity cache file provide extensive impact content popularity cache file lack popularity compare file profile rest survey background section present popularity profile bound performance wireless introduction limited capacity wireless popularity consider huge content decide limited space cache form leaf cache node cache optimally place cache consumption study wireless transmission code transmission storing end device user several storage cache studied show user move across wireless slot cache code content content cache content popularity result place content cache providing mab maker action balance instantaneous consider slot machine time instant arm
switch tailed pathway subject classification collecting observation deterministic deterministic nature phenomenon deterministic thereby random go decide appropriate show cycle specific type cyclic monitoring year slow several local peak situation minus convert observation make part phenomena production consumption residual output production output reaction produce particle may situation certain span period medium yet produce situation create triangle proportional another adopt distribute simple situation type sum independently variable density identically elsewhere act scale dispersion scatter suppose repeat successive sufficiently apart graph like take simplicity location generate location close maxima spike production cyclic pattern arise scale location happen residual input strength contribution come negligible contribution q within spike divide combination laplace create point govern mixture b cm asymmetric laplace case laplace behave graph type eq sometimes sometimes tail due area develop author another gamma consider normalize since integral correspond density gamma gamma cm constant eq figure gamma cm simplify convenient specialized well constant simplify computation cm switch three form density current stay generalize change extend beta go gamma capable switching pathway pathway family pathway note situation tail cut close cut origin go go model beta gamma pathway tail cut tail gamma brownian boltzmann stable ideal physical neighborhood stable form cm statistic extensive mechanic also derive unconditional density see various variety situation start author mathematical reaction situation non tail etc integral aspect since integrable evaluate convolution convolution evaluate reaction correspond moment gamma statistical integral eq structure ratio transform transform integral pair integral author early pathway generalize integral pathway consider keep negative positive eq integral integral integral generalize beta form different could whole model know transform know also integral cm paper author recently integral fractional operator kind side fractional integral operator convolution pre function arbitrary type beta arbitrary right give derivative cm reaction result reaction diffusion differential solution situation come equation sense integer cm thank department sr centre mathematical cm nuclear rate reaction nuclear detect cm super generalize lead reaction science arise fractional science science
yield policy reward lie multiplicative round multiplicative distribution decision eq take absolute value substitute modification monte carlo update use value set round maker get dm accord distribution dm next weight sr observe epoch give follow repeat avoid uniformly sampling cast nature context slight modification bind impractical game play discuss link idea mdps procedure approximately calculate policy seem potential inherently amenable solution method suffer specify belief solve stochastic sum work problem interact computationally efficient manner definition markov tuple action horizon utility additive take mdp utility utility mdps distribute horizon write mdp mdps finally define depend maker policy bayes belief convexity utility obtain reveal view mdp draw expect nature select wish q policy policy belief vice game couple well fact set action set policy policy mixed set mdp e u deterministic history policy know exist policy piecewise achieve state outcome fair policy deterministic mapping marginal action observe h b optimal mixed game horizon distribution robust define generalise achievable maker distribution oracle probability equal statistic cumulative particular policy choice optimal mdp policy define explain minimax bayes optimal previous finding optimisation oracle good response zero asymptotically via even expert literature weight majority maker
vary bayesian monte carlo method computationally intensive difficulty lag selection var turn shrinkage impose restriction reliable tractable early take perspective use intercept depend context prior incorporate belief lag informative lag lag structure coefficient lag decay lag coefficient lag toward shrinkage incorporate consider lasso need exhaustive space explicitly encourage lag lasso penalty lag attack lag selection force correspond shrink toward lag coefficient lag enforce lag desirable fitting datum order lag corresponding allow flexible computationally study advantage forecast lag vector regression intercept noise uncorrelated series square fit minimize convenient express var compact procedure minimize denote frobenius norm appear model challenging unless sufficiently space tractable small author use lasso building assumption performance even large structure arise dynamic describe lag structure notational convention lag define define maximal lag particular numerous lag simple pair easily lag structure equation maximal imply hierarchical lag lag structure add series lag informative lag lag hierarchical lag long self hierarchical illustrate finally completely c propose aim shrink lag introduce tailor group euclidean norm encourage nest hierarchical sparsity set multiple interaction estimation transfer estimation covariate decay lag aim lag increase group towards lower identical addition influence lag influence thought ensure structure hierarchical lag occur flexible strength across expect example expect objective unify solve appear detail differ simplification observe row kp solve via proximal view method nonsmooth f differentiable proximal gradient cf operator take nested euclidean operator l backtrack iterative fista minimal implementation three middle period ahead forecast forecast method observe three lag scenario row row use magnitude depict figure show scenario r lasso lag lasso walk perform weight lasso similarly total coefficient greatly lasso exploit exception orientation toward modeling behavior suffer lag estimator create manner lag lag I magnitude decrease simulate view autoregressive next row lag lag magnitude scenario lead method forecast slightly bad perform lasso r var walk lag scenario allow sparsity pattern magnitude var lag row method interpretability matrix matrix lag order define optimal prediction procedure modification intend tendency select regression standard favor parsimonious approximately sum table report relative mean good benchmark fact lag good performance follow var lag scenario incorrectly constant lag selection estimation series indicator information include stock price exchange full list variable nest basic dynamic stochastic equilibrium modeling plus consumption exchange medium additional aggregate variable plus consist primarily component production etc initially focus forecast code approximately comparison convention one ahead forecast summarize least square random ahead indicator medium lasso lag become realistic economic application core include forecasting table perform period forecast hold application component lag likely lag vary lag economic activity lag series period lag lag economic several economic activity product taylor hence aid forecast rational rate year serve see growth causal price economic analysis price appear exhibit degree three row series component important activity taylor suggest aid david university fundamental component increase quickly tool number incorporate structure var traditional attempt low lag among component short assumption universal order information lag relationship base approach notion propose selection regularizer lasso autoregressive inherent focus three computationally component forecast lag highlight improvement var seminal var widely number parameterization intractable system var infeasible except lag order model scalar component dynamic impose regularizer adapt specifically inherent lag
sum probability substitute inequality q hoeffding sum validation union select drawing example near hence empirical using theorem range side draw partitioning subset nearest near outline statement I drawing q subset imply treat complete proof slowly expand exponent convert exponent q research bound practice allow directly stronger open solve leave estimate bound use positive term odd versa range much less equation indicate requirement contribute neighbor nearest neighbor interesting paper node sometimes node yet add collective adapt local draw neighborhood neighbor relationship challenge classification local node setting refer show sum partition return input return generate sample example bound cube center origin label odd odd depend nn classifier example expect partition near integer depth error average test deviation estimate difference plot statistically figure figure curve small bound validation count sample decrease beyond show minimum exponentially neighbor exponentially increase practice replace length bernstein hoeffding reduce range remove coefficient tradeoff truncate term tend require neighbor near condition separate union alternatively combination subset rewrite q use probability define q sized validation optimal value derivative partial optimal sum simplify symmetry straightforward close well approximation small binomial expansion q bind valid convert corollary thm near error integer good decision correct error develop label develop perform goal evaluate want generalization want classifier focus nearest classifier input determine close input label near possible use validate probably pac bound actual rate effective pac small failure pac bound vc likely assignment give overview comparison type bound sometimes condition hence distribution neighbor average show converge twice affect neighbor see book classifier difference classifier half bound classifier remain call sample disagreement hold classifier plus disagreement classifier bad disagreement draw replacement sum disagreement classifier one near randomly disagreement minimize disagreement use set together rate classifier use bind full cause range must disagreement cause near occur select produce bind range bound disagreement combination combination grow sense size classifier validate come bound discuss develop inclusion let draw input draw output draw example bind base average draw condition let validation union validation example index otherwise call quantity wish however expectation example subscript error validation call rate inclusion sample close figure illustrate vx sx sc validate subset r f sx sx ss validation set index classifier set agree classifier illustrate rate decompose term condition near r g sx index validation illustrate sign label apply apply definition g sx set area pr pr pr system diagram result theorem later sample error near randomly rt minus sign upper bind rest bind probability rate wish side range rhs tends make close selecting
gaussian proportion ensure identifiability many within mixture mixture skew partition density concentration mixture distribution lin lee mixture cluster skewed mixture parameter focus mixture lin distribution suggest concentration skew extent concentrated affect tail mind contaminate skewed specifically mixture contaminate skew herein variate vector parameter definite skewness k modify third index covariance clear write mixture contaminate g bad contaminate membership otherwise introduce indicate bad bad mixture contaminate distribution carry expectation maximization variant incomplete iterated likelihood compute maximization expect maximize extensive algorithm likelihood contaminate eq extensively literature work log likelihood detail appendix employ skew skew normal distribution definite scale skewness pp generate relationship normal write furthermore show frequently eq classification compare adjust rand index rand index chance agreement perfect agreement partition account random ari rand perfect ari skew two appear ht component skew skew mixture fit contaminate fit contaminate skew normal mixture skew mixture fit good average ari ari belong body sn record chemical physical package available discuss seven specie bank second component pc principal implement ht mixture contaminate distribution contaminate skew distribution give ari ari contaminate correctly contaminate bank contaminate mixture contaminate contaminate mixture artificial contaminated skew mixture mixture contaminate skew previously base contaminate contaminate skew extension mixture skew mixture detail give method good performance skewed concentration introduce early award science grateful provide normal herein corollary contaminated base asymmetric well spurious contaminate shift contaminated controlling proportion spurious outlier contamination specify outline
incorporate reader setting comment consider ar ease variable four distribution reflect independent modify curve tail approach word reflect information maintain modify uniform gamma prior I application research likelihood close statistic force summary sufficient marginal beta univariate statistic summary statistic influence implication simulation portion dataset table generate extremely acceptance second mse repeat simulation offer observe correlation abc mse denote distance preliminary simulation variability variety adjustment consider statistic small capture dependency z eight describe size tolerance together dataset run mse interested observe behavior tolerance prior set dataset mse result know simulation present abc attempt broadly sometimes result rate heavily tune htb setting third add scatter parameter bias effort increase prior implement dependent variability clearly bias present reduction variability improvement mse similar conclusion table decrease expense increase mse require similar result mse furthermore require substantially overall summary quality estimation improve introduce bias reduce variability observe introduce additional individual correlate e p e e b write context simulate bivariate must cell numerous define absolute mh tolerance abc ar accept mh walk proposal deviation respectively yield initialize auxiliary accept b gamma prior guide equation simply near desire carlo estimate choice allow exploration effort estimate yield albeit cost ar require seem suffice iteration result see computationally demand near surprising positive ar ar mh result difference observe ccccc observe cell count accept abc ar compare apparent see decrease stem ccccc total correlation end partial table abc bivariate prior subject correlation specific monte run ar mh summarize though variability original perhaps would suffice carlo lower accept correlation ht ar ar mh introduce provide grateful two anonymous valuable improve manuscript second partially c bias mse mse c mse c c c bias mse iteration bias bias mse mse mse mse bias mse c mse mse mse c bias mse bias mse mse mse mse mse bias department california edu several beta bivariate allow latter accommodate however come expense intractable research carry case prior tolerance real serve binomial comparison bayesian bivariate reject bivariate become use variable incomplete use reader extensive bivariate beta along bivariate continuous bivariate beta contain bivariate marginal beta limit flexible negative correlation flexibility simulate closed form maximum mle refer modified maximum approach approach distribute estimating equation final obtain via expectation unstable zero propose result difficulty free approximate first describe element abc algorithms generate candidate base generate auxiliary close candidate plausible accept parameter accept approximate posterior notion close decrease mse presence introduce select know exist finally make application bivariate beta beta binomial beta serve proportion serve abc hasting base follow beta selection summary study finding define beta easy marginal beta parameter parameter construct beta pair construction variate density parameter technique promise combine marginal suppose n respectively set equal theoretical solution negative yield negative equation parameter influence discuss bivariate observation ar sufficient table summary set distribution heavy c evaluate due impossible calculate condition markov knowledge nonetheless likelihood spirit class algorithm perform sampling possible fundamental
region visit statistical extract information visit duration profile recommend information contextual however recommendation learn exp describe document high estimate observe reward open refer make randomly first iteration select greedy call decrease strategy document estimate select except select adopt ucb price reward distribute reward mean add additional confidence interval number document high reward encourage computationally contextual bandit recommender context reward document contextual document maximize click reward document author bandit combine greedy dynamically value set uniformly initialize update click begin technique describe model bandit situation critical situation situation exploration consider situation exp risk aware yet recommender define reward total reward uncertainty first parametric instance process mdps propose sensitivity term inherent stochastic model state develop cost applicability exp propose greedy introduce situation contrast indicate criterion adjust study risk study none mention new handle semantic concept express risk associate situation help adaptation environment exploitation resp line ucb focus introduce situation external semantic enable specification human behaviour nc consider dimensional preference preference click document spend reading structure situation model contextual bandit include situation bandit algorithm proceed discrete trial situation compare metric concept similarity depend relate compute path node root observe trial recommendation one great document click algorithm observation situation document obtain reward depend similarity sim current situation preference sim base predefine compute document time recommend ucb select confidence uniformly choose correspond return element exploratory adaptation ucb alg level compute situation strict lead document multiply allow maximum exploration avoid document rs rs rd ucb rd ucb system user critical perform exploration decrease situation increase environment directly situation risk art risk variance approach similarity state describe use detection h aggregated concept approach semantic situation situation variance reward click aggregation click click recommendation normal eq clicks threshold constant accord gauss time click document recommendation situation concept risk weight associate arithmetic mean level associate concept system permit situation situation situation threshold use situation increase centroid risk situation aggregate eq risk recommendation feedback idea risk concept situation computed idea make give result company company time application situation contextual location social use spatial place paris finance paris entry user click situation illustrate entry analyse situation manually depend risk level interval h take age gender click depict heat situation h situation mostly office mainly home situation situation level level situation click suggest content management click considerably compute threshold opposite impact end situation overall performance obtain identify value situation similarity take different figure click display insufficient exploration consequently fail click lot click well collect step randomly algorithm select feedback simulation converge goal evaluating period time retrieval exploitation work ucb ucb ucb decrease exploration exploration axis axis parametrize test ucb starts reduce every iteration small regarding decrease ucb algorithm converge exploitation ucb static exploration interesting dynamic ucb algorithm average consider uncertainty r ucb ucb factor baseline r ucb improvement come exploration exploitation consider finally expect ucb ucb risk approach r take advantage cs give good critical cs different test algorithm different risk level describe comparison first r outperform exploration exploitation high risk ucb come exploration base size visualize comparison fig refer level notice decrease significantly ucb exploitation begin ucb parameter lie exploration beginning base line promise r ucb environment time participant hour split three week record recommendation week equip group system run ucb run ucb easily follow user rs user usage usage compare new recommendation impact recommendation comparison week week visit document document week respectively significantly week visit week introduction document would chance recommendation appear recommender list recommend first without find exclude recommendation week exclude recommend new week group discover exp visit document recommend document discover recommendation benefit discovery ucb ucb ucb look recommend spend figure spend group exploration trade impact spend use document exploration
alternate slowly employ direction derivation propose admm significantly formulate affect score remainder hinge find justify devise solve leverage align matrix leverage non show provide assume sum side leverage score indicate score count index find matrix less expect penalize find uniformly row leverage hinge function descent coordinate access exact score obviously completion problem help adapt solve score observation begin algorithm hinge size close theorem series row matrix coordinate desire order follow suppose desire leverage descent hinge precede property hinge hinge hinge reason hinge provable optimize get configuration detailed comparison optimize essence descent need provable score leverage rank truth rough svd leverage norm approximately score score observation superposition conduct leverage score perturbation completion motivate leverage sample additive perturbation ensure leverage score additive identify rank uniformly enough theorem ensure necessary provable row weight leverage leverage within consider leverage mn leverage give leverage row sampling ideal leverage score row theorem I gradually analyze except take accord tb c ki index ik practical optimize hinge know leverage directly non uniform sampling estimate score except score step pick index notice leverage additive index kind weight provable provable objective objective decrease condition turn increment score increment leverage ensure hinge nonempty iteration iteration nonempty corollary indicate greater well great provable hinge loss select r hinge weighting theorem theorem desirable number much practice fact get sequence figure condition time n challenge seek cost reduce svd basis replace n help rough svd use matrix way drop practice gradually leverage leverage leverage may appear additive row weight ideally attain even accurate leverage purpose heuristic score gaussian noise entry method tune attain figure noise intensity show coherent fail large number entry observe method strictly succeed free datum heavy rank matrix matrix know plus show coherent recover world superposition rank observation surveillance surveillance stack background foreground robust perhaps tool recovery define l recover coherence low rank coherent weighted matrix l matrix coherence low weight compute input input solution increase completion weight input compute finally weight ht sparse set entry algorithm input vary result figure clearly row make coherence recovery accuracy relative recovery way subsection error fix error matrix highly coherent fail weight accordance achieve high accuracy computational ht column weight adjust leverage recovery well model describe discrepancy score leverage score algorithm leverage score algorithm objective condition problem coherent matrix complete weight nuclear coherent apply uniform matrix quality leverage score clear accurately score non sampling leverage application non law leverage uniformly weighting power real equivalently slack augment minimize maximize contain otherwise term update update ascent ht hinge optimize row region configuration hinge nd row hinge coordinate descent st hinge decrease happen scale nd row leverage score nd decrease toy compare generate synthetic use three kind decrease loss lead fast decrease fast small hinge problematic local hold define hinge write th lead decrease loss score increase decrease contribute loss know r I hinge loss equality scale loss increase follow directly decrease hinge term hinge loss mc u simplified pt leverage score additive leverage omit leverage score value inverse great increase since theorem ensure objective directly empty r basis leverage score respectively basis exist let r c c u dr leverage zhang zhang problem uniformly underlie require incoherent model practical weighting leverage score become weight effectiveness recover matrix whereas unweighted method free completion extensive world collaborative completion nuclear minimization model variant solve portion seek recover incomplete entry cardinality great factor sample coherence requirement active coherent column world impossible access miss column impossible demand user pay coherence eliminate assume independently sum obviously restrictive use previous reverse adjust let diagonal diagonal instead complete compute th proportional dependence coherence eliminate score motivate none previous offer potentially sampling setting complete add e observe number unobserve complete coherent estimate provide leverage near interest derive admm weight nuclear minimization machine recovery component recover heavily noisy rest completion nuclear model solving complete perturbation apply weight practical empirically evaluate synthetic dataset apply
density would parameter choice class hide show represent hide visible bit hidden exponential unit compactly feedforward reveal intrinsic prevent think particular combination feedforward net combine recurrent temporal rbms study focus binary proposition distribution statement item statement strong statement appear disjoint contain form complement subset develop jacobian parametrization jacobian z pz jacobian dimension span kronecker ij kl l operation input rbm remove without span vector low parameter space partition region piece wise map thus piece represent function state geometrically namely visible preferred row multiply indicator classify positive classifier hamming ball disjoint hamming ball time plus remainder block minus dimension column number statement contain point ball radius hamming sphere choose hamming center hamming apart ensure contain maximal cardinality code large constructive target adjust obtain hide adjusted hidden jointly input dependent difficulty construction successively compose lemma proposition two finite probability build support disjoint vanish represent rbms term hadamard model precisely strictly word hadamard multiplying distribution rbm product strictly n sharing take strictly two hamming consist immediate hamming distance coordinate intuition vertex ham corner unit cube exist probability idea realize add rbm sharing step ball dirac delta positive conditional let share take joint conditional share distribution support hamming ball center contain sx x c qx word restriction product hamming proportional vector entry implication strictly joint conditional share conditional dirac delta vector consider dirac delta enumeration start dirac delta th step ty sharing whereby equal verify satisfy condition specific note row get trivial sharing measure trivial joint transformed share conditional share depend sharing construct algorithm generate accurate sharing intersection ball center initialize leave leave row corollary sharing strictly joint readily evaluate algorithm sharing step length star pack star intersect star call packing star packing input star length star packing show state star pack certain length size bad star packing sequence illustration star packing construct star packing procedure define pack site star far star sequence initialization split set star radius ball star dimensional set share iterate th terminate initialize create branch produce split generally branch star split branch total star time whereby number branch create precisely illustration packing sequence figure branch show dash green clarity star highlight star branch translate highlighted distribution well whenever give universal evaluate expression coefficient appear yield except universal c r evaluate universal explicit ir rr sr monotonically leave direct obtain yield ir evaluate evaluation million indeed remark proof strategy select adapt restriction unit restrict conditional want conditional care case pack ib xx xx rigorous machine family without dimension denote input finite conditional intuition conditional arbitrarily conditional exactly imply distribution family point point jacobian map zero universal latter family p universal conditional consist v fx exponential family contain k k product integer partition analogous correspond set collection share lemma proof binary number qx qx generalization present consider rbm visible unit unit observable rbm element x w first cardinality choose appropriate repeatedly conditional give first joint desire conditional approximate probability proposition analogous compute deterministic feedforward threshold network number deterministic arbitrarily feedforward x z z w yy x yy xt v bx statement precisely note give feedforward marginal regardless strictly build combinatorial deterministic approximate policy well entry wise mixture py py w union py eq decrease arbitrarily maximize proof directly function parallel hence fact feedforward hide unit linear bit lemma policy approximate arbitrarily fix composition linear threshold nz b policy arbitrarily input know policy thus bound mn n small share policy learn initial work observation mm expressive power machine boltzmann machines undirecte neural hide network parametrize interaction bias prove restrict support universal maximal contribute investigate restrict conditional restrict boltzmann universal kullback leibler dimension restrict rbms bipartite interaction visible infer distribute network connectivity rbm define boltzmann states network weight bias expressive attract study numerous treat particular universal address expressive exist theoretical work analysis rbms rbms bias influence input bias bias visible substantial distinction rbm power theoretical focus lie class possibly represent fix hide unit give rise desire distribution derivation incomplete generalization discuss reference list depend concrete follow map distinct small suffice universal tolerance select well unit extend non unit organize formal definition subsection dimension distribution universal deriving unit purpose analyze assume derive minimal unit suffice tolerance theorem subsection natural ability distribution way defer nonetheless fair detail entry set denote style transform circle inner sep cm minimum dot distance cm divergence conditional whenever hide unit within plain unit one next ask class conditional compactly distribution familiar distribution express term allow develop second part specific nonetheless contain certain conditional represent field main idea appear universal approximation field form n output arbitrarily field define rbm random rbm architecture represent
learn capture relaxed dictionary ensure note vector atom atom dominant eigenvector initialization process mp I accord r code manifold helpful dealing manifold practice high reproduce rkh value perform code explicitly work like solution text achieve let p trick principal matrix considerably row pick problem embed eq code tb initialization q q I ti l manifold problem rkhs dictionary update atom atom independent code r maximize orthogonality constraint form define maximize orthogonality give eigenvector eigenvalue want load kernel dictionary manifold initialization ni r r classification atom label use determine query dictionary learn discuss video model subspace video order simply demonstrate information follow sequence appearance image video frame represent like svd specifically take account image frame however capture extended image formalism vector covariance speak one appearance spatio feature vector feature index two subspace angle extend give finite video span orthonormal gram schmidt manifold column see discriminant hull discriminant discriminant intrinsic gender recognition scene maximize discrimination image image point affine characterize affine feature kernel discriminant maximize measure inter intra consider manifold local similarity dissimilarity sparse locality code extension respectively preliminary define determine classification query discrete dirac function measure recognize gender human gender constitute individual angle show recognition gender video capture result select individual table consistently big margin well burden sometimes tb lc dataset image sequence class perform subject primitive obtain medium perform setup experiment relax tangent scenario tangent space experiment euclidean perform poorly compare log euclidean tb code sc intrinsic code experiment sample normal tangent reflect scenario class tangent tangent le sc classification face texture gaussian code fed classifier intrinsic analytic extension conjunction discriminant hull recognition still extensively image popular choice image face video resolution create face extract region region describe histogram local linear order ten report dictionary percentage dictionary tb video dataset pattern action hand leave turn movement describe gradient descriptor set set split six tb example dynamic video move certain video human comprise contain training video train time accuracy use take dynamic video video length frame histogram descriptor compare two design spectrum learn dl see component volume capture view volume dl descriptor descriptor well discriminate texture overall obtain recognition tb dataset tb dl load code logarithm size multiplication thin thin require add reader computational efficiency experiment assume constrain expensive unconstrained tangent geometry randomly table run intrinsic core coding manifold manifold show code locality perform problem manifold dictionary atom use coding manifold linearity classification gender recognition scene analysis recognition texture classification show achieve notable discrimination art discriminant embed code learn minimize necessarily benefit propose reconstruction manifold interesting devise solution geometry induce acknowledgement department communication digital research centre discovery dp arc fellowship proof symmetric u v p x sufficient x n diag u u diag pa kb kb w matrix distance form analogous rotation nm date date hyper extra date nm hyper nm extra date open figure token school university representation notable various riemannian deal aim bridge coding manifold space enable extend manifold furthermore algorithm atom atom lastly linearity code dictionary embed hilbert task gender recognition face texture considerable improvement state art discriminant past decade term compressive sense suitable basis overcomplete nature decomposition notable visual recognition subspace develop theory code linear vision example art matching video spatio include filter adaptation tracking despite wide appealing property subspace lie riemannian conceptual learn represent accurately develop analyze image sparse signal topic like efficiently superposition subspace code video datum coding manifold study opt intrinsic sparse code riemannian intrinsic exploit due complexity logarithm manifold sparse code demanding term base logarithm later logarithm analytic manifold contribution end manifold preserve accomplished devise dictionary atom furthermore linearity coding manifold version dictionary embed space linearity apply computer vision recognition scene tb conceptual diagram conceptual work represent code manifold green red triangle query combination geometry curvature manifold manifold red triangle geometry color geometry provide technique manifold vision loose emphasize word capital letter bold letter one euclidean manifold grouping admit right orthogonal consist orthogonal furthermore thought element form detail element specify order column span write riemannian formally product tangent bundle metric shall concern geodesic manifold allow many definition manifold smooth geodesic short curve manifold embed may define consequently length path length curve give distance short equivalence geodesic small angle angle column principal subspace recursively word angle pair second subspace logarithm map switch tangent space logarithm close manifold map paper however logarithm map previous code notion query combination satisfy constraint may express small alternatively constraint restrict combination reconstruct dictionary atom combine reflect energy encourage locality yu wang determine total cost q good dictionary structure dictionary write jointly minimize choice coefficient solve alternate treatment similarly generalize code general space riemannian manifold metric n every affine metric encoding shall concerned dictionary manifold embed default natural choice point q manifold tangent hand code q notation tangent refer step manifold follow terminology definite geodesic distance true riemannian manifold dictionary ty elegant intrinsic approach tangent bundle tangent accord root dimensionality riemannian encoding write extra trivial turn learn riemannian euclidean code along update tangent represent x fx reader manifold logarithm interest work manifold propose dictionary specialized vision dimensionality manifold dictionary alternate atom admit method intrinsic logarithm manifold code non linearity think possibly experiment manifold matrix subspace embed form smooth embed riemannian metric path isometry riemannian curve length metric geodesic work represent action projection embed x note geodesic use term establish link underlie concept use interest code dictionary code address combination way element generalize prefer generalization n point rely verify multipli minimize affine combination manifold metric mean mean geodesic metric metric contrast mean weight close I term geodesic metric closely furthermore give conceptual illustration slightly call code step reason later expand explicit manifold see store encoding similarity atom offline symmetric common package specifically let code initialization processing dictionary I ip manifold unit sphere albeit subtle code vector space result solution f x conceptual diagram sparse address work surface four atom red square describe query circle atom green step atom space might unit favor locality locality vice versa show sparse however free parameter neighbor fast locality coding wang q
code indeed much generative aim query model conditional fixing building focus code present force impose name believe building model co bilinear word loop learn usage notation motivate capture hierarchical structural bilinear traversal combine natural bilinear incorporate reasoning efficiently far outperform nlp previously code b programming focus specifically decision readily available recently c easy data processing challenge building process motivate representation terminology throughout code sequence token serve syntactic element code flat lead inefficient description token increment body fundamentally compactly represent loop instead code process abstract sequence code token correspond syntactic internal node token subtree primary source code example determine reason generative distribution generate root leave repeatedly tuple parent leave token first traversal tuple independently rest independence assumption produce weak contextual lose name construction dependence people limit see example write nest loop name outer loop inner loop dependence come code program evolve sequentially traversal variable distribution child tuple initialize stack stack root element stack line child child stack line token node fashion traversal update internal evolve produce internal traversal desire token distribution prior traversal variable child condition node joint equip stack probabilistic first token particularly suit traversal right leave internal circle token circle traversal stack state computation tuple indicate conditioning brevity encounter uncertainty generation child avoid child tuple use bilinear simple log bilinear representation pair child tuple energy tuple child normalize child tuple observe child notion index value denote tuple similarly look representation pair sum represent variable matrix child log bilinear grow high traversal exponentially extension certain traversal depend arbitrarily element rich type let combine traversal variable tree token latter traversal deterministic traversal generative traversal satisfy tree replace variable traversal inference explain unique compute type give token generate elaborate deterministic object let node take value problematic cardinality annotation annotation uncertain choice value evaluate bad token decrease annotation improvement token source great child parent token name build language keyword currently scope signal program variable variable scope vector string along key tuple vector string token decide whether token accomplish internal binary proceed global token token smoothing device although scope token pattern scope logic three include method available option many scope select token child proportional I normalize currently scope stre token probability straightforward second latent traversal allow traversal use set traversal traversal computed token total log learning problem production stack bilinear generally traversal latent traversal traversal variable couple across tree simplicity restrict restriction deterministic token correspond depth traversal algorithm adapt learn bilinear detail supplementary describe exist child token terminate discrete traversal child equivalent english many variant explore annotation annotation aside traversal make special weak bilinear widely language tree traversal bilinear novel believe include traversal bilinear parameterization logic general token inductive traversal rank gram effectively analogous issue factorize similar way nlp recently explore sophisticated non program repeat applicable language rule sophisticated language specification programming com program k line code program programming program identity validation overall split easily interpretable divide token report token choose strength smoothing epoch stop validation setting gradient stochastic dimension test token child unobserve assign locally smoothed tuple low log detail smooth additional material novel scope unobserved zero baseline bilinear model gram additive smoothing hyperparameter smoothing choose performance bilinear bilinear parameterization traversal result bilinear equivalent dominate gram allow context generalize appear train gram gram gram gram ccc valid seq augment traversal node parent store upon reach sequential token variant hierarchy hierarchy alone perform alone contribution r ccc model em ccc scope scope traversal latent traversal consider latent latent result gain train bilinear bad tried add traversal training slow step scope model train scope scope use list sort de index appear sort understand room improvement experiment value total log report contribution incur generate token seq properly cost token supplementary material far good reporting come parent kind cover local next model token seq qualitatively drawing loop ask simply token token initialize traversal reasonable value scope scope source code file supplementary loop capture particularly organization learn subtle thing like variable often square largely build appear key leverage great result yield improvement quantitative baseline qualitatively produce realistic many challenge notion source structure relate statement naive sophisticated child tuple apply compositional scope tuple would extend scope model handle call level piece briefly great potential properly find simple generally focus modeling popular generative extract might apply argue probabilistic code rich potential hope help acknowledgment grateful helpful work supplementary material generative model code traversal latent deterministic traversal traversal union firstly compute become use forward backward algorithm free energy brevity drop term e forward emission log bilinear weight handle bilinear correspond add unweighted sample step experimental validate minibatch initialization subsample set manually
parameter train learn learner basis able reconstruct bootstrap identity bootstrap nearly reconstruction may high bootstrap implement pyramid pool omp predict mnist tune summarize mnist perform baseline provide bootstrap bootstrap cca bootstrap suggest mass commonly class column common expression perhaps expression art suggest useful supervise weak label build pre full bootstrappe heuristic annotation large confidence drop come confident location bootstrappe confident modify top bootstrap curve figure bootstrapping end imagenet propose apply way network predict image proposal deep classifier region post classifier describe section l baseline baseline bootstrap baseline hard bootstrap bootstrapping detection datum mainly bootstrappe develop training weakly multi output method engineering effort purely supervise improvement even simple suggest move research attention achieve gain collect price label scale extend agent promise unlabele label benefit consistency table token edu google ca usa google art use purely depend label assumption often label may localize general labeling work generic noisy incomplete consistency similar deep computed substantial robustness mnist handwritten digits label case label recognition achieve art face modification challenge approach image output currently recognition purely dropout overfitte system account miss label annotate large image complex object image localize recognition human agree become noisy become argue vision noisy incomplete labeling usual objective consider notion incorporate consistency world match incoming output learner justification label effectively accurate lead label clean carry balance experiment robustness several mnist handwritten digits robust corruption case achieve benefit challenge improve single shot network improve discuss describe probabilistic supervise vast key paper paper weakly semi supervise deep bootstrapping unlabele build seed iteratively unlabele extract seed expand repeat algorithm recently co similarly pair iteratively additional label identify training object detection weakly self demonstrate comparable system work bootstrappe network share motivation label loop incorporate directly noisy annotate network robust handle label share motivation noisy label semi encourage notable beneficial image similar generative dimensional extend language learn language label newly collect training effort build robustness noisy deep rbm use hybrid generative deep machine multi network simplify training model enable backpropagation much deep supervise generative unsupervised imagenet image still behind term way benefit probability observe q label perform discriminative ascent purely discriminative bottleneck develop rbm multinomial energy conditionally energy lead hidden multinomial unit arising give via divergence prediction learn j assume rbm generative exact due mcmc generative certainly feature binary make rapidly exist activation exact descent analogous version rbm autoencoder approach consistency objective feed version experimental develop explicit reconstruction dynamically target result target current model improve prediction label incorrect eventually highly inconsistent predict coherent ability consistency noisy bootstrapping bootstrapping entropy target mini bootstrapping bootstrappe directly target show entropy regularization entropy regularization encourage enable semi learn bootstrappe mini stochastic step estimate target parameter well predict bootstrapping instance model target recover bootstrappe two operate may noisy structure state object system annotate box image label however category miss annotation modify object approach box cluster centroid prior predict object bound bounding appear location proposal enable efficient quality score attractive section predict objective detail write cross entropy note section
representation image variability preserve geometric function contour image probabilistic traditionally distance sized transform close contour template variation express interpretation give parse parse generic object infer fine pose human rich prior graphic simulator handle extreme variability domain ease denote range rotation express give popular graphic take define cross axis extremely due variability consist flexible profile graphic simulator mesh object simplicity circular cut along axis beta distribution span cut since hyper kernel gps point gps pass graphic simulator mesh generation reconstruct amount calculate formalize truth show box super buffer color consist result depict roughly viewpoint collect image illustrate challenge suggest object beneficial compositional gp infinite hierarchical shape human scope program compositional mesh body group graphic generate result mesh prior center center mesh underlie axis mesh part fashion ta mesh whenever define smoothly mesh illustrative inference invert resort markov model variable mix noisy invert graphic simulator propose hasting affine rotation mesh belong affine transformation affine mix proposal discriminative despite minima sampler make latent variable since time couple exploit hamiltonian denote model obtain baseline figure outperform pose detector set contour stick result failure primarily resolution fine contour mid texture descriptor improve get seem reasonable failure show infer position contour section utilize strength bottom aid pose explore act generator pose feed pose pose dataset pose local kde generate proposal rapidly reasonable leave fine effect proposal discriminative number prior fit pose detector image stick result treat bound pose detector retrieve datum give kde get proposal kde close posterior fit via kernel run speed world vision inversion probabilistic shape address computer graphic category appearance handle compare mid vision handle monte standard site hasting move yield quantitative human pose reconstruction compare computational additionally proposal synthetic pose detector inference research seem result handle incorporate shape decomposition mixture augment integrated potential practical build image rich appearance illumination via rich state modern neural many generative probabilistic graphic program programming system vision long go bottom vision yet understand scene perform recognition good offer rich suggest potential produce obtain good quantitative thank feedback singleton fellowship partly google ai project recently formulation computer graphic natural account via realistic generative seem intractable inverting version computation evaluate address show solve world generative output probabilistic program generate plausible affine place scene likelihood similarity base mid vision site hasting proposal hamiltonian discriminative datum proposal datum pose achieve quantitative qualitative formulation generative graphic attract single low comprise shape make heavy use temporal continuity mit edu microsoft com mit edu accounting object appearance via graphic primarily graphic engine lead identify engine seem challenge prior propose evaluate address challenge solve challenge vision generative model tool bayesian geometry probabilistic engine environment stochastic sample prior affine scene mid formulation rich model object shape reconstruction image quantitative baseline mesh parametrization discrete couple graphic computation site locally hasting proposal hamiltonian monte proposal learn discriminative successfully strength generator define affine engine express change scene prior generic
popular preferred interestingly also network backpropagation indicate design topology neural three loading table x necessity layer three neuron eight represent order idea cross neural stem datum name step intelligence important relationship model suitable purpose framework well exist dominate economic highlight way like school economic provide dataset mm mm ac cs ac modern far carry show offer experimentally result flexibility offer elaborate framework discovery neural lot community explain nature question factor affect answer latter reveal diverse factor economic deep factor mainly reveal economic deal characteristic real possess make unable incorrect limitation raise regard accuracy guarantee conduct consider quantitative prediction apparent economic benefit variety computational intelligence offer neural capable remarkable successfully linearity flexibility topology step datum numerous tackle relationship combine discovery real neural economic work evaluate data transformation deal high advantage topology novel topology derive technique assess improve great classification prove therefore work serve comparison great network mining method exploratory extend real characteristic nd predictions rd briefly together perform identify present proceed analyse conclude section amount separate logit suffer low way model factor predictor exhibit probit around surprisingly achieve explain build estimate whether regard finding model exist receive argue linearity learn mining handle lot ability handle large number performance interesting measure sale usefulness exhibit predictor capacity able encountered application outperform economic successfully stock possess topology network concept logical among neuron exploit result order include derive network purpose many achieve support economics model dataset attribute year order overcome financial category interest age status status person status financial car total service total total total total self employ detail outlier time tackle aforementioned difficulty series transformation perform beneficial unsupervised homogeneity map categorical datum coordinate together financial attribute item reduce attribute remove outlier provide interpretability nine transform coordinate discriminate financial factor financial necessity cluster seven characteristic characteristic dataset objective derive exploratory classification level employ employ old average old status linear simple explanatory variable explanatory try error input straight line estimating coefficient estimate parameter ordinary try aggregate create tree sample construct decide build aggregate vote specification forest simplicity allow overfitte node input layer predictor layer arbitrary node intercept fed activation pass activation non activation tangent simple neural perceptron n output easily generalised take parameter weight randomly mean model try learn backpropagation try difference calculate difference output adapt weight accord specific argue subtract gradient reduce predefine update big update keep sign tend raise concern common avoid validate fold get fold choose layer topology designing extract unsupervise perform neuron idea behind neural factor factor variable factor depict factor widely input incorporate hand characteristic variable introduce extra class combine relationship final something class fuzzy neural name define economic context neural amount rest reason whether series incorporate develop validate fit fold compare rmse fold method evaluate model neural representative take perfect fit root difference actual I well rmse random contain miss linear calculate equal initial choose order hide produce ten case cross evaluate one good appropriate transform create four build test transformation cluster table iii original network classification design check quick indicate forest clearly almost backpropagation networks rmse perform regression seem improve build transform specifically transform attribute dataset especially train backpropagation random forest around case forest around backpropagation neural classification provide backpropagation network forest backpropagation compare classification interestingly increase proportion explain backpropagation closely random backpropagation neural exhibit big backpropagation build transform verify suitable purpose neural economic traditionally dominate computational intelligence broad tool technique use combined framework forest beneficial preprocessing despite return case provide combine
draw normal covariate make incomplete randomly imputation imputation scientific rule mis calculate cf exclude variation interval calculation proper completely apparent information contribution degree freedom eventually variation conventional rule total simplified pooling coverage percent conclusion illustrate situation essentially observe precision estimate imputation inference find scientific field sciences big useful application study evaluation decade essential multiple infinite population simulation properly compare account miss sampling make generation much multivariate van imputation blue van multiply infinite unit cover population exist pooling conventional rule situation standard lead amount result imputation simplify pooling implement essentially study address medical lead bias incorrect inference straightforward approach imputation datum complete dataset analyze combine pooling multiply sample infinite unit rare play yet observe affect precision situation confidence interval long statistical
learn subject number know target exactly ensure try try fit matter represent predictor merely know large architecture excellent reality even large property make question back understand play base hidden increase behave capacity capacity play deep understanding inductive analogy understand analogy regularization infinite sized bound demonstrate implicit decay infinite give rise convex net feed find minimize input linear learn soft entropy correct cross write otherwise cross margin deviation negligible always dominate label training increase necessarily decrease loose estimation tradeoff train size mnist stochastic descent expect initially decrease network achieve continue predict control mnist attain allow well error add beyond generalization go cifar momentum test phenomenon artificial decrease hide cifar train agree think censor represent exactly still continue decrease reach force overfitte add datum network fit figure percent significant error continue decrease increase past achieve cifar explanation implicitly try even huge furthermore thus infinite control want explicit regularization modify drop weight pass convergence zero try identical get vast many final try add form still increase network help go simple feed forward single activation model capacity limit number sensible computationally last decade much instead constrain example frobenius norm use eq norm lead regularizer trace network high contrast constrain local trace norm factorization justify sensible bias ensure trace realistic factor suggest inductive activation reality explain light perhaps target unit implicitly toward inductive bias really fitting view fitting infinite common matrix norm model capacity purely indeed improve sized start decay regularization weight approximately implicit regularization neural network top regularization aim instead try match deep g simplicity focus e network regularization layer hide unit example arithmetic geometric mean attain input rescale h v v reason mapping piece piece rescale finally since rescale establish learn hidden connection convex represent part net regularizer unit layer nn even discrete support equivalently versus select limit unit merely select norm allow decay equivalent hide regularization equivalent decay implicit regularization stochastic descent equivalence network output regularization regularizer indeed activation feed forward relu perfectly provide enough succeed trivial polynomial network super sample correspond description return learner
value miss sensor specific diameter determine node euclidean sensor relative local positive couple neighboring near distribute inaccurate analyze distance invariant slightly application near neighbor predict iterate input feature property specific rich use dt resolve challenge reliability identify critical corruption build optimal tree construct chain unit radial recognize due computational requirement network management centralize network learn solve challenge localization network angle distance measurement receive node measurement receive indicator difference arrival node value coordinate reference therein neural big example use classify point label detect use give observation point feature part I read classified gap include optimize network unconstraine security e discussion please algorithm adapt uncertain datum belief give assess investigate statistical process wide structure available outcome k algorithm recognize learn widely cluster resolve node centroid cluster b close membership stop valid g predefine perspective centroid dimensionality aim set new orthogonal component order first correspond discard content uncorrelated linear combination simplify problem thorough theory important note pca interact action maximize reward experience reinforcement reward value state use learning determine fast easily seek maximize challenge memory sensor change failure management adopt challenge wireless energy processing aggregation designing protocol various challenge sensor provide memory bandwidth traditionally wireless represent set represent channel model reach span vertex leaf node child np hard machine sensor previous dynamic benefit summarize path dynamically divide simple problem consider achieve efficient meet method sensor network span path exchange figure machine learning neighboring node procedure decide assign transmission prove near sensor problem require communication single exchange machine protocol comparison protocol imply large scale network wireless sensor protocol adopt c regression limit sir flat som limited hybrid yes flat multi moderate good yes moderate rely measurement node execute facilitate refer framework exploit fact sensor overhead detect structure serve develop wireless linear utilize sensor intelligence sir som illustrate sir modification short path learn second high accordingly neuron update match pattern highly execution run hybrid som take account requirement throughput cycle process update neuron weight overhead set som construction sir throughput enhance hoc basically guarantee reliable resource allocation mobile hoc heterogeneous capability addition maintain network join forward join backward create learn overhead search energy efficiency requirement need e g consider communication dedicate band ghz spectrum technique reinforcement protocol reward use technology detect equip device moreover use simple maintain location benefit achieve acceptable solution pr enhance select high rate past period protocol importance constraint pr bayesian next node transmission introduce reinforcement novel source message disadvantage reinforcement highly environment sensor inefficient pass local cluster head typically works discuss head head process classical node network incorrect operation technique node cluster aggregation cluster extract node sensor machine efficiently head head enhance aggregation cluster architecture working compare protocol intensive energy aware network mechanism moderate yes high yes head yes low yes sensor moderate moderate yes som moderate moderate yes online compression acquisition compressive sense yes transmission consensus pca high moderate yes moderate surveillance moderate low decentralize datum yes neural network target efficiently transmission service head critical iterate input cluster centroid cluster hierarchy gp parameterize probabilistic base gaussian regression regression focus energy consumption broadly speak process preferable smooth function however scale space low lee self classify win neuron represents number win network traffic network lin adaptive quantization retrieve sensor node historical pattern code book transmission original reading compress crucial disadvantage use aggregation neuron far away competition develop token set use pca aggregation traditional explore original simple compose step expectation em function fix expectation cost method estimating produce compressive ability spatial correlation direct transmission collect distribute technique execute combine compression likelihood cb eigenvectors matrix pca cb method consensus predict hence global communication cb cb tune provide quality adjust increase consensus round increase pca transform collect time cluster head cluster head compressed eliminate achieve ignore throughput cope dimensionality collect keep important reduction li address track collaborative signal environment additionally track multiple target nearest surveillance collect massive surveillance complex introduce therefore mobile surveillance wireless mobile sensor enhance surveillance cluster cluster mobile sensor idea appeal straightforward implementation sensitive outlier role free wireless sensor clique perform clique enable node achieve combination address topology locally central control efficiency network transmission overhead energy consider requirement sensor introduce schedule human intervention classify drive query drive event fundamentally machine offer restrict area assess processing mechanism benefit efficient mechanism requirement storage resource event machine development processing query without machine learn assess controller spread intend node detect node sign area attention research rely define strict phenomenon recent query process complicated develop advanced query processing solution present processing solution detection detection activity recognition nn low query space enhance drive real distribute detection dt drive detect environmental phenomenon manner decentralized percent result important correction bayesian summary correction enhance present activity accurately body initially spread body detect sensor axis negative hmm sensor sensor rely informative description naive maximize human naive challenge solution yu detection processing aggregated maker beneficial environment core interpretable introduce highly query processing technique develop nn aware location k search region correspondingly knn process neighbor bind within snr refine primary concern memory collect delay develop tree event recognition sensor network area vote traditional overhead dynamically detect I e dominant set step language request attribute send database management management system query component optimize wireless sensor attribute mac mac enable base localization acoustic specification gps support localization network feasible propagation limitation gps develop system surveillance monitoring sensor surface unit central recognize site application spatially system delay major request moreover achieve failure ambiguity sensor collective employ regression predict mobile computational complexity execute give introduce som thousand propose execute som layer connect neuron formulate spatial anchor coordinate disadvantage equally spaced location introduce localization algorithm som develop limited resource require gps centralize central processing adjacency node similarly lee provide localization service node algorithm som propose network unit transmission overhead li develop reinforcement localization call path mobile management mobile mobile mb aware movement number brief state position mb cover sensor message mb run save resource fail mobile transfer mac protocol pose challenge wireless network energy consumption cycle fraction sensor node therefore protocol comprehensive protocol provide recently enhance mac protocol adaptively cycle transmission history able predict channel energy consumption design mac transmission concept mac protocol mac security able attack brief mac protocol review synchronization protocol assume indicate ability handle node comparison mac protocol mac dt hybrid present mac protocol active continuously medium bayesian learn channel allocate save network protocol sensor mac mac mac mac division protocol employ periodic frame access transmission change wang transmission schedule fuzzy distribute maximize length mac prevent attack type attack huge traffic useful case limitation capability neural prevent network traffic investigate network request consequently mac layer exceeds predefine importantly site design employ mac mac protocol basically mac reduce usage increase throughput mac mac rl mac transmission schedule frame mac determine slot length cycle active traffic load bandwidth new base mac inform achieve benefit simple resource frame nod transmission map slot node attain slot allocation transmission demonstrate initially initialize upon transmission slot update update rate upon successful transmission reward equal fail transmission certainly example three employ medium access reinforcement appeal requirement may initial phase share collect user mobile challenge communication service requirement propose adapt layer design mac switch mac protocol mac engine mac current inter e requirement energy usage traffic mac pure mac mac though mac environment introduce design mac architecture functional specification operational capable date comprehensive machine advance adopt security highlight effort specialized researcher improve learn security anomaly implement security limited resource constraint moreover attack observation network figure present monitoring sensor classify read region inconsistent attack consider anomaly phenomenon monitor system employ attack detect basically security adopting save significantly expand enhance reliability avoid discovery convert often intervention attack explore various address security review indicate summary wireless sensor machine detection belief outlier nn moderate distribute outlier detecting attack selective attack outlier detection online outli centralized system distribute adaptive analyze som som behaviors bayesian node temporal correlation infer observation collect evaluate branch detection near replace value nearest nn base black attack request message indicate accordingly source discovery assume drop attack classifier capable attack selective attack information bandwidth use origin requirement could sphere distinguish anomaly minimize communication overhead design svm svm outli detector inspire biological body chen detect algorithm furthermore zhang temporal outlier main svm scalability requirement address detect attack wireless ad map determine som attack large sensor network service event potential query node suffer energy furthermore issue couple topology important reliable art requirement review review effort machine achieve advantage recognize stream thus need flow aware guarantee detection depend network service handle ensure resource power review column service datum dynamic low link assess reliability moderate processing sensor converge aware power management management low tool grow performance estimate availability reliability failure dynamic capture dynamic behavior effect propagation idea requirement link inaccurate unstable variation interference wang link quality method protocol adopt offline method learner indicator feature classification tree receive signal strength indicator size load forward receive reverse communication experiment time method present handle assess provide iterative experience distribution environmental mean historical update consider sequentially collect adaptive sensor base learn technique throughput observe learner able site stage mechanism conversely power management energy capability base aware management level capability employ attain reinforcement mesh cc structure tool basically adopt reliably might examine impact load whole present constraint minimize consumption sensor intend consider application fig fundamental reading receive incoming put must execute network static schedule node go move take knowledge management environmental monitoring employ accurately behavior inactive movement velocity advantage implementation since design consider predict device storage power network measure air level effect air quality neural server computer server radial rbf extract measure fundamentally control digital crucial affect develop scheme compare wireless although machine open maintain several management percent compression reduce transmission hence traditional compression extra consumption requirement tradeoff transmission efficiency even extend basic concept compressive meet resource decentralize compressive please technique include device centralize enable rapidly tune current reason learn processing algorithm include adaptive weight improve ellipsoid soft develop efficient two communication protocol mac protocol design detecting activity first mac protocol discuss survey technique study enhance second focus minor energy I machine management circumstance basically sensor broadly cluster technique cluster network temperature monitoring cluster combine hierarchical spatial temperature monitoring energy activate hierarchical balanced wireless protocol tool consequence wireless sensor energy aware scheduling localization datum machine wireless sensor network table summarize adopt machine learning challenge challenge sensor network several extensive study adopt wireless network resource pattern numerous open effort hierarchical adopt learn resource management wireless lin school computer engineering wireless sensor network environment dynamic either cause external sensor eliminate unnecessary solution literature review period address wireless sensor advantage provide comparative aid suitable machine challenge wireless localization cluster aggregation processing medium compressive sense wireless cost node nod forward unit base sensor equip various acoustic chemical pressure weather optical diversity building characteristic develop scenario aggregation aware scheduling security shift robust last decade machine extensively task area bioinformatics vision come mathematic definition essence development acquisition enhance develop machine describe exploit
velocity consequently differential quantifie coupling measure wind wind take wide present challenge research datum develop method reproduce velocity measure wind speed non fluctuation power prediction energy production wind reflect wind contribute determine model well understand production focus fluctuation wind extreme anomalous responsible load additional show derive wind end wind power wind north show approach ref apply describe detail comparative sec conclusion topic htb velocity illustrate normalization three wind full month wind wind n measurement rotation angular velocity operating rotation hz hz rate measurement exist hz analyze protocol scientific requirement wind measurement accord velocity fig together right fluctuation responsible wind wind see increment statistic time wind hour increment particularly power several quantify wind velocity increment plot shift vertical axis visualization illustrate coefficient drift equation framework propose develop range modeling medical eeg stock market ref therein method wind ability properly power characteristic langevin briefly one full period extract apart multiplicative conditional namely dash interpolation correspond plot instance fit range offset moment evidence stationary langevin equation moment address narrow wind condition represent wind velocity section range wind velocity diffusion top drift diffusion velocity low value velocity lack range check fig drift diffusion depend linearly well see functional coefficient wind euler langevin wind integrate reconstruct plot fig together clearly reconstruct series real measurement increment time scale b clearly conclude ability langevin evolution langevin time reconstruct rate wind drift text fig increment fluctuation within lag second unit deviation wind also increment notice condition necessary langevin evolution still fulfil condition langevin stochastic evolution drift coefficient first order correction consider drift diffusion point evolution velocity velocity measurement less couple wind langevin straightforward forecast nearest properly neural
interpret learn function decoder bias e tangent project onto activation free matrix tie bias order function base error choice depend typical xx appropriate e l dimension dimension encourage learn underlie interpretable simply overcomplete kind autoencoder autoencoder among reconstruct attempt corrupt solution learn distribution noise gaussian noise corrupt probability apart representation regularization corruption procedure move must project corrupt manifold useful pre especially stack recent variant locally characterize generate corruption process predict form corruption markov autoencoder distribution sec sample generate generate spurious region around determine corruption allow place amount spurious mode large amount noise naive divergence reconstruction define series reconstruction random subsequently reconstruct final spurious mode manifold autoencoder relational autoencoder extension pair give define learn involve indicate indexing encoder store infeasible weight cubic roughly restrict project quadratic weight need factored model w develop neural training denoise autoencoder apply although training typically examine corruption procedure sample interpretable form alternate algorithm spurious define converge generate distribution argument x make geometrically datum like correct manifold multiplicative learn class structure share class digit example tail may relatively share tail correspond weight importance tail interpretation factor factor class label generative conditional mnist database intensity image scale thresholded mnist unit unit relu visible activation epoch via mini descent initial nesterov gradient noise apply pixel corruption average reconstruction markov example sample conditioning depict define chain zero corrupt begin generate chain spurious sample digit expression pixel gray intensity intensity experiment relu hide sigmoid epoch descent epoch nesterov noise train factorize activation variation mnist likely size training provide unlabeled generative way learning separate share light act practical apply rich explore sampling autoencoder operator unimodal limit capacity complex order extend work autoencoder purely train massive much attention last generative autoencoder transition markov theoretically empirically capacity datum
run step burn cluster probable generation diagram upper region ht capable mix tree performance real example first sampler examine breast cancer repository originally subsequently utilize numerous breast nine clinical covariate uniformity cell uniformity cell shape size miss outcome discard log wu majority suggest homogeneous contribute misclassification assess performance select set make remain ensemble calculation repeat ten separate different seed low wu support gibbs rapidly heterogeneous illustrate percent force continuous measure patient clinical visit via penalize cubic spline inspection aic short use longitudinal datum subject entry utilize clinical gender pa bc status longitudinal roughly carry concern several variable since importance covariate value covariate choose form tree importance quite variable forest rank decrease purely result forest gender role empirically group ensemble review tree random forest boost average interesting enough machine resort bootstrap calculation size bayesian averaging method demonstrate ensemble capability help self behavior average tree show important provide worth compare model component unimodal importantly modal construction fitting imagine gibbs sampling rapidly fit grow chance develop empirical possible greedy randomness implementation difficult user exist cart package grow cluster mention acknowledgment author grateful foundation patient comment breast cancer university author thank availability supplementary forest pt utilize ensemble cart subset similar heterogeneity aggregate approach develop cart classification breast cancer regression patient key bayesian cart heterogeneity regression cart nonparametric binary intuitive relation covariate aside simple cart affect cart conditionally independent simplicity preserve cart derive generate bootstrap tree utilize aggregate bag boost stochastic create generalize cart tree sum difference multiple create diverse fitting therefore combine source variability utilize hope tree fit nonetheless rather bootstrappe control tree tree performance three setting breast study benchmark heterogeneous patient record outcome cart assign record region identically independently distribute predict one origin probability th estimate impossible calculate later define conditional mixture infinite correspond dirichlet node unit wu child node node simply integer part child node node one leaf therefore least one splitting correspond draw element leave iterate leaf node follow q bernoulli multinomial distribution small partition guarantee distribution proportion certain constructing time utilize name change dimension create include explore jump model auxiliary new stick process gain popularity decrease burden stick break dirichlet straightforward illustration infinite indistinguishable large number slice slice sampler carlo sampling lead slice sampler rapid assignment scheme iteration grow clustering allocate rapid change update provide find choice grow metropolis mh grow yet grow tree therefore scheme conditional mh restrict result step compare use micro every node convergence force wu efficient facilitate jump mode tree change useful prevent switching uniform truncation effect keep posterior analysis marginal costly allocation joint type estimator ensemble estimator former assignment latter define assignment posterior estimator often know observation capability single partitioning
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v v v v v stroke v v v v v v stroke v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v stroke v v v v v v v v stroke v v v v v stroke lt v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v stroke v v v v stroke v v v v stroke v v v v stroke v v v v v v v stroke v v v v stroke v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v stroke v v v v v v stroke v v v v v v v v stroke v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v stroke ltb def rgb exch exch def exch def mul roll exch exch sub mul mul sub mul mul def mod def ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch exch ifelse def mul exch mul add constrain roll exch mul constrain roll exch mul add constrain roll def exch exch exch roll exch roll exch exch mul exch constrain roll copy mul exch mul mul add constrain roll exch mul add exch mul exch add roll def eq rgb ifelse ifelse ifelse gidx get gidx add def gidx gidx gidx def gidx get mul gidx gidx get add gidx gidx gidx mul gidx le gidx gidx def def def def mul sub ifelse pm def pm exch def cf constrain exch exch cf constrain ifelse pm pm def ifelse ltb stroke ltb v stroke ltb stroke stroke ltb r stroke ltb stroke stroke ltb ltb stroke stroke ltb r stroke ltb stroke ltb r stroke stroke ltb stroke stroke ltb ltb ltb v v v lt v v v v v r stroke lt v v stroke v v v v ltb def rgb exch exch exch def exch def mul roll exch exch def mul mul def mul mul mod def ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch exch ifelse def copy mul exch mul add constrain roll copy mul exch mul constrain roll mul exch mul add constrain roll exch exch roll exch def copy mul mul add exch mul roll mul add exch constrain roll exch exch add constrain roll def ifelse ifelse ifelse gidx gidx gidx gidx gidx sub def sub get gidx mul gidx gidx mul gidx gidx add def gidx get le gidx gidx gidx ifelse def def def pm ifelse def pm def stroke pm exch pm constrain constrain constrain def ifelse stroke pm pm exp ifelse ltb stroke ltb stroke stroke ltb v stroke stroke ltb stroke stroke ltb ltb stroke ltb v ltb stroke ltb v stroke stroke ltb r stroke ltb v stroke stroke ltb v r ltb stroke ltb stroke ltb stroke stroke ltb v ltb ltb ltb lt v v v v v v v v v stroke v v v v v v v v v v v v v stroke v v v v v v stroke v v v v v v stroke v v v v v stroke v v v stroke v v v v v v v stroke v v v v v stroke v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v stroke v v v v v v v stroke v v v v v stroke v v v v v v v stroke v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v lt v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v stroke v v v v v stroke v v v v v v stroke v v v v v v v v v v v v v stroke v v v v stroke v v v v v v v v v v v v v v stroke v v v v stroke v v v v v v v v stroke v v v v v v v v v v v v v v v stroke v v v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v stroke lt v v v v v v v v stroke v v v v v v v stroke v v v v v stroke v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v stroke v v v v v v v v v stroke v v v v v v v v stroke v v v v v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v stroke v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v stroke v v v v v v v stroke v v v v v v v v v v stroke v v v v v v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v stroke v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v lt v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v stroke v v v v v stroke v v v v stroke v v v v v v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v stroke v v v v stroke v v v v v v v v stroke v v v v v v stroke v v v v v v v v v v stroke v v v v v stroke v v v stroke v v v v v v v v v v v v v stroke v v v stroke v v v v v v v v v v stroke v ltb stroke r r show five plot vertical help side denoise distribution plot five super laplace pdf logarithmic behave plus signal neuron scalar theoretically approach variance corruption denoise scale step readily closely serve statistical modeling translate connection help high level relevant component independent source however span source limitation scale distribution leave sample five five sub autoencoder zero hide nonlinearity denoise simplify essentially mean mapping matrix denoise assumption ica incorporate denoise covariance gaussian verify mixture scale apparent loading contribute hide unit load product measure contribution recover activation scale length dominant loading row angle vector dominant source correspond dominant source sign determine super sub source unit preference source represent structure perform happen miss unit pressure possible mode operation primarily span covariance align independent happen align principal pca ica hide retrieve ten loading ten loading without mapping around std turn require time converge ten loading loading network tendency extract see span eigenvector whereas note pre whiten autoencoder recover component allow perform principal normalize information shall expand dependency hide ica usually impossible produce truly statistically projection even activation lack normally high dependency example activation correlate activation represent recall q activation variance translate connection strength unit activation flexibility layer network representation connection allow high unlike ica whiten cost function dataset linear source ica source determine word high source variance source source independent source distribution level contain super pre mapping observation ica perceptron mlp encoder q activation operate separately make def dl mul mul add def vpt vpt def begin title subject plot author ifelse def def def vpt vpt mul mul stroke show ifelse ifelse ifelse vpt exch def exch mul vpt vpt mul def dl solid ifelse bl stroke def exch def lt mul exch def pl stroke def lc lc def lc def lc lc lc pl ltb bl dl def al mul def pl lc dl pl dl dl lc def pl dl dl lc dl pl dl dl lc dl def lt pl dl dl lc dl def pl dl dl dl dl lt pl dl dl dl lc lt pl dl dl dl dl dl dl lc dl def pl dl dl dl dl dl dl dl def vpt vpt vpt vpt vpt def pls vpt vpt stroke vpt stroke stroke copy exch exch vpt vpt vpt stroke exch vpt add vpt vpt stroke stroke copy mul vpt v mul vpt def star pls stroke exch exch vpt vpt fill def mul vpt mul vpt mul def copy vpt mul mul vpt mul stroke vpt sub vpt mul vpt fill vpt add vpt vpt vpt vpt stroke copy translate stroke translate fill copy arc arc fill def bl copy vpt arc def bl copy vpt arc vpt bl copy vpt arc bl copy vpt arc fill vpt arc def bl copy arc vpt arc def bl copy copy arc vpt arc vpt arc bl vpt arc vpt arc bl copy vpt arc arc def bl vpt vpt def bl arc vpt arc def bl copy vpt fill copy vpt arc fill vpt arc bl copy copy vpt copy copy vpt arc vpt arc def bl copy vpt arc fill vpt arc bl vpt fill copy vpt arc fill vpt bl copy copy vpt arc def bl copy vpt fill vpt arc roll exch sub exch vpt exch bl vpt bl bl bl copy vpt sub exch vpt def bl copy exch vpt vpt vpt def bl copy exch exch vpt vpt square bl exch vpt exch vpt vpt fill def bl copy vpt exch sub vpt vpt fill def bl copy exch vpt exch vpt vpt vpt fill copy fill def bl copy vpt sub vpt bl copy vpt vpt vpt bl vpt vpt copy exch sub exch vpt fill bl vpt vpt exch vpt vpt vpt bl exch vpt exch vpt sub vpt vpt def bl exch vpt vpt sub vpt vpt copy vpt def bl copy exch sub exch vpt vpt vpt exch vpt vpt fill bl translate def translate stroke translate stroke stroke translate stroke def translate translate def stroke translate def translate def translate stroke def translate translate translate translate stroke stroke vpt add vpt v vpt vpt stroke def stroke exch vpt vpt vpt stroke vpt add vpt mul mul vpt mul stroke stroke vpt vpt mul mul stroke def translate repeat stroke stroke vpt vpt vpt v vpt vpt v def stroke exch exch vpt add vpt vpt stroke vpt mul mul vpt mul v def stroke vpt vpt mul v stroke def stroke translate repeat stroke stroke def fill exch exch def exch exch add mul def add def def fill roll add add translate mul get mul def get translate add mul ne get mul roll stroke def ifelse lt def ifelse def l stroke l stroke exch l def exch def l stroke exch l stroke stroke exch def l stroke stroke def pattern landscape ifelse def pattern landscape ifelse def landscape ifelse pattern landscape ifelse def fill ifelse def pattern def pattern pattern def pattern def def ifelse begin ifelse translate rgb exch exch mul roll exch def def sub mul mul def mul def mod def ifelse ifelse ifelse ifelse ifelse ifelse def constrain ifelse mul add constrain roll exch add add roll exch constrain roll rgb sub exch roll roll exch def add exch mul exch constrain roll copy mul exch mul add mul constrain roll mul exch mul exch mul exch constrain roll ifelse ifelse ifelse def def gidx def gidx gidx def gidx sub gidx gidx gidx gidx gidx gidx gidx mul gidx gidx mul def gidx le gidx def ifelse def def pm mul ifelse mul def rgb def stroke def pm cf constrain exch constrain exch constrain def ifelse stroke pm pm ifelse ltb v stroke ltb ltb ltb ltb v r level image def translate index f f e c c b b b width ifelse ga dp ltb stroke ltb stroke reflect loading appropriately dl mul mul mul def def vpt def def vpt vpt put title denoise plot ifelse def v def def fill def def vpt vpt mul show ifelse ifelse def stroke r ifelse vpt mul def mul exch def mul def vpt vpt solid solid ifelse bl stroke def stroke def mul exch def def pl def def lc def lc lc def lc lc lc def lc lc def pl ltb bl def mul def lt pl lc def lt pl dl dl dl lt pl dl dl lc def pl dl lc dl lt pl dl dl dl lc dl lt pl dl dl dl dl lc dl lt pl dl dl lc dl pl dl dl dl dl dl dl lc dl def pl dl dl dl dl dl dl lc dl copy vpt add vpt vpt vpt v vpt stroke pls stroke vpt vpt stroke vpt r box exch vpt vpt vpt stroke exch exch vpt vpt stroke vpt v def stroke copy vpt mul add mul mul v vpt mul stroke def copy stroke exch sub exch vpt vpt vpt fill stroke vpt mul mul mul v vpt mul stroke vpt vpt mul mul v vpt mul def stroke vpt vpt mul vpt fill def stroke vpt vpt vpt vpt vpt v copy translate repeat def translate stroke stroke def bl copy vpt arc bl copy vpt arc fill vpt arc def bl vpt arc def bl copy vpt arc vpt def bl copy vpt bl copy copy vpt arc copy vpt arc fill vpt arc def bl copy vpt fill vpt arc bl vpt arc vpt arc bl copy copy vpt fill vpt bl copy vpt arc vpt def bl copy vpt arc fill vpt arc def bl copy copy vpt arc fill copy copy vpt arc fill vpt arc def bl vpt fill arc bl copy copy vpt arc fill copy copy vpt arc bl copy vpt fill vpt arc bl vpt arc roll index exch def vpt sub exch vpt exch def bl copy def bl vpt fill def bl copy exch vpt exch vpt square def bl exch vpt fill def bl copy exch vpt exch sub vpt fill bl vpt exch vpt vpt def bl exch vpt exch vpt vpt bl copy exch vpt exch vpt vpt bl copy vpt vpt fill bl vpt vpt vpt bl copy vpt vpt fill exch vpt exch vpt square bl vpt sub vpt exch vpt sub exch vpt vpt fill bl copy exch vpt vpt vpt def bl exch vpt exch vpt vpt vpt fill copy vpt fill def bl exch vpt sub exch vpt vpt vpt fill copy vpt exch vpt copy translate translate def translate stroke translate stroke translate stroke translate translate stroke translate def translate translate def translate stroke translate stroke def translate def translate stroke translate stroke def stroke vpt vpt vpt v vpt stroke exch exch vpt add vpt v v vpt def stroke vpt vpt mul vpt mul vpt mul vpt mul stroke repeat arc vpt add vpt v vpt vpt vpt stroke stroke exch sub exch vpt vpt stroke vpt add vpt vpt mul vpt mul vpt mul mul vpt stroke repeat stroke arc stroke def density exch def exch exch def def mul add def mul add fill def fill def roll translate mul mul def translate add mul ne mul roll stroke ifelse ifelse def def stroke stroke exch l fill exch stroke exch def l stroke exch exch def def pattern landscape ifelse def pattern landscape ifelse pattern landscape ifelse def landscape ifelse def fill ifelse def pattern def def pattern density def def level ifelse symbol index eq def ifelse translate scale def rgb exch exch def roll exch def sub mul mul mul def mod def ifelse ifelse ifelse ifelse ifelse ifelse def ifelse copy exch mul constrain roll constrain roll mul mul constrain roll def exch roll roll rgb copy mul exch mul constrain roll copy mul mul exch mul constrain roll mul exch mul exch mul constrain roll ifelse ifelse ifelse def def gidx def gidx gidx gidx add loop def gidx gidx gidx sub gidx sub mul gidx gidx gidx sub add def gidx gidx get gidx sub add gidx gidx gidx ifelse pm mul ifelse pm color stroke pm def cf exch constrain exch ifelse stroke pm ifelse ltb r v r r z stroke ltb ltb lt v v v v v stroke ltb def exch exch def mul roll def exch def def mul def def ifelse ifelse ifelse ifelse constrain ifelse mul mul constrain roll copy mul constrain roll mul add roll def exch exch sub roll exch roll copy mul exch mul exch mul exch add roll mul mul exch mul add constrain roll mul exch add exch mul add constrain roll ifelse ifelse ifelse gidx gidx add def gidx sub gidx gidx get gidx gidx def get gidx add def gidx gidx get gidx gidx le gidx gidx ifelse def def def pm mul ifelse def mul stroke exch stroke pm cf constrain exch constrain constrain ifelse g stroke pm exp def ifelse ltb v r v stroke stroke ltb ltb lt v v v v v v v v ltb exch exch def exch mul exch def exch sub mul def mul sub def mod ifelse ifelse ifelse ifelse ifelse ifelse def constrain ifelse copy mul constrain roll exch mul roll mul constrain roll def exch exch sub roll exch roll exch def copy exch mul exch mul exch constrain roll mul exch mul exch mul constrain roll mul mul exch mul add constrain roll ifelse ifelse ifelse def gidx def gidx get gidx gidx def loop gidx gidx gidx gidx gidx gidx sub mul def gidx get gidx mul add def gidx sub get get mul add sub gidx gidx gidx ifelse def def pm ifelse pm gamma def stroke pm exch pm cf constrain exch constrain constrain ifelse pm pm exp def ifelse ltb v v v r v stroke ltb ltb lt v v v v v ltb def exch exch exch mul roll exch def mul mul mul mod def ifelse ifelse ifelse ifelse ifelse constrain exch exch mul constrain roll mul roll mul add constrain roll rgb exch exch sub roll exch roll exch def rgb copy mul exch add exch mul add constrain roll copy exch add exch mul constrain roll mul exch mul add exch mul roll def eq rgb ifelse ifelse ifelse def def gidx gidx get gidx def gidx gidx sub get gidx get sub gidx gidx gidx def gidx gidx add gidx gidx get gidx sub def gidx gidx gidx ifelse pm def gamma mul def def pm exch def pm exch constrain exch cf constrain ifelse pm pm ifelse ltb v v v v stroke v ltb ltb ltb lt v v v stroke ltb rgb exch eq exch exch def mul roll exch def exch mul mul def mul sub def mod def ifelse ifelse ifelse ifelse def constrain lt exch exch ifelse def exch mul add constrain roll copy add roll mul exch mul roll rgb exch sub exch exch exch roll exch mul mul exch mul exch constrain roll mul mul exch constrain roll mul exch exch mul exch add constrain roll rgb ifelse ifelse ifelse def def gidx def gidx gidx def gidx gidx get gidx def gidx gidx sub gidx add def gidx gidx get gidx mul add gidx gidx sub gidx gidx get le get gidx def ifelse def def def def mul ifelse def pm gamma mul rgb stroke exch def cf constrain exch constrain def ifelse exp ifelse ltb v r v r v stroke ltb lt v stroke ltb def exch exch mul roll def mul mul mul sub mul sub mul def mod def ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch ifelse mul exch mul constrain roll copy mul add add roll mul exch mul constrain roll def roll exch mul exch exch exch constrain roll mul mul exch constrain roll mul exch mul exch mul add roll ifelse ifelse ifelse def def gidx gidx get gidx def gidx gidx get gidx sub def gidx gidx gidx get add gidx get gidx gidx mul gidx gidx sub get gidx sub mul def gidx le get gidx gidx ifelse def def def pm ifelse pm gamma stroke pm exch def cf constrain exch constrain def ifelse stroke pm pm def ifelse ltb v r v r stroke z stroke ltb lt v v v v v v v stroke ltb exch def def exch def sub mul mul def ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch ifelse rgb exch mul add roll exch mul add roll mul exch constrain roll exch roll exch roll def copy mul exch mul exch add constrain roll mul exch mul exch mul add roll mul exch mul constrain roll eq ifelse ifelse ifelse def gidx def gidx gidx def loop gidx gidx gidx def gidx gidx mul def gidx mul def gidx gidx gidx sub mul add gidx le gidx get gidx def ifelse def def pm ifelse def pm mul def pm exch def stroke constrain exch cf constrain def ifelse stroke exp ifelse v r stroke ltb v v v v v stroke ltb def exch eq exch exch exch mul roll def exch mul sub mul mul mod def ifelse ifelse ifelse ifelse ifelse constrain ifelse def mul exch mul add constrain roll exch mul add constrain roll mul exch mul constrain roll sub exch exch roll exch sub roll exch def exch mul add mul constrain mul add exch mul constrain roll roll rgb ifelse ifelse ifelse def gidx def gidx gidx gidx add def def gidx gidx gidx gidx gidx gidx mul add gidx get gidx def gidx mul add def gidx sub le gidx get gidx get ifelse def def mul ifelse gamma def stroke pm def pm cf exch constrain exch constrain ifelse stroke pm pm ifelse ltb r v stroke ltb ltb v v v v v ltb def rgb exch exch exch exch def roll sub exch def mul mul def mod ifelse ifelse ifelse ifelse ifelse ifelse def lt exch ifelse def copy mul add roll copy mul exch mul add roll mul exch add constrain roll def rgb exch exch exch roll exch sub copy exch mul exch mul exch add constrain roll copy mul exch mul constrain roll mul exch mul exch add constrain roll def ifelse ifelse ifelse gidx gidx gidx def def gidx gidx gidx sub def gidx gidx gidx get gidx mul def gidx gidx mul def gidx le gidx get def ifelse def def mul ifelse pm mul def def color stroke exch def def stroke pm constrain exch exch constrain def ifelse stroke pm def ifelse ltb v r v v v v v ltb v v v v v v v v ltb def exch exch exch mul roll def mul def mul mul sub mul mul mod eq ifelse ifelse ifelse ifelse ifelse def constrain lt exch exch ifelse def copy add add roll copy exch add roll exch mul add constrain roll def rgb exch exch roll exch roll def copy mul mul add mul constrain roll copy mul exch mul roll exch add constrain roll ifelse ifelse ifelse def gidx def gidx gidx gidx add def loop def gidx gidx gidx gidx gidx gidx def gidx sub get gidx add def gidx gidx get gidx get mul add get le gidx gidx gidx def ifelse def mul sub ifelse pm gamma mul def rgb color stroke pm exch constrain exch cf constrain exch cf constrain def ifelse stroke pm exp ifelse ltb v v r ltb lt v v stroke ltb def exch exch def exch roll exch exch def sub mul mul def mul mul def ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch ifelse rgb mul constrain roll mul exch add constrain roll mul mul constrain roll def roll exch exch rgb mul mul exch mul exch add roll mul exch mul exch mul roll exch mul mul constrain roll def ifelse ifelse def gidx gidx gidx add def loop gidx gidx gidx get gidx get gidx mul def gidx gidx sub gidx sub mul add gidx gidx sub gidx mul add def sub le gidx gidx gidx ifelse def def def pm mul ifelse def pm def g stroke pm exch stroke constrain exch cf constrain def ifelse stroke pm ifelse ltb v r r v ltb lt v v v v v ltb exch exch def exch mul roll exch exch def mul def mul sub mul def mod def ifelse ifelse ifelse ifelse ifelse ifelse constrain lt exch exch ifelse copy exch roll copy mul exch constrain roll mul exch mul constrain roll def exch exch roll exch roll def mul exch mul mul exch add constrain roll copy mul exch exch add roll mul exch mul exch exch add constrain roll ifelse ifelse def def gidx gidx get gidx gidx def loop gidx sub gidx gidx sub gidx get gidx mul def get sub mul add def gidx mul gidx gidx gidx def ifelse def def def def mul ifelse pm def def exch cf constrain def ifelse stroke pm pm exp ifelse ltb v v stroke ltb ltb v v v v v v stroke ltb exch exch exch def mul roll sub def mul mul mul sub def mod def ifelse ifelse ifelse ifelse ifelse ifelse def constrain lt exch exch ifelse def mul mul add roll mul exch mul add add roll mul add roll exch sub exch exch roll exch roll exch copy mul exch mul exch mul exch constrain roll copy mul exch roll exch mul exch exch add roll ifelse ifelse ifelse def true def gidx gidx gidx loop def gidx get gidx sub gidx get sub gidx gidx sub sub mul def gidx gidx gidx gidx get gidx gidx add def gidx get sub le gidx gidx ifelse def def def def pm ifelse pm mul def rgb pm exch stroke cf constrain exch constrain exch constrain def ifelse pm pm exp def ifelse ltb r stroke ltb lt v v v v v v stroke def exch exch def exch mul roll sub exch def mul def mul mul def mul mul mod def eq ifelse ifelse ifelse ifelse ifelse ifelse constrain lt exch exch ifelse def mul exch mul constrain roll copy mul add roll mul exch constrain def exch exch roll exch mul mul exch add roll exch mul constrain roll mul exch mul exch mul exch constrain ifelse ifelse ifelse def gidx gidx gidx add gidx gidx def gidx gidx gidx mul add gidx gidx gidx add def get gidx gidx gidx gidx def ifelse def def ifelse pm mul def def def g constrain cf constrain def ifelse stroke pm exp def ifelse ltb v v r v stroke v ltb lt v v v v v v v v v ltb exch exch def mul roll def mul def mul mul sub mul mod ifelse ifelse ifelse ifelse ifelse ifelse constrain exch exch ifelse mul exch add constrain roll exch constrain roll exch add roll rgb exch exch exch roll exch roll exch def mul mul exch constrain roll copy mul exch mul add exch mul constrain roll mul exch exch mul exch constrain roll ifelse ifelse ifelse def def gidx gidx gidx add def gidx get gidx sub gidx sub gidx get gidx gidx mul add def gidx gidx sub add def gidx gidx gidx mul gidx le gidx gidx gidx def ifelse def def def ifelse pm def g exch def pm constrain exch exch def ifelse stroke pm pm exp def ifelse ltb r v v v ltb lt v v v v v ltb def exch exch mul roll exch sub mul mul mul mod eq ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch exch ifelse def copy mul exch mul add roll copy mul mul add constrain roll add roll def rgb exch exch roll roll exch def mul exch mul exch mul exch add constrain roll copy mul exch add exch mul constrain roll mul exch exch mul exch constrain roll rgb ifelse ifelse ifelse def gidx def gidx gidx gidx def gidx gidx get gidx get gidx gidx gidx mul gidx gidx mul add gidx gidx sub gidx mul def gidx get gidx gidx get gidx def ifelse def def pm mul ifelse mul def stroke exch def def stroke cf constrain exch cf constrain exch constrain ifelse stroke pm pm exp ifelse ltb r v v r v ltb ltb v v v v v v v v stroke ltb stroke r r r r r r r r r r r r nonzero top row change belong whose verify separate individual source model figure second mlp result square scale one generate appearance loading rotation rotation feed ica readily depict learn denoise plot belong behave neurons sigmoid neuron activation function activation readily activation learn activation sigmoid appear activation subspace sigmoid learn assume second reconstruction combine alone change high ten unit learn would also need detail recover lowest initialization network turn function speed considerably stage crucial combine proper success help turn network alone autoencoder seem reasonable autoencoder close sensible finding third seem despite whereas speedup speedup phase initialization gradually optimum cost extra term therefore gradually decrease could zero simplicity kept present verify hierarchy abstract efficiently although six representation abstract layer feature roughly nc promising result necessary far verify really deeply compatible therefore support discard information semi autoencoder another split high use feature decoder recently combine overall resemble autoencoder encoder approach extend include interaction interaction motivated autoencoder target propose include exchange information encoder mapping appeal approximation mlp extract dependency independent analysis tailor mapping mapping density make multiple round corruption denoise particularly ability useful make possibility corruption simple add support also involve completely type corruption possible extend denoise type information corruption denoise function might keeping previously mapping essential hierarchical line complex inference take kalman study implement dynamical give rise like fashion study already structure model elegant feedback loop unit top denoise learn connection reliably capacity stochastic latent hierarchical level information abstract invariant feature support unsupervise deep cost key contribute term receive training network also pay match prevent competitive additionally otherwise biased pca preliminary verify abstract verify claim acknowledgment thank parallel version manuscript certainly create work research intuition unsupervise importance algorithm pca rule ica would cm standard autoencoder network autoencoder corrupt denoise cm clean support autoencoder connection autoencoder analogous combine denoise autoencoder framework contribute produce level invariant ever publish hierarchy increasingly abstract visual cat researcher hierarchy stage artificial neural extraction lot propose learn deeply somewhat scheme produce exist seem obvious unlabeled information statistical structure image label carry bit compare carry certainly order argue reason able version supervise learning try irrelevant chapter fit learn base stochastic network continue learn discard leave detail represent approach relevant unsupervise come new explain add autoencoder give abstract invariant vertical path regular autoencoder decoder decoder encoder slow even since explain add hierarchy novel combine level discard invariant representation target high layer discuss extension argue unsupervised need unsupervise learning stand exception hierarchical variable model complicate simple alternative autoencoder feedforward promise candidate combine unsupervised learn autoencoder normally layer stochastic discarding summarize role learn variable model propose connection capacity role supervise learn unlabeled sample output unsupervised representation task obvious act pre key unsupervised important fully learn continue representation supervise start tune filter unsupervise semi argue early unsupervised discard find select recognize face face unlabeled find detector detector feature miss specifically keep reasonable compatible know supervise follow seem common variable latent denote eqs try everything datum go piece would many reduce alone orientation benefit reduce latent discard abstract invariant fix variable eq need represent everything represent high level abstract hierarchical represent inference posterior intractable inference amount intractable simple tractable approximation minimizing proceed would require involve approximation limited structure mathematically tractable layer propagation autoencoder mapping unit model encoder observation model decoder mapping decoder mapping latent analogous mapping autoencoder minimize remainder chapter omit like latent variable autoencoder together observation take define connect encoder decoder path new add add layer mapping layer feedforward actual hierarchical autoencoder eq hierarchical variable intermediate call require call variable matter prior deterministic network mapping stochastic latent inference representation receive path fix add bottom path high layer also combine abstract information orientation low layer l autoencoder complete representation fortunately denoise autoencoder add input explain abstract invariant seem task hand standard autoencoder information bottom path layer signal propagate activation top need lt picture show autoencoder layer need detail activation receive high layer represent autoencoder feedforward part far signal train fashion autoencoder difference connection chance reconstruction leave shrink autoencoder layer nonlinear slow variable share hierarchical reasonable try introduce combine remove rather gradient turn unsupervised study component utilize learn independent ica denoise derive estimate going show nonlinearity interpret expectation learn combine efficient latent tune ica development denoise source separation mapping operate alternate step assume fix reverse mapping derivation assume q depend noise noise approximate amount substitute estimating completely step equation yield algorithm essentially nonlinear simple multiplication nonlinear expect noisy observation crucial input layer contribute training nonlinear rule additional implement competition require nonlinear instead possible require denoise particularly useful denoise propose autoencoder feed autoencoder ask force autoencoder corrupt sample denoise autoencoder iterate corruption denoise sample original distribution denoise learn diffusion corruption force average sample density denoise start corruption denoise flow cause exactly flow sampling model corruption place network surprising denoise possible information relative turn representation energy readily reconstruct turn input corruption normalize probability denoise normalization factor particularly important feature denoise autoencoder representation include connection copy output input possible ready function denoise autoencoder recursively autoencoder internal implement mapping denoise alternate mapping minimizing add refer cost mapping layer consistent alternate recall framework mapping practically mean cost constant optimize autoencoder role combine essentially hierarchical offer forward cf gradient propagate backward path supervise layer output thing care amount error possible mutual predict long avoid go sufficiently loss assume matrix equal assume activation zero generality long mapping kronecker word distinguish viewpoint style representation fact eigenvalue former infinitely viewpoint sound content determinant eigenvector determinant logarithm determinant equal analytical applie eigenvalue power expansion eigenvalue logarithm logarithm element equation small content turn sensible zero relatively relatively analytical require computing gradient chain rule straight form twice behaviour close close chapter cost sure activation really mean input autoencoder high influence mapping solution projection pca type desirable ready collect cost use mapping batch quasi properly could corruption keep apply add corrupt activation term clean activation reconstruction depict computational going share mapping corrupt network add activation reconstruct activation computation observation cost along gradient cost direction along correspond signal denoise path path take denoise autoencoder activation require close gradient present key represent pressure layer allow invariant speed section gradually put hide add variance automatically eigenvalue hyperparameter bp dl mul mul def def vpt def vpt vpt def title subject ifelse def def def def def g def vpt vpt def stroke def stroke show ifelse ifelse def vpt mul vpt exch exch mul def vpt vpt mul def def solid ifelse def stroke def stroke mul exch lt mul def pl stroke def def lc lc lc lc lc lc def lc pl ltb bl dl al mul mul pl lc dl pl dl dl lc dl def pl dl dl lc def pl dl dl lc dl pl dl dl dl dl lc dl def lt pl dl dl dl dl def lt pl dl dl dl lc def lt pl dl dl dl dl lc dl pl dl dl dl dl dl lc dl def stroke stroke stroke vpt vpt vpt vpt vpt stroke pls vpt box exch vpt vpt vpt stroke stroke exch exch vpt vpt stroke vpt stroke def stroke vpt mul add vpt mul mul mul copy def exch vpt vpt stroke vpt add vpt mul v mul vpt mul copy vpt mul mul mul vpt stroke def stroke vpt mul vpt mul vpt mul v stroke vpt add vpt vpt vpt vpt stroke copy translate stroke def stroke translate fill def circle arc stroke def stroke arc def bl vpt bl copy vpt fill vpt arc c bl copy vpt fill v stroke v v v stroke v v v v v v v v stroke v v v v v v v v v v v v v v stroke v v v v v stroke v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v stroke v v v v v stroke v v v v v v v v v v stroke v v v v v v stroke v v v v v v v v v v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v
duality path dataset duality randomize gap dataset small size difference compute duality hand counterpart progress furthermore exhibit sampling lead stop degradation measure reflect accuracy phenomenon typical fw iteration suggest technique problem prohibitive medium updating exploit well choice european european union paper reflect author union contain project g base grants medical ii office optimization macro lemma ia electrical engineering frank wolfe recently gain community application however linear strategy sampling researcher experimental alternative frank hereafter denote fw solve fw cm direction vertex community show fw variant procedure attain convergence fw problem arise application find subproblem often easy solve impractical handling motivating stem cost proportional fu random never systematically study effort fw try identify avoid fix motivate approximation suggest pick least small fw algorithms duality gap applicable without entire randomized possible simplification entail tradeoff consider context solve impact investigate iteration exploit keep fw naive update fw conduct dataset code execute gb ram average run obtain tolerance computation recently point pt acc acc acc ex acc size first dependent fw fairly appear case cutoff considerably trend monotonically expect monotonic
particular get bind tell special provide explicit value distinguish direct undirected undirected neighbor undirecte large adjacent independence bandit view expert always view reward recover structure hold turn bad computing acyclic independence np algorithm tractable fix entail independence compute independence unlikely approximate moreover either trivially lead regret similar bandit set less issue exp rely efficiently computable quantity turn graph direct namely choose motivate one whereas second click thm clique namely number partition result thm early rely clique key undirecte graph trivial bad problem regime bandit setting provide low characterize feedback system open mention interesting feedback would prevent direct construction unbiased unobserved loss upon nice provide sequence generate inform strategy achieve bind identify achievable information feedback action period affect observation delay affect leave analyze finally assume I result acknowledgment grant core sm support european community fp grant agreement science foundation united foundation technology center os science integration section theoretic lemma throughout shorthand exposition condition later take take remove q exp sum moreover together rearrange give take side q finally expectation remove claim follow direct induce graph acyclic prove initially vertex minimize along incoming incoming iterate neighborhood vertex step minimize node arcs graph cycle follow use armed bandit bind non bandit beyond establish probabilistic adversary condition whose round condition consider sequence namely equal whose loss e iterate I conclude next lemma relate dominate set operating graph cover algorithm see bind graph remove cover algorithm arcs theorem iterate thereby dominate eq lift graph arc let independence number dominate appropriately upper k discretization concern version unique single node shorthand I recall give continue hand make clique clique size arc draw arc lemma clique satisfie recall ready derive upper statement quantity occur therein occur occur see throughout appendix denote I first condition history expectation preliminary importance p recall direct understand neighborhood arc set acyclic subgraph eq hand dominate valid assignment return linear might left acyclic contrary include would create node direct acyclic subgraph relations q shorthand inequality assumption conclude distribution compute round lemma put establishes next martingale difference sequence also positive ti tp tt round item get lemma distribution run eq ti item expand item simplify exploit eq assumption lemma q back follow round round pick get solve union key use sum result series fix combine slightly rearrange simplify thing asymptotic notation notation logarithmic ignore factor assumption least I simplify substitute simplify pick plug thereby obtaining claim thm lemma size adjacent node include positive hold nonnegative allow take convention contrary since adjacent entire putting repeat guarantee independent respect original configuration expression split adjacent adjacent neither way explicitly function indeed weight adjacent node early immediate corollary possible clique definition universit di microsoft research university institute information repeatedly loss relate know moreover set loss choose player reveal address variant combinatorial abstract weather forecasting need devise forecast day well forecast goal time model adversary round assign action discrepancy forecast player choose randomization incur regret excess incur compare round associate consider choose ad ad forecast sequentially choose well fix ad ad know choose ad display abstract framework observe pick refer bandit refer player observe loss goal bridge setting create spectrum quantify expert available action play round good regret achieve perturb bandit inf variant achieve bad switching expert crucial action setting get single rather round square setting receive loss example web bandit assume whether ad action ad information ad package display ad ad unlikely click sort capture online social reveal interest product friend connect select drive consumption linear could arise type expert action also system arc action action play action round loss round expert obtain complete action playing reveal loss regret trivial describe set consider side undirecte asymmetric preference person person ad person game situation modeling handle case easy feedback system choose action know ad relate inform might player feedback choose party recommendation social around I send contribution lie providing summarize brief algorithm exp achieve acyclic set independence result factor fix round exp attain feedback exp direct inform regret inform another guarantee turn direct case exp find approximately dominate always gets recover expert observe loss action bandit standard bandit logarithmic bound scale lie depend graph log base framework unbiased key challenge design scheme exploration small loss variance feedback key quantity combinatorial recently factor without protocol exp upper base acyclic subgraph case handle inform algorithm section bound demand high conclude text question claim state adversarial pick incur unlike problem reveal subset action reveal observe loss bandit expert action write playing sometimes play role notation v call feedback arbitrary regret measure fix adversary would measure player loss action player depend past section variant system horizon round advance easily goal actual depend actual begin step informed set select adversary make learner prediction convenient adopt arc I e order loop ignore number notable feedback system word playing reveal playing reveal symmetric define undirected arc direct cycle symmetry feedback distinguish direct symmetric depend play dominate set connected edge proper independent still associate ignore arc direct bandit setting dominate property remain dominate proper dominate direct graph denote dominating example action loop reveal action light blue dominating dominate bottom light maximal orientation bottom acyclic subgraph action compute minimum dominating direct cover associate system cover greedy large also lift notion independence undirected graph acyclic subgraph acyclic cycle either arc length cycle graph see investigate set action current feedback algorithm exp logarithmic correspondingly regret exp regret exp ifelse action accord distribution set I similar exp exp importance sampling divide observe loss observe action recall bandit recover exp precisely recover exp quantity view condition event analysis irrespective suitable add inform building regret set expert hence bind large constant take form equivalent exp regret point hold undirecte follow note non return expand issue whether powerful unfortunately graph independence number acyclic confirm total order arc bound achievable regret undirecte adversarial strategy regret adversary standard bandit play action bandit bind case whether exist exhibit regret exp feedback stochastically via direct arc probability loop default exp expectation occur r arm bandit r regret theory see fact low theorem something refine way
synchronization evident eliminate cpu calculation update work throughout execution consider expect cpu essential bottleneck achieve speedup heuristic show previous empirical dimension process shrink shrink heuristic shrink multi core machine version shrink section limitation evaluation name description evaluation handwritten dimension box dimensional white convert represent digit odd digit census web page collection handwritten recognition united service description correspond represent conference net experiment cluster dual run ghz gb memory node gb mix result single multiple node evaluation connect modern cloud provide programming array dataset previous shrink evaluate lin et elimination line shrink proceed shrink similarly read shrink half call proceed shrink throughout shrink name none single pc pc random multi multi multi pc pc default htbp name acc acc speedup x training algorithm branch selection active problem part decompose quadratic approach include optimization decomposition seminal svm sequential al primary method separable simplicity implementation choice solve classification problem multi system cluster brief overview setup al cascade parallel primary divide combine vector load imbalance finish individual sub target architecture research conduct create paradigm primary approach paper large restrict projection method solver qp leverage svm consider solve problem incomplete parallelization use active decompose svms approach elimination sample system address limitation parallel support machine shrink intuitive shrink art programming pass interface array design storage efficacy algorithm work involve shrink heuristic evaluation heuristic consider work elimination study shrink architecture elimination kernel cache cache intel fusion machine wide range domain finance identify care student college role social challenge designing scale core machine cloud improve adaptive elimination elimination case might reconstruction structure heuristic publicly available improvement improvement time baseline produce grow dramatically machine mining algorithms extraction volume domain finance social rely algorithm supervise learn supervise learn due excellent broadly space category use surface svm excellent core machine svm extremely albeit limitation parallel extend previously optimization however good entire special calculation though vector contribute calculation shrinking eliminate shrink core system literature address limitation utilize theoretical framework shrink speed format observation world sparse effect heuristic elimination stage execution approach art programming message interface array design communication storage scale cloud algorithms processor make analysis time category elimination efficient compressed kernel cache make attractive scale dataset multi large efficacy speedup execution parallel shrink organize work maximal separate hyperplane formulate generality slack allow possibly dimensional space convex introduce multiplier lagrangian wolfe eq primal maximization tucker kkt treatment svms refer contribute separate hyperplane remove small sample package reduce datum maintain essential relationship show objective maintain optimize explain section equation working selection selection step derivative refer address possibility second evaluate two loop avoid loop index nature eq threshold condition termination algorithm numerical user specify sample optimize bind sample shrink mechanism svm hyperplane eliminate decision heuristic eliminate belong one paper programming array global array model provide array load store semantic distribute array side global array programming domain sub array useful store easy array design asynchronous read array array use communication work tolerance avg row array begin presentation training organization introduce parallel algorithm present follow shrink pdf shrink prototype design shrink iteration calculate update require need structure lagrange multiplier early computation formulate cache several avoid cache prohibitive target cache low temporal pattern trend exhibit memory unit core unit intel graphic compute hardware support wide multiply add movement individual row across hence approach paper avoid cache organization structure critical dataset dataset less compressed sparse algorithmic relate reduction core several calculation datum among read organization structure significant improve cache rate leverage locality read write design choice job co load balancing process feasible contiguous movement approach semantic compress row ga semantics semantic system cloud algorithm show inner execute hardware representation inner line algorithm simple algebra trick shrink shrink present reasoning shrink must shrink parallel variant also section primitive operation process independently new integer loop expensive calculation require several locally operation update update
interval dark circle dash line reward although notice considerable variation reward mean significant slight difference game attain determined examine percentile subsampling subsample exhibit proportion run attain proportion exception skewness go see vary considerably game generator difficult determine generator find also normalize generator generator attain algorithm across generator percentile sample mean generator generator generator good generator frequently generator h lp algorithm good despite generators portfolio fail algorithm would interesting run experiment determine one generator game large possibility complicate slow decrease game generator generator algorithm reward intuition many generator reject reward lower exist negative negative negative kind generator tie reward make desirable game show reject support large game bad coefficient positive sensitive anomalous consider algorithm reward opponent play make star achieve opponent broad reward play profile appear performance issue portfolio avoid know opponent construct opponent percentile call opponent response assign overlap percentile claim apparent frequently response never p interpret single shot correspond algorithm mean attain algorithm mean always response confident get bootstrappe check subsampling game check algorithm dominate dominate proportion game show strict weak pure equilibria equilibria ever occur play occur involve restrict generator generator nash equilibria self nash equilibrium seven generator pure equilibrium game play symmetric strategy equilibrium arise equilibrium pure nash equilibrium pure strategy nash equilibria equilibria yield e yield reward game exactly weak nash equilibrium attain reward pure nash game dominate indicate nash equilibria asymmetric equilibria twice dominate never play performance opponent dominate dominate playing play trend avoid dominate play interesting specific observed ambiguity opponent tendency self play reward self play play run dominate self run self self play significantly low play percentile interval self run play self play despite occur whether play equilibrium offer nash equilibrium keep reader answer equilibrium convergence similarity achieve major likewise follow latter addition also opponent regret dominate particularly generator avoid generator connection expect reward largely support reward link generator induce make outcome reward reward regret run dominate positive occur fashion attain reward dramatically run algorithm nine phenomenon arise single generator opponent omit discuss base action frequency strategy interest right necessary strong record stability always strategie successful criterion detect criterion stable match stable self match play difference tend small resemble particularly self instability stability test change stability match step indicate produce good stationarity always profile occur discrete behavioral state generator different generators difficult rare vast find stationary run game equilibrium nash equilibrium convergence converge run pareto optimal nash pareto nash frequently dominate equilibrium likely pick left equilibrium equilibrium find whether converge pareto dominate non algorithm converge nash play converge nash surprisingly game nash look qualitatively generator likely produce equilibrium optimal ne converge receive play self failure high equilibrium use play equilibria equilibrium improve play maintain exploitation aim stage equilibrium goal convergence generally correlate obtain high proximity game generator correlate close equilibrium furthermore generator notable especially reward converge equilibria algorithm play repeat repeat payoff achievable repeat game profile determine profile game necessary agent meaningful payoff profile repeat nash equilibria build achieve payoff game nash examine consistent game equilibrium overall repeat worth profile agent agent like standardized researcher experimental implementation conclusion idea modern would repeat game environment considerably suggest area effort experimentally drive many include performance game size detailed investigation generator algorithm game line sophisticated differently partly explain set setting match important nash equilibrium equilibrium equilibrium also third dominate algorithm portfolio exist promising empirical see portfolio switch different opponent portfolio algorithm portfolio empirical play track situation portfolio improve acknowledgement thank stage project feedback anonymous helpful suggestion david coding provide appendix metric seed uniquely instance could either match play quantile avoid formally sample yield sample yield sample estimate sampling equation claim point albeit pt exist many justify term guarantee little empirical claim literature experiment new tool design facilitate remove baseline implementation many test equilibrium confirm piece conventional discover surprising agent outperform algorithms road system system see system analyze algorithm environment prominent example well high approach algorithm qualitatively reinforcement learn fundamental difference classical reinforcement learner policy attempt identify environment policy opponent way opponent action opponent conceptual hard claim algorithm tend aspect intend stand game nash equilibria case self regret performance basis property ability achieve generally compare comparison literature g newly design survey landscape consequence small consider make overall centralized standardized exist centralized public decrease achieve difference implementation publicly implementation offer easy reproduce platform run platform several advantage hope facilitate analysis offer connection explore sophisticated extremely competitive performance rich investigation compete distinct know reward reward opponent action nash computationally expensive game survey experimental evaluation describe broad disagreement aim order access reward opponent costly game property able compare capable upon choose player repeat restrict instead game reason game experimentally interesting play two thus mention aside generalization probably game repeat essentially potentially mixed opponent collect count opponent iteration need response assume payoff converge equilibrium game property see eventually extend way see provide known asymmetric game randomization remain problem solve adapt signal name simply adapt strategy game nash equilibria maximize play idea play variant instead action payoff nash equilibrium move simultaneously move instance action equilibria lead equilibrium every reward costly game useful like modern focus class opponent attention updating behavior assessment track opponent try play nash implementation equilibrium issue act fashion depend simple strategy style play nash history opponent action period stationary agent exceed security level situation portfolio algorithms style mdps opponent discount encode step possibly decay strategy straightforward adaptation single agent opponent payoff eq essentially opponent part stationary entirely work environment explicitly idea function learn profile mix calculate eq q sensible aim payoff worst note game actual fail reflect security modification play game nash something similar except choose equilibrium use observe action reward necessary ascent maintain mixed payoff updating depend high action opponent environment gradient learner make strategy compare strategy strategy make perform guarantee regret limit great nash action version opponent use proof convergence experiment merely assume reward gradient start adaptive payoff mean unlike update current nash equilibrium therefore complete computational power one nash equilibrium additionally self class game way ensure unconstrained probability map simplex reduce less aim primarily use scale performance algorithm survey repeat consider include mixed simulation see table additionally test play varied version version great baseline seven investigate come small game version limitation partly create diverse instance game generator generate set instance recent paper finally substantially iteration range repeat burn period phase algorithm behavior record final generally code tailor consequence instance special new experiment spend run call available platform player repeat game experiment include version introduce version programming matlab variety gain overall hope researcher repository setting upon game stochastic work hard make add list engine term order pair two asymmetric payoff player payoff column pair concentrate generator generator instance payoff order generator heterogeneous example distinct call play performance due randomization hold pair characterize algorithm play different case characterize empirical run iteration feedback reward action choice opponent mixed strategy separate platform step platform three configuration engine piece visualization turn algorithm must pick must step generate job desire match reference game file file job generate primarily engine implementation nash equilibria job cluster facilitate job create sample action reward receive metric file plain file specify calculate metric batch across cluster analyze visualize result make easy validation implementation action action singleton well otherwise break action still belief unit virtual count ne break high opponent see repeatedly nash high equilibria opponent equilibrium tie one pseudo code largely use game encounter nash game need decide would play equilibrium compare computationally expensive pick equilibrium implementation implementation implementation involve ten instance agent show quality use kolmogorov game dominate implementation e reward track however considerable implementation code unable difference period tt security stationarity threshold switch window implement pseudo distance even norm follow pseudo table step set draw original iteration simulate compare map operation distance variant assume play arbitrary know reward implement experimental produce formula eq equation play vector reward table suggest resemble update reward vector sample control parameter give show use poorly setting schedule discount keeps perform probability decaying drop period decay end discount arbitrarily future action take solve linear strategy see removal preprocesse check exploration discount baseline uniformly action platform conduct scale performance metric setup statistical algorithm setting nevertheless
dominate median c sec sec sec run appear decrease linearly ex objective plot residual sparse bilinear real world bilinear sparse bilinear multi test performance bilinear eeg competition concern eeg record arm presentation hand hz mark choose temporal slice train tune tune select grid prediction regression logistic regression sparse bilinear show accuracy roc roc tune regression logistic solve logistic fista observe logistic bilinear well bilinear logistic logistic logistic bilinear regression class one eeg cognitive subject three category try decision record hz use channel hz table consistently three bilinear outperform regression bilinear logistic far improve benefit bilinear content various camera image camera record plane image camera logistic bilinear logistic bilinear logistic regression comparison sparse bilinear bilinear sparse bilinear achieve bilinear video visual video dimensionality histogram descriptor figure illustrate vocabulary codebook sift descriptor frames video construct histogram codebook technique performance side side front pick discrimination ex logistic bilinear regression sparse bilinear bilinear regression achieve good bilinear algorithm method reveal convergence bilinear regression dimensionality traditionally curse component ica commonly preprocesse bilinear reduction logistic logistic regression ambiguity bilinear lead feature spatial importantly complexity improvement challenge minimax bi nature bilinear introduce bi boundary still bilinear follow factor form write interpretation bilinear bilinear equivalent due critical bilinear spatial difficulty pose bi convexity bilinear extremely guarantee empirically bilinear boost generalization binomial bilinear logistic generalize multinomial hyperplane model py n ic minimize set sample form multinomial bilinear take eq r corollary section xu department electrical introduce concept logistic explanatory common vision brain computer style factor study coordinate descent theoretical inequality sparse logistic history vision bioinformatics gene sparsity introduce logistic curse dimensionality explanatory correspond informative therefore lead logistic attractive logarithmic complexity bound recognition task feature transform histogram result histogram base eeg channel bilinear explanatory two generalization standard logistic learn bilinear learn boundary show logistic outperform logistic include visual apply bilinear outperform dimensionality principle pca bilinear logistic regression content separation improve nuisance orientation bilinear identify informative feature nuisance thus generalization bilinear contribution lead interpretability result sparsity bilinear demonstrate classification three fold first propose bilinear behind logistic descent solve numerical bi nature conventional block coordinate solve subproblem proximal provide estimate rate bilinear logistic improve generalization classifier various task convexity bilinear consider give label explanatory categorical variable seek logistic transform explanatory pp x category binomial illustrate idea h assume class conditional empirical function logistic decision assume laplacian reduced minimization problem regularization form logistic regression multinomial regularize development efficient lasso ce lars fista bilinear bilinear regression preserve explanatory construct block coordinate follow subproblem coordinate summarize choose accelerate proximal stepsize specify independent subproblem b proximal descent summarize solve reduce subproblem subproblem convex solve dimension large elastic term close component computationally algorithm attain sufficient typically choose b continuous dependent gradient constant straightforward calculation b n f eq cauchy schwarz therefore dynamically integer let integer inequality constant inequality update allow namely small previous define global bilinear logistic regression asymptotic establishe method update iteration result convergence follow accord bound least solution subsequence stationary give observe subsequence pass say satisfy limit subdifferential f converge boundedness intermediate assume
multidimensional reliable estimation shape check effect hour slice explain appendix period day surprisingly law illustration display thing conditional law behave long flow law roughly behave exponent law kind kind time believe conditional kind appear reaction event law wiener second comment intensity intensity interpret source represent dimensional simply h c c intensity ratio low percent directly event theoretical process one mid price jump event ratio provide mid one involve external market limit order surprising show thus limit order market order order mid price occurrence asset discuss precise kernel comment self book simplicity blue correspond interpretation quantity positive linear lead reliable estimation even bid recover empirically matrix shape homogeneous input input stock index expect discrepancy side jump slightly jump market negligible anti shape plot order attribute less diagonal kernel roughly law decrease mid jump behaviour price guarantee absence correlation price agreement link change influence price change dynamic else come impact price impact recall impact event type wants cascade whose precisely introduce proposition correspond event number display ij price mainly cause jump indicate price mid asset price move mainly move price asset variation event former study estimation involve order flow link cause splitting order description focus kernel market mid high display kernel appear behave exponent high kernel loose fit range though display stress behavior kernel rich law fact clarity kernel plot represent normalization dynamic process indeed proportion flow move mid price appendix one localize delay ask price move market average delay law scale negligible order appear localized price move opposite bid conversely move order change order bid oppose order order market order almost order significant order impact price limit order kernel impact jump flow correspond normalize mid order ask contrary effect far bid ask selection price want execute order ask bid interesting execute bid move move bid limit ask ask order bid order localize average long short second stationary state time price occur kernel correspond impact normalize influence order price kernel negative stationary far order time influence become proportion see impact proportion concern reverse impact limit flow flow kernel resp bid bid order trading rate resp bid flow correspond display kernel market dominant also short influence limit localize order law correspond influence order numerical wiener kernel behave significant rather dimension estimation natural interpretation financial micro allow type first occur book retrieve event impact influence allow rich influence type subtle event example limit order localize time thing probably purely event directly book spread arrive side spectrum book mention markovian flow believe reality approach market book stability increase improve different account finance field measure acknowledgement support market finance growth laboratory universit ed aim empirical choose grid l affine grid one good estimation scale one law scale grid point represent result h b b proposition remark modify simulation least decade trade limit relationship mid process estimation order impact price formation challenge modern financial agent stock price execute limit use limit order execute formation book market large anonymous great practical theoretical book intelligence purely book markovian book simple asset price impact market order recent account book work share joint price influence mutually adapt count example successfully domain finance arrival order variation level impact account price variation book causality concern finance lead law decrease exponent slightly wide range scale capture book improvement illustrate us kernel scale recall property multidimensional explain principle adapt slowly book correspond front comment event concern work conclude process briefly intensity jumps intensity matrix simply stability kernel one admit increment remark stability remain equation lipschitz interesting satisfied describe statistic satisfy covariance intensity dirac average q notation time see e build arrival birth appear individual type individual type type appear proportion since jump process also law measure link proposition wiener law equation wiener structure autoregressive see counter equation wiener admit unique respectively average covariance use wiener matrix definite theorem set satisfie thus thus try wiener course solution necessarily interpretation point wiener system predictor square contrast proxy explain numerically wiener nystr method realization fine enough grid time interpolation quadrature wiener quadrature get system appendix estimate use display display theoretical perform kernel section quadrature follow thus three empirical kernel quadrature theoretical green dot kernel instead error causality direct interpretation average cause try give intuitive method quadrature thank change quadrature quadrature captures vary quickly bad around previous quadrature quadrature capture let time grid adapt quadrature scheme consist piecewise hypothesis k k k kt quadrature empirical integral linear law quadrature procedure perfectly quadrature point estimation quadrature step curve correspond blue theoretical kernel quadrature perfectly match theoretical green dot blue describe wiener quadrature obvious inverse point use empirically model account event various type arrival future event classical impact g accounting impact market dynamic price market extend multidimensional event
fluctuation motivate distributional conceptually suggest additive shape way modify relatively consider version model state high decrease burden associate smoothing consideration product state incorporate model gray series regression covariate control building b spline additive predictor common generalize model feasibility approach demonstrate simulation exchange spline model instability pattern employ vary finitely control markov refer markov regression seminal paper state residual chain state stochastic serial persistent regime active long regime classic economic effect explanatory economic simple glm exist relationship form little investigation goodness predictor build strength hmm evaluation spline obtain arbitrarily flexible functional estimator target penalize generalized validation select control goodness smoothness model include parameter comprise possibility reduce case conventional parametric switching nest flexible consideration switch additive ms simply time parametric subject regime restriction decide present consideration identity link density dependent structured formulate describe efficiently spline functional predictor approach potential conclude switch nonparametric target interest value denote chain finally distribution exponential covariate link canonical family link via map state essentially model use shorthand additional depending specifie whereas distribute specify dispersion parameter conditional probably popular dependent model assume homogeneity desire probability usually stationary switching underlie chain dependence covariate efficient importantly irrespective efficient forward variable generic symbol recursive derive analogously comprise numerical large moderate notably density switch regression g predictor glm concern predictor comprise function one state express finite combination represent simple numerically piecewise fuse together smoothly cubic use spline determine flexibility basis function allow curvature adjacent spline need increase basis long impact penalty second integrate ms characterize function dependent reflect smoothness parameter increase emphasis smoothness parameter dominate lead straight line similarly difference nest directly advantageous functional way allow obtain constrain drive choice model observation clearly powerful software already give determine markov chain spline predictor estimate maximize calibration remain constitute calibrate calibration treat datum forward subsequently validation assess calibrate convenience fit stage treat calibration pre meaningful experience computationally alternative intensive aic calculate aic likelihood denote freedom product information inverse fisher penalize freedom result penalization smoothing consider one distribute target markov regime covariate functional display dash curve function go covariate run ms link fit optimizer spline implement cross choose smoothing fold validation integrate q estimate report aic f use dash green state transition obtain monte shift fairly value target variable chain display figure go draw run choice lead marginally counterpart aic criterion straight notably parameter value case curvature smoothing scenario dash dependent predictor leave panel predictor monte shift estimate encouraging lead overall scenario encourage ms clearly occur form circumstance identification induce run exact change autocorrelation series modify fairly autocorrelation bad run fail overall wrong chain reflect mean probability sample value use scenario dash red datum collect exchange financial make assumption relationship two also probable motivating nonparametric predictor function illustrate flexible ms analyze predictor lin ms lin nonparametric meet restriction mean htb lin pass region around residual switch formally ahead fit lin ms lin ms forecasts ms ms lin result
variability behaviour epidemic frequentist infection epidemic model individual kernel via removal infection period meanwhile infect epidemic parametric survival contact become infection removal removal model dependent author adopt find homogeneous removal rate affect removal contact focus infection enable assume removal within involve assign wherein use chain monte epidemic population three epidemic time epidemic bivariate transition infection removal transition sigma process population process become remove period play epidemic end population period infection removal assume removal epidemic end infection individual denote infection time density proportional left limit purpose least epidemic infection removal rate temporal disease consist removal time think infection integral high dimensionality trivial nature overcome miss become intractable choice augmentation infection inference infection markov monte relax parametric force infection proportional knowledge augment priori time infection removal exponential remain infection discuss formulate form total pressure course epidemic trivial compute priori poisson condition distribute prior alternative priori follow nd continuous knot I ht b spline order function recursively assume interior even need sufficient condition fix infection epidemic assume cover full spline coefficient much purpose unobserved infection carlo conditional n reversible jump hasting use assume make three update birth death change poisson rate maximum interval new ratio jacobian death probability inversion probability propose birth death take repeat improve mix move uniformly accepted q change height height distribute q priori previous mechanism update acceptance birth death change height acceptance prior j death probability change acceptance height eq involve infection initial infection infection infection propose exponential infection infection value infection accept update infected infection minor assign nd interior knot infection infection time propose infection infection time accept illustrate propose dataset generate mass epidemic start among uninformative martingale prior induce case around may explain infection slightly notice close infection show instead set offer spline dash line percentile infection curve dash percentile posterior solid infection curve modify vary contact rate epidemic start population number fit parametric dataset informative although spline work obviously action infection percentile posterior infection spline hour hour table typical fit dataset removal day mass action several parametric infection initial infection mass run quickly infection rate simple fit spline severe consist population track number epidemic sufficiently remain epidemic martingale fig day removal spread similar length period day dataset want death removal individual period infect tendency uncertainty begin epidemic spike spread super event infect patient cluster explicitly could indicate curve infection per flat fit infection curve peak middle infection may power large modelling could intervention intervention spread epidemic epidemic assumption instance incorporate place essentially
conditional hide visible parameter visible small conditional feedforward joint top feedforward feedforward network intend feedforward feedforward layer hence feedforward independently apply nonetheless resolve able marginal bottom compound feedforward pass represent namely marginal feedforward feedforward resolve marginal order feedforward transformation way bottom marginal desire tune depend bottom require arbitrarily bottom account fraction distribution regardless second approximate element strictly arbitrarily approximate distribution ss prove subsection study feedforward put proposition arrive approximate arbitrarily bottom marginal arbitrarily layer interesting feedforward define feedforward approximate probability visible unit arbitrarily feedforward ns approximated obtain mean rbms call adjacent every bottom marginal pair feedforward layer previous paper well tool flip state invert approximate intersect along p np n describe length string entry form existence condition argument conclude material paper deep feedforward develop advantage vs undirected architecture seem undirected train initialize universal narrow thereby intuition verification surprisingly long address rbms narrow feedforward counterpart narrow compositional present trick activity part regard feedforward network pass high multiplication feedforward layer acknowledgment grateful institute article th l l k k k l z p kp k l p hold versa map output assign unit take state unit divide visible input output give relation implication note universal map deterministic regard map universal map rbms discuss input joint distribution would interesting corollary softmax main article hold unit feedforward layer value formulate case layer state minimal universal narrow bottleneck narrow visible hide fact odd follow visible conditional approximate mixture mixture assign string one odd even without detail imply odd conditional string odd even visible narrow direct towards bottom except layer state layer although layer narrow stem essentially feedforward layer hide exactly kind exploit rbms rbm rbms provide minimal hide rbm visible narrow minimal sufficient form rbms narrow interaction weight restriction interaction weight backward activity product arise input model pass feedforward desirable obtain universal detail help understand take close look future pt pt pt bp pt pt pt pt bp pt pt pt pt arc bottom double white mark width pt theorem theorem institute mathematics sciences deep narrow boltzmann machines universal many visible layer within boltzmann feedforward depth various undirected show narrow boltzmann compact narrow sigmoid restrict currently available power neural compare network compare undirected direct connection respect represent reach endowed unit refer universal property various feedforward feedforward deep narrow undirected architecture problem prove narrow boltzmann universal layer visible machine undirecte boltzmann boltzmann machine whose pair interact layer visible unit conditionally illustrate appearance practical especially regard node line black sep fill bl dot cm right dot node transform circle draw black cm fill bl dot dot scale shape draw sep bl dots node distance cm dot transform shape circle line black inner bl dots distance dot h v cm distance right h node l transform shape transform circle pt black sep cm bl dot right dot scale circle black fill bl dot dot line width bl distance dot pt black inner sep cm fill bl dot distance dot right node distance h distance right h right h leave left style shape transform pt version rbms exponentially organize fix narrow result section compositional probability feedforward share section elaborate study perspective present trick distribution follow feedforward universal boltzmann probability l l l unit exponential embed strictly assign bottom layer visible unit right panel figure restrict boltzmann visible set distribution top panel provide rbms universal distribution kullback visible unit enough precisely implication property remain unit input output visible minus layer universal unit universal unit unit softmax unit unit visible next compositional feedforward look compositional composition wise hadamard hadamard distribution definition hadamard product element natural fs gr g style circle bl scale circle width minimum size fill bl dot right end dot draw black sep size cm bl cm v
vector latter plane meet plane intersect must q happen consideration couple interval coupling conditional simulate exact determine certain time step associate calculation section time interval method distribution bridge association compare copula mu matrix complex invariant matrix reversible symmetric obtain symmetric reversible covariance matrix invariant summarize ergodic time distribute bridge expectation argument give simulated process diffusion euler scheme satisfy first diffusion approximate distribution compare mean bridge level copula two distribution copula distribution two simulation bridge since bridge fit exact mcmc distribution approximate bridge therefore nice diffusion diffusion unlikely bridge rarely rarely likelihood approximate copula lemma compare marginal distribution simulate marginal level copula compare copula curve plot compare empirical copula copula draw q marginal dimensional approximate exact marginal curve copula exact draw exactly distribution bridge bridge fit marginal metropolis hasting essentially example compare copula time produce metropolis run compute second generally compute generating metropolis varied dimensional hasting exact curve copula dimensional distribution compare exact compare copula time alternative rejection produce large output algorithm plot empirical marginal produce alternative level copula full diffusion bridge diffusion compare exact time approximate copula dimensional compare extreme bridge bridge produce surprisingly bridge fit excellent tend become tend mathematically generality marginal level copula dimensional full draw restrict simulation bridge simulation give exact whether value burn bridge expectation geometric conditionally bridge variance constant reciprocal simulated slope check conclusion variance ratio deviation vary unlikely bridge bridge ratio simulation diffusion close however show g bridge eq contribution bring close contribution therefore approximate start deterministic unlikely bring depend likely constant bridge mainly contribution depend end bridge good bridge randomly draw multivariate characteristic multivariate third diffusion ergodic reversible suppose set partial observation full path draw need simulate continuous path conditionally diffusion path likelihood expression integral apply detail q exponential family distribution normal bridge sample two run sampler sample method interval posterior obtain draw introduce wiener wiener pair span obviously function clearly dimensional wiener process projection orthogonal complement wiener eq wiener independent wiener characteristic v quadratic characteristic I td give event event bp tb transition bridge establish calculation conditional trajectory joint density wiener wiener dominate marginalization lemma straightforwardly multivariate give start check z bt nz nx formula g apply mu grant national foundation two grant university cm lemma theorem corollary remark de sigma mathematical university mathematical bridge fundamental role inference novel coupling generalize variate set first accurate propose proposal exact diffusion applicable work length simulation usefulness multivariate inference coupling inference stochastic address mail propose applicable multi diffusion bridge motivation play fundamental role simulation include inference diffusion diffusion volatility end state start start time go process equal start diffusion meet suitably make often tend infinity obtain apply couple generalization two independent ergodic dimensional intersect go application coupling efficiency meet diffusion bridge repeatedly two euler implement diffusion process bridge algorithm diffusion pseudo rejection bridge simulate new bridge diffusion bridge exposition thought sampler try rejection acceptable rather meet bridge literature metropolis hasting proposal distribution force go diffusion brownian boundedness relatively complex method simulate path spirit advantage ergodic understand transform equal transformation refer transformation multi variate exists rarely require exact important advantage particularly length time interval bridge surprising order time interval coupling process bridge apart bridge publish work interval note mainly short follow challenge approximation estimator time long accurate density approximation kolmogorov pde numerically expansion alternatively simulation approach back seminal incomplete datum continuously process observe continuous miss either em sampler continuous path simultaneously realize several base bridge simulation several author bridge crucial diffusion time observation simulation inference possibly functional cover volatility crucial observed measurement idea bridge process simulation approximate approximate proposal diffusion improve solve point meet two point know approximation distribution diffusion except extremely surprisingly couple usefulness observe consider briefly diffusion stochastic wiener coefficient function dd regular ensure strong solution ergodic invariant measure invertible specifically conditional call bridge bridge go start start intersect meet bridge equal approximate bridge proposal bridge couple sense diffusion equal subsection bridge question correspond initial u equal distribution bridge process variable wiener play bridge depend depend start let initial base theorem simulate simulate diffusion reversible simplify euler increment drive increment wiener discretize simulated method wiener process simulate approximation bridge rejection keep copy couple whether coupling coupling interval define probability detect usual considered lemma hence reversible follow time reversible diagonal simulation usually diffusion proposal exact bridge bridge diffusion bridge continuous induce probability bridge dominate bridge differential e give drift bridge end one carefully must correspond similarly bridge diffusion bridge diffusion wiener intersect definition approximate bridge corollary give bridge b random wiener associate bridge associate equation give expression quality diffusion bridge distribution bridge crucial detail usually time simulate time euler discretize z increment simulation equal wiener increment process meet equal hasting marginal bridge bridge proposal accept rx diffusion mh produce diffusion simulated type idea replace ratio estimate chain bridge draw irrespective randomness simulate index diffusion result draw mh go diffusion bridge conditionally xx path simulate independently simulate conditionally value rx I exact
fast perform slightly detector outline subsection use integral detector slide improve adopt classifier arrange adopt approach detector pass gradient speed classical slide object paradigm discard patch propose selective order magnitude detector patch underlie idea detector generic detection modification original detector resolution aspect bit integer integer use training detector aspect scan per sp consist level statistic exclude co discriminative exclude coefficient co channel color carry parameter weak classifier shrinkage adaboost final bootstrapping final cascade soft cascade rejection node detector train table beneficial pooling increase robustness percent sp bad sp bt sp sp pooling sp original descriptor eigen learner propose feature train learner learner highly compare original descriptor report sp significantly detection original covariance window sp fair increase combine result sp table sp test bad par set combine detector sp sp sp display illustrate ensemble radius angle angle draw uniform asymmetric testing baseline cs adaboost asymmetric adaboost assign asymmetric restrict high partial good asymmetric vertical horizontal train strong classifier evaluate decision asymmetric observe emphasis part cs worse optimize bt marked classifier l svm protein bioinformatic consider protein protein prediction predict interact use type publicly protein interact protein label interact evaluation form weak learner baseline svm asymmetric optimize either attribute linearity detection report tb deviation datum set iteration repeat time average ccc face comparison boost adaboost lda post adaboost train tree algorithm repeat report cross approach choose evaluate digits pixel even digit odd divide scene set descriptor visual histogram intersection manner sub windows face face extract principle component preserve table us mt mt motion indicate head pixel weight learn svm fig b near contour shape detection vision capture camera mid city graphic format fully along axis patch collect care detector resolution feature channel feature orientation sp propose learner depth decision cross validate divide validation bootstrapping around detection greedy maxima apply website curve evaluate usa usa drive traffic pixel scale include medium far start training exclude along expand negative detector resolution magnitude bins sp sp depth tree bootstrappe weak previous experiment pixel height visible publicly compute auc experimental compare approach exist detector spatial visual feature pixel head apply intel detector generation place detector set svm show pixel white human contour weight detector vary detector stage vary region plot roc detector fig start reduce similar performance demonstrate detector discard compare region proposal stage exclude processing maximum stage usa table threshold result reduce improvement log average value classify fail soft soft low level visual experimental keep number classifier bootstrapping etc rejection cascade various threshold classifier perform bad detector benchmark window table cascade weak classifier coefficient repeatedly rejection threshold increase window time cascade small important come achieve average scan soft average scan scan bt c window avg avg discard per bt cascade avg avg scan approach effectiveness level object proposal generation optimize extensive demonstrate plan scale order acknowledgement centre part fellowship analysis algorithm learner weak learner primal value solution unchanged current another add master solve objective proposition guarantee subscript index project positive j objective reformulate iteration know iteration continue definition simplify objective bound calculated ensemble easily apply computer master interest pattern recognition machine research interest computer study university receive future fellowship van university innovation production medium receive law science vision fig eps fig corollary many object detection operate prescribe situation detector basis area roc full label partial curve method positive directly optimize partial auc object detection low spatial pooling spatial robustness detection structure spatially pool report usa boost ensemble pooling gain great attention past decade topic vision visible give gain set progress decade area due computer surveillance interaction difficult appearance author evaluate detector boost apply promise recent success pooling feature type train commonly evaluation compare operate roc illustrate classifier system threshold human researcher false area characterize need world vision fig positive would preferable moderate positive researcher report partial area range name calculate roc curve specify summarize detector often optimize approach ensemble area range area calculate accord upon propose ensemble classifier directly auc boost algorithm predictor ensemble mechanism pass unlike place emphasis incorrect ordering optimize partial auc summarize novel extract low spatial pooling spatial code generic classification descriptor optimize curve method wide roc proposed term tight approach share conventional differs method optimize multivariate structure conventional boost visual transform efficient training cut plane solver auc directly optimize partial auc arbitrary good detector experimental set effectiveness propose new benchmark work cascade optimize rate visual spatial single structured tight proposal generation evaluation detector careful compare report detector several detector part convnet mutual propose newly benchmark reader excellent framework section briefly cover recent work consider pooling vision system state performance benchmark scene imagenet method extract visual summary patch window feature sub pool form generally spatial recognition convolutional achieves scale max layer form invariance spatial matching sift significantly outperform linear pyramid max statistic pooling mean value process selective image descriptor boost recognition dictionary know computer vision machine miss negative exploit weight heavily report classifier validate parameter order cost sensitive boost boost cost descent address need carefully false positive several directly optimize bioinformatics modeling boost develop optimize exist building ensemble optimize criterion range knowledge principled optimize auc difference structural ensemble recently detector detector instead auto automatically hierarchy multi scale capture detail shape component result detector benchmark level aspect author low processing image major benchmark improve performance detect align frame camera motion feature five fold reduction false positive well apply object work capture favor cast cutting assumption quite form histogram texture descriptor descriptor texture adopt filter binary contain transition vice versa pool descriptor pool common strategy use window patch feature pixel within translation spatial pooling pooling covariance pooling pooling summarize matrix pool region refer extract spatially pool extract compute efficiently trick sp descriptor pool ignore geometry matrix stack upper pooling simplicity carry image rectangular likely normalizing descriptor patch whole detection coefficient correlation coefficient return gpu extract sized pooling implementation sp scale patch multi enable capture human body part patch pooling pixel experiment pool sp divide window patch extract histogram frequency occur well translation perform spatial histogram pooling region spatially pool sp feature implementation neighbourhood pixel extract histogram sp patch pixel although pooling differ unsupervised problem instead encode pre train sp pooling extract remove encoding advantage conventional much word generic visual word thus classifier computationally infeasible structured learning review concept ensemble build auc auc equivalently minimize auc false write train sort negative sort negative obtain j sample empirical compute rank prescribe zero adopt consider pair label j q pair instance rank consistent define ordering produce optimize score summarize ensemble project learner boost optimize false rate vector learner assume train j include new write dual variable lagrange generation ensemble condition apply column generation duality kkt optimality restrict work globally hence subproblem weak learner
mode ex acc mode current scheme simple superior cm lemma prop thm corollary definition section conjecture thm electrical computer california false cluster centroid mean shift ideally centroid pattern representative datum nonconvex algorithm mode combine assignment estimation centroid estimate shift encourage assignment nearby separate centroid find estimate challenge manifold nonconvex centroid representative unlike predict soft assignment representative assign besides meaningful centroid valid input space representative challenge nonconvex digit represent nonconvex pixel digit image digit mode bandwidth require mode valid digit shift centroid regard centroid valid minimize non euclidean typically slow besides often noisy c remarkably binary assignment high representative cluster kernel centroid representative nice outlier disadvantage use centroid find handle nonconvex shape manifold unlike shift nice property idea modify rule become much give alternate mode nonparametric like shaped cluster shift spectral give datum achieve work minimize assignment bandwidth gaussian define kernel estimate kde apply iteration start datum mode initial converge mean round low image segmentation kde centroid create singleton mode computationally mean shift per desirable force one mean mode sum kde separately alternate fix l discrete constraint fix optimization maximization z nk proportional kde truly outer mainly objective leave unchanged local loop step obtain optimize indicator interesting objective difficulty approximate mean partition process multiple optima laplacian qp also combination nonnegative nmf basis coefficient produce part term regard objective coefficient directly cluster directly optimize optimize rescale however multiple solution relate rotation straightforward procedure approximate data cluster walk divergence affinity matrix point move augment stochastic laplacian mode laplacian mode obtain issue equivalence onto simplex mode handle complex shape idea cluster g near heat add nk nn nk trade sum assign nm mn kb nk graph laplacian constraint assign assignment mode control kde note case become mean point code first force assignment purpose intermediate call alternate take concerned second therefore identical cluster mean solve separately step laplacian semidefinite problem qp standard qp provide algorithm lipschitz gradient proximal first project counter nesterov identification smooth objective indicator simplex onto program fortunately efficient initial ks mm accelerate projection laplacian power stepsize determine right laplacian acceleration maintain auxiliary improved iteration neighborhood construct number nonzero account account project onto simplex qp per independent step despite sublinear clear costly implement kde leave unchanged step accelerate gradient counter well step convergence threshold use iterative number precision cost empirically moderate efficient inexact run algorithm valid assignment laplacian objective optima nonlinear term homotopy homotopy follow laplacian mode homotopy optimum scenario algorithm hyperparameter automatically number cluster often uniquely mode intuitive hyperparameter membership smoothness value neighbor kde near well kde bandwidth neighbor hyperparameter homotopy slowly good minimum geometrically computationally supervised section problem efficient solve avoid drop follow quadratic program follow q projection onto dominate cost since laplacian average neighboring training assignment nonconvex kde term centroid distinct rule give assign nearby point assign define mapping useful mean iterate another solve alternate compare laplacian mode popular shift clustering mode valid valid nonconvex yes yes yes yes yes yes mode laplacian mode demonstrate laplacian smoothing fig consist point denote partitioning assign nonconvex shape achieve differently mode even mode centroid lie build near neighbor heat weighting laplacian perfect one centroid step show show colored mixture color contour kde red cluster manifold kde localize shape density soft assignment flexibility kde point vary kde shift contrast mode major mode kernel small small centroid achieve laplacian throughout explain spectral nonconvex point around perfectly plot mode problem cluster know create merge heat weighting mode homotopy hard centroid kde outlier perfectly area density plot color colored color blue assignment centroid boundary assignment fine grid use combine nearby centroid complex cut mode l normalize spectral intensity cause connect intensity nearby heat width kde laplacian prediction background negative fig narrow mode fix much segmentation
discriminant channel locate max low short connectivity carry tendency stay matrix improve fusion element match improvement approach ec compare general obtain perfect ec fig optimal channel ec match fusion highlight symbol channel pair connectivity notice rapidly vertical plot pattern short line single discriminant reveal symmetry ec genetic eeg activity thus highly specific role distinction influence environmental factor comprehensive analysis report detailed table obtain identification short connectivity characterize brain notably range connectivity file superiority long connectivity volume effect coherence measurement remove important interaction generator study volume depend electrical subject structure regard eeg instead trait exploit recognition claim recognition performance volume spectral supplementary file sum herein identification subject consider head match level outperform eeg notably present scenario eeg stationarity wavelet mahalanobi possible reduce polynomial regression classifier third fusion sensor place good sensor electrical contact come technology computational spend min computer aim element validation l f tp fc cp tp pz f f fp f fc fc fc fc fc p pz ec investigate brain past grow trait signature etc eeg measure despite classification performance recognize cause rely method extract eeg signal majority method consider brain account point brain specialized area continuously exchange information stable present exploit eeg sensor exhibit strong class invariant specifically fusion score level approach number subject record eeg obtain coherence improve performance standard notably recognition close open consider eeg region eeg base although connectivity much attention probably eeg eeg purpose automatic receive extraction feature activity brain spectrum possible temporal dependency generate extract brain couple spectral connectivity different subject ec condition notably ec integrate ec power take suggest connectivity effective improve eeg eeg spectral coherence fusion eeg provide related brain trait point isolated attempt discriminate people electrical brain activity perform community investigation eeg trait potentially eeg protocol purpose automatic implement range open ec state advantageous subject eeg reduce occurrence eeg activity range hz share support cognitive furthermore eeg genetic eeg activity effort efficacy recognition recognize protocol study eeg single complementary information dependence activity area region exhibit coherent activity suppose key organization tool statistical brain different principle causality frequency domain capture tool allow brain connectivity interestingly specific connectivity hypothesis eeg feature recognition compare eeg activity channel amongst connectivity subject obtained reach maximum cross correlation mutual regard eeg describe content frequency change eeg subject occur technical contrast univariate power spectral bivariate connectivity sensitive amplitude change eeg fact signal therefore play critical overall classification intra subject scale issue eeg tendency power spectra less include priori parsimonious certainly justify eeg technical consequently integrate element beneficial aim robust brain integrate score obtain spectral coherence coherence exploited distinguish epoch characterize psd element study functional calculate frequently intuitive coherence quantify level frequency channel coherence frequency acquire channel respective maximum psd improve eeg range epoch characterize feature approach identity observe belong analysis assume vector transformation logarithmic psd mahalanobis epoch consist simplification equal pool merge remove normalize normalize template represent assess cross remain epoch perform identification phase framework mahalanobis distribution accord class pool misclassification confusion recognition instance eventually run percentage psd connectivity separately channel channel pair group correctly try complementary activity brain brain activity suppose subject fusion score sum element channel misclassification evaluate describe n forward retain sort accord subset retain remove code score fusion find supplementary file leave element specific step relate performance compare
repeatedly pick expand discover stage algorithm regression oppose feature selection intuition scenario beyond product simple illustrate underlie progress towards heuristic rigorously analyze despite guarantee follow variant support current budget magnitude stage stage hence degree exposition total enumeration degree enforcing ensure case fall minor overhead good case succeed rapidly since really address implement note support henceforth henceforth conduct parent keep track scoring element parent traversal reasonable empirical evaluation assess large end associate repository execute good quadratic expansion dynamically store read disk consideration deal dynamically map bit online always feature large marked parent parent zero choice upon recursively expand parent product base example learn heuristic collection medium publicly challenge uci repository common resource tune rate reliable apart hashing square evaluation regularization leave aggregate diverse refer algorithm among baseline ratio aggregate various baseline plot cdf dataset entry plot replace heuristic show set dataset aggregate table figure relative large uniformly dominate statistically inclusion baseline example key baseline setting relative relative overall quite baseline exception extremely improvement slight cost dataset performance expectation finally implement version building repeat build run approximately example roughly task pass average baseline prohibitive intermediate per select promising epoch set union parent locally find across parent pass average base maximally expressive roc auc highly report albeit cost finish dataset c like generic statement behind neither make upon upon derive satisfy convex respective strong smoothness existence suggest progress distance core display analogous smoothness boundedness wolfe last whereas naturally iterate incur extra parameter desire smoothness establish induction adopt grant case complete simplify eq simplification choice provide simplifie desire within algebra information way expand lipschitz non dataset problem news binary binary census hard binary target letter binary three bar performance baseline present tell time despite competitive much average example coding include baseline view baseline color baseline b bb proposition comparable describe base representation design experimental show tradeoff ability compare observe simple superior possibly question execute large I baye offer simple real large scale computationally achieve superior start learn explicitly adaptively add high order interaction learn guide increase power baseline gram approach appeal avoid additional negligible improve baseline marker coordinate dataset bar algorithm heavily influence consideration computational fail adequate outperform aforementioned compare baseline propose give dominant tradeoff ability illustrative aspect amenable regret effectively grow feature exhibit enable nonlinear learn scalable learning starting improve computational resource g batch speed statistical run massive arise boost learner exhaustive search batch algorithm challenge parallelization alternative polynomial expansion employ kernel trick generally nystr suffer drawback implementation typically test scheme embedding product recently create example lead substantial complexity exhibit linear reduction result dense construction information primarily challenge I expansion sound run polynomial batch suffer online variant describe algorithm expansion adaptively define justified epoch receive stochastic restriction order highest low expand k x k regard gradient space coordinate finite proceed expand current order magnitude create care pick track high term grow computationally add expensive statistically corresponding entire update bound loss justify gradient expand bound tt substantial budget opt carefully frequently carefully pick small effectively allow jointly converge stage ask well adversarial unknown restriction place stochastic evaluating implication direct way tracking might epoch immediately possibility rate obtain use expectation conditioning round differentiable convex loss regularization definition fix remarkably dependence unlike sort amongst good
weight figure consist actual weight adjust give rate training example perceptron compute accord perceptron desire weight adjust distance quantum execute step quantum training simply classification follow consist perceptron quickly since neuron weight layer feed network however process miss block superposition set quantum perceptron superposition quantum perceptron quantum perceptron present simulate quantum computer neural research especially quantum quantum perceptron scheme superposition process acknowledgement upon research south department national foundation basic model activation output income classifier several attempt make theory quantum introduce quantum perceptron activation perceptron resource application quantum inspire neural assume state consist feed neuron input neuron neuron govern word step neuron k field artificial intelligence subsequently input compare adjust accordingly image reveal classify respective figure function artificial also multi see application two decade investigate quantum law exploited effort intelligence network approach build relatively influential proposal direct formalism physics namely k replace unfortunately proposal challenge procedure inspire classical k operator provide severe quantum system positivity increase literature remain actual quantum mechanic rigorous exception introduce perceptron evolution quantum idea superposition introduce resource nonlinear classical reproduce device learn resource classical lie quantum perceptron entire superposition block feed neuron activation map quantum perceptron circuit write normalise quantum apply return binary precisely encode j j digit phase indicate big quantum perceptron activation classical quantum perceptron initial n encoding value represent state hadamard superposition x jj j j copy unitary transformation front input add result phase useful represent give quantum perceptron apply quantum fourier exactly amplitude except accurately interested value allow quantum transform require need distribution peak perceptron resolution consequently precision parameter simulation reproduce classical perceptron precision binary digit precision resolution order deviation course consideration necessarily distribute around quantum perceptron quantum therefore precision grow standard deviation consequently increase number neuron perceptron resource multiplication quantum transform
select purpose visualization incorporate cross validation serve stock return relevance choice select consistency five error determine cross surprising observe follow instead model identical explain practice either near boundary nature intuitive sense augmentation preferable classification summarize year material care technology neighbor predictor evaluate neighbor company describe know double triple stock purpose partition value cross error first minimized search visualization minimize reduce nine stop grow forest substantial tree variable dynamic inspection termination however undesirable true force require testing validation fortunately methodology forest essential third decision naive voting create prediction present estimate true validate case fall grow achieve true approach theoretical perspective say justify individual vary second stock constitute stock vary error column achieve formulate l benchmark technology careful reader notice exclude health care discover operation explain individually particularly scoring differ individual performance case health care threshold result return period health care major division market area distinct motivated explanatory say full fail stock belong field algorithmic standard train include execute aggregated nearly factor complete second suggest runtime realize implementation advance gap execution aggregate partition stock accuracy outperform classification seem stock price necessarily certain financial reproduce produce forecast low effectively explanation reproduce near stem model achieve recognize superiority neighbor question concern severe apply methodology nothing l suggest subtle attempt return far beyond train accurately predict learn stock subsequent corresponding neighbor ensemble model confirm hyperparameter prediction subsequent parameter keep involve partition model discover good financial notice instance sometimes exceed achieve remarkably characteristic uninformative remain interval stock interval explanatory power return remarkably advance information case remarkable year financial cause stock ensemble case model forecast stock price far would also train forecasting stock predict stock price subsequent previous result reproduce care suggest stock price financial maintain approximately effective prediction consistently error present price consider denote negative model forest vector neighbor ensemble present explanatory range poor financial recommend explore learn price prediction directly weight boost furthermore attempt representation autoencoder rather rely input advantage represent stock discover preferable able stock return classification daily daily stock return encourage immediately year incorporate phenomena efficacy model author like thank valuable collection general project portfolio trading formulate algorithm field capability identify index prefer portfolio allocation stock characterize consist technical reflect market classification decision classifier relevance machine classifier ensemble reason economic forecasting employ network cluster construction incorporate rate area outside finance addition model augment selection efficacy field include material information technology choose range market circumstance accuracy advance uncertainty outcome historical record motivating behind explanatory stock train price important contribution financial forecasting first recommendation conduct automate allow important force accept explanatory variable attribute parameter time incorporate rank component approximate financial formulate scoring intend undesirable stock capability concept yet hierarchy preference stock return uncertain game portfolio benefit confidence brief classification incorporate intend intend merely identify whole reader delay machine classifier relevance classifier classifier meta boost formalize trading must relatively motivate ensemble model sense fast strong parameterized learner forest nearest parameterize present advantage still maintain ability individually learner subset ensemble decision tree key node split measure entropy measurement class label binary split something probability hereafter almost svm elegant resemble apply ideally linearly text use non support summation label exist augmentation choice radial basis rbf resemble matter principled intercept separate feature machine assume form input prediction essentially analogous bias svm construct iterative optimization process rather take sign sense boost rely nonetheless experimentally learn comparable grid efficacy learner conduct extensive choose subset exclude division use rate record good average meta h total basic exclude share exclude capital total net loss share core capital balance expense total capital per include item core eps core eps basic preliminary core preliminary operating expense common price close high counterpart tune represents intuitively extent linear often feature determine assignment validate train fine tuning work datum service year interval history relevance reflect extent learn circumstance explanatory work index reference description effectiveness financial material technology service discover certain possess divide numerical calculation tend low set successfully partitioning learn explanatory explanatory power vary across capability model
fine tuning epoch validation report combination epoch low number tune da grow layer sentiment review amazon adapt experimental review six domain amazon com whether review reason keep transform indicate presence validation label example label domain mix example number unit train raw obtain yield relative raw improvement da explain da test learn da da yield diagnostic check da sensitive learning method learn work denoise autoencoder heavily coarse grain corrupted fine grain start find result significant boost denoise supervise fine tuning model achieve ever cifar feature level span pixel background portion feature might topic big tv feature contain denoise autoencoder provide autoencoder version corruption parameter affect final digit recognition notice detector obtain detector part digit learn autoencoder tune understand network encourage train schedule corrupt force coarse grain low learn reconstruct fine schedule combination coarse grain grained idea neural goal feature learn learn noise experimentally autoencoder autoencoder supervise autoencoder train supervise fine ever cifar among invariant idea corrupt recurrent neural purpose layer wise representation appear intuition contain corrupt original input autoencoder hide encoder encoder analogy interpretation principal component encoder decoder amount corruption reconstruct corrupt encoder sequence gradient mini batch autoencoder result autoencoder subspace hide large autoencoder error denoise contrast force large indeed denoise autoencoder representation parameter transformation continuous paper common include sigmoid encoder decoder encoder pair sigmoid important denoise autoencoder p common use independently corrupt training level effect learn hand distant autoencoder denoise autoencoder map representation learn another autoencoder stack autoencoder combine good aspect learn denoise aim initial final level da e hold stochastic level take apply meaning mapping encourage cluster centroid conceptually learn network train task less early task achievable actually early high level observe cf fast cf panel start minima easier understand insight da match density noise implication learn hard da learn distribution much smoother easy hard tb c tb evaluate two cifar text data amazon product review material similar learn representation fashion learn quality unsupervised use corruption encoder decoder library optimisation mini batch though good noise performance learn low optimisation epoch datum high cccc cifar da become training learn noise learn initially schedule filter visually detector filter learn detector ccc hide hide learnt mixture various value put optimisation achievable learn da da contain detector feature learn diverse detector learn detector noise yield unit da train result da initial level da necessary investigate network da epoch large use work optimally optimally use use schedule little bad da train yield yield put optimisation observe start training examine whether learn contain explore train da set set representation representation within yielded classifier representation piece evidence hypothesis learn help learn help fair train unit da good confirm learn noise help hypothesis train learn intuitively active activation item total total eight autoencoder da result close feature learn da da da cosine feature da j da describe learn starting find learn da confirm expectation da c level subset define right cm denote corrupted level preliminary even level da perform par standard sgd update perform much bad version sgd extension justify superior ability diverse denoise autoencoder learn autoencoder single exploit global retained level unsupervised help set learn sigmoid
three day pay special measure measure include several commonly test kullback pearson chi express examine interval sample small empirical need important base mean cm r r cm r r consider construct normally take mention family simulation table divergence n see show coverage simulation accordance interval test discrimination kullback explain coverage likelihood test characteristic modify ratio another possible strictly usually happen cc conduct simulation subsection shift shift shift shift observation shift observation follow distribution table introduce shift interest less shift quite non shift pointed statistic robustness difference well shift dispersion higher close nominal likelihood ratio furth e respect shift comparison test agree conclusion obtain cm cm n r r r r r r cm r r r composite hypothesis rest proceed follow introduce devoted estimating equation new parameterization obtain e consequently become know must satisfy solve subroutine compare interval separately since continuous student size see read statistic divergence early section statistic empirical lagrange multiplier obtain e replace consistent unconstrained system take finally lagrange htbp htbp focus two continuous exact probability confidence distribution lagrange multipli statistic comparative purpose test satisfactory power statistic underlie chi statistic underlie read normally theoretical coverage empirical statistic empirical multipli little chi slightly superior empirical practice know empirical read test statistic table distribution empirical ratio test chi coverage nominal seem superior empirical coverage broad empirical composite nan interval think coverage presence shift power base sample insight tend yield interval empirical test sample multi currently work hope finding axiom exercise notation summary proof university department university nan two simple hypothesis divergence carry likelihood construct respective modify test robust empirical test presence hypothesis divergence function ratio empirical powerful currently widely develop introduce paper appear varied contribution inferential functional population purpose statistic testing empirical ratio empirical vector unknown dimensional unbiased essential difference adopt method moment let empirical give I atom maximize empirical log restriction estimate base apply lagrange multiplier subject empirical log function test maximum q furthermore nr fact degree freedom test testing derive refer hereafter divergence statistic nan empirical divergence derive power approximation test illustrative monte carry respect likelihood statistic likelihood contamination propose test composite test conclude remark measure sequel since clear equivalently x test refer empirical statistic statistic generalize important regard main testing satisfy observe know especially maximum asymptotic example assumption theorem eq eq central formula van page equivalently give probability vector I f f restriction consider subject exponential instance subject restriction member et nx e detail valid replace expression empirical estimator share regularity integrable neighbourhood integrable neighbourhood integrable empirical divergence taylor expansion hold accord enyi write present expression ccccc q nan square degree chi square freedom n hx taylor influence vanish van expression q second come previous one nan member reach divergence present reject alternative freedom test explicitly eq order taylor expansion eq thus asymptotic extend f n hx present rejection first less power analysis le attention neighborhood tool e le rely direct contiguous fix asymptotic statistic n degree freedom non centrality obtain contiguous extend e contiguous precede obtain power want take note use well illustrate divergence hypothesis pass laboratory mirror back contain
traffic group partition independently encode entirely another desirable collect overhead bs one replace penalty inside square bs bs represent traffic relaxed solve obtain coincide sec partition belong fully receive central unit cluster head furth latter case bs head traffic cluster proportional cardinality former intra traffic proportional regardless system static cluster appeal thank implementation proximity scenario cell within radius connect locate center mcp user interference help drawback overhead proportional number nonzero bss cardinality characterization suitable regularizer bb special singleton cut vertex comprise bss direct associate give static mcp solves adapt dynamically interest partition tradeoff feedback possibly however regularizer degenerate full inter unbalanced size undesirable intra consider cost size denominator joint formulate even np complete graph aim w note sec optimize undirected graph care problem ratio cut state drop problem directly except q correspond small find row initialize vector eigenvector small stop far bs access full gain nearby bs provide centralized implementation need collect individual bss central processor appropriate feed bss however scalable processor overhead decentralize simply bss bs independently mcp entire clearly bs consideration individual bss bss decentralize static mcp algorithm develop graph bss undirecte twice assume mean node bss sequel extend algorithm finally bs know gain bss start randomly initialize nonzero bb collect decentralize fashion bss run consensus average bs factorization subsequently upon magnitude root corresponding eigenvalue magnitude obtain table execute desire cluster choice ensure definite desire decentralize employ exchange raw among bs possess decentralized execute obtain line sec penalty comprise bss radius bss show triangle drop channel bss deviation db scale distance km path bss high long channel bss assume fig depict traffic varied ratio bs ms locate without accounting cell interference denote thus represent bss small instance traffic necessary attain mse full greedy greedy summarize greedy pick bs sequentially bss assume depict average distribute total traffic incur network snr db solid represent dash clearly see cumulative plot traffic amount curve greedy scheme comparison greedy size yield traffic adjust amount traffic traffic level cluster partition edge experience interference depict c ms inter turn inter cluster distribute per cell amount plot cluster curve range similarly marker cluster traffic inter plot traffic cluster bs member head without inter cluster outperform intra traffic improve traffic practical interest gain central per bs partition major portion mcp instance dynamic bi intra link obtain check cluster scale formation solely bs ms mark circle mostly cell user ms cluster cluster inter mcp exploit compressive sensing reduce sparsity inter formation formulate decentralize significant traffic edu department computer electrical engineering university usa mail edu multi cell overhead sparsity regularize multi design formulate clustered base form tight solve via decentralized implementation cluster scalability robustness verify efficacy propose wireless traffic due mobile device development boost cell recognize inter interference major bottleneck link quality service mcp multi cell interference idea base bss transmission mobile place link heavy interference boundary overall trial bss interference cell user channel scheduling speed tighter locate bss transform array symbol signal bss together jointly exploit inter link adopt expand achievable burden bss complexity gain mcp issue address many impractical practice token connect bss central limitation range localize uncertainty due quantization maintain synchronization across challenge mcp become prohibitive solution address mcp analog sample albeit compress pool overhead digital share capture capacity theoretical assume bss finite capacity link share bss successive interference limit bss traffic propose bss collect feedback bss bss receive filter cluster scenario bss form dynamically channel direct bss context distribute greedy clustering maximize clustered cell network bs transmission consider distribute theoretic mcp place bss conference decentralize clustered work decentralize cluster mcp isolated resource centralize piece bss decentralized implementation component group eigenvector discuss extensive verify user cluster introduce sec constrain sec static cluster decentralize implementation sec conclusion cell network bss bs bs ms possess bs ms set bs symbol slot band define receive bs represent channel entry channel th bs output express compactly interference network vector justify normalization ms power assume mcp bss receiver bs traffic estimate bs need receive bs square one
adopt conditional generate draw involve contrast conditional conditionally second conditionally independent conditional differ latter multivariate challenge constant scale q normalize present integer interest rejection generate table sampler initialize prior density indeed plot column minor carlo surprising among freedom prior chi square degree column concern prior column propose propose identifiable loading invariant intend possible default impose loading loading differently concern scenario loading mention latter situation term issue address arise generally preserve largely prior need sample square setup efficient sampler work derive spherical normal loading situation merely considerably scenario acknowledgment work support science foundation dms theorem loading matrix identify rotation identifiability take loading prior loading normally diagonal loading truncate normal associated loading order minor identifiable low loading maintain centered factor load comprise model loading exploratory contrast refer situation entry model factor see discuss load orthogonal rotation orthogonal exploratory computation impose identifiability loading restrict entry uniquely paper also triangular loading conditional variance conclude numerical example discussion section identifiability loading natural would spherical clearly invariant induce prior matrix permutation describe come identifiability assume matrix uniquely decompose triangular imply low prior joint triangular haar density proportional triangular distribution q tuple
compute give line propagate activation compute refine refined refine compute root square mlp description rating item description netflix rating twitter set friend last fm create focus art recommendation take hour netflix use validation contain rating movie item description binary movie belong use phase training equivalent technique collaborative filter mf recommendation use learn rating preference user na I commonly neural backpropagation also neighbor range mf mf variable regularization node range rating set parameter fold absolute mae bold within demonstrate item engineering difficult cccc ccc mf cv improve variable similar matrix factorization widely state recommendation compare mae power problem mf collaborative technique address rate suffer item rate capable address still factorization remove rate movie movie remove use item challenge induce create description rating beneficial mutual user item share latent quality receive contain rating lack utilize new represent level near neighbor induce neighbor fed train rating new item weight mode predict rating rate weighting number neighbor rate choose mean outlier result detail counter keep track rating near neighbor distance enough neighbor good description distance play item use great rate choose content rate help quality induce latent item variable line find close neighbor index line rate rate rate count rating help discount item rate index count mode top table represent mae recommendation produce mae suggest recommendation item score mae individually exception movie lowest mae previously rate mae latent create rating user use collaborative technique recommend emphasis produce item recommendation utilize power alg efficiency iteration second require use previous recommend use near complexity mf present item user call item description recommendation hybrid advantage collaborative filter content achieve similar result art filtering address start filter achieve previously item art recommendation item rate build datum many factorization may inherently future examine incorporate look item description input address user single department engineering recommend item user without outperform suffer item yet rate rate incorporate additional user description collaborative start present model latent hybrid technique address outperform broad content recommendation description hybrid maintain accuracy art collaborative filtering technique technology access abundance datum make find try product amazon recommend movie netflix recommender system find look commonly recommender system recommendation item user description user rating description predictive user item user rating infer rating recommendation description accuracy suffer recommend unless rate rate particularly domain new user newly movie rather old movie old away recommendation profile one address hybrid recommender leverage advantage recommendation system develop hybrid hybrid approach combine rating recommendation present address start use description train induce input matrix allow flexible order unsupervise induce input internal latent input variable dimensionality technique hold network language train input incorporate input item description address profile latent portion item rating item user fed network unit refine gradient refine item description description description backpropagation compute input commonly train presentation know express gradient derivative input intrinsic affect unit input value algorithm unit additional backpropagation perceptron equal layer single hide term backpropagation error network hide strict output integrate use phase train first phase compute estimate intrinsic second three train intrinsic vector likewise chance quality three phase phase produce together tb weight single layer perceptron value multi layer perceptron layer
learn remove prior stack variety objective recent autoencoder additional noisy autoencoder autoencoder work analyze characterize unit activation reconstruction multiplication corrupt version denoise activation reconstruction costly approximate corrupted test autoencoder due noisy autoencoder yield effect encoder nonlinearity perform order taylor encoder fw encoder effective multiplicative bernoulli use result approximate allow noise activation unit autoencoder decoder use square independently activation apply noise penalty input accurate allow relate autoencoder framework regularize autoencoder network autoencoder autoencoder tie gaussian encourage unit penalty type penalty learn overcomplete representation ica encourage noisy autoencoder building compute layer encode clean input autoencoder second layer activation representation hide representation autoencoder learn insensitive frobenius jacobian encoder fx f additive alternatively hide activation recover autoencoder activation encourage activation activation activation many unit activation experimental neuron dropout multiplicative effectively link adaptation neuron motivation generalization dropout computing h projective field shrink sensitivity reconstruction dropping layer noise yield system representation semantic intermediate learn binary binary code add fully analysis autoencoder show implement autoencoder lead segment digit input activation error initialize perceptron hide importantly activation h c dropout c c noise evaluate hidden input dropout noise hide activation mlp standard backpropagation use optimize also autoencoder error backpropagation noise perform perform backpropagation dropout additive dropout dropout low error capacity require overfitte noise improve help decay validate dropout hide activation extract cifar yield accuracy slightly high autoencoder accuracy different representation classification performance mnist cifar mnist model train dropout activation domain maxout dropout noise activation additive fix call shorthand sgd momentum error maxout understand dropout lead less noisy layer far understand influence activation type sparsity neuron yield dropout representation noiseless row spectrum slow row act sparsity propose principle namely robustness auto level internal wide different lead design supervise use achieve mnist full systematically huge deal lie ahead understand automatically make explore noise supervise interact intuitively deep ball unless class ball effective invariance properly introduce compressive projective map hide contraction beneficial stanford edu autoencoder unsupervise internal representation conceptually extend autoencoder additionally nonlinearity wide variety framework practical benefit strategy new internal designing autoencoder outperform denoise autoencoder denoise competitive technique mnist deep information representation noise neural dropping backpropagation performance neural pooling convolutional net noiseless explore recent layer autoencoder show yield autoencoder success input layer autoencoder systematically explore layer learn see unify unsupervised prior autoencoder call autoencoder derive penalty relate autoencoder
k v p relation hence vector relate bandwidth illustrate semi clearly bandwidth semi interesting benchmark scale determinant extend determinant dense extend denote row write matrix matrix determinant row correspond last triangular b complement dense I arise permutation matrix extend algebra recall correlation lie monotone algorithm rely happen ingredient sparse use recognize separable separable see set ai j ji early spread large exponentially sparse issue analytic linear equation eq embed obtain appropriately carry semi separable numerical appropriate q extended matrix numerical benchmark semi form apart factorization infinity residual purpose choose sort benchmark sparse factorization eigen sequential perform use method system scale linearly store triplet eigen row format exact benchmark compare eigen fix rank htbp gray c time taken take sparse solve versus linear relative log extend illustrate scale factorization increase separable htbp stage residual illustrate scaling add equivalently rank solve scale stage equivalently semi illustrate algorithm separable factorization semi separable rank article discuss numerically stable enable determinant matrix entry publication formally sparse semi implementation available university remark determinant semi lemma corollary discuss invert arithmetic introduce solve enable semi illustrate determinant solver semi large storing perform dense finite one sparse study statistic detail reader refer al throughout separable article term semi q upper triangular separable linear library discuss implement york david put author anonymous detailed
length buffer size epoch find accuracy several dataset buffer cut exhibit method hence intermediate necessarily accuracy offer progress problem good intermediate advantageous scenario acknowledgement helpful comment thank anonymous suggestion thank google fellowship lem lem corollary lem lem definition lem lem fact lem lem title lemma claim fig microsoft modern sensitive frequently precision offer grain time challenge decomposable solver enjoy nice amongst sublinear provable learning function popular loss sublinear regret novel lemma descent solver uniform provably converge known proof extensive method order cut plane modern frequently require grained prediction hinge mild imbalance spam wherein spam constitute task diagnosis sensitive include rank precision top rank recall specifically express point unlike area roc despite success domain nearly understand decomposable counterpart loss function lead deep behavior online decomposable popular online first contribution instantaneous penalty decomposable canonical principled way objective desirable online admit online satisfy offer convex surrogate namely roc indeed achieve sublinear regret involve structural sort continuity list might analyze property designing solver offline introduce variant provably converge minimizer hand proof style require conduct extensive real life cut fast dataset take achieve comparable decomposable auc demonstrate imbalance interest indicator dataset due measure seek arguably hand hand perform start seminal receive interest average formulation design solver theory also interested provide additive regret focus show however implementation learn pair dataset crucially dy label shall simplicity p evaluate decomposable measure auc surrogate term construct simplicity drop point predict score rank position valuable situation positive formalize top q structural surrogate calculate use modify consider label area roc allow range medical application detection label number express replace indicator hinge hinge framework decomposable measure gradient method function prove much large function several player incur point surrogate definition instantaneous penalty clear remain challenge guide process learn framework point continuity indeed crucially sort list product let ic I I j z k see lipschitz gives show prove recall handle positive negative previous obtain appear resp stream use regret bind adversary may naturally see bound generalize slightly point compose batch penalty l population normalize sequence point batch q note rt appendix guarantee model rapidly become infeasible descent non decomposable loss motivation mini amenable computing environment offer scalable assume access limited memory buffer stream stream epoch stream descent step buffer epoch describe computation batch pass batch collect b unable point epoch help loss life scenario label imbalance table store label exploit utilize pass pass label stream pass stream restrict non perform goal sn eq convergence common decomposable true decomposable bridge show surrogate exhibit proof require technique proof use arrive loss demonstrate arbitrary set predictor return buffer stress assume order see regularize formulation improve explore consideration instead look uniform convergence nature area roc curve surrogate result cover large family surrogate logistic rank introduce challenge structural surrogate exhibit uniform version extend performance minimizer wide variety performance section verify propose gradient performance auc proportion positive auc
require adapt behaviour cnn introduce deep selective selective cnn bottom top connection implement learn reinforcement usual enhance capture initially aim usefulness filter manual inspection separable evolution agent reinforcement learn etc million require cnns cifar cifar difficult instance de certain maxout network combine underlie object recognition task outperform convolutional network reduce favor maxout cnns consist stack alternate convolutional image width map convolutional parameterize filter index map convolutional wise pooling dimensionality layer take partially reduce dimensionality width maxout pooling layer consecutive map map keep maximum every layer form pool large softmax activation maxout reinforcement sequential decision reward signal receives agent state receive reward objective policy future discount space parameterize attention space close policy rl evolve use strategy black algorithm multivariate gaussian parameterize epoch update natural gradient fitness instead use theoretically substantially power attention maxout net augment allow filter weight differently pass image strength activation learn order sequentially attention discriminative change cnn result follow vector describe net already classify x policy image sample represent trial maxout pass set maxout would normally net activation output layer vector average activation meaningful allow softmax note softmax action regular ensure filter activation process pass pass correct constant loss misclassifie classified misclassifie input process assign fitness q classify pass network maxout output weight maxout finally update natural gradient fitness repeatedly fitness stop meet system visual level construct area visual connect connection numerous bottom connection think play primarily analysis response newly stage visual fast feedforward follow due feedforward basic orientation categorical role foreground salient support top play extract guess category connection rely feedback computer vision learn selective face also combine vision processing reconstruction face localization rl lead novel apply simplify aim state perspective feasible case cifar cifar interesting case approach already aim cifar compose color training assign cifar similarly rl enough step enough practical meet could serious limitation experiment l cifar column maxout maxout model cifar show several method improve state art reference connection add augmentation consist convolutional maxout follow maxout softmax input vs method cnn establish art classification cat activation de show probability cat receive feedback cat dramatically subsequently drop bit successfully layer emphasis filter almost map high correspondence lose code final layer simple increase map complex emphasis network value run figure training cifar peak reduce stay stable even step
recover compressed firstly measurement operator condition rank condition guarantee matrix rank guarantee norm prove isometry property sufficient quasi provide rip affine nuclear isometry rip constrain affine minimization numerous processing case aim recover arm collaborative filter quantum arrival affine equality formulate measurement operator measurement usually np solve relaxed version replace convex relaxation condition form superiority quasi replace norm pp norm nonconvex update recover rank constant eq though minimization paper guarantee quasi author quasi discussion show indeed provide sharp condition prove uniquely large rank exploit condition restrict isometry rip quasi minimization generalize minimization dominant singular recover generalization sufficiently recover singular measurement rip value organize introduce notation present devoted proof conclusion denote obtain sort p always svd n n exploit nan necessary successful reconstruction minimum condition gap introduce lemma sufficient uniquely equal large uniquely rank recover sufficient weak formulate condition restrictive inspire arm simplify remarkably rip stable sparse directly prove equivalence condition recovery nevertheless one essence rip vector rip equivalence utilize aforementioned formulation quasi rip use vector noisy integer small eq vector def integer show denote likewise rip f sufficient g rip g threshold find work cover accurate nearly noisy simply rank mean proposition organize presentation herein prop let constant corollary knowing rip accurate c solution recovery obtain fix respectively satisfy decrease guarantee tend clear passing original range
similar trick already reinforcement classification task well netflix demonstrate online exploration tend towards user bad artificial surprisingly perform approach netflix difference strongly netflix protocol top movie movie user ucb user suffer strategy factorization candidate item deal suffer regret favor item old one hence old recommender policy new user item strong exploitation computational additional square total factorization consequence factorization stay idea principled effective way conceptually large publicly furthermore extension currently study extend contextual might translate want regret large full work item good bandit point translate plan ellipsoid user item sum odd perspective lead artificial fr version ad music video movie book system cope user visit website item user exist estimate handle side available perspective perfectly aware utility side information consider item mix fit perfectly decision set traditional continuously though hundred arm million make bandit asymptotic approximation look efficient effective way cope web obvious strategy consist repeatedly present seem exploitation problem exploration eventually problem netflix challenge recommendation factorization square rmse however along heavy rmse make item rate rate user user well rate wants recommend illustrate rating netflix range make qualitatively different correspond rate rating real allow precise rmse item already outcome recommendation user information regard interaction leave aside average birth death often past item really rank rmse aspect handle recommendation address since history order perform information minimize recommendation idea paper original tackle start system cast play optimize balance problem focus familiar bandit contextual bandit hypothesis introduce methodology assess recommendation introduce matrix factorization approach introduce bandit sec new user item line sec face transpose row bold denote scalar letter sub compose element contain accordingly small line index matter notation dedicate rs evolve user user number drop bound number ever necessary represent truth obviously application whereas rate user size denote finite majority submatrix make row actually available item rate user likewise rate symbol user rating respectively thing let assume triplet item know observation rs receive stream rating rating netflix challenge click sake omit subscript netflix challenge interested reader done approach convex alternate problem value svd compute compute minimize user item respective rating limited rmse consider arm receive follow good arm parameter aim cumulative consecutive reward player want unknown except player accord arm vs ucb play ucb equation exploration exploitation arm tend arm extend contextual arm assume assume arm represent rating item introduction item user item recommendation rs rs never return obviously objective rs maximize context scenario approach rs predict rating user pure exploitation suboptimal rs balance rs recommendation soon compare along fast drawback describe recommendation aim item trade exploration order hold fix ucb I possibly consequence uncertainty word mostly express iteration fix system coordinate axis mid coordinate axis mid fill blue blue ellipsoid circle circle circle mid mid circle dot area indicate associate optimistic rating user scalar ellipsoid contour scalar product equal optimistic recommendation item strategy select amounts tune ellipsoid closed name stand alg presentation optimize clarity item ellipsoid use regularization regularization reward please matrix exactly ji ji ji ji tr proof care context matrix decomposition degradation trick observe order unbiased description fact instead uncertainty occurrence criterion ellipsoid rating user item presentation clarity efficiency item lead modify axis mid axis mid color red fill color red red fill red red color red color red red mid mid blue live optimistic item ji ji tr j evaluate empirically real greedy greedy always item ucb ucb consider reward word ucb information comparison approach highlight context ucb user recommendation strategy netflix yahoo item ucb item user random rate request yet rate compute rating item user user rate difficulty real
conditioning var aim paper context short policy minimize ensure expectation level stay bound solution govern recurrent horizon mdp separability risk constrain reason globally risk return mdp risk constrain complicated expectation govern optimization constrain mdp complicated tail reduction speed first propose prove converge locally principle mini policy gradient line estimate policy scheme base mini batch spirit propose proximal gradient develop four approximation scheme along conjunction mini batch gradient ratio gradient estimate fast scale negative descent ascent multiplier operate employ batch incorporate close well interesting come variable concern scheme trivial variant sampling sum contribution synthesis stochastic sampling provably convergent propose another idea simulate latter novel neutral mdps rest formalize constrain present mini variant later section previous conclude remark formalize constrained variable continuous var low drawback coherent measure coherent variant define coherent finite terminal feasible incur follow specify stationary state continuously differentiable identifiable reach transition state class parameterize proper outline constrain randomized specify constraint trick convert solve operate follow inner episode along estimate loop lagrange multiplier converge white rectangle minimum height fill circle coordinate thin cm simulation align green cm label align center block green align green fill red height align ex right thick triangle thick triangle update triangle triangle thick triangle update chapter lagrange multipli assumption proceed recursion close expression gradient moreover observable k hence policy follow section differ establish describe component var well input output pg sa episode underlying end visit state visit episode action gs cs likelihood tuple th episode obtain let parameterized hold estimate alone technique estimate mdps policy simulate since know approximate approximation height width em fill white thin align fill cm align n fill red cm align pg mb obtain empirical mean negative mini batch policy q output batch approximation asymptotically scale lagrange scale pg sa variant know existence lyapunov follow martingale suppose exist continuously differentiable govern contain involve convergence pg sa fast quasi var serve lyapunov fact size iterate bound recursion set iterate establish recursion plain average intermediate scale main argument govern stable equilibrium follow ode effect static ratio almost discretization lyapunov recursion stable ode lagrange multipli update step argument mdps general view almost scale converge equilibrium converge multiplier use suitably keep evolve pg sa minimum claim argument particular former recursion ascent require envelope theorem economic mini variant var gradient mini large estimate converge rest manner pg sa importance estimation scheme continuity translation eq variance use recursion ensure result function double translation var formally classic concavity differentiable write px ba piece growth assume control hx scheme approximation hx b episode let straightforward one density pd ratio episode transition dynamic pd scheme rule result recursion estimate var attempt estimation approximate part place tuple accord would horizon separability cost separability devise constrain optimize employ convergent locally policy paper novel policy stochastic motivated energy market incorporate along line risk
payment call denote similarly via portfolio abstract span allow discussion beyond scope consist process choose portfolio hold move portfolio decision portfolio selection trading market context clear write variable letter use letter functional without preference preference functional asset specific theory utility theory economic risk measure see introduce measure put detailed justification name risk measure understand potential certain asset satisfy eq understand monotonicity asset translation invariance map asset asset asset asset domain fortunately extend invariance eq thus risk generic measure combination high hold word encourage risk measure loss predefine risk moment function kl hold measure market portfolio prefer portfolio rational choose trade rational prevent market machine aim objective design market implicitly global contribute objective section build multi period trading maker allow trade maker maker introduce simplify market trading market mechanism market share security one want amount share simplify market maker maker trade security allow maker make pay maker pricing pricing market maker step functional different asset maker price market maker portfolio asset restrict rational agent choose optimal portfolio portfolio selection rational agent pricing multi involve maker market maker market maker joint trading since agent subscript distinguish agent collect bring asset maker agent portfolio trading maker agent keep asset like initial pricing x maker update use market agent portfolio communication maker pricing trading choose trade initial portfolio measure receive pricing rule w request maker x study pricing market maker class mechanism market later axiom price trade maker trade market maker maker follow natural total pricing rule market analyse market machine want market find describe involve eq part functional share multi market define machine learning problem pricing motivate meet pricing market share cf transform could relax could replace convex conjugate ready show market involve duality pricing rule market maker derive generalise substitute back dual match cf dual duality way market market primal solve interestingly could market market distribute environment extra benefit build market discuss significant past agent model market base market agent model equilibrium author belief agent event market mechanism market market progress market scoring rule market mechanism agent draw demand market implement solve partially certain also convergence market dynamic market belief justify risk measure asset standard operation expect abstract I agent additionally help asset portfolio highly inconsistent dramatically measure amount accept asset sensible asset example convex risk functional risk illustrate connection multi period trading market opinion pool opinion pool log opinion state opinion pool market introduce agent maker primal problem market apply proposition simplex optimal eq introduce market maker aggregate bias towards however sufficiently maker ignore end aggregation observation aggregation bias due biased market maker upper agent increase much market sign quickly aggregated aggregation lead belief close expect reproduce biased coin setting market market build define security asset moment generate maker scoring could market implement map update market maker univariate statistic clarity exposition care conjugate goal transform agent market mean convert market let interested trading th share security hold rhs define agent deeply costly relax accept portfolio well current agent towards portfolio rule could choose backtracking line instead introduce agent match logistic discuss prediction market instead analytical global objective connection market tool interesting topic condition converge come agent involve nature support microsoft rgb ed ac uk markets introduce market describe trading process modelling bring convenience objective despite agent additionally market sensible objective market analyse global solve machine certain market valuable direction machine towards scalable market abstract learner system market distribute additionally relationship market probabilistic spend building still market
dnn utilize increase corpus heuristic offer performance understand build acoustic dnn explore optimization acoustic model function difference across example whether improvement due network architecture technique concern systematically explore several improve dnn acoustic dnn component dnn classifier draw dnn many task dnn acoustic simply speech rate dnn acoustic unclear acoustic ultimately across acoustic far understand aspect dnn rapid dnn acoustic new speech corpora language understand variant various understanding principle artificial intelligence study component perform effect dnn task hour increase dnn fit include architecture choice convolutional locally evaluate alternative convolutional dimension dnn fitting corpora speech research corpus us explore dnn ten corpus also impact choice layer final compare across dnn architecture process section outline question address neural network paper corpus focus dense convolutional choice combine corpus explore deep dnn act acoustic recognition use approach system resemble speech mixture gmm acoustic overview scope refer article speech focus component approximate distribution span acoustic acoustic represent ms audio hmm cluster dependent state hmm use gmm network distribution network acoustic use rule neural distribution approximate occurrence count usually acoustic span acoustic difficult feature fix scale drop hmm term form acoustic acoustic model decoding introduce scaling empirically adjust introduce un normalize recognition hybrid modern component progress speech recognition acoustic work refine framework signal building performance extension gain task complexity purely capacity gmm acoustic parallel interest new apply initialization learning provide interesting forward acoustic offer path capacity dnn early recognition recognition demonstrate state innovation couple capacity yield challenging task acoustic gain challenge microsoft google factor attribute modern hybrid acoustic specifically total number number hide initialization modern hybrid researcher hybrid dnn weight purely nearly many beneficial hide layer size define hybrid system neural network build understand aspect building network acoustic understand define set acoustic hmm early neural context state dnn acoustic critical success generally modern system variant context try acoustic model create baseline hmm gmm force alignment originally contain assign label acoustic force alignment ground generate consistent force hmm hybrid speech creates force hmm gmm align use train previous dnn system dnn produce performance yield dnn start force repeatedly use forced alignment produce hmm build dnn acoustic structure use acoustic modern dnn use success recently standard hide gain architecture counterpart task hold deep obtain dnn time translate modern dnn hmm system continue performance fundamental neural architecture aside series densely connect densely network intend leverage meaningful audio input task addition dnn acoustic architecture densely architecture perhaps layer window temporal neural recurrent modern recurrent acoustic task task available term hmm continue variant model loss acoustic function also control default dnn observe standard classification dnn account system loss gmm acoustic dnn acoustic acoustic begin strong function step combine standard view discriminative acoustic act component choose additionally function regularization especially simple widely weight penalty develop area regularization effective dnn apply regularization dnn acoustic combine change quality dnn far solution dnn stochastic practitioner default optimize researcher advanced method yield dnn well dnn acoustic quasi sometimes processor recently like accelerate task outside sgd still newton require consideration dnn procedure utilize graphic hundred computer persistent throughout history utilize modern approach final capable time network acoustic design component hold baseline acoustic building variant difficult assess decision systematically vary critical variation baseline question neural hide corpus overfitte build ten total drive build dnn acoustic much broad locally versus densely dnn recent combine type acoustic generalize still feature share reveal overfitte dnn early many training acoustic towards improve large small processing dnn acoustic yet dnn acoustic task ultimately far encode change large deep question use hour corpus corpus dnn architecture address corpus use present experiment large size dnn depth code property corpus understand encode integrate architecture amount equation predict class architecture utilize cross apply regularization many research entropy always task specific serve instead acoustic entropy objective single training take entropy acoustic short acoustic wish minimize mistake word step conversely acoustic frame experiment always frame error metric dnn loss dnn series fully connect transform act conditional dnn layer follow dnn layer vector activation first layer dnn partial effectively layer activation hide apply wise nonlinearity computation traditional neural network unit hybrid speech dnn nonlinearity final dnn output final nonlinearity softmax nonlinearity output loss state dnn computation equation loss non practice apply gradient dnn dnn speech benefit neural acoustic issue dnn study connect dnn serve acoustic modern speech vision deep convolutional neural bank architecture relationship convolutional code shift localize acoustic evaluate specialized combine replace network dnn feed fully layer image vision restrict hide connect spatial control localize stationary vision location rather separately convolutional time connect equation apply move across produce meaningful activation map operation pooling act code slight localize connect contiguous region convolutional layer apply pool local region feature feature map contain activation select activation hide separately use pooling pooling alternative pooling replace pooling often consist layer convolution follow densely softmax classifier pooling act input dnn dnn input densely convolutional layer densely hide frequency relationship architecture dnn layer input convolution pooling hyper size neural architecture combine connect hide sharing unit idea utilize locally region architecture apply frequency feature unit use unit slight occur frequency architecture convolutional convolutional connect weight sharing united behave grouping pooling post similar tb pooling layer behave architecture define wish minimum gradient impractical gradient many convex difficult general statement optimality heuristic classical momentum probably algorithm modern momentum denote accumulate momentum velocity vector close expect information update ill condition problem momentum might actually cause fluctuation parameter turn slow nesterov issue encounter network optimization help past sensitivity avoid ahead gradient detailed optimization question cm acoustic establish hour baseline force create open phone dnn estimate likelihood hybrid dnn setup input context adaptation globally dnn overall setup evaluation report subset subset perhaps direct improve large add unit dnn increase capacity scalability interest apply large dnn acoustic model architecture ask question improvement large frame serve improve size vary hidden total number unit hide network size million output layer network typically study output often layer comprise contrast rarely fraction modeling work explore additionally context use nonlinearity optimization nesterov accelerate momentum schedule update pass stop cross hold development tolerance threshold distribute gpu capable training restrict parameter task dnn day frame acoustic baseline gmm show vary context substantially dnn dnn small substantial dnn absolute classification accuracy improve large window context overall frame dnn window frame always proxy final acoustic reduce suggest increase gain translate large set dnn model beneficial task acc train gmm understand dnn acoustic dnn epoch fairly dramatically continues nearly set realize epoch training implication acoustic beneficial fitting performance become increasingly utilize utilize finding epoch window metric gain translate provide training dnn tb dropout recently prevent dnn dropout randomly unit activation training prevent observe activation demonstrate processing use dropout acoustic hour news dropout additionally yield gain convolutional network hour dropout impact alone dnn dropout whether large train dropout training preliminary set generalization possible otherwise evaluation build force hmm dnn train dropout improve acoustic beneficial dropout seem insufficient selection critical poor preliminary bt layer er er layer early converge early dnn technique dnn acoustic network capacity produce generalization par well network back propagation early capacity analyze stop improve low test dnn dnn early stop dnn beneficial perhaps insufficient benefit large acoustic label dnn acoustic force alignment level labeling training partially dnn improve acoustic span label supervise force alignment force alignment system force speech recognition ability variation version independent corruption correct label outline improve dnn performance optimization exhibit dynamic early capacity exhibit capacity early phase combine dnn hide suggest dnn dynamic categorization coarse class generate force acoustic baseline acoustic model train iteratively improve ann hmm hybrid fully dnn acoustic system new force alignment continue label begin predict label much early save day converge completely dnn fitting alignment dnn gmm force far dnn hmm dnn force alignment dnn proceed dnn newly regularization dnn measure epoch experiment epoch result low quality model randomly initialize dnn bad dnn occur anneal schedule adjust newly alone without performance well dropout epoch beneficial dnn bad epoch early lead train dropout stopping make evaluate corpus dropout train dropout figure curve acoustic train early dnn briefly surprising dnn follow begin label labeling computing label change early change label change dnn five match capacity dnn train corrupt extremely capacity dnn perfectly dnn take suggest early bias characteristic phase dnn require benefit train epoch five epoch eight dnn translate day implement modern system force alignment dnn overall early technique dnn acoustic minimal additional bt acc gmm dnn dnn far dnn regularization deep train use train acoustic experiment facilitate across bank dnn five train accelerate smoothly increase momentum schedule epoch acoustic overlap pooling region layer second filter use select preliminary densely hide unit map unit convolutional map convolutional layer dense hide layer bank run filter along frequency region run experiment context report frame system acoustic bank improvement connect acoustic tie weight localize field frequency outperform indeed dnn lack meaningful flexible architecture able useful parameter well bank much leverage acoustic run bank feature randomly axis convolutional fairly experiment indeed leverage localize promise compare filter bank bank bank bank transformation may automatically gender specialized post appear neural superior complement dnn competitive increase amount summary replace reliable conclude acoustic hour limited benefit dnn acoustic modeling dnn substantially large corpus experiment explore dnn acoustic maximize task combine fisher corpus approximately hour accurate gmm acoustic train feature obtain frame current project dimension discriminant lda normalize mean obtaining semi tie feature linear estimate per fisher contain system train fisher fine back probability sentence preliminary build web page text gain derive language alone use system serve compare build alone rt evaluation train baseline table dnn hold dnn optimization five roughly free typical acoustic literature nesterov key hyper momentum decrease annealing preliminary overall anneal pass quite find epoch lead solution table accuracy algorithm evaluate effect evaluate performance optimizer optimizer important appear train absolute thus remainder use optimizer optimizer somewhat robust optimizer bt produce performance setting momentum report test contain evaluate rt rt corpus systems optimizer acc rt gmm cm performance keep fix improve hidden layer hide hide layer total free add hide unit show frame classification hide reproduce dnn dnn code dnn comprise facilitate comparison framework parameter perform frame evaluation experiment dnn lead gain limit compare dnn move dnn dnn size rt evaluation small corpus overall believe corpus challenge acoustic induce quick dnn improvement acc rt gmm n n dnn system keep total vary layer dnn architecture total model change priori reason believe heuristic dnn layer multiple performance gain deep versus layer hidden layer model hide deep depth performance
relate six function formula ft place middle remove plot rule partitioning form length cost cost ft ti cost ft tr segment neighbourhood section neighbourhood method segment neighbourhood prune introduce segment search neighbourhood search enable set c position idea segmentation shall update describe fully needs specify satisfie tn tt nc k nk fit prune strong c prune large overhead detect univariate prune implement compare extent insight method figure amount store pruning optimisation figure illustrate rarely contrast evidence prune fact pruning pruning state condition c solving optimisation use program functional pruning pruning also slow inaccurate round middle profile assess performance confirm speed speed depend number change cope change need look size dependency clear pruning signal expect efficient efficient range simulate point varied signal repeat code simulation repository see fast interestingly benchmark accuracy annotate region visually copy benchmark training annotate segmentation annotation segment set vector annotate compute benchmark small test introduce detect question answer whether solve solve optimisation give disadvantage slow particularly interactive solve constrain preferred interactive scenario know purely efficiency pruning always prune empirically difference pruning apply require currently even prefer always detect particularly large observe computational comment discussion support centre mm mm em mm ex ex em definition lemma definition proposition em cm de universit abstract accurately long detecting description formulate dynamic programming exactly tend least binary segmentation computational segmentation suggest true cost extend pruning method new efficiency detect variation keyword optimal partitioning segment series model effectively segment model detect efficiently eeg datum speech signal section dna copy microarray reduce relate cell area crucial classifying widely moreover microarray basis many estimate zhang formulate define call segment specific cost dynamic programming variation segmentation circular offer slight decrease way prune technique pruning technique functional pruning optimum pruning take pruning try former pruning inequality always suggest regardless number segmentation structure introduce constrain optimisation review exist dynamic pruning optimisation optimisation algorithm empirically theoretically exist pruning empirically simulated assume order trivially order attribute observation distinct th segment consist let statistical consider infer type wish segment infer consist datum q depend detect segment segment likelihood identically segment detect across segment normally segment simply error depend segment segmentation get location unknown approach minimum datum often monotonically either call calculate choice aic linear penalty optimal note solve criterion program neighbourhood optimal insight segmentation vary advantage incorporate model pruning pruning method two functional pruning discuss prune stronger hold hold solve optimisation programming optimal partitioning first reduce computational denote position early use simple recursion order calculate q calculate increase partitioning discuss exact equation never location hand value discount time update rule restrict denote update pruning pruning algorithm expect basic partitioning show computational bound show optimisation solve constrain optimisation segment neighbourhood search approach prune segment neighbourhood derive relationship thus recursion obtain extract segmentation position calculate segmentation calculate value equation calculation total developed technique neighbourhood call dynamic generic use segment search assume segment split store candidate recover far return allow split c interval correspond idea example change value need store c square criterion line contribute line interval correspond previously contribute formally interval recursion bold long give criterion analyse four introduce corresponding change middle plot function long optimal plot time empirically towards overhead segment neighbourhood search
weighted square label instance computation unchanged unchanged except wolfe gradient challenge video annotate annotation kind provide truth construct dataset annotate action movie manually add build annotation annotation annotation form video sequence end range temporal frame feature vector recall frames aggregate decide pool feature long center compute video descriptor feature restrict channel improve run aware yield hellinger feature normalize experiment split dataset supervise annotation set sec hyper practice contain annotation use cost evaluate carry frank wolfe union three set annotation constrain rest annotation please assignment five split bar may task use yet another prediction ground standard compare ensemble slight big annotation low annotation take averaged align annotation accurate segmentation long predict label within ground interval compare three baseline train use baseline scheme sec proximity appearance interval j x square distance cut use frank wolfe intuitively search segment chi replace appear adapt size problem instead minimize frank algorithm scalable method graph compare sl supervision completeness classifier square annotate interval square setup baseline except sl supervision illustrate baseline weakly temporal constraint signal baseline recover alignment annotate increase blue mark make lack order fully annotate expensive movie easy appear manually video necessary good supervision baseline stand b figure right axis supervise error bar baseline annotation whole dataset annotation sl learn low improve supervision quality assignment z instead quality hold classifier treat use compute section classifier hold make data one split error correspond compare sl classification experiment sl blue always explain fact propose weak annotation sufficient train et task fact access supervision constraint weakly semi supervised red sl blue recover consistently well fully semi annotate example constraint improve one constraint scale sup video annotate list action walk extract seek action formulate weakly constraint time assign annotation discriminative manner dataset video total movie recognition video year cast detection fully annotate boundary give exploit order video stream annotate video time label quite consume fully supervise use weakly semi method promise easy video poor localization movie localization ignore supervise set discriminative optimal assignment constraint learn temporal order action constrain kind activity video infer detector surveillance laboratory limit action explore challenge video length movie relate composite activity learn give supervision composite activity atomic action action annotation without order action explore form action priori temporal order individual supervision explore uncertain temporal annotation action movie contrary multiple simultaneously incorporate label movie dynamic match speech like improve pre detector discriminative clustering unsupervised partition formulation discriminative explore vision successfully apply co approach order work frank wolfe gradient minimize classical permit continuously differentiable domain receive temporal assignment address illustrate set short define contiguous frame movie divide video annotated order action consist stand phone assign original annotation list fig contribution model supervision order localization video propose efficiently solve improve new localization publicly discriminative video annotate specify action order annotation formulate assignment individual cluster parametrization assignment discriminative lead preserve predefine way transition possibility extremely algebraic constraint stochastic matrix define use notation lead w frobenius computed inversion concatenation objective rewrite matrix ridge minimize done close plug back yield matrix p center p find optimisation z share video recover form sec combinatorial replace hull appropriate define convex large admissible assignment kind operation projection tractable convex frank wolfe sec choice frank wolfe rather approximation edge convergence interpolation iteration counter implementation frank provide duality refer linearization duality relax tb k programming wolfe linearization current view actually minimize linear depict hyperplane seem shift argument prop minimization mind wolfe amount solve ta plain representation equivalent well temporal let becomes solve indeed k sp admissible recursion p kp dynamic maintain frank wolfe optimum z find nearby simple rounding scheme consists find
argue previously monotonic inner unless experimentally compare hash report well scheme normalize example norm set lsh impossible locality locality hash inner exist lsh suppose hash function product similarity exist event lsh inner possible great show function satisfy totally lsh monotonic basic lsh requirement lsh algorithm step candidate use identify increase behind lsh still preprocesse hash increase traditional lsh runtime locality hashing locality family along query preprocesse transformation instance hash recover lsh hash back thus counter high note query processing create hash table efficiently section asymmetric locality randomization hash query preprocesse transformation nn use lsh modification preprocessing query retrieve element table definition asymmetric lsh probability element original lsh query q one ready transformation qx fix constant suffice normalize distance un inner shrink instead interesting connect show become negligible purpose perspective since instance e transformation define respectively last decrease nature root second follow complete proof respectively explicitly rp structure eq sublinear algorithm guarantee parameter hash rd main query constraint grid search parameter like lsh depend aim know threshold threshold choose high conduct corresponding optimal show actual unknown reasonable convenience recommend use choice use neighbor way small choose small consider figure eq h tradeoff hashing transformation compare hash lsh neighbor euclidean lsh optimal un product indexing capability outperform lsh product surprising inner product interest two hash function top task gold hash vector hash hash query indicator sort vector hash function consideration lsh asymmetric hash subscript draw ideally hash scheme precision item sort list suppose rank belong top top increment count already see item vary obtain continuously look precision recall higher recall indicate choose user important parameter lsh largely parameter l lsh lsh lsh lsh algorithm hash hash code show big improvement lsh lsh indicate task clearly asymmetric transformation higher vary color lsh present lsh lsh label curve different lsh top vary solid red available significantly outperform lsh l lsh l lsh lsh achieve interesting confirm precision rest demonstrate indeed choice sensitive unless ii sensitive search numerous scenario collaborative filter task find normalize instead challenge provably lsh locality study develop lsh generalize exist lsh input vector repository propose novel space provably useful line high product similarity even due pilot computation find fast hash special binary powerful binary bit hashing hash propose hash projection hashing rich make fast improve runtime application efficiency scheme application interesting work consist mention detection svms department university usa department department science nj usa provably sublinear approximate search un find hashing scheme locality sensitive hashing lsh insufficient lsh asymmetric interesting mathematical phenomenon convert approximate near neighbor sublinear hash lsh provably lsh independent propose simple implement collaborative item recommendation netflix focus query interested search inner problem two value throughout scenario place variation control directly solve subroutine large structural prediction recommender past behavior past rating model collaborative filter factorization latent characteristic vector rating user concatenation recently svd rating outperform exist neighborhood latent item instance control characteristic wide recommend solve recommendation web linear scan image art object detection activation various image image product filter million test costly score activation identify collection filter high activation image top product object cut plane popular method cut identify violate current exponential heuristic scalable class predict basically class vector class fine grain class compute multiplication predict e costly deal massive make branch technique come provable runtime guarantee technique partition suffer curse current linear search dimension locality hashing lsh randomize near unlike technique well accuracy guarantee lsh way dimensionality make lsh large deal common day lsh make ideal modern focus hashing suffer curse lsh lsh efficiently approximate hash hard lsh lsh negative lsh asymmetric framework lsh lsh interesting mathematical phenomenon problem neighbor asymmetric hash sublinear un hash framework independent experimentally item filter hashing netflix dataset evaluation theoretical asymmetric hash product well function surprising break bottleneck commonly near space construct datum near neighbor distance near
hasting propose metropolis go otherwise stay repeat carlo inclusion individual base easily obtain conjugacy proposal exactly position simplify ratio implement correspond case theory two theory choice near boundary unknown modify residual square lasso lasso experiment present multivariate unit give practical justification correspond configuration employ complexity allow move space mix reasonably carry selection exceed sense summarize mean inclusion expect latter bayes across three discuss method method credible denote show inclusion outperform argue assume work lasso extension distributional strategy subset entry specify depend intuitively idea likelihood track datum application explore parameter mixture mixture denominator involve suitable great big calculation equal accord square theory proportional exponential state write numerator make inside divergence available since rank square centrality formula function chi square integrate clear resemble multiplicative put everything take proof idea equal expectation work separately old consist like bound integral corollary remark sense concentration distribution relevant bayes selection variable assume design recently considerable drive primarily challenging application indeed study trait subject feature consideration association trait confirm non negligible association reasonable zero give literature set variety method base loss equip penalty include smoothly absolute selector give selective perspective selection include review recently establish selection approach distribution ask true rate show frequentist include compatibility matrix work allow conjugate prior enjoy posterior concentration special posterior minimax high describe concentration bayes truth rate consistency identifying propose markov chain monte sample study compare key provably concentration strong often schmidt dimensional specify prior incorporate decompose non zero assign put require suitable concentration prior restriction support primary impose practical e seven insufficient rank admit equation number interpret reasonable restrict support prior least square estimator parameter center square center summarize datum bayes restriction hold clear dependence obviously assume subset sparse alternatively could rewrite model modification regular inverse inverse stick restriction likelihood ny x proportional identity feature center greedy closely interesting approach fractional center center fractional regularization tool variety dimensional theory arbitrarily close see make flat conditional fractional subset empirical bayes bayes fractional bayes q fractional prior bayes dependent explain rescale help namely track property though want asymptotic array setup something provide notation array size call problem follow call exponentially datum generate cardinality quantity generic event probability denominator useful throughout fix cs result characterize concentration mean though simple specify notion consider equation would term act concentration empirical remark provide numerator see intuition make big justification lemma argument lemma write expectation lemma lemma eq empirical vanish bayes sufficiently concentrate big condition concentration discussion admit condition inside assume next take rates concentration distribution ordinary detect adjust high bounding phase parameter interesting open question behavior plug phase condition satisfie claim calculation prior satisfy expectation trivial bind could dimensional formula sum confirm bound prior bayes minimax ordinary equation constant condition remark claim complexity low ratio inside vanish p putting conclude prior yield posterior consider constant calculation binomial coefficient q last easy confirm corresponding rate next mass previous example equation fall subspace concentrated question consideration let effective posterior probability vanish numerator lemma ns see sufficiently upper since sufficiently tail condition upper dimension satisfie effective proportional claim light second order summation bound particular summation clearly therefore vanish prove claim learn provide light tail concentrate posterior bayes neighborhood directly norm dimensional eigenvalue possible maximal diagonal quantity call scale dimension definition facilitate norm get present follow observation control formulation put trivially indeed result frequentist ready concentration prior hold eq follow last upper vanish deal extra place first compatibility important identify
ascent prox sdca guarantee quantity result rather optimization improve reduce specifically study sampling sdca importance prox sgd adopt throughout process algorithm employ variance prox sdca propose importance achieve extensive rate importance suitable prox prox sdca verify traditional sample coordinate sample although uniform simplify insufficient introduce result sampling reduce rate proximal stochastic prox mirror ascent prox prox traditional sample since vary improve convergence propose corresponding gradient estimator analyze distribution proportional gradient simplify computation use improve exist uniformly prox sgd special prox sdca traditional ascent sdca pick show sdca converge cyclic order appropriately importance strategy optimal sampling distribution parameter convergence rate exist uniformly prox sdca special paper organize review work importance list empirical evaluation stochastic proximal coordinate proximal stochastic study approximation asymptotic term finite sgd prediction study general prox achieve rate loss recently researcher previous average return average average may issue old polynomially dual ascent zhang enjoy rate loss researcher non ascent obtain convergence investigate sdca dual primal learning recently zhang provide show convergence rate duality sdca loss sgd prox sdca study noticed importance stochastic similar variant iterative equation select pointed algorithm objective mirror descent cover furthermore prox sdca addition zhang version stochastic mirror convergence sag smooth sampling algorithm shall mention researcher result directly proximal fold prox sdca paper rate duality rely structure primal descent distribution prox primal addition distribution notice mini batch sdca version prox sdca version use inner sdca therefore apply accelerate convergence inner paper focus effectiveness prox non online regard extensively sample selective purpose selective sample label goal reduce label need certain importance key definition throughout vector function respect function lipschitz respect dual strongly dual norm define differentiable value taylor expansion example usual strongly optimization predictor regularization regularizer problem fall optimum analyze rate respect iteration stochastic mirror importance descent proximal prox sgd abuse mirror descent directly full stochastic descent satisfy desirable however efficiently proximal iteration randomly bregman minimize iterate regularizer trade objective unbiased efficiently assume proximal mapping operation wise proximal standard introduce variance prox importance adopt proximal descent sgd importance iterate np derivative follow implicit rule subgradient attract indicate prox define update norm strongly np variance np np combine inequality p second side combine f plug equality side p term full f f inequality due equality plugging conclude next study reduce rigorous result value choose easy stochastic inefficient issue calculate keep change parameter still inefficient solution relax introduce right inequality suggest firstly suggest smooth finally summarize proximal importance update section provide algorithm present assumption well importance convex norm strongly convex take firstly satisfy assumption p p divide conclude smooth p p achievable factor remove property easy derive use strategy improve variance explicitly v plug part prove part equality accord sampling strongly norm fact assumption corollary n page conclude derive set n ig plugging conclude first lipschitz plug observe bound bound lipschitz n schwarz importance hold give follow hold page side conclude lipschitz tn plugging proof plug adopt r r r l importance improve r ascent sdca importance prox sdca proximal ascent sdca uniformly pick dual maximize follow however prox adopt vector sdca prox sampling pick th element pick coordinate prox sdca main interested optimally accelerate sdca introduce prox sdca r update dual simplify nr n therefore respect choice ns note therefore plug ds many easy accord maximize dual ascent solution p relax r I optimize omit set optimize inequality guarantee ascent set combine fact inequality I inequality duality gap optimize p il sdca importance sampling ht n nn np ti option objective option option iii remark easy option bad option iv loss option option optimize option iii choose optimize three option lemma iv replace option option ii choose proposition subsection convergence present follow duality n ni sdca suffice lemma combine equality tn furthermore n n adopt uniform I accord conclusion importance improve especially smooth sdca propose sdca duality suffice n tell indicate n expand n inequality third hold inductive rearrange term imply eq combine small also n overall proof remark replace valid sampling theorem I max convergence numerous propose list mirror descent task solve typical large categorization bag sgd proximal sdca regularizer loss convex optimal svm project iterative solution theoretical analysis still way easy get form ny hinge solution euclidean distance analysis set sdca r plug equation suppose interest optimize hinge improve interpretability still regularizer previous approximated approximated adopt
projection inaccurate task neighbor demonstrate bit outperform hash answer retrieval cosine prove lsh establish cosine term cosine similarity binary datum view cosine similarity clearly illustrate freedom fortunately bind purely bind high similarity upper low note overlap high handle interestingly six dim news every query query rank point plot median among similarity separately dash solid dot query list cosine similarity panel plot among similarity cosine solid together news match panel figure similarity train cosine number compute vary curve cosine clearly formalism approximate approximate near neighbor parameter report neighbor notion deal similarity near point theory lsh lsh uniformly typically distance analogy requirement lsh family link lsh us structure provably time nn family measure lsh lsh lsh lsh report neighbor additional lsh depend hash lsh hashing permutation hash hash I lsh sign random utilize sign highlight show comparable similarity measure preferable vice lsh gold x immediate consequence corollary combine lsh hash cosine output integer recently provide simple informative section theoretically datum bit hash three study nevertheless similarity comparison outperform case obviously replace go lsh cosine note bad case still gap confirm outperform even z z figure less competitive similarity expect analyze hash even bit performance bad conservative call surprisingly confirm outperform somewhat optimistic parameterized lsh hash hash table hash top every gold neighbor use cosine underlie measure dependent similarity threshold hash consideration task threshold vary actual ideal implement combination hash function mean gold neighbor report number need percentage recall gold choice plot fraction retrieve good retrieve total consistently case irrespective choice standard computing similarity favor lsh cosine similarity despite disadvantage outperform cosine confirm add mnist evaluate retrieval similarity place figure although improvement figure hashing originally detect page widely adopt numerous web spam online web graph hash substantially preprocesse cost make practical however take theoretically decide base desire cosine provably lsh cosine provide provable compare different experimental evidence indicate computational advantage lsh wide practitioner view ph student nsf dms support fa nsf less obvious step qp pf achievable rational cauchy rational continuity c pf
address estimate unified deal quantile nest randomly truncate support central limit extensively heavy soon nest relate mass quadrature multidimensional generalise divide subset twice first derivative bound estimate arise rare complex necessarily continuous failure writing go split property interact furthermore gain link process indeed connection nest remain unclear fill core tool value link general carlo apply carlo central limit stop enable possible generate conditional law identify nest carlo propose one deal implementation numerical study heavy tailed carlo non random truncation appendix consider variable case reaction coordinate assume negative common increase walk define recursively sequence conditionally great arrival event x estimator especially generate marked exhibit asymptotically rao compare naive monte c monte pt quantile manner bit interested refer detail value variable idea optimal build point simulate associate mark poisson process markov sort marked consider come precisely nest nest one use expand poisson unbiased proposition finite moment order especially heavy tail much weak globally monte one monte require choice estimator define simulate infinite sum nest sampling propose sum criterion unbiased stochastic sde path recently two address issue idea biased convergence reduce construct unbiased base biased one basically simulate chain combine final truncate integer random independent nest stop sum randomly seem use keep consistency notation aim estimate order reasoning give independent bring give u measure interest forward expect go relatively probable nothing go process furthermore sequence corollary xx ie bring proof end generalise imply light tail remain especially interesting computational call simulator chain simulation bring convergence n furthermore power expansion dominate rewrite bring together close shall far limitation behind effort show product non optimisation assume decrease pareto distribution context auxiliary show bring solution c contrary bring constraint solve optimisation iteratively q q estimator recall marked process necessary tail optimal exponentially low present exact optimisation pareto finally present framework optimal resolution furthermore pareto decrease decrease turn theorem computation demand computer play key sequence decrease shift parametric e n e argument function choose truncate one especially power optimisation become hence reach one rate argument apply consider sample context chain relatively big already nc start basically change account update ns procedure stop become time estimate simulation computational budget computational know advance budget end budget mean compute implementation evidence show error estimate evidence nest hold seem choose simulated remark twice infinite fact compare usual notice tail consequently sufficient become approximately nested approximate hand still perform nest enable increase variance give new heavy tailed identify often address index estimation tailed sequel give pareto analytic formulae ideal monte carlo pareto pareto variance third one clearly visible moment density estimator monte pareto central limit generalise central limit law characteristic pareto decrease state proposition decrease q n exact give asymptotic approximation growth rate pareto rewrite bring bring hand equal order result display comparison distribution monte carlo variance explain get expression instead denote derive value furthermore instead denote dotted dash good variance far implement truncation estimator geometric well combinatorial turn computational scope parametric much simple aim depend parametric parameter fig pdf budget also deviation chain require infinite call certain competitive solid parameter almost optimal monte optimal implementation soon implementation soon confirm tailed variable proposition speak condition result test result one nest pareto nz nx multiply parametric gain chain latter initial pick ht proposition mh call simulator call simulator cf remark give formula especially quantile perform estimate standard deviation uncertainty quantification triangular moreover reason coefficient bottom bank width interested quantify estimate water variable vector independent transformation analytical quadrature approximation bring nest stop budget nest algorithm stop simulation monte nest nest negative easy limitation chain run previous expect handle leave tail split budget tail b g
finish discussion communication protocol present constant family compact instance bernoulli take hypercube receive single observation universal machine proof estimation scale bind receive generate bernoulli bit fusion center fusion center compute square bounded note family gap describe minimax square interactive turn feedback interactive message machine freedom powerful begin uniform family p direct nearly uniform q conversely dc eq receive minimax centralized scale bit require centralized machine bit whether show nearly identical receive universal see slightly weak corresponding interactive bit distribute allow interactive attain minimax logarithmic factor bit nearly dimension gap solution compute distribute manner specifically describe hand nearly scale dramatically scale sharp require careful logarithmic minor open rate constant whether scaling differ problem previous though closely complex section low estimation low generalize probit fix machine store independent goal unknown vector small large rescale matrix linear universal classical e scale corollary budget achievable protocol logarithmic factor bind achieve separately solution lower allow low design semidefinite regression solve reduction protocol pair expression turn binary particular draw denote cdf cf universal turn lower bind show probit least regression probit linear response probit regression inspection probit problem error construction low shorthand point pack conditional x determine send pack upper mutual shannon source yield slight refinement proof somewhat variant distribution member define suppose fix hamming distance pack hamming index suppose fix find distance uniformly infimum range observation set recover testing flexibility identify hamming variant control chain uniform pack neighborhood control challenge allow bit tight inequality sequel receive sample note moreover condition independent quantitative ratio contraction precede quantitative processing context privacy chain likelihood ratio require valid protocol appendix valid remainder break namely information since inequality follow code put piece upper mutual lemma low proof let hypothesis construction finally machine demonstrating sufficiently sized throughout provide section first quantitative processing analogous conditionally packing denote iff leave indexing implicit state precede paragraph involve indicator fix lemma number pair satisfy p mutual proof prove upper mutual use theorem shannon simplify communication relation lemma le imply complete protocol packing packing invert local minimum coordinate machine accuracy round machine initialize global machine operation index update list bit I bit clear global quantization yield minimax protocol step machine bit bit send index j piece machine draw across machine addition sample allow interactive protocol message dependent measurable machine simply value say nothing require analogue lemma protocol sequel reduce multi dimension statement abstract bit specific likelihood provide indicator state building dimensional make concrete bind apply pair negative apply intermediate bound chain conditioning reduce turn mutual equality conditioning reduce iii analog entirely analogous involve minor setting follow paper establish amount problem theoretic rely quantitative characterize bit constraint information message question argument differ logarithmic would interesting inference protocol improve require insight believe perhaps support laboratory research office office research science grant support facebook fellowship body paper rely inequality contraction develop present proof lemma direct analytical lastly width graphical eps write combine pa bb bb db version lemma likelihood absolutely generality bb bb bb e bs express conditional shorthand correctness indeed bind recall figure product combine three display likelihood bb bb auxiliary negativity require argument build technical independence expand kl bind lastly argue message conditionally give shorthand role definition collect must notational prove complete proof turn prove conditioning consequence recall final follow end moreover assumption apply yield cf proceeding eq classical conditional kl eq prove inequality two variance consequence addition condition recall lemma challenge however regardless yield note q choose choice desire collect restriction measurable set mutual marginal marginalization q kl divergence ratio kl divergence claim bound q kl inequality rule mutual complete remain establish variable verify hold must independent conditional q give ki k hold satisfy precisely obtain condition everything event standard cf imply integrate lemma em zhang engineering berkeley berkeley edu conference often machine minimax compare quantity amount communication achieve centralized protocol channel interactive server message novel quantitative inequality characterize effect communication rapid growth modern set computer natural involve computer yet machine expensive slow power intensive survey parallel system consumption bandwidth limitation inter impose significant algorithmic important study require machine large class estimate unknown classical minimax rate characterize bad centralized machine intermediate processor fusion try answer minimal realize minimax g characterize range decentralize g substantial communication though relate formulation bivariate protocol bit classical protocol bit randomization theoretic guarantee contrast characteristic difference communication however communication imply setting decentralize communication low bound distribute certain conclusion message send rise consider machine bit processor receive single dimensional centralized study estimation particular work focus store machine encoding sequence rate converge formulation communication attain centralize say finite ask statistical rate paper decentralize minimax protocol single message pass interactive protocol must protocol fusion center past message depend protocol simplicity use indicate send independent enforce shorthand contrast protocol interactive protocol stage message particular message fusion public machine reading incur communication think may processor centralize message message measurable message interactive protocol define risk central achievable message classical quality protocol estimator
brevity vector correspond complement eq compatibility eq bound compatibility later certain law ready direct consequence oracle eq implementation ensure asymptotically problem et although need consider value p nc oracle corollary suggest method maximize although compatibility sufficient compatibility would correlate index index whose corresponding thus low theorem compatibility corollary satisfied size xy irrelevant parameter follow construct drop column index implement find objective x large optimization quasi newton prefer fast super necessarily algorithm optimization bfgs memory test smooth bfgs hybrid result implement bfgs cp eliminate remain minimize coefficient column adapt use informed involve unnecessary rt true example select elastic next positive elastic net well correlate overall perform application cancer et al sample patient I severe patient stage mass scan disease could protein fit peak deterministic peak algorithm al obtain z peak location severe reasonable intensity peak available www mixed stage peak peaks choose et al elastic choose peak peak al peak common peak exist sparse explanation useful implement without impose assumption notable restrictive number example fan al extended analyse use regression propose know art predictive seem high dimensional situation promise sparse toolbox valuable comment insight manuscript new variable variable mean exhibit grouping screening result theorem standard deviation performance grouping penalization screen dimensional great predict response set add least predictor wide achieve one essential goal regression identify literature area early high dimensional minimize penalty ridge regression elastic combination penalty include selector predictor sure root involve minimize plus call recovery require noise euclidean base norm gain group property elastic net distance variety particularly signal variable matrix regression matrix set associate whose observation could however reconstruction parameter solution independent challenge detect minimize euclidean penalty minimize distance minimizer function norm one combine manner net I criterion ridge elastic net combine features ridge penalty lasso like square root grouping transformation design response note covariate nonetheless positive number large minimization exclude exact solution denote angle global must complement index build objective circle distance highly estimate standardized objective enable impose restrictive concept simple relative component relative grouping relative f p theorems minimizer penalize euclidean consider important highly base relative minimizer standardize response tx jx ix grouping perfectly correlate special grouping detect
lexical relation rather design lexical need lexical cover division company sub part mention herein also cover lexical water death directly whereas lexical directional specifie entail hold agree part whole lexical issue break category category ten claim eight lexical cover water death water instance cause relation handle table believe argument category incorrect offer lexical category relation hypothesis semantic relation treat algorithm research even wrong reason treat relation use relation evaluate train lexical case lexical readily expand discover place system include module early typically use lin understand inherently asymmetric symmetric rough replace without change sentence specific general paper measure degree entail describe lexical lexical lexical include accept kind lexical exclude lexical context natural paper lexical body relation classification semantic part semantic semantic relation nine task relational nine semantic lexical paper relation important involve e algorithm semantic relation supervise although lexical offer elegant training issue noun phrase noun entail head noun cat entail cat little effort seem applicable pair label amazon cover wide range semantic measure value entail generate score give example belong ap measure value score measure accuracy precision distributional inclusion partly inspire precision useful ap retrieval system query engine return rank document degree relevance assume relevant fraction rank list document label ap range ap several ap typical typical document irrelevant emphasis entail example scoring challenge equal text sentence entail therefore ap precision respect pair manually dataset measure assign word sort pair bottom rank label otherwise let document bottom experiment increase sensitive top happen bottom list low ignore prefer top list poorly list ap originally design retrieval precision truly query system truly tradeoff precision cost harmonic design natural size equal sized difficult class well balanced ap measure depend class practice two variation weight average finally usual accuracy fraction percentage discuss describe introduction behind word survey development tune optimize describe matrix development context choose vector svms supervise svms radial rbf experimentally find similarity measure asymmetric measure lin achieved mean eq lin terminology terminology terminology algebraic terminology reader easily believe helpful connect view notation word matrix correspond word row raw occurrence transform represent importance negative survey association raw set notation word correspond think word correspond nonzero cell value range thus normalize range close interpret importance feature information retrieval context inclusion tend context tend broad term include feature q range among feature analogous ready originally measure retrieval consider ranking include otherwise word weight word seem algebraic lin lin combine varied remove ranking impact low pair classified entail describe support utility row row correspond gram whether appear context gram window percentage engine raw occurrence frequency matrix gram th matrix calculate frequency retrieve detail matrix truncate value truncate equation density context column sum behind sparse assumption false well smoothing vector label entail enable cell th word th experiment context smooth svd svd three matrix orthonormal I unit top singular matrix k minimize error k norm vector vector represent concatenation normalize need singular power familiar information retrieval use word matrix lexical semantic evaluation follow mention say tendency learnable occur contexts context tend give concatenation lexical minimal probability binary value class fit output regression try kernel dataset polynomial length lexical thing know lexical reliably recognize pair appear datum broad range domain domain matrix design measure similarity topic domain measure similarity similarity relationship usage function near word occur part varied work context combination domain tune domain function matrix corpus column correspond entry gram complex column aspect matrix remove row value angle likewise let recall difference hypothesis tendency correlate learnable difference tend sim domain similarity reference difference spatial death suggest involve whereas death involve suggest similarity perhaps similar similarity death perhaps death see make reference english represent english vocabulary english wide range concept may inefficient hand look example supervised development function setting normalization generate class describe dataset lexical semantic lexical report evaluate word pair label entail create add pair labeling detail lexical label pair agreement two three size entail every word time bank education state fortunately use describe entail label noun noun pair pair validate although balanced semantic relation word pair appear vector create measure relational package gold rating dataset contain word label convert label word relation ten level category five ten nine distinct type come add inclusion schema song phase song amazon worker phase nine ask word semantic nine pair relation example original lexical improve ten low rate original pair reduce pair word pair label example car object create car increase pair map entail mapping word pair belong either none map word label label balance make balanced dataset remove pair label pair interpret label see table f goal entail f text however temperature due nine semantic functional class inclusion collective inclusion collective inclusion car whole member mass moment stage activity item room whole object ice j ex car similarity similar similar sound contrast contrary contrast reverse directional right slow g walk attribute attribute attribute object attribute attribute act attribute typical live act act attribute slow attribute act object act example object attribute act attribute act act act object patient relation act relations act relation object speech relation relation cause cause cause object purpose agent peak g cause cause purpose time product activity drive location reference representation person create mapping agreement table consensus label consensus five class class independently annotation entail paradigm instance relational schema interpret pair light schema interpret category likewise entail pair entail wrong proceeding compare table table mention assume none pair label label relational lexical lexical table percentage agreement manual automatic annotation percent versus percent agreement versus percent percent percent lexical manual varied assumption word pair belong reasonable manual agreement level lexical discuss table manual relational support hypothesis report pair entail support relational lexical capture automate manual relational definition manual labeling accordance evaluate lexical first word equal test size maintain test ten level table class inclusion class category inclusion pair inclusion entail category class inclusion whole contrast attribute attribute relation total five optimize set maximized measure accuracy tune tune try try increment pair achieve setting use data tune domain value respect test three measure accuracy usually column confidence f acc accuracy ten category set substantially expect approach near attribute substantially g attribute relation category part similar attribute cause explore contribution exclude similarity reference difference different individually tune acc difference accuracy significant together fisher level space two base discussion general dataset select set matrix choose bold font seem might difference difference significantly pre f acc word pair way experimental splitting evaluation validation evaluation pair common supervise easy pair share fold cluster evaluation evaluation cluster ten fold fold ten share two give fold allow rarely pair fold clustered evaluation entail balance clustered step remove pair fold train dataset balance randomly remove label measure threshold four fold fold validation whole split tune supervise cluster balanced challenge believe evaluation realistic system module pair field evaluation come field usage acc standard cluster cluster balanced evaluation achieve accuracy statistically accuracy supervise standard testing tune threshold gap question qualitative training testing helpful qualitative gap qualitative gap challenge quantitative face past comparable evaluate recall measure set use word class way balanced setup already evaluation table evaluation accuracy fisher test f acc cluster standard use setup likely minor setup accuracy dataset tune give context accuracy setup accuracy setup seem dataset nonetheless accuracy summarize evaluation bold accuracy accuracy nine type accordance relational definition lexical explain design mind perform poorly surprising well lexical difference learn dataset dataset cope qualitative difference positive relation relation dataset reach qualitative able bridge argue entail also summary report instead put emphasis entail clustered setup bold difference similarity difference model lexical suggest beneficial construct lexical manually lexical tractable supervised effort involve designing indicate supervised yield manually design evaluate application module well competition room contextual substantial future lexical derive score show novel hypothesis achieve believe progress come wide hypothesis lexical semantic relation handle hypothesis reject algorithms three degree task combine one voting value focus idea phrase promise phrase table see density class whole strong lexical relation relation relation find particularly inference rely lexical find difference three suggest difference make feature recognize lexical lexical learn semantic result lexical build bridge lexical semantic hope field acknowledgement thank copy answering thank provide answer thank natural engineering helpful comment lexical r national lexical identifying entail one construct asymmetric treat learn recognize machine relation experiment strategy similarity context word second relation represent pair concatenation supervise feature instance word feature similarity semantic extensive three similarity dataset dataset past make connection lexical semantic language relevance question answer translation involve sentence determine whether entail gold establish text entail mean infer mean text typical interpretation text entail rich challenging paragraph many recognize entail recognize text entail entail lexical definition lexical useful lexical semantic apply lexical three matrix represent word context context consist distributional distributional hypothesis occur tend algorithm average precision distributional inclusion attempt context capture inclusion word broad distributional inclusion prefer call distributional inclusion
goal learner modern make effective decade great broad quality intensive design code feedback ii scalable ml base learner learner learner obtain content content question automatic author recently sparfa sparfa knowledge give typically term leverage concept value response correct incorrect sparfa estimate association via concept profiles ordinal sparfa sparfa enable first ordinal sparfa tag exploit often sparfa tag exploit tag keyword characterize exploit question framework new pre nonetheless response superiority ordinal sparfa tag method ground truth real ordinal sparfa tag outperform art technique miss ordinal learner abstract response provide question sparfa characterize learner incorrect response question I question association learner question difficulty extend sparfa underlying score learner order label response encode question model positive quantity uncertainty learner answer incorrectly gaussian learner answering question slack variable set learner order accord quantization bin satisfy upper equivalent relation x ss represent normal matrix intrinsic form furthermore matrix low bin emphasize propose original sparfa ordinal affect statistical parameter ill pose since ordinal three accounting redundancy assessment response live dimensional question e question chance score value good vice mm reasonable context discussion particular often question learner mm tag question association tag single significantly improve limited interpretability sparfa rely ad hoc processing provide tag concept contrast tag oracle explanatory predefine tag tag complete association predefine association develop sparfa ordinal tag enforce maximize subject account prevent norm norm freedom overfitte different preference observation problem problem hold third coordinate variable normal variable set iteratively factor hold optimize hold hold outer loop ordinal sparfa iterative optimize precision instead bin boundary intrinsic difficulty bin instead emphasize optimization straightforwardly via keep ordinal sparfa directly bin keep fista fall fista constraint penalty regularizer otherwise algorithm two py shrinkage step suitable size form aggregate eq singular decomposition suitable constant step value ordinal sparfa tag tag question tag tag penalty predefine predefine might support reduce discover concept solve analogously except part operate separately index index give ordinal tag necessarily converge global optimum result ordinal sparfa tag optimum furthermore close optimum sparfa tag optimum sparfa tag synthetic ground truth leverage tag constraint two real ordinal sparfa collaborative ordinal response ordinal sparfa estimating ordinal sparfa tag priori synthetic experiment trial retrieve ordinal sparfa scale consider concept arithmetic simplifying expression concept concept geometry slope concept polynomial concept concept matter water heat energy concept circuit force formation motion concept concept concept property environmental energy force synthetic test evenly concept question size fix concept impact size first ordinal sparfa svd vary corresponding learner fix ordinal sparfa ordinal sparfa oracle k svd ordinal sparfa variant sparfa know accurately impact quantization bin bin quantization need value outperform svd decrease conventional sparfa sparfa approach svd quantization bin increase superiority ordinal sparfa tag sparfa particular tag impose nuclear constraint matrix tag oracle provide tag algebra dataset school carry crowd multiple question cover geometry graph manually domain expert manually map bin follow totally wrong correct show concept ordinal sparfa tag circle concept question label difficulty connect line line association sparfa tag entry red dash green solid association entry discover ordinal sparfa tag sparfa tag concept enable interpretable learner directly tag learner tag profile use pls learners association tool expert tag association course ordinal sparfa tag analyze answer incomplete entry label tag sparfa tag nuclear match pre association association discover pre tag domain expert association school algebra pre specify tag indeed question interestingly eigen learner investigation phenomenon learner response collaborative filter treat ordinal number rely ordinal logit ordinal optimize ii bins learner iv nuclear constraint test train fold rmse demonstrate nuclear norm ordinal sparfa ordinal sparfa suggest consider ordinal enable accurate prediction response variant sparfa boundary identical emphasize sparfa state predicting response also interpretable key application bayesian belief analyze trace learner concept
q vanishe insight overlap neighborhood via overlap bias intersection bias conservative bias g guarantee e together factorize intersection r ss q v c f conjecture institute cs university sample great large problematic grow scalable planning factored exploit apply planning reinforcement able solution planning planning planning achieve performance gain carlo solving become intractable grow problematic system specifically observable agent view action centralize agent intractable novel online planning exploit structure base exploit mass agent interact subset value factor factor applicable important adaptive translate problem planning planning planning planning efficient non factor able effectively exploit locality bayesian plan every agent receive allow team agent act centralize communication free cost delay tuple state reward immediate observation horizon controller joint receive joint remainder section inherent many focus full specification determine belief policy extract rs pz space continuous scalable monte plan current belief expensive create simply action history root use trajectory visit search relevant action history reach time domain consider know effective rl call bayes ba planning enable advance plan particular ba utilize intuitively observation result take represent see action transition state actual count count ba ba extend yield adaptive ba ba state count vector reduction possible intractable sample planning scale suitable planning elaborate sketch locality agent problematic branching though theoretically use severe must often plan previous sample particle filter bad act independent action confidence joint exploration principled may completely try action individual illustrated amount water house action factorize approximation certain compactly represent interaction paper shot cg specify payoff interpret select cg follow cite suitable factorization easily identifiable payoff function exactly factor maximization perform algorithm apply advance necessarily require seek joint close maximum unknown technique contrast factor directly try estimate maximize action action predict mixture joint scope use maximization remainder integrate simply expert expert particular keep mean payoff efficiently additional allow keep track turn integration describe algorithmic factored planning exploit expert variant address joint second address joint mass achieve complexity factorization joint method beneficial factor remain retained maintain maintain set component accord factor u q history store value count style parent style thick distance black draw parent thick circle mm draw child edge style node circle statistic via application ba directly address joint possible realize limit call try overcome joint factor conduct maximize upper bound set agent relevant generalize maintain node chance produce particle distance node draw child parent thick distance child edge parent draw thick mm child edge style node draw thick level circle circle child circle parent mm black child style draw edge child draw thick sufficiently factored factor suffer local depend future policy well past include long ft monte ft practice problem reinforcement network produce high exhibit locality factor good complexity modify elimination complexity width effectiveness compare factored planning sensor house also spread house align along axis reward track agent nothing agent correctly break factor side break along agent sensor agent agent experiment number core ghz gb compare flat code difference use planning already factor fs factored agent form simulator set compare action poor bt horizon poorly simulation increase converge ft number simulation able near well ft fs continue reach ft see sensor outperform fs simulation ft low planning target position factor episode illustrate benefit exploit mass agent ba end episode state count observable environment extremely observation indistinguishable therefore hard compare baseline apply simulation proxy expect ba agent fs ft number ft learn quickly fs horizon due horizon ba factor outperform visible grid ft fs outperform ba episode learn effectively game game continuous action action available many also design observable approximation contrast promise branch factorization agent individual reward reward incorporate consider decentralize factored distribute nd strict assumption action past observation nd impose factored function restriction instead know action factor nd method mapping factor locality factored conditioning central information perfect locality factorization function apply factored fashion simulator upon perform factorization shot relate since replace obvious integrate focus minimize issue factor relevant receive centralized maker potentially relax exploit approach joint observation grow agent na I agent exploit structure novel factor factored tree greatly increase scalability planning investigation scalability four ba space self interested support fa describe start root empty sample current comment algorithm filter simulate pseudo factor comment highlight joint initialization select maximize action return component e n simulate e factor ft particle filter update computational
fact distinguish large necessary large enough anomalous object detection introduce mmd construction test present computational brief introduction idea mmd include distribution rkh mapping refer hilbert reproduce p embed element many gaussian laplace study embedding without distinguish discrepancy mmd rkhs shown namely achieve maximum unit sample given construct interval compute expect anomalous differently determined corollary follow characterize anomalous test successful constant threshold boundedness many gaussian kernel theorem imply minimum candidate anomalous interval exclude anomalous require event get asymptotically small corollary threshold constant unknown priori satisfie minimum corollary prior capability anomalous resolve anomalous length resolve big anomalous first dyadic interval dyadic let dyadic show interval dyadic interval dyadic union dyadic error occur u x go infinity test demonstrate unknown numerical two mixture anomalous object bernoulli distribution plot normalize see converge agree state ht ht test size change minimum size error although respectively test htb affect study plot probability error versus correspond similar first dropping gets imply guarantee mmd unknown mmd choose laplace likely threshold go infinity minimum anomalous suggest theorem run mmd advance set average unknown mmd demonstrate asymptotically agree minimum length mmd much fast case importance prior knowledge mmd paper investigate anomalous arbitrary unknown mmd embedding rkh anomalous equal successful infinity reduce mmd guarantee successful technique study believe involve distinguish compare test anomalous line anomalous interval distribution generate node anomalous interval sample generate anomalous reproduce mmd show go anomalous interval asymptotically reduction show interval problem goal detect existence anomalous anomalous network take anomalous exist object nan compound object may sensor sensor take measurement anomalous typically occurrence arise anomalous dna detect anomalous detecting existence node embed embed structure line node multiscale analyze model study scan statistic result inference mean mean node structure anomalous subgraph detect unknown network graph combinatorial geometric small cluster anomalous connect subgraph structure detect anomalous incorporate successful majority variable application case differ mean either advance hence anomalous interval priori detect anomalous although simple already study deal reproduce introduction distinguish evaluate embedding approach sample discrepancy mmd embedding distribution shift existence anomalous size become model anomalous accurately interval object successful goal characterize length anomalous node infinity order successfully existence anomalous summarize main nonparametric model detect anomalous interval network length characteristic must successful large remark anomalous artificial test mmd test infinity candidate anomalous interval scale successfully detect exist anomalous interval length depend mmd adapt efficient algorithm test mmd performance propose numerical organize performance guarantee provide numerical result remark future consecutive node length interval denote associated variable set candidate anomalous minimum candidate anomalous impose requirement explain remark distribution line random variable put context scenario noise activate anomalous arbitrary instead one sample practical system collect stage detection activate occur initial serve hypothesis scenario node detector
user item rank triple item tweet predict item receive past predict user f rank user prediction base learner multiple tree gradient adjust forest subset split dataset user prune keep tree leave shrinkage early additional validation round feature split rely upon implementation directly goal ndcg summarize feedback express rate simple recommender ndcg factorization plot rating relationship rating tend receive item outperform fm fm tweet rating alone engineering fm include baseline write place team challenge fm competition ir metric ndcg result collaborative variability find factorization item work well square also help improve outlier help individual ndcg tweet ndcg threshold selective sampling successfully knn model rate try improve collect movie profile g release tag actor find year release improve paper base dataset provide evaluation protocol reflect tweet receive compare user assess triple tweet receive triple identify predict level g require user user popularity twitter access characteristic velocity recently author make effort characterize exist approach introduce aware recommender additional contextual company people recommendation rich collaborative art context recommender primarily collaborative rank cast recommendation rank absence rich feature learn rank rich user tweet part ranking construct neighborhood collaborative collaborative user twitter twitter build feature tweet capture rating express tweet collaborative attractive tweet insight business success public neighborhood performance direction future datum city usa ie web service amazon netflix twitter recommend item present would history cf focus assess recommend recommender rather characterize optimize item cast tweet transaction rating learn scoring optimize user ndcg conduct version challenge effectiveness information storage artificial filter deal information user base netflix amazon twitter customer recommendation rating prediction recommendation cast among item page article book place retrieve list advance year supervise application gain relatively little potential reason rank usually rely characterize maintain knn prefer cf item collaborative extract user tweet create amenable leverage directly ir ndcg user task training tweet provide tweet interaction possible tweet user construct rank rank test sense ndcg test unseen ranking return rank tweet relevance preference tweet comment movie video give tweet movie tweet create relevance indicate tweet document triple training number triple rank value learn feature item tweet respect I history boolean rating friend tweet give movie otherwise rate ratio friend aggregate tweet aggregate boolean mention update tweet tweet tweet infer tweet actually use include tweet tweet additional observe count extract field item tweet extract user tweet tweet parameter ndcg optimize ir ndcg additive tree pairwise learn method capable ir metric change iterate detailed scope refer comprehensive dataset part challenge challenge summarize table note one respectively figure interaction user item tweet triple note user outlier
cnn convolution size convolution pooling follow pooling extend first regularize tie train similar rate momentum epoch ten tune get final cifar cifar view arbitrary become scale indistinguishable large equation scale activation canonical filter image describe section transform filter norm original take filter cnn report filter invariance measure q take random set row filter learn filter learn row pooling usually invariance filter accordingly size clear sensitive column transformation need consider back propagation normalization become apparent transform scale pattern achieve invariance scale comparable filter adapt consequently invariance filter get example feature map pool normalization original middle filter clear applying preserve characteristic filter scale fix significantly train scan activation pooling layer scale pick filter visualize dataset scale cnn cifar statistically gain cifar drop drop central scale cifar verify pick different central area cnn compare drop whereas htbp lc dropout cnn cnn maxout maxout network error gain cnn network nevertheless address goal maxout simply extra result encourage benchmark higher suggest take suffer severe error reach model cifar error scale cifar incremental invariance come current cost linearly column refine expect explored name cnn begin refine epoch name continue baseline cnn use build refine softmax small incremental summarize row half cost fourth although get well cnn maxout unit able reach result incremental help balance gain efficiency incorporate flip cnn learn column learn detection localization task preliminary nice trade training remain concatenation column wise column instead canonical column plan imagenet zhang microsoft microsoft zhang convolutional human popular make big data augmentation extensive convolutional neural design incorporate column column focus scale filter transformation deal experimental scale exhibit classical vision primarily amount training convolution network competition cnn human image localization trying learn contribute field allow recognize regardless pooling contribute slight cnn deal shift invariance capture plain cnn rotation introduce filter cope magnitude proposal deal size scale use explore observe filter detect adopt design column scale cnn unlike filter among column exhibit deal indeed scale become dataset well previous complementary technique find incremental refinement dramatically reduce organized present foundation analysis one scale free stack convolution filter build complex representation representation mean activation layer job exist cnn architecture jointly scale variant map unit field save filter learn popular inspire exist cnn scale independent cnn specialize one crucially parameter convolution stay augmentation intuition mathematical htbp column stack scale image feed max pooling conventional filter share filter column keep canonical call canonical filter detect architecture top softmax case discuss canonical canonical convolution image scale expect another transform generate another convolution property want satisfy easy filter invariance preserve layer reach feed separately relu nonlinear max keep recursively keep relationship canonical image column generate fit column scale fall scale two neighbor column end eliminate variance column convolution transformation give canonical equation transform filter system problem make easy equation
perfect portion energy try capture energy important tolerance solving unique solution full give desire framework base mode supervise include experiment design optimization category sampling notion dimensionality measure signal variation manifold laplacian maximize try relate zhang experiment design consider way embed approximate try whole datum local reconstruction generalization global relate note eigenvalue speak function maximize try ensure unlabeled cf section graph work node maximize maximize unlabele agree give justification cut also motivate partition base heuristic say node graph compare three active allow prediction implementation method good cut learn package filter method fix accuracy also randomly sample rate effectiveness circle toy figure comprise connect ensure connection x unweighted see circle additionally evenly select circle accordance handwritten digit letter computational complexity scalable dataset graph typical experiment semi supervise classification handwritten digits pixel select digit vector digit construct w ij intensity rd th heuristic additionally restrict e remove node near neighbor construct label report prediction use semi supervise repeat illustrate notable criterion maximize text partition experiment document graphic os ms mac hardware clean remove document frequent feature tf statistic capture word corpus frequency document document feature document pairwise similarity feature vector error inherently letter consider letter alphabet set alphabet create consider alphabet digit construct weight node rd neighbor neighbor criterion perform semi label chance select point class effect repeat accuracy remain largely unchanged dataset slight improvement result agree membership frequency well cut word loose cut membership fraction maximize tight pick capture novel batch active semi signal active uniquely signal lead efficient intuition try efficient conjunction term cut hope desire accuracy useful batch batch improve offline graph vertex undirected graph similarity capture sample graph shannon signal reconstruct base selection frequency effectiveness classifier design pattern many real available effective learning technique inherent expensive learner pick informative representative label goal ability give small label query propose novel advance sampling graph stream problem pool semi supervise datum crowdsource would label focus batch label leave semi formulation node connect weight feature task function scalar depend whether belong choose label conversely space membership e unlikely lead membership semi view signal many technique harmonic consistency graph approach quantify quality methodology capture low graph pick node unlabele node pick lead semi inversion decomposition matrix pose implementation active suffer give uniquely significant well establish processing contribution wavelet depend local interpolation newly develop unified perform provide uniquely subset interpretation numerically make semi closely tie theoretically justify large show test rest review signal section derive active summarize experiment present section conclude remark formulate undirected context respective degree connect graph graph shall laplacian set eigenvector subset index submatrix sake brevity scalar value discrete ease notation paper signal membership interest graph realize signal value sample vector reduce sample include membership upper signal uniquely reconstruct notion fourier indicate variation high eigenvector basis signal smooth pass frequency vanish define restrict space signal note frame adequate graph label penalize thus equivalent relaxation nature study empty e include node energy si tends maximally find cut adapt cut discuss signal linear correspond recovery condition solve least square square eigen expensive iterative onto propose reconstruct case problem constraint pass graph graph domain operator iterative actual depict non asymptotically regular converge pass exact complexity possible use truncate spectral distribute fashion chebyshev smooth sigmoid limited supervise expect slowly decay end improve slightly signal semi supervise propose edge membership illustrate experimentally vertice membership see cut vertex set recover membership active aim sampling maximum good set data amount reconstruct strategy cut frequency maximize detail multi class class true membership class indicate membership signal membership predict node label unlabele supervised summarize label set q solve add summarize label finally membership function convex set connect simplify
inequality partition many union inequality eq b nm formulate loss distribution obtain generalization probability substitute substitute c cat mt max bad attribute front open cover blue negative one ac algorithm relate advantageous work process subsequent jointly find task criterion optimize expect classification able discover task study label achieve expensive consume application categorization information several relate experimentally allow transfer multi learning correspond relate parameter representation concentrate similarity distance propose corresponding task lie prototype effectiveness several treat realistic outli task similarity task negative scenario learner task source show vision detection hand image categorization though domain scenario multi area relate propose decompose adaptation school suppose learn process meaningful student gradually increase accumulate learn inspire propose manner previously learn process datum student school crucially performance question pac theory prove generalization representation use quantifie solve world multi jointly reliably discover advantageous idea approach literature represent linear combination allow overlap far extend experimentally subspace base performance underlie feature grow type vision problem fast method experiment setup clearly baseline even require method vector original particular relax task achieve introduce graph chen method amount regard similarity sequence study mainly experimentally gradually lead similarly automatically choose solve optimize pairwise preference sequence user scenario label one read vector term multi task object predict attribute share share assume learner predictor performance learner expected propose task domain adaptation specifically process sequentially order symmetric subsequent task domain method svm computer vision adaptive svm optimization q standard simplify vector equally need subsequent automatically define beneficial examine average svms algorithm task predictor inequality sampling set harmonic z gauss function hand inequality learner however distribution unknown contrast bind right I x monotonically specifically separate close wrong corresponding may subsequent task ensure lead right hand hold fix require search incremental procedure perform successively minimize already include order minimize every low fit human intuitive simple proceeding every summarize input minimizer return order task join task within subsequence iteratively learner solve continue learner empty transfer find continue compare subsequence continue task please refer claim task manner use publicly dataset difficulty svm solve svm accuracy baseline semantic human annotation inspire diversity instead diversity regularization trade experiment validation experiment figure outperform mt case task advantageous support claim task effective jointly task equally baseline improve single baseline generalization perform mt method example expect explain hyperplane unable share mt hyperplane g lead task solved reporting finding figure svms differ outperform learn baseline task order class semantic par case hardness coincide opposite task machine task medium random learning per easy medium define human visualize finally baseline visualize horizontal slice reflect competitive well fix clearly learn algorithm term combination achieve never sometimes even order beneficial strategy annotate attribute attribute top rank bottom class rank see balance randomly sample descriptor descriptor baseline attribute information transfer subsequence attribute order option ht average result main see baseline confirm advantageous equally strong affected transfer unable one perform poorly sequence compare task row task diversity baseline however perform well baseline conclude one affected transfer task diversity bad baseline pattern relate frequently subsequence next follow attribute always often end subsequence attribute either two task transfer attribute half relate solve task influence overall theoretical propose principled sequentially effective jointly solve effect overall able beneficial limitation transfer solve plan pac generalization sequential learner observe sequence I sample task share output use task arbitrary fix posterior prior task predictor associate randomized learner task minimize classifier task directly compute empirical base bind quantity fix learn follow nm kullback
set correspond support literature state form lasso asymptotic author though assumption net author impose weak restrict special variation belong partly relative instance well norm composition operator instance whose fit framework lastly similar infinite dimensional recovery interesting finding compute degenerate identification illustrate mutual coherence mutual degree ill conditioning correlation low frame fine variant cumulative propose account vector recover introduce recovery norm complement view weak criterion derive subdifferential check equivalence sign proposition order criterion coherence elaborate discussion establish lasso indeed ensure exist degenerate indicate error whose equivalently non degenerate pre rank incomplete nuclear norm decomposable regularizer measurement non degenerate measurement matrix identify correct comprehensive jx jx sensitivity minimizer sensitivity seek ensure lipschitz assess stable manifold unique remain start work smoothness show broad function enjoy powerful calculus sensitivity convex partial closely decomposition fact minimizer restriction manifold hence sensitivity partly perturbation feature variation function analytic analyze small perturbation see important risk argue performance solve sensitivity involve sensitivity perturbation sensitivity partly reason smoothness additionally regularizer notable manifold non respect actually single smooth hope observation see characterize precisely outside actually set point locally minimizer motivate coin transition boundary respect perturbation check hyperplane crucial able write explicit derivative theorem solution mapping every restrict hessian surely goal show lebesgue everywhere structure function algebraic broadly various area wide applicability largely semi algebraic possibly function semi algebraic minimal instance section qualitative algebraic share big function minimal structure correspond sense prominent algebraic rational minimal formulate framework result section algebraic stable algebraic algebraic practical adjust statistic density though sake white realization quadratic risk solely nice risk reliably reliable instance risk show value mapping choice see mapping quantify complexity statistical estimation empirical differentiable lebesgue everywhere freedom define lebesgue everywhere get intuitive notion close valid lebesgue appropriate main lie show mapping fact lipschitz argument sensitivity rather formula validity lebesgue e subtle partial smoothness rule hold every turn empty reasoning regularizer precisely exist constructive build hand risk differentiable lebesgue sure turn stein sure together theorem stein algebraic lebesgue minimize hard problem remarkably extended risk prediction risk define section review relevant smooth semi partly smooth prove feature toward sure sensitivity manifold appropriate key put emphasis unbiased generalize thus extension sure exponential family extend well sure extensively various lasso read q hold full extend analysis formula projection prove extend treat case close give sufficiently formula group orthogonal heuristic prove group denoise norm derive challenging address divergence expensive approximation prohibitive may difference monte analytical serious think approximate sure recursively though problem type mention attention proximal splitting one possibly quasi newton detail regularizer intend non replace subdifferential assume finite value close convex guarantee assumption make global get optimization handle regularizer cast cone constraint barrier interior enjoy fast quite costly become prohibitive dimension increase study frank wolfe solve forward backward variant projection many solution subproblem structure rank total rank recovery signal processing see e homotopy minimization regularization regularization see crucial path affine lar accelerate homotopy compute homotopy increase monotonically compressed sense homotopy empirically ensemble threshold bad case thus homotopy take cost per homotopy like ad hoc large imaging solver medium machine homotopy need extension homotopy method change see five idea pass pass form regularizer nuclear comprehensive review behavior broad ensemble split iterative tailor structured essentially optimization increasingly concrete general closed g smooth proximity operator form proximity splitting algorithm possibly approximately individual g proximity operator operator separately iteration never sum function composition iteration rigorous guarantee quantitie popularity image sublinear scope huge research field instead brief popular optimal fista guarantee scheme good elaborate section minimize either space projective dr interpret minimize admm dual conjugate dr apply whereas initially close bregman primal objective starting read sequence smoothness whose result favorable case degenerate typical circumstance compress local exhibit global sublinear term manifold regime general manifold identification degeneracy result partly cover linear fidelity satisfy convex decomposable partly smooth general variety manifold author show identification manifold associate partly projection newton extend identifiable surface smooth non simple partly necessarily prove identification remain review work work see unify namely smooth one regularizer low recover analysis chapter list believe important focused fidelity regularizers result chapter extend proper generalization regularizers regularity need deal difficulty tackle instance property hold impose bottleneck result present recovery less non recovery synthesis set input stand trivial adapt dictionary extensively regularizer scheme vector space extend infinite di far stability constant bound scale regularization hilbert banach banach space measure stable highlight compressed norm synthesis regularizer difficult smoothness exact turn iterate raise hope acceleration study guarantee extend splitting importance acknowledgement european research project sigma would like unify universit paris universit chapter dimensional problem linear measurement w observation offer acquisition processing acquisition hardware imaging regression problem assume section explicitly assume rest knowing measurement much ambient general ill condition entail ill pose might think convolution camera spread account low sensor medical imaging operator possibly partial transform propagation sensor imaging amount wavelet impulse response approximate wave propagation medium column covariate argue ill pose inversion plausible solution include adopt though class elaborate chapter reader refer discussion school notion conditional introduce refer solving stand fidelity fidelity use instead underlie noise instance stress provide fidelity replace smooth strongly focus sequel quadratic recover play give brief overview tackle class optimization penalty counterpart chapter performance guarantee brief account non approach fidelity theoretical hereafter optimally automatically offer neither strictly turn existence minimizer minimizer exist mild constrain formulation parameter share solution view problem reference detailed valid chapter one literature value increase fidelity perfect limit jx transpose pseudo hull interior topology boundary manifold tangent say say otherwise subdifferential function jx normal support tangent set compact subdifferential reflect subdifferential differentiable singleton illustrative subdifferential value separability subdifferential I differentiable processing use collection structure datum manifold high collection idea manifold smooth regularizer detail turn natural unified description thus manifold element penalty typical simply notion underlie description recover noisy accordance notion key whose statistical theory see set inverse problem restrict inversion correspond jx sparse signal subspace I index support use combinatorial pseudo jx intend way instance piecewise manifold parameterize location thus selection literature approach bottleneck close thus operator alternative point hard thresholde iterative scheme consist instance reference algorithm manifold actually pursuit comprehensive reference therein chapter regularizer complexity manifold regularizer prove hull convex penalty design necessarily surrogate remainder give convex formally subspace terminology tangent property belong differentiable essentially affine illustrate formula subdifferential obtain sparsity partly function partly smooth behave smoothly move move hereafter finite contain smoothness around continuous partly manifold exist neighbourhood partly relative uniqueness around partly take example describe partly use image machine sparsity anti calculus create pre quadratic regularization check partly interpret prior restrict pseudo hull restriction pseudo norm unit norm sparsity trace back decade application rigorous recovery appear mid regularization literature pursuit name dramatically capture pattern non norm partly manifold first learn typical structure see reference natural image model audio structure channel partly whole ambient example impose partly separable smooth fundamental attract year linear image texture wise part cast texture solve variational favor principal pursuit decompose superposition component finite partly observation contaminate important recover assess decay present ensure recovered body vanish terminology solution precisely note equivalent convenient first optimality ensure stability minimizer inside subdifferential precisely non degeneracy previously degenerate establish valid regularizer without particular assume consider choice minimizer detail plain tell noise terminology dimension likely restrict smooth regularizer fulfil reason show slow uniqueness condition might find depend choose constant get close intuition degenerate source condition literature problem overview implication check degenerate trivial give compressed particular linearize partly smooth regularizer full constant fine show quadratic decay extension hilbert smooth lagrangian dimensional banach ill inverse convergence rate non degenerate degeneracy convergence isometry rip homogeneous result equivalent regularizer recovery compress suitable widely isometry rip restrict isometry small show exist degenerate discussion sense generalized frame sparsity completion quantity random instance entry rip remain subgaussian constant expense comprehensive rip give rip base uniform recovery wave rip recovery guarantee claim uniform mean rip gaussian measurement obeys holds depend gaussian low completion minimizing hold high lead bound inverse measurement overhead necessary handle norm imply existence closely call dimension aspect exact
thorough topic simple intuitive mathematical algebra technique svd apply world thorough foundation machine learn paper spirit rigorous mathematical appendix proof complete understand work provide idea avoid please contact I suggestion correction comment spectra appear unclear redundant problem fundamental obstacle complex web indexing toy physics study system mass ball ideal along motion direction explicit express let alone axis measure decide live movie interest movie camera indicate position projection axis camera arbitrary angle angle might minute big remain simple equation priori axis camera world system question record need deal world problem toy air toy challenge keep mind understanding systematically use basis express new reveal goal pca axis goal determine redundant precise definition datum individual record etc set point camera ball position column contribute ball position entire record hz record us term equivalently lie orthonormal algebra know vector unit basis orthonormal toy camera reason naive record position camera mean camera algebra row row vector matrix construct naive effective record another reader notice linearity simplify restrict set pca moment let matrix record quantity row column equation represent change interpretation transform geometrically rotation obvious writing explicit dot note recognize dot product row set basis linearity find appropriate change exhibit beyond linearity arrive section build answer question two lie matter absolute strength common snr snr noisy camera camera straight line line motion signal noise indicate line diagram cloud include thin snr bad direction large interest thus dynamic exist variance snr naive direction variance intuition indicate large good naive direction motion dimension redundancy evident record ask record reflect range leave panel depict apparent depict plot nearby plot clearly panel meaningful calculate vice versa response express variable behind reduction identify line generalize notion dimension sample number individually define forward absolute magnitude redundancy uncorrelated convert express covariance dot I slight generalize arbitrary vector additional interpretation one trial arrive element dot measurement summarize property type covariance reflect redundancy diagonal diagonal magnitude redundancy option want section state goal redundancy measure optimize diagonal say successive order arguably vector orthonormal orthonormal pca act rotation maximal normalize maximize save restrict direction direction save select order principal algorithm true benefit assumption notice gain importance direction order variance implication arrive behind linearity problem area extend regime see large belief sometimes incorrect simplification algebra decomposition technique highlight aspect straightforward understand important pca eigenvector measurement goal follow n identify last line recognize eigenvector symmetric provide eigenvector arrange degenerate subspace constraint orthogonality situation fill finish evident summarize pca entail subtract computing eigenvector demonstrate matlab code mathematically involve continuity name basis quickly derive decomposition interpret pca let square fashion positive singular r cl theorem say bit multiply eigenvector scalar summarize vector multiplication prescribe construct order set singular likewise additional orthonormal deal degeneracy issue representation piece fit form multiply side arrive motivation decomposition matrix orthogonal second express q scalar understand new matrix stack place diagonal position generate look like respectively solving solve equation thick orthonormal speaking span formalize precise column equation transform infer transform orthonormal column basis span matrix quantity orthonormal transform transpose span understand implication though information pca fall framework evident svd original dimension derivation equal matrix principal component matlab include b must orthonormal column svd also principal quick pca type calculate covariance pca reveal structure solution summary implement pca quantifying dimension describe set along mean compare hope behind employ variance along component type reasonable precise intuition behind reduction vast majority variation direction record pca ask fail remarkable feature completely answer come parameter record plug play feature answer independent perspective pca agnostic track point variable angle pca would fail deep require primary motivation explore unite run big
angle canonical angle need suppose part last inequality use theorem least unitary put nearly unitary henceforth prove diagonal decomposition matrix eigenvalue real discard retain derivative complex eq show remainder full denote bound theorem angle subspace span take appropriately second imagine isotropic must account slight multiply accumulate depth claim appendix satisfy recurrence error final suffice estimate matrix latter suffice nearly four term particular ty tx need failure small argue get thereby give high algorithm derivative large gap pick find randomness tx ti tx p expectation overall appear large recursive scale smoothly matlab code gaps family gaussian gap degree proceed clearly true grateful anonymous suggestion nsf section perspective performance recursive purpose implement use one blind speech run gb ram run matlab synthetic fully model pick unitary first gaussian column speech speech run unitary source simulate natural blind impossible make isotropic construct bipartite product also desirable property invariant column sign extension synthetic recursive run unable c isotropic source dramatically dramatically raw non isotropic difference fourier us recover author distinguish even achieve still unit roughly speak ica fourth exploit second moment information fourth moment smoothly second fourth conclude note experimental section comprehensive account consideration implement highlight practical construct parallelization split approximation pick algorithm type dramatically improve outside thesis school computer science school science technique complexity apply distribution give ica improve decomposition decomposition unsupervise mixture obstacle datum expensive recover unlabele become classic ica become diverse area vision recently deep net input vector unknown observation write unknown component ica estimate hope subspace span gaussian assume component distribution fashion common fourth nonzero gaussian ica fourth improve polynomial broadly tensor decomposition tensor sample analyze spherical extend corrupt state precisely major ica latter guarantee paper improve fully result fourth bound away give deal difference gaussian set rectangular algorithm technique hermitian decomposition decomposition powerful tool theoretical generalization tensor decomposition np hard tensor provable decomposition polynomial dependence conditioning impractical moderately find gap eigenvalue core need large gap subspace gap project ica technique recover assume model surely find sign satisfy I ia use singular value decomposition symmetric substantially previous good recursive variant proceed first isotropic eigenvector accurately eigenvalue adjacent pair e gap idea simple estimating gap shot group eigenvector either necessarily desire recursively need gap thus sample motivate point sample grow square inverse go go apply characteristic call function precisely eq tx derivative derivative follow complex hermitian hold unitary isotropic unitary random vector robust algorithm spaced mix anti sample guarantee tu large eigenvalue fall use low isotropic transformation every compute compute reweighted svd large else return decomposition block carry span recursive three part gap isotropic gaussian partition column subspace version perturbation accumulate gap define function large gap successive polynomial iid gaussians least repeat type rough intuition tail work differ firstly quantitative pick fix quite bit long easily analyse maximum gap pick independently pick pick replacement pick wise chernoff bernoulli pr proof trivial chernoff take union
singular value intersection space kind include quantile express probability density pass easily quantile depth assumption refer contour contour half construction mesh straight contour dash curve function affine median affine mean singular matrix unique symmetry survey uniqueness uniqueness case strict positivity point indicate show section skewness deviation angular angular wise median median th square random distribution solution quantile yy partial moment distribution strictly support imply set q calculate obvious use indicate refer depth contour kind technique elliptical axis proportion however depth set shape level eq technical e comprise mean satisfy implication scenario depth satisfactory multivariate quantile scenario scenario es relate risk consider skewed canonical invariance suffice skew skewed tail freedom function student freedom later tail characterize shape student recover multivariate result give appendix introduce canonical important skewness family equal cauchy generality component contour axis let denote two whose first differ space index sign define p contour simplify next contour circular distribution indicate quantile distribution u element st du observing euclidean unit unit htbp panel example e right illustrate contour contour certain st skew cauchy depth contour contour elliptical elliptical calculate half space depth use construction letting panel dot line approximate obtain depth contour contour panel panel dot contour htbp e ic ellipsoid measure skewness variable angular symmetry median affine imply affine measure skewness canonical also alternative skewness wise skewness curve increase towards deviation angular symmetry closeness angular symmetry indicate close result angular symmetry wise directional ix express ix I arbitrary conclude angular symmetric median median family explain follow representation tend hence argument apply constant draw large approximation ellipsoid misclassification positive misclassification intractable quadrature integral let small misclassification case misclassification next observation consider approximation mass report misclassification increase high case negligible misclassification reach quality family year figure bivariate skewness indicate skewness angular symmetry estimate misclassification low evident scenario distribution simplify one component asymmetric additionally represent skewness closeness distribution angular symmetry propose angular misclassification approximation invariance availability mean sufficient skewness curve underlie although misclassification dimension elliptical construct elliptical attractive financial risk factor stress application introduction skew close affine admit given skew skew many material skew depend existence moment consider elliptical corollary location namely center angular symmetry lie investigate whether depth use canonical acknowledgement student sciences university supervision k generalize modify kind constraint mm mm examine scenario skew distribution interest motivated set stress test financial set half notion ed ed set elliptical coincide region contour skewed equivalence contour contour skewness heavy skew make form skewness contour elliptical contour exactly elliptical skewness angular symmetry elliptical contour keyword angular symmetry depth school university infection health university uk department statistic university mathematical science interest original study issue financial management represent index interest exchange factor financial portfolio give portfolio derivative bank book compute information portfolio formula quantify factor value question distribution plausible scenario opinion set space presence multivariate short tail many outer tail one change year quantify portfolio portfolio bivariate fit contour plot line divide half space form intersection close point depth grey line boundary space
dna structural dna sequencing length ignore information section alternative simplify read map equally informative existence describe model structural introduce observed intensity read sequence coverage index genome read location map within local intensity function single location length become apparent representation technical role scan copy seq moment section specification alternative intensity map length examine detailed simplify certain notation baseline rate general raw scan index differentiable detect pair read differ read pair least map template map position plus minus map dna sequencing read reference assign position read may sequence mapping due inclusion dna match minus fail call experimental conservative pair reverse orientation easily broad notation narrow definition thing definition pair read read map let read either plus position mathematical convenience process intensity integral second account unobserve read marginal intensity simplify map read minus proportion width reference window partition non w u u pair window window minus read map map uninformative whether broad definition remain simplify read set separately come way contain pair mapping non latter read sequence due reasoning form log belong b signature specific respectively coverage simplify fs fs weight length modify u u u v replace minus eq read peak genome combine peak give peak small since alignment accomplish statistic must read detect little ideal true maximization shorter long well section cut corner function ease within segment ignore formula priori one try combine separately scan long correct simplify possible develop variability length map begin statistic beginning whenever read begin determined chance appropriate modification detect suggest base section approximate p value rely transformation towards scan window zhang cite ratio transformation approximation calculation nan intensity define still field k z simplify expression call local field p field eq q denote partial vanish hessian evaluation consider vector clearly allow set increment special case variance x statistic threshold zero local theorem propose use order marginal magnitude marginal transformation rely interest summation respectively note rely quantity rely localization zhang condition asymptotically q note local brownian former latter increment respect smooth hold genome location fix add brownian notation brownian smooth element equation omit genomic location notation introduce w df r df rt control detection amount may range boost sensitivity study zhang zhang specific function constant window alternative calculation equivalent define suggest various scan approximation simpler appear tail zhang rx purpose numerical value improve identically equal calculation rarely per maximize complicated r precede w x maximization become negative similar paper effect maximize adequate result implication function approximation scan see fix nuisance threshold maximize respect threshold define easily interest event put choose leibl inside particularly integration denote minimum derivative eq asymptotically w occur maximization simplification upper correction carlo evaluate repetition increase experiment realistic consume approximation use helpful evaluate grid base multiply unchanged incorporate monte effect use example poisson figure homogeneous apply slight expression sum instead multiply throughout approximation moment calculate numerical formula smoothly twice w dt approximately simplify justification may maximize g toy pair read score derivation formally consider dt discrete important family determine choose minor formal asymptotically let find asymptotic natural ask approximation precede expression fact place equation x var identity possible upper z var replace conditional dy maximize seem reasonable term usually small section use approximation procedure power four statistic maximum rescale shift amount correspond base genomic example appear statistic significance tail correction max produce threshold statistic significance opt power statistic indicate actually remarkably well c appear far powerful considerably powerful quite seem less actual affected hoc method slightly consistently less powerful drive poisson process determinant small value significance question concern power value threshold simple describe statistic approximately marginal approximately appropriate appropriate slightly example power level would number calculation report power moderately threshold seem try vary accommodate change consider detect signature practice scan combination power improve power individually adjust tail combination individual consideration false score score nominal except read align peak detect triple corresponding b gx case simplify applie range range change scan line incorporate behave toy scan distinction worth decrease substantial fraction span genome read detection score piecewise smooth path account maximization describe remark value require change fashion complicated examine power pair maximize sequence library influence length read sequence coverage read heavily probability mapping read nan nan estimate typical publicly last sample quality gold sample standard read length trend increase length shorter read map increase recent fall sequence study period widely across goal wants currently usually less example population dna extremely desire refer sequence experiment frequency examine scenario coverage first coverage sequence case occur seem additional insight toy base pair statistic true relatively detect na na na first case coverage power large somewhat well length read table show detect coverage unlike monotonically capture comparison power size whether preferable large surprisingly adequate power substantially short small read case detect sequence examine length threshold scenario scenario read detect power power relatively complicated contribute row would bring tail power read add would produce find precede add lead row read case lead seem require combination dominate extent power take adjustment significance significance power fall read different genomic would illustration scan european nucleotide read base length map empirical show bold density estimate dash normal dotted normal density scan step control family wise deviation deviation distribution map read stepsize threshold visualize massive quite plot dash represent score analytically model show score skew lie nan minus read bottom plus length place call support overlap score merge call call make call robustness statistic combination factor generate boundary read right region end call statistic evidence overlap concern inspection read suggest call show minus read end na scan length statistic read read percentage overlap end region iii two pass ideal read shift precede support peak region contain top plot map plus bottom read read roughly bp precede half yet significant minus like datum read evidence pattern exception rule roughly map read region bottom show score genome scan produce many region carefully ideally read scan emphasis useful signal generation sequence detection seq variant pair formulation significance framework embed family technique zhang characterize simplify read simplified read suggest approximations carlo also study picture regard assume homogeneity nuisance statement regard aspect fail accurately illustrate row well simplify variant detection pair sequence formulate incorporate read map log likelihood scan call long increase read depend detect read report statistic poorly map reasonably powerful sum although row respectively marginal alone read even empirical substantially library tend exhibit skewness increase gain substantial analysis assume coverage read genome score vary threshold genome position appropriate substantially increase amount computation conduct likelihood ratio issue avoid option read coverage scan threshold value simply implement one genomic one simple lost adjustment threshold sum score individually reason dominate incorporate contribute mainly individually combine weight zhang focus mainly error genomic false discovery fdr often mode multiple testing control boundary fdr describe zhang characterize genome sequencing genome content throughput sequence p j e h cox identification acquire cancer genome pair sequence zhang scan weight observation american resolution copy p linkage complete identity descent molecular trait linkage discover generation sequence nature approximate smoothed poisson maxima peak statistic j zhang poisson application copy next generation dna sequence unknown ann zhang maxima fields ratio zhang simultaneous ann zhang statistics trait three analysis field united support national foundation foundation stanford genomic throughput dna sequence poisson derive false calculation accommodate deal example pair compare power current illustrate application modern biology especially great deal research local signal al typically represent datum standardize neighborhood detection magnitude location field central search local achieve theory field control normal provide adequate approximation threshold especially involve contain condition zhang dna sequencing possibly homogeneous process involve signal detection copy sequence bind site follow sequence seq rna sequencing pair sequence brief motivating section also dna structural although context relate detect local scan precede cast derive ratio scan field study variant directly consider tailor specific pair read dna sequencing motivating application give framework field first variant
efficiently matrix describe matrix uniformly simplest fast bad propose randomized row normalize hadamard choose fast version embed enable linear map guarantee theoretically leverage without replacement error uniform replacement compute approximate obtain x u attain way error bound arbitrarily previous prove sake prove bound theorem define u lemma w inequality singular q inequality hold leverage union least technique two column matrix obviously hold finally u u two third set least cost column induce time space discuss sampling solve efficiently previous score attain probability find linear seek square compute cholesky solve big prohibitive fortunately one portion instance use datum base efficiently speak projection instead projection construct theoretically construct projection error hope without technique self contain sampling still simplest efficient though score true perspective score sampling uniform still study theorem uniform sampling attain
problem convex time convergence derive dimensionality allow grow required convergence maximum norm strong condition weak inference modify subsequently estimator limit reach piecewise mild assumption multivariate lead precision regression acquire combined form normality aim element approach present build invert tucker paper demonstrate linear generalize fully invert kkt condition consequently normality section main result contain proof matrix notation ex ex ex e zero analogously cone e ps p model throughout estimator gaussian perhaps correspond rest paragraph come associate undirected edge structure precision correspond estimation graphical non selection precision translate gaussian cardinality loop allow shall cf certain allow grow presentation define operator I second shall impose restriction index consequently define e ex ss ex hessian theoretically one could estimator lasso sensitive besides procedure consequence zero graphical derive assumption parameter appear influence base concern condition matrix covariance depend roughly speak show pre assume model tail introduce follow given define tail tail probability sub tail condition variable interest restrict random mean random follow covariance sub event tail give remark positive characterize tucker condition belong subdifferential evaluate inverting condition lead de remainder converge probability hold uniformly sample additional vector satisfy furthermore wishart shift sub asymptotic theorem individual fix keep dependence doubly index parameter satisfy suppose converge next hold exception brief graphical fully simulation instance graph star may maximum degree vary specify corresponding matrix one diagonal chain graph precision cardinality active star graph implement interval cover number calculate obtain respectively calculate interval parameter average confidence interval construction estimator pre big cc graphical lasso specify star mle covariance cc cc value definition cm theory correct regularization I suffice empirically optimize average roughly experiment kkt optimization read subdifferential side rearrange obtain asymptotic expectation remainder equality since ex ex ex obtain q mx ax ex condition ex ex ex ex u ex b ex b ex together give ex ex ex ex ex note ex ex ex ex since uniformly theorem sub hold claim scale converge distribute nk e nk nk finite sum sub since nk statement satisfy nk n nk n z n remains apply follow hand tail de c show term limit substitute n satisfy x p nz ij ij
whose transition mdps compactly several develop exponentially however factor factored mdps factor optimistic confidence heuristic factored mdps make exponentially upon aspect discussion reduction reinforcement approximate many effective planning method extend optimize finite horizon action mdp p operator indicate mdp episode scalar transition make learn deterministic function agent th episode reinforcement learn eq episode mdp mdp internal history random reward regret factor mdp structure formalize introduce factored define scope mapping mapping factor factor draw within factored factor lr rx ir ix individually transition factor mp factored mdp mdp factor reward factor transition write bind regret factor factor I span optimal factored high high factored factored factor z dm factor cd factored span policy mdp may hold optimistic formally replace analogous factored instance shorthand simple clean clean factored factored satisfy tight factored mdp corollary appendix confidence estimate underlie contain literature exploit additional graph bound empirical tn sequence confidence depend q shorthand interpret nan follow bound measurable counting factor j ts confidence j k j k td factored refer generally either mdp optimistic bellman return expect state law place write break add subtract reward agent clarity change nothing relate reward mdp near plan factor dynamic programming I factor hoeffding remain bellman actually h aim create deviation triangle build confidence set factor factor deviation z also j conclude apply factored transition lie posterior finite relaxation j td j ensure td use fact equal law complete factor theorem corollary substitute upper bounding factored mdps reinforcement previously important question access may prohibitive priori algorithm seek mdps reward value belong stanford support award foundation elementary argument concentration contradict x corollary delay visit imagine let length finite act start episode episode define notion radius tx w tx times episode set td kx h expression time repeatedly say composition act low integer finally complete kx ir kx tt tt tt complete suitable trivial look trivially certainly bound last claim van stanford suffer mdp space interest curse dimensionality possible polynomially factor satisfy near sampling learn confidence factor uncertain cumulative environment mdp
visualization comprise video environment category sequence action video select video datum division video multiple include testing extract densely video training descriptor video fig manually histogram dense sift histogram binary histogram optical heat maps cl ap ap validate kernel baseline svm selection mkl connection mkl define bin histogram performance ap assess selection measure achieve precision outperform experiment feature major paper selection interpretability select codebook fig feature discriminative visual unclear mi art method use recall localization overlap quantitative fig rs chi perform linear however video segmentation explicitly assume video get result category reason lie segmentation since motion good segmentation see pr result comparable segmentation car cat train propose select convex problem commonly classification task plan address limitation beyond classification well art feature region tool region future selection weakly supervise tool visual pt tr abstract typically extension despite success unclear video discriminate among answer visual answer question present region visual allow visualization feature region jointly optimize parameter classifier benefit approach linear additive intersection spatio video unify selection scalable method learn illustrate decade great advance performance although much progress visual exist labeling involve category discriminative interpretability mid car image contain car goal weakly discover region train car successful weakly rely build art space discrimination aim weakly supervise discriminative aim discriminate image also temporal activity fig discriminative avoid consume manual informative train due importance popular exist learn mostly cope allow would approximation linear generally inefficient region contribution kernel discovery visualization word spatio volume video work include experimental illustrate address et weight space classifier construction exist method jointly induce build et al feature weight linear svm add kernel computer vision approximation purpose histogram differently come weight impose unclear convexity hold address region mkl image model class bag one instance positive weakly mi svm series mi arguably popular bounding localization limited aim jointly instance np heuristic heavily use instance moreover essential liu region visualize method use unclear convex feature additive kernel bold letter letter g work sample histogram codebook histogram satisfie I dx ik jk weight x assign bin bin histogram feature maximum hyperplane margin correspond margin two space directly normalize consider nc training error transform property additive factor jk interpret bin priori concatenation ik ik variable substitution additive optimize optimization see section note selection formulation mkl rewrite simplex constraint drive bin differently bin mkl exploit non optimize reformulate set lagrangian primal variable identify dual k ik dual eq objective problem ensure negativity constraint reduce positivity constraint take descent method however visual difficult classifier discriminative video segment video region spatio encode codebook learn video assume region classifier homogeneity long selection allow image histogram importance q map index simplex sparse region trade model bag bag segment image negative localization region belong b ik ik ik region connection
word track research public available ground surveillance surveillance program section period event specify developed simulator signature simulator going adapt purpose base streaming appropriate scale data http com research acknowledgment project north operational national reference framework european ci acknowledge european project p cm dataset laboratory artificial intelligence university dr pt surveillance continuously daily stream indicator detection detect day clinical laboratory datum usually dimension recent method search multivariate signal alarm find propose bottom bottom search track detect change reveal art false alarm keyword stream surveillance goal surveillance public health clinical laboratory come kind event usually made event activity like attack epidemic like west etc event make identify early stage save prevent early event purpose system stream diagnostic health room counter sale school internet health simultaneously trace ht demonstrate surveillance see aggregated daily generate complex alarm straightforward anomaly chart impose alarm diagnostic indicator apply detector individual reject hypothesis false alarm univariate process base series processing base surveillance besides event category method pca hmm var multivariate operate look via area surveillance spatial statistic fluctuation operate spatial account scan statistic operate datum adequate surveillance mention drawback static effect offline group suit bayesian pattern event real update network main limited exist handle temporal stream baseline vary raw historical day historical environmental attribute use oppose network historical many surveillance opposed overall change track consequently present delay alarm bottom rule middle bottom top track level approach take purpose cause overall surveillance turn epidemic early even hour detection require problematic signal emphasis surveillance rarely take study alarm inverse effect detection study concern reveal people deviation increase besides oppose anomaly assume process occur isolate static surveillance dynamic environment attribute week weather etc affect whole illustrate individual kind surveillance detection complexity environmental issue surveillance false alarm detection contribution time apply surveillance problem dimension correlation criterion event detection introduce novel baseline baseline environmental rest organize propose datum performance analysis last conclude exposition present fundamental rely tracking unless match recent streaming setting dynamic environment item strategy take stationarity demonstrate illustrative method receive stream stream slide window size top cell correspond region slide environmental fed historical tensor previously slide window combine decompose subspace pairwise observe baseline vector principal eigenvector correspond eigenvalue receive eigenvalue principal vector solid alarm considerable eigenvalue close consider dot eigenvector vector describe present slide day environmental weather significance slide phase slide format assess slide window need utilize window previous another environmental combine take window baseline present apply high principal diagnostic however perform important high alarm happen article name production p stock market ten period selection automate matching phase eigenvalue eigenvalue baseline define historical eigenvalue eigenvector q deviation purpose want transform ease use explain decompose specific match principal eigenvector spatial dimension may kind change match overall change infection reflect unseen environmental environmental algorithm receive include current historical tensor instant environmental vector environmental baseline tensor search historical match item illustrative snapshot day environmental color cube compose assume environmental dominant frequent history figure include baseline occurrence dominant day receive historical setting find input unchanged day receive match time one rewrite element match baseline tensor match still dominate therefore baseline compose preference day matrix stay repeat environmental setting update add already matrix unseen add distance adequate keep historical environmental basically data clearly infeasible anomaly research hand multi recently semi automatic event ensemble background access background include simulated disease network simulator namely base factor weather manually mention simulator produce extremely publicly online set way multiple environmental setting distinct temporal day also environmental cardinality record sample record xy ne gender feature environmental week environmental weather environmental receiver operate roc trade specificity roc method evaluation ability critical surveillance positive heavy delay surveillance characteristic like surveillance proper evaluation monitor characteristic curve evaluate specificity method surveillance release occur alarm correspond period consider detection alarm release reality event release release detection delay day delay figure false detection alarm day release mark alarm mark specify alarm release one day delay optimum alarm indicate depend alarm alarm recent performance use totally temporal day use day day whenever agent second year year contain release slide move window match output alarm delay detection axis month delay close curve pattern perform false alarm bad delay curve make overall instance false alarm delay specify need area false area outperform considerably separate look detection delay see detect event release difference oppose subsequently less small slow ability dimension suffer delay detection runtime surveillance system process receive huge processing hour efficiency runtime superiority version fast factor whole track structure factor relate compute exploit tensor decomposition offline tensor decomposition baseline tensor temporal size tensor tensor three dimensionality develop instance dynamic analysis streaming require mr mn case I I fast track slide window element 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correctness counter result set choose vertex choose go game two player token place token vertex token player move token edge go player result call formally infinite specifie formally function represent history end vertex strategy player multidimensional payoff map edge th dimension path k inf resp sup vector average multidimensional boolean atom example boolean always multidimensional payoff win satisfy condition player strategy infinite multidimensional boolean formula occur determine sequel abuse sure mean determine mean counter machine payoff counter first one counter tailor state counter right left allow increment right since state transition change counter standard machine game one counter reduction consist figure state final machine sim player counter reach player player play dimension game simulation sim sim sim sim sim sim sim sim sim sim construction win side counter objective sim run player game graph simulate simulate describe role sure sim read definition section way right q whenever state sure leave left transition round sim role sim entire value counter current sim happen simulate illustrate follow stand g explicitly role transition transition loop loop want assumption side side transition qp keep sure testing test four note win win go eventually game assign way enforce player player construction give player player act differently game player correctly win sure stay stay state get remain either negative sup player positive bad bad g g sim sim sim claim win win give win player make sure state enough round fulfil test player fulfil otherwise negative least win satisfied player winner prove converse direction never win strategy win win condition trivially stay sim otherwise eventually win winner correctness namely describe win win case subsection extend two subsection round player play begin play subsection counter denote play g player sufficiently reach state since get player maintain left machine right denote show leave maintain simulate number play current simulate sim round begin begin current transition step complete every play proof sim hence complete player step simulation player assertion recall four assertion condition violate assume case invoke end loop go round violate claim indeed sim sim change versa sum exceed round condition violate analyze violate get observation begin sim remain value every round maintain infinitely condition testing proof leave symmetric show visit actual counter counter simulation g prove claim sim get contribute contribution visit contribute g x g c player right maintain c item lemma round play current sim ng counter machine winning win counter value maintain leave violate correspond stay immediate lemma stay violate negative weight hence simulation right simulation whenever leave invoke counter maintain negative every violate transition properly player since suppose number visit subsection c loop fulfil sim simulate loop player zero strategy lemma accord sim sim initially sim first item item sim initially simulation g use player play whenever visit number round play self loop round every round hence round proof proof ready win win strategy enough play constant distinct case strategy invoke many play player sim sim value change sim run satisfied consider always thus sim infinitely sim proof first never sim fact never certain round either infinitely often counter win c ii counter proof proposition hence problem winner payoff payoff game deterministic input counter decide end counter zero counter player simulation know test counter second counter
increase sufficiently follow show denote nonconvex nonconvex stepsize kl respect uniformly coordinate block iteration little extra computation new cyclic greatly save nonconvex fix update nonconvex smooth addition nonconvex increase mirror stepsize sg kl km k f k compare block gradient stepsize reciprocal lipschitz k test start sg independently follow follow solution time perform start randomly use compare another calculate empirical repeat average empirical table sg sg report well sg take less gaussian sg sg compare belong sample sg epoch epoch bilinear logistic interface visual recognition test eeg concern eeg arm subject hz marked randomly slice size start depict behavior epoch epoch reach run return logistic default run method run choose give accuracy epoch plot give however take eeg epoch also analyze nonconvex sg convergence clear sg mirror gradient method acknowledgement nsf dms grant lemma second first eq complete proof prove lemma q cauchy schwarz second projection first prove hold q secondly complete exist choose integer addition way therefore complete xu theorem remark apply mathematics university mathematic sg stochastic block descent update combine sg multiple block paper propose program sg block gauss update previously outperform sg latter small constant establish expect optimality case nontrivial convex significantly sg early near local minimizer benefit update especially engineering involve way sample keep mind constraint partition block sn differentiable sparse throughout omit without confusion logistic bilinear sparse sublinear objective convexity establish expect condition convex solve accurately calculate expectation subgradient risk sg method assume gradient oracle kk stepsize compare sg competitive problem utility flow sg deterministic deterministic require nonsmooth descent update variable much low iteration together find solve benefit sg size integer mini batch order block sample randomly update prefer typically proximal iteration see reference use nonsmooth constraint one certainly take incur begin block especially nonconvex sg sg become require sg sg numerical list method coordinate gradient sg stochastic deterministic block back consider concave programming original format coordinate respect fix recent nonconvex example work original understand propose method method randomly update analyze show smooth sublinear expect strongly linearly analyze case cyclic smooth also history sg become popular stochastic programming classic require convexity great robust sg obtain asymptotic iterate propose mirror descent stochastic convergence mirror accelerate composite sg handle term optimality relevant propose combine mirror descent approximation propose randomly update depend early update demonstrate practical intuitive sequentially processor update resource partial gradient randomly cyclic block gradient thus require assumption specifically boundedness appear system assume update row vector explain update variable block benefit stochastic form huge amount sg method establish condition synthetic world deterministic sg problem restrict euclidean norm partial subdifferential scalar constant history st expectation set eq onto without loss generality order hence define depend big difference make challenging objective lower bound lipschitz every namely lipschitz assumption bound k update boundedness proper nf I continuous namely singleton pf kp give last inequality method vary literature need analysis boundedness together partial boundedness lemma gradient mapping scalar obey large integer next algorithm nonconvex case convex sublinear sg nonconvex optimality analyze problem strong ergodic non k furthermore usually play exact differently phenomenon sg numerical test page biased partial see I become sg although result bad sg generally per sg read gradient partial sg computationally dominate need update rest little performance numerical summing together note inequality let last substituting let convexity iteration establish sg l assumption strongly modulus modulus impossible subgradient reference therein convenience discussion become modulus result
wise sample take integer encode variable query take bit set equal ar appendix material remark differentially private answer number high dimensional task however package hard define integer solver prove demonstrate experimentally netflix multiple become concern task often operate netflix challenge privacy netflix movie compete mechanism competition great success team improve hoc able lead subsequent query release attempt strong privacy thing prevent identification release answer private release extensively differential query many query safe version dataset unfortunately size exponential dimension run necessary bad especially produce runtime evaluation release notable exception thorough experimental find quite accuracy dimensional nevertheless seem attribute query scale feature partition query never critical bottleneck maintain universe quickly impractical record grow complex alternative algorithm rather object represent np step require private exact exist quickly practice part extremely efficient require practice time query like query release strong way marginal table demonstrating solve release include hundred thousand perform query release convert differentially private complexity numerous efficient problem parallel preserve extremely strong synthetic exponentially notable achieve theoretically exponential bad mechanism answer evaluation base inefficient heuristic multiplicative attribute seem scale family release base runtime mechanism algorithm family view synthetic generation propose interpretation solve game repeatedly well optimize player regret main problem record record differential privacy become record database requirement removal outcome mechanism database abstract possible record often consider bit database differ symmetric differential database differential query private sensitivity fundamental tool bind algorithm privacy private rely interpretation player present database close may player player action universe player set let payoff sum datum try query payoff distribution player play von intuitively von advantage go minimize force payoff force suggest well play opponent equilibrium interpret strategy payoff player game mix nash equilibrium eq query query place least player play equilibrium query player play precisely release need calculate approximate equilibrium update receive update normalize material maintain action payoff average distribution payoff multiplicative response nash multiplicative response role solve query release algorithm response round database sensitive change mechanism quality cost round composition eq differentially define advanced quality privacy privacy material form let let query define answer least note chernoff hold release game synthetic database answer query exact privacy associate actually state delta privacy first query query next approximate game find suffice response release game union condition event sample aggregate payoff accordingly convention treat plug response depend query discuss query try player minimize actually play query maximizer guarantee accuracy look precise query record binary universe mean let query integer though everything general marginal query include query sample eq associate satisfy many clause convert optimization problem conjunction form integer clause clause result program solve exist attribute netflix collection several census uci repository binary movie binary lower find algorithm error predict satisfy collection marginal several census repository movie wide preserve beyond frequently set guarantee evaluate netflix report query range run differentially private actually privacy small example also could differentially small use netflix heuristic dataset gradually netflix allow runtime query set million marginal experiment privacy parameter stable grow show remain mostly demonstrate perturbation laplace rate privacy query evaluate behavior privacy report runtime attribute axis include measurement overhead common answer query synthetic uniformly synthetic datum record satisfie realistic pick separate attribute equal way answer expense differentially minute exclude experimental implementation mid machine core processor ram experiment reach discretization binary discrete attribute attribute randomly marginal query sensible take different attribute query
express artificial change forward reference paper indeed observation reference thing mention david topic model people claim select separability define hull problem offer novel formulation utilize submodular later establish cone find cone row separability trivial time via backward removal convex matrix separable factorization inner dimension propose procedure remove row row convex row run output dimension achieve run polynomial dimension find constraint problem lastly cover large lie row index hull problem verify constraint equal cover separability separable submodular cover generalize separability general nmf find hull need require point point unchanged anchor ensure finitely generate finite another lastly encourage among cover finitely generator say algebraic ki k hull separability aim find anchor minimum hull find cone row hull define critical one case variable separability contain point fortunately additional separability assumption row two sum stay identify apply substitute eq element ground rank side uniquely model equal respectively select guarantee uniqueness avoid satisfy note assumption much limit separable nmf identifiability achieve vector inequality hold uniqueness variable achieve minimal hull gmm view feature observation triple observation respectively sphere l hull factor tx lda word occurrence interpretation index cluster real hmm solve normalize anchor reduce general matrix general minimum technique besides nmf mf mf variable deterministic assign maximum map separable nmf mf otherwise finite recover intuitively small prior exist dimensional subspace g angle sc various segmentation cluster model costly lasso row reliable general separability sc reduce general minimum impose n ik ix k group associate sc cone cluster cover nmf special separable cone might difficult dim show reduce cone hyperplane solve efficient see reduce equation moment probabilistic variable hull stand hmms conditional independence also ix third write matrix operator mode tensor outer product operator mainly focus recover moment moment necessary estimate training side rh unified space matrix example gmm rhs either rank column basis rhs leave rhs let tx ta hull anchor assign index solve computational suffer thank retain merely ok still successfully acceleration completion nonzero entry happen model simplex mean topic separability treating common text moment sparse general widely filter hide continuous distribution linearity similar moment hull hmm therefore equation case diagonal even fall discuss usually separability separability separability transpose e select let matrix immediately filter allocation bag vision semantic topic document firstly proportion document drawing topic conditional probability h j linearity lda number topic word number word gram statistic co occurrence matrix word fall latent separability assumption occurrence simplex occurrence document show recover anchor reduce mixture noting prove fast accord mf hull divide multiple easy sub extremely dimension separable largely rich problem design insight observation geometry cone project hull low hyperplane partially preserve geometry problem sub handle solver hull secondly pick without iterative pursuit significantly solely subroutine cover due cone hull cover cover hull generate projection eq since merely minimum hull hyperplane rarely return hyperplane projection problem bad flat anchor hull surface anchor hyperplane anchor flat face span adjacent still hull robustness flat htp ref latent anchor set ty nmf trivially hull leave critical reason converse uniqueness hyperplane could violate fortunately minimal hyperplane proposition reveal p htp hull generator minimal hull iff angle minimal hull separability lead identifiability case sub anchor right anchor anchor green marks intersection hyperplane hyperplane dim hyperplane h dim minimal hull proposition immediately angle identify hull region anchor identify angle verify angle computed specific subset flat increase interior angle intersection turn flat anchor right anchor intersection hyperplane anchor hull anchor lead approximation hull aim failure still rich ensemble random ensemble datum sparse ensemble bring acceleration projection point large unique true compare estimator ty iy therefore chernoff flat proposition suppose introduce binary eq anchor chernoff randomness random corollary guarantee success I factor matter solver choose sub learn note use projection reduce variant subspace projection still project gain original sub furth speedup divide randomization base unified distribute solver subroutine sub solver iterative algorithm pursuit address although solver hyperplane always geometry hull plane sub problem plane cosine find max max plane min max angle large angle close otherwise vertical horizontal plane plug extremely note large axis nonzero broad span hull lead generalization novel divide low hyperplane solver present solver cosine max subroutine check improve dimension apply gmm lda nmf subspace cluster show performance rich maximization factorization commonly produce posteriori estimate use wide collaborative rely update initialization moment contrast relate observation thus yield suffer large estimating poor moreover simplify recover uncertain column latent extreme hull obtain matrix separability represent call non factorization simplex extend negative generalize build identifiability generalization
entropy approach moment derive clear consider ise spin variable principle probability goal ise observed make inverse quite partition derivative spin system inverse approximate infer ise approximation isolated spin expansion obstacle overfitte affect inference noisy fitting moment exactly incorporate sample obtain provide good description dataset general become serious exceed effect example ising regularize add penalty describe fit alternatively empirical moment modify accord ij work total error square therefore predictive typically parameter perform randomly mutually exclusive usually fitting parameter predictive model agreement model contain third focus ise study ability reconstruct sample g performance inverse ise define ise various reproduce work compare different ise performance quantify agreement datum easily ph p angular average spin ise ignore act configuration easy calculate model consist letter see letter letter diverse letter mutually exclusive comparison ise accord letter indicate bold use case estimate l naive nmf isolate spin direct performing method include variational magnitude run ignore spend penalty parameter regularize negative regularize nmf demonstrate ise direct variational perform many variational contrast produce result performance train model good simple learn letter variational specific ij ss pattern describe energy infer energy pattern unconstraine transformation break initial letter pattern unconstrained nevertheless feature present art modeling include process fine tuning step network order nevertheless demonstrate pattern data variational variational outperform consist letter computer attractive practice structure extend variational inverse spin helpful provide award consist letter com image simple define intensity spin vector variational
discuss compare indicator algorithm gaussian output functional consider finite heat nine parameterize reference nonzero specify functional integral domain h w boundary ax depict boundary replace u contain freedom discretize space fidelity output reduce maximum set less replace reduce basis analyze type error u output regard finite element reduce supplementary compute surrogate error employ residual learning ex validation use relate set indicator true polynomial interval span purpose large interval polynomial font legend legend align label font align center xlabel axis align west legend title ii e index size anchor legend name title anchor legend title anchor north surrogate ex method depict ex ex comparison display remark arise inherent uncertainty ex interval include train indistinguishable trend large parameter attribute see focus norm dominant surrogate polynomial polynomial compare due superior validity surrogate ex uncertainty mean behave sample report validation label style font legend style column legend align width ylabel histogram west legend title bs legend name title histogram anchor legend title histogram west anchor ex ex ex curve depict probability table report actual lie infer interval within set within model increase discuss section infer moderately sized consist correction surrogate e improve low fidelity order surrogate reduce validate validation quality increase moderately sized training converge surrogate style width align legend indicator anchor west name legend error input anchor south east observe former relationship amenable construct process quickly surrogate nine curse difficulties ii depict error improvement error correct surrogate surrogate report expect evaluate distribution error surrogate depict almost always remain output alone hand expect always great mean always fact approximation improvement produce correction greater suit far legend align center align center width restrict legend title correction v basis gp error style width legend font cell align center style cm center v west legend name title correction anchor west anchor north surrogate sample red curve depict report interval test correction validate assumption surrogate histogram mean density associate interval surrogate correction align depict confidence validation remark interval closely number training effectively addition reasonably converge hand surrogate exhibit training use style font text width cm align center legend style legend column legend align style width align anchor title correction north surrogate gp point often actual lie infer surrogate imply surrogate eqs ex important quantify bound imply rigorous ex legend font legend column legend align font cm align title style width align center bs anchor west legend name linear anchor north legend align label font text width align cm bs anchor ex ex report surrogate compare plot state surrogate convergence small dimension close mean figure correction surrogate small however dimension require factor error produce large I conclude fidelity assess performance discuss output denote dirac delta separate surrogate error indicator computation dual dual offline infer side coincide mesh reduce counterpart generate assess ability weight residual see remark bases fidelity tolerance tolerance tolerance surrogate fidelity figure depict indicator fidelity first exhibit good label style font width cm align center title legend font every xshift pt anchor south west sep west middle title index size size anchor west iii dual figure necessity employ dual residual indicator actually accurate yield error order utility residual indicator style font legend align title font font align cm ylabel small improvement anchor name legend ylabel e title dual anchor west legend ylabel e title improvement legend name ylabel well anchor west legend log west name legend ylabel title vi reduce space report result infer confidence interval converge correct accurate surrogate confidence interval basis present modeling error quantification employ learning mapping computable indicator error distribution reflect uncertainty validate surrogate lead one exist surrogate bind general allow output yield uncertainty modify employ input indicator input error model prediction demonstrate dimensionality although characterize nine indicator validate combination surrogate powerful number future analyze bayesian different surrogate algorithm basis near surrogate acknowledgment thank support understand selection support part national fellowship national security science engineering national energy contract ac acknowledge office scientific research contract section reduce parametric affine parameter dependence detail discretize pde convergence ref read follow interest input express degree solve function projection basis dependent full span reduce p selection transformation provide pde eq reduce interest assume bilinear parameter bilinear form functional q dependent quantity quickly via combination f eq compute offline manner complexity dimension ref offline operator approximate measure q ex output functional eq analogously bound bound ex ex ex constant freedom residual representation residual dimensional pre offline surrogate dual dual lead error residual norm far brief overview exposition generation primal previously reduce accurate find low distance measure candidate reduce state optimally achievable manifold kolmogorov kolmogorov know manifold possible reduce often define finite solution measure bound allow construction following maximize verify construct converge kolmogorov width exponentially algorithm bound allow therefore expect rigorous gain constitute area investigation yes definition reduce technique regression computationally indicator introduce employ norm indicator numerical experiment near expected residual improve prediction magnitude exist curse surrogate comment correction email address quantification computing power system answer question guide becoming rigorously quantify uncertainty view uncertain decision measurable assimilation employ collect sensor uncertain input via thousand require fidelity model avoid turn fidelity yet rigorously incorporate quantification context quantify measure fidelity markov carlo costly high fidelity appear employ output output represent bias posterior map evaluation surrogate error practice employ fidelity fidelity statistical surrogate computable exhibit e introduce uncertainty numerically validate various surrogate surrogate fit fidelity order fit employ gaussian process high prediction associate query suffer curse access physics fidelity high fidelity model mesh employ remain physics correction develop primarily global fidelity first trust region center exhibit fidelity dimension order employ high fidelity implement fidelity model fit limited primarily error satisfy often I actual magnitude equip complex computational burden discretization fidelity model quantification problem stochastic useful correction often input exhibit fidelity correction non rigorous tight rigorous surrogate correction efficiency reduce compare correction approach aim datum fit surrogate mapping key physics computable residual rigorous discuss mapping process fit constitute correction indicator depict propagation construct system probabilistic specify htp split west align reduced order anchor south west north west split indicator anchor surrogate input output indicator bind quantity interest correct next introduce introduction objective choice particular summarize relevance construct surrogate rely technique statistical analysis basis dimension nine system input error reduce computation supplementary section high fidelity reduce surrogate formulation quantification consider solve define arise element fidelity system output first query thousand output evaluation aim reduce employ execute g trial basis capture computationally approximately trial implicitly predict state affine reduction decomposition require low residual incur become output output count reduce method linear pde quantify incur equip rigorous residual also exist quantify close tighter control tight constant accomplished difficult lower result various effort improve bound et successive method lower depend offline entire space time improve solution dual aim reduce method offline often fidelity implementation useful quantifying employ rather reflect knowledge error would would boundary correspond uninformative interval demonstrate correlate observe structure reduce apply pde logarithmic scale true error exhibit residual fairly section correction wherein nine input font text align legend font correlation bs anchor west correlation bs anchor south ex ex space ex ex norm map employ strength surrogate indicator addition output norm state training point construct deterministic mapping invertible logarithm interpret statistical model indicator methodology indicator practice g modify fidelity ensure interval employ error essence validation indeed behave distribution predict describe propose methodology indicator employ relevance merely tool accord validate class correction framework equivalent propose identity function mapping highly reduce dimensional ng demonstrate indicator practice exhibit scale indicator equip error strong well bound output error always negative treat error error even employ logarithmic lie within range p bind true expect affine gaussian capture employ transformation permit surrogate assume employ expensive candidate indicator simply computation expect indicator produce residual costly compute constant model variation approximated return depict energy log log accurately model detail sec experiment strong candidate expensive result behave include unfortunately applicable error output strictly positive error log probable might probability scalar section model dual weighted commonly adopt reduction adjoint accuracy adjoint indicator main drop approximate arise
hierarchical increasingly demanding field approximated field although computationally set work approximation mat construct mesh mesh choose extreme extent follow extreme model likelihood sampler structure yield prediction quantile extreme resolution grid covariate outline give explore paper raw hour site year datum base correct account effect correction observational basis accord solid daily wind temperature site set htp observational site set area choose exhibit htp improve predictive wind analysis european center medium weather take account dynamic water output contain calibrate five year information km km across domain km grid reasonable extreme physical spatial furthermore extend calculated knowledge lead quality refer hereafter grid point km km covariate observational site grid point order construct observational site spatial furthermore mean observational site tuning distance illustrative htp use grid decay cumulative parameter two interaction shape neighboring covariate observation parameter smoother tune site close distance calculate covariate use observational site field mat mat field flexible dense become computationally demand spatial field markov field precision predefine spatial address issue mat mesh partial spatial mesh linear mat mesh behavior extreme scale vary extent model hierarchical continuously spatial present model assume hour shape site distributional assumption generalize belong distribution asymptotic condition parameter conditional maxima everywhere affect apart spatial simulate year unobserved site year structure design one mesh product capture spatial variation random mesh triangle figure basis sparse mat matrix enter hyperparameter mat field hyperparameter variance location vertex mesh observational site approximate linear basis spatial project linearly triangle mesh mesh line analogous spatial structure implement scale logarithmic scale covariate dominate distribution field fix weakly relationship approximate mat half km exploratory exceed standard field lead effect capture covariate deviation mainly scale interpretability two spatial field figure deviation lack shape assume assign due mcmc make posterior oppose method converge slowly heavily inference split datum notational dot zero condition zero gaussian precision correspond eq type outline order conditionally q posterior denote extreme index site ai u ai ai logarithm conditional step outline htp f f symmetric eq proposal k proposal calculate calculate k spatially grid th effect parameter spatial triangle mesh however every grid triangle mesh th sample spatial grid combination vertex mesh serve mcmc run posterior particular standard regular location grid analogous th quantile calculate regular generalize plug th scale thus quantile main objective spatial log scale quantile mcmc briefly base iteration modern bridge intel gb ram hard hour calculation base four set statistic mcmc chain show location covariate figure hold correct moreover figure indicate convergence plot four chain evaluate show trace site autocorrelation plot covariate autocorrelation lag autocorrelation highly claim amount highly burn log observational site site axis site labeling show suggest average south low htp b quantile correct measurement construction mean sd sd density suggest effect indicate location time posterior yield point simulated extreme indicate extreme moment finite correlation point near parameter km scale logarithmic standard variation leave spatial might observational site locate stationary behavior observe scale indicate hyperparameter b b observational site compare correspond observation behavior due observational site lower observation correct parameter fit value observational apply indicate difference overall time vary correct row spatial raise prediction surface south spatial south part observational site expect standard deviation near away observational site interesting inside triangle vertex standard regular figure south side spatial gradient rapidly nearby top air root north middle country law see figure spatial arrange manner see raise south part south second row deviation along south spatial see discuss year reflect high predict year south lowest predict interior approximately datum west observational site mm data set b g panel correct htp b leave panel correct methodology beyond framework
choice mh mix approximate multiple computation could big diagnostic monitoring space univariate use summary robust looking room example future capture heterogeneity cluster behaviour allow care alternative would use euclidean square large dense try dispersion cluster non spatial fact mark remark probability region miss realistic historical cluster interesting try incorporate source seem interesting incorporate supervision grant ep name ex model support study complementary name complementary obtain hypergraph metropolis hasting consider efficient proposal develop allow arise careful convergence diagnostic allow dataset around ad without strong intra dispersion interaction organization complementary name college consist location ad fully kind form information historical role dedicate moreover expect form approximately context coherence organized indicate within tend involve variety figure indicate plausible dedicate dedicated name hypothesis typical cluster together list historical period lot neutral historical avoiding assumption help historical already particular work topic question mark realization type variate process available simplify requirement assumption represent role inference model analyze loose provide single visualization historical interpretation nonparametric inference flexibility specification complementary approach process cluster cox process cluster center process seek explicit inference partition method mark mark search mark seek spatial prevent extended interaction point would provide explicit complementary cluster specification point target association measurement track problem perform complementary type datum association interested assess cluster interaction quantifying estimate modeling aspect careful whether significant type common k cross complementary appropriate see section intractable express term find classical association assignment metropolis hasting obtain posterior overcome develop scheme allow hypothesis cluster explicit inference historical interest discuss material include extensive calculation plot datum preliminary result spatial make different list involve refer ref date ref evidence sp sp db db db stand location express survey os national os grid great letter letter location accurate amount c precise term column record see discussion count merge analysis concern datum process entail subject project place variable variation actual record vary amongst treat convert os assume locate os triple triple record see primarily historical interpretation whether separate merge change present merge km record cross great package fall approximately buffer km include inaccurate region investigate type point bivariate interaction type bivariate function divide intensity g type weight intensity classical rely stationary therefore use contribution couple function whether show significant labeling type I location arise poisson dataset spatially concentrate stationarity pattern nan hypothesis poisson intensity potentially vary realistic type additional approximate monte intensity figure multivariate finally deviation significance version km summarize single value gray area simulate pattern dash black red dash value independently strong test function preliminary clustering include advanced provide answer exploratory indicate motivation use moment preliminary communication partition model process discussion nx disjoint trivial subset cluster accord depend global intra dispersion thus nh j abuse exchangeable respect arbitrary cluster unobserved observation q l expect independently value euclidean sample calculation dirichlet dp random conditioning model enforce almost expect alternative inference poisson preferable cluster graphical structure section intractable meaning inference inference complementary move precise little hope solve satisfactory application two induce blue red green type set admissible connect partition hypergraph correspond remainder paper treat formulation color reduce give proposal without change balance infinity help rate derive ordering argument asymptotic omit favor demonstrate diagnostic complexity proposal increase poor need mix property require little derive try obtain continuous tune high acceptance propose proposal long scale move mh hypercube integer randomly sample scale possibility choose move propose l j move roughly move seem perform implement multiple significant consider approximation great cite truncated region grid proposal scheme accept move parallel fashion perform mh factor note bound region parallel would increase especially dataset require need various diagnostic assess indicate section p blue accord code available qualitative look plot sample occur summary match real run configuration estimate autocorrelation integrate autocorrelation ess real use package see version diagnostic overview particular version statistic context summary informative compare look association probability consider proximity consider empty mode mcmc individually diagnostic severe one summary none method indicate except ess step step acc sec configuration proposal multivariate scale evaluate software diagnostic agree mix note performance computer case proposal commonly speed g keep unchanged matching cycle configuration maxima reach configuration cycle move need configuration configuration potential implement local maxima complete case configuration nevertheless application complete therefore exhibit sufficient use simulate diagnostic arise maxima mh long move target cluster pairwise kernel mh group color eq binomial select color project color color configuration follow color replace point color number merge together move know move balance prove indeed equivalent never merge color proposal cluster induce point birth move propose create move application among meaning ij informed color computationally see performance two informed project subspace contain hypergraph state would extremely therefore difficult efficiently proposal choose subspace perform inform space uniformly mix poorly inform inform proposal expensive complicate one care long run convergence diagnostic reach mix high two nevertheless color properly complete section section obtain partition cluster complementary permit range parameter correspond section c reduce respectively lie support posterior association region concentrate fit synthetic without would figure reduce consider posterior km therefore fit approximately km accordance project coherent historical density
show bottom opt rewrite opt combination denote right procedure represent unsupervised utilize mkl combination mkl dependent weight red weighting assume multivariate perform contrast intuitively guarantee regularizer binary mkl classification path denote transpose decision b mkl dependent predefine regularizer rademacher induce mkl classifier g definition prove equation target hold prove mkl l learn lagrange multiplier binary label optimize wise opt equivalent provide follow path product node exist utilize update regularization order utilize therefore node solve fix multiclass fixing multiclass fixing fixing summarize optimize show learn mkl equal path go accelerate speed generate correspondingly mkl label algorithm multiclass mkl list alg regularizers mkl describe structure accordingly encode structure distribution convexity regularizer efficient world information scene decompose scene part pixel answer like useful mkl tool aim combine mkl strategy induce attention mkl mkl direct acyclic dag et formulation information combine account construct regularizer formulation connection node make rather research sum product describe combine kernel product consider mkl create multiplication negative kernel describe embed directly taylor series still kernel regularization path weighting encodes entire involve strong connection regularizer rademacher complexity update organize weighting regularization include comparison direct acyclic dag
ai ai n bi determine eigenvector tv nc nx ix eigenvectors note unlike machine application information eigenvalue dominate dominant step graph consist compute final computed operation overall compute recommend value usually constant graph treat complexity bad fact sparsity adjacency word spectrum walk short e vertex actually application pre five step independently preprocesse pair permutation invariant algebraic consequence path point observation component simple correspond length multiply normalization come among number length length interpret path difficult aggregated statistic path length section capture structure kind indicate relative kernel path path walk kernel common subgraph relative capture summary exploit functional expressive vector deeply beneficial application deal graph graph expansion reduce skew spectrum walk kernel short kernel choose benchmark consist size label mean around protein ec around active anti cancer node maximum focus remain evaluate capture label procedure follow evaluation run split fold identical fold validation set train c fold fold fold act fold act whole classification error result average partition stable accuracy unlabeled graph kernel short count skew optimize noted tune keep thing easy dataset dataset previous skew huge dataset accuracy perform short path perform kernel achieve accuracy representation capture structure kernel power consist skew spectrum capture expressive functional well compare outperform short path kernel representation expressive superior surprising base count common path subgraph small subgraph run see vertex compete except gain time competitive wise superior except success capturing preserve near probability two graph perturb version compare determine graph uniquely determined hard practice behave dynamic loose kernel permutation invariance node long require different behavior less adjacency usually small perturbation operation like edge perturbation kernel thus perturbation kernel graph verify dataset randomly evaluation edge randomly process increase perturbation plot clearly smoothly perturbation compare relatively big clearly perturbation jump value kernel functional significantly power expressive functional simple huge scope possible flexibility room derive expressive row estimator power kernel another explore iteration demonstrate provide interface deal gain kernel lot partially nsf dms fa functional adjacency functional remain unchanged handling form construct functional significantly dataset superiority approach methodology make kernel cubic becoming span bioinformatic social network search natural etc meaningful operating similarity vary design graph incorporate rich structural spurious transformation like certain additional edge annotation domain paper structure extract kernel graph harmonic analysis technique extract set dot product kernel alternatively design kernel graph walk count common shortest count vertex distance although path still widely adopt disadvantage walk count subgraph possible lead kernel subgraph node kind recently subgraph facebook count common walk path subgraph etc instance relatively embed graph represent expressive dynamical construct impose summary graph distribution power benchmark node compute methodology graph matrix node node vertex adjacency always unweighted otherwise matlab therefore invariant capture information dynamical embed simply iteration matrix power sufficient summary power recursively normalization input x x x tx x starting node row use difficult along fact require general force limited degree freedom compare intuitive imagine associate graph start tell sequence generate node go preserve unit treat node kind update
end simultaneous become tensor maxima thus version eigenvalue stochastic ascent discrete rule modify neuron modify fire one component triplet triplet mixture rule rule selective step stable show triplet mixture ensure neuron selective component modify triplet align poisson name emphasize interpret latent underlie input distribution slowly spike period neuron need spike I class maximum somewhat limited sample sequence triplet triple tensor triplet perform ascent expect complicated pre fire multiple interval order spike full triplet r subsection triplet low triplet rule q ball cr see subsection detail projection practice sufficiently projection rarely occur make p linearly mean triplet identical proof update go unchanged view triplet view triplet w stable selective mixture say e co linearly dependent poor k suppose selective intuition emphasize extremely independent must triplet regardless regardless often additional fact transform transform may rule vary bounding thresholding easily either domain produce useful rich possible research proof theory stochastic decompose part update martingale ode previously point lyapunov triplet follow martingale take triplet classical triplet converge slight modification unconstraine space actually lie may act point algorithm behave biased walk toward perturbation algorithm stable infinitely infinitely difficult check set ever small find biological limit fire neuron define lyapunov finite tend hx xx vx x x require algorithms compact region infinitely deterministic lyapunov completeness step converge continuity dot taylor series slightly go go zero fix open neighborhood v nr n nu nu na un smaller disjoint therefore must go converge start simple full ball projection q let rank open replace immediately variance boundedness bound increment requirement martingale bound us requirement q q requirement satisfied note stability zero case somewhat rank instead increment drift randomly undesirable neuron randomly span slight would slowly decrease increment stable expect update denote fact w process note measurable stable martingale increment behave precisely like extra increment control lyapunov note expect lyapunov trivially decay rapidly directly column shrinkage ignore increase variance increment stochastic drift modify arrange stable remain selective mixture however prove kronecker canonical matrix fact property follow trivially kronecker update stable triplet say neuron selective interaction neuron ht vector neuron ik pp computation p nk nd fire neuron triplet fire neuron fire weight fire connection assume l identity hand neuron hand neuron notation kronecker prevent stability stable conditional meet lyapunov neuron selective one neuron selective component iff depend vector fire calculation zero critical point unchanged stability connection jacobian neuron selective analysis occur selective selective eq kronecker semidefinite neuron selective expectation member network feed neuron connect network converge distribute gaussian network neuron triplet mixture randomly unit neuron converge selective initialization per even cause neuron encourage selective call triplet multi provably independent implement mechanism sliding maintain also triplet combine neuron connection tensor decomposition information circuit publication dependent thm lemma thm remark thm plausible learning rule triplet generalize novel kind decomposition substantial incorporate triplet sample spike dependent rather mixture distribution biological fashion backpropagation signal modify incorporate refer provably broad class learning presence interpretation specifically show classical function input prove requirement implication spike arrive spike train learn spike adjacent stimulus biological fully spike dependent however much posterior requirement issue provable form presentation formalize decomposition triplet rule show network triplet neuron outline triplet decomposition definition triplet finally article notation tensor tensor product denote application tensor application matrix rule fire fire fire slide firing rate formulation ht variant rule purpose article define step rule system input draw linearly convenient expectation stochastic
previous possible require spectrum decay furthermore check sufficiently note claim satisfy definition svd standard sketch appendix low necessarily sort sketch randomize see sketch rank approximation satisfy satisfies definition follow projection rely diagonal select proof satisfie orthonormal satisfy furthermore splitting ensure rank frobenius singular selection need compute want approximate diagonal reduction start show us condition together satisfy sketch understand subspace know error several family refer family write transpose sketch independently uniformly except embed position position choose except position sign column hash embed alternatively sample I bss algorithm guarantee family construction family follow family stable requirement ensure f frobenius matrix prove via lemma c moment remark family f frobenius meet entry row preserve preserve preserve k frobenius decrease probabilitie sufficient column norm entry norm family list suitable svd apply purely matrix preserve definition fast tradeoff cost simple avoid establish lemma without go follow thus r apply multiplication moment family note generalization follow case approximate multiplication thus svd sampling produce easy interpret sketch maintain substantial benefit perform obtain error first column subspace satisfy probability norm suggest lemma alone could allow additionally suitable identify nearly without formally data matrix orthonormal constant factor md lemma routine norm column transform column norm preserve give desire family norm runtime md issue computation requirement analysis svd guarantee produce argue frobenius norm trick probabilitie potentially singular direction square spectral newly satisfy frobenius sufficient define effectively singular norm put everything f f ok I connection sampling row norm project onto span leverage respect refer norm residual project first round shot avoid step recover projection cost singular sketch project dependence rather satisfy weak span column finally give selection sketch introduce however extend stable furthermore substantially reduce runtime produce ok ok overall technique sketch multiply project short way project subspace albeit choose give dependence multiply single matrix let satisfy family reduce ok ok cost preserving sketch simply rotation frobenius sketch show cost sketch projection complete low orthonormal matrix satisfy approximate svd follow require combine give column orthonormal basis actually alternative multiply find row onto let whose orthonormal row within span first project complement give suffice svd sketch frobenius norm multiplicative frobenius frobenius give requirement appendix give note sufficient completeness illustrate application spectral project sufficient approximation sketch size interesting use dimension let constant may achieve cluster clear insufficient constrain projection column identity cluster achieve project dimension selection column least cluster optimize give multiplicative substantially write b frobenius norm simply row center multiplying preserve distance preserve alone combine inequality fact decrease preserve preserve give algorithm scope black improvement dependence streaming give streaming computation aside immediate application approximately subspace row compute approximate svd wish give streaming process server row necessary bit streaming stream word failure sketch bit specify give streaming row streaming approximate give matrix arbitrarily assume able central require communication failure probability recent line seek apply svd top vector server locally use communication improve additionally projection result preprocesse entirely inherently pca stem amongst matrix server project proceed could non technique communication logarithmic dimension connect server clustering succeed probability bit ok ok bits lemma column ok draw family server ok orthonormal basis server basis rank proof server row adjust factor communication ok open sampling svd sketch approximate coarse even svd refinement eventually relative approximate svd exact possibly lead extend schmidt lee also david discussion support science foundation nsf fellowship grant grant fa decomposition necessary also constrained mean choose let first prove orthogonal place form cloud drop notation place cluster center simplex center centroid gaussian near choose cloud simplex optimal cloud cloud rather cluster significantly cost optimal low define slightly row cloud lemma value gaussian comment turn follow yield exponentially therefore lemma fraction cost cluster gaussians naturally cluster lemma prove theorem first projection one put cloud simplex cloud origin rather centroid cloud incur sum square gaussian f repeat origin argue incur centroid total square point k mn include claim cloud proof origin cluster course cluster centroid point well claim notice term cost origin gain origin cluster I high gain prove gaussian bernoulli concentrate eq probability contribution sum geometric yield remain bind concentrated around since number high carry desire unit vector product n enough union nc function prove extend analysis svd orthonormal satisfying condition projection kk km svd give equation follow lemma f definition remainder row f r triangle norm substitute gm result conceptually result rely frobenius inequality alternatively compute set section extend preserve frobenius motivation guarantee give f rank orthogonal use notation give give span rewrite cauchy schwarz give finally combine derive bound give easy version theorem draw family error probability cost requirement preserve sketch constrain width width title corollary definition email mit edu sketch solve mean approximation reduce accelerate heuristic many svd streaming additionally give subset cover datum give first dimension sublinear reduction attention fast algorithm reduce usage decrease multiplication rank similar tool heuristic provably seek accelerate reduce cluster original datum approximately analysis nearly reconstruct e start note problem problem nonnegative concept independent ensures solve approximation obtain preserve multiply sampling well focus heavily implementation runtime amenable acceleration underlie svd embedding inexact preserve future randomize significant year embed preserve column cost summarize show compare prior construct apply constrain projection prior c thm thm small preserve project identify improve rank nearly due expense suffice application svd method svd typically lack sketch spectrum dimensionality reduction would preserve useful unconstrained setting rely problem allow generalize work address reduction use selection approximation via sided cost preserve orthogonal projection f k lemma except lemma place seek characterize
input mse estimate coefficient evaluate broadly divide set second measurement mse correlation measure percentage identify inefficient efficient implement matlab code request criterion production process table criterion variation production production pc present obtain model variable pc correlation type operation accordingly robustness variable varied exhibit consistently fluctuation percentage identify percentage efficient correctly identify three pc inefficient experiment e production low production efficiency result experiment production dimensionality production outperform test production mse efficiency production weak production decrease performance improve study robust variation covariance input production fast choice technology three roll surprisingly study overview output likely likewise specification must concern report selection benchmark examine correlation production base obtain experiment evident benchmark method parsimonious selection envelope sparsity lasso group multiplier admm york ny york ny introduction seminal linear powerful quantitative management research single comprehensive generate decision year economic range song efficiency technology stock interested reader popularity certainly research despite publish accumulate paper web decade surprisingly attention variable literature selection often experience economic matter major concern irrelevant omit relevant negative impact misspecification instance al irrelevant position production rank misspecification relevant addition misspecification production space increase tend shift lead power essential limit include consensus extend lasso selection design group derive version tailor multiplier thorough measure parametric production program output output input deviation inefficient originally output sample time input constraint impose ensure estimate case output input lp orient primal dual output orient augment convexity account production return production basically differ assumption production technology radial approach measure efficiency radial assume change assumption radial additive take efficiency orientation formal dual production technology respective associate give however note formulation note introduce situation optimization regularization add absolute geometric e select variable although linear extend additive model respective sparsity solution shift entry one readily application model reader guarantee variable select care selection consistency across stack goal variable extension induce e correlated achieve regression regression grouping limit joint variable solve guarantee study extensively machine efficient solve unconstrained problem section tailor method multiplier admm additive variable apply element sufficiently bound variable regularization drop sign introduce slack transform write stacking column matlab low letter elementary writing ccc ccc x vector similar apply matrix variable likewise describe next alternate direction multiplier admm belong augment method solve lagrangian ax ax b term follow structure unconstraine introduce constrain lagrangian admm find desirable solve augment lagrangian sequentially subproblem simplify cholesky leave hand side cache substitution subproblem solution subproblem lasso convergence admm way splitting function full rank tucker pair problem one decrease simply stay constant optimal treat way rank apply admm selection selection method variable literature contribution measure principal bootstrappe four approach result among pca bootstrapping pca replace pc retain true curse issue bootstrapping involve computational four select benchmark variable efficiency candidate particular scalar quantify marginal measurement essence test statistical significance contribution mean consist selection elimination removal support radial technical population random cumulative density underlie irrelevant additional represent represent proportion whose change consider production associate change statistic reader respectively test estimate production include variable statistically give significance proper output candidate production add process repeat include test technical orient radial behind radial orient radial production mostly observe estimate production contain measurement importance matter overcome study use carlo production process production input represent produce role production production production production efficient intuitively importance production thing denote additive production distribution study half variance uncorrelate generate production show increase
cache probability word representation long input however work contextual nlp part contextual stochastic present possesse unit change slowly see layer gram change similar cache precisely denote unit rule note nonlinearity apply contextual hide decay bag representation trace propose integration force neuron evaluate result observe show big gain strong unit unit activation unit identity matrix size show structural modification constrain equal identity diagonal reason fix constant force unit allow weight delay precisely contextual diagonal diagonal element diag stay strictly help self language corpus division part art achieve combination language lstm language dataset moderately text million character split first character development last character report construct replace less token speedup finding model recurrent contextual allow representation history unit various cache hide short recurrent weight seem text corpus fix text significantly long term current play illustrate cache gram drop model hide show contextual result add contextual unit drop hidden hide lstm contextual increase lstm slightly versus lstm much significant actually lstm lstm paper perfectly introduce structural interpret quickly change short pattern slowly update context short term lstm gain recurrent tune similar outperform lstm margin practically model thousand hide neuron help researcher understand greatly simplify recurrent long pattern published reproduce none model nature store long symbol reproduce would become net controller need increasingly task com recurrent learn time recurrent difficult gradient descent vanish gradient long pattern language perfectly slight encourage hide state slowly part close form kind memory evaluate short memory lstm core variety recently obtain state automatic modeling mostly feedforward recurrent feedforward architecture delay usually time history make hard done increase architecture represent recursively recurrent layer previous store complex period memory theory architecture perfect memory simply powerful recurrent model widely vanish simple simple time simple network memory term pattern practically ignore long reason happen sigmoid zero partially deep relu recurrent empirically backpropagation recurrent term pattern architecture deal vanish long term lstm recurrent neural recurrent promise write recognition lstm fairly sophisticated make neuron information another interesting direction exploit vanish gradient non objective hessian well empirical partially solve vanish recurrent close hide unit behave cache long term model modeling dataset h cc character contain able token past see figure connection hide token token predict store token see sequence token apply token embed vector token max dictionary type architecture replace hierarchical hierarchy token word soft loss mention descent back propagation gradient practice rarely reasonable hyper detail strong nonlinearity appear world neural along vanish recurrent vanish gradient back magnitude quickly pattern difficult fail capture simple extension yield retain
mean mean particle directly accurate within deviation prior become magnitude indicate base well case one guarantee walk surrogate indicate physics tb panel via panel estimate far assume predict informative away evident feasible construct mc posterior problem dominate require pde construct gaussian quadrature parameter locate problem failure lack accuracy reduce suggest mechanism sampling show beyond prior require unless polynomial rigorous impractical constructing base find costly acknowledgement office technology department contract ac grant dms dms dms normalizing laplace indicate substitute equality eq term equality delta follow substitute tu berkeley laboratory department mathematics university california berkeley mathematics university expansion reduce inverse beyond assume surrogate posterior different posterior inaccurate adaptively effective compare parameter incomplete pressure approach yield pdf sampling see require evaluation repeat expansion representation problem approximate surrogate result sample approximate accuracy informative behavior sufficiently sampling limitation quantification inverse analyze surrogate small study sampling numerical summary proof derivation appendix describe affect represent uncertainty pde pde computationally bayesian prior combine give pdf simplicity throughout gaussian prior identity relax simplified nonlinear mc monte posterior involve computationally expensive cost truncate solve pde prior polynomial orthonormal I assume convergence depend regularity p remainder regularity quickly replace model truncate posterior nd kullback hellinger moderate expensive g introduce unless represent truncation surrogate method surrogate approximate wish interaction mechanism due truncation poorly must construct significant unlikely surrogate region significant locate polynomial moderate locally lack base inaccurate tool posterior depend well truncation assume surrogate inaccurate move regime introduce small regime assimilation allow rigorous situation pick choice eq grow small small grow derivation interpretation small posterior informative informative problem example geometrically getting get around obtain q similarly surrogate posterior surrogate different surrogate singular respect posterior surrogate large accurate truncate increase sufficient rapid become increasingly expensive stochastic quadrature point constructing estimate increase minimizer truncate e make small wise eq exponent far mass informative accuracy grow increasingly informative effect experiment inverse choose parameter estimation understand realistic integrable wish datum see experiment length element mesh pde symmetric quadrature solve discretize first eigenvector square multiply squared function rapidly decay spectrum capture expand use gaussian quadrature effect vary could equal decrease length perhaps realistic capture impractical global figure show prior approximation require focus grid eigenvector assume two deviation restrict finite element correspond almost
similar prediction straight method neighbor count serious index tendency similarity algorithm prohibitive completely call broad refer dependence pearson adjacency challenge mean lot sparse extract similarity long large similarity poor outcome high order path method combined method substantially exist especially begin briefly representative unweighted simple self connection measure suppose top exist way score node ref accuracy similarity neighbor cn resource local path simplest overlap method drawback obvious number number neighbor future representative cn insufficient base global ii literature q pure lot easy large denominator tendency degree node tendency cosine index resource pair directly assume need neighbor play neighbor case resource similarity eq q symmetric similarity aa replace although aa index form contribution aa take considerable heavily aa previous study common network iv introduce ref local consideration wide cn eq cn connect extend path uncorrelated fast exceed around positively short prediction node common neighbor propose calculate similarity node rank node similarity coefficient mathematically attribute directly go consideration set fraction link error realization independent short length p cm email area receiver operate auc miss give order list latter consume record time score auc calculate score independent identical auc high cm p cm c compare method four representative email detail set similarity cn lp prediction extract path order compare measure apply move probe link correspond auc fig interestingly advantage fraction path achieve cn resource allocation method enjoy dense predict link link method suppose dense validate dependence real improve lp lp indicate problem address actually reasonable method generally network consider local cn cn account auc well auc outperform cn path auc cn link know literature item well cn accuracy node largely improve degree way auc substantially increase change cn information accurately address node prediction auc probe link cn lp auc indicate connect moreover indeed cn auc though result generally path auc lp extracting employ pearson accordingly predict future common variant prediction extract similarity path pearson combine resource method outperform little new issue remain open compare study pearson direction coefficient study also show also valuable especially already recommender system semi improve recommendation important paper possible way salient investigation partially project link aim node miss evolution link prediction investigate far prediction node base high finally resource allocation substantially epidemic coverage certain model citation
nucleotide incorporation incorporation significantly cycle variance nucleotide incorporation probability flow cycle incorporation calculate variance eqs normal cycle fix sequence cycle eqs q show length number cycle discuss ignore expression exact distribution fix nucleotide flow nucleotide incorporation distribution distribution calculate eqs distribution cycle sequence discuss exact flow exact introduction incomplete nucleotide incorporation determine cycle nucleotide incorporation variance respectively normal slightly long tail compare normal distribution discrepancy find nucleotide incorporation situation ij mean calculate show discussion incorporation sequence may previous generalize account dependent nucleotide incorporation cycle nucleotide incorporate nucleotide correspondingly nucleotide incorporation function nucleotide incorporation incorporation eqs eqs seem form incorporation incorporation generalization nucleotide complete flow cycle exact various formula incorporation probabilistic incomplete nucleotide incorporation although thing avoid traditional bring increase incorporation high resolution region template individually potential throughput become biological software development sequence work support school sequence synthesis generation dna sequencing especially explore allow nucleotide incorporation cycle sequence synthesis incorporation flow statistical nucleotide sequence incorporation nucleotide cycle distribution generalization incorporation significant variance approximate handle sequence incorporation useful software sequencing generation sequence technology aspect technology many available development sequencing repeatedly determine complementary template incorporate usually presence absence signal step nucleotide distinguish modify sequencing read simple relation cycle equal reaction sequence sequencing rather length nucleotide nucleotide incorporation complete possible extension cycle ideally nucleotide incorporate complementary include statistical situation paper nucleotide incorporation dna sequencing technology sometimes nucleotide incorporation cycle nucleotide incorporation incomplete nucleotide incorporation mathematically generalization obtain previously nucleotide incorporation dna sequence development testing software machine dna sequence technology define derivation cycle length nucleotide incorporation sequence technology become employ principle sequencing unlike sequencing target single sequencing sequence significant synchronization identical lose lead signal decay sequence reaction incorporation completion cycle increase individually exist reaction adjust incorporation sequencing reaction incorporation dependent nucleotide define nucleotide illustrate lc cccc cccc cccc nucleotide expansion power detailed recurrence equation close form normalization sequence flow cycle nucleotide look example assume nucleotide incorporation recurrence refer understand q nucleotide incorporation solve however form nucleotide incorporation probability recurrence equation need identity transform solve nucleotide incorporation put compact form four nucleotide sequence incorporation incorporation nucleotide nucleotide incorporation nucleotide flow cycle together unnormalized probability flow cycle treat work obtain get eq obtain normalization fix denominator become denominator dominant part expansion expansion come normalization cycle stand derivative cycle part expression compare availability length variance formula nucleotide
correspond kk k pc original pc solution coordinate many iteration along less al algorithm panel minimizer f give transformation affect composite monotone pairwise query pc show constant great pc positive pc objective strongly convex gradient whole relax convexity twice continuously differentiable hessian f kp oracle ensure correct high repeat reliability pc x subroutine request use arbitrary require repeat query response pc oracle coincide pc find component direction investigate parallel conduct ghz core run compute indeed dimensional showed compare quadratic generate positive use parallel computation overhead accuracy line stop hand tend optimization stop limitation tp several moderate scale optimization parallel implementation original e require implementation line except search assign core core parallel computation approximately serial great practically overhead among processor may scale tp quadratic ann positive matrix quadratic assumption algorithm find simplex significantly standard method depict solid cpu indicate efficiently even pc pc upper efficiency parallel outperform serial implementation cpu communication overhead parallel conduct stochastic pc correct query repeat serial extremely panel slow iteration implementation fast cpu algorithm pairwise value direction search hence effectively large practically implement outperform important direction include kind mm paper provide unconstrained require pairwise comparison estimate pairwise us function estimate pairwise along computation bind find exist engineer field kind tune infeasible treat information widely decade algorithm search function trust oracle tell value evaluation derivative pairwise collect estimate prefer among alternative comparison information stochastic sign oracle stochastic early simplex receive namely reduction order close guarantee problem poorly show positive hold constant convergence objective pairwise stochastic pairwise binary f call oracle affect meaning change al convergence optimization stochastic provide choose uniform solve
profile investigate demonstrate capability multivariate system paper follow ii background family divergence end iv conclusion reduction sequential incremental version technique dimension datum range generalize factorization perspective rank vector value eigenvector rectangular diagonal value sort pca less widely probabilistic draw low large family example exponential family parameter distribution ensure integral pca dimensional happen lot bregman introduce quantification q family divergence equal logit case bregman place efficient bregman divergence approximate work mainly bernoulli random logistic hence logit thus optimize quadratic alternating define iterate call q version streaming every e update e base base gradient vector sequentially equation investigate full td l discuss mainly variable exponential random batch loss function relationship take locally solution define opt surrogate lipschitz continuous regularity thus martingale within within constant appendix theorem recognize sequential converge within however note probably firstly use simulate illustrate focus since principal straight update ht ct cc try equal show sequential step interesting finding firstly within stochastic phase period initialization phase characterize decay whereas phase stand secondly behave differently place hence another regularization summation loss function note unbounded ft could behavior last important mention many completely ignore building modeling end energy attract dependence model bottom work energy pattern individual generate efficiently characterize whole consumption tt energy size want small collect minute obtain pattern enough consider achieve reduction fig good consumption demonstrate adaptively update model interestingly give pair periodic pattern whereas probably result non ht online address streaming extend sequential optimization capability storage sequential application end sum rhs proof lemma n nc tc want decay similarly prefer research berkeley education building
replace sigmoid good available cifar configuration file well classify mnist lstm introducing model objective decrease smoothly resolution investigate follow match begin end want parameter parameter sgd early sgd point coordinate span plot vertical axis objective vary much tell far sgd shape plot dimensionality walk plot residual norm converge similar geometric dimensional different maximum whole fig keep mind give information plot subspace whether behave path direction explore point primary investigate line explain sgd factored fig predict neural curvature curvature solution sgd connect globally neural equivalent rescale parameter multiply divide factor shape linearly high middle manifold kind achievable via span factor lstm feedforward maxout narrow text local saddle fig sgd pass point early trajectory seem explanation sufficiently avoid saddle analytical sgd descent simplify hessian view time gradient hessian taylor gradient go big encourage curvature visualization objective function necessity interpret visualization rich function trajectory instead multiply subspace intend side reduce trajectory circle point almost variation cost subspace sgd trajectory intermediate mp axis allow circle department electrical engineering stanford stanford com involve solve scale non minima motivating however modern achieve negligible task technique network optima initialization never network generally regard optimize train theoretical nevertheless commonly successfully art result variety simple roughly involved neural training intend quantitative answer enter minima pass variety saddle point answer question suggest exist could single break sgd behave subspace main text review case evidence saddle point suggest conditioning neural examine add training sgd ever act stochastic approximation could examine remain cost due induce seven model examine fully connect supervised feed model analytically factor qualitatively outside remainder qualitative factor competitive interpret sgd neural sgd structure training consists initialize extra momentum reach early high trajectory visualization simple learn repeatedly sgd rapidly minibatch gradient remain long period way technique qualitatively analyze line parameter objective behave line search job consistent work begin neural feed forward connect dataset maxout adversarial momentum see specification solution minima saddle fail minima solution break saddle rather fundamentally perform neural network feedforward network verify advanced convolutional network barrier network initialize correspond initialized weight barrier reasonably detail look behave barrier mp model purpose good easy business secondary visualization trajectory sgd pass learn mp may visualization technique explore area e interpolation lstm regularize dropout see experiment convex appear cause difficulty recurrent mathematical deep mathematical network deep form transformation learn transformation expressive capacity factor dynamic fit deep non deep suffer saddle point vary quality link interpolation carry analytically regression problem square error qualitative network interpolation
lp reasoning coincide differ solve impossible column privacy enough constraint time satisfy impossible sensitivity column need accuracy general program program classify private affect program efficient natural private program g semidefinite program multiplicative certain crucially compatibility projection seem differential privacy privacy strong privacy record database range differential record database output definition function database differentially pair record differentially private use laplace laplace differ laplace draw differentially tail laplace mechanism laplace scale exponential mechanism discrete value exponential output maximize mechanism differentially private suppose mechanism combine mechanism composition composition private consider constraint easy negativity approximately repeatedly feasibility unless attention privacy find whether lp roughly private database scalar constraint lp neighboring want satisfie differential notion equal equal additional algorithm operate select action favor write multiplicative maintain dense weight roughly algorithm project action dense distribution step point approximately satisfies arbitrarily instances define bregman let dense multiplicative multiplicative combine ht dense weight follow measure distribution represent hence public feasible independent lp oracle concrete oracle fractional set cover section oracle program maintain pick intuitively loss lead violate lead point feasible taking full least multiplicative pair constraint see projection db ax multiplicative density run point union succeed least condition sx ty ax q let contradiction make depend private point final point since public one privacy parameter oracle neighboring sensitivity private whole directly composition oracle private add constraint lp know project check neighboring satisfy bregman projection reproduce completeness identical respective bregman dense follow constraint privacy fractional though argument private width cover packing covering wish collection cover person fractional select whole set cover degree cost degree least variable degree constraint exactly cover otherwise optimal goal wish individual cover constraint cover contain person valid cover people cover constraint people constraint private solving since vertex select vertex mechanism suitable oracle adjacent return eq sensitivity neighboring database sum neighboring contribute sensitivity contribute differ lp formally randomize input sensitivity normalize multiplicative oracle loss feed laplace mx normalize run private oracle private sensitivity operation operation private private oracle exponential eq since mechanism follow private loss multiplicative weight distribution loss satisfy solver lp program feasible produce point let exponential choice leave x tp bind satisfie take union loss event guarantee independent desire unfold definition like result quite row private entire change neighboring differ private database objective pair neighboring constraint trivial randomized input vector column row slight b mx oracle find private private dual oracle neighboring satisfy private mechanism dual oracle eq sensitive low column sensitivity differ laplace suffice mechanism choice composition let exponential mechanism oracle find point proof nearly everything previous tight coefficient differ leave tight row amount privacy randomize solve private objective simple response solve throughout neighboring change randomize input vector sensitivity concrete lp change laplace solve get exactly lp optimal solution lp objective private solving perturb lp q privacy composition accuracy single laplace bound eq lp perturb add optimality perturb find exactly feasible detail consider various sensitivity turn section show high solve exception constraint private relaxed low reconstruction attack show differential reconstruct non fraction reconstruction key due differentially q restricted entry round zero also desire impossible private lp neighboring database neighbor lp non q likewise say eq bit change private feasible round exactly privacy zero bit objective similar private arbitrarily private find exactly feasible lp objective find objective place mass share private database zero column lp coefficient set coefficient private want satisfy private find feasible public consider find reasoning coincide e produce correspond two impossible possible allow however time produce satisfy single accuracy impossible column relaxation privacy linear program program affect give program approach multiplicative solving feature algorithm use crucially compatibility extend extend rgb rgb keyword claim systematic program privacy introduce several class private program incorporate class program give solver differential differential privacy strong database belong randomize output differential change single record database database private record private basic differential privacy laplace database record mechanism sensitive laplace mechanism differentially private follow tail laplace mechanism mechanism produce element range exponential approximately maximize let quality mechanism proportional exponential differentially private satisfie follow suppose score private mechanism combine mechanism composition theorem composition adaptively mechanism differentially private consider b negativity lp repeatedly solve search restrict feasibility feasible want private database scalar private lp neighboring dataset except solver vector constraint private standard approach algorithm brief algorithm operate action loss perhaps favor aa use multiplicative maintain dense place multiplicative project action probability neighbor give bregman distribution define dense combine multiplicative guarantee arbitrary loss subset public bind eq lp concrete fractional cover see give find present dense multiplicative pick satisfy intuitively lead feasible program multiplicative weight onto dense approximately ht ax loss via dense weight accurate point union succeed step event sx ty ax define eq constraint contradiction first depend final note minimize since hence public follow q private neighboring distribution except row entry density remove lp exactly except neighbor satisfy st reproduce identical respective bregman treat clear divide except privacy example fractional though example packing cover cost select relaxation instead whole cover decide set degree open fractional weight cover program variable cover cover goal fractional wish individual approximate person contain person valid cover people find people private since point lie vector I zero vertex exponential suitable vertex oracle return sensitivity neighboring database extra neighbor since take neighbor contribute since source contribute guarantee show select probability fractional exponential find constraint constraint let unfold applie q guarantee demonstrate fail constraint imagine cover approximation guarantee guarantee output implicit set cover private interpret weak rather differential apply turn adjacent database individual grow simplify form constraint private feasible note rescale find get kind first multiplicative receive loss update aa dense multiplicative maintain response approach ht db mx dual oracle multiplicative accuracy running find point linear public side private map vector neighboring database think decrease guarantee lp feasibility implicit matrix sensitivity scalar generalization offline private weight influential express differentially private solve differential privacy throughout private neighboring database norm look private accordingly oracle private private run private differentially private private query release appropriate vector differ neighboring private oracle private combine private sensitivity private guarantee linear feasible low private sensitivity private quality desire synthetic query universe privacy neighboring far differ neighboring want single sensitivity private neighboring technique equally matrix assume feasibility leave vector vector low sensitivity private basic primal selecting fed weight give ht mx private loss I normalize sensitivity operation operation private private whole neighboring private mechanism distribution follow private analysis regret multiplicative loss satisfy sensitivity lp program solution private sensitivity probability mechanism find oracle hand event tail laplace mechanism take union loss show hold side noisy exactly multiplicative independent exponential mechanism small since feasible desire remain unfold private entire neighboring differ differ neighboring row decrease formally input vector sensitivity private slight modification algorithm ht mx set compute normalize exponential mechanism dual oracle neighboring satisfy distribution dual distribution sensitive private private sensitivity differ norm add noise differentially privacy step private choice
evaluate point large population prohibitive exchangeability agent agent exchangeable invariant permutation main utility depend compete un agent outcome outcome rule realize report behavior h array initialize jt enable pair exchange pair one two medical one test sensitivity accept reject pair issue however pair exchange currently operate centralized exchange mechanism resolve exchange fit paper follow assume mechanism former whereas agent randomly assign mechanism per month usually round pool pool patient medical test easy thus report report pool patient mention compatibility medical literature respectively assume perform exchange study pool report compute mechanism apply along detailed define patient truth behavior adopt armed specifically try maximize pair match simple track utility play pt internal pair tt inference game theoretic observe agent report multi bandit band respective causal round armed believe trend point simulation causal take mechanism make distinction goal experimental collect interested term prior game prior former put exchangeability payoff matrix case obtain expect utility would average match mechanism iii strategy show ccc utility payoff proceed describe collect effect long ground simulate run agent report informative behavior separability take around report show method theoretic former inspection perform informative agent likelihood exchangeability bias long method center clearly biased towards payoff able capture evolution system extent underlie behavior report practically report respective figure histogram estimate method weak separability empirical center empirical overall difference theoretic method equilibrium towards evaluation mechanism challenge dynamic challenge strategy outcome former use report distributional assumption likelihood report agent equilibrium improve multiple way good however ignore aspect agent game theoretic crucial practice base calculation principled hyperparameter third assumption realistic substitution I case agent switch mechanism long independently agent sharing require sophisticated section theorem section economic allocation interested agent process type analysis operate orient good evaluation usually ignore nature raise outcome interest interaction interestingly methodology effect use equilibrium mechanism exchange improve ignore agent causal inference system effect mechanism determine allocation online determine appropriate report designing mechanism good mechanism appeal agent report desire resource price face intuition agent affect report mechanism high high bid initial bid even mechanism property practice typical ii iii model iv interaction get desire participant item bid truly round participant price light design outcome ad want able change decision economic property whole population adopt notation causal mechanism agent agent randomly assign view treatment report raise technical since outcome agent observe report reach interested sensible data consideration body work outcome study assignment potential outcome agent round fashion potential mechanism outcome strategie distinction whereas outcome realize realize potential mechanism denote show causal mechanism strategy compare option median summarize mean also dependent adapt strategy round long capture dynamic evolution agent literature inspection one challenge report observe outcome omit brevity report mechanism hence depend justification main estimate equilibrium accurate economic illustrate compare imputation uniform fully approach serve
increase property streaming encode single adaptively send code match reconstruct optimally dropout neural training iteration drop network dropout main drop unit drop sample subset result importance particular unit rely choice exponential architecture resemble parametric composition encoder lie via assume drop structure autoencoder representation truncation vector remove equivalently truncation function truncation take truncation subset contribute distribution truncation nest mutual representation truncation distribution p l mutual nest connection choice establish intuitively dropout idea index long autoencoder assumption index proof allocation unit autoencoder encoder decoder rigorously property nest dropout autoencoder subset class introduce quality second class characterize restriction last constraint eigenvector magnitude arise input order contrast autoencoder pca linear encoder apply encoder omit clarity similarly define matrix whose consist composition decoder semi autoencoder seek denote frobenius add continue truncation drop truncation define truncation truncation truncation let diagonal eigenvalue arrange respective similarly eigenvector arrange eigenvalue magnitude truncation place autoencoder prove invertible bb correspond eigenvector reformulate notation observe semi connection autoencoder great linear include rotation permutation identifiability undesirable nest problem assign seek dropout justification begin appendix dropout autoencoder problem lead inverse minor inverse combine establish tight lead principal truncation row let truncation inversion element nonzero couple effectively non add rotation nest feature unique optimum solution discuss dropout deep specifically deep autoencoder million image dropout introduce challenge proceed strategy overcome image process subtract conjugate select wolfe relate seek encoder feature motivation unit epoch independently element layer dropout nest dropout minibatch mask virtue decaying become index training phenomenon vanish curvature raw mean slow call stem example word latent unit index upon gradient omit speak iterate neighborhood cardinality decay terminate retrieval pre specify terminal marginal retrieval reduce retrieval retrieve fraction dataset retrieval complexity independent share consistent produce demand study property autoencoder visualize neighborhood query nest variation loss train dropout autoencoder invariance choose retrieval retrieval hamming scan database mean semantic perform neighborhood semantic great force scan addition increase likely query order retrieval carry terminal plot retrieval terminal neighborhood size similarity bit retrieval well representation continuous degradation degradation message give rise quality combination correspond continuous degradation property appeal digital video estimate bandwidth receive pose minimize formulate give minimize online streaming signal quality attain advance various seven different definition select order offer utilize high variant needs order advance length transmission correspondingly fashion minimize distortion qualitatively degradation compression autoencoder dropout cifar reconstruction column represent represent quality image row look original bit reconstruction code autoencoder architecture dropout apply truncation approach truncation un order bit optimal remove decrease influence second taylor units disjoint training compression quality low reconstruction suit image study spectra highly image lose content quality unit dimension autoencoder generalize deep enable learn representation adaptive truncate shorter order retrieval cardinality idea approach knn competitive optimistic combine future insight practically spirit variance idea complicate grateful partially award appendix proof every nest necessarily optimal solution autoencoder autoencoder recall nest dropout different truncation truncation pca decomposition exactly minimize nested dropout nest dropout dropout mixture particular truncation autoencoder problem lead corner truncation inversion apply mean zero nest truncation consider optimal dropout truncation proof hold must true equation give principal bb bb truncation solution nest set top order namely must principal minor orthonormal sake must identically theorem degree remove coherent linear autoencoder rigorously application deep number learn retrieval logarithmic independent allow long currently feasible avoid quality code perform speed order promise learn compression automatic discovery increasingly aspect learn consideration extraction often critical procedure representation enable feature engineering deep find representation hand analysis find low interest unsupervised discover code structure hash deep result encoder decoder autoencoder boltzmann code equally transformation parameter permutation give kind autoencoder invertible degeneracy pose due attain representation architecture freedom impose constraint learn include permutation propose structural specify dimension representation choice us representation intuition behind propose representation index pre decay dropout apply mask individual assign nested unit space mask early depend lead inherent ordering representation motivate order solution strict dropout
provide original split hold investigate typically help attribute semantic traditional descriptor indexing category descriptor reveal vs rank pick moderately domain shot learn invariant demonstrate e straightforward combine place descriptor want periodic variable like pose ii semantic observe acknowledge support style none text height begin ac uk multi descriptor framework semantic descriptor shot datum practically domain analogous domain generate descriptor demonstrate outperform alternative multi establish share domain address distinction subtle method distinguish relate capture device office amazon pose across multi individual category address domain un address knowledge neural address multi learn perform simultaneous concept descriptor exploit improve share classic descriptor implicitly task classic school pose student school school year group represent semantic descriptor tuple school sharing exploit variate semantic share know task paradigm address construct category unseen interestingly lead shot adaptation appropriate unseen suppose audio variety acoustic variety first shot address jointly discover various linear task another common column low encourage share share grouping framework task disjoint share low task fundamental task middle linear predictor think column predictor predictor decomposition ibp prior vector dp entity categorical variable study notice drawback task structure school year group school replace categorical variable impose variety rank suffer adaptation da study propose unsupervised mention typical amazon imagenet video despite categorical generalise continuous angle paper alternative formulation share adaptation knowledge way encourage knowledge direction exist review previous tackle classification camera problem versus eliminate time category construct mid information exist semantic refer attribute illustration go shoot shot see novel descriptor shot issue partially modality see domain despite title actually consider adaptation label domain task denote feature vector descriptor j effectively indicate task loss generality task two figure side start original descriptor weight train back propagation perform calculated ground every neuron try neuron neuron hide input miss neuron neuron align leave right output neural side inner efficacy approach middle length descriptor prediction next clarity task setting w ls notion keep correspond semantic descriptor categorical semantic descriptor constant domain available improve share simple state fashion contrast form task descriptor improve information share exist multivariate efficacy framework interpretation simple well share multiple task task simply descriptor task instance dominant task regressor descriptor word semantic shot match e present testing category turn category zero shoot f adaptation address rather encode descriptor domain effectively thus construct apply datum demonstrate help term relu place encourage model preliminary satisfactory task logistic lr four ii iii within verify original descriptor encoding setting except hold time hold descriptor baseline transfer lr fair descriptor vector baseline include plain tensor completion tc store model categorical always domain rank low rank dataset student predict note student year school year student domain distinct categorical variable domain leave one domain strategy base hold descriptor case hold performance hold turn outperform alternative h r lr
component simulated observation vary stagewise different top row frank wolfe frank wolfe computing coordinate descent update frank share stagewise one frank wolfe regression start warm frank wolfe uncorrelated simulation termination terminate stagewise interpolation estimate rule frank stop frank wolfe stagewise rule frank wolfe uncorrelated second setup count wolfe iteration case number meaningful frank wolfe stagewise frank wolfe match accuracy especially change frank wolfe converge compute default second stop frank setup stop situation frank wolfe overview completion solution trace regularization use implement proximal decomposition svd dimension generally expensive scheme alternate package truncate full svd partial svd roughly solution repeat iteration problem explanation emphasize proximal descent converge desire stagewise vector r computational example simulate add discard entry value warm start run stagewise step curve draw stagewise estimate identical exact solution suboptimal stagewise measure square large yield basically albeit square curve exact gradient solution stagewise estimate stagewise wolfe frank wolfe mean stagewise frank wolfe repetition frank wolfe proximal descent iteration across average quite rapid descent default moderately stagewise run type compute truncate svd become stagewise throughout rank current bottom average spend second compute second per stagewise translate square routine develop per stagewise reflect runtime leave standard implementation somewhat advantage naive stagewise specialize top big stagewise gradient collect project examine set movie rating estimate use rating error hold end warm start stagewise right plot stagewise solution slight advantage exceed stagewise error curve begin drop strongly continue size stop size reach slightly minimum descent compute average per simulated explain long second stagewise construction take second singular stagewise step beneficial stagewise frank wolfe frank solution value wolfe compute pair description frank wolfe frank wolfe problems particular implement frank wolfe algorithm start regularize use warm frank wolfe figure rule frank wolfe stop achieve proximal second stop square maximum value stop frank wolfe meet limit regularization frank wolfe stagewise accordingly step stagewise run less second versus message frank wolfe serious difficulty solution level proximal gradient descent frank wolfe run much compute second point stress performance termination actually somewhat mean begin figure rule cause frank wolfe warm trivially terminate take iteration trivially overview cast gaussian fuse total variation compute solution apply flow maximum flow elegant highly fast exist fuse fuse line stagewise trivially simple comparison require line code fuse stagewise sparse multiplication denoise example add pixel noisy display flow solution direct warm stagewise square corner bottom draw noisy record computer exact slightly towards end stagewise take stagewise roughly second majority spend solution mean square visually reasonable job stagewise surprising htbp stagewise step stagewise runtime maximum solution stagewise estimate stagewise cc noisy version stagewise step compute consider stagewise large front new color channel green separately lie noise pixel stagewise achieve rise red blue visually reconstruct remarkably noisy stagewise total recall produce fuse lasso discuss issue course less large stagewise fail path portion therefore across proper seem tendency practice monotone progress alternate back behavior encounter response decrease say continue stagewise htb uncorrelated case regularizer plot across size begin small step continue way end attempt stagewise offer suboptimal estimate completion problem current mention somewhat helpful estimate stagewise regularizer general appendix differentiable convex lipschitz pair stagewise value result stagewise estimate denote remark stagewise take approximate regularization still simplify e norm gradient optimization usually make norm utilize naturally suggest namely example square spirit extend regularizer nontrivial q one theorem go bound stagewise apparent stagewise repeat stagewise update case unbounded update constrain version theory motivate conceptual help procedure forward stagewise stagewise large absolute residual direction sign intuitively step inner residual inner change increment coefficient product variable residual component stagewise increase monotonically eventually occur thought decrease large variable seem unlikely especially variable many setting recover exact stagewise like therefore make successive update explicit backward stagewise routine significant amount arguably effect step stagewise parameter stagewise importance update towards even mechanism repeatedly achieve maximal absolute inner seem implication entirely speak frank wolfe step discard frank absolute e maintain section regularization stagewise wolfe trivial conclusion look regularization regularization use frank wolfe procedure practice insight gain introduce typical wolfe frank wolfe warm start frank wolfe choose iterate feasible minimizer far wolfe basically end run frank wolfe parameter prevent control place frank wolfe warm stagewise share great successive stagewise differ control history helps think stagewise increment another frank wolfe warm meaning construct iterate component correspond absolute adjusted say word frank entire start empty seem inefficient certainly stagewise step default give frank wolfe history depend distinction remain wolfe less history stagewise rely adaptively course wolfe guarantee solution arbitrarily stagewise goes also argue frank wolfe compare stagewise warm start consider specialized strategy provable achieve control duality varie depend frank frank iterate admit implementation run frank wolfe warm start sequence dense reasonably choice spread converge draw frank wolfe follow strategy stagewise subsection reason section run stagewise frank across variety summarize comparison follow frank wolfe adaptively start make efficient use previously guarantee stagewise wolfe algorithm speak solution solution predictor component predictor enter leave set strongly frank wolfe sequence contrast previously estimate construction one constrain additional type circumstance monotone stagewise necessary history carefully end sometimes momentum stagewise place claim perhaps surprising regularization frank set solution repeat frank approximate solution follow simplify entirely construct begin compute initial duality gap verify path visit property mean visit begin quantity serve gap representation construction mt q frank wolfe path follow case uncorrelated choose duality path mean exact stagewise maximum regularization solution interpolation wolfe stagewise leave square curve vary produce parameter value span path start path meet duality total iteration wolfe iteration stagewise cut stagewise frank tune performance frank wolfe frank wolfe estimate competitive square error stagewise estimate plot frank wolfe visit value number huge fuse setup noise display fuse stagewise path construct basically rough surprising problem limit path path ensure stagewise stagewise parametrization interesting implementation iteratively shrink adjacent component possible prove stagewise rely monotonicity fuse signal piecewise corner image fuse lasso grid adjacent format fuse stagewise see stagewise appear good emphasize figure stagewise stagewise build contrary stagewise outside generalize stagewise accurate path otherwise desirable concern want certainly low image fuse lasso usually underlie complex image stagewise simulated ridge ridge offer highly optimize coordinate implementation package ridge net utilize pure ridge compare moderate amount close choose example meet rough stagewise regularization iteratively amount gradient logistic implementation stagewise routine section meanwhile sophisticated coordinate thousand line code solve regularize regularize broad stand complexity predictor coefficient vector predictor differently entry row independently diagonal setup uncorrelated positively correlate predictor case run path warm run stagewise size figure look uncorrelated simulate draw observation sequence minimum path record early stage exhibit misclassification stagewise estimate turn estimate table share average draw record computer coordinate stagewise second stagewise second compute estimate uncorrelated stagewise stagewise estimate descent stagewise stagewise coordinate repetition simulation model dot show table stagewise perform uncorrelated computational require step produce meaningful right stagewise curve draw deviation stagewise stand uncorrelated represent close failure perfectly though entirely stagewise step full stagewise small show stagewise track closely display odd trend test error slow stagewise progress rough contour positively correlate contour exactly elliptical origin stagewise add update direction gradient thin pass norm iterate norm increase progress stagewise plot display distinct achieve jump may behavior square stagewise competitive probably similarity issue course stagewise thorough understanding topic future htb base serve feasible convexity helpful dual lastly rewrite stagewise q arbitrary express term old inequality third eq rely calculation prove stagewise update generality argument similar obtain bind separately apply lipschitz inductive use term b decompose term triangle argument assumption converge apply write upper together finish complete lemma fix follow limit hold limit verify taylor eq q taylor around generalize binomial forward stagewise start zero update residual interesting forward stagewise path furthermore essentially outside minimization differentiable stagewise even stagewise success modeling motivate stagewise apply broadly current stagewise structured completion keyword forward regression boost variable linear candidate sparse stagewise produce coefficient decrease inner residual precise description repeat q commonly element stagewise select standardized add possibly stagewise reasonable forward easily active assign small htb model predictor panel forward stagewise plot color stagewise algorithm axis path also stagewise path visually appear identical study intuitive difference stagewise closely name procedure iteration choose ia far iteration produce old early accord forward stagewise inefficient useful forward backward perhaps keep mind resource time modern present considerable benefit across stagewise regularize iteration product could trivially connection sequence stagewise review panel essence stagewise appear lasso counterpart step exactly consideration stagewise hold stagewise lasso yet situation former still point stagewise general convex follow stagewise regularization parameter vary initialize intuition behind stagewise algorithm update repeatedly implicitly adjust imagine solve htbp setting detail stagewise forward stagewise dedicate stagewise algorithm derive specific stagewise problem stagewise theory conclude discussion argument center stagewise stagewise procedure matrix completion nonparametric stagewise simpler stagewise highly actual though share actual stagewise component actual path competitive essentially case even actual path view comparable metric across setting stagewise proximity solution third favorable property stagewise estimate regression display hold norm square control tt tt zero path tt enough panel discuss early stagewise seminal purpose path limit stagewise path tt algorithm coincide stagewise lasso path monotone confirm stagewise estimate example regression detect truly less understand cast lasso favorable stagewise forward stagewise forward stagewise estimate path monotone level somewhat remarkable simple produce stand relatively sophisticated practice component rarely monotone stagewise theoretical answer know correlate enter leave repeatedly stagewise finding stagewise along lasso decrease sum rate limit stagewise speak arc length account entire history stagewise produce estimate tend lasso predictor lasso see point stagewise tool apart link loss fortunately stagewise beyond begin analogy step opposite component large reduce stagewise routine author path imply order expansion twice position path path cover loss position stagewise simple really advanced part logistic poisson outcome connection optimization encourage outside stagewise regularize backward stagewise forward general take towards would amount forward backward backward stagewise distinction result monotonicity path suitable globally lasso latter prove stagewise lasso along path forward stagewise path backward fairly extensive work connect stagewise stagewise still end degree similarity stagewise path mathematical much fact simple forward lasso next start stagewise maximal value norm q minimizer stagewise stagewise regularize stagewise direction minimize among stagewise balance small increase solution regularization intuitive aside stagewise regularization already present discuss differentiable stagewise procedure motivate stagewise small regularizer stagewise trade path iterate subsection stagewise special stagewise procedure stagewise beyond present make several remark initialization termination easy stagewise criterion general solution path successive upon reach iteration iteration last reach justification triangle g norm successive give justification step parameter modify replace linear taylor element modification different stagewise stagewise choose choose perform necessarily get taylor tight imagine close direction write subgradient norm admit come section invariance minimizer provide express common statistical follow stagewise update evaluate add analogy operator nonsmooth express simply use unbounded stagewise z stagewise modification stagewise problem initialize stagewise update lie replace eq modification g semidefinite work call lagrange probably optimization lagrange rather solution path vary respective necessarily mild nonempty hold visit versus focus intuition formulation relate paper reader familiar identify stagewise descent convex iterate minimize inner iterate one usual descent descent special meanwhile stagewise stagewise usage seek eventually nonetheless stagewise algorithm minimizer path minimizer stagewise iterate statistical property gradually balance path researcher method stagewise frank wolfe algorithm minimize differentiable project descent minimize approximation frank wolfe minimize approximations wolfe modern problem frank wolfe frank wolfe look general stagewise distinction frank wolfe rather iterate general discussion appendix linearization cutting bundle regularize minimization frank wolfe problem value one linearization history iterate conduct bundle stagewise though would another class boost iterative think stagewise descent wolfe vast review closely forward stagewise fitting consider setup weak boost factor select match scale unit equivalent selection stagewise express stagewise boost add search meanwhile stagewise loss boost greedy stagewise practically especially fact slight stagewise forward stagewise boost forward stagewise suggest stagewise universe weak learner boost transformation variable iteration add weak early stage dense problem specify universe straightforward completion image denoise learner broad stagewise offer learner intuitively reasonable group completion section lead boost arbitrary though form work apart method regularizer extend update utilize variable learner perspective stagewise regularizer regularizer similarity proposal stagewise block index partition group weight generic differentiable function stagewise invariant term computational lasso predictor variable admit grouping define group outside set see study logistic relate distinct consider regularization fit collect task write outcome task coefficient jj regularize default importance group stagewise distinction initialize regularize update w stagewise update let omit proof follow straight kkt simply block scale move coefficient opposite group visit leave identically stagewise match intuition coefficient nonzero value group actual intuitive examine kkt back compare solution stagewise problem stagewise step similarity path highly path behave stagewise later large give thorough comparison place group approach q stagewise problem stagewise compute follow cover recall self recall define update direction broadly stagewise update regularizer wise compute dual norm norm consideration nuclear sum singular trace norm partially observe define trace regularization trace multiple consider stagewise initialize stagewise procedure leave vector respectively proof trace stagewise stagewise singular matrix vector assume let method depend either recover multiplication computing method iteration qr second stagewise path completion problem stagewise fairly note hard path role clear coordinate path correspond interpret slight stagewise present large get effect square stagewise several regularization ridge regularizer component towards predictor estimate identity spline p express let spline basis cubic knot knot typically knot across ix difference set stagewise follow matrix stagewise check kkt stagewise quadratic computationally trivial reduce yield estimator compute system generally system across operator compute cholesky relate operation require operation certainly naive entirely desirable bandwidth initial cholesky operation successive importantly spline regularization case spline local spline care singular semidefinite strictly stagewise albeit slightly deal iterate spline difference projection onto stagewise arbitrarily must constrain stagewise check instead denote generalize computational stagewise rank spline apply computationally solve multiplication stagewise update p spline regularization exclude regularization display solution stagewise notably stagewise path surprisingly step number need stagewise large cover rough appear regularization stagewise path trend uniformly seem ridge really
describe useful exploring beyond neural autoencoder enforcing rely heavily perturbation generative stochastic traditional robust optimize output pseudo boost meanwhile child share member bag forest prefer diversity member output member average regularizer member ensemble motivated intuition robust perturbation perturbation input example produce programming seek perturbation optimization generally seek solution g perturbation several machine robust svm regularization correspond rkhs optimization support notion statistical appeal directly rather proof I closely ensemble work input globally optimize poisson input particularly relevant work effect quadratic ridge estimate several pseudo ensemble input layer input part use fairly network operate control distributional property layer output child generate parent unlabele observation variance activity importance act ix ix hold reasonably architecture dropout help prevent co encourage e hide activity remove local provide regularization kf never act parent accuracy perturbation layer regularizer sec ability regularizer reproduce support co empirical act perturbation fx influence dropout optimize logistic regularization indicate logistic penalty layer bring regularizer train network regularization apply penalty operate element wise far expand recall layer penalty performance test three digit mnist digit supervise dataset learn full implementation reproduce available mnist comprise digit test activation layer layer bias output bias inter early e measure state sde five initialization penalty parent network encourage match dropout course sde test protocol label set split set split denoise autoencoder pre penalty layer pre output latter gradually course bias layer sgd supervise table compare previous aside specific technique comparison nn pl regularizer respect manifold tangent manifold gradient tangent pl pseudo ensemble prediction unlabele treat label carefully training except pl benefit add I reduction label htp r nn pl sde sde manifold dropout plus protocol pl remain sde former agreement mnist label available challenge domain label cifar color unlabele comprise among neither class image target domain winner challenge convolutional coding pooling pool ignore convolutional pre cifar deep comprise max pool convolutional fed output remove hidden layer train domain layer dropout achieve training strategy pre unsupervise train feature activity allow improvement show adapt ensemble stanford task sentiment phrase extract movie review com label phrase amazon ensemble bilinear form recursively indicate indicate vertical stacking training condition full norm constrained pre train code publicly online parent weight dimension parent phrase computation significantly could training subspace time phrase tree process phrase tree perform weight phrase perturb implicit recursive recurrent may objective curvature ill training process code htp grain compact sampling paragraph dynamic suggest measure prediction fine grained sentiment sentiment fine grain class strongly sentiment similar neutral phrase negative class four test past noise past performance task propose pseudo dropout feature model conceptual ensemble regularizer perform empirically behind dropout success ensemble competitive real sentiment benchmark rapidly evolve line especially figure notion accord train child randomly parent examine
misclassifie otherwise would conjunction option load color graphic terminal graphic macro ltb lt lt lt lt lt ltb lt lt lt lt lt lt mathematically induce overfitte instance still instance impact early difficult instance negative hyper quality part hyper hyper effect instance induce loss support loss limit affect set hypothesis without hyper gray hypothesis bold instance reduce mathematically color conjunction option explanation load graphic explanation terminal graphic ltb lt lt lt lt ltb lt lt lt lt lt lt bp quality low quality lower search probable hypothesis searching subset result power training remove obviously induce mathematically instance induce package conjunction terminal explanation load package graphic terminal graphic ltb lt lt lt lt ltb lt lt lt lt lt bp r r fully search computationally infeasible induce determine way induce inducing investigate remove training algorithm probable classification interest maximize input contribute class start presence misclassifie commonly follow use diverse learning sum would hypothesis hyper course diverse estimate free weight hyper equal treat zero set run fold seed algorithm select algorithm use classifier output diversity different agglomerative default set dendrogram connect distance value py perceptron tree nn I rule learner forest repeat incremental pruning algorithm candidate filter set empty initialize accuracy return cross accuracy filter use entire filter dynamically allow set combination adaptive ensemble filter construct add binary also threshold discard iteration baseline accuracy approach without filter line stop algorithm increase add since maximize allow actually practical setting insight potential gain improve compare sign alpha addition report average percentage lr gd reduction acc gd set interested optimization percent likewise percent reduction accuracy capture time equal percent percent variance percent percent percent percent accuracy accuracy validation use search hyper selection hyper learn hyper considerably considerably hyper amount grid provide supplementary material high validation filter identify instance construct learning accuracy produce instance hyper six commonly nb memory hyper mlp ib rf rip red red red compare default classification increase hyper variance filter percent large percent also generally mlp ib rip red red red count red na count compare hyper learn f use default setting optimize adaptive compose hyper accuracy accuracy examine compose optimize algorithm hyper result gain exhibit less filter hyper filter filter gain filtering could accurately gain choose appropriate training datum provide motivation improve algorithm without variance compose hyper optimize increase na I forest algorithm cccc mlp ib nb rf rip mlp ib rf set hyper great filter bold base instance filter filtering previously examine efficacy filtering characteristic rule filtering include well efficacy filter learning determine set hyper instance examine existence hard classify significant hardness instance induce improve handle instance examine training survey noise classification quality improve technique instance misclassifie instance outlier induce negative beneficial discard produce model instance training seek clean instance valid weight instance discard instance consider binary beneficial corrupted correctly label noisy induce properly significant thus previous set grid manual type hyper technique machine combination hyper alternative approach discuss hyper mathematically mathematically hyper reduce instance process filtering remove completely effect estimate potential benefit also choose learn maximize filter hyper increase learn algorithm filter parameter optimize optimization reason requirement instance determine instance filter motivation quality dependent
uncertain uncertain link connect presence uncertain spurious reduce iterative propagation uncertain link stage designed formalize collective introduce uncertain connection exist uncertain network associate network undirecte easily label represent ease notation node integer set node special uncertain collective subset label uncertain label collective assign label bayes perform labeling belong adjacency incorporate uncertainty continue build augmentation classifier overall label rest refer give unlabele denote note uncertain probability particularly edge likely much description particular label contain use label node successively label far initially expand set adjacent expand label node propagation probability iteratively estimate propagation label expand repeat remain label label terminate remains perform uncertainty begin termination compute expand end end step propagation examining decide successively refine iteration examine edge fraction estimate consider edge reduce probability always prevent problem label probabilitie value default adjacency reasonably unlabele neighbor unnormalize bayes compute li li p incorporate sum determine noisy low impact classification example white must assign existence probability low htb unknown augmentation collective subset edge activate modeling adopt selection link would determine configuration training ratio aid selection check precise value set accuracy note avoid overfitte pick edge optimize start expand enable inactive probability ratio ideally want contribute positively unlabeled configuration goodness node value highest denote uncertain iterative good basic labeling particular vary evaluate efficient identify accuracy uncertain uncertain network probability node ratio network sample uncertain set high iteratively active edge estimate label probabilistic labeling sample begin expand graph configuration correspond frequency use conditional probability maintain third subset node classifier relational follow li collective equally linear combination classification discuss probability cardinality immediate unlabeled neighbor node represent node neighbor unlabeled probability require require sum iteration cost uncertain maintain list automatic decompose expand summing terminate cost simplify algorithm library intel ghz processor gb ram times average interval comprehensive record scientific relation co year author publish paper period research computer area author rest corresponding verification graphic human software bioinformatics security htb name verification testing graphic interaction software engineering bioinformatic compute security retrieval citation contain least edge category category report name node truth although manually category pick one two label word class algorithm robust lack name chemical computer electrical perturb set advantage test vary inherent performance edge edge normal interval parameter deviation large eventually control add edge remove exist edge datum retain criterion also know node label remove label whose noisy default equal assess use repeat sampling training label remain refer truth limited compute confusion count diagonal repeat statistically result sample sample node identify vary experimentally dataset algorithm remain variation sampling since trade run spend method method node probability thus weight relaxation version relaxation accuracy far algorithm deterministic sample estimate membership probability voting variety ratio edge report worst achieve dataset percentage accuracy worth note due high report eventually experiment deviation edge figure consistently nearly explain capture correlation different processing well ignore contribute overall classification set label default retain edge report figure perform well dataset slightly percentage improvement low sampling stress test consistently consistently retain edge uncertain automatic robust dataset percentage confusion confusion insight especially misclassifie cell report ground classified label label bioinformatic lead label network refer information example mining due class versa fact example chemical confusion c c c positive indicate bold true positive indicate accuracy vary control overall classification process always never configuration omit belong video category computer communication north american medical label chemical turn label improved answer efficiency perturb cpu algorithm noisy slow due automatic bad vary noisy see label complexity iteration contrary iterative become successive visit network propose high level accuracy time require execution baseline include remove iteration examine accuracy processing increase reference execute depict figure increase stop set htb algorithm graph becoming use protein network link collective relevant determining describe efficient effect probabilistic treat label parameter diverse classification conventional directly support fp ip project grant agreement author laboratory agreement pt claim real network graph probabilistic classification affect final link accuracy underlie network focus structure treat automatic show incorporation effectiveness efficiency collective node represent example real list co label network area desirable information classify interact protein movie actor actor edge actor actor movie correspond category fraction label collective study present many link well uncertain probabilistic
increment achieve parent child flat omit fig high code four compute root bellman block perform visit ensemble step accomplish third top way dag also dag nod compute analogously contribution decay linearly dag represent simple consistent flat dag include algorithm tree exploit child dag propagate top specific hierarchy positive prediction pass propagate structured assess effectiveness propose hierarchical cm cm di di universit di mail di range text biology label present tree direct contribution structure dag dag dag present accord hierarchy gene acyclic go gene flat prediction prediction independently classification relationship accord general step step connect learn basis yield learn machine learner provide classifier combine consider hierarchy prediction flat method apply protein categorization music classification annotation classification different structure dag contribution dag dag path design structure direct acyclic represent represent relationship parent child class unique dag add unique root add node flat eq classifier classifier example flat continuous classifier belonging case flat label set predict label true path rule say multi label score valid easy example flat classifier prediction without label class classifier hierarchical obey word propose hierarchical dag top level correspond maximum node root flat straightforward brevity I begin max bellman visit else v fig code root path end bellman use find sign obtain contain per visit row root process top dag ensemble first row bellman vertex graph dag version traversal step bottom visit traversal perform fashion traversal version top necessary true v begin algorithm compute bellman visit top visit
wang case pt informative screening mode result bound fan also assume correlation theory come screen eliminate uninformative mode later screen test excess test test multimodal uninformative mode informative present method interest provide cluster relate penalize version bic use pairwise none penalty provide consistency relevant relevant consistency np hessian denote denote closed review mode shift detail let feature population ascent flow satisfy iff mode function cluster assignment algorithm second hausdorff distance mm screening coordinate density estimator let mr empirical distribution feature reject nan multimodal critical want unimodal versus unimodal unimodal choose u suggest multiple bind least rejection refined test building simplicity available bandwidth scope accurate suggest rs deviation coordinate al note three finitely mode function critical point exist finally relevant multimodal particular slowly function unimodal cluster smoothness need sure appear set dimensional restrictive close axis middle show possible selection make assumption curve imply much version assumption well mass boundary furthermore derivative n b except error second low large vanish bandwidth pair near exponentially boundary decrease hausdorff mode relative mode separation tend hausdorff distance negative event close n false testing recall note lemma nx x bx extension bridge reject large previous include property mode separate eigenvalue finite take density supplementary material let mx latter gx xt bm et start mode lemma relative hold basically omit condition hence suppose already I cx jk condition word mean px hx expression b x nc near boundary third cardinality case hausdorff distance follow bound brief type false implement package figure show test fail value power appear logarithmic plot multivariate fraction combination incorrectly multimodal instance case surprising test conservative method dimension mixture correctly recover multimodal problem hausdorff signature quite strong selection assumption prove top loss probably boundary hausdorff think acknowledgement
letter modern advance thousand million impractical limited promising deal dataset relevant statistical difficult feature often redundant relevance employ powerful unified perform unified selection commonly unify unfortunately method monte bayesian describe model py notice flat maximize distribution maximize usual laplace impose penalty helpful perform logarithm energy strongly relevance computing strongly affect go assumption effect useful surprisingly likelihood mild feature strongly practically two rapidly irrelevant e feature apply intensive infer model consider variable furthermore assume denote specify position define contain definition observe throughout prior work factorize prior strength regularization twice minimize commonly plug subset dominate prior log extensive regularization much saddle evidence log likelihood physics commonly reason work strongly regime series expansion eq twice effective regularization strength spin ise proportional small weak model ise ise require expansion converge letter letter selection regularization expect distinguish bs informative label red part logistic commonly statistical model categorical data simplify notation extra feature variable always parameter take transpose supplement expression hessian plug except multiplicative constant ise use classify bs dataset diverse publicly computer use number letter comparable suggest expression describe logistic regression number feature square pixel b well visualize agree pixel distinguish b general approach vanish prior even limit show selection mean model ise model aside mild regularity independent inference new give algorithm many employ machine approach modern set potential number selection outline stage irrelevant variable reduce dataset apply comprehensive physics gradient model exponential exponential statistic notice denote connect glm restrict scalar write
performance dataset suggest difference composition candidate metric median percent percent percent candidate super compare competitive actually want one serve table super easy see challenging evaluation percent candidate candidate c super decomposition combination testing training dataset merge focus dataset merge versus testing significant standard training alone good merge significant benefit dataset candidate percent top percent candidate candidate super training include evaluate super explore candidate explore super explore much super encouraging section multiplication approach multiplication improve find super achieve multiplication achieve well top top achieve score super seem substantially account super explore decomposition approach due decomposition correct super versus statistically baseline train super compositional hope thus improvement super achieve take domain adaptation insight table super thing ability ability handle work attention noun noun consider pooling sentence generation recognize relation analogy rely section hand eight super way avoid use selection super instead first pass could use highly version super expect pruning reduce still super set future limit adaptation recognition unlike task avoid ability new beyond recognize extend distributional composition noun composition generate limit pseudo distributional task decomposition indicate considerably composition increase candidate table generation allow composition accuracy domain model training insight limitation achieve level expect super compositional suggest section may supplement standard main extend composition approach pass generation semantic meaning aspect component mean distributional semantic context consider decomposition tackle simplicity semantic noun noun noun semantic noun generate noun candidate decomposition pass generate initial candidate slow solution highly include top highly distributional semantic hypothesis similar represent extend beyond phrase sentence work phrase sentence sparsity noun explore phrase noun consist head noun noun mean head noun noun noun test whether recognize noun list recognize give list model recognize many noun noun example head noun composition promise noun decomposition noun composition seven candidate noun choice task candidate different generation avoid choice resource corpus page university text corpus approximately gram gram include gram majority majority noun phrase composition target fast unsupervised high scoring slow supervise super top candidate top dataset unsupervise score list top top combine concatenation every head list candidate slow super top candidate variation unsupervise learn super supervise main contribution together recognition decomposition dataset evaluate super describe present composition cover discusse relate look work composition noun thorough survey generate knowledge focus method much effort technique corpus distributional phrase distributional hypothesis phrase occur context tend consider shared phrase compositional phrase phrase power vocabulary possible gram word phrase excellent suited phrase eventually diverse phrase sparsity compositional include baseline noun phrase context composition measure similarity noun phrase noun calculate cosine angle relatively although order since representation word remain order meaning composition select context word vocabulary composition operation word seven compositional element multiplication word construct frequency train example map vector context vector distributional phrase propose extension distributional semantic involve algebra tensor product proposal similarity context phrase conversely distance product frobenius capture search tensor wise instance follow various composition composition similarity mean equation build call composition sentence unsupervise autoencoder softmax classifier dataset composition question split question training show noun seven belong question heart disease heart seven noun question unsupervise training question yield get correct suffer create noun phrase pair phrase noun task distributional noun double star binary word electrical noun composition evaluate seven noun rank candidate experiment noun phrase either noun noun noun phrase modify thus believe noun phrase noun phrase noun noun noun phrase noun comparison also noun production human ask short target understand scoring answer measure semantic gold reference similarity create noun gold definition show four definition narrow child child dataset term label evaluate classify definition noun definition must class wise attain noun noun noun rank list candidate attempt combined noun experiment create package derive noun test table dataset include compositional force non compositional difficulty problem decide make dataset easy avoid find compositional compositional contain noun character five character act noun neither compositional force well field head noun occur character match five character word force compositional extract choice noun check compositional pass composition solution composition divide testing whether test seven decomposition extract seven noun solution check compositional select target standard decomposition decomposition divide whether target seven noun question effort pseudo composition pseudo true idea red composition standard analogous red although trivial three correspond construct target dataset match size four solution give successful guess red red good major stream price room target decomposition composition five type pseudo argument return five fs group description log pointwise domain five cl variable description right fs power measure ds fs frequency zero frequency corpus cover pseudo experiment hash google web frequency pseudo never argument web dataset much ram store berkeley rapidly pointwise mutual measure strength association positive indicate negative indicate useful semantic improve force logistic sigmoid shape follow positive infinity infinity rescale infinity logarithm base yield normalize pointwise store detail gram correspond mark approximately gram correspond pointwise mutual observe side word normalize treat stop mark assign zero phrase contain look phrase rare phrase relatively phrase likely phrase likely mark never correspond opposite never pseudo domain ds word domain row gram near gram gram corpus phrase process part speech noun close noun leave row frequency column word column noun context density convert process svd svd domain specify truncate singular generate raise singular power zero factor weight similarity extract correspond gram cosine tune wide use since domain gram pseudo extract calculate similarity design experiment except nearby hypothesis role pattern nearby word function row pattern context functional function compute extract gram cosine present super use refine initial super build take input rank super take input rank list generate list scoring consider domain fs negative let zero noun noun general head noun example thing red functional every sort high scoring package rapidly score calculate quickly process candidate work candidate show follow experiment trial composition training come super factor domain singular exponent value function target parameter list use score head noun like similarity fs term candidate corpus red corpus phrases score preference side appear high scoring scoring candidate head define sort score highest design candidate occur frequently follow four set trial decomposition super description factor space exponent domain factor exponent target equation explore candidate consider target decomposition fs million target considerably web gram split form pt pt triple super refine input super view triple range target generate super triple range triple vector represent triple regardless triple come super let triple super possible ds fs order pair matter cosine symmetric kp feature except log positive pointwise super triple train first target appear output leave training triple adjust target standard candidate solution triple label triple class incorrect candidate imbalance target triple randomly triple class triple every triple triple triple triple apply four dataset lack class minimal provide output super target candidate super standard setting four description ds ds fs class super supervise construct super use eight include computation time spend calculate super speed accuracy advantage target red composition solution mark candidate score rank term candidate blue candidate red rank super candidate answer rank median rank row label top percentage solution percent candidate generate possibility metric rank candidate percent top percent percent candidate composition calculate rank include rank define super answer target make median rank confident six broad preferred percent percentage answer super work super vector treat pseudo optimize smoothed element multiplication section point element multiplication contain ds truncate modify form element multiplication target target vector tune approach form merging optimize smoothed matrix show domain multiplication noun recognition decide multiplication output apply vector approach percent multiplication percent benefit significantly confidence level super super percent top candidate candidate percent top percent percent candidate candidate super compare baseline approach target vocabulary composition test section learn expert like past noun vector dataset investigate apply composition dataset use construct ignore rich table super composition subset size composition target target standard dataset challenge dataset testing metric candidate candidate percent percent percent percent candidate look performance combination column super training merging distribution candidate median percent percent percent percent candidate c training carry standard achieve significantly fisher standard testing achieve benefit merging algorithm train good merge noun phrase super noun merge dataset remove appear noun target noun phrase mean candidate median candidate percent top percent top percent percent super noun noun noun compositional noun dataset heuristic seem noun phrase noun noun copy evaluate seven noun dataset column dimensionality nmf value dim rank median candidate dim nmf nmf nmf noun addition introduce wise
modular organization adopt confident correctness modification david school sciences markov carlo probabilistic failure outline write code modular unit implementation mixture gaussian often justify correctness mathematical ultimately implement software arise correctly implement mathematical specification outline correctness type monte sampler believe widely factor test correct reason mode often matter correctly work conv deal cause mistake still sensible look prediction problematic mean might yet high number percent researcher far concern job simply prediction run inner affect vary change hyperparameter setting focus sampler partly good challenge involve highlight specific also implementation distinguish kind test check correctness small piece test possible fail overall behavior implementation test whether different interact produce kind discuss test check distribution review integration modular enable run example isotropic gaussian q toy analogue strategy state alpha alpha k self self mu self pi pi know evaluate def mu self mu sigma log np pi sigma self sigma np sigma conditional probability def count alpha def np mu return evidence h np np h sum size else def self self gibbs sampling routine sample mu x section way unit test code must modular modular happen automatically define keep modular unfortunately encourage program neither modular student update make easy project box fortunately formulate principle modular organization formulate solve purpose function model fed library check finite implementation optimize computation optimization code separate purpose sampler update rule routine mention sampler conditional decompose directly correctness undesirable unit instead recommend eq wrong would fail value verify suitable inference reason suggest write mass function conditional routine replace distribution number exact modular organization support also project modify sophisticated lot use elegant keep separate main additional def return alpha k pi pi sum mu check mu np mu model substitute self b np z enough fail fail sigma n file package file code testing py run fail failure sigma test ok mistake unit unit subtle test interaction sampler chain two goal somewhat mathematically fail good algorithm mathematical correctness powerful technique testing mcmc algorithm simple generative want posterior mcmc resampling operation sampler correct yield variety two indistinguishable frequentist hypothesis account determine significance less simple could ccc pass unclear figure synthetic representative output indistinguishable pass middle unclear
dot os demonstrate hessian weight choose voxel fit equation acceleration theorem applicable os lack acceleration practice show reconstruct scan acceleration acceleration early os appear still well os momentum os use standard os algorithm window convergence support tolerance th os os subset dot os subset baseline electrical ann mi supplementary material detail linearize lagrangian inexact update additional composite q proper inexact linearize separable quadratic l scale lagrange multiplier al show inexact linearize solve consider convex denote inexact linearize method linearize al linearize minimization split corresponding lagrangian eq al definite column iterate yield identity admm iterate linearize convergent linearize convergent satisfie proper saddle since inexact linearize method kk j inexact linearize inexact choice substitution inexact inexact linearize inexact inexact applicable solve inexact mapping mapping inexact inexact linearize reduce proximal update inexact inequality side sum primal gap
haar rotation union put together q putting together hence find correct minimize polynomial constant prove find neighbor fully statistical step limit depend fact choose valid eq complete eq replace last equality coordinate manifold equip metric follow jensen put triangle inequality claim theorem conjecture edu university mail proposition subspace cluster near subspace recover identify subspace estimate subspace geometric linear many specific paper new algorithm statistical affinity subspace weak standard datum demonstrate segmentation simple cost subspace classic dimensional ambient like linear contain jointly unlabeled setting phenomenon separate application cluster motion segmentation face system application point person illumination move lie dimensional subspace mix model subspace reader reference many theoretical performance two find cluster neighbor subspace contribution devise new two near neighbor point subspace subspace neighborhood conjunction main statistical subspace considered cluster weak exist always provide art much simple fully random semi neighborhood lrr intersection ssc omp neighborhood intersection none none none dl refer reader formulate canonical axis arbitrary mostly community result justification provide theoretical dataset name curvature good theoretical guarantee remarkable algorithm call ssc ssc find match pursuit ssc call lrr norm nuclear present construct inner product use spectral paper focus subspace exact ssc ssc omp correct neighborhood exact cluster intersection exact see denote point lie point near set subspace letter scalar frequently denote index span v indicator construct matrix spectral near spectral greedy fashion likely step lie subspace span describe neighbor dimension j u n neighbor dimensional span point close th collect lie subspace u ok pn implementation describe ssc lrr share orthogonal pursuit recovery pick dictionary correct assume point currently sparse close omp theoretically empirically one work provable perform collect neighbor lie lie natural close span span true subspace lie span subspace recover subspace subspace successfully n n matrix contain subspace obtain schmidt give store basis compute projection onto near already subspace consensus select point collect receive analyze noiseless algorithm nonetheless case use subspace subspace iid arbitrarily iid basis subspace eq principal two subspace identical every define lie draw iid uniformly random unit subspace lie dimensional subspace polynomial constant constant explain consistent find neighbor subspace look lie note subspace distinguished increase become subspace compare condition require literature correct neighborhood become worse guarantee exact wise correct neighborhood table semi general subspace choose cluster semi condition ambient lie subspace depend close difficult distinguish explain intuition easier guarantee importantly condition improve performance algorithm neighborhood fast lrr ssc fix compare term ce incorrectly label cluster index disagreement label performance generate uniformly subspace norm fix figure ce average figure theorem however subspace believe tight correct theoretical computational algorithm motion segmentation video use individual illumination raw exist code table ce average method algorithm ssc omp significantly c ssc ssc omp ce ce avg ce ce c ssc spectral mean ce ce time sec mean ce median avg time sec ce ce avg sec ce ce ce median ce avg onto norm implementation reduce find projection square norm u th number ji k u w j step spectral consecutive number group subspace lie subspace subspace also intersect practice extend step main theorem whether correct neighbor exact subspace lie good subspace subspace one neighbor point lie subspace pick subspace probability step consider correct union fact establish cluster neighborhood trivial exact fully find surely subspace form fully subspace find neighbor fix replace find neighbor subspace every proposition find correct neighbor subspace lie construct subspace true projection correct lead subspace index oracle whose success easy concentration random provide iid let draw iid ball row exist constant prove
without share well click display rank position try unbiased click model user behavior click almost click simplify user interact sequentially user examine model analysis click engine user interaction confirm assumption contain click click order click xlabel consecutive click xlabel near ylabel fraction clicks ylabel click conjecture scan small page track linear search apply search click model click page search short word display font look might page contain want user somewhat click distinguish estimate attractive query information accurately present novel statistical reverse click separable model search position past click contribution summarize user multiple click click furthermore share display response user click post derive click empirical engine document ad search click separability examine view independent independent ad factor product separability cascade scan top user ad eq cascade click cascade click extend click modify previous click ic model relevance bayesian differ relevance click click incorporate click document display essentially number display naive bayes sharing user interact skip click incorporate behavior distinguish post click allow kind transition probability depend location examine decay current click document page certain display recent try click ad show position affect click context recommendation basic account modular gain score affect relevance sake brevity discuss deal reverse click rank document query multinomial denote click c I select position click satisfied stop yes west stop east east node click choice return click document user find result bernoulli random satisfied turn ps ahead search click expect decrease click increase multiply pre click instantaneous click document ad click click previous click relevance user continue position click document vector click influence position next click capture transition allow click list specification htbp fill black inner right ci draw ci draw sep ti vi inner si draw sep right si ti vi ti si si ci vi ci inner circle pt ci si right vi circle fill inner ti matrix capital click describe follow user capture click well post click relevance estimate efficiently pass click novel method display three description display word exclude commonly occur stop appear title compute occurrence occurrence click week training query frequency great restrict word time occur due normalize share information response likely relevant query method give able word copy collect large use second week filter retain display close commonly split particular retain purely statistic query four click baseline parameter via independent click click position click position one click position baseline click section position attractive click click click mainly motivation study click since model post relevance dynamic predict click happen dataset accuracy click ad gain understand outperform gain substantial hard learn h xlabel query xlabel ylabel ylabel symbolic east legend pos north bar width table bin forward click table bin click txt bin table bin base click txt clicks txt xlabel query xlabel near ylabel ylabel symbolic anchor east legend pos west bar click txt bin clicks txt bin clicks txt clicks txt base forward click txt xlabel xlabel symbolic x label anchor east pos north west bar click txt bin click forward click table bin click txt bin clicks txt xlabel volume symbolic style anchor east pos north west bar width bin click txt bin clicks click table bin forward click click txt click sequence click sequence click predict click click resp resp pm accuracy predict click sequence fraction datum although unable predict solely click play account click pm figure consistently click prediction click perform well query tail top well overall focused predict click correctly rank actual likelihood permutation click sort sequence likelihood check actual click summarize see rank click click sequence click rank click sequence click frequent click click low query xlabel query xlabel ylabel click ylabel symbolic style bar anchor north bar bar bar bar black click txt error click table click txt bin forward click scale xlabel xlabel ylabel ylabel east legend pos north west bar width bin click bin clicks txt table click forward click txt clicks txt xlabel xlabel ylabel ylabel symbolic x style anchor pos north west bin forward click txt bin click x click table click bin click xlabel xlabel ylabel ylabel symbolic anchor east legend north west bar bin reverse click txt clicks txt bin reverse click txt scale xlabel xlabel ylabel symbolic style legend pos click txt bin rank reverse click txt bin clicks txt xlabel query xlabel ylabel symbolic legend pos north west bar bin click txt bin reverse click bin reverse click txt reverse click bin reverse click txt xlabel xlabel ylabel ylabel symbolic label east legend pos north click txt reverse click bin click txt bin clicks table reverse click txt ignore focus click click table predict top accuracy h click pm click observe reverse click table click predict location click position click click expect prediction reverse reverse around multi click reverse click click observe reverse click pm c click rank c click pm click user interact search reverse click extensive empirical
neuron fully connect neuron keep image architecture image resolution convolution exception specifically convolution layer max unit layer indicator convolution layer difference x convolution third convolution layer size pooling layer convolution size size perform convolution resolution except max pooling third architecture simplification convolution second convolution grouping overlap convolution layer architecture follow modification fully connect layer layer remove response normalization layer convolution layer worth note convolution architecture study convolution fourth twice exist package network dropout rate momentum intensive study network range heuristic us experiment understand configuration validation image great cost set diversity sensitive resolution reflect recognition image hand scene level recognition category entire nevertheless resolution always performance convolution depth impose due pooling conv conv conv conv conv x conv dataset claim human obvious visual recognition art usually imply computational cost grow investigate visual range either show consistently relative convolution bad performance image resolution pool otherwise extremely bad degradation resolution stem purpose dataset design object dataset tag tag concept may nevertheless increase consistently well enable usage deep resolution different add layer would possible additional contribution minor deep grow monotonically depth layer map depth layer top convolution layer number convolution turn yahoo fc conv fc yahoo fc fc conv fc conv fc training conv fc conv yahoo fc conv conv yahoo fc conv fc yahoo fc conv fc fc conv fc fc conv fc conv conv yahoo fc conv fc yahoo fc conv fc yahoo fc fc conv fc fc fc video static dataset subsample million mix dataset new static video video dataset convolution fig video successfully ignore ignore examine mid initialize video unchanged convolution unchanged learnable avoid convolution regularization learn pattern line corner unchanged table pre train update fully layer avoid problem identical connect totally convolution therefore tune convolution layer dataset different kernel change fine dataset helpful dataset different domain combine initialize image network video benefit additional precise provide supervision enable indicate benefit supervision target supervise recognition annotation entire intervention annotation overhead unconstraine preliminary requirement sample overfitte important video hard overcome train transfer video image corpus weakly medium rich sample video frame even domain transfer training supervise pre collect meta performance image resolution computation result indicate resolution image always yield object scene additional helpful sometimes bad select meta future facilitate would unsupervise current far eliminate level important topic great past recognize concept unconstraine show annotation video corpora robust network obtain datum video image corpus pattern pattern ignore video video learn image weakly process less visual lead enable video deep transfer learn complex event video much research recent year technology real popularity device sharing site generate video internet become need video management recognition specific recognition video come movie record event require various concept various spatio capture diverse work learn deep convolution suffer video trivial overcome problem weakly label collection learn image video corpus help learn feature appearance corpus intervention extensive effort make image capture appearance information frame aggregate addition effort develop video overhead empirical show image may spatio complex concept design video remain video recognition recent recognition video show domain natural speech etc great imagenet attract far make apply difficulty extremely imagenet source image collect ground corpora static effort million video benchmark irrelevant video annotation noisy content video impose pixel scalable meta svm meta search extensive requirement grid meta infeasible previous setting overcome lack avoid overfitte apply approach knowledge static image improve unseen video perform correlation meta basis heuristic video competitive previous basis aggregate frame meaningful improve recent work frame benefit verify efficacy transfer semantic image effort fully human annotated image corpus contain supervision collect internet boost recognition practice dataset annotate frame annotation effort much image contribution transfer visual recognition systematic study g resolution knowledge network review propose describe use study convolution show learnable knowledge work evaluate architecture difficulty deep train complexity depth large pre overcome pre fine tuned capture pattern lead transfer computer vision deep architecture multi perceptron mlp manually architecture mlp series transform follow gradient descent extremely matrix illustration essentially response tie connection field tie number learnable convolution visual layer reformulate entire position show small part small dependency mlp learnable learnable enforce share still learnable overfitte unsupervise pre fully technique therefore easily moderate recognition come imagenet report significant improvement traditional architecture significantly group acceleration partially explain develop receive recently power learnable except learnable layer include layer manually fix layer pool previous usually convolution locally pool purpose local certain degree translation handle shift pool output input therefore translation invariance reduce cost pooling overlap size convolution operate consumption pool number include architecture initialization specify depth detail great configuration popular architecture research configuration far successful architecture explanation lack configuration provide high motivation provide systematic experiment justify architecture factor optimize field convolution layer empirical hard layer evaluate various recognition network architecture provide information exist network significantly motivated static grow video intuitive video spatio convolution short extension successful benchmark perform well event fusion million event suggest frame similarly complex recognition frame exploit image properly frame wise static image combine wise feature static propose approach motion appearance information motion frame capture optical process frame spatio volume capture usually semantic concept use top high concept boost frame pooling frame exist work learn approach image help learn case collect solve domain network belief network stack sense necessarily backpropagation capture supervise domain transfer transfer train video frame video learn network overfitte learn video equivalently train frame dataset intermediate pattern additional help well middle transfer learn learnable use train consider supervise pre analogous network visual pattern outside convolution image convolution natural image dataset dataset dataset despite learn overlap kernel visually support image share fine tuning optimize especially high capture corner appear network lead learnable pattern corpus suggest process previous focus pre representation parallel utilize supervise pre address training problem work complementary label truth video rather also conclusion property static architecture boost video recognition certain include target process recognize semantic video
exploration walk proposal popularity method year limitation hamiltonian problem involve streaming instead estimate datum hmc surprisingly stochastic introduce langevin dynamics validate provide task neural hamiltonian hmc powerful chain monte carlo mcmc term parameterize momentum hamiltonian dynamical system enable proposal distant discretization continuous system need metropolis attractive property hmc rapid hmc popularity limitation hmc necessity compute simulate hamiltonian million inference recommender ever scenario massive batch infeasible since utilize entire big datum development maintain desirable attempt apply langevin langevin crucial momentum hmc explore space hmc big enable scale rapidly hmc assess noisy long dynamical system one noise mh costly computation entire practice mh correction acceptance deviation hamiltonian efficiency recent hmc momentum update appeal analyze enable maintain stationary noise discretize small fix computation tradeoff material hmc ii stochastic hmc incorporate langevin finally standard effectiveness suppose independent hybrid monte propose metropolis mh efficiently explore state introduce sample hmc generate simply discard result define identity hamiltonian energy momentum hmc hamiltonian dynamic concrete imagine slide ice energy height momentum mass surface move velocity positive decrease energy hill increase energy direct whereas momentum artificial construct rr u hamiltonian dynamic define map importantly reversible showing leave invariant likewise preserve practice usually simulate system outline alg introduce discretization mh rate long however acceptance tend development make setting turn sampler method propose tuning simulation hmc geometry enable sampling attempt hmc direction potentially implication implement gradient hamiltonian scenario observation appeal theorem stochastic depend parameter abuse introduction random accord multivariate accurate small wide consider minibatch hundred limit hmc hmc introduce momentum continuous differential return order property analogy sec imagine ice wind wind theorem nonzero long invariant dynamic govern vanishing infinity furthermore vanish entropy since semi intuitively noise preserve entropy reasonable fisher increase toward dynamic proof material must introduce correction step consider discretization dynamical entire argument hmc datum splitting likewise consider simulate hamiltonian dynamic importantly result full rate reduce gain fig noisy system minibatch herein different high resulting provide deviation poorly behave extremely intensive mh short run rate mh step rejection variant alg future use sec modification hamiltonian gradient invariant continuous hamiltonian require frequent costly mh alternatively run low problem remainder omit analogy imagine play ice introduce wind surface prevent away decrease reduce type dynamical refer physics langevin use view second follow gd symmetric follow decompose govern supplementary verify calculate furthermore stationary show invariance original dynamic second langevin momentum partial langevin partial momentum show greatly hmc case demonstrate crucial gradient refer previously discuss relate langevin particular dynamic langevin demonstrate large much rapidly case fast lead stationary b serve decay gradient langevin dynamic momentum hmc hmc gradient gradient conduct standard hmc alg without mh correction hmc gradient alg mh correction finally compare mh sampling see imply negligible unless correction add finding validate theoretical maintain distribution stochastic gradient correct costly mh simulated hamiltonian noisy scenario associate hmc illustrated consider hamiltonian sampler fig trajectory path significantly dynamical add correct resample momentum though fig mcmc naive maintaining well behave hamiltonian I million hmc correlate positive five decrease million per calculate sample average absolute autocorrelation versus five stepsize autocorrelation inefficient autocorrelation sampler indeed explore distribution sampler momentum instead move contour handwritten digits classification instance split remain instance classification network hide sigmoid four sgd momentum base regularizer network fully place weakly informative regularizer resample sampler discard burn mcmc iteration burn report momentum converge converge backpropagation dominate computational show scalable collaborative filtering application predict movie music pmf due rating matrix versus recommender system severe issue conduct pmf million rating movie compare base approach rating update item minibatch neural large pmf difference consider hyperparameter use model discard burn rmse sgd result online pmf key mcmc online set high distant build hmc costly surprisingly natural gradient hmc poor address langevin term effect maintain target modification next explore technique broadly technique herein enable scale bayesian acknowledgement fa intel discussion material sde sde nz evolve upon hamiltonian position momentum r p evolution govern two free hamiltonian change vanish contribution equality assume vanish p statement immediately behave r also fisher full note conclude increase langevin dynamic follow temperature usually set apply verification give particular substitute imply compact desire generalization consider case depend problem adapt simulation correction dynamic govern stationary reverse associated r r tp generator generator reverse q g sde reverse tr together detailed balance allow backward property hmc symmetry rely balance efficiency trade efficiency case nonzero fast relate choice relate sampling inaccurate stationary indicate indeed sde correspond p correspond inaccurate divergence distribution evolve divergence govern decompose change inaccurate dynamic size mix rate bind correspond unclear leave bind relate time process prove refer reader detail sgd momentum analogy momentum learn momentum equivalent estimation
iteration yield incorporate ensemble maintain persistent round team cascade round introduce new warm start classifier ten five weight iteration five train variant private evidence utility benefit reveal comprehensive empirical cascade regularization em em causality author common significance physics solve close manner moreover improvement discovery significance complement derivation significance weight classification cascade derive maximization optimize measure challenge let w w n represent assign label b quantity g n w g employed equally increase differentiable close convex make conjugate hand fa fa ac fa c lemma expression representation representation significance apply f e minimize strategy hold optimize carry support furthermore optimal optimization scheme consist series weight cascade optimize illustration weight cascade progress ht input minimizer classification weight classification g ht minimizer weighted procedure whenever achieve small respect monotonicity characteristic maximization h unnormalize analogous derive optimize divergence section turning support maximizer procedure couple effective ensure adequate generalization describe team incorporate
stochastic rich history begin thesis loss information geometry li convexity take hypothesis er chernoff method control generate particular value tool inequality exponentially erm later er chernoff erm let copy since apply show excess constant must take value subproblem excess pair constant erm prefer conduct nice measure measurable space general moment term let g equal choose need let form curve segment clearly equivalent program exist need satisfy result stochastic find stochastic let element value apply corner perturb nearly nearly perturb erm would pick slightly closeness erm pick common setting random yx random maximize previously xy xy oracle various vc class logarithmic class polynomial exact oracle vary excess let z f excess random hyper concentrate show exist arbitrarily arbitrarily empirical great risk hyper erm replacement function latter apply recall erm select hypothesis purely inversion erm excess present vc type class definition cover minimal ball cover far constrain ensure stochastic argument localization result class vc subset contain union center separable function cover bound net long point present finite vc compose oracle l fx minimizers x bb jensen question necessarily minimizer sense bad minimizer minimizer arbitrarily ball minimizer x stochastically stochastically stochastically z decrease minimizer show loss index sequence slow size consequence poor connect good constant target measure poor bernstein rely bernstein f fix excess high erm mistake e modify n q straightforward result excess certain amenable localization control individual straightforward extend result open result vc result class bernstein condition question bernstein show loss bernstein offer problem whether bernstein constructive bound loss condition minimizer true bernstein regardless classical motivate great interest discard assumption ignore metric loss difficulty extensive wise function understand concern mild thank initial chernoff visit without thank serious department communication research centre g value interior moment objective pick yield replacement q definition q condition small either compute eq eliminate sufficient hence far pick constraint minima root substitution root root increase eq consider yield c arrive suggest attain fix verify true eventually limit put regime exceed objective bound z z v less increase arbitrarily arbitrarily consequence separable satisfie ball loss loss bound select large large taking zero trivially setup large proof center follow jensen inequality appear equation convenience countable measurable function positive g separable admit inequality every f take control control state current yield put concentration incorporate step avoid issue operate assumption step expect nd define sense radius life make step state term careful inspection reveal argument cover moreover rademacher process convenience let sub countable paragraph concluding apply result rademacher l step jensen follow assumption thus everything replacement amount yield set coarse bounding yield approximation separable consider countable slightly number observe cardinality hence cardinality probability function analysis factor term particular observe set f cover union imply rhs q inequality exact class convenience begin notation abuse next introduce play recall norm f concentrate excess nf apply concentration theorem hence er chernoff imply failure obtain last since failure guarantee statement erm erm excess corollary thm empirical minimization erm statistical accord n excess exist result joint property prediction expert phenomenon entirely role notion build bridge reduce special fast rate erm phenomenon exploit old suggest recent contact area statistical online include unified bregman
linear complex introduce process idea observe transformation expressive successfully function gps enhance capability include robot contact modeling acknowledgement department college european community fp rgb rgb rgb assumption complex space often learn lead novel feature gp regression gps process nonparametric encodes assumption hence suitable stationarity common smoothness violate overcome limitation approach combine covariance transformation covariance one implement transform input transform subsequently stationary periodic reduction transformation heuristic suboptimal overall regression base devise expressive gps gp space gp function space property incorporation model uncertainty attempt learn objective combine unsupervised supervise unlike motivated discover within experimentally validate model scale ground challenge g full would integrate analytically task discover subsequent learn mapping review use provide gaussian jointly regression representation input dy function latent mm integrate often common mapping consecutive learn reduce dimensional representation simple high input discriminant reduce curse case nonetheless gp unsupervised mapping nothing input reconstruction ica unsupervise learn insufficient regression optimize objective necessarily match unsupervise maximize marginal likelihood guide mapping toward representation overall intuition mapping jointly objective figure gps probabilistic jk measurement use square relevance ard ff weight possess select negative likelihood thus relate gp model show manifold decompose overall overall objective marginal parametrize map input e kernel operate valid gp test distribution gp matrix construct x covariance function train jointly optimize mapping gradient objective equation compute parameter feature dimensional gp parameter approach deterministic multi layer number neuron layer layer backpropagation sigmoid se ard covariance function blue nn dotted sigmoid dash capture well se ard discover input non mapping world demonstrate function assess datum cause gp embed gp capture thank transformation uncertainty still assume require care effect example encode easy space jointly underlie well gp set c se ard ard use possess length anti map I substantially model horizontal slice spectral problematic ard assume learn hyperparameter length need trade frequency preference short scale generalization standard se ard gp ard point sigmoid transfer use show outperform evaluate believe transform intensity transfer intensity map sigmoid transfer notice tend initial transformation frequency visible log sigmoid spectrum non transformation transfer translate superior model physical especially good force evaluate model inspire covariate regular interval covariate leave angle right two signal contact sensor remain five consecutive datum use structure inspire sigmoid real data se ard data gps either se ard nn show gps se nn predict angle area movement occur due regularity degree uncertainty angle fully preferable trajectory smoothly per point set method rmse se ard sigmoid gp ard gp ard learn representation neural successfully feature gps feature discovery feature often similarity neural network gps unclear exploit good deep neural network gps stack also regression framework similar
attention core figure core varied core speed lda yahoo amazon set number core machine dramatically outperform memory disk version yahoo solution time lda number document topic appropriately update handle propose asynchronous framework lead core result yahoo lda able handle million ability stream document disk yahoo idea black conjecture cs edu university edu amazon com california edu meaningful massive document collection contain million token challenge deal topic scalable efficient way paper novel simultaneously handle appropriately modify moreover change computation across processor asynchronous inspire lda significantly art massive topic topic provide way vocabulary corpus topic dirichlet allocation one meaningful massive token challenge typically second need across multiple resource develop scalable tackle package yahoo recently award win distribution effort towards computation across processor early effort work processor word vocabulary partition processor subset document synchronization fact idea lda count across arguably effort scalable recently trend towards asynchronous algorithm synchronization iteration large tree datum allow multinomial item time topic count order computation across novel asynchronous collapse technical key various single processor present tree encode sampling maintain lda type modeling utilize avoid parallel asynchronous communication moreover scalability method million briefly allocation lda number denote vocabulary denote document topic include denote draw hyper generative draw collapse accord technique lda collapse bayes follow wider unnormalized many sample initialization compute generation first p arrays construction scheme generation generate comparison head tail shape style normal style head tail text label grow style level label label child b node child normal child child draw none densely corner thin fit transform every style mm normal style head blue black grow style distance label head node child node child name thick child head draw thick edge parent draw thick none start west east densely dash corner thin transform mm style center style distance distance cm child child node b normal child name thick label head edge child label parent parent draw thick draw e densely corner thin describe multinomial sampling initialization f maintain parameter use accelerate lda without simplify generalized sampling update regard version leaf internal leaf value two binary internal representation use index store child node tree use addition carry simple traversal z consider f go right child ensure half remove cost time toy htbp ti efficient routine deal slight support routine single th simple tree f carry leaf delta update q see procedure deal normalization construct table clearly update method generation procedure operation htbp tw www tn td tw f tw w sample td tw tw td tree yes yes yes yes apply tree sampling step current document current decompose implication sampling sampling fast element increment time tree document document document document element change switch one word document propose cumulative initialization sampling word lda word fact level element change maintain occurrence generate required word performance lda document document expect consider sum dense non zero third implementation yahoo lda follow change blue rectangle node node node node node node node rectangle rectangle rectangle rectangle red rectangle rectangle red rectangle rectangle green rectangle green green rectangle blue blue rectangle blue rectangle rectangle rectangle rectangle node node node worker work area begin scale red green node node node rectangle rectangle rectangle red rectangle rectangle blue rectangle rectangle rectangle node node node node red rectangle rectangle node node node node node red rectangle green rectangle green rectangle green rectangle rectangle rectangle rectangle rectangle rectangle rectangle node node node node node node node j process worker rectangle rectangle rectangle node node node node rectangle rectangle blue rectangle rectangle rectangle rectangle red rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle green rectangle rectangle rectangle rectangle node node node node node node node node node parallel memory processor partition split document corpus unlike document worker grain split correspond occurrence worker illustration split denote occurrence word block big partition rectangle stand worker asynchronous computation suffer work aim worker maintain job queue without synchronization characteristic worker update occurrence access guarantee worker access keep th difficulty parallel execution overcome difficulty token token access resource token share resource access token token dedicated token pass token token tuple token worker token mean activation result guarantee always date access token far successfully keep token pass update require make depend base summation deal issue token copy worker always modification arrival delta arrival worker local illustration unlike case parallel mechanism distribute yahoo update snapshot synchronization conduct yahoo central server server communication yahoo avoid expensive network yahoo sampler significantly could follow copy machine close utilize variable completely concentrate completion graph need processor core show performance lda core demonstrate tree sampling handling compare approach distribute amazon among bag uci repository many paper fact capability implementation demonstrate scalability algorithm amazon amazon million product review amazon project home review typically stop processing discard review review leave process discard result document corpus collection process process stop word follow approximately document amazon parallel platform advanced gb job node core yahoo deal experimental evaluation yahoo scale fair yahoo
table report none semantic cardinality symbol occurrence unbounded equivalent unbounded equivalent offer define schema name impact call consider library describe kind item book art define element publication report book paper article name publication element name title type title name element name element publication element article publication publication thank want global element prefer shared node schema relationship element share loop concept consider library publication additional material like software prototype suppose specifie point site contain specify material software program use paper specify publication present conference element specify title number web prototype paper dataset experiment report name schema element reference additional circle pt inner sep thick scale auto publication b diagram offer definition schema source schema target schema attribute way schema schema assume concept suppose file specification book example generate book book book book share book book book library student schema http www http www library http www library book library element book element sequence schema describe may opt include new schema schema reveal semantic two share exist book ideally style adopt reasoning discover semantic shared challenging template reason goal author receive schema could state schema element high tool degree two state two overlap nice definition author introduce concept schema repository scenario want able query encode schema rank repository schema second researcher work approach appear instance heavily exist attempt one describe solve outline template applicable template schema element regard approach component template act enough template domain indicate set set one processing instance convert step representation definition template match er template tuple format domain return element ki k array receive two schema return real play introduce similar schema resp belong confusion symbol schema denote resp aggregate score schema one convenient aggregate similarity extent two describe piece next subsection template observe handle generic review module management mapping coincide document onto array call array token design deal kind token token token element token specify hierarchical schema approach map array token pair array root name aggregation direct acyclic direct name sub attribute approach name element sequence name parent sub child name element name element schema similar schema map tree sub element presence repeat share graph loop rule loop schema tree map schema parent relationship report type cardinality constraint map name approach onto graph root node uniquely schema tie encode relationship like aggregation finally procedure onto direct acyclic original schema path schema direct uniquely associate relationship attribute al label uniquely node follow return return iii return parent return v returning element say label schema element encode function process detail accord define fall like name element call group concept context element element context context belong pay special attention ability implement implementation merely linguistic match structural principle extend type metric first category metric classify name name last two schema name purpose gram provide overview string e implement receive first end phone exploit receive string compute short character transform latter result call syntactic distance string character character different gram character string house gram use approach rely gram distance gram schema gram report experimental string matching classify like elimination language metric element name parent contribution child two schema element let define child child resp coefficient element similarity higher share induce penalization since element parent q resp parent language dictionary context adopt case semantic formulate schema semantic former neighbor exploit naive bayes look tf score schema far capable consider include third category metric similarity category cardinality similarity discussion compatibility usage cardinality usage language constraint tf elimination yes yes yes yes yes distance yes elimination yes yes name child elimination context node represent resp close closeness exploit interpret element behind element context similar schema match furthermore popular classified approach ii approach approach tree coincide similarity leaf leaf classify child similarity child two child child compare coefficient leaf similarity subtree similarity non leaf compare leaf leaf node array cosine node place node pair high matching pair similarity similarity particular node root node correspond refine efficiently encode structure document map onto vector apply combine leaf classify sub fashion leaf schema element highly mutually similar consist attribute element link leave root element semantic formulate reduce tree matching problem formulate node word word suitably formulate deal complexity dynamic often schema model structural like converted schema neighborhood whose less threshold take integer schema resp neighborhood iii exist syntactic reveal maximum matching solve corresponding threshold similar parametric semantic solve schema approach structural semantic constraint feature schema element sufficient poorly reason similarity schema element multiple score similarity global aggregate partial similarity score improvement aggregate score propose suggest classify aggregation homogeneous see iterative human expert line expectation adjust threshold decide similarity function remove modify aggregation introduce belong interval belong say aggregate aggregate aggregate vector equal totally agree similarity two schema aggregate return monotonic monotonicity agree pair equal similarity way aggregate sum interpret confidence correctness produce system configuration weight exploit adopt option max high similarity available return lowest consider non rely appear similarity approach among similarity accord partial score score equally contribute latter term partial threshold add mean similarity coefficient instrumental interval investigate definition later therefore approach apply linguistic schema linguistic score similarity similarity obtain weight sum similarity score semantic used map onto detail appear involve collapse map reduce two detail consider two list list filter element similarity recognize proportional however happen think describe university describe correctly handle normalized degree computing simplify onto mapping use mention implicitly primitive operation call change node subtree cost sum form consist cost operation uniquely equation tree coincide q th input map map onto map j review tool handle schema handle microsoft server template system specify discussion aggregate implement element vs match external microsoft name yes structural external open element source structural name source relational structural structural researcher base prototype mapping support schema match nice company either source worth observe conjunction datum service subsequently acquire case source file also deal provide use represent match column table focus operate structural discover structural cardinality server discover discuss section specify detailed discussion offer capability semi automatic fashion simple technique check schema share matching support external apply string pre procedure also interesting observe server candidate function heuristic former lexical similarity latter implement aggregation exploit aggregation max operator discuss end people manually specify schema element advance provide report line ask line element mapping format support enhanced interface visualize report auto mapping core intermediate output variety help user datum integration two integration server effort area schema put issue address properly strictly match deal discuss uncertainty match cluster task traditional integration technique source unfortunately span multiple many service handle country social public health service education clearly virtual schema virtual schema heterogeneous ultimately clustered domain domain whole cluster abstract represent repeat represent service schema proceeding schema classify kind conceptually involve ii cluster significantly small source iii activity effective schema classify strategy mapping approach low similarity belong operate stage available linguistic reduction activity map mapping perform array array belong accuracy preliminary play tree linguistic conjunction user schema arbitrary require reason substitute degree schema target one obtain involve approach schema engine search repository one query keyword schema require repository candidate likely user query report degree element schema produce finally combine candidate rank multi affinity keyword schema filter removal schema construct pair coefficient keyword schema uncertain introduce consider contact detail book contact mail phone user contact phone mail assume semantic similarity uncertain preferable identify absence select yield want contact detail student book depend obtain contact mail loose contact phone opt contact phone retrieve user contact mail address schema map score discover matching state study uncertain relational database analyze join et extended e count refer reader book receive knowledge approach et issue source schema schema author element correspondence schema correctness number pose storage fact mapping suggest unfortunately require management graph still demand cost bipartite graph recursively partition merged mapping store mapping mapping high overlap consequence share mapping store call node schema involve block query uncertain mapping purpose recursively query take tree fact opinion fact highlight exist improve overall effort require existence standardize observe schema offer business popular define semantic match similar explore emphasis repository repository initially business actor integrate fashion domain match consider next compose detail perform composition public business life consider standardized usage constraint relational schema analogously key pair schema reference multiple context recognize explicitly model connect argue schema model structural schema differ kind match open computation schema similarity observe year computation schema similarity integrate recognize compute schema similarity look implementation implementation consequence approach generic reason hard achieve different approach theory schema open problem next strategy combine produce semantic step ensure multiple detailed depth construct specification available diagram think hierarchical schema element introduce describe main component role interaction template compare popularity focus finally relate cluster collection management despite schema match perspective external domain dictionary play schema handle expert intervention correctness advanced tool auxiliary employ great system deal hundred schema future plan analyze scientific anonymous thorough suggestion quality manuscript top white drop xshift ex yshift color bottom color em white red draw red color draw version proposition corollary schema discover intelligence many largely database relational field researcher match semantic well know originally exploit schema research describe impact schema template template template introduction template useful future implement introduce related source uncertainty schema schema management de standard representation exchange wide scenario widely scientific domain like order make exchange easier wide web increasingly advanced language content schema schema build schema schema schema schema availability schema simplify exchange procedure software program impose schema exchange interested capability name attribute name denote semantic attribute name content document identify schema despite name share semantic long artificial intelligence schema community alignment vast review survey format represent model diagram subsequently grow researcher hoc schema match offer advanced capability usage schema research contribution survey schema regard narrow schema match describe extent schema hierarchical diagram provide call implement popular help template act appear totally act discuss template classify challenge schema management survey schema match aim section summarize notion schema template systematically schema provide cluster management conclusion schema context like integration distribute answer matching issue bernstein recognize relevant originally application domain se classification researcher work exist development schema decade suggest list current schema excellent schema therein cover survey perform schema evolution schema merge survey devote adopt assess tune optimize discover problem valid specific matching survey several respect show exploit broad area management schema web huge impractical manually automatically classify schema source web organize group task integrate belong domain basis relevance benefit answer get likely sound correct query irrelevant semantic focused alignment matching resemble schema relationship schema match review book find differ r relational flexibility level cardinality hierarchical organization available r secondly schema specific piece reality exist relational schema relational schema generally human expert vocabulary become decentralized effort discussion task resp relational match hierarchical matching discovery big axiom semantic trend schema match uncertainty schema excellent uncertainty schema book various aspect alternative representation schema uncertainty narrow area namely source area publish survey survey exist agnostic survey focus schema match develop subsequently influence schema match problem detail schema business pay discuss management specific schema match impact uncertainty wide range world schema match effort deal specific schema match schema find semantic web literature schema also alignment refer user exploit next subsection size become business imply explore pose relevant challenge surprising design capable rely idea filter element form match one schema recursively partition partition schema base recently entity resolution research size call author framework prototype entity resolution recent approach incorporate describe interact query interest analyze frequent attribute two attribute frequently co query likely match aim attribute attribute co occur aggregate several resp attribute exploit find attribute similarity score database scenario search engine quite traditional schema source time bad quality matching behavior ultimately behavior handle limitation adopt user query available semantic available poor activity extent schema throughout sake simplicity resp encode resp matching matching encode resp subsection encode show present far find usage part generic r diagram relational schema convert internal oriented schema database
initialize message empty line satisfy iff incoming message neighbor strict disagreement original ii reduce bp inaccurate marginal example fix apply bp give true false false markov inference recover probability q markov certain unbiased become correspondence gs variable gs update sample message bp gs require idea bp gs perturb message message bp gs linearly get final iteration bp gradually linearly change summarize initialize message tx marginal I combine gibbs incoming message strict inherently avoid contradiction bp bp repository complex extensive format use dense factor remove instance discard instance represent form remove many factor bp probably instance representation perform variable initially bp fail apply reduced attempt perturb start factor failure repeat bp final perturb bp number iteration iteration bp compare bp result appendix perturb solve bp hundred efficient iteration average run fold iteration bp perturb bp appendix detailed report study combinatorial related spin physics follow hamiltonian spin resemble problem interaction allow sp neighborhood analogy extend relate dynamic behavior solve spin translate dynamic phenomenon focus geometry space rigorous algorithmic rigorous rigorous cavity confirm picture work instance instance situation characterize constraint select control col random col generate sequentially select generate equivalent distribution analysis several phase phase increase reflect r col symmetric r blue regime neighbor one solve belong replica analyze characterize solution distant distant member phase dominant roughly bp converge regime valid bp transition identify phase finite total picture summarize replica symmetry break geometric perturb bp message initialize message neighborhood resort perturbation message bias towards solution continuous message focus short ensure absence message remain equation completely existence several point fix point bp quasi solution uncorrelated result incorporate equilibrium specifie weight implicitly back true true false false distribution cluster uniform solution construction represent bp bp message bp cluster requirement distribution make practically infinite simplify bp limited apply bp solve product eqs message use max posteriori high assignment I represent message message initialize ix ix incoming trivial define allow assignment correspond consider false false false false allow assignment particular update max message large sp j denote sp update equation aggregate sp message marginal perturb combination gs perturb bp iteration perturb gradually increase perturbed sp reach implicit marginal advantage perturb apply single search perturb perturb solve perturb bp solve factor knowledge general sp sp tailor various sp second use sp search iteration different instance power use sp report col satisfy different portion satisfy help break variable use threshold variable perturb bp perturb sp per fail increase factor attempt sp iteration iteration instance col row report bp local show fail attempt sp time figure instance disk iteration control close chance require computationally inefficient bp solution instance large sp trivial I allow col point bp sp similar col sp success attempt sp search bp easier see bp result col col support advantage bp perturbed sp perturb instance factor cardinality variable col col sp perturb sp impractical dynamical besides report perturb packing cover clique cover min max col success rate average successful transition pt pt l pt pt bp sp perturb perturb avg avg success avg success avg success transition transition l transition n transition n transition n n n corner table west table south check produce bp sp eqs product bp eqs whole assignment formation distant bp focus reduce bp sp bp well reach ignore analysis similar effect variable form fix assignment loop lead alternatively long lack marginal perturb bp non avoid adapt choice variable unable backtracking attempt sp simultaneous message towards region bp valid prevent formation correlation experiment bp sp bp fails exponentially update meanwhile negligible local accordingly sp limited applicable perturb attractive experimental conclusion produce assignment hard combinatorial col message pass perturb sample perturb bp solve tractable factor perturbation sp produce sp outperform sp anonymous constructive center sr technology use compute resource compute benchmark report iteration attempt assignment fail series perturb instance avg satisfied avg avg geometric aim c c c school c n n n n book job c c c c c c c c efficient message pass perturbation belief propagation bp propagation satisfy smoothly end solution perturbation sp bp hundred perturb bp compare state sp bad cardinality sp outperform make incomplete solver regime science neural physics code pass successfully solver constraint produce suggest assignment subset sequentially marginal give failure guide procedure back track backtrack branch rely solver purpose propagation solution well bp fail survey message bp typically convergent message hard produce single pass avoid alternative bp apply bp gibbs gs update message perturb start end smoothly change produce change bp marginal bias bp bp sometimes bp fail random perturb bp sp difficult instance sp bp sp perturb sp experiment perturb sp sp assignment bp particle perturb bp directly gs bp introduce perturb gs compare bp bp fold solve geometric review order replica break perturb sp present instance discuss
non fully model al include clinical interpretation fit etc survival log logistic researcher parametric analyze cox et issue prominent grow survival observation censor attempt obtain covariate wang life power along automatic estimator likelihood inefficient provide pure datum develop estimator censor covariate need law central limit censor suitably covariate note consider usual semi regression cox proportional cox robust model respect etc develop robust propose accelerated failure location without parametric survival censor stochastic covariate brief background parametric general censor propose context censor performance illustrate property density divergence suitable remark tune propose section end one derive iterated number central etc mainly work wang wang theorem assume life censor censor denote give limit assumption life censor respective whenever life covariate censor precisely form strong distributional consistency integrable mean atom write define coincide consistent far strong hold corollary measurable function length consistent estimator replace definition respective assumption strong assumption censor see write statistic censor censor biased censor covariate move like cox regression strong efficient see fully set paper deriving result propose power divergence inference quite day robustness high parametric smoothing density two dominate measure parametric power minimize datum equivalently correspond coincide mle drive apply suitably lee index lee extension identically distribute generalize extend censor efficient censor generalize estimator parametric censor covariate set covariate note early focus give f estimate property mle know drawback lack robustness previous density efficiency pure common motivate et suitable estimator joint optimality give objective nothing generalization routine u estimate simple form substitute equation divergence datum simultaneous root equation objective clearly inference objective particular estimating extend concept define however make censor estimator note estimate may suffer root need technique define examine simple response generally variate variable response unknown regression coefficient symmetric distribution group reliability distribution response incomplete censor observation inference belong consider frequently robust solving work example family family distribute covariate p auxiliary variable normally ph px dx function random numerical technique simply minimize function simplify particular px simplify see simplify estimating simple subsection life science model censor short support identifiable become need say suitably cover routine medical science widely popular science variate multiplicative ensure positivity life application exponentially covariate exponential error belong robust normally covariates ph n minimize ne tx use equation simplify erm form equation simplify linearize natural logarithm scale model log covariate density general extreme distribution early censor section minimize objective objective easily highly robust presence accelerate robustness comparable alternative advantage et al censor normal previous simulation exercise scalar covariate simulate response exponential censor consider censor censor keep expect censor exponential censor censor respectively study numerically bias bias mse mle contamination compute total along absolute censor proportion efficiency absolute mse increase censor repeat simulation contamination covariate censor covariate simulate observation total small ignore outlier generate inference prop prop prop prop base censor correspond define empirical prove consistency normality result replace far coincide replace finitely finite proposition consistency asymptotic integrable parameter distribution wang normality result censor extend present censor response censor huber routine application wang assume distribution estimator property wide equation population root eq strong estimator extension wang page replace hence simplicity presentation assumption satisfy estimating sequence converge I complete censor data covariate estimate fall strong really root numerical estimator strongly attention normality regard first proof similarly continuous neighborhood continuous real sequence set singular converge multivariate value yield q follow convergence fact distributional convergence estimator convergence wang extensively appropriate censor complete present case wang replace particular derive particular however particular closely assumption censor score existence second moment condition assumption lemma asymptotic normality require rather assume require strong simple al property extend many researcher censor relax family define parameter function equation condition support parameter space differentiable integral f derivative finitely singular third bound assumption sequence estimate tend fitting normality part show tend local interior tend respectively present get tend combine proving equation sequence root well complete routine assumption although particular estimator namely divergence estimator general suitable equation optimum class huber fact class estimating location odd wang optimum might similar censor presence reason variable belong family asymmetric censor
train radial regression validation follow optimal kernel forecast approach calculate forecast conventional ap integrated move ann table fig summarie numerical ahead power ann svm hybrid svm conventional ann smoothing ann base machine prediction evaluation machine forecasting ff ann sa ann sa performance indicate rmse mae optimization ann stationarity wind note processing field modification ensemble component original hybrid algorithm employ system forecasting series hilbert transform algorithm random tree technique examine rank model employ hybrid forecasting employ radial support introduction follow approach forecasting power hybrid machine forecasting parameter fourth describe variable experimental flow price wind speed direction two major market lead inter trade neighboring region source system often non limit efficiency effective market participant whole past decade power effort novel short flow wind forecasting forecasting addition difficulty power researcher publicly make delay consequently pattern possible novel approach power combine effective stationary series machine organize follow forecasting hybrid machine short forecasting power system balance maintain consumption may power scale intend forecast forecast refer short forecast forecast trading term system aid analysis aid intelligence purpose include machine machine svms random forest etc forecasting note often forecasting show hybrid potential include ann wavelet ann fuzzy expert system reader characteristic account influence everything price often information scientific series impose forecast performance consider improved series task initial investigation improve forecast machine preprocesse forecasting collection merely two mode hilbert transform ht intrinsic block random bt rank ann employ obtain forecast ann svm forecasting system hybrid forecasting increase forecast limited forecasting wind major power need competitive technique wind power energy price decade algorithm system forecast system success success forecasting forecasting mainly reveal close reality plausible important cause forecasting algorithm predictor believe researcher identify subset another separately take value root split item binary node lt reach tree constructing maximize decrease forest decrease proportion reach use examined decision tree tree decrease introduce modify machine realize develop demonstrate forecasting pool market wind hour wind wind year decompose hilbert hour calculating illustrate wind machine forest frequency wind speed comparison exclude train construct test rbf rbf phase rbf scheme rbf network neuron briefly svm xy svm regression g svm determination reduce follow capacity trade deviation parameter loss base fx condition negativity represent case validation follow rbf feed ann candidate ff ann time wind times wind direction time normalize ratio validation set hour search algorithm network ahead wind use pruning involve define maximal intra neuron connection parsimonious neuron use bias backpropagation htbp candidate hide neuron hide neuron predictive inter activation
consider admissible model size nonzero entry later consider subset theorem implie subset respectively subset complete cm pt lemma example corollary generalize glm link assume subset proof use necessarily similar kullback risk include correspond exist equal p repeat difficulty prove generalization g
fast though potentially evaluate face five uci demonstrate cluster consist unlike exist completely setting scene camera use different people leave background acquire feature result face stanford different feature affinity via kernel dataset dim diabetes gene uci measure via kernel cluster evaluation represents assign cluster ground assign alternate determine truth define entropy completeness result harmonic homogeneity completeness member harmonic mean homogeneity completeness weight measure active strategy method number variation baseline multiple active variant active reduce gradient without gradient parametric baseline comparison include active pair constraint fed seek seek nearby cluster use guide constraint query base query maximum reduction value compute uncertainty multi active label request constraint run variant list compete parametric set uci coefficient lead consistently particularly notable minor relatively consistently meet exceed performance scale consistently perform par well particular validate show combination scale way selection problem drive compare technique uci reasonable binary outperform al competition generally plug cluster uncertainty learn clearly superior seek global impact uncertainty idea query consider entire active visually appear applicable present overall compete method far exceed dramatically match also run constraint fail somewhat competitive nature limit usefulness method winner though nonparametric leaf diabetes reliably work active rule make dataset amazon experiment query rate dataset slow experiment overall active passive clustering actually discover cluster effect face certain unknown result initially converge discover test indistinguishable novel sample online active select pairwise query problem estimate expansion decompose scale two entropy pairwise query uncertainty support demonstrate state initially burden adjust iteration naive select uncertain redundant adjustment active powerful particularly crowdsource grateful grant nsf nf ap finding reflect david seek cluster side via semantic side randomly require could redundant unnecessary semi maximize human human great impact select proceed principle taylor decompose step assignment result uncertainty sample different image uci validate show superior noise number cluster semi uncertainty play machine top external effort pairwise face supervise cluster categorization surveillance grouping person particular location may problematic recognition human label might realistic probably image action make context human crowd amazon probably specie even semantic cluster image visual identify specific apply semi supervise clustering rapidly approach large redundant expensive improvement circumstance overcome explore constraint constraint base method interest query constraint select propose intermediate explores initialize grow cluster large semi active min criterion utilize exploration also seek min max informative parametric provide encourage however complexity semidefinite sdp limit constraint process xu et al wang problem suit multiclass recently seek prove meaningful criterion method cluster suffer drawback offline select thus incorporate cluster decision online overcome limitation novel base online great pairwise query base user cluster cluster discover human interaction proceed section yield great identify uncertainty component estimate perturbation eigenvector assignment uncertainty formulation baseline active cluster dataset face leaf common uci gene see show state art cluster ultimately relationship thus relationship pair sample may highly clear relationship relationship decision ambiguity relationship semi supervise remove ambiguity relationship reduce assignment uncertainty measure contribution local entropy detail uncertainty propose uncertain uncertain uncertain beyond limited presence every inherently complexity advantage naturally via proceed pairwise result increment active active typically control generate domain output face dataset initially available application conduct evaluate method selection encourage recall proceed iteratively informative sample pairwise similarity matrix partition eigenvector via effective whenever constraint modify produce new affinity pair link value proceed define dataset similarity current therefore direct uncertainty remove ambiguity cluster uncertainty result consider select sample greatest though estimate must simulate answer could predict oracle would expensive worst require iteration active adopt estimate impact sample perturbation entropy select present briefly describe select pair base query certain generate sample within close l record record sort correspond connection certain sample relation create certain sample add certain regardless correspondingly add reflect newly discover certain select human describe use eigenvector query eigenvectors taylor expansion decompose eigenvector ambiguity represent ambiguity eigenvector result ambiguity reduction reduction first spectral change ambiguity approximate represent incremental change jx iteration reconstruct eigenvector correspond
ps big ml single memory pool machine read simplify distribute program ps systems interface arguably interface challenge application ml program stochastic subsampling dependency parameter parallelization work introduce relaxed read throughput promise consistent success nature play relaxed relaxed synchronization throughput possess relaxed affect ml stability improve progress iteration throughput execute per recent focused system various start principled insight ps ps consistency angle read impact stability learn insight design outperform issue new ideal gold progress ml problematic bounding amount synchronization different empirically scheme parallelization develop attain guarantee provide deep particularly exist ps theory outperform distribute carefully throughput ml still mean optimal answer convergent ml converge environment enforce reflect quickly lead frequent consume synchronization limit speed parallelization therefore synchronization parallelism closely approximate sequential execution inconsistent algorithmic imply still happen read carefully explain trade introduce asynchronous parallel approximate strong magnitude value single view represent worker method worker fewer visible worker update update worker variable condition vary worker analyze algorithmic broadly speak either within base sgd worker operate ps popularity later well sgd expectation optimum successive implement update worker pose worker condition need update structure difficult achieve design consistency provide correctness ps implementation impose work algorithm worker assign initially zero operation share parameter store ps ps worker progress fast worker parameter achieve fast update make visible crucially communication meet implementation propagate beyond require reduce average age axis observation count factorization minibatch experiment bar computation communication show reduce priori predict complex empirically read ps draw conclusion consider worker worker could behind ahead worker last cache update tail profile salient advantage convergence tight bound base analysis strength achieve comparable excellent ml topic show throughput thank exploit scheduling affect place ml descent popularity prove sgd involve miss follow product constant update put access introduce computation produce apply order worker start upon update different worker server network view update I update r worker mild worker sufficiently bound drift global schedule expectation via component size tf measure expectation use speak execution bound assume var var optima capture randomness condition imply argue amount synchronization motivate analysis noisy system noisy update view worker start condition spc decrease suffer theoretical threshold usually unnecessary ml gradually execution read close threshold use grain update implement inside server one process type distinct logic key interface system increment worker define library locally parameter fit memory request cache cache read request send server mean worker worker generate update send server request time server make read request server server advance get row read request cache call exploit often convergent server explicit request cause accumulate usually parameter regardless specify threshold burden server update batch separately request quality collapse gibbs factorization robust additional use gibbs sampling gradient interface minibatch call measure quality mf minibatch record convenient step mf topic york token topic netflix dataset use run core connect via lda connect iteration leave consistent progress pointing help much less robustness tuning slow problem stepsize distribute aggregate deterministic dependent mf investigate profile introduce produce low improvement parameter addition per provide reduce reduce chance update right speed per second lda server state art consistent parallel ps frequently big commonly ml framework yahoo might special generality machine factorization small world lda lda vocabulary size memory lda data scale speed count matrix mf kind ccc header cc split header header software tailor single scalable roughly group category library framework purpose tailor category constrain mf yahoo lda topic primary solver restrict application program improve code communication synchronization protocol careful design consistency iteration specific category distribute ml consistency benchmark fair match algorithmic benchmark ps practically ml framework specialized algorithmic tailor purpose framework ml way purpose enable application research unlikely solver many nature ps propose implement present herein enforce assume transmission could delay read delay inconsistent parameter wider popular framework ml application sometimes knowledge superior ml framework ml alone solver salient consistency support bound consistency side ensure program yet give via update term fact divide gets desire answer search x dx l divide tt eq hessian close expand use take gradient bound invertible optimum var capture condition tf similarly variable condition represent
error occur representation multiple concern latent dimension mean social phenomenon contribute express space help decision feature believe easy simple appropriately world seek follow characteristic repeat learn similarity member generalization community view membership satisfying requirement stream walk originally walk combination denote walk root k vertex walks content recommendation sublinear structure motivate short walk tool extract community information desirable machine explore part secondly rely walk possible small learn walk sub linear walk primitive capture suitable method capture power law observe appear walk follow word frequency behavior law walk text contribution work idea language symbol distribution community law distribution language grow representation appear formally vocabulary maximize network representation language beyond goal present language stream short walks walk think short sentence phrase language analog give visit walk goal distribution occurrence latent vertex later walk grow language modeling turn head use secondly compose right give word remove required appear offset give vertex optimization find relaxation social representation random walk time problem representation neighborhood citation similarity machine learn walk representation social exist encode intermediate adapt change topology b walk vertex softmax factor occur variant require walk vocabulary know walk ahead walk update window walk initialization iw sample vertex random walk walk neighbor visit walk experiment return advantage practice specify walks start start random think walk pass order strictly know vertex generate representation use accordance language word sentence appear window line vertex representation figure give walk use model logistic huge could million resource could whole speed iw expensive leave tree maximize vertex identify root speed assign entire graph maintain enable web web c label interest overview dataset use reproduce website figure pt network relationship topic category website label user popular label group validate baseline generate laplacian utilize eigenvector assume cut generate representation modularity eigenvector encode modular modular graph partition use mean adjacency vote relational neighbor neighborhood appropriately iw surprisingly real sensible frequent micro r micro macro r r nodes micro majority macro majority sensitivity several facilitate comparison method baseline node use repeat report micro result vs logistic implement classification liu experiment bold perform consistently label perform prove much still datum macro micro label b vary approximately label outperform baseline micro additionally micro word baseline perform perform improvement size prevent run much close real result present table scalable baseline create micro improve macro increase lead end micro macro experiment benefit classification order change conduct experiment label sensible emphasize local walk start determine impact available figure vary performance quite dimensionality dimensionality stable accomplish start walk second consistent interesting various show walk see increase start dimension amount datum initially result effect quickly learn walk machine easier dense thing method dense walk extract place limit hope walk distance way good emphasis effectively hope limited difference summarize representation statistic centrality extend procedure collective kernel scalable local information feature relational link collective np solution approximate guarantee converge relevant add nearby relational substantially propose learn inference include representation relational approximated complementary encoding directly method concept propagation instability decade recently compute allow growth distribute diverse speech language novel learn walk learn encodes variety different task cs edu novel representation vertex representation encode exploit generalize modeling sequence use truncate random treating walk sentence demonstrate multi classification task result baseline global missing compete outperform scalable build trivially broad class anomaly mining artificial pattern network strength sparsity hard application detection prediction must able introduce prove successful develop stream walk capture membership dimension generalize generate neural language semantic syntactic logical b representation exploit generate representation community vertex color input take produce output apply well study force show beyond linearly modularity graph demonstrate real scenario
proceed note many variant momentum em algorithm update repeatedly perform dropout parameterization would z dropout hereafter optimize dropout example differently hereafter wise dropout mask unit layer set dropout test validity optimization problem vector informative dimensional informative obeys informative informative irrelevant variance performance evaluate work variable approximate gaussian q dropout rate optimize dropout optimize dropout describe properly decrease dropout inverse rate dropout dropout rate dropout dropout optimize maximum several study except theoretical state cost propose dropout maximize hidden variable parameter infer infer pd interpret bayesian solve nothing assign achieve well assign suffer one one look discrete value parametrize light improvement standard dropout posterior distribution adjacent parametrize freedom expect discussion parameterization obeys also match consider var number quickly distribution diagonal reduce hyperparameter structure intend parametrization allow resolve sophisticated consider conventional dropout overfitte modify interpretation enable dropout beneficial encourage dropout neural kind task language likely fail huge difficulty one important overfitte researcher study perform understanding explain kind dropout think dropout solve input hide train accordance optimize weighted marginal learning model model dropout benefit likelihood close already paper dropout learn training dropout dropout dropout involve feature selection note dropout machine input unit output respectively activation function sigmoid w optimize selection describe model unit mask diagonal sometimes z mask determination determination correspond problem may rewrite represent explicit architecture redundancy huge challenge optimization good mask binary mask take stochastically mix possible stochastic summarize follow initial mask determine every dropout rate decrease properly denote increment termination output denote mask matrix bernoulli pz pz w pz pz fast apply also weight sum independent upper denote transpose unit treat approximate utilize lyapunov improves show beneficial idea artificial corruption dropout adaptive artificial kind regularization viewpoint regularizer generalize linear equivalent transform fisher input transformation regularizer cost dropout function function bring dependent also typical treat extend dropout train dropout bayesian share dropout depend variable mask input whole try mix work clear optimize discuss treat variable infer dropout hyper mask mask take omit simplicity introduce call trial log kl q kullback trial negativity kullback leibl trial respect maximization gradient descent lead dropout mask sample mask explain assume pp dropout dropout determine dropout also output infer intractable calculation summation variation consider trial last explain precede mean
ed mmd mmd rbf mmd neural net mmd word count tf domain represent bag ignore review split task target label source without target domain unlabeled prediction evaluation hyper average prediction random cross domain adaptation connect hide unit relu mmd penalty gaussian mmd mmd source domain train descent gradually decay rate count mmd get boost tf mmd method feature domain adaptation even word count baseline taylor hidden unit dimension dimension recover auto mmd dimension feature expansion equivalent variant contraction tune mmd hx hide momentum mmd filter tend localize variant b distribution feedforward relu units final sigmoid unit mmd sample edu transfer learn develop factor new domain learn readily salient factor important unbiased definition bias bias discrepancy mmd suggest mmd representation apply across formulation include domain adaptation invariant insensitive autoencoder generative suggest formulation transfer focus deep formulation learn scenario task
dot add nb worse exist nb find method compare statistic accept small performance mean sided rank significantly small level set discover attribute datum switching rule test side significantly nearest way control instance inaccurate unweighted datum attribute discrepancy weight whole sound theoretically improve correctness make eps stroke locally naive classifier li chinese university li department electrical engineering stanford consequence strong violate bayes nb classifier favorable size relax nb local ignore weighted special unweighted weight intuitive handling imbalance learner naive bayes base near show parameter seven keyword weight naive nb computational competitive let class nb violate independence categorical call realization label unlabeled whether comprising instance unlabeled instance test instance nb estimate common laplace modification application nb achieve violate behavior observe data nb multinomial characteristic nb confirm available correct classification nb classifiers assumption set attempt network average dependence use estimator extend replace naive classifier parent add paper modify fit locally weight impact neighborhood remain apply nb call seven appropriate choice conclude respectively weight random least weight reduce assign call avoid vector weight estimate weighted relative probability I frequency weighted classifier weighting instance instance datum make compatible b depend compatibility weighting classifier failure sensitivity condition former estimator hybrid associate approach nb partition axis surface utilize test propose compatible nb call cell weight realization hamming total hamming weight use convention weighting cell cell multinomial nb multinomial compatibility hx mx probability encounter laplace law unweighted laplace unweighted desirable sample weight weighted importance effective size importance scale weight weight multiply I choice constant label number weight multiply make total common training class search dominate unnecessary l cm x size training select calculate multipli class q return look candidate appropriate choice exist nb augmented average weighted dependence v vi locally naive instance local uci repository website description summary training attribute kp letter breast cancer breast tumor diabetes heart rate cross fold miss unsupervised attribute preprocesse percentage correct balance breast diabetes heart heart vs kp letter tumor l degradation analysis comparison large size bias towards impact picture statistic interpretation mean list second average rank large method mean pair bar chart arrange bar chart permutation bar lie performance reject significance nb likely inferior two appropriate value choice conclusion side lie method rank differently rank commonly rank
variance scale ok gradient summation term expectation contribute variance undesirable scale number independent variance assumption generative structure energy weakly correlate discuss nature respectively whereas correspond configuration large variance additionally variance former later variance marginal likelihood completion pixel recognition reconstruct recognition iterate respectively computation imputation write miss transition model constitute see eigen marginal immediate consequence apply fundamental practice complete pixel norm stationary recognition q marginal sense apply obtain true fix variational specify bayes consider use compute gradient variance mini batch energy objective maintain remain use size use variance recognition describe appendix show joint explicitly separate deterministic view work transform match provide view clarity interpret formally time determinant jacobian co transformation visible generative model equation explicitly consist single gaussian layer linearity use st international conference china cp author derivation theorem integrable twice gradient eq identity product integral term evaluate support line differently general exponential base parameter show bx also lead derive rule would simple search section self distribution rescale base distribution propagation
costly proximity require gradient dominant well gradient dual operation proximity step see subgradient could fast proximity stop desire example method necessarily obtain desire formulation provide applicability penalty gradient technique valid hilbert problem useful machine machine svms motivate step reflect singleton step past affine weighting still subdifferential let variant extensive literature k attain reference therein gap estimate order mirror variant algorithm view consequently computation become q algorithm domain imply quadratic continuous lipschitz formally assumption every simple therein norm penalty norm regularization composite composite penalty well remark smoothing involve smoothing envelope define smooth smooth proximity u gx property g proposition control tradeoff hybrid adaptive smoothing lemma smooth smooth view conditional algorithm algorithm besides also algorithms lipschitz objective method primarily nesterov smoothing envelope connect nesterov smoothing unlike nesterov function proximity subgradient single dominant singular feasible large power rate bound convergence objective exponent term regard mostly side similarity fista suppose descent twice obtain yield add theorem finite eq convexity iii ax theorem obtain every add lemma assertion every easily computation depend translate iteration rate convergence flexibility penalty standard involve term become method impractical rate smoothing method proximity problem matrix subgradient subgradient require full decomposition addition build parsimonious fashion solution simplify desirable parsimonious dependence square loss denoise prediction proximity norm soft operator sign multiplication absolute matrix symmetric matrix may u k value k share convex relaxation q sparse prescribed indicator bound subgradient eigenvector input k u rank semidefinite propose semidefinite initialization example fall penalty prescribe positive semidefinite prescribed arise fit favor norm norm total hybrid specialized show subgradient etc large compute latter problem stop match penalty dictionary pursuit gradient absence term similarly extension omp penalty hierarchical yield scalable variant omp jj z j simultaneously aim proximal scale recover simulation random draw entry uniformly denote rank corrupt matrix fraction entry solve simultaneously frobenius ij n constrain equivalent comparison intel gb memory evolution sake comparison apply cpu include singular termination termination algorithm change refer per iteration figure time whereas entry observe simulation efficiency uniformly uniform distribution q nesterov optimize matlab core sufficient tolerance rescale order keep nesterov smoothing computational change optimize optimize verify hence run nesterov smoothing optimistic observe nesterov study composite optimization example problem norm penalty nonsmooth technique proximal order operation exhibit unlike proximal benefit advantage advantage matrix trace penalty relaxation acknowledgment lead receive union fp agreement european fp agreement support grant kp grant project structure decomposition g grant policy fp mc theorem proposition lemma question paris fr study frank wolfe programming much optimization formulation algorithm past field control statistic computational currently gradient problem example involve sparsity use learn inspire chapter method solve convex focus function continuous lipschitz domain available proximity subgradient conjugate particularly cover closed term present review gradient whenever composite conjugate many interest alternative smoothing proximal nesterov smoothing involve approximate besides modification show suitable choice objective claim theoretical recent hybrid interest applicability denoise sparse pca penalty trace rank require subgradient computation whereas require expensive computation vector practical mean thus though exhibit rate gradient backward scale large chapter space endow inner convex take constraint
project validate contain validate comparison pass involved loss appropriately website available website acknowledgement helpful ep via centre training pt detecting region property behaviour possibility present potentially model show enable independent distribution motivate application variation individual subset bayesian give evidence copy variation pass outli segment behaviour time kind behaviour could change correlation work concern proportion number dimension detection segment recurrent segment application relate image take detect copy copy dna show account within cell log r probe probe normal would ratio away give substantial noise genome detect cell pool individual complicated observe datum portion affect series segment individual research method subset dimension less variant segment dimension change able rare whether region recursively statistic point region region able estimate number possible simulate posterior describe partition interval segment normal want tractable location observation follow come likelihood define hide full assumption segment segment likelihood form likelihood normal segment likelihood draw segment segment independence completed find practice numerically need specify segment follow model normally also study present calculate give calculate challenge conjugacy numerically integral calculate numerical value reasonable task computationally develop hide start segment filtering eventually full posterior distribution enable calculate widely markov two b p equation segment kt filter c term computational storage cost store filtering thus calculation prohibitive filtering many negligible remove point without much filtering potentially keep greater remove resampling done store approximately posterior straightforward simulate simulate assume simulate repeat go time simulate segment early posterior hyper use hyper parameter monte em rapid look section fast initially segment method detail posterior easily want estimate loss mistake seek segment overlap segment detect small indicate overlap impose generate affected proportion detect accuracy false positive ex pass pass ex ex especially worth proportion detect ex consider clearly mis specification accuracy positive pass apart detect pass mis specification position segment seven segment intensity value normal varied randomly dimension set mean ccc method proportion false positive ex pass pass ex pass ex ex see still pass positive robust misspecification draw keep mean normal indistinguishable replicate segmentation distribution segment plot simulate either two geometric fit take partly pass potentially segment long occur cdf straight line quantile propose fit generate actual think segment datum suggest reality tail distribution distribution shift take dimension take affect give simulate measured observation segment histogram mean segment simulate varied affected dimension set c ccc affect proportion pass ex proportion normally
random technique decompose ica ar ar ar information identity mutual cross q optimal dedicated solver subproblem thank flexibility sufficient switch solver entropy quantity base one base meta function member cm unify formulate solve theoretical whose template aspect scheme spectral ii computer million ica element extensively test file work alternative source toolbox interested user mathematical available example project european co agreement ac computational centre university college house ar estimator free open platform toolbox mutual association measure distribution modular support additionally combination ii theoretical application prototype central problem subspace association analysis extension modularity matlab platform machine entropy provide quantify measure offer tool define distance central objective party relevant ica cluster ica state scale prove iv observe nonparametric dynamic recent exist package quite specialized fill gap come modular free platform toolbox package estimate kind association kernel offers construct exist optimization problem overview toolbox capable numerous quantity complex generalize kernel generalize schmidt shannon quadratic base copula dependency multivariate version hoeffding distance approximate kullback leibler shannon jensen jensen k pearson divergence bregman extension center
operation maintain library hope benefit community least way practitioner elastic net researcher facilitate performance ss program china education grant u ns nsf grant center grant author suggestion chen chen university st arguably past decade rise demand implementation classification easily voxel genome million meanwhile availability gpu multi parallelization however easy convolutional neural machine svm multi although originally utilize ascent drastically lasso outperform optimize single core implementation imbalance parallelization handwritten truly utilize graphic intensive also software elastic net special design recent bias reduction result vast svms immediately elastic non trivial equivalence hinge equivalence relationship elastic net box square loss world set eight four fast date efficient across almost bold scalar capital bold matrix scalar convention contrast refer remainder briefly elastic net regression real value response normalize sparse minimize elastic net constraint encourage effect strictly convex unique highly correlate stable large squared separate hinge denote please separate hyperplane pass origin solve dual duality directly derivation mi formulation connect inner rescaled ij remain run formulation commonly achieve decision boundary help trick product matrix hardware therefore peak hinge svm simply elastic formulation state constraint substitute rescale entirely follow negative represent negative rewrite non please long l always tight large solution constraint carefully classification p p p classifier denote elastic scaling solution add become equality become constant drop remove affect solution difference design without obtain highlight reduction highly summarize refer mention primal formulation svm complexity choose fast versa dual implementation default trivial remove experiment svm formulation small running depend implementation adapt would allow recent solver might exact practice elastic net many effort parallelization dominate core implementation elastic mostly language strategy know net extremely hard propose run include memory constraint transform solve newton popular implementation library optimization tailor svms hinge modern gpu acceleration update operation tend recent contribution svms extend trivial net soft margin validate gpu line match conduct extensive experiment set brief online gpu core cpu cpu cpu baseline core al solver implement et processors ghz ram core gb different solve path slowly evenly spaced setting path select particular procedure compare implementation setting identical tolerance gpu paragraph original evaluate gpu eight clinical volume weight matching budget gpu eight dataset set gpu markers diagonal cpu gpu baselines elastic net eight concern mass predict scene car area whose financial report tf figure depict cpu gpu correspond comparison gpu path training gpu budget marker correspond run fast marker diagonal gpu observe trend across eight gpu marker transfer parallel even cpu fast baseline baseline gpu markers slope
color ground produce effect relatively popular mit bottom row comparison mapping ground produce collect dataset choose image image enhance participant range little experience static website image enhance enhance et al right enhanced ask vote image enhance enhanced choice enhance receive vote category produce enhance user verify whether capability enhance statistically significant conduct effect section ask join design left enhance image produce ask assign enhance enhanced image look visually ground receive score participant discretize range look ground truth score enhance conduct pair significantly enhance desire effectiveness mapping layer descriptor descriptor contextual descriptor build top parsing conduct include conventional propose able automatic spatially vary adjustment parse object contextual challenge vision recognition propagate contextual affect adjustment show foreground contrast incorrectly increase scene parse rapidly develop failure effect semantic label area incorrect semantic labeling highlight correspondingly area receive incorrect result failure mit group mit learn adjustment ground high distance dnn train adjustment semantic object treat correctly system spatially vary transform exist choice dnn layer activation give rise consume search dnn architecture dnn behave box predict neural grateful discussion suggestion support research general adjustment neural paris yu image email address edu cs microsoft microsoft enable invoke consume advance beyond automate alternative manual face many rely subtle content spatially characteristic exist limited cover subset challenge machine learn unique motivated explore deep context automatic adjustment descriptor account semantic experiment technique semantic yield qualitatively ht enhance deep adjustment region effect digital device social medium popular try exposure invoke visual even traditionally well extensive study color wish novel automatically enhance several reason adjustment empirical relate color enhance image need process quantitative relationship nonlinear especially style vary nontrivial capable relationship accurately learn computable scale semantic pixel meaningful human type improve appearance likely appearance region would semantic challenge semantic image specific content automatic accumulate speech semantic motivate explore context cast adjustment neural highly spatially enhanced dnn arbitrarily complex continuous scale design key issue dnn sure learn color color design informative yet descriptor pixel feature semantic contextual global descriptor whereas descriptor understand image semantic advance object detection pixel semantic incorporate novel context descriptor automatic adjustment achieve superior framework yet discriminative image contextual descriptor exploit multiscale improved region effective choose subset test deep use standard learn possible descriptor demonstrate effectiveness comprehensive descriptor system traditional correction adjustment google auto microsoft office automatic automatic operate manner content consideration automatic first include salient determined well exposure code achieve style shall inherently achieve effect especially practice technique outside global semantic category handle deep provide universal style receive much assessment actually adjustment al process mapping statistic method image adjustment context spatially local wang approximation semantic contextual automatic interactive soft infeasible automatically enhance collection propose scalable search within combination work neighbor challenge create slow thereby mid level color sift high semantic difference impact adjustment cast regression dnn represent pair image mapping function color image enhance color pixel complex color parametric feature color training function regression connection blue backpropagation start line force frequency noisy relatively tackle color color basis color space use transform l I ia I ib I ib vc vary frequency pixel much minimization pixel dnn multiple architecture sake network prove able acyclic neuron hide input input output layer map color transform precede neuron relu activation function tangent relu induce unit neuron layer neuron output linear input precede neuron output color span neural architecture architecture classic useful capability contribute backpropagation typically actually enhance pixel color transform dnn smooth descriptor pixel serve feature represent contextual entire detail reflect high resolution vary represent neighborhood position within practice attribute intensity enhance image feature representation six feature give contextual car result region pixel semantic scene parse scene train highly parse good labeling category shape texture detector category characterize appearance foreground two semantic parse fusion parse semantic feature descriptor annotation scene parse algorithm set scene parse road parse obtain parse map pixel receive label indicate cover category state cover predefine type person predefine object high fuse map pixel predefine threshold since vote image merge frequently segment image segmentation map map final contextual feature descriptor pixel multiscale point nest I sensitive nearby consecutive eight semantic histogram sum histogram nine small contextual descriptor concatenation histogram multiscale descriptor inspire shape context unlike shape descriptor either facilitate calculate descriptor contextual complex spatially adjustment contextual simple region large contextual able local adjustment ground adjust color transform three pixel potential even specific local million largely risk finish neural medium hundred nevertheless train enhance neural stage first scene parse object segmentation obtain extract centroid every apply color transform within adjusted compute cover enhance foreground effect enhance effect enhance pixel remain testing image wide operation adjust include select object area variation tool contrast foreground salient background foreground salient region selection production effect foreground object visually make less three enhanced refer material training enhance generalize popular tailor category scene parse object profile series color channel adjustment name tool region apply image region within avoid boundary adjust color profile profile style profile additional minor heavily show effect enhance effect ask try style also apply foreground foreground foreground create layer small foreground background together mode complex color force network enhance testing example transform pixel calculate look visually produce enhance rigorously simulate visualization color enhance successfully effect important make practice find helpful involve always define style category scene parse consequently semantic action tool semantic transform apply color truly spatially vary verify collect pixel region draw scatter pixel image able visualize vary transform clearly transform differ build road region successfully spatially complex conduct approximation due contextual adjustment parameter input enhance enhance enhanced enhance close image map local plot semantic region scatter plot plot horizontal coordinate vertical capability base adjustment mention early far number use thousand pair show image significant image top appearance bottom configuration car people different despite dnn adjust input example effect enhance visual demonstrate contextual subsection calculate color image ground truth reflect magnitude image result contextual third contextual test enhanced mean foreground include contextual feature error indicate necessity input enhance enhanced simple contextual pooling enhance contextual obvious enhanced enhance contextual feature cm truth te foreground effectiveness multiscale spatial pooling design simpler intuitive contextual descriptor pooling region contextual helpful reduce take drop multiscale feature multiscale region rotation invariance histogram visual contextual region enhance might severe color transform help frequency variation dnn spatially nonlinear part highlight benefit use color transform train different dnn color directly dnn similar enhance increase indicate beneficial cm w transform foreground dnn primarily layer inherent complexity dnn sufficient able learn exceed novel datum training show feedforward single regressor layer vary inherent easy small training error deep neuron hide layer dnn keep hold validation vary train repeat experiment report inferior deep exceed layer execute
exploit besides persistent localization variety desirable e due runtime streaming time requirement task arm robot weather online gp constant field plan algorithm richer high like camera localization gp exploit gps krige lastly gp method employ improve scalability full gp marginal equation gp online likelihood compute maximize online detail future mit r localization simultaneous localization determine environment past measurement map representation know field evenly grid grid cover access environment become map know problem localization firstly sense representation information robot reduce summarize environmental grid need maintain secondly environmental field since point exploit assume give localization difficult costly information robot need maintain national mit technology edu sg mit central robot exploration persistent environmental characterize paper exploit field measurement take robot gp observation online capable achieve feasibility persistent robot localization dataset robot focus develop modeling spatially environmental characterized measurement monitor phenomena concentration span light concentration field affect towards environmental different overlap wireless field environment operate assumption widely gps device gps robot exploration environment usually probabilistic update robot state measurement take robot preserve impose location conditionally violate environmental strongly issue integrate rich filter model field probabilistic gp uncertainty gp persistent incur cubic computational availability exploration localization assume location current past measurement gps datum step localization train relaxed hold easily violate environmental consumption permit relative large environmental spatially distant making train gp uninformative motivate fundamental training localization environmental spatially contrast work spatially take robot rely gp capable believe towards demonstrate feasibility employ gps persistent robot localization empirically gp outperform localization gp field environmental realization environmental realize unobserved latter define common choice exponential control intensity noise measurement diagonal controlling similarity horizontal field kronecker delta environmental capability perform robot visit gp predict unobserved location correspond uncertainty predictive pz mean component full filter persistent robot localization poor require invert incur improve scalability gp measurement matrix yield redundant turn rank exploit set utilize predictive measurement variance sn reduce either invert approximated matrix incur matrix employ incur grow increase size impractical repeatedly train localization conditioning field take perform location visit realize field measurement take denote tx robot maintain possible denote column measurement time step track robot realize prior robot posterior bx pz bx px robot moving location describe likelihood preserve efficiency bayes filter impose markov robot action past action exploit learn work model parametric offline former extensively discuss latter already say realize field pz pz ease discuss impose restrictive observation robot normalize constant robot location location visit robot pz z gaussian provide derivation computation constrain motion integration denote location sampling motion past action ensure entire path motion accounting constraint spend ignore constraint theorem exploit considerable poor scalability derive incur persistent impractical incur per gp gp newly slice summary slice span size slice far slice n measurement take robot slice slice tuple manner sx localization offline c c n definition memory keep require independent summary regular step online sparse pz n c theorem equivalent formulation offline incur offline compute incur slice incur time per equivalence structural offline measurement conditionally time slice frequent large slice summary improve slice summary motion motion time step draw belief maintain motion interval particle belief update consequently motion often cause sample locate occur independent offline generalize online gp generalize point gp online variant offline slice summary slice e update robot little predict current summary resolve incremental thereby pz transpose computing pz incur theorem online incur constant localization simulate produce access throughout intel berkeley research km h road network speed km km mobile weather fm office localization road road speed direction field model relational previously whose exploit road hyperparameter gp field gp interested refer technical filter particle represent belief mobile take gp train use action mobile move another learn use along generate localization performance robot location scalability step method possibly unobserve location localization exploit evenly compare localization employ access table error run exploit field well perform poorly area capability produce inaccurate gp achieve small error explore densely relatively make localization localization robot explore area highly
uninformative block uniformly moreover identify estimation enhance close negligible interpretation count improve reweighte outline sequence form identically cf data bias sequencing specie introduce spurious functional part sequence sense protein family partially weight pass value elimination sequence score contact ranking pair position strength mention interact one direct di mutual support information follow frobenius interaction column gap symbol cf reach di achieve good clear interpretation indicator compare gaussian di result gain field achieve l di direct omit moreover matrix position represent characterize interaction interaction precision positive definite marginalization exploit formula block inversion matrix kullback algebra recall help tell invertible factor l l l equation identity notice constitute latter observe identity equivalent substitution denote necessarily scheme introduce computation convenience pre similarity weight average average pair constant refinement threshold neighborhood sequence identical carry protein count least factor mean use estimate eqs cm carlo science sciences human foundation universit es universit et paris biology paris france centre national biology paris france mail author protein remarkable constraint aim extract constraint rapidly datum thereby infer protein recently global direct coupling towards prediction successfully prediction however due nature variable require protein efficient need propose multivariate gaussian modeling coupling problem superior mean field coupling signal implementation website biology sequence complex property protein empirically evolution distant along contact work review recent lead precision sequence alone evolutionary analysis find valuable insight specificity protein interaction progress algorithm principle naturally lead protein initially recently evolution behind global occur two multiple alignment couple show couple aim indirect evolutionary empirically inference challenge biology infer protein homology model single protein highlight predict auto functional protein well subsequently confirm experimental ray guide protein prediction resolution structure molecular protein biology cite signal system constitute way sensor protein signal rr rr typically act external pathway pathway avoid evolutionary pressure evolutionary interesting infer reflect physical come interact protein rr obvious protein evolutionary analysis understand allow sampling simplification allow empirically correlation approach share mean field simple allow comparable aforementioned result briefly material section fast implementation model gaussian sequence briefly come highlight support input protein align form contain alignment gap multivariate bias probable produce observe turn provide infeasible sequence space approach constraint number bayesian introduce convenient prior normal wishart conjugate prior choice posterior parametrization result analytically uninformative interestingly pseudo correction term amount inversion one strength direct predict protein candidate interaction gaussian contact rely matrix rank numerically identical field test di introduce norm di mutual information direct expression support prediction hand therefore contact context score yield however di invariant physical therefore assess contact aim original publication protein recently evolution intra development efficient wide availability like co protein whereas pass limited require hoc protein ten efficient explicitly computation relevant coupling di likelihood analytical formulae analytical major advantage run include algorithm analyze pf sequences di pf intel core cpu computation pair model blue gaussian describe count field information pseudo correct mutual arithmetic thin deviation gain aim first prediction intra fig database family select allow statistical average average sequence cf determine di di average correct mi computed approximation model pair pair apart pair evaluate proximity protein structure cutoff heavy overall note well mean di underlie turn see surprisingly also overall pseudo count explore optimum di code except check insufficient positive mean family list thin deviation cccc pf pf pf pf parallel matlab r version code test run data score show marginally infer accuracy slightly predict positive size fast order magnitude candidate protein structure produce software panel predict contact map obtain predict green last grey positive contact occur predict visual inspection reveal bias protein first pf protein use rr curve rr apply contact prediction report positive show use di substantially true prediction specificity highly little improve gaussian interaction protein use external complex signal mechanism strongly mechanism rr protein pf rr pf closely relate interaction pathway correct recognize rr belonging pathway co localize correct call rr protein isolate genome signal rr major system identify act co evolutionary pass cc bs experimentally interaction cc correctly obtain co evolutionary scoring signal bs interact visible evolutionary experimentally red green dot overall method cf equal log odd infer rr interact infer independently family set rank fig mention present clear cc neither able interaction prediction interaction kind b great protein display red maximally rr evolutionary whereby cast interact interact protein formalism parameter major advantage distribution analytical likelihood posterior efficient demonstrate test gaussian comparable superior field interaction comparable furth example di could kind use suitably design informative relevant enhance prediction prior notably advance interaction mapping interact interact inter protein material multiple domain protein database successively sequence generate family statistic family experimentally assess quality purely length family pf profile list structure l l c page page benchmark together discard gap large choose position alignment gap remove processing improve identification directly summary detail datum hmms domain pf rr rr find code gene sequence rr protein family row align
mostly english word vocabulary character substantial lm lm lm character correct occur lm lm demonstrate train attain without rely lattice good list speech dnn acoustic whether recurrence directional recurrence essential evaluate recurrent compare roughly architecture bt parameter dnn recurrent substantial recurrent report dnn parameter worse fitting total free conversely bi directional recurrence single recurrence recurrent speech recognition present language train decoding remove hmm system find pass decoding demonstrate capability space lattice result recognition system create multiplication dominate suggest recurrent bi recurrence help recurrence together quality hmm science stanford university stanford stanford university stanford ng department university stanford vocabulary acoustic hmm system recent feasibility discard hmm sequence modeling extend way neural level modify pass speech journal corpus competitive error bi directional recurrence vocabulary continuous speech modify model sub hmms carefully design complex difficulty modify isolated advance demonstrate speech use predict audio approach modern favor treat speech recognition train network sum able predict character character journal yet exist system speech often heavily upon decode lattice hypothesis list introduce factor well additionally system final language decode train neural space pass decode enable rely exist word lattice remove speech enable network act acoustic model acoustic dnn place hmm sequence dependency reason handle temporal dependency lstm lstm network architecture originally design prevent vanish gradient recurrent neural architecture amenable graphic unit gpu hmm system without vanish train letter sequence acoustic single maximize full exposition function character fix define character audio feature function basic dnn dnn hide activation matrix wise nonlinearity choose layer character output subgradient parameter dnn utilize integrate temporal extent extend form representation propagate backward backward via recurrent forward backward entirely nonlinearity obtain representation recurrent aside change recurrent layer length network character alphabet audio character character ts map language propose capable incorporate acoustic input seek language language attempt find maximize bt string character alphabet character extend character incorporate language constraint otherwise extend incorporate active include probable probable list nb ta b nb pc nb tp nb pc nb p nb nb maintain first audio maintain time never probability product word probable sort give variable character sequence character character language include propose character word act constraint character string consist hour news available hour transform corpora subset system instead drop
generalization many domain include novel closed scoring examine exponential scoring construction restrict maximum give take rule q pareto score agent belief th lead follow score gamma factorial measure real number interested gaussian density rule stress score construction generalize rule depend weakly second moment density expectation instead rule write agent scoring agent belief market belief seminal market scoring appeal thin thick market section adapt market close function agent belief claim approach statistic interpret payoff function security security portfolio share hold share occur inner example outcome security outcome occur statistic security payoff amount contract literature centralize market maker maker maintain convex collect vector share portfolio give neutral payoff equal portfolio move market vector way choice reveal risk neutral incorporate market information budget examine relaxation later remainder focus arise market course exactly family sufficient statistic recover indicator statistic portfolio share give economic interpretation correspondence partition interpretation market maker family state parametrize share belief score agent report expect report portfolio share reasoning rely assumption share c c recalling leibl family bregman divergence kl rule market entity share consist security q share security stay market maker enforce property share see moment volatility payoff log natural parametrization effective possible share exceed however arbitrary amount increase market dimensional sphere lead alternative dimension security outcome entropy statistic von refer function quantity positive parametrization von rule expect component several price perform aggregation final price expectation aggregation agent take account form belief require belief agent exponential suited reasoning px tt sufficient direct belief maintain conjugate family map partition think prior size observe empirical random agent respect base px dx x dx line recall log market exponential utility belief natural market trade move agent trade maximize equivalently strictly argument private grow agent make reduce exposure stay market weighted market adjustment centralize maker allow capture price adjust parametrize cost c price mean share need reach price transformation know perspective adjust mean risk neutral mean q high share must market price scale follow result respectively theorem accord agent market state rather share update directly right section analyze pose future exist portfolio belief incorporate portfolio reason utility parametrize share first market market give move vector market maximize market subsequently market behave utility maximize belief exposure market financial exposure understand utility pick market state compute draw result game regard strategy family market equilibrium utility utility I final equilibrium convex market belief weight consider market budget multiple market vector measure standard log amount market experience eventually unconstraine together market suffer ill inform also make informative run budget suppose budget share move state want budget market final move market budget maker result informed entity instances market thus exposure market maker loss share px market uninformative budget round round move tx x impact round thus incremental change budget round market evolve never fall round interesting loss quantify characterize budget informative market parametrize move market limited belief differ market function equal expect net bregman divergence base expect belief need belief belief formation growth positive sequence sample belief state accordance belief budget limited notice move utility utility case market exponential utility adjust payoff least dual supremum mean exponential backward mean parameter px px nice dual negative entropy multinomial market maker supremum natural mapping px result nice essentially distribution maker thus exponential bound market maker special case alternative market scoring market learn connect payoff associate point expect likewise entropy equality dx dx equation market scoring kl security quantity particular cost initially imply market price distribution incur give expect kl outcome outcome dx family exponential dx unbounded entropy unbounde derive conjugate primal definition supremum may rewrite q achieve market outcome sufficient loss market maker fx follow expression distribution px dx recall dual negative log term mean market maker maker sign observation design proper scoring scoring case simply different implication scoring rule agent agent expect fundamental statistic variety suppose rule precede space rule provide broad pareto density one mapping alternatively parametrize mean give follow proper scoring pareto stress rule know pareto density e exponential outcome parametrize score median eq q median expectation highlight circumstance subsequent market state give movement px x x follow rearrange share current ct c expectation exponential parametrize thus choose share dx utility maximize exposure behave identically utility maximize exposure market exposure change belief achieve argument belief initial function market aggregate prediction informative want informative receive draw belief able initial market prohibitive market growth budget eventually move restriction idea dual use recommender section belief cost payoff impose market requirement budget market restrict movement payoff market budget cost movement budget market maker maker directly limit allow would market represent belief enough share market market budget state market c move market move optimal trade rational move case move close maker might entity multiple exposure market maker several use loss initial budget define share hold eq prediction share security market make report assume outcome reveal receive market track budget tx eq q incremental budget active capture incremental market evolve fall cause expensive market budget increase expectation prediction input move result net theoretic interpretation informative market increase budget round belief informative result market market informative belief parametrize budget limited expectation belief whenever budget differ theorem belief payoff payoff last market exactly belief positive sequence belief eventually every accordance without limited require combination maximize utility market parameter theorem trade respect optimality update formation growth extend true introduce bind prediction set share payoff first note px tx move market write px aa utility parameter payoff conjecture thm thm example thm market mechanism template market price agent analysis market price equilibrium assumption exhibit aspect budget constrain behavioral artificial market aggregation mechanism market assume private belief payoff market probability sequentially private sense market aggregate consensus forecast prediction focus heavily mechanism compatibility market fluctuation name literature correspond price interpret equilibrium market underlie aggregation price bayesian incorporation posterior classical tool market statistical attractive interpretation relate via family market price
theorem solve consequently least confidence strictly satisfy prove first statement generate number theorem probability let get statement remark numerical practical difficulty require define approach issue convert define augment large potentially exploit solver solver natural angle major low minor minor high residual angle low major alternatively deduce remain pick remain orthonormal na k component half make know poor potentially dimensionality reduction versa combinatorial combinatorial effort introduce slack polytope dimension row argument bound consequently dimension general appear satisfy inequality inner fix combination preserve inequality minor number iteration path different matrix result problem generate first table residual angle however map residual monotonically dimension residual decrease furthermore angle interestingly number minor high major iteration quick notice uniqueness solution representative report table representative carry section solver solve solver directly point compare minor total report obtain final point major additional minor report solve high initial point notice minor solve minor high problem recover solver compute motivate big reduction feasible approximate form appropriately project solution solver thereby exact numerical author dr research systems engineering technology affine compact feasible randomized produce dimensional choose quality subroutine generating solution lower appropriately choose solve solver original recover validate substantial time collection datum change interact store report accumulation concern challenge large computationally polynomial whereas context rather quick approximation spirit essence accuracy large focus generalize convex optimization saddle nash nash game amongst affine region include quadratic feasible quadratic subproblem affine competition game dimensionality high affine substantial high derive solver low error satisfie get project made unity choose inner product translate high deterministic high norm quadratic program remarkable recover approximate require deterministic exact generate solver plausible run nature emphasize assume plausible improve theoretical computation appear try considerable ambient may conceptually speak work exploit convex embedding metric result obvious convex analytic euclidean embed inner optimality preserve follow introduction present background formally introduce concept require proof discuss low conclude f problem solve amount ensure range solution vi nash equilibrium game simultaneous vi equilibrium capture besides contact continue equivalent k quantify rare solve solver work solve variational inequality structure reference therein direction splitting decompose large small subproblem deterministic work probabilistic introduction subroutine science community research control two aware first zero differ rank vi solve nonlinear whereas linearity formally interested seek expensive involve low exactly proceed projection embed deep mathematical study limit operational involve lie dimensional need type randomness introduce construct study uniformly realize orthonormal call manifold distribution schmidt surely formalize random form normalizing column uniformly zero subspace probability induction follow produce r manifold invoke lemma value standard qr uniformly rank column paper automatically multiply expect shall ahead multiply depend upon random arbitrary dependent projection call think vector distance improvement show project multiplication pairwise preserve original distance preserve expense concern preserve differ exact fix argue random lying sphere fix projection coordinate deterministic depend project uniformly random operator easily convert apply union bind finally project norm phenomenon product preserve mapping vector try norm pa pa p since fu fu fu fu fu result result finitely simultaneously random section mapping fu union require random projection high inequality feasible region vector generate solve approximately ax kk dimensional polytope norm linear vector final main compact low claim b k sufficiently give sense inner range least computationally run invert hence deterministic solution second produce correctness part establish show showing solution prove approximate proceed analysis lower low x vi polytope n generate combination
efficiency randomize fourier accelerate large approximate arise approximation integral shift invariant kernel approximation evaluate discrepancy measure discrepancy adapt explicit minimization empirical offer mathematically well model wide machine learning series modeling testing rather general non associate embed feature model provide input representation construction dimensional however elegant generalization assume complexity solver requirement kernel nonlinear counterpart however require involve gram none particularly conclusion apply algorithm rather appeal setting adapt strong hypothesis empirically potential large data domain necessary method intensive improve kernel recent randomize approximate distortion approximation kernel complex inner though real technical exposition simplify adopt scalable solution back technique successful accuracy characterize definite function positive definite point dm borel measure henceforth scale shift put one notable member shift invariant kernel integral subscript denote goal quality work need observation density approximation approximation kernel mc discrepancy quasi overview theoretical provide clearly demonstrate superiority analysis potential subscript rely use scalar k nz see mean randomly typically unit cube definition rkhs reproduce possess reproduce f reproduce equivalently space functional informally hilbert space nice I word control pointwise scalable scalability identify dominant construct either randomize map kernel classical nystr om nystr om deterministic seminal harmonic map invariant considerable effort extend technique feature deterministic random wide class group invariant suggest random intersection generalize kernel gd gd suggest product classical result suggest feature kernel show subspace observation feature laplace feature work effort fast original construction devise random random fast convergence amount apply distortion make map rather try scalable scale method year start early day optimization gaus specific expansion optimize svm objective draw scale reformulate broad restrict subsequent section excellent computing follow integral computed respect central monte carlo carlo converge rate discrepancy illustrate dimensional graph scatter random see empty little region empty space lack uniformity fact design correlate avoid phenomenon fast integral theoretical designing form variation integrate dependent measure uniformity remarkable classical sequence variation star discrepancy measure actual volume discrepancy construction discrepancy notable example decomposition construction detail however mention notable sequence also mention star discrepancy decay base convergence rate improvement past integration however notice integration literature leverage discrepancy classical measure variation direction provide reproduce behave term detail follow set kernel g cauchy applicable integral cube integral form generate discrepancy drawing sequence convert integral unit cube cdf low yield procedure summarize map analyze tb made develop approximation show q unbounded integral unbounde new characterize integral throughout convention interest behavior reproduce space relate interested rkh derive integration particular integration bound proposition rkhs rkhs follow vector admit integral associate support transform fundamental shannon sampling product space wiener constitute wiener admit inner rkhs kernel notational discrepancy suppose write univariate function box star discrepancy box discrepancy notational wiener state integration unfortunately member directly discrepancy measure integrate function uniform spirit similar q explicit multivariate important equal discrepancy minimize decaying go hence pairwise separate distribution unit cube drive cube tradeoff compete expect decay behave formula follow density zero subscript analysis density characterization measure typically sequence discrepancy behave box discrepancy needed leave future unlike box function formula candidate low specialized via optimization proposition discrepancy namely global greedy pose discrepancy q rank lattice rule integral generate local box discrepancy non greedy recently present notation series minimize p ps equivalent restriction unclear restriction restriction hold compact case since report sequence learn discrepancy examine behavior discrepancy fouri digital implementation www com lattice digital net publicly people generator low recommend literature net introduce randomization generation compare sequence around net long long second report essentially work gaussian fold favor monte fundamental exact examine randomly example dimension subsample exact reason lattice cc trial clearly classical sequence gram lattice yield sequence may mc whether sequence yield high sequence quality gram use build ridge mc summarize cccc lattice mc cpu census truth execute deviation list behave lowest significant almost follow digital mc sequences model regression worth analysis connection nystr om gram kernel examine normalize box box base range inside far box value scope rule discrepancy bound yield plot discrepancy box error part box concentrate predictive see quality discrepancy cc sequence box discrepancy provide proof concept sequence describe demonstrate sequence produce well gram matrix running sequence less experimental flexibility adjust one long shorter force bounding box scaling feature turn applicability variety generate learn sequence number dominant term scale learn name global adaptive set optimization initial use gradient examine unit use mc see mc concentrate near significant expand integration give sequence control box adaptive sequence make ccc examine adaptive metric gram plot examine behavior various plot various square squared norm error evolve perform dataset evolve iteration examine optimize box discrepancy initially go box box continue go plausible explanation box entire increase box actually concentrate handle box concentrate try improve reasonable box hard subsequently discrepancy translate monotonic metric gram however
essence measure utilize hypergraph underlie relationship propose hypergraph partitioning characterize intra inter cluster separability maximize vertice maximization pairwise nn hypergraph algorithm datum hypergraph show pairwise two near neighbor display hypergraph structure highlight color consist left hypergraph incidence hypergraph hypergraph similarity three hypergraph partition affinity hypergraph hypergraph pairwise hypergraph reflect relationship vertex nn hypergraph neighbor cluster among type construction mechanism explore relationship easy exposition denote mathematically graph correspond return affinity measure word easy cluster similarity correspond hypergraph hypergraph equal traditional hypergraph compose many mathematically hypergraph incidence ne order measure belong hypergraph pairwise affinity hypergraph represent diagonal hypergraph vertex hypergraph near nn define indicator hypergraph assign th e similarity nn compose centroid vertex near vertex similarity hypergraph n n capture hypergraph characterize n nn similarity nn hypergraph similarity interpret cosine similarity feature essentially explore nn hypergraph nn hypergraph construction hypergraph sample right display highlight encode vertex often vertex mutually effectively discover community propose hypergraph vertex vertex fig case belong mutually influence context loss vertex community convenience community order incidence indicator vertex similarity relationship index similarity hypergraph formulate cross within vertex reflect affinity relationship vertex th therefore hypergraph view cosine vector similarity obtain group information order hypergraph matrix diagonal similarity matrix hypergraph draw hypergraph hypergraph close truth hypergraph combine type aware hypergraph encode local hypergraph show bottom part hypergraph capture manifold datum sample accurate therefore aware hypergraph keep easy emphasize hypergraph associate hypergraph hypergraph context aware hypergraph partition hypergraph partition hypergraph hypergraph partitioning aim disjoint eq diagonal th diagonal element point typically relaxed trace trace maximization decomposition capture optimal hypergraph partitioning eight pairwise hypergraph list experiment dataset configuration c face dataset comprise person near face convenience face image dataset service handwritten digit subset handwritten digit constitute digit ten dataset trajectory shape dataset uci repository contain face digit mnist corresponding descriptor computer trajectory fourier dft uci repository j nn hypergraph exist cluster spectral cluster vertex cluster refer issue iv sec ground cluster configuration pairwise kernel refer accordingly nn hypergraph construction cosine similarly cluster hypergraph cosine partitioning eigenvalue newton complexity partitioning therefore spatial method lie hypergraph incidence hypergraph incidence compare recently convenience classic tune spectral noise robust spectral hypergraph nn hypergraph spectral actually special different evaluate context hypergraph together normalize demonstrate aware hypergraph make quantitative similarity hypergraph datum cluster nc nc quantitative introduce evaluation criterion configuration obtain cluster cardinality intersection cardinality dataset large truth th cluster clustering evaluate original comparison perturbation effectiveness cluster report accuracy eight accuracy regard seven dataset nc report weight cluster performance sensitive configuration weight perturbation additive fig bar choose high accuracy noise accuracy gain nc respectively element display performance regard outli corruption level corruption ratio consistently show accuracy corruption average gain nc hypergraph corruption weight mnist corruption configuration different configuration nearest nn hypergraph fig number community cluster hypergraph clearly see three type hypergraph capable intrinsic optimize hypergraph partitioning criterion intra inter separability robustness aware hypergraph similarity type nearest hypergraph high hypergraph capture neighborhood grouping vertex capable explore intrinsic vertex robust capture intra inter separability discriminative hypergraph maximization spectral develop cluster experimental corruption effectiveness adaptively combine weighting mechanism hypergraph mnist fig author li zhang laboratory institute chinese sciences china mail ia ac cn li school science york usa powerful tool analysis aware hypergraph type hypergraph hypergraph neighbor hypergraph hypergraph pairwise hypergraph hypergraph capture neighborhood hypergraph encode dataset affinity intrinsic dataset intra discriminative hypergraph spectral theoretical experimental propose hypergraph graph measure play important unsupervised range circuit load balance image motion segmentation video affinity aim cluster local still issue traditional discover number graph construction noise enhance issue design local scaling explore intrinsic laplacian discover mechanism cluster resolve propose via eigenvalue gap vertex intra separability sample video densely therefore map topological information cluster cluster around origin may separability hypergraph address issue iv hypergraph analysis affinity among construct hypergraph video
domain review large heuristic technique multi particularly bag label positive hx lx lx equal indicator belong bag bag distribution hand arbitrary bag statistically hard pac pac learnable side approach model bag manifold generalization set combine paper regression iii determine true realization bag bag I might bag specific bag several guarantee problem iv vi feature extraction bayesian adapt bag come probably hausdorff metric compact finitely exist hausdorff sensitive applicability hausdorff design minimal hausdorff instance hausdorff contextual hausdorff instance unfortunately lack task might consider however highly standard negative notation section formally define notation paper cm integer hilbert closure complement intersection direct set denote logical topology borel measurable product algebra weak topology b functional operator shorthand hilbert call schmidt schmidt know iv compact cl trace schmidt operator identity rkh embed reproduce canonical map average separable norm reproduce kernel let conditional k belong embed adjoint operator value kernel property hold cx xx requirement assumption see task endow topology algebra measurable I lx learn involve wherein sampling goal notational classical regression rkh random construct composition word map embed measurable regression empirical determine lk remark cm cm tackle stage difficult enable stage scalar equation bit generally analyse risk scenario specify establish f detail fit problem ridge focus stage present specify derivation illustrative example serve compare cm theoretically justified regression avoid specific alternative experiment concern section toolbox technique latter goal entropy whose uniformly construct rotation validation e select goal learn marginal display test typical estimate square confirm figure need density estimation sample pose optical prediction I image correspond consider radius sensor use bag bag bag baseline em achieve accuracy experimental protocol testing repeat hard validation set first linear pick ensemble polynomial quadratic mat ern smoothness parameter summarize obtain l nonlinear summarize one drop prediction decrease far set despite precise choice however poorly output separable topological endow study analytical allow parallelization parallelization difficulty well specify case assume model class case old kernel family expand table focus quadratic loss kernel distribution regression I relax plan question whether present focus section somewhat demand certain present detail concern excess specify proof concern excess eq without modification provide empirical xt make kk consequently bernstein q uv kf g cm ig boundedness c bernstein inequality tt bernstein see kf adjoint kf k cm kf cauchy schwarz analytical eq hence apply arbitrary ii triangle arrive bound let boundedness old hilbert schmidt self adjoint cm cm k adjoint operator countable f cl j exploit identity k bound supplementary technical detail derive k al ac uk college house department state pa edu school pa value fit point analytical hyperparameter entropy quite observable rely good guarantee distribution estimation step often perform poorly study analytically hilbert regressor output scheme stage setup mild condition answer old classical kernel kernel include stage ridge embed instance address bag distribution output learn statistic way case bag analytical expression entropy hyperparameter intuitive suppose meta distribution I I l health patient infer observation test health indicator hope observe test large mapping solve consistently work mapping analytical depend candidate analysis regression belong bound goodness high use hard excess convergence vast learn address response consider case nonparametric regressor act reproduce kernel appear throughout variance regressor consistency construct regressor rate regressor old propose handle scale bag set bag distribution kernel call multi kernel kernel average point similarity little learn introduction bag bag allow increase however distribution valid kernel mean hilbert characteristic use embedding reason embedding operation paper consistency mean distribution regression basic relate herein break arise difficulty cm specify case bound excess prove parameter triplet obtain large particular large process obtain occur rigorously intuitively particularly process easy regression rate guarantee open topological domain endow p compact dimensional string separable value suffer work estimation step proving sample consider use focus short upper verification care embedding kernel rkh alone take invoke fundamental challenge cope combine rkh technique construct operator associate distribution section convergence specify section although illustrative numerical give idea supplementary
bc conduct well bold nmf bc nmf vs domain work conduct preference user click comment system reasonable scalability recommendation propose transfer pattern pool together rating domain rating pattern co cluster capture propose novel probabilistic recommendation art recommender recommendation belong movie historical item preference record recommender system suffer case item even large rating matrix prediction content number user domain treat item acquire domain refer domain recommendation show domain recommendation learn study rating pattern codebook cluster codebook codebook call combine codebook expansion membership matrix existence codebook multiple domain share common rating diversity common rate improve strength cross recommendation learn simultaneously enhance across flexibility sharing capture specific rating prior item show domain offer evidence suppose rating item multiple isolated rating datum rating matrix z r z r cross domain collaborative predict miss rating domain knowledge across domain world scenario item simultaneously example movie music classify movie science describe music region movie affect result movie domain rating rating specific sharing cluster item clearly co user belong cluster user belong exact membership previous represent feature z domain item describe specific item cluster user co rating rate common cluster rating rating element expectation user cluster rating define function rating cross domain specific rating cluster rating recommendation combine cross domain balance weight cross define rating prediction movie result introduce pool rating expectation ease loss k kl tu v equation compute pool rating rating pc pc l simplicity pc update eq em name adopt get model rating also compute accord rating rating q predict domain examine rating domain recommendation recommendation nmf matrix single domain nonnegative method factor model single single separately rating matrix transfer rating pattern multiple experiment evaluation dataset movie rating scale rating movie experiment million user movie user rating normalize scale contain scale book rating scale comparison examine respectively rating start user keep ml vs vs discover different domain conduct repeat mae mae predict mae use good k pc c manually report setting observe show mae vs domain performance clearly show cross recommendation
derive forest tree tree hierarchical convert randomly height take forest kernel algorithm interpret collaborative filter movie recommendation cluster movie two view cluster splitting similarly word understand kernel select forest sampling forest train leaf explanation binary necessary tree height give random toy stationarity distance hyper normal phase gps svms replace forest computationally unsupervise nothing inherently train overfitte supervise dimensionality plot fast pca cluster kernel fast partition cluster center assign largely result piece wise note fast easily training tb compare forest radial basis without automatic relevance detection uci repository train evaluation likelihood posterior learn variance covariance kernel cluster outperform kernel dataset graph scale discrepancy often simply mse forest improvement improvement standard prediction kernel kernel value low random forest perform nearly well whereas might convergence wise matrix fast square exp execute entirely fast partially execute intel ram log gradient predict scale fast processing point dataset theoretical comment future scaling worth fast random forest algorithm trivially present connection kernel intuitively trivially algorithm forest cluster kernel show excellent regression dataset gps rgb rgb section demonstrate kernel construction forest fast show consistently outperform kernel problem inference vector often cite success world algorithms kernel never single task hope kernel intuition commonly kernel generally derive despite practitioner periodic radial spline rbf far away almost kernel argument datum towards smoothness may leave operator automatically search structure difficult general intuitive originally design find partition partition trivially kernel real simple svms regression semi psd indicate machine choice view implicit issue element gram operation find solver solver gram store offer great quadratic form common use free solution svms generate take partition representation allow partition eq partition induce two cluster constitute eq psd psd psd psd far since psd psd valid however partition possible evaluate fortunately approximate partition bernoulli kernel definition use crp ibp party affinity naturally naturally evolve people join good state class efficiently store space operation use analytic require
remove choose probability crucially equivalent select cl la w nice markov event low constant denote intersect intersection hold contain form e np n substituting solving complete proof g c suffice chernoff upper q intersect complete shall together bound primal constant geometric e addition universe keep exposition self discrepancy shall sum set original subset axis plane half etc behave primal dimension system set suppose plane follow dimension belong discrepancy discrepancy match size less discrepancy e throughout apply vertex get notion discrepancy reason sensitive discrepancy theorem set rest partition colored discrepancy continue process constant depend use inductive recursive bind eq try eq triangle constant imply proposition packing additional answer question another sensitive bound system imply size begin primal play primal nm om system grind exactly member even dimension system look bound subset hamming cube packing vertex tight bind packing packing constant packing probabilistic look vc dimension book simplify theorem refine size size onto primal sensitive dimension set separation could remove make paper packing packing primal scenario thus extra remove imbalance system discrepancy space set let set big generalise size sensitive primal make set discrepancy considerable case pack discrepancy primal recent improvement appear discrepancy conjecture low entropy constructive see constructive proved discrepancy constructive discrepancy relative system proportional subset inequality approximation tackle range previous recent primal sensitive constant geometric section upper universe wish minimize technique fraction leave low colored universe colored recursively develop major extensively yield discrepancy imply subsequently constructive lemma say set let recursively remain element choose sufficiently round denote discrepancy final decomposition refine separate clearly exist member member contradict notice lie hamming denote close call sensitive refinement truncate chain truncate close assigning chain sensitive family denote later apply
admissible value real appear blue initialize text nm fm pixel pure art estimate inversion step perform constrain besides extraction denote namely fm nonlinear nm inversion achieve nm fm taylor inversion achieve subgradient base scheme gaussian generation initialization algorithm stop successive function algorithm evaluate average spectral abundance square measure table first algorithm two namely improve pure analyze flexibility various scenario ability kind nonlinear preserve analyze mixture hyperspectral discuss hyperspectral availability truth acquire visible imaging water remove lead band range bandwidth nm compose report image consider acquire france range consist sub image mainly compose additional unknown plant hyperspectral divergence divergence noise assumption physical support divergence measure grind unfortunately public hyperspectral perfectly spectra abundance way select ability unseen datum require truth study nmf interpolation remove hyperspectral image pixel various pixel uniformly image describe fit entry outli miss entry outli entry datum identifiable use minor mm pixel reconstruct complete initialization set pixel performance study standard hyperspectral kl come optimization equally implement technique image consider divergence spectra abundance depict abundance visually description regard explain display residual component regard pixel accurately mainly locate pixel probably correspond water water confirm image rd regular vertical surely post present mix hyperspectral datum model denote extend standard include capture effect nonlinear active hyperspectral require specification simple provide penalty specify illustrate various acknowledgement feedback manuscript receive engineering degree group et de laboratory university department technology company paris la universit nice interest generally process separation member technical electrical france sc institute ph post associate computer ann mi since national institute university associate communication group laboratory also member laboratory sense hyperspectral several signature generalize mix handle rely mild assumption regard constraint spectral constraint impose nonlinearity factorization fidelity express take kullback leibler case minimize minimization result data state nonlinear hyperspectral factorization analyze hyperspectral datum comprehensive various sensing monitoring consist hyperspectral propose commonly provide observation result interesting inaccurate nonlinear need multiple scene lead take account bilinear bilinear differ impose nonlinearity incorporate interaction demonstrate nonlinear feature consist supplementary major drawback choose nonlinearity limit practice nonlinear detail build supplementary account nonlinearity merely motivation valid number pixel contribution observation reflect pixel impose sequel article organize coordinate real hyperspectral preliminary conference generalize use square general update obtain rigorously minimization rule choose weight efficiently finally propose pixel observe lk outli term accounting formulation symbol dissimilarity section abundance coefficient sum hyperspectral nonlinear well bilinear constructive introduction general pixel become energy define objective nonnegative defines nmf problem appear nonnegative free spectra refer feature fitting square regular article hyperspectral regular robust entirely next take divergence scalar introduce auxiliary separable w r thank order inequality lead detail brevity lk kp approximation problem eq may lk kp lp lp lp lp kp kp lp definition tangent hand r concavity root write essentially replace quadratic tight involve effect within square result lead exponent value follow constraint induce extra optimization handle multiplier set special corresponding leibl resort approach nonnegative turn approach unfortunately able long resort function ensure nonnegative descent resp positive experimentally decrease value denote update simply kp kp turn update implement operator operation fraction bar term matrix tolerance function
combine obtain prove generate algorithm maintain condition divide argument q k substitute expression substitute inequality maximize obtain k complete proof induction condition relation indeed f q estimate indeed third corollary function strongly let write optimality condition cl p concave g g parameter eq k g maximization side finally estimate inexact augment lagrangian divide function smoothed sense q estimate kf k k estimate write k lead eq start e plug estimate pt pt ed de primal dual algorithmic rigorously efficiency analysis structure fashion primal method instance choice smooth lagrangian alternate direction multiplier case primal primal feasibility gap iterate cm alternate separable convex minimization programming concern constrain convex capture surprisingly broad sequel rigorously characterize assumption efficiency limitation eliminate onto understand smooth minimization barrier point method use smooth unconstrained simple overall strategy curse dimensionality well numerical formulation medium simple function despite approach capabilities numerical alternate scalability rely three structure stand among say set f ii pn parallel implementation hardware architecture convex problem pose significant numerical smooth nearly constant canonical feasibility algorithmic iterate many smooth proximal function possess enhance efficiency highlight aid design momentum parameter full rate composite minimization complexity unfortunately penalty approach block ideally characterization solving value iterate feasibility primal significance primal gap since constrained time trivially demonstrate far ideal ergodic averaged iterate feasibility reduce scope applicability rate dual primal residual feasibility necessarily feasibility convergence function function necessary parallel admm rate cm decomposable gap decomposition decomposable admm decomposable dual decomposable linearize decomposable gap primal augment lagrangian inexact decomposable k development algorithm special scalability away foundation flexibility restrict solely unfortunately decomposition often complicate backtrack computational selection efficiency well handle solve proximal characterize primal primal feasibility separately positive solution still exploit favorable exploit decomposable sub optimization trade residual feasibility gap crucial numerically numerical dual smoothing technique optimization primal dual rely duality saddle point monotone approach mixed use develop smoothing technique replace non smooth bregman augment lagrangian technique lagrangian smooth property solve norm bregman smoother rely proximal nesterov dual solve nonsmooth unconstraine characterize combine three unified convergence mild theoretical primal constrain cover decomposition prove variant cf theorem framework residual particular inexact algorithmic case subproblem maintain control appropriately proximal class characterization different importance well bregman parallel manner feasibility act consensus practical trading feasibility front well numerical synthetic real state art source enhance performance trading gap gap result advantage interest unfortunately impossible comprehensive expand reasonable algorithmic framework representative method subgradient provably rate e sensitive rule overcome difficulty instance augment smooth study establish nesterov accelerate recent feasibility f separately several primal variant hybrid primal study several ergodic leverage instance variational also belong tailor may come offer convex define lagrangian produce global k accelerate scheme method variant alternate admm recognize split separable cover problem use nf admm solved iteratively admm update subproblem except interestingly notable completion sub entry differential computational difficulty use one efficiency significantly penalty guarantee well convex drop term forward backward splitting optimality inclusion approach idea structure whereby preserve moreover formulation algorithmic study augment lagrangian bregman smoothing dual problem primal dual boundedness simultaneously odd four objective function hold constrain feasibility gap oppose primal variational formulation bregman also formal function property section main solving specify connection devote implementation provide bregman sequel close n f denote lipschitz constant notation nonempty nonnegative prox bb prox smooth bregman project prox diameter range b distance write base lagrange call continuous nonsmooth numerical weak duality guarantee duality require nonempty either assumption dual strong dual solution lagrange goal solve numerical specify approximate give accuracy say solution feasible onto absolute objective residual mf tucker gap indeed augment lagrangian principled smoothing technique choose center nonsmooth bregman convexity center projection primal characterize smoothed smooth function lipschitz continuous smoothed simply smoothed function specify note case choose g augment dual augment lagrangian concave gradient lipschitz continuous constant augment smooth short smoothed smoothed smooth summarize define diameter respect solution gradient ml bf md hold condition form dual gap nonsmooth smooth add distance however however overall gap function optimality explicitly goal become follow gap g k scheme nesterov function call nesterov smooth convex note gap basic analyze allow show find sequence definition definition k g smoothed bind objective design primal update template scheme subsection assumption metric might lead nonsmooth require replace linearization follow q decomposable whose defer k switch primal dual new k update maintain whose respectively choose point g follow obtain lipschitz residual primal feasibility simultaneously exploit introduce gap primal dual convergence residual primal feasibility gap inexact lagrangian objective feasibility nesterov accelerate scheme case characterize feasibility quantity drop close result corollary strongly procedure ff f c estimate corollary theorem conclude choose standard knowledge algorithm multiplier minimization multiplier admm linearization alm instance separable primal subproblem alternatively many scheme primal indicate arbitrarily residual feasibility gap scheme k instead obtain primal function boundedness set consider still feasibility stochastic case set result closely relate look author joint constant result combine feasibility gap take arbitrarily value residual primal feasibility separately joint primal algorithmic prove criterion residual trade quantity primal discuss enhance observe enhance parallel distribute implementation bregman propose option c employ bregman euclidean variant discuss point unknown k bregman simplex b decrease smoothed decrease might improve increase primal feasibility gap ks default option define correspond figure link request neighbor ij iii pi link number number coupling end problem coupling constraint separable constraint bregman distance either pd pd main need primal scheme step separability solve precisely subproblem form dual k neighbor next iteration request send need store copy dual matrix neighbor feasibility consensus communication approximate operator expect characterization important future present section extend simple slack transform modify update eq indeed variant solve numerical simulation several machine image compressive sense numerical mac os ram terminate algorithm feasibility state variant bound basic group solver k theoretical performance randomly iid ii use algorithm tune calculation twice variant iteration empirical feasibility deviation increase improve adaptive proximal suggest outperform hundred illustrate vs augment technique augment lagrangian smooth fista true suggest actual basis close primal feasibility exhibit strongly convex elastic select generate randomly iteration configuration enhance version backtrack converge ht relative line variant one relative correspond compute theoretical plotted figure line ht cm corollary variant iteration basic require procedure end verify justification done use pursuit indicator compute variant adaptively obtain cm figure variant relatively reach hundred variant smooth variant deconvolution eq isotropic oppose tv additional coupling constraint solve result implement per point lead method case tune recent admm surprisingly periodic condition norm solution subproblem fouri hence class problem illustrate resp done suggest exact code use move cm admm solver admm rule calculation choose common imaging problem norm self image regard result inexact computation optimization variant paragraph solve subproblem background logarithmic fortunately ht approximately prox inner randomly randomly group error n show many time profile noiseless computational rest inexact fista slow step linearly system inexact present basically noiseless group increase increase converge happen penalty significantly follow eq norm suggest perfect video surveillance camera video matrix tuning compare source admm inexact report algorithms k svd operation admm reach value gap admm many plot algorithms ht plot object human low similar solver reformulate probably basis generate generate tune lagrangian admm admm tune smoothness work three paragraph subproblem matrix carlo fast cf ccc size multiplication prox multiplication
rademacher random observe coin identification obtain I accord generalization coin identification let coin identification q position complete reduction identification task coin identification subroutine need input clearly hypothesis return coin add q hand complete sequence recall return set function vector da di lk inside time symbol mean standard basis ab goal subspace expect sample attribute low bound involve variety application recognition instance measurement correspond outcome medical relatively attribute patient measurement partial show example let subset find rank distance subspace pca general arbitrary assume usual learning learner sample allow reveal squared analyze number particular set sample propose three subspace matrix stochastic strategy attribute upper sample algorithm inferior really inferior analysis pca regime small bound observe regime family include bandit low show balance suffice follow small optimal log dependence mention hold achievable accord subspace formal example think partially entrie one strong come free guarantee completion treat subspace problem estimate correlation partially rely resemble bandit obtain partially attribute g matrix main difficulty stem example vector therefore think prove single attribute give correlation build control exploitation subspace bandit formalize rx x subspace function every dimension exist call satisfie randomness follow summarize next section complexity bandit bandit learner namely learnable family learner pca bandit inner maximize learn full replace pick learner eq eq call matrix independently r v v approximate rely follow spectral surely translate complexity subspace pca mention part bandit bandit descent gradient approximately optimization descent descent replace implementation require implementation convergence sgd essentially completeness getting convex combination set guarantee detailed index independently ss iw runtime accurate comparison use complete correlation loose nothing runtime observe sgd recall xx conclude short difference sophisticated strong subspace learner include learner complexity deduce dyadic describe zeros learner learner maintain attribute bandit dyadic dyadic learner assume prove bandit subspace advance random sample observe distinguish employ low information sample partial case provide part bandit q divide part rely suffice satisfie close pca c I run combine surely along every trace old maximize return section subspace partial argument sp u uniformly let subspace allow attribute order return fix subspace accordingly denote computation show let environment recognize recognize output subspace successful use identify I mention attribute learner distribution since attribute sample expect least recall definition concrete distribution view denote output obtain l l task environment role recognize reduction maximize identify r zero negligible distinguish zero draw subsection successful subspace learner distribution concrete make learner equal zero subspace prove optimality complexity information eq first idea behind draw random coin integer th coin successful learner bias coin identification subspace statistic bias lower follow free attribute whenever distribution know interestingly also e
vertex hence generality assume henceforth index set eq follow chernoff tv v inequality index index ball bin stochastically follow chernoff tv tv tt tp ta tc let prove stochastically dominate eq denote scheme c straightforward suppose sake contradiction randomized polynomial arbitrarily follow moreover definition contradict holds plant dense dense random vertex plant approximation least notice plant subgraph plant exist solve construct sequentially let replace connect vertex take unit run bind type ii bind bernstein plant subgraph chernoff exactly vertex plant dense bernstein inequality follow pm notice trivially assume separate depending value probability pm pm ms mh stochastically fx px rest shall intermediate dominate stochastically last function satisfy proceed toward end definition pm ed pm pm bt mp complete hypothesis chapter edu detect plant enyi graph within exceed assume hardness plant clique community exhibit grow become exist computationally intensive procedure hardness recover dense approximate often community many edge vertex numerous science exposition therein work study detect community random recently detect new event theoretical interest understand statistical algorithmic community model formulate plant dense enyi independently plant subgraph include connect connectivity elsewhere plant subgraph model deterministic dense subgraph detection henceforth distinguish hypothesis difficulty intuitively subgraph decrease decrease become recent obtain condition plant subgraph certain unclear whether procedure show test maximal subgraph relaxation highly suboptimal limit problem sharp admit test vanish conversely detect plant dense subgraph reliably community factor rest graph adopt approach parameter hard plant clique parameter regime plant clique vertex form clique henceforth refer distinguish plant clique extensively state solver pc pc require pc constant pc hypothesis hold require detect enyi asymptotic regime dense subgraph either pc depict detect reliable detection thresholding subgraph solver plant subgraph h font right cycle hard hardness transition moderately detect procedure graph detection total regime therefore surprisingly linear base edge procedure reliably lead polynomial term parametrization boundary beyond reliable detection analogous plant succeed satisfie sophisticated spectral method succeed hardness result recent community scale linearly density scale resp sharp regime slowly achieve exponent demand plant clique hardness plant subgraph ensemble bad clique hardness well particular plant subgraph subgraph constant light plant dense subgraph research e follow randomized polynomial appropriately previous highlight technical theoretical pc hypothesis g approximate nash independence pc hypothesis investigate incur complexity principal submatrix strong pc positive pc use open work theoretical computer science literature hardness instance certain reduction pc generate feasible prior space establish hardness pc problem reduction close start dense arrive whose pc tradeoff graph rather complicated enyi problem refer find subgraph edge view hardness np hardness subgraph vertex fraction subgraph hardness comprehensive discussion bad result average case behavior plant subgraph scaling plant dense subgraph recover polynomial simple region approximate within bind subgraph high average hardness detection consequence scan achieve vanish probability succeed scan succeed cardinality intensive unclear whether exist solver question plant problem pc reduction formal problem bound problem constant equivalently adjacency equivalently resp draw draw close reduction vertex vertex parent child number vertex cardinality uniformly remain give ideally want construct edge clique unfortunately provably nonetheless accomplish long nan distribution match core match node distinct parent plant desire plant though desire average negligible let consequence follow show induce solver probability theorem establishe hold satisfy polynomial test type error bound hold regime computational barrier limit incur significant phenomenon line noisy submatrix submatrix exceeds plant dense subgraph plant dense subgraph size bipartite plant computational implication hard dense recovered subgraph result plant recover green regime consequently conversely polynomial otherwise density pc subgraph plant subgraph hard open scale font thick right left impossible cycle node plant subgraph hardness deterministic deterministic enyi graph alternative vertex dense subgraph distribute entirely clear extend approximately lie variation hardness extend subgraph monotone imply whenever obtain intuitive likely plant dense contain scan monotonicity statistically also monotone scope bind hard solver plant subgraph least interesting bound without restrict test monotone conjecture establish bipartite deterministic plant subgraph intractable regime statement bipartite vertex plant plant subgraph vertex refer bound use bipartite pc test bipartite plant plant bi clique refer constant succeed analogue bipartite low graph much verify variation hold computable regime carry limit distribution conditional since chernoff inequality subset copy
preference influence environment face perform environment identify appropriate satisfie preference work video perform task internet user preference encode trajectory optimize use robot arm robot alone human environment oppose cost function drive trajectory preference show activity human environment prefer arise human activitie tv prefer minimal block tv agent activity cost generalize environment activity challenge spatial crucial activity distribution plan planning trajectory object work limited planning tv activity informative move propose planning develop web service short video rich environment human activitie feedback segment video bad neutral previous feedback require environment come cost preference room environment generate environment feedback aware validate use pr human environment section plan overview complete discuss evaluation evaluate human pr initial collection routine future human implement object extensively wherein preference learn task preference preference preference human human object prefer minimal agent share environment user agent important second preference object object act agent human tv music prefer minimal environment object rich environment understand plan accordingly social tv stand front planning trajectory action human differently object instant people book people room state execute environment successfully path environment pr preference rich environment planning serve tv object serve agent environment capability previously human way exploit design bridge interaction provide strong path planning tv behind human front robot understand tv label planning learn jointly user environment exhibit depend demonstrate tv read book though environment unchanged rich multiple interact tv room human interaction design trivial relate trajectory perform environment distinguish preference base object activity mean plan differently around object e tv object trajectory indicate undesirable robot configuration learn expectation trajectory number particularly environment human short robot video ask reveal preference interactive interface setup non feedback segment come tv preference activitie cost trajectory location object human activity plan rich environment human activity paper e context rich human object activity robot interaction trajectory design expressive accurately reflect preference environment ccc object away trajectory cumulative cost preference activity separate activity denote cost along multiple associate activity human activity activity define membership allow decompose activity activity trajectory preferred short trajectory latter tv discriminative trajectory human interaction stand behind move multiple human robot contact human environment human human object arise produce stand behind multiple human interact robot contact hand code environment become arbitrarily introduce human therefore learn human activity environment goal robot plan robot robot configuration problem challenge configuration state hand challenge environment object spatial learn activity context environment define human environment correspond environment mixture per graph preference likelihood design environment arbitrarily human interaction drive approach collect rich parametrized family take learn advantage furthermore adapt available heuristic set arbitrarily human object heuristic tb user iii preference build easy preference wherein robot maintain human activity robot observe environment distribution trajectory cost present engine user observe feedback robot book color interval carefully avoid human colored expert user rich principled achieve drive task environment trajectory drive collect preference trajectory cm distance normalize interact preference symmetric activity prefer approach previous reveal preference approach accomplish expert preference clear instead demonstrate demonstrate relevant approach iteratively multiple preference consume expensive collect preference environment main preference environment approach specific interested situation large develop video video particular keep user provide segment pass tv label segment neutral trajectory b good neutral effort reveal environment multiple activity capture learn trajectory human environment currently database trajectory environment interact activity reveal preference idea environment three generative preference activity give user collection easy neutral good segment pass affect activity incorporate give follow activity vary activity prior obtain environment consider bad use solve likelihood solution calculate activity assignment update step keep posterior activity consist von von von first moment von provide detailed derivation supplementary expert idea planning path approach interact trajectory collect preference goal optimize heuristic optimize guide work approach planning room leave reconstruct environment type often human view human pattern recent human joint differ activity preference arise ambiguity maintain cause e robot study effect entity value train relate action work plan preferred trajectory environment interact context work scene planning robot et al user specify robot human et al gradient trajectory function learn rich cost expert similarly preference understanding whereas preference preference perceptron rich environment either create environment google reconstructing correspond activity environment six activity activity generate video receive preference feedback video feedback trajectory trajectory segment assign trajectory mcp near trajectory trajectory stay away human base human activity rule hand encode opinion plan trajectory map euclidean heuristic show heuristic baseline learn user online set use trajectory feature apply trajectory cost preferred quantitative assign feedback ground score minimum score segment segment trajectory neutral ground score score metric quantify chance incorrectly trajectory trajectory trajectory grind truth assign normalize number trajectory quantifie execute sort train good bad learn heat match plan good reliably good trajectory bad trajectory strategie train room vice test room bad order incorrectly misclassification baseline testing improve room human activity environment converge optima rank environment training environment preference well design also observe model robot discriminate trajectory bad one present trajectory choose accordance misclassification rate cost encode planning chance however drive well learn improve room bar cccc activity object heat heat activity heat interact work heat reach activity spatial activity interact human unless contact preference front human critical human towards object however activity move activity human object reach book work spatial work human region spatial front environment multiple planning map use activity planning map region
gmm allow different deviation contiguous center mixture belong intersect bregman solver com compute scalar disjoint interval extend choose model selection illustrate refine bregman maximize complete cluster programming mean divergence primitive homogeneous mean seek minimize intra cluster one solve heuristic locally hardness center well center centroid center etc surprisingly mean program seminal dynamic dp optimally element contiguous maximum appropriate dp require correspond cluster problem refine dp optimal rely area table bregman application identically iid maximize use exponential family series bregman mean optimally dp gaussians graph intersect laplacian belong family space totally nk nx contiguous among contiguous partition ask minimize function intra cost calculate inter counterpart etc sort I ie matrix find contiguous indeed say contiguous recurrence q store position top yield denote require require recover storing index cluster solution iteratively retrieve index also note satisfy partition may potential auxiliary look solver contiguous optimally fact far run consider add constraint add non empty th great constrain balanced obtain kk appropriate clearly costly monotonically depict explanation choose cluster compute good entry column range regularize last avoid computation check entry last row range center store prototype center dissimilarity prototype intra cost rx l j rx dp return contiguous cluster cell prototype dx potential induce dissimilarity diagram display illustrate refine bregman bregman bregman triangular asymmetric bregman variance bregman bregman prototype lx j bregman area table contiguous cumulative time preprocessing evaluate cell x p since bregman diagram cell cluster contiguous directly bregman center exactly memory element bregman solve often mixture dominate lebesgue simplex indexing px space usually locally maximum need proper reach iid amount maximize q maximize dissimilarity prove proportion solve
often roughly tie emphasis interesting distinguish effect standard illustrate issue normalize equation bayes choice reversible jump consume view follow compete random label evidence q around section density normalize integration estimate g g g integrate routine evaluation manner factor enable model previous suggest systematically prefer computationally intensive promising composite recent composite likelihood currently acknowledgement acknowledge insight centre science foundation grant support science foundation grant mix integral simulate network auxiliary network exchange close case flat therefore two reason simplicity random extend heterogeneity framework yield model paradigm estimating factor develop feasible calculation bayes mix analysis statistic publish article development refer introduction field represent adjacency connection vertex symmetric triangle equally matrix cross distinguish explain existence model first model edge replace see unit principle actor consider heterogeneous heterogeneity observable lie generalize mixed estimation software network modelling call exponential number star see term graph therefore early approach advance fully develop interpretation focus node change statistic deep discussion exponential equation equation model possible heterogeneity actor include homogeneity lead want exercise latent heterogeneity exponential form term likelihood statistic vertex fit node effect accounting heterogeneity fall family random unlike specific latent author effect issue extension bayes calculation suffer numerically calculation selection extend fully develop routine package http package organize fully model routine deal bayes factor result propose mind infeasible calculate small numerically impose prior independent identically distribute accordingly use denote unity prior q prior choose flat hyper note doubly intractable firstly possible marginal secondly infeasible normalize draw entire vector drawing augment proposal accept p normalize easy direct step detail follow
goal rotation independent branch mathematic distinct suggest section reading function measure independence function say hoc present idea theory close independence generalization information multiple reach independent example ica estimate underlie multi therefore ica reach zero middle example matrix form rotation variable calculate multi zero statistically imply rotation optimization appear abstract validate source rotation recover unimodal gaussian bottom add integer minimize practice interpretation ica entropy amount sum entropy entropy joint employ relate determinant rotation rotation information constant rotation maximize statistical connection mention calculate ica strategy interpretation distinct equivalent interpretation ica solution equation find transform kullback leibler variance minus sum rotation maximize assumption statistically interpretation permit turn reconstruct component rotation angle multi plot rotation angle recover source rotation respectively bottom grey marginal distribution rotation degree unimodal quick summary datum optimize code ica mix recover assume statistically toolbox find popularity signal processing perform mixture basis independent component source interest party recover independent selecting music recover music success analytical approximated numerically inherently minima objective equation estimate quantity optimize extremely challenge entropy identical analysis bar part employ remove ica employ rotation remove high order pca inference understand observe apply whiten distribute covariance factorial whiten pca achieve factorial implication ica try ask prominent correlation high small pca recover independent source empirically measure correlation check rarely issue minima optimization rigorously calculate procedure overcomplete exceed middle underlie source lie superposition exclusive reader recognize ica readily highlight reader ica back source clearly suffice mathematically intuition match exceed overcomplete discussion employ additional form regularization latter perspective situation multiple ica blind separation focus handle arise sparse source representation popular journal ica relationship elegant achieve performance writing experience I hope ica assumption behind technique please I write orthogonal transpose orthogonal associate eigenvector let eigenvector proof part eigenvector independent matrix eigenvector complete part e degeneracy place column therefore little provide complete distinct eigenvalue relation unique eigenvector symmetric orthogonal final observe require covariance datum note popular package employ notable speed attempt direction algorithm although elegant demonstrate demonstrate ica introduction linear algebra intuition ica principle behind reflect thing piece measurement reflect city etc sometimes clean arise distinct identifiable subtle measurement estimate source corrupt fluctuation additive white instead distinct source broad topic separate source name source bss writing solve arbitrary bss often last decade ica solve blind ica filter reduction retain e person operation retain project filter ica g eeg biological e micro array audio ica naive treat technique black box one ica appropriate believe necessary ica write thorough goal algebra topic pca truly concrete example building intuition mathematic insight please I comment correction blind separation bss intuition signal interact world make measurement addition make bss entail intuition well bss sound fundamental physics sound linearly record pressure multiple source amongst background person signal party sound environment highly applicable base idea process bss remove image due camera try camera steady pixel sensor record light integration intend camera motion recorded light original camera pixel image pixel original require camera party de ill additional vision highlight signal interaction world complex exist recover individual combine bss arbitrarily generally progress interaction superposition problem ica bss address ica party problem ica mixed music likewise arise solely depict source record term component whose amplitude weighted magnitude magnitude source formation top row unimodal bottom middle component blue figure might correspond imagine party record music axis record figure happen music play look alone play datum would lie music since music fix record must volume volume sample panel arise middle panel note audio merely reflect source basis label ic would sound add depict panel color accord contribution give diversity bss middle row axis namely unimodal appear orient row situation analyses solely sum examine piece arm direction extract desire music benefit visualization become difficult might salient salient middle figure might middle technique fail find understand recover basis goal mathematical highlight type distribution play understand framework sample unknown e keep thing interesting variable observation party amplitude behind underlying source component ica find inverse construct ica appear black point constrain number observation challenge hope find essence red correspond operation composition operation divide strategy problem simultaneously rather piece fashion cutting divide piece svd decompose simple operation axis svd parameter infer diagonal matrix figure provide familiar recover piece individually exploit rotation appendix successive stage order independence ica consider failure concrete suggestion might appear motivated provide strategy emphasize reader general explore three operation equation start covariance capture correlation capture dimension outer product discuss plug linear exploit property arrive choice independent extra equation look familiar student algebra aside tell decomposition refer operation multiply orthogonal orthogonal stack whose covariance orthonormal basis compare equation matrix decomposition underlie ica purely propertie matrix symmetric behind
variational work core computation supervise also term analytically rewrite scale transformation element analytically gradient gradient contain efficiently gradient conjunction sgd adaptively training procedure experimental algorithmic minibatch variate distribution layer average classifier neural complexity complexity form label gradient stack generative algorithmic provide complexity make appeal since encoder low model fully wide range inferential query approach semi h I figure please benchmark supervise vary label ensure number create use confidence mean repeat draw construct hide activation use layer unit pre deeply still optimize solution semi machine label svm obtain reason space generative variable image comparative semi near kernel performance generate discriminative model original present supervise classification cccc knn knn cccc knn initialize objective optimize minibatch ascent momentum bias momentum normalise pixel intensity image weight decay prior introduce extend full variational provide ground perform selection particularly supervise image task network current model readily exploit general connect promise future exploration limitation expensive potential reduction use truncation combine truncation mechanism code label approximation supervise generative amongst competitive currently hope supervise classification model much scope grateful e experiment google google ever modern label significant datum supervise develop label scalable deep inference exploit recent advance consider subset observation interest wide search language parse entire ask datum improve decision classification accurate label datum answer question building advance scalable amongst scheme additional obtain confident prediction repeat termination reach poor svms aim ensure approach difficulty extend efficient open amongst popular similar information label eigen laplacian scale unsupervise feed forward classifier additional penalty auto encoder manifold classifier train learn manifold lie follow train invariant local perturbation manifold use combine amongst currently problem hide successful mixture either process scalability variational explore small set generalise scalable semi supervise still gap new framework employ rich parametric estimator form fusion develop inference optimisation scalable demonstrate approach qualitatively generative intra allow fashion variety dy omit index clear subset class exploit label alone model separate relate latent feature limited auto encoder generative able generative form linear transformation essential choose network variable regression latent dimensionality embedding form linear approach performance propose probabilistic describe generate latent treat latent marginally digit separate writing transformation label unobserve integrate data inference perform inference miss see mixture two learn subsequently supervise instead generative stochastic deep exact intractable variable allow tractable scalable advance variational approximate variational principle derive likelihood form posterior variational variational
fast star asynchronous performance performance conduct another master perform loop master obtain require subproblem master information back need master master obtain master allow increment ensure master level synchronization master decomposable conservative master step master step master atomic increment preferred period case vary stop three add artificial delay complicated setting show speedup fully method suffer star asynchronous method suffer achieve although speedup star topology second vs parallelism randomize base dual state sequential cd outperform descent selection rule randomize clique svm use clique introduce sparse box constraint describe one coordinate descent however expense computation without increase maintain basically increment increment increment fact ij prove equation induction ij prove choice last statement therefore statement induction substitute prove definition third last descent require solution intermediate find vector formally prove vector element iteration small let absolute albeit ni ar decrease least terminate termination otherwise algorithm always pick continue multi manner remain part essentially similar detail completeness step recurrence relation form simplicity line consider gradient orthonormal column g use less rewrite q become clear preserve preserve wise gradient denote minimum zero singular note however operate variable much much conditioning operate tight compare bind technology cd constrain scale separable couple knowledge cd iteration constraint present four key convex smooth separable asynchronous detail illustrate coordinate cd conceptually unconstraine study greatly root successful statistic therein cd randomization generic randomize cd though optimization nesterov global iteration complexity improvement nonsmooth term randomization cd cd problem allow constraint develop problem differentiable lower coordinate separable necessarily lc specify see however fit cd framework nonsmooth part separable usual alternate direction multiplier latter treat familiar special fuse constrain recently study set assume feasible ensure descent maintain feasibility scheme well convex problem recently generalization separable block cd obtain present dependence study couple cd gauss apply cd linear present light background rate randomize block case tight composite sum asynchronous solve contribution exist art proof claim lc prox parallel yes yes yes yes yes yes yes cd gain full refer reader thorough analyze author family gauss like analysis gain combination randomize frank wolfe though frank wolfe approach projection cyclic descent global lin recently cd nesterov prove linear sublinear slightly refined pure et rt problem processor update speedup overlap strategy study read fix liu al prove convergence dd allow inconsistent read bc arithmetic cost well similar stochastic advantageous size closed form stepsize tuning duality gap handle inexact linear easier parallel minibatch distribute delay minibatch sg nesterov parallel minibatch subgradient sag asynchronous also read asynchronous vector store asynchronous fashion rate speedup control max processor able impractical delay notice contraction method essentially linear general assumption decompose communication constraint node present exchange denote restriction column column identity place typical coordinate lipschitz gradient make block critical composite also pt ready asynchronous stochastic variant idea maintain ensure pt begin nonsmooth pick minimize around iterate maintain formally involve stepsize bound present result update always however condition update space constraint group typical size apply describe smooth theorem nonsmooth assumption result I k kx kk solve iteration inherently disadvantage develop asynchronous smooth difference processor solve subproblem asynchronous execute without require asynchronous presence non ensure feasibility throughout require sequentially consistent swap increment update despite convergence asynchronous base gradient iteration bound sublinear gradient parallelism asynchronous conclude discussion asynchronous algorithm extend nonsmooth function suggest maintain feasibility respect constraint convex iterate retain feasibility future loss many separate add separability innovation addition randomly pick also update since calculation involve gradient random random integer later price convergence size generally outline describe hence lack key constraint loss generality rank rewrite form transformation ease exposition experiment assume also unconstraine introduce diagonal induce quantify far take reader impact laplacian graph consider simple update fx prove attain generate algorithm convergence iteration prove follows desire decomposable worth improve upon involve towards read context like assume enforce asynchronous expected value define equation sketch ease exposition analysis carry manner iterate iteration existence consistent read fx k u derive prove mathematical method expectation large stepsize decrease rate convergence rate oppose iteration factor tradeoff algorithm slow believe generally algorithms careful analysis nonsmooth sum assume separable write coordinate totally impractical set uniform algorithm graph clique linear improvement theorem form oppose
tb set pose quality report mutual result table embed cauchy score reach accuracy base mean furthermore k matlab demonstrate benefit tackle tb accuracy utilize locality hashing perform video projection low hamming encode key neighbor query time sublinear contain hilbert familiar euclidean kernel ability approximate superiority previously cauchy keep kernel open future could give answer equivalence pl embedding follow definition intrinsic metric curve l dt intrinsic infimum length intrinsic metric induce metric define respect intrinsic metric length conjunction conclude proof token li act communications digital centre program image set prove beneficial incur arise fact subspace type riemannian leverage develop support subspace study hilbert make positive unfortunately two kernel none show introduce positive superiority code embed positive definite subspace nonlinear euclidean video visual subspace prove vision despite success suffer drawback euclidean subspace lie manifold consequence popular euclidean space reproduce hilbert rkhs exist euclidean recent study report manifold intuitively attribute fact geometry manifold capacity capture nonlinearity manifold rkh preferable rkh cauchy former ability approximate universal well new include universal kernel two embedding cauchy pl embedding yield distance distance conjunction analyze kernel ten summarize evaluation demonstrate benefit gender categorization kernel bc rbf ex r ex laplace universal k binomial bi bc universal bi p universal logarithm log bc ex k review property paper use capital letter letter column identity trace space euclidean note become projective orthogonal column pp dd denote slight abuse notation whenever represent subspace riemannian length short connect call geodesic geodesic geodesic give let subspace recursively principal angle principal correspond addition geodesic metric similarity metric manifold hilbert us real kernel nonempty symmetric p kernel pp arguably radial basis radial replace euclidean unfortunately symmetric define verify counter digit nevertheless manifold principal subspace bc successfully problem hilbert space receive little bridge discuss space help devise kernel pl concept algebra alternate multilinear vector space g copy multilinear slot alternate g multilinear copy k generalization arbitrary projective pl space describe pl dimensional close embedded indeed pl taking row pl pl inner importantly meaningful inner need two realization correspond point like dimensional hence computing goal compound arrange compound cauchy rectangular matrix pp hence pl coordinate suggest pl indeed linear kernel verified may sign issue problem design define induce principal angle invariant follow show pl nice relate geometry scale turn well pl embed projection embed one differentiable smooth choice subspace induce discuss projection isometry length curve reader thorough discussion embed see order induce space induce pl actually exploit kernel derive pl embed cauchy kernel define kernel create consider kernel readily kernel often crucial impact pd learn express importantly kernel arbitrarily sufficiently develop universal negative definite us kernel nonempty symmetric kernel definite form example nonempty inner function f fx I therefore hilbert important measure half designing cast probability kernel euclidean rbf dirac delta discard scalar rbf embed positive neither prove theorem employ laplace pl embedding extend kernel binomial kernel translate noting see us kernel bi important positive formally conditionally kernel nonempty conditionally relation kernel study kernel translation equally instead kernel invariant position origin svms separate
degenerate prevent directly apply recursive summing u lemma get vanish hold eigenvalue zero write equation reduce equation iff condition lemma asymptotic asymptotic involve quite remarkable penalty extend idea yx x l e center spatial let f update spatial estimate yx cosine bc b cauchy schwarz inequality estimate choose upper bind bx ax x apply p ip j ib j cosine rewrite equation estimate bind ax bx ax equation l enough take expectation variable pseudo initialize ix g implementation master momentum initialize batch image batch training center precisely local worker general use datum read memory map file raw always send worker request worker request different datum data cycle file worker receive request datum reach start file worker rate summarize rate explore htp initial explore rule initial rate divide decrease htp experiment sgd htp figure log computed experiment always begin experiment start average show compute discuss dependence trade exploration exploitation learn rate experiment observe test fluctuation worker conjecture high lead impose opposite learn rate interestingly train lead bad performance experiment trade exhibit get energy avoid decrease figure seem sensitive increase period crucial result experiment seed fast package interface gpu communication htp experiment various communication decrease gradually base run load experiment whereas process run whereas negligible large ideal communication leave right correspond experiment summarize need worker level method time level time level counter though large learn meanwhile outperform achieve consistently achieve level h bar denote achieve htp leave never achieve section lemma facebook deep environment process worker elastic force link server enable worker allow variable reduce communication demonstrate many optima improve asynchronous asynchronous variant compare stability asynchronous asynchronous convolutional neural deep compare baseline approach large use descent deep consist devise large yield run gpu variant method interpret idea worker among elastic link store update move time worker contribution provide convergent simultaneously reduce worker maintains measure deep organize asynchronous momentum analysis conclude supplement contain environment master parameter distribution reformulate assume worker refer center equation fall far communication master worker emphasize highly trivial resp descent resp gradient rule center take become local equation choose elastic symmetry rule force influence stability small allow center worker exploration master differ set asynchronous worker master center maintain update master update refer see whenever worker master request center worker entire capture send update communication worker trade exploration exploitation rate communication period randomly move randomly x g ix v momentum capture nesterov worker equation momentum overhead computing parameter explore asynchronous momentum variant asynchronous realistic study show dimensional lead analytic show analytic stable supplement property strongly defer minimax master worker master round lagrangian multiplier give let fx tx become lagrangian multipli ascent update center multiplier equation choose convention multiplier update algorithm similarly worker activate perform follow equation focus state dynamical compose map simplicity write map simple absolute batch momentum moving rate start examine period comparison also outperform figure supplement rate explore table supplement time well small achievable sequential training center different value conclude achieve error small unstable significantly outperform also become tendency characteristic algorithm simultaneously htp local worker convolutional neural achieve compare relative error equal htp layer decrease factor loss speed simultaneously overhead communication explore different experiment experiment result result experiment converge fast lowest achievable potentially explain exploration supplement exploration exploitation period section advantage supplementary loading communication supplement achieve training quickly stable plausible communication method behavior momentum future work acknowledgment li implementation helpful l valuable feedback focus center local quadratic generalization strongly quadratic observe noisy definite eigenvalue strictly positive covariance square center error center variable stable verify root iff iff symmetric due relation stability asynchronous elastic symmetry substitute rule local worker eq q lemma explicitly linear rewrite capture diffusion eigenvalue satisfy eigenvector projection eigenvector therefore last expand recursively substitute
prior superior test histogram mean percent percent gmm percent histogram bayes nb histogram eight particularly poorly censor believe due issue h flat mean ci ci gmm ci rate gmm sbm indicate superiority well gmm pair methodology wikipedia graph graph represent wikipedia page either page neighborhood algebraic vertex class article label available analyze label exclude isolated induce adjacency pair adjacency spectral figure red green figure indicate wikipedia sbm nonetheless wikipedia class green blue adjacency wikipedia adjacency spectral wikipedia illustrate bayes generate bootstrap embedding depict gmm embed estimate figure adjacency spectral common reasonable justify sbm gmm provide wikipedia induce constraint empirical color class gmm membership depict prior yield statistically pair sign empirical improved gmm represent statistically significant absolute misclassification bootstrap gmm statistically significant perhaps improvement chance performance optimal misclassification class gaussian yield nonetheless gmm neighbor indicate conditional gaussians clear despite real dramatically stochastic robustness formulate empirical methodology motivate theoretical advance regard embed blockmodel within gibbs algorithm block latent consistently outperform gmm alternative notably dirichlet wherein wikipedia graph though sbm extension interest automatic eigen case sbm embed justify adjacency dimension bayes misspecification importance dot model extend methodology challenge sbm position simple bayes conclusion adopt empirical bayes approach estimate membership blockmodel embedding assignment acknowledgement work support security science engineering fellowship technology advanced project fellowship corollary apply mathematics stochastic blockmodel interest statistical various diverse social citation brain network etc dot product position formulation stochastic normal adjacency spectral theory bayes membership vertex draw blockmodel practical utility posterior inference conduct within theory carlo studies blockmodel wikipedia blockmodel sbm position use various diverse may indicate citation network neuron connection vertice edge indicate connectivity statistical statistical become area sbm vertex edge depend membership edge give membership entry represent blockmodel memberships important approach membership maximization maximization modularity parametrize latent vertex particular dot product dot vertex absence dot vector blockmodel define dot vertex share common dot step often estimate position position position consequently describe latent truncate eigen adjacency dot latent position adjacency multivariate gaussian mixture sbm identically distribute multivariate paper mixture methodology block memberships blockmodel section formally blockmodel dot motivate empirical prior methodology estimate blockmodel mcmc implement section experimental demonstrate bayes methodology discuss conclude vertex may adjacency symmetric undirected edge loop multi edge let dot product random product latent vector furthermore condition position ij tp decomposition u p nu ds diagonal along introduce obvious identifiability loss generality suppose adjacency embed dimension blockmodel dot accord semidefinite blockmodel sbm block distinct mass standard purpose block membership stochastic blockmodel assign memberships sbm typically specification include special often sbm advance describe empirical position prove position converge gaussian express follow corollary motivate eigenvalue second moment principal row corollary setting suppose theorem likelihood position membership probability inference membership base gibbs behind vertex calculation assume block latent gold prior position represent corollary center theoretical spectral correspond gibbs thus position sampler gmm metropolis distribution k denominator use multivariate limit adjacency gold thought present empirical bayes membership probability unknown posterior simplex gmm choose conjugate dirichlet distribution q block follow marginal metropolis initialize utilize metropolis prior eqn k compute special parameter alternative flat gibbs sampler identical present metropolis position proposal however initialize provide modeling initial k point k k illustrate various blockmodel sbm dot stochastic blockmodel three wikipedia graph block via compete two percentage vertex calculate inference base calculate assignment times blockmodel parameterize probability entry membership graph sbm parameter spectral adjacency position subsequently gmm cluster embed membership component variance prior latent position know scatter replicate color denote membership symbol cluster membership curve estimate gmm gmm scatter estimate carlo replicate sbm denote true membership vertex sbm
length dimensional employ normalised radius randomness normalise uniformly know provide unknown use classifier model independently select li form surface fraction reduce surface pdf introduce follow dimension unitary dot product examine concludes select uniformly independently surface di ip second determine circle matter cdf general expression give neither pdf formulas sphere radius coordinate eq hyperplane distance large point lie sphere hyperplane cut distance small portion cut hyperplane consequently height height q htb maximum height recently prove li surface hyper surface angle beta angle sphere radius follow cumulative sphere length eq stand proposition refer radius replace length dimension sub htb length dimension cdf recursively formula sphere second term right cdf reduce similarly always score length end radius eq approximation suggest q eq propositions infinity figure length around order quantitative difference show table length sphere interval length around distance intuitive change dimensional sphere fix north point concentration expense high area ideally almost lie mean within radius point dot product centre q distance dot matter unitary dot reformulate correspond symmetric around odd moment among formula length estimate pdf basic property start dot two unitary definition theorem study length end
truth truth underlie subsequent particular suggest condition exponent assume initialization circumstance uniformly unit sphere normal zero instance obtain random amount heuristic condition additional available information unfold unfold cf gaussian necessary sufficient tensor power initialization suggest unfold magnitude suggest procedure tensor initialization power natural tensor within machine main tensor assume away vanish consider characterize convergence bound allow unfold message amp prove successful reconstruction retrieval appeal dimensional characterize technique evolution develop tensor focus publication sophisticated iteration unlike normalize multiply simple expression conclusion two side signal amp remain number theorem evolution exact sharp characterization location next summarize defer journal publication denote tensor follow finally produce decomposition proportional uniformly recursively state coincide apart iteration describe power depend apply amp power show consequence obey evolution require information optima say optimum surely converge local large special optimum numerically note emphasize practical suggestion tensor unfold superior tensor unfold produce improve performance warm decomposition power approximate simulation describe unfold tight compare side dramatically simplify product semi psd suggest cone vector belong cone ni mat eq special however rigorous efficiently project onto psd cone compare correlation n unfold psd psd power initialization psd initialization line confidence consistent theory tensor poorly approach unfold dimension psd principal slightly plain unfold initial unfold component either recursive unfolding performance simple algorithm close behavior heuristic argument tensor random initialization work unfold plot superposition correct experiment concern simultaneous tensor noise fix addition noise varie experiment triple lead report predict apply theory otherwise appear superior capture difference already matrix green pca acknowledgement partially grant fa fa introduce operator eq ai symmetric f recall packing cardinality nm b g q symmetric tensor objective eq function dramatically whose prove lead quantify phenomenon characterize growth rate minima lemma far monotone negative strictly informally exponentially maxima value obtain last indeed maximum next maximum theorem let unique asymptotic eq term variable get asymptotic follow obeys turn show around lipschitz modulus euclidean tensor modulus notation proper n consider tensor loose except case matrix loose factor optimality modulus ns kn enough vector exist triangular inequality borel note eq gaussian prove exploit symmetry argument process process follow x aa op q imply prove induction assume inequality fx conclude notice inequality note imply inequality strictly positive monotone prove satisfie order statement parametrization algebra discard point read continuously differentiable calculus stationary maximum point imply latter inverting parametrization since evolution mr lemma conjecture claim corollary arbitrary spike rank noise establish sufficient computational resource turn soon remain dimension tensor power pass idea unless none succeed fundamental limitation tractable initialization tractable initialization statistically finally unknown signal allow iterative replace approximation understand pca tensor exploit include collaborative context information hypergraph hyper find bottleneck np last greedy bad perspective know regression rank resource arise whereby unknown noisy multilinear precise observation read notation analogous immediate np summarize observed tensor recover therefore natural dramatically unbounded computational resource probability threshold sake well polynomial tensor unfold pca provide heuristic unfold factor odd conjecture confirm conjecture rapidly argument necessary iteration substantially observation warm power initialize output unfolding appear unfold variation unfold method pass amp compressed sensing estimation behavior amp qualitatively fail computational barrier weak information measurement relate matrix noise amp threshold amp rigorous require tensor unfold random theory paper insight believe insight throughout paper defer low ordinary nk n nk frobenius euclidean furthermore maxima develop picture translate guess algorithm initialize section intuition popular estimator perform operation unfold vary expect affect qualitatively summarize sake amount unfold succeed heuristic argument tight expect generally unfold essentially construct achieve remarkable behavior point method construct perform introduce refer mat j relaxation problem natural expect signal latter minimal phenomenon integer universal bound minimize gaussian modulus mat op concentration standard triangular standard mat q consider identity since concentration
correlation obtain section mt jt serial long z describe argument r formula dim criterion pc bic ic max dim true max restrict loading nan role specify argument simultaneously however convergence slope iteration control trend modify df lt unobserved individual perform consumption price balance column consumption price dim pc price dim pc dimension number inference error choose corresponding argument presence serial correlation choose realization appropriate inference correct call r consumption dim pc slope std e code none unobserved factor degree report factor effect log price sale effect log well graphic factor reason explicitly effect see hand effect consider follow time effect order restriction variable eliminate effect eq x n restriction affect effect procedure variant model appropriate control formula refer intercept purpose model ignore illustration continue section concerned existence factor merely specification appropriate datum rather existence factor classical dimensionality hypothesis dimensionality test statistic reason simplification reject test function test consumption testing interactive song factor consumption price presence interactive effect h equal level model significance level intend outline interpretation panel estimate convenience choose attractive form unobserve heterogeneity beyond great individual valuable literature stochastic leave additive variable right panel common factor common importance reflect loading convenient quantity share loading variance loading explain two rl red middle figure difference effect factor loading parameter explanatory visualization vary visualization individual effect model propose eq usually cover rotation factor interpret appropriate rotation scheme set rotation package preferable loading parameter instead factor introduce package function package available elsewhere remarkable package usage function demonstrate many helpful comment paper research science trend package procedure dimension heterogeneous procedure complement heterogeneous factor case common whereas focus bound factor additionally range dimensionality criterion unobserve remain model difficulty time appealing advantage panel heterogeneity classical try unobserve heterogeneity structural assumption e unobserved heterogeneity time within unit apart trend fairly availability panel model focus advanced panel individual heterogeneous time trend basic explanatory time vary individual vary specification package individual loading consideration intercept intercept vary center intercept individual center classical panel individual effect choose individual loading factor identifiable rotation ensure uniqueness require ex replace certain sign consider factor strongly auto correlate well stationary way semi identically error allow weak iterate stationary vary stationary deterministic trend rule stationary integration moreover usually package refinement package criterion factor dimension compute estimator see allow test classical fix panel section function longitudinal package panel exhaustive package publish panel toolbox possibility heterogeneity effect well package criterion publicly code ng demonstrate explore american price g sale real price per year dataset htbp price devote short common usage recently discuss well relatively common panel effect parametrize term asymptotic rely second difference discrete series restrict purely functional interpretation estimation first common vary semi component use describe eq time continuously differentiable denote spline imply possess expansion spline basis st see rewrite formalize kt dt follow usual cubic smoothing spline specify explanatory allow specific around intercept common also possible factor factor define eigenvector factor normalization individual loading ordinary square regression lt il crucial testing vary consistency obtain determine smoothing propose cross observation unfortunately computationally costly determining factor advance overcome disadvantage discuss follow theoretically cv cross validation costly explain specify critical get quick objective updating formally calculate start initial proceed step advantage approach inversion update moreover rapid routine formalize goal rather factor cv note smoothing dimension explicitly dimension argument devote argument r formula dim dim cv convergence restrict factor restrict loading symbolic specificity balanced panel replace processing imputation logical dimensionality compute ask default maintain argument adjust dimensionality logical computed default function discuss restriction alternatively loading variable store necessary function variable equal library l consumption r r factor loading default inference slope consumption summary consumption residual pr intercept price code additive effect none unobserve r price real sale significant summary individual effect provide summary factor right effect figure six estimate panel figure correspondingly eigenvalue obviously extend effect logical argument dimensionality criteria consideration slope propose follow test statistic lf l lt significance begin test hypothesis reject estimate rejection dimensionality stationary tendency weakly correlate criterion cause formally fit propose specify residual condition impose restriction proportional requirement fulfil bic large practice bic error cros correlate arbitrary kind case problem criteria ic ic q improve performance ic ic strategy different tuple value grid refined criterion ic abc ic respectively modification affect maximize criterion er gr theory pc bic ic abc ic ic er gr stochastically number panel introduce threshold distribution dimension iteratively ed eigenvalue argument pc pc pc bic ic abc ic c ed er seq seq criterion character default tuple appropriate construct sequence sequence give standardize matrix imagine input consumption call pc offer selection procedure give automatically compare procedure consumption criterion er gr consumption criteria pc gr ng pc criterion er gr criterion user method display percentage criterion er gr come criterion dimensionality ask consumption price dim estimation ng pc pc ic ic ic abc ic
also per somewhat simple projection challenging interestingly also compressive course seem target influence main exact characterization equip result fairly perturbation first analogously characterize bernstein inequality adjoint dimension self adjoint resp onto top eigenvector capture informally say principal provide appropriately small bernstein write deviation suffice setting two high rank measurement draw uniformly invariance think project onto standard ba geometric argument angle orthogonal uniformly random subject basis vector orthogonal sphere compare increase dimensionality bottom see measurement column target although play role plot careful explicitly lead exist result lastly error fix improve qualitatively data compressive insight independent operator preserve practically appeal simulation bad finding secondly mention compressive column justification immediately would understand show compressive column lastly fundamental compressive measurement column achievable algorithm achieve achieve hope acknowledgement research part nsf award nsf research fellowship eq dd expression rational first plug principal large number compressive number suffice principal insight exploit average effect projection preprocesse wide signal processing application give subspace capture vector large singular minimize reconstruction dimensional situation obtain motivate different theoretical study range column burden concern acquisition motivate line datum compressive completion focus approximation adaptively entry compressive column translate usually project principal strategy use sense approximate projection phenomenon compressive per column column exist require rank proceeding mention motivate analysis inferential observe point scientific measurement overhead extract principal sensor network suppose sensor record make cost compression share synchronization acquisition avoid need synchronization compression good proceeding recover principal norm x subspace span eigenvector th st eigenvalue principal subspace large hard matrix large compressive vector unit sphere equivalent projection column connect two column span compression kk call conceptually algorithm vector covariance top estimate span eigenvector appropriately normalization necessary mention algorithm streaming sensor early quite appealing sensor observe I communication overhead compressive need projection acquisition overhead two vector make overhead achieve good property approach even method main turn remark order term active thus term dependence relationship since compressive suffice sharp
compression try apply autoencoder solve sound input autoencoder appropriate source music signal encode distinguish separate unknown source description autoencoder fourier audio frame index autoencoder rectangular window spectra rectangular autoencoder set frame axis design autoencoder unit respectively source mean whole window cluster window also classified source reconstruct propose separation carry mixture source speech music acoustic music generate audio autoencoder frame ms window decide connect node output speech third speech six compose straight line frequency music signal mask middle music peak fourth htb htb autoencoder separation time use represent spatio temporal autoencoder source main complete effort national foundation education technology science department advanced institute technology ac unsupervise deep source signal properly autoencoder coefficient investigate representation audio domain coefficient layer original reconstruct mask speech work extract noisy
later assess effect longitudinal treatment potential outcome tt history make identify dynamic potential outcome assumption outcome correspond actual outcome assumption within randomized manuscript law definition markovian drop subscript discount factor immediate consequence reward specify inferential take severe taking say however say high regime action equation inner quantifie quality policy interpret reward treatment would optimal ensure optimal also note action estimate turning recurrence update density large infeasible explain section need estimate represent parameter vector pair accordingly discuss first bellman science account feature multiply depend transition observe q sometimes solve deal take similar technique objective manuscript improve minimization algorithm generalization w optimal treatment respectively estimator make assumption list statement asymptotic estimate consistently replace expectation estimate objective non process often fail incremental stochastic greedy tool point forward gradient descent gradient introduce show objective introduce toward minimize start individual trajectory obtain tuning step continue size rule distance consecutive constant empirical inside update simulate diabetes construct treatment consider manuscript include reflect treatment extract patient treatment drop take treatment simulate consist individual start decision option continue decision bernoulli augment interval soon treatment variable generate multivariate pressure continue although ideal raise concern feasibility need treat treatment regime patient treatment control whose treatment patient continue take augmented patient either treatment depend variable treatment add c bernoulli augment treatment treatment pd pd pd take treatment avoid effect percentage effect similar effect regime bp treatment operational definition otherwise reward help identify efficacy treatment turn recurrence relation rule rs function summarize similar chapter note transition usage large cat bp cat axis cat respectively axe percentage treatment action construct radial gaussian specify function satisfy list multiply triplet find transition approximate benchmark depict treatment discretize oracle discount average report vertical side leave horizontal axis represent plot propose classical moderate fourth plot efficacy specifically present optimal value size indicate error theorem interval decision tool another important asymptotic whether estimate whether contain adjust option interval result state suggest sometimes dynamic regime horizon priori decision assume base temporal asymptotic property study work raise derive distribution practical provide violate may type may major second try minimize alternatively regression diabetes cyclic point decision happen decide visit patient request easier require visit patient discuss visit award da institute drug abuse content solely author grateful satisfy follow deterministic radial basis quantile variable second first third quantile trade decrease decrease estimator besides b b b part prove complete normality vb vb normally distribute continuity cauchy ga ga listed van show result assume full decomposition eigenvalue satisfy van small condition automatically enough sufficiently definition fa ga ga thus van decompose part n w n b rest follow subtracting complete eq enough identity hence optimal regime simulation scenario discuss manuscript suggest effect treatment get large policy long effect immediate effect present proposition em minus em cm pt title pt email utility manuscript inferential residual dynamic regime horizon priori treat throughout life large necessary inference diabetes third wave national health survey examine
design feasibility z follow application algebraic extension proposal directly regularize proper initial univariate unclear directly mention justify hypothesis interested extension chi take subsection match group structure projection result rotation scale subspace essence inferential rewrite eq match e approximately projection g condition moreover g g g convert elliptical mapping g consistency statistic testing least estimator course need relax requirement orthogonality k relaxation theory true term g moreover standard freedom chi square projection proper upper g motivate summarize suppose provide geometric insight mention equal call try sp projection complement addition g follow immediately remove g g v k somewhat moment feasible feasibility describe subsection iid subgaussian row let orthogonal space well supplementary let r iff r k kb k k v since ball bound n k n inequality complete discuss g g ty subgaussian g v orthogonal g g project indicate g g g g remain view due group theorem prediction weight may use define u j bound view event eq os result eigenvalue kkt j rearrange h g kkt multiplying event g h n follow optimization scale account individual could proportional literature see iterative convexity joint limit give provide freedom practice discussion optimization characterize derivative profile objective minimizer l involve claim scale proof gaussian setup theorem ii matrix cone fix take certain corollary upon prediction g fold j prediction mix obtain lasso g j g bb g g scale proof suppose j kkt h tt group estimator theorem rescale error imply due pn limit fu n fu fu concentration fu e result section lasso simulation experiment design independently true zero design regression wise jj scale lasso penalty replication lasso dot fit replication setup deviation show plot convergence correction frobenius simulation scheme simulated group sparse specifically group nonzero provide base variable group contain suggest group size normality group ps author blue blue theorem remark support grant dms small projection group dimensional estimator chi inference group develop chi inference sparsity condition group benefit inequality scale group q p py unknown inference test approximate version procedure potentially group dimensional possibly low projection regular sufficient asymptotic normality group design match efficiency efficiency imply large group condition expansion moderately large certainly beyond chi type statistical scale group combine extend idea lasso group estimation consideration attain efficiency super characterization informative adjustment min attempt select regularizer done consider subsample consider conservative alternative project residual adopt correct generalize treatment effect consider precision matrix upon brief discussion literature index effect regularize prominent g group variant among many lasso assumption develop reference follow statistical feasibility subsection work availability work verify subsection formulation variable subsection provide subsection solution use paper vector norm u jk denote column indicate column complement span additionally coefficient vector inherent pre group
index x header header header txt index header txt table index header index plot index header txt title td xlabel ylabel ndcg header td td minor xlabel ylabel ndcg header td txt header td txt xlabel ylabel legend pos east x plot txt header minor title xlabel k ylabel ndcg legend pos south east true plot txt header minor title xlabel ylabel ndcg header plot txt index title xlabel ylabel ndcg index header txt table x header true plot txt estimation initialize seed blue line ndcg truncation though bfgs reliably solution try line td initialization give baseline identity replace identity medium dataset million song baseline td xlabel ylabel ndcg k legend pos color thick bar cd title k xlabel ylabel ndcg legend pos south explicit thick mark option index header txt thick mark black mark option index header txt scale minor title xlabel ylabel ndcg pos east color cd mark plot txt thick color mark option solid mark xlabel ylabel ndcg pos south blue explicit error color option solid header plot mark option fu v fu fu fu u vx fu fu contexts partition mutually exclusive exhaustive subset partitioning find qx fu fu fu fu fu fu divide mutually exclusive exhaustive partition let previously iteration instead define set parameter access every progress fraction guarantee optimum accord expectation stochastic optimum objective function retrieve partition way parallelization therefore linearly default com ex definition conjecture axiom assumption department science university west ranking observe close metric competitive extension feature across machine require rank learn literature perspective ranking namely ask machine logistic outlier capable observe requirement standard metric try evaluate discount cumulative ndcg metric therefore algorithm metric show ndcg loss maximize ndcg convexity robust easier suggest reliably converge though non optimization bfgs use large efficient latent collaborative retrieval unlike rank stochastic attractive characteristic interaction therefore scale serial popularity computing service amazon services collaborative million song record rank objective interaction amount art lift metric training attempt vector predict denote dot mistake call function mistake difficult solution machine optimize x easy optimize hinge upper outlier intuition large negative x solution decrease outlier expense transformation follow loss function much slow derivative rapidly loss behave classify ii therefore base function difficult optimize often successful intuitively reduce fraction rank example movie recommender movie subset document set adopt literature context aim learn important feature use write function see g top word motivate objective weighting factor xy reflect metric discuss sum upper bounding function minimize although objective user pay top desirable gain quality top intuition discount metric ranking achievable metric rewrite rewrite objective function monotonically increase bound logistic apply construct loss logistic enable model give discuss transformation difficult transformation generate type eq q twice minimize regularizer add norm conduct standard small follow rank source yahoo benchmark consist fold validation split provide parameter ndcg value test dataset report optimize implementation bfgs advanced framework minor title td xlabel ylabel ndcg header td txt index header td txt index td txt header td index header td minor title td xlabel ylabel ndcg x index header plot td txt x index header txt x txt td txt plots index txt index header td txt index td txt header txt truncation space td plot seem insensitive truncation ndcg hand perform truncation note ndcg consistently outperform inf ir list dataset list dataset td ndcg table complete period assign context item difficult pair collect nonetheless may realistic movie recommender movie somewhat relevant lot user netflix stream leave rating movie express preference consist feedback attempt without euclidean latent embedding otherwise score summation inside instead range avoid overfitte regularizer add frobenius norm become evaluation stochastic optimization pair still summation however stochastic nonetheless guarantee descent attack introduce iterative minimize code algorithm calculate fu fu easy stochastic calculating exist fu fu uv computation spend step linearization trick enable across space detail appendix subsection million song song go ndcg applicable evaluation scale machine across second cpu linearly processor line exhibit speed scale mark xlabel number legend legend pos south east mark header col comma header col sep header sep comma plot txt table index header col sep minor xlabel ylabel west mark option index header col comma header sep comma plot header false col comma txt false col sep txt col comma plot txt minor xlabel second ylabel legend north solid comma txt header false col comma comma plot table header col sep comma txt header col sep comma plot computation art optimize objective compare fast solution code structure librarie comparison identical grid figure machine competitive convergence clear consequently significantly objective basic optimize ndcg objective metric immediately relate construct upper bind improve convexity also nonetheless ndcg metric propose optimization collaborative explore attempt objective advantage differentiable therefore gradient algorithm bfgs convergence guarantee since natural linearization trick necessary guarantee discuss unclear insight binary scalable optimization task collaborative retrieval large care experimental learning collaborative retrieval experiment arguably towards derive parallel averaging gradient machine dataset dataset well therefore dataset principled appendix could constraint similar comparison provide run ndcg obtain library name ndcg c td td yahoo yahoo name td report ndcg medium song report precision htbp minor xlabel ylabel ndcg header txt index header plot td txt plots td txt td xlabel ylabel ndcg k x plot td txt x index header true td index table td header plot txt minor title yahoo learn xlabel ylabel legend pos south east index txt header txt index txt header txt scale yahoo xlabel ylabel ndcg legend pos south east header txt header txt index index header txt index minor title xlabel k ylabel ndcg pos south east table header txt txt index header txt header txt table index header true plot txt minor title xlabel ylabel ndcg legend pos east plot header txt table header txt index header txt header txt title xlabel ylabel ndcg table txt index header plot header txt index index header plot txt true minor title xlabel ylabel ndcg k header true plot index index header txt table txt header true txt xlabel k ylabel ndcg header table plot txt table header plot txt plot txt header plot title xlabel ylabel ndcg index header plot txt index index plot header txt index header plot index header inf ir title td xlabel ylabel ndcg table index index header plot td txt header plot td txt header td table index plot txt x header td index plot td index td index header plot txt header plot td txt title ylabel ndcg header td txt index td txt index td txt index header plot td txt plots td txt table true td txt table x header plot td table index header true txt minor title yahoo rank ylabel ndcg legend south east header true txt table index header txt index header true index txt true plot txt index index header plot txt txt index txt title yahoo xlabel ylabel ndcg pos
guarantee optimality converge exactly rank semidefinite alternate numerous suitably provably range dictionary prefer naturally architecture reference therein reduce present differ way consistently regularizer approximate generality us loss regularizer moreover perspective enable extend idea consideration broadly first must misclassification categorical say predict handle regularizers nonsmooth infinite generality consequence give algorithm variation minimization old pca minima organization first variation reader notation factor return heterogeneous technique abstract type dimensional algorithm row example column example boolean encode false consecutive encode categorical variable represent principled deal number principal warm variant seek approximation square solve k square square factor variable product use rewrite interpretation associate think use compressed original map onto objective na iy form xy reduce well obtain compact svd give orthonormal rewrite rank approximate diagonal well rank svd keep orthogonality u k x pca statement find svd value value familiar svd solution unique form pca analytical exist technique computer machine example iteration find record iteration converge mention method solve readily extension hold minimize hold fix guess na ij l ij condition unique objective iteration iterate stationary solution lie span vector column span vice versa orthogonal arithmetic iteration see appendix matter alternate minimization work minimization ij parallel ease exposition update compute gram outer compute streaming split entire matrix computation operation trivially worker add form cholesky factorization row row factorization worker require scale j entry find matrix ij entry regime miss simply exclude instead affected row strength regime relatively missing become rank surprising recover noisy certain incoherence minimize nuclear penalty fit rank regularize large nuclear norm encourage completion like predict customer consist rating customer vast majority customer product miss customer rate analytical alternate alternate quickly satisfy recovery value carefully hand sample alternate none use achieve minimization logarithmic reason plausible expect alternate minimization recover regularize pca admit interpretation course interpretation regularize feature vector use interpret think space example example row capture maximally might profile every example represent combination give coefficient th axis simple interpret representation intuition example cluster group might rather consist noisy miss column column represent combination might redundant row example matrix discover provide explanation summary building pca matrix generate normal noise sample posteriori explain recommendation simply map observation encode back good auto encoder bi linear impose low rank fit pca bottleneck interpret solution generalize critical ensure match error need get offset exactly whose extend pca arbitrary row form regularizers pca reduce pca regularizer express compactly matrix notation regularization separable infinite enforce example impose nonnegative depend regularizer satisfy vary choice regularizer wide regularizer turn pca convex bi fix regularize case minima however column parallel ix minimize I regularizers problem convex regularizer convex many analytical nonconvex see concrete subproblem program quadratic discuss factorization variation detail clarity strength course mix match regularizer different different indicator define indicator factorization np hard analytical specialized code replace easier interpret small understand penalty together wide variety could enforce entry column vector define denote nonzero regularizer pursuit branch bind reduce small relaxed convex regularize know component choose impose due matrix orthogonal conversely column mutually addition nonnegative eq let geometrically row ray pass nonnegative along orthogonal entry penalty norm matrix define equivalent require sometimes bound rank completion rank factor bound function encodes cluster assign minimization well know update update assign solve assign assignment indicator approximate subspace also interested datum think generalize assign centroid frame pca indicator nonzero solve block define correspond compute th subspace square sparse provable understand variation label predict procedure solution regression make sure model use supervise regularization regularize regularizer encourage column uninformative sense say feature regularizer feature dictionary design representation supervise learning dictionary linear atom represent atom fit solve pca pose notation usual dictionary regularizer one sparse nonnegative eq factorization go pca nonnegative make subtracting introduce allow offset r j trivial case offset included regularize slightly offset term modify regularizer extend regularization regularizer penalize first q form l j ij reduce problem bi alternate still speed subproblem discussion extremely problem alternate minimization direction algorithm allow generalize nearly detailed objective importance fit rank hard despite sensitive outlier replace least less problem interpret robust component large use rewrite huber huber place loss huber small result outlier previously matrix decomposition huber make ingredient observation huber log error regularize pca transformation replace huber obtain use similarly huber vice versa function loss th percentile may interested matrix less logarithmic may loss nice fit note least solution formulate exponential give family family pca kl correspond poisson model loss function property loss loss mean audio fractional error generalize recover loss limit recover kl divergence datum datum subtracting matrix preserve loss importance automatic scaling time loss nonnegative easy generalize column sample median deviation column form offset may trick encode offset regularization simply generalize table consider column draw discrete represent entry ordinal incur feature value give function ij variable regularizer convex bi separable minimization objective subproblem suppose observation low surprisingly accurately technical take regularization sum minimize minimization repeatedly name margin factorization also logistic measure quality loss fixing minimize label xlabel ylabel pos pos xlabel pos framework generalize low poisson family differ ordinal variable encode wish rank give ordinal hinge loss generalize ordinal ordinal loss loss encoding degree agreement question might fit ordinal increment bad agree bad neither agree ordinal learn ordinal determine agree neither agree suppose tuple denote interval miss sometimes map equal loss huber make sense abstract must still think boolean pca approximate boolean solution boolean say boolean boolean fill compute minimize lie domain losse huber type general boolean imply value use original datum lie interpretation pca interpretation think capture embed understand intuition may arbitrarily complex plot cluster generalize convert vector think non give agree say form value map representation back imputation think discover good benefit approximation latent good explanation generalize generate accord matrix random take entry maximum posteriori solve matrix encode auto encoder impose bottleneck auto fit bottleneck reduction giving maximize bottleneck efficiently store previous offset describe function note scale instead type much little adapt regularize alternate explain fully run alternate model identical normal compare square boolean truth round close show error run regularize show boolean pca draw entry hinge advantage boolean htb boolean htb boolean pca boolean consider boolean might customer medical disease negative unobserve positive rank draw let constant positive observation consist entry matrix draw random fit model ten observation usefulness predict top unseen figure path range generate see error reach improve increase since compute insensitive shrinkage introduce grey identify significantly particularly round auxiliary axis line leave width xlabel regularization ylabel legend north west view axis xlabel ylabel axis cs boolean censor grey line probability entry boolean column map thus correspond ordinal heterogeneous also fit produce fitting heterogeneous ground round close third regularize pca show boolean misclassification define notational convenience let truth draw truth heterogeneous mse heterogeneous pca mixed recovered block boolean removing describe truth block round entry close fourth show heterogeneous similarly perform censor compare difference ground ground truth table misclassification ordinal h miss datum missing regularize generalize represent abstract example permutation ranking loss embed loss approximation column formulate regularizer vector entry section suppose categorical q fixing optimize per separate label sometimes multiclass optimizing identify svms accurately predict categorical boolean miss entry project onto span simply interesting column lie interior column restrict use classifier categorical model corresponding function function imply acyclic one vs separable mean feature function fit boolean example combine rather perform well sophisticated perform nonconvex example minimization cluster rank see mode ordinal embed onto scale ordinal much similar infer label integer multi ordinal optimizing hyperplane separate level hyperplane fix optimize place appropriate hyperplane simple ordinal loss loss index generalize find permutation permutation pca example interpret deviation level strongly choose set ranking observe ranking modification literature correctly rank loss include area weight pairwise order allow scale loss american community survey fit heterogeneous population unite economic exclude year variable home boolean boolean house boolean water boolean health school currently school boolean high education high attain status categorical force boolean worker boolean year ordinal work week look categorical huber hinge hinge ordinal categorical categorical parameter offset select two minimize exclude group water california intuition hour work per week work per education similar california water hour work work education feature generalize rank computationally global optimum low rank matrix completion optimization implement per alternate notice variant likely saddle point explore alternate sense see discuss strategy provable sometimes show approximate show extend alternate generalize ny naturally algorithm loop may lot optimize iterative minimum sense early replace rule towards empirically local perform computational rule write fact switching role still execute parallel example might replace size rule globally subgradient value regularizer take say represent use proximal method indicator onto set globally optimal fix sufficiently small technical step update globally critical mild f proximal prox prox prox minimizer proximal gradient global minimum prox prox admm initialize objective see prox update make iterate admm mean subproblem alternate allow step fast progress towards convergence ensure size motivate grow row step choose objective decrease decrease reported size increase increase decrease check prevent poorly decrease adjust compute operator sum update per iteration job onto replace gradient among ij less implement prox objective exactly update take ease invert gram prox prox objective gram prox prox take advantageous many form example nonnegative square project alternate converge objective example explore serial alternate regularize entry draw initialization normal entry come trajectory convergence factorization zero plot objective optimal initialization trajectory converge fast somewhat substantially ylabel xlabel coordinate coordinate coordinate alternate regularize pca htb ylabel xlabel coordinate coordinate convergence proximal alternate converge converge differ significantly initialization problem discuss scheme extensive provide provably completion algorithm minimization show value choose carefully initialization consist svd datum give construct motivated key insight zero construct take expand categorical column boolean interpret scale propose insensitive sensitive ordinal make encoding initialization mean whose version change column decrease truncate row th instead triple compute triple time compare initialization initialization census detail six five initialization iid converge substantially behaviour indicate random view width xlabel ylabel coordinate coordinate initialization svd mean initialization scheme initial centroid centroid choose minimum previously choose centroid quadratic even alternate expectation randomization initialization initialization center arbitrarily bad proportional loss distance metric important non convexity argument one find factor nonconvex regularize efficiently sdp rely subroutine lagrangian problem rather saddle nonconvex consider global present solution compare z following constrain factored svd u xy x use inequality rely third taking orthogonal frobenius note orthonormal still solve solve optimality globally value least problem solve subgradient subgradient nuclear equivalently function nuclear rank result particular globally objective pick globally suppose entire frequently regularization achieve initially fitting value norm fit correspond call regularize huber draw normal entry draw uniformly huber quadratic normalize error eq fit decrease reach interestingly second path view xlabel ylabel normalize error regularization path needs specify regularizer rank domain expert intuitive fit hand regularizer choose consideration model generalize unseen consideration balance discussion regularizer table compression pick give ratio low observe error well high low error rate achieve require analogy aic degree freedom rank compute example regularizer transformation number degree propose dimensionality observe perform cross observation small entry true contaminate may choose l distinguish method dropout linearly eigenvalue increase decrease parallel objective compare fitting model draw cross simple apply denoise leave entry leave present understand explore wish similarly loss consider fitting miss neither discuss resample validation cross indeed distinguish rise chosen row case index make sense reasonable rise scheme validation resample row matrix far resample bootstrap residual number resample generate pca reference therein example explore vary draw draw outli draw pick huber result draw perform validate qualitatively interestingly difficult rank xlabel
theoretical precisely rate worker recent line generalize allow subroutine coordinate optimizer result different update moreover variant entirely trade choose optimizer internal procedure duality fair discuss empirically loss convex possibly regularization class ordinal problem become iteration access understand associate conjugate define dual conjugate come convenient dual primal optimality configuration duality serve useful criterion form prove primal sdca make completely software package prove suitable primal superior defined stop loss problem worker dual efficiently merge worker disjoint h distribute k k kk arbitrary apply local internal try observation worker state compactly represent single ever machine dual initialize sdca single h subroutine local round dramatically worker outer suggest dual coordinate sdca optimizer practice partition across worker write na rescale locally worker iteration determine trade block optimum geometric alone scale block coordinate outer iteration machine let dual objective q satisfy eq eq show tight special interestingly subproblem optimality let serial main g recent generalize distribute practical internal descent dual coordinate inner notion sub address without local round communication result solution rate full loss experimental encouraging yet provide quantitative gain communication efficiency compare analogous experiment section first parallel subproblem solve essentially rate journal quantify show local improve potentially interpret include update mini datum example iteration mini update add worker study naive variant essentially identical mini descent difference define sub suffer immediately furthermore average instability illustrate coordinate sdca difficult choose especially parameter range many convergence safe convergence method linearly per worker practical order small mini one consider local author compute communication matter accurately subproblem machine sometimes communication online divide inner perform outer early framework update delay insight study sum directly case spirit implement communication process coordinate descent analogous time communication machine require communication experiment large convergence strong sparsity understand communication round classifier mini ascent stochastic locally mini batch sdca mini mini batch update iteration update size mini batch sgd machine scale tune dependent hinge machine implementation amazon ec instance though smooth analysis remarkable converge regularization r imagenet compare analyze progress primal compete size performance locally size process batch batch mini batch sgd averaging become quickly account correlation spend process number indicate accurate ability avoid still computation communication node describe communication also affect convergence attempt scale small mini communication framework dual ascent solve minimization distribute machine communication amongst costly show real world dataset rate update safe remain rate efficient describe discussion institute give block everywhere outer worker procedure local improvement function following rate outer concavity subtract side maximizer consider ready procedure choose improvement inner optimizer core primal structure coordinate ascent change restrict notation index worker us valid reader status active block idea constant observe duality th dual formulation local dual k optimality derivative inner plug back minimization write precisely dual coordinate duality gap contribution active collect iteration identical keep subsection h I k strong convexity hence q primal eq prop function smooth q local subproblem
algorithm concave fairly weight right center expect flat interval concave symmetric consider measurement north water describe capability low loss previously study amongst place sequential normal posterior related figure old left bin log flat modal density method center think component flat measurement log component flat symmetric concave next population study di interested identify isolated population population genetic homogeneity uniformity style valuable diabetes disease simplification isolate population gene genetic homogeneity individual relative environmental isolate necessarily large estimating model component alone concave component maximum plot relate plot around estimate multi component approach capture near include individual use plot center label center value need especially near center density shape make log component semi location unknown motivation choose totally bandwidth kernel kernel consume offer section enjoy easily active implement likelihood know mode enable put concave estimator concave mode constrain location mixture probability location estimate parametric find estimate log concavity would find maximum component impose challenge derive convergence involve expression unfortunately likelihood sophisticated overcome difficulty conjecture remain alternative validity introduction may tail appropriate concavity however development concave density barrier concavity heavy tail concavity sensible result present numerical keep reasonable introduction location identifiable identifiable symmetry certainly impose concave natural log concave argue non class hard price pay necessarily piecewise flat transform consider function piecewise clearly admit e logarithm mle necessarily piecewise flat maximizer exist vector n order u u imply u u put respective put x decrease big satisfied integer let small j j increase reasoning conclusion dt last contradict definition perturbation x x z f dx nx nx become straight line enough z nx nc pc denote fix convex simplify notation respectively g fact continuity imply converge nc u g nc g nx gx nc pc nx gx p denote g consider g imply complete proof b function decrease large finite show apply lebesgue dominate need support enough take infinity integrable integrable since condition large integrable respective mode component respectively restriction easily modify h c function monotone dx dx admit function decrease fix probability monotone generality increase bound case handle monotonic ga gx gb class fx nx hx nx many proof go proposition first restriction respective c x respectively h dx c dx dx admit fix symmetry decrease reasoning side large convex sequel k c find write event n u n q hence enough decrease include occurrence imply turn argument event nx u occur great event e occur great depend g x dx g dx use fact concavity logarithm log put g pm c proposition last subsequence extract subsequence integrable f inversion inequality turn dt indeed dt dt f inequalities dt increase hence extract subsequence weakly assertion proposition surely follow arbitrarily net second hellinger u u j h j u fact conclude complete k n u mle page section criterion claim concavity logarithm n g associate nr n gx gx n nr obtain fulfil density choose maximization take approximate probability let proposition consider j n occurrence n n pg r pg n enough hence hence put r note turn imply triangular rate normalize write pn proof triangle n p pn g pn inequality decrease symmetry f f inequality property dt u increase hand precede display handle give expression switch role take switch switch role ratio stay stress fact ratio remain configuration handle ft dt rate dt u u f dt dt pn assumption ft ft dt ft ft ft ft dt ft ft dt ft ft dt ft ft ft dt ft dt ft dt ft dt ft dt ft reasoning definition dt dt h ts use author calculation j j definition mixture concave nonparametric log concave mixed shift location establish converge distance location probability unknown package mode article symmetric concave mixing nonparametric mle hellinger location mix establish mle truth shift unknown use package supplement technical material concavity number theorem statement unified document independent identically cdf q common modeling part framework problem reason popularity flexible major difficulty work multiple avoid thereby leave question another add restrict model restrict symmetric focus largely many application restriction identifiability make mixture although identifiability detail main parametric parametric asymptotically however property build progress focus take constrained modeling concavity gain major require contrast nonparametric make smoothness successful inference tuning parameter statistically different choice mle tune difficult verify practitioner believe unimodal distinct assumption appropriate practitioner choose poorly specify parametric log concavity often surrogate log density unimodal log additionally familiar family approach unified approach handle mixture benefit concave estimator density concave turn distribution converge mean log converge log heavy tailed admit smoothness parameter impossible concavity concavity base concavity assumption effective require modification concave estimate component concavity consistently log concave sample global mle behavior estimator includes derive use construct furthermore programming assign symmetric concave log transformation log density log mode mode active set proportional probability em able mle almost symmetric hellinger supremum true provide mild assumption although use true fast kde concavity modeling mixture concave impose perhaps fundamental pure feed implementation maximize component likelihood difficulty already primary property rather make presentation add detail establish hellinger form however deal mainly bound maximum likelihood likelihood symmetric special compactly support stem exclude convergence concave smoothed version first section absence mode critical bootstrappe concave ratio trace log concavity end include cluster performance log concave algorithm assign corresponding great application conclusion proof base minimization distribution estimation symmetry parameter zero kde give mixture u consistency rate density writing fourier estimator mix q support maximize overview cumulative cdf explain follow dirac probability let imply location symmetric x minimize empirical simple equation system drawback three aforementioned consistency rate assumption show regularity condition obtain smoothness rate convergence estimator supremum location kernel estimate smoothness choose optimally specifically notation I nx x observation nb b symmetric concave nonnegative concave concave hellinger distance dx integrable g fx gx measurable come location subscript commonly fix symmetric mixture maximize class penalty proposition estimator estimator admit maximizer ready symmetric concave mixed density f pn pn three mixture consistent encounter estimator adopt approach pose package latter author david private communication description estimator u theorem identifiable adopt write associate rewrite identifiability equivalent origin u u minimize distance origin jt thus estimator mixture sm code indicate function use generate number minimize maximum n n decrease initialize start center variance symmetric n case moderate replace simply value mode z z transform concave maximizer z z I maximize stop smaller choose equal need method density simulation follow ny ny k independent u u knot dx nx dx x ny j ny ny j ny ny ny ny ny dx calculations n ny ny ny ny ny j smoothed concave mle choose smooth mle always concave occur mle characterization univariate extend explain bandwidth pass yield j dt u absence component definition mixture symmetric hypothesis exclude concave mix ratio statistic nan around denote statistic hypothesis reject use bootstrap concave detail usual statistic quantile aim nan underlie concave log concave density identifiable around mode possible density expect differently indeed assess standard beta give nan hypothesis ignore powerful detect tr whereas tr easier distinguish lr power especially less scenario trace anti conservative behave level differ size unclear cause normal notice log concave tr type lr nearly laplace concave difficult regardless counter distinguish follow g leibler divergence concave density nan follow leibler exist tend
previously categorical majority patient concept categorical dataset work identify characteristic particular time look yet appear history broad presentation component source manually massive chart imagine present raw infer factor rich result plan acknowledgement work grant foundation national health lm clinical support grant asynchronous nature medical away categorical modeling longitudinal function several infer event intractable duration paper infer interpolation twice accurate method regular finally demonstrate abstraction amenable algorithm medical pattern problematic record categorical get record contact activity laboratory visit stay event would like thing clinical contact disease often apply away stream stream stream abstract longitudinal intensity contact density process intensity raw standard previously unfortunately infer applicable categorical intensity monte intensity event approach find flexibility scalability datum event stream unable adapt flexibility good property datum continuous clinical process event assume independent iid drop iid add longitudinal intensity event model work gamma time q homogeneous interval think event time space draw well simple poisson specifically process produce clinical behave highly regular intensity ft square smoothness covariance generally must inference assume homogeneous similarly ft place uninformative prior set desirable avoid degenerate intensity parameter slice surrogate compute incomplete interval end challenge direct require certain log gaussian integral numerically smoothness efficiency bottleneck compute need instead time driving scale intensity interval work nearly medical cubic period fine resolution year inefficient bin poisson infer intensity neither intend suit form abstraction raw discrete hazard efficient intensity span stream intensity big medical range several flexibility exist refer integration avoid computing intensity shape function stream clinical event parametric function piecewise event intensity interpretable raw intensity compare kernel smoothing method assume poisson compare assume test intensity burn inference intensity shape efficient magnitude rt accurate true might sensitive available prior mode near tune slightly mode follow advantage next synthetic generate medical amenable event intensity gamma consistently accurate htb patient record infer marker aid clarity division clarity confidence lastly direct sequence represent clinical code five great mirror medical record arrange stream disease broadly level stream event include code division event strictly still group event informative intensity figure
sample value base reject acceptance construction enough sample large augmentation begin increasingly rely become convergence stationary figure I ambiguity branch barrier energy convergence initialize period check converge stationary converge check leaf grow monotonically internal node barrier monotonically barrier update initialization match match node penalty omit definition important figure decrease mixture two influence separability supervision ii behavior performance maximization wang synthetic mixture derivative landscape partial derivative computation restrict inverse gradient definite equal need restriction possible step gmm I matrix symmetric project space symmetric decompose ensure point possible range component upper unbounded sample gmm run walk location stay boundary gmm separability separability gmm overlap often measure difficulty true show minima landscape separability prominent landscape supervision assign ground truth label portion label separability become much label minima labeling decrease landscape supervise lm behavior separability expectation popular pattern separability condition wang cut generalize probability em step energy approximate learning include comparison synthetic run time energy vary em gmm landscape local minima energy em cut sample low separability good converge random almost always find outperform separability majority local high confirm showing inductive bias separability energy observable gmm distribution energy gmm encourage low gmm easy separability run repository gmm represent linearly separable remain linearly visualize minima minima energy first merge local minima overlap rd nd rd run percent label assign landscape figure scatter pattern illustrate template faces lattice denote typically sketch image template template template simplicity assume template energy boolean experiment propose em template template experiment take value use template represent face face align grid cell cell contain cell location sketch straight connect cell possible detect edge binary generate minima landscape heat map expect minima increase particular noise landscape convex local experiment face swap face head thereby template degree overlap generate various template degree minima overlap template face extract prominent eight filter different corner filter strong fixed correspond element dimensional response face cat equal version image model template template minima corresponding identifiable minima face across explain energy face mixture run close count minima display bar minimum face bi process bioinformatics e use find movie occur phrase bi multiplicative share element see graph mixture component conjunction co theoretical co show observation bi bi matrix bi matrix bi bi identify bi goal bi explain therefore instead adapt element row bi term coherence minimal proportional correspond prior bi add exclude entirely zero bi element varied random background point construct overlap maxima cluster bi mark red circle bi mark gray maximal marked green regime learnable bi minima regime dominate biased bi weak energy level true although approximately difficulty difficulty difficulty visualize choose landscape cluster convex energy problem separability level supervision level strength landscape algorithm worth explore repeatedly adjacent branch gradually structure landscape low supervision previous next conjunction dimension study challenge thank suggestion wu discussion acknowledge project statistical highly non difficult analyze paper leaf barrier adjacent mass volume correspond construct adopt wang dynamically classic mixture template ii study landscape separability cluster supervision visualize behavior k em step wang cut optimize research replace regression designing algorithm local property inspire spin model problem bi figure barrier adjacent characterize landscape energy intrinsic problem either task bi hard impossible regime complexity separability percent algorithm show frequency various minima find highly separable work well less separability htbp htbp illustrative component denote construct range visualize keep map landscape cause component like little finite show identify mcmc sample cluster minima leave minima landscape b energy work multidimensional efficiency notably generalize wang effective barrier bayesian segmentation belong fall example upper collect construction kb adjacent collect consecutive across eq next iterate barrier barrier ridge descent structure energy modify agglomerative initially leaf coordinate minima new parent whose energy barrier regard barrier merge energy merge complete structure clarity remove less energy
diagram bottleneck distance range individual yield wasserstein et linear speaking establish old wasserstein distance map apply operating string persistence induce kx kx yx lipschitz continuity particularly exist hyperplane perturb separate persistence diagram define half plane motivate persistence diagram possess persistence uniquely dirac delta dirac functional hilbert adopt unfortunately induce account perturbation diagram motivate dirac heat diffusion dirichlet condition diagonal equation persistence scale formula wasserstein differential sense rigorous persistence diagram persistence empty diagram linearity map partial differential equation replace diagonal restrict original initial closed simple q derivation visualization solution wasserstein wasserstein denote persistence diagram diagram persistence diagram assume achieve wasserstein leave decrease adjust accordingly influence cause beneficial question extend call diagram additive choosing say sharp stable persistence diagram non sec investigate conceptual difference sec texture sec persistence diagram banach intend computation space hilbert structure kernel induce wasserstein persistence diagram eq persistence range think let diagram point point move diagonal increase consequently unbounded mean persistence value response compute persistence diagram detail persistence diagram handle margin svm implement ten fold validation average ten validation split distribution result choice explain smoothness scale allow stage extent capability rely suitably choice carefully adjust undesirable consume even additional c performance synthetic shape evaluation allow assess kernel induce distance list measure query shape neighbor scale list similar classification specific time input induce another unstable top drop retrieval list synthetic rank top five entry topological persistence alone assess performance improve elaborate fusion c top training informative train normalize histogram response table evident perform margin gain apparent topological alone histogram persistence persistence conventional representation lead nature topological feature combination lead gain include assess illustrate curve selection cross validation discretization drawback shape could operator strategy would change contrast adjust c texture svm validate theoretically exhibit task texture tune parameter prove practice future would address scale include leverage error summation wasserstein useful persistence wasserstein persistence diagram lead topological topological machine area topological ne set nest nest offer rich source vision theoretically kernel pca establish design persistence diagram summary topological wasserstein texture compare persistence visual surface analyze homology roughly persistent homology capture birth death enable theoretically sound representation kernel computer vision etc processing task appearance descriptor sift high activation convolutional feed popular vector technique progress extract discriminative recently start become readily demonstrate capture characteristic often along study persistent homology birth death etc homology input lead change wasserstein persistence topological surprising persistence diagram wasserstein metric employ persistent homology obstacle typically hilbert space possible persistence diagram wasserstein main propose positive diagram fig feature idea theory map wasserstein thereby maintain stability persistent homology robustness applicability shape texture benchmark method topological vision medical group first identify utilize topological specific identify information topological input representative adapt surface shape drive topological persistence diagram persistence cycle extract segmentation chen propose topological segmentation category investigate surface contrast persistence directly feed discriminant control patient disease topological typically instance diagram regular grid eventually density kernel unclear induce bottleneck wasserstein directly recently bottleneck wasserstein distance employ correspond complementary persistence combine bag inspire focus development persistence propose diagram motivate algebraic geometry algebraic function death persistence conceptual probably close another diagram persistence design machine sec show admit persistence attractive alternative bottleneck matching review fundamental notion result persistent homology relevant topological grow sequence shape growth shape gap etc give rise point formally persistence diagram detail every diagram
proceed precisely eq old chain bind equality hence equality claim attention fix imply qp qp eq union one subspace bind first valid let put thing proof make small step shorthand whose z generality finish attention sufficiently q q observe combine noting conclude observe prove converse follow prove finish pick q hence arbitrarily characterize regularization uncertainty correspond parameter magnitude let except scenario easy compact non interior analyze another regularization variety rely decomposition leave singular natural question claim find equality inequality strict bound gap proposition consider subject generalized know similarly eq general regularization strict low proof proposition summarize function equivalence always qp rp proposition equivalence g throughout row connection regularization sort residual quantile despite hard property possible provable hour advance robust formulation structure nominal formulation recall common formulation remainder separability norm summarize p ng g c norm norm separability maximization form summarize particular norm likewise part complete easy duality therefore q observe recognize convex conjugate assumption conclude implication norm norm proposition operate analysis begin state main theorem choice exactly reformulate mixed choice norm linear integer program formulation mix order nominal use formulation nominal uncertainty norm satisfie reformulate program formulation formulation bit care solution maximization unique q direct tucker condition equivalently write reformulate programming eq reformulate exactly formulation substantial problem core development involve variable prominent completion principal pca common nominal uncertainty expand exist regression novel substantial class introduce uncertainty completion e interest constrain appear wide netflix preference give important parsimonious description user preference term convert nuclear convex uncertain f frobenius pca arise low plus well truncate popular application robust low rank entry pca form penalty rank spirit sense recovery small sufficiently explicit expression certain uncertainty additive uncertainty concern model uncertainty different assume uncertainty measurement ny ij entry matrix note nominal subject linear ij ij form direct analogy clarity interpret albeit cost avoid vector uncertainty induce induce many choice interpret hence depth truly directly continue theorem model uncertainty uncertainty note result exactly therefore remainder q restrict induced uncertainty begin therefore without note ng h ng case conclude theorem subsection follow implication matrix frobenius corollary recover completion note shown arise directly sparsity nuclear envelope ball nuclear directly may arguably induce detail penalty nuclear appeal convexity attention begin note f eq continue present linear observation combine imply uncertain pca entirely model usual replace surrogate respectively regularize uncertain pca appearing impose penalty summarize always regularization completeness regression state precisely proposition n qp qp bind strict long gap low attention non arise model plausible one column examine j matrix multiplication completion netflix treat true rating address within allow user rely early equivalence vector regression characterize modification theorem proof q jj p uncertainty possible regression variety long conclusion section equivalence q equivalence problem modern take directly understanding least quantile regression completion emphasis modern massive scale statistical modern work broadly rely estimation engineering science inherently desirable effective inform corollary conjecture office science engineering fellowship penalty beyond contrast appear plus regularization method reliably solution paper extend loss show regularization completion modern lead one especially nuclear minimization compress practical scalability due advance variety statistical linear early goal median matrix contribution include uncertainty consistent provably appropriate known restrict nominal appropriate method perform median demonstrate problem regularize integer principal characterize uncertainty regularization paper section background focus equivalence turn regression matrix variable consider depth conclude regression homogeneity dual transpose norm define analogously eq denote norm also frobenius spectral induce definition reference norm norm entry analogous spectral norm value eq contain singular true penalty precise summarize p n imply without replace continue another fix eq q q replace completeness eq subsection comment learn relevant domain
mechanism relation law many physical follow clear experimentally measure provide description behavior understand interaction material study phenomena e behavioral single hope nature implement successful search important regard automate typically law science find law demonstrate truly automate match study dynamic understand key underlying mechanism law law unknown grain description dynamic mapping interaction typically system one specify hard simple dynamic sir optimal proceed important center study national laboratory support grant laboratory program grant supplementary ref hierarchy maximum fluctuation never take away arbitrarily sufficiently search multidimensional model single predefine path suboptimal true fall even guarantee eventually dynamical done power system dynamical govern ordinary differential production degradation call rewrite variable correct e taylor integer value predefine space find model biological example outperform typical production degradation grow bound force selection discard negative network among biological often system show approximate variation rescale switch linear combination traditional advantage system existence hierarchy simply interaction consist add hide without connection specify dynamic variable ii ji g ii example form law x dt I dt h dt x x dt dt x h dt g g seven fix connection parameter vary time add dt x dt dt x dt x dt dt I full prior normal standard deviation motion mass evolve specific angular momentum velocity vector connect initial measuring distance unit rescale equation system represent exactly system result never construct adaptive input cover circular elliptical determine time fig ensure would hence requirement reliably motion mean initial time observation deviation equal maximum value typical adaptive fitting hide certain transformation leave initial parameter rescale loss second shift parameter perfect dynamic compare performance fit mechanic result network still generalize range contain range fit true limited multi site treat input measure single treat total dotted circle error circle vary motion include correspond know likely well show datum contain leave dark blue part divergence correspond right half blue part trajectory explore complicated biological relatively imagine five site arrange rate site affect state nearest neighboring site reaction specify total easily site log min measure site uniformly minute add noise min compare evenly space guess functional total hoc quite well amount well dynamic site example system typical performance fig class qualitatively correct close interval notation section metropolis carlo multidimensional hessian fit ratios monte step remove avoid condition every inference complicate originally automate dynamic seven molecular parameter value ref ref mm min min mm mm define solid line circle interaction right fit measurement circle indicate strength clarity self show c ic sample ic mm initial noise visible specie ref range range reference set cycle ref visible specie choose range list ic table visible minute mean measurement table input uniformly sample ic evenly actual separately visible set condition plot assume aside measure assumption relax assumption variation learn ref engine infer roughly detail attempt specie specie time costly integration match remain unconstrained ref fitting constrain dynamic entire space produce main range ref ability tested compare sample initial range note appear see infer sample twice range plot choose range narrow range plot standard symbol derivation set measurement aside prior constant normally error goodness square residual normalization constant pm p since fit ref sufficiently constrain limit dominate near approximated derivative jacobian measure generalization term constitute overfitte goodness penalty individual selection know general section residual parameter integrate parameter use ensemble routine parameter sequence fit number consistently decrease stop small calculate metropolis monte encourage acceptance ratio integration treat starting else default candidate space isotropic residual hessian previously less ensemble start member perform stop detect norm step member ensemble fit temperature temperature model full model member full conservative achieve use multi example fast evaluation search model calculation evaluation gradually depict size case evaluation scale model plot compare parameter number infer plot direction constrain prior expect stay realization dynamic drive network molecular understanding limited extreme hoc define propose construct coarse grain dynamic automatically adapt adaptive insufficient computationally dynamical variable software realization sir unobserve motion overfitte biological produce half interact specie unobserve mail edu biology field complicate vast amount demonstrate encounter physical datum cat success generalize insight resource sometimes impossible large yet unobserved molecular structural involve intensive easily challenge unlikely solely accurately dynamic bre predict response system perturbation disease agent system time early day natural engineering result successful continuous dynamic dynamic recurrent nonlinear common biology unnecessary approach possible effort especially couple move note complex parameter structural guarantee perturbation hope accuracy coarse grain propose adaptive attempt well interpretable possible account restrict hierarchy complete gain theoretical meaning able adaptively able happen search believe infer complex dynamic search space polynomially computational resource construct interpretable much effort experimental variable unobserve sir input intrinsic dynamic random system repeat condition series save field trajectory g elliptical familiar statistical representation form create gradually complexity fit sufficient ideally much one hierarchy complex desire material som criterion dynamical along two add variable study within two match time som degradation form power mass law class interaction similar reaction represent hide perform bad input show deviation adapt red fit second rely intuition manually parameterization salient see som exponential third create fig approach varied fit fitting bad prediction less subtle accommodate subtle axis b time model blue overfitte red prediction median dark line median behavior sample parameter som confidence site demonstrate sir rather true prediction qualitatively different infer reasonable overfitte evident detailed average complicated interest system inform knowledge pathway consist specie perturbation make ideal sir circle measurement observable
conventional cloud processing quantization theory interest detection operate clutter detection interference optimal pearson np sense np theoretic criterion kullback leibler design set perform receive place environment sensor fc capacity channel cope limitation communication sensor receive transmission fc operate receive refer system cloud cr mr tackle problem jointly optimize code operation receive adopt quantization fc base investigate seem optimization code quantization bold bold letter determinant random column denote symmetric mr system clutter receive fc limited capacity present accommodate capacity letter available communication fc receive capture channel share form code large control clutter sensor type return envelope fix interval target envelope observe clutter homogeneous receive match q respectively represent absence target resolution cell useful part receive target clutter clutter complex amplitude accounting interference distribute respectively fc e receiver fc receiver know whether target facilitate standard effect quantization quantization signal receive th quantization sake quantization communication fc form receive additive sensor n c alarm aim code quantization end sake resort theoretic detection capacity requirement therein adopt receive leverage zero n make explicit assumption approximate argue distortion discuss requirement mean theory operate asymptotically hypothesis evaluate adopt measure bit fc dependence mutual maximize metric vector covariance capacity formulate minimization distance n convention exceed ensure total transmission receive theoretic difficult solve obtain global locally accordingly th code iteration feasible mm repeat step still require resort mean purpose converge convex convex locally bound iterate convex function feasibility iterate local optimum emphasize use identify outer loop index discuss application goal end mm locally b h easily x x lead attain step matrix desire evaluate follow convex iteratively find equality vector use algorithm optimize jointly table throughout receive length code variance clutter n k fig gain properly quantization beneficial plot operate characteristic alarm versus probability bit evaluated implement detector remarkable gain
key gradient simulation adaptive sequentially mean noise signal model element associate canonical adaptive subspace dictionary update typically fix instantaneous instant coefficient cost coefficient ed n nd rkhs respect optimize optimize rkhs subspace short optimize filter span singleton thus derive follow gradient induce respectively inner element linearly definition observe j correspondence note correspondence follow q give ascent direction subspace plane gradient j n natural make impact selective serious selective work theoretical mse stability kernel multiply side nj rewrite regard descent q cross modify respectively weight guarantee make recursive integer natural spectral less say square tend state regardless initial condition complete steady positive verify tend mse mean mean stability remark simulate learn conduct generate additive parameter mse selective update parameter theory steady mse selective selective mse depict steady dot respectively simulate obtain average mse estimate steady computed theorem input correlate experiment consider dictionary selective generate dictionary element coherence criterion experiment depict validity multiplication natural selective see simply mean hence drastically mse selective exhibit mse drastically low stochastic descent method steady meet outcome serve basis present behavior analysis give ascent direction provide mean square include stability normalize initially chen et validate reproduce attractive rkh adaptive filtering classify perform rkhs ii normalize example steady square mse present rkhs counterpart natural natural distinguish primitive gradient descent theoretical eventually relationship two adaptive orientation give short rkh
accurate accurate term leave give value occur binomial square write distribute doubly variable draw must binomial network small activity unlike good nonlinearity though attempt reduce walk show vector give produce unbiased predict compute start well generate blue red variance walk predict numerical equation show nonlinearity average variance nonlinearity provide value due growth use vector show solid dash provide practice right panel scale propagation randomness world far initial final need adjust separately affect output layer summary walk scale generally tune entire far important training deep mnist equation mnist multiclass mnist auto frame depth generalization bias width actual value value possible deeply also narrow encoder layer middle pick increase total number first lead integral layer layer per compare depth varied ensure deep varied essence function mnist result agreement figure good agreement analytic calculation middle panel goal second assess error focus nonlinearity believe utility scale value tie place train auto encoder achieve demonstrate scheme mostly layer classify mnist mistake random walk initialization figure result mnist good training among test tie depth improve task examine nevertheless deep real world classification mnist auto hyper mnist show epoch hyper deep network curvature average discussion imply correctly network layer one simply decrease derive equation correct avoid regularization reason walk initialization different deep bias initialize always allow modify bias care bias quickly happen careful initialization forward progress schedule huge landscape layer optimization indeed deal improve nevertheless allow network go broken hyper zero extremely reconstruction depth whether difficult task use deep feedforward way apply regardless deep feedforward network initialization accord value sensible initialization acknowledgment thank le york ny usa deep learn difficulty decay feed forward difficult previously unlike recurrent amount matrix scale initial result vector compute make walk square vanish gradient optimize relate mnist since early neural suffer vanishing refer fact feedforward back increase final recurrent rnn back error exponentially vanish add layer rnn network vanish recurrent recurrent network back involve matrix repeatedly process lead produce lead vanishing achieve lost process go rnn suffer vanish gradient magnitude scale square network feedforward apply initial network train also address vanish gradient mathematical training analyze successive analytical hold propagation equation procedure norm feedforward activation transformation bias depth wise nonlinearity normalize scale network layer length far initially otherwise initialize define input assume back propagation eq evolution square back become apparent entire magnitude across network layer magnitude gradient keep order appropriately adjust adjustment experimentally gradient variable random variable product approximately log case fail mean interested avoid issue instead equation walk walk unbiased equivalently describe logarithm norm back back output like vanish independent happen back rather vector
definite attribute numerically unique transpose subsequently negligible loss numerical kernel normalize value call traditional inductive overfitting ec post ranking make symmetric average transpose algorithm compare ground apply commonly retrieval ranking return argument negative roc order class accuracy reason firstly unlike ec hierarchy account determine different ranking ranking rank count comparison optimize software use characterize important traditional kernel traditional ranking know evaluate bipartite ranking e ranking relevant area roc auc discount gain ranking convert bipartite ranking retrieve relevant ec section truth compute datum set validation fold use individually allow comparison query remain ranking average fold fold three every fold cross validation neither database demonstrate loop implement control contain power final fold train cb fp ndcg map ndcg ii cb ndcg supervise ndcg give global summary obtain rank difference cavity despite considerably hard fact certain site active site expert active site choose cavity mistake functional similarity hard cavity rank supervise cavity cause accord cb data though relatively well cb clear explain easily cavity representation information moreover maximum subgraph node loss resolution drawback though solve size subgraph bind site lead slight hand transform moreover treat every equally emphasize rank approximation optimize auc coincide latter make distinction ec use severe performance measure set auc close theoretical cavity similarity case curve apply cut digit ec scalar roc information ranking immediately supervise rank unsupervised rank former left corner show certain relevant high sensitivity specificity detect without mistake step indicate offset curve fp roc section slope hard detect unsupervised straight detection concave ranking score ndcg auc ndcg explain map I ec cavity require perform figure quality measure low ndcg bad top depend application might interest since bioinformatic vast protein reliable geometry fold regard alignment template also biological close return program despite powerful become nevertheless inefficient similarity score focus protein protein account protein protein network try neighbor optimization connect protein share graphical markov field conceptually might prediction base cavity site ec annotation substantially rank take truth ec contrast rely heavily similarity annotation account focus cavity sequence work meaningful similarity could demonstrate indicate highly nevertheless supervise unsupervised cavity similarity match query ec well preserve ranking supervise powerful alternative retrieval traditionally bioinformatics acknowledgement ms acknowledge university bioinformatics financial foundation cm structure biological science database usually look surface biological annotate ranking kind improve approach similarity active function approach annotate training outperform measure ec hierarchy annotate training experiment consistent improvement similarity measure surface modern throughput molecular biology generating ever annotate remain challenge hard despite automated decade service often tool rely notion vast measure calculation abstraction solely take fold bind measure able condition prediction protein sequence alignment reliable sequence comparable exhibit sequence reason structure become database protein gain increase attention secondary know biological contain miss calculation fold protein coarse often responsible show diversity show functional many local structural evolutionary appropriate surface know valuable information similarity help bind similarity particular drug discovery provide family allow protein highlight bind site approach relate substantially improve training build mathematical ranking construct demonstrate cavity machine often learn popularity bioinformatics discovery protein solely rely ranking without utilize annotated search annotate database make amount transition four cavity base base input algorithm improvement use learn ranking ec hierarchy commonly ec adopt detail importantly focus chemical homology ec number truth subsequently truth ranking annotate fair comparison traditional approach way evaluating characterize engine algorithm ec number work unable output nonetheless instead predict ec query ec provide generally ec encounter build automated detection storage bank chemical first predefine spatial characterize geometric particular spatial bind functional group protein structure rule mix pi interaction property regard compressed surface protein encounter consequently representation distribution chemical ec experience expert choose ec generate I retrieve protein ec set protein ensure unique set protein server protein pairwise homology filter result cardinality extract protein assumption large bind site contain center take bind site maximize ray protein remove result drawback bind centre determine bind site protein sufficient resolution select low rely expert ec datum set resolution structure condition eliminate ec accept ec ec ec ec ec ec ec ec ec ec specific ec ec ec ec ec ec ec ec ec ec ec introduction restrict measure multiple transform unfortunately boolean flexible paper specify way greedy programming unfortunately quite kernel gain lot realization less protein poor representative category measure spatial denote transform remarkably calculate superposition derive alignment apply geometric hashing several point unfortunately cope biological also represent cavity feature geometry cavity see subsequently protein experiment representative base also protein five different explain processing label directly transform another spatially specifically approximate superposition structure fix cloud second cloud well cloud match point cloud small fitness maximize fitness taken label suggest similarity representation label cloud transform label become chemical geometry consider adjacent measuring common subgraph define relative graph define recommend make consider large subgraph large common subgraph use determine means protein post rule derive use protein use comparison bind site transform protein bind site moreover feature namely label weighted label label perform edge graph represent protein bin pattern mean protein bind site alignment sequence sequence subsequently perform introduction explain service construct rank similarity annotate annotate database similarity
allow give known support know solve obviously equal require allow little tolerance distinguish indeed vs show version conjunction poisson binomial albeit moreover recent applicable vs interesting seem optimal evidence binomial observe identity single latter try solve problem whether answer allow subtle identity turn make big hope distribution enough identity identity proceed first learn constant decide restricted support identity tight support overall complexity look reduce testing test feasible binomial sample consider translate poisson translate poisson get heart perform estimate indeed possibility distinguish logarithmic total variation reason argue use solve boost repeat majority basic logarithm interval distribution place mass bernoulli distribution pi variation allow unknown binomial use output binomial deviation within standard deviation run author poisson estimate mean variance sample poisson binomial indicator chernoff sum indicator bernoulli variable poisson random obtain chernoff poisson eq binomial deal interest next translate part bound poisson binomial bind via let random eq variation translate poisson follow translate distribution provide start run theorem outline give sparse heavy variance separately first identity describe finite compute bind enable simple give output distinguish imply exist length check mass outside identity distinguish level plan argue variance mean translate enough preprocessing follow close closeness proceed within factor heavy know probability go good variance compute estimate give enough triangle give go choice ensure hold pre distinguish step explicit distribution albeit somewhat involve term hand term use cauchy schwarz show side suffice recall enough claim lemma markov exceed distinguish appropriately threshold claim conclude correctness heavy conditioning various correctness account algorithm continue pay factor algorithm continue heavy factor account factor account sample majority answer desire easy convert low construct uniformly satisfie succeed distinguish sample straight check whether succeed probability along even vector specify prove unimodal suffice equally point since behave varie smoothly expect typical lot mode unimodal eq suppose increase q unimodal simple chernoff bound consecutive location high along enough far distribution proof pick uniformly distribution low w length random number probability binomial q objective inequality concavity follow elementary calculus simplify get therefore unless distribution pick author thank helpful valuable close part suppose find run pick choose large use chernoff deviation almost surely let high underlie translate close estimate interval small triangle q schwarz inequality times appear sample shall keep mind large variance part also suffice q spirit distribution consist succeed distinguish least success satisfied use distinguish succeed prof absolute specify later prove pick since poisson unimodal unimodal pick equally behave smoothly expect typical lot mode say mode unimodal triangle prove mode suppose unimodal randomly choose string consecutive location show enough far unimodal probability pick generate r occurrence except concavity calculate elementary simplify last therefore unless distinguish pick mit mit poisson sample near access bind complexity improve upon follow applicable distribution whether test besides matter stem
improve coverage interval value raise bias underlie approximation euler adjustment sufficiently ignore multiple provide well neighbourhood additional regressor variance see specification recall dr relative moderate via pre filter assess bias whether attack nature determine estimator table record bias subscript favorable highlight bold adjustment estimator already analytically adjust bias gain evident size full ba estimator estimator regressor unity save magnitude regressor proof follow directly ba ba lemma standardize standardized bias adjust analytical adjustment low bias absolute highlight c mse bootstrap analytical ba mse highlight bold k analytical ba analytically adjust ba report close nominal highlighted c coverage adjust analytical adjustment lowest highlight bold ba mse highlight bias adjustment ba experimental analytically report close highlight bold c coverage justification center bootstrap long integrate filter preliminary parametric justification bootstrap adjust simulation evidence performance bootstrap analytical bootstrap analytical already interval bias adjust estimator nominal reasonably adjust measured bootstrap correct analytical secondary log local long strongly process come play wide characterize decay absolutely rate stable dependent popular long memory process interpret via q short stable coefficient mean process model run integration particular impulse reference behaviour empirical seminal article area thorough fractional finance notably financial return highlight paper issue statistical memory method asymptotic gaussian normality mle fractional provide sample usual however correct inconsistent autoregressive average incorrectly specify parametric placing near semi estimator short broad applicability parametric asymptotically distribution place parametric asymptotic slow true despite asymptotic semi parametric exhibit particular substantial see correction area explore log hereafter least square ol n regressor local estimator denote cumulative estimator monotonically locally assign small whereas entail bias bias present seek reduce example memory broad band range zero even power frequency regressor approach present demonstrate usefulness adjust estimator particular bias semi correctly expense correction think parametric semi scheme rely parametric odd estimation nonetheless require capture salient importance series employ find poor coverage probability one side compare achieve attractive alternative bootstrap whitening fit basic property linearly minimum invoke likely although extension number past minor autoregressive coefficient kronecker delta mmse order unknown suitably process autoregressive model infinite validity regularity condition prove covariance estimator coefficient asymptotically aic commonly employ bootstrap asymptotically efficient sense select via plausible parametric raw bootstrap fractional reproduce realization adopt adjustment bootstrap standardize ht h tt mass h tt realization autoregressive uniform support integer crucially fractional index formalize follow regular stationary assumption place proof bootstrap see regularity achieve rate anti persistent regularity range filter series specifically employ modify wherein preliminary value filter value approximation filter apply construction filter short coefficient fractional binomial preliminary filter filter ar approximation generate filter pre filter bootstrap distinguish draw produce raw show choice shorter induce accordingly pass draw short memory proceed bias adjustment justification adjust choose proceed preliminary process evaluate estimator correct bootstrap copy magnitude surprisingly dependent proximity employ namely autoregressive use datum say main content bias correct asymptotically distribution admit expansion normal expectation original space realization process use pre filter sake integrate bootstrap step approximately strictly derivation produce report proceed construction obtain approximation bootstrap estimator expansion process reference precede require bandwidth estimator see term assumption estimator choose closely make induce quickly e n e provide justification estimate expansion c c j similarly construction yield algebraic give obviously depend bandwidth entail correction ar mmse coefficient derive apply preliminary approximation eq conditions express preliminary autoregressive proximity preliminary employ true choice optimal aic estimate aic mean model sense aic increase behave deterministic autoregressive appropriate pre filter value require give guide choice exactly n borel converge pre filtering sufficient convergence need nevertheless case basis denote ba probability pre simulation experiment initial actual adjust latter perhaps necessary consistent detail proof provide paper expression adjust common concern bootstrap adjust serve valid role pre iterative adjusted remain severe biased counterpart propose correction initialization assign tolerance go ba step ba notation tend bias adjusted estimate bias replace value b k value iterate determine accuracy achieve add iteration criterion correction think move estimator current b bootstrap filter conditional plus infer variance successive recurrence formula var nb iteration linear asymptotically evaluate percentile point similarly criterion accumulate correction tolerance level percentile convergence criterion decision probability next reference test conjecture iterate strong little change prefer terminate evidence substantial correction therefore go iteration follow ba ba comment far discuss follow process operator component early highlight set calculate modify estimator bias ba adjust modeling yield optimal seem use bootstrap autoregressive base square comparative purpose also analytically ba improvement analytical improvement via analytically bootstrap nominal interval plus interval bootstrap tu coverage estimator coverage calculate proportion cover true interval replication coverage bootstrap estimator record bootstrap produce two stop criterion criterion whereby iterative retain one iteration ba result estimator include omit brevity record bias mse bootstrapping adjust relevant ba subsequent table favorable highlight bold column modify key
consider repeatedly use decomposition case q observe na enough term argument enough enough cut coordinate cell informative variable sense theoretical make coordinate support european valuable suggestion lemma pt minus assumption minus universit es paris paris france fr universit paris paris france de france fr university centre f france paris france paris france forests combine decision despite usage randomization consistency regression random forest sparsity high dimensional forest randomization forest construct tree publication seminal become practice apply tune aside recognize ability size complex forest involve air win object microarray extra forest quantile focus version step subsequently forest explore year consistency perform simplified move ever practice prove consistency online forest forest difficulty nature procedure subtle analyze forest bag hand cart influential cart individual cut select optimize cart split notion bag cart theoretical done simply ignore bag cart split protocol leaf terminal individual tree pre specify total leave call difficult analyze rigorous mathematic author focus simplify create gap practice motivate property context forest theoretical guarantee consistency cart grow regression function proper number forest analysis adapt ambient carry organized notation technical framework goal predict integrable response aim prototype respect forest query denote variable set prior grow individual successive candidate direction splitting forest study forest limit infinity justify sc sequel forest rectangular cell form select sample leave resample resample word random datum choose maximize stop reach therefore contain exactly nm level level replacement optimize cart cut split call cell level xy resample different resampling bootstrappe point replacement replacement mathematical induce bootstrap offer establish precise forest regime forest occur fully consistency subsample make cart split properly cell position along th limit possible cut x cart independent variance extend univariate easy interpret provide calculation year play dimensional analysis successfully involve lasso various aggregation infinity random forest eq hold play tree sufficient forest cart noise control situation replace variable term turn account let examine regime e tree subsample complicate z nz finally coefficient whenever follow sequence surely technical statement h interpretation understand show h mean influence probability connection tend zero random vanish enough case satisfied verify noiseless partition strongly unfortunately whether let knowledge bootstrapping subsampling theorem consistency forest far independently thorough mathematical force action also interesting behavior ambient small constraint model assume loss generality informative independent ambient believe representation value proposition set split informative convention strictly forest select along variable everything happen forest upon variable support good difficulty assess forest process individual tree local averaging estimate chapter upon thing complicated proof adaptation theorem tailor forest result partition rely forest variation cell aim control let forest proposition offer control forest separate analysis regime allow standard thing subsample requirement term requirement every tree probability become datum inconsistent fact reveal forest small small probability ensure connected forest diversity difficult analyse subsampling come price assumption know theorem practice towards forest imply tree consistency require terminal tend infinity forest individual tree provably architecture also interesting pointwise diameter highlight forest inconsistent particularly mention extend context sake proof notation notation represent recall cut cut generally consecutive cut build understand build particular tuple cut proximity accordingly cell similarly cut later x forest follow eq large almost surely cut cell forest cut forest divide change instead cell point impose theoretical stop fix random randomness cut cut tuple equip clarity firstly lemma theoretical theoretical cut establish consequence proof within cell fall theoretical aim tuple cut pl x ensure stochastically denote k let stochastically lemma cut ready prove fix sure eq uniformly still need notation achievable accordingly terminal z hence finally index subsample tree estimate eq work tailor sake completeness equip theorem let n iii arbitrary pick h eq terminal assumption complete q contain thus may jensen large enough term double product handle lemma recall large subsampling observation cell subsampling combine enough inside satisfied equality thus surely statement surely return recall h connect select subsampling procedure
manifold set extract latent topological diffusion process start compute numerical subsequently feed diffusion amount classify subset high case bag get assign stationary except low domain distribution subset great enable diffusion part domain graph gender form decrease average appear function retain construction diffusion use procedure classify df fix metric token intuition project amount domain graph compute heuristic use select correlation g correlation graph gender wise wise graph suggest compute vector generate make robust vector get subsequently show gender well reduction bag subgraph extraction compare establish elegant conceptually principle improve quality context believe useful detection many domain domain construct df detect shift study co network online df profile df enable universit classify directed start collection reach topological original successfully apply get art pathway illustrate present classifying leaving make helpful context combine way either item walk provide explicitly matrix merge direct diffusion dimension token text undirecte make propose construct kernel graph walk collaborative walk user recommender system work set operating enable deep insight biased random walk diffusion reach time finally vector present formalism successfully extract pathway illustrate collection free association association define compute different appear call cardinality necessary consider corpus document consist token stop omit token position token fix omit association token every position token every token token successive occurrence occurrence still token appear conduct measure distance two token frequency frequent change somewhat want matrix density certain adapt weighted bring direct reflect association diffusion particular give document fit node df process start represent diagonal compute df recursively call define document default property well study nothing prevent bias random walk vector extract pathway idea adapt throughput biology quantity produce dramatically increase decade interpret result whole represent simple bipartite link draw chemical reaction case specie species chemical relation refer species context pathway transform source propose develop rank connect source node accord possible protein first pathway random walk pathway connect search short pathway mean path reach pathway whereby total pathway df explain call either annotate pathway weakly nan degree pathway general pathway reconstruct set source df df highlight hadamard multiplication direct application pathway connect effectively short two propose direct elimination degree arbitrarily fix relevant characterize path keep node account use full reverse multiplication division intend wise boost effectively pathway connect increase subgraph connect weak pathway large entry belong connect find pathway extract annotate chemical apply pathway minimal short target node pathway reconstruction pathway scale xlabel ylabel coordinate plot coordinate choice tune generally surprisingly variation parameter behavior explain constant pathway opposite pathway independently neither exponentially decrease source short pathway preferred one behavior explain geometric influence infer annotated pathway number annotate infer pathway dependency free appear bioinformatic relate walk pathway database quantify simplify nonetheless even since diffusion start cardinality whole annotate pathway cover result geometric give accuracy diffusion give terminal pathway compare select pathway analyze therein version df pathway reconstruct value database difficult publish inference df dedicate apply df binary problem give gender node form connect direct diameter scale xlabel ylabel red densely marker densely reach part line mark start decrease computation costly random average accuracy adaboost algorithm classifier comparison vector note moreover domain xlabel ylabel accuracy coordinate coordinate projection solid red note token split select token
exactly difficulty candidate detector show fusion sensitivity measure predefine build ensemble detector detectors result ensemble appear small circular dark detectors way channel extract enhance characteristic coarse extraction rest detect fine algorithm usually remove false former detector candidate apply preprocessing method technique slight increment false decrease voting detector combination exhaustive quantitative superiority prove competitive screening preprocesse section propose discuss discussion candidate extraction component comparison preprocesse dedicate publish yet select candidate characteristic image unlike generate noisy histogram medical processing preserve summary preprocessing aim gray max f max intensity enhanced transition popular technique make salient part visible split region histogram apply boundary eliminate bilinear complete remove base part fill removal preprocesse illumination pixel intensity original intensity local appear enhance also consider preprocesse formally operation l histogram removal near removal illumination aim show characteristic principles brief overview involve preprocesse add candidate future performance measure candidate extraction accomplish dark radial follow application match threshold candidate representation grow implementation publish et circular obtain detect circle circular circular extract candidate construct accomplish coefficient standard maximal response thresholde pixel wise cross directional map assign height describe map thresholde fp diameter transformation circular transformation matching framework ensemble preprocesse preprocesse preprocesse extract ensemble contain distance small centroid ensemble perform evaluate ground truth predefine otherwise ground truth candidate ensemble currently preprocesse combination would resource demand anneal search find proven procedure latter evaluate configuration function competition predefine positive rate preprocesse candidate pair ensemble phase namely detect fusion candidate build final decision output thresholded confidence combination candidate ce p dc h good evaluate detection dr present capability propose competition detector publicly available overview database competition dedicate roc mark compressed image set image marked expert database consist receiver curve average false sensitivity false level thresholded output detector ranking serve ranking addition calculate normalize calculated way likely auc uncertainty high also evaluate dr determine image presence contain indicate yes provide database train quite strong measure circumstance publicly database compress image range clinical patient dr r mild severe appearance classify contain sign dr detector measure specificity detector level thresholde candidate use follow correctly recognize receiver operate roc auc database present dr include row preprocessing list removal illumination rank result roc competition current performance ensemble see ensemble auc individual algorithm ht team auc htb publish database yet ensemble htb sensitivity specificity measure detector different fit roc detector area auc fit curve recognize case sensitivity specificity perform circumstance figure use removal miss absence thin preprocessing create diversity member multiple diversity ensure diverse different mistake false receive confidence voting detector outperform high flexibility ensemble high database roc consensus expert fp database level achieve good thresholding detect dr r recognize dr affect detector sensitivity severe recognize appropriate specificity suggest specificity dr screening threshold value specificity achieve level specificity result database dr dr case auc minimum sign
distribution omit primary univariate follow induce use query well query estimation artificial test discard serious setup first good exponential mat follow likelihood process optimize distribution demonstrate methodology copula distinct task copula primary middle secondary merge result toy omit identical performance input increase input root mean average number query significant cd ni ni f co krige induce l time ni cd ni ni ni ni ni cd provide completeness rough baseline contain type chemical element lead region primary sample secondary hard expensive divide sample primary cd primary ni secondary variable furthermore use mat ern cd ni square model marginal extreme mae task input less prediction omit gaussian use less induce process approximation second completeness l rmse mae opt show deviation concrete water fine cm day compressive compressive secondary ern generalize variable interestingly happen optimization optimizer sometimes well change process extremely different inter show copula learn address derive furthermore experimentally synthetic public learning centre centre rgb process ex plus minus ex ex minus minus plus pt em em em spatial monitoring address convenient dominate non gaussian likelihood copula process elegant handle capture structure cumulative distribution rather task use prior hold task expression copula model compare artificial publicly resource many environmental fusion problem advantageous dependency appropriate base tool problem gp predictive incorrect gp comparable root flexibility informally cumulative function cdf couple handle variable distribution help appeal address cost copula multi task problem analytical process copula machine fundamental process finance copula stochastic volatility propose heavy tailed robustness copula field krige process share approximation individually predictive depend location demand memory computational divide process representative approximate later recently residual mixture problem computational handle grow significantly simultaneously consequence copula decompose marginal cdf actual copula map though call integral transformation copula meet joint copula possible create huge marginal copula analytical cdf cdf root density create distribution get though leave toy txt index marks black dash mark toy txt toy txt scalar aim task broad improve secondary situation occur environmental extend gaussian copula task gets reduce inspire krige convolutional cross result kernel task mat ern process ordinary merge input different output univariate usually parameterize way denote advantage standard going minimize log explicitly minimize simulated require numerous numerator cost dominate rapidly element task introduce attack cm axis axis xlabel ylabel sample pi densely mark densely pi table index toy txt toy txt index toy data txt vs height x bottom axis line xlabel ylabel legend north index toy coordinate width height axis ylabel
application semi relationship deterministic inductive unbiased otherwise constant scale small singular ground ground informative subspace inductive completion underlie recall inductive optimal solution deterministic solve inductive clean eq case matrix expect related transformation continue extend inductive bound row typical proximal bx singular use power fast procedure thin order rewrite solution store residual computed remain computed solver solve million nuclear sufficiently trick thus descent cd scale link hard constraint relax bound cd complexity synthetic meaningful effectiveness real theorem link consistently orthogonal set linearly deterministic setting zero fix fix interestingly decrease linearly decay plot poor behave range inductive completion include recommender system semi supervise latter demonstrate usefulness semi supervise problem low norm recover biased dataset sample feature present inductive inductive pair minimize classification ground vertical rate q well approach motivated modern application completion bridge even bit quantization well sided measurement setting similar recovery insight bias past theory effectiveness evident link real principled select exploration want apply ij ij change chance fact want rademacher ij q universal mn minimizer therefore eq q case hand q ij r therefore get need hand side side convenience argument get bind ij ij take trace l discuss element exist value enforce constraint equation complete claim bit measurement zero modern application recommender system social network learn positive unlabele binary classification positive entry reveal assumption provide recovery guarantee shift subset propose completion recover binary denote sample scalable procedure effectiveness consist node million link semi recover arise netflix predict motivate rating observation low variant completion underlie one bit quantization modern completion link recover snapshot social consist pose recover adjacency otherwise negative context call unlabele short study completion recovery paper completion answer minimize observe solution popular treat good positive motivate positive completion sufficiently nuclear learning bit binary consider completion non deterministic show estimator square estimate motivated shift obtain scalable reveal low end scalable differently inductive bilinear row extend two contribution paper propose inductive recover imply insight completion efficient scalable simulate social consist million million superiority link establish hardness describe give extend inductive completion describe synthetic world last work completion remarkable low observation also recently motivate domain recommender heavily draw motivation recommender seek case matrix albeit sense field closely completion measurement bit quantization consist sign remarkable prove assume matrix independent state completion problem straight goal recover sided basic however world unlikely setting special non generality normalize partial randomly precisely uniformly let denote deterministic specify q bit apply subset uniformly bit unobserved bit completion satisfactory completion subset assume obtain substitute recovery error recovery high dense drawback completion complexity make large moreover average affect vanish deterministic indicator subset uniformly impossible entry recover therefore underlie good completion trivial way deterministic completion observe fix incoherent suppose location sample recover matrix deterministic matrix completion bias deterministic matrix proof defer want lipschitz traditional trace one use instead unbiased formalize mn interestingly rewrite follow want
meet pair visit infinitely learn outline repeatedly sample cr cumulative measure receive denote discount select maximize secondary exist queue length existence strong interaction use q learn state computer wireless center communication cognitive organize cr user assume cr cr user queue buffer queue moreover queue channel empty assume transmission policy cr arrival channel cr energy primary optimally cognitive usage utilize cognitive cr users environmental exploit reasoning dynamically communication diversity communication attention cognitive primary secondary cognitive e investigate cognitive cr cognitive terminal queue maximize stable author cr terminal primary spectrum inactive assign queue cr admit fraction fraction secondary incorporated management management allocation horizon throughput wireless programming online average arrival channel state cognitive energy g markov mdp secondary policy spectrum cognitive terminal explicitly involve queue author cognitive throughput region author investigate service rate secondary capability add consist cr user investigate impact node traffic well protocol randomly begin slot sense maximum throughput queue propose cognitive select duration activity throughput queue network utilize spectrum inactive one decoding outer secondary cr second primary action take slot exactly sense inactive cr end slot unity reference make queue capacity assume length nonempty e furthermore arrival arrival identically queue arrival bernoulli generic apply adopt share channel channel cr low inactive energy queue queue fig buffer incoming datum denote buffer energy token cr user store traffic store primary store environment finite length precisely queue maintain duration slot second bit arrival bernoulli process process evolve accord queue assume two fig queue slot slot queue slot slot assume variable queue queue time gain node gaussian random read primary secondary link cr cr user perturb model additive independent channel specifically slot slot channel unity link channel capacity know channel gain slot primary channel send primary dedicate narrow phase spectrum medium control begin cr channel begin activity record binary inactive cr queue queue cr user slot correctly cr decide accept negative decoding drop primary queue cr primary primary overhead negligible size feedback receive accord description distinct action begin slot cr high early cr four cr optimal slot follow slot cr user energy energy queue cr receiver service term unity queue empty queue empty nonempty cr queue cr access channel receiver inactive arrival queue queue primary queue nonempty queue channel arrival energy queue either either queue queue process energy queue channel receiver queue accept follow whenever energy queue primary queue nonempty occur queue size begin slot queue argument rl maximize value payoff user cr adaptive accord rate mdps framework uncertainty bellman many dynamic programming mdps discount immediate discount small investigate satisfie bellman discount cumulative agent cumulative beginning maximize sum service rate
time decode half measurement count sketch demonstrate measurement major scan sketch compress sense entry instead design propose decoding utilize estimator absolute theoretically analyze estimator estimator ratio zero ij happen motivate nonzero tie long make detailed need easier particular false certain still practical preferable see general estimator estimator introduce theoretical exploit prior estimator recommend estimator recall I residual major computing absolute iteration positive irrelevant care express ij ready lemma practically convenient numerically understand substantially hand convenient theoretical lead complexity convenience set convex upper resort poisson consider define confirm poisson two small basically away perhaps choose reasonably news practitioner confirm ratio suffice express datum construct absolute sort value examine difficult ix measurement error easy suffice recover nonzero nice property reveal nonzero recovery consider need pm eq q kt e compress crucial theoretical page see count sketch l available achieve count half sketch show contour plot decode count recovery compare become replace dense sensor cost perspective decode department statistics department science university nj usa department nj usa sparse projection compress sparse recovery design fraction major scan coordinate absolute use minimum combine practical decode exist scan algorithm decode nonzero entry positive method l compress sense become popular field computer science mathematic recover number adaptive nonzero location coordinate database naturally formulate compressed sense sensing may back paper compress moment linear lp
width axis style col comma auc accuracy utility dash red fill red col sep comma bb smooth thick table col comma auc bb nb xlabel ylabel recall legend legend pos east font sep comma sensitivity utility thick style red col comma bb sensitivity accuracy utility col central bb nb prior plot set well impact thin another indicator recall indicator sensitive htbp classification return classification behavior available reason aim close wide thank knowledge group one safe class prior one safe meaningful vs vs vs fraction instance recognize safe instance drop close accuracy safe perform dependence phenomenon know observe detect cross checking prediction induce model quite done recognize safe dependent qualitatively yet independent eventually recognize dependent note include accuracy xlabel ylabel legend legend pos east every font table col accuracy traditional accuracy quadratic function respectively denote compare accuracy valuable classification grow close figure follow assign classification utility high situation become fully reasonable robustness function risk differently yet rank despite explanation utility assume costly miss presence assess art uncertainty yet set especially small deal single sensitivity instance version represent valuable however variant matter partially cost error interesting analysis detect check automatically opinion acknowledgment grateful center application master student partially nsf grant support project moreover grateful anonymous show ib contain include set instance belong definition address pc j pd includes minimize maximized interval k j numerator polynomial complex interested solution together boundary solution constitute maximum candidate retain beta beta bb treat prior include covariate beta pm b explain beta k beta give combine uniformly lemma remark empty di di data result address prior sensitivity building regressor characterize show tune probable return compare variant presence recognize dependent predict outcome categorical basis several call include covariate plausible drawing conclusion conclusion average inference model sophisticated model yield inference specification conclusion especially often report sensitivity present model characteristic value return single class safe generalize thus combine classifier firstly naive bayes express weak classify posterior sensitivity identify namely probable depend prior predict presence environmental covariate slope analyze logistic analytical condition near moreover present preliminary presence consider yet knowledge class prior previous new variant expert publish paper specie master also extend two informative uniform prior expert informative also another posterior inclusion organize algorithm study expert covariate covariate denote I covariate train th covariate address combine inference weight probability model give respectively marginal respect model parameter log number approximation viewpoint adopt bic parameter probability huge approximate summation computational limited covariate often interested binary include otherwise prior ib ib covariate denote include covariate probability obtain assign informative viewpoint covariate include ib far flat flat prior adopt bb ib bb less bb recommend handle bb treat prior choice probability uniform analytical derivation formula ib distribute bb covariate eqn distribute compare ib bb htp xlabel ylabel legend cs anchor west draw every font col comma beta bin col sep comma differently specify inclusion covariate generalize prior inclusion probability thus recall call nb nb ib independence inclusion covariate set model call discuss generalize induce ib induce nb ib ib specify vary ib apart sharp remain identical word prevent ib prevent ib exclude condition prior follow inference probability compute probability sensitivity probability minimize covariate covariate covariate otherwise analytical nb describe nb specify permit collect us admissible value covariate want use one represent get vary curvature near posterior varie compute dimensional complement low inclusion covariate include analytically package interface symbolic solver far assume prevent sample strategy space discuss accommodate inference formula whole summing accord interval dominate class vice generally class lemma class deal equivalent prediction probability probable class upper probable depend consideration induce interval prediction match regard collect distribution explore area cell cell consider range introduce covariate third piece aspect namely maximum aspect temporal covariate concavity namely cover digital mobile use huge reason average cell ask inclusion report publish paper specie l master belief expert table label expert assign belief expert covariate two expert skew inclusion htbp expert aggregate two way firstly use within specification secondly take represent knowledge later nb among slope expert different inclusion appropriately represent substantial central inclusion covariate point uncertainty characterize ib informative bb informative informative call probability covariate hull report consider three variant originally represent assumption covariate equal configuration wide variety prior third partial knowledge inclusion low hull expert belief inclusion covariate prior probability inclusion recall binomial prior include reason outside slope curvature consider huge remarkably probability curvature recognize estimate inclusion discard covariate interestingly achieve assign curvature upper thus conclusion sharp bayesian inclusion depend yet inclusion exceed much comprised showing repeat great create comprise presence set shape ylabel post legend none west major
explore mode mode whereas penalize mode benefit hyper penalization leave bayesian method tail hamiltonian carlo restrict mode focus report investigate fully elsewhere lasso choice cauchy fit relatively stable therefore unnecessary greatly increase mcmc tail hyper play role shape structure lasso laplace illustration technical investigate report apply method microarray set article logistic class integer feature vector datum column feature look consider infer prior article genomic however rare df gene regression model find gene marker great mcmc clear difference laplace freedom well high express stand inverse shape term number generator parametrize penalty superior dimensional prior cauchy uniformity notation prior article table description generate generate look huge look tail value figure move change moderately tail great well belief df tail heavy either tail htp distinguish redundant feature moderately heavy tailed separate path constrain maximizer ie contour find shrink contour tangent df generate map contour origin axis path go laplace look also find divide conceptual illustration datum contour figure penalty two end contour explain divide correlated make selection among correlate automatically within large contrast gaussian penalty constrain middle contour explain absolute discussion difficulty problem minor unstable require see full advantage choose mode mode describe technical letter index index letter subscript index denote integer integer collect class case take integer logistic coefficient hyperparameter define convenience useful predict provide control variability variability ig equation bivariate assign coefficient shrink signal assign half cauchy cauchy half various induce propose coefficient assign exp mean obtain form notational simplicity call name coefficient well towards study confirm little regression rejection adaptive ig example hour hour denote recommend fix contain empirical stable choice especially heavy may avoid treat hyperparameter capture fix hyper differ greatly freedom consequence stick region likelihood sampling accommodate recommend fix issue model coefficient identifiable add class identifiability implication may inference class common identifiable equation c markov naive sampler long chain symmetric transform identifiable transform symmetric transform distributed indicator likelihood parameter integrate prior element normal variance hierarchy asymmetric cause difficulty useful label normalize deviation select look standard deviation alternatively sample joint put conditional alternatively transformation deterministic leave invariant state state still reversible transformation compose transformation valid full explanation sampling procedure page gibbs full last sampling ig alternate pt carlo transformation leave straightforward prior use differently distribution concave sample give sample key high hmc hmc greatly walk due common regression long trajectory sampling prior local redundancy sampling correlate probably fairly hmc chance ordinary move contour mode dimensional thousand bottleneck challenge greatly important gibbs coefficient detail trick notation setting recommend iteration compute rank discuss pool mode true feature appear frequency markov correlate ranking correlate totally note recommend useful feature light choice mcmc number discuss alternative generate predictive across nd therefore class pt run ie ig setting chain since take estimate lasso package see give lasso second stand path see estimate stable bias large marginal skew absolute explain predictive feature pt compare path predict importantly predictive contrast choice weak conversely validate even nearly difficulty look follow equally draw mean feature within group class draw absolute differential differential class however relate feature higher generally select discard obviously million run prior choice choose choose hyperparameter reason automatically n large chain setting prior validate lasso set shape stand group point horizontal pt separate times nd weak recognize another useful discriminate believe useful base consistently stable apply almost large separate absolute set think hard tend discrimination entirely coefficient tail flat tail therefore heavy tail good likelihood tail table summary ie relative prediction feature choice note moderate chance weak therefore choice critical figure see prediction performance measure attribute nonzero pdfs without statistical degree freedom almost complicated logistic regression hand negligible high problem chain take whereas chain take merely posterior rather ig large however possibly eliminate well microarray relate cancer set website look f statistic rank statistic use leave fashion always standardize lasso gene take hour prior prior use top pt run perform compare thousand small omit top rank gene rank narrow around value small rd check report predictive substantially method test performance prior statistical figure improve htp r r er pt use case select plot show useful joint skewed absolute fairly classification slope hyperplane function indicate necessary hmc ordinary mcmc weakly redundant separate gene different absolute gene clearly indicate redundant multimodal prediction prior result gene figure probability label shown think biological see gene separate gene pt pt statistic introduce bayesian prior investigate microarray demonstrate feasible high hyper retain selection hyper similar superior appear also problem choose demonstrate light hyper modal average particularly group highly coefficient fit divide markov look separately list markov coefficient totally skewness demand development simulation future drawback still slow penalize room computational difficulty lie exist transform feature mcmc simulation crucial step solution devise transform li support engineering foundation li discriminant vector produce estimate suppose leave log ie independent p physics momentum particle qp way qp qp give discard transform qp hamiltonian move keep unchanged crucial hamiltonian metropolis hamiltonian dynamic discretize stepsize transformation several alternative transformation I transformation independently denote property nearly hamiltonian series add transformation reversible jacobian leave exactly sampling transformation qp reject algorithm carlo current step independently transform qp p time transform qp qp trajectory connect along call decide accept qp last implement hmc appropriate hamiltonian poor may slowly even low rejection hoc close reciprocal square root nd automatically account adjust factor adjustment choose hmc beyond work thing hamiltonian trajectory nearly adjustment appropriate phase sampling value frequently make look problem fairly well another initial trajectory run need value hmc
always primal unknown ax center know potentially simply invert find solution unknown treat vector regressor imputation one among alternatively compress interpret accurately e concept cast task name partially concatenation position entry noise name reconstruct decay strategy capture target ac imply elementary circuit circuit target vector say circuit circuit correspond dependency regression circuit scalar multiplication let minimal circuit pick define circuit circuit either circuit circuit formal circuit circuit formal associate circuit cd emphasis circuit circuit circuit vector circuit respective circuit circuit see differential accurately efficiently circuit circuit let call contain circuit circuit rich circuit vector equality span suffice circuit statement equality one circuit circuit algorithmic benefit treat circuit set circuit vector reader think circuit term circuit think notion make advantage describe close short circuit affine equivalence short circuit general circuit element additionally important particular particular circuit circuit regression circuit generic uniform special generic generic circuit particular circuit support rank circuit require name cycle homology compute completion find special circuit combinatorial non behind technical circuit generic non kernel span generic vector hypergraph construct evaluation form view outline circuit special circuit decrease regression unknown want circuit estimator let circuit expectation equality circuit converse optimization major formulation circuit involve inversion nothing gain inversion usual circuit circuit analogue induce circuit circuit circuit circuit matrix circuit entry circuit bilinear positive variable cholesky observe definite let minimize kx slack term minimizer satisfy exactly give independent minimizer algorithmic good generator keep track circuit vector compute highlight advantage pseudo matrix oppose pseudo inversion naive advantage scenario estimate depend choose choose circuit circuit circuit needs change bilinear j variance error circuit circuit circuit matlab concatenation circuit circuit circuit calculate computation huge circuit small scenario increase row negligible stay algorithm compute multiplication circuit evaluation circuit covariance return estimate circuit covariance variance bind return matrix circuit matrix column compute multiplicative column circuit basis fairly near consideration circuit submatrix believe invert merely structure imply small circuit problem simple setting much circuit well attempt circuit e solve choose circuit row combinatorial property completion concept graph homology may local neighborhood also example combinatorial circuit identify circuit sparse dual highly structured analyze network circuit basic identity regression circuit element square multiple observation copy circuit contain regression circuit type copy copy general circuit prevent multiplicative suggest pool observation single denoising occur circuit observation row denoise occur row improve merely observation difference basis orient node characterization circuit circuit show form contain circuit one edge circuit circuit therefore element graph homology edge sparse sum row graph assignment regression circuit path length contain circuit vector start circuit cycle length circuit potential search homology provide cycle efficiently write cite concatenation entry unobserved case sum scenario discuss always neighborhood principle apply search homology arbitrary tool equation structure measure matrix scenario reality matrix format case recognition hermitian potential scenario circuit circuit phase non symmetric product see otherwise include namely circuit disjoint form inversion exponent estimator blue variance unbiased row relate furthermore estimator drop compute without estimate provide employ important add circuit improve incur error conversely add circuit estimate system make relate inclusion wise maximal conversely c proposition estimator unbiased imply blue analogue gauss estimator way complete minimum optimality estimator need information signal noise first gaussian statistic respect I remove contain sufficient virtue follow thus see transform prove statement prove complete prove write ax follow straightforward exponential similar conclusion signal f bf immediately imply statement px n px px include
schmidt difference projection operator similarity involve w mixture relatively separate measure whenever bound mixture relationship translate mixture recall set simplify clear suggesting remove important relationship stem kernel square root concentrate tight spike illustrate colored ratio color latent analysis limited population infinitely relate version laplacian embed sample certain geometric call reveal I subset draw nu notation embed label pair distinct almost illustration establishe embed cone additional parameter mean low decay decay finite perturbation root close difficulty parameter previously require essence small notation apply finite angular depending hold probability normalize embedding right display laplacian embed orthogonal component overlap result high illustration way proportional small depend dominant simplify tail dominant increase increase tail truncation literature chen von de restrictive hold add increase performance practice performance standard apply embed h normalize ix work proposition quantitative apply embed notation random initialization enough latent fall within angle angular occur close let initialization cone close case q single step fall angle angular level provide perturbation approximate invariant recall analogously three ideal overlap mixture generally perturbation long schmidt relative problem long sep apply follow hilbert schmidt mx spectral consequently material combine moreover theorem remain projection operator write give consequently continuous calculus expansion n put piece shorthand subset orthogonal angular structure subset element diverse tuple break step number tuple diverse construct diverse construct select form laplacian embed copy diagonal nx lie follow tuple q hold return explicitly claim end combine establish finite sample intermediate entry satisfy n q aa transform involve write note therefore find inequality b bb mn expectation generation selection diverse least v condition auxiliary result set simplify whenever consequence kk tuple thereby complete remain hoeffde bernstein control put piece nx ny laplacian matrix principal principal prove relate namely bridge intermediate e k supplementary material lemma must handle fluctuation denote reproduce similarly satisfy condition k x lemma triangle identity return term note schwarz inner product logic bernstein nc e q eigenfunction eigenvalue note eigenfunction form orthonormal therefore r g ig analyze spectral context nonparametric mixture level square coupling parameter kernel undesirable necessary guarantee identifiability follow sense rich mixture component component mixture conversely function representation could optimize cone symmetric find family building characterize normalize collection take structure component almost angular angular perhaps fact minimize believe bandwidth distinguish vanish however project shrink population spectral mention provably shrink leave bandwidth cluster interesting acknowledgment wu helpful discussion grateful associate suggestion manuscript give overview background material symbol nsf grant dms grant dms dms grant nf information technology agreement cluster many area spectral eigenvector normalize laplacian recover label finite nonparametric difficulty label overlap divide compare root embed cluster past decade spectral learn information attempt answer spectral np hard partitioning cut spectral cluster machine application clustering set decade normalize division back modern cluster eigenvector laplacian apply embed cluster ng al well separate recently expression analytical eigenvector perturbation away ideal laplacian study manifold primary convergence limit connectivity reconstruct eigenfunction sample analyze laplacian laplace von embed much laplacian provide proof convergence part unnecessary kernel nonparametric mixture study characterize difficulty eigenfunction top eigenfunction eigenfunction integral sign fraction laplacian embed nonparametric normalize laplacian operator component overlap span square characterization result certain nonparametric refer mixture orthogonal structure perturbation theory remainder follow separate component result support distribution set hilbert schmidt supplementary material symbol introduce result discussion consequence involve span square root provide description mixture give play important intrinsic difficulty kx cluster index symmetric measure overlap respect density precisely distribution analogy kernel component coupling laplacian decompose mx small measure difficult split splitting identify since ambiguity define one measurable shorthand kx dy infimum measurable subset mixture coupling illustrate triangular function correspond triangular similarity
sum rank component svd decomposition term analyze detail decompose rank unlike tensor dimension fall less overcomplete regime regime rich option extract early base incorporate discriminative subsequent show discriminative model feedforward far unlabeled function label present mechanism relate assume component ica rbm conditional correspond reasonable rich element element previous work estimate new transfer estimate proceed due sample domain vision framework supervise transfer popular domain nlp vision list extensively study sample bootstrappe label dataset train label main general tie investigate transfer use various work transfer unlabeled propose fisher score yield argue learn mechanism score cross moment extract attempt discriminative feature consume feature learn ica machine rbm argue overcomplete latent representation latent dimensionality good develop guarantee ica code various information transfer argue incorporate act improve learn label dictionary learn unlabele label sample coefficient fed framework score review probabilistic model base score score function involve partition intractable compute compute far nice maximum presence rbms extract superior learn auto encoder input encoder map penalty see review special case establish score zero argue employ establish yield derivative derivative stein construct familiar multivariate polynomial polynomial polynomial review semi assumption establish describe setting example general label contain specify cat assumption unlabele label input instance human task cat cat gain another access imagine set image human label use classify transfer original unlabeled instance contain label unlabele image internet framework conjunction score feature setting self draw mild incorporate discriminative probabilistic randomly mild regularity variety unlabeled g mixture boltzmann distribution unlabeled cover assume learn previous g early contain internet observe correspond share probability draw observe image include random marginal represent datum show mechanism explain semi consider unlabele limited learn task unlabele many challenge general work extract entirely extract conditional useful variation gx e derivative derivative w expectation picture vector depend derivative component question score score th perform manner sample see mx estimate score empirical moment investigation work propose task show yield differential operator discussion formal lemma yield derivative fisher differential start stein building original version stein regularity scalar usual part introduce decompose derivative derivative tensor gx order principle eigenvector high tensor array framework tensor represent outer tensor decomposition computing form high similarly perform power multilinear form definition different vector clustering ensure converge vector tensor property guarantee dl multilinear tu update remove center algorithm base tensor tensor orthogonal analyze decomposition rank unlike overcomplete tensor large dimension incoherence impose soft orthogonality component non dimension application power perturbation whiten tensor multiply whiten tensor orthogonal decomposition discuss framework connection score expression distribution exponential family energy let vector know order give kernel joint kernel gaussian manner score posterior center contribution mixture center score attempt encoding decode call unsupervised unlabeled argue approximately learn go describe framework match fit analysis let good score minimize two sign equivalently laplacian nice interpretation fisher divergence relate leibler px sense robustness note estimation ml kl density regularity score minimize change amount function introduce derivative minimizing operator function sign score px form efficiently close compute nonlinear density compute ti self self unlabele label assumption sample unlabele information model joint function use assumption unlabele unlabeled unsupervised way characterize notation include convenience restrict identify array rt limit tensor also multilinear tm j eq multilinear similarly multilinear combination slice rd tensor rank cp write tensor th derivative fx permutation states stein score yield stein lemma function continuously integration scalar scalar provide recall expand early collection probability integrable along line px x dx map derivative characterization stein characterization definition function high order score function denote denote score recursive differential relation induction prove score stein high generalize order order operator let random suppose function consider continuously gx regularity gx px prove recursion stein identity see relation induction q yield parametric exist thank nsf h support support microsoft fellowship nsf award nsf award award award first regularity inequality iteratively recursion stein lemma formula stein identity entry appropriate mode tensor prove permutation step affect symmetric mode gradient lemma derivative apply necessity argue gradient tensor permutation put last tensor require last prove score induction hold show hold substitute rgb corollary conjecture proposition claim observation example bold time challenge vision processing extract train label sample high establish theoretical characterize nature extracted label employ tensor advantage employ value rich discriminative information form employ discriminative feature good critical achieve performance domain speech vision language traditionally engineering carefully tailor towards task automatically feature various framework deep sparse coding approach unsupervise thus vast incorporate almost probabilistic incorporate important explanatory input incorporate boost discriminative behind input unlabele learn label one self framework transfer adaptation involve interest mainly due natural language process syntactic access unlabele human learn common without goal human design capability unlabele general extract relevant answer class pre unlabele present pre discriminative nature consider high derivative pdf local manifold pdf input rich access feature characterize precise nature work extract moment main expect word moment capture informative employ spectral representation moment algorithm suffer spurious optima typical problem backpropagation construct overcomplete representation tensor argue overcomplete crucial get spectral method extraction scalar discrete handle regression classification multi present efficient end extract overview figure fill shape corner sep sep purpose score mx moment derivative discriminative pt line width width corner line dash corner green label extract supervise discriminative equal informative indeed derivative vanish input distribution carry either nearly degenerate vanishe average scenario cross contain useful model mixture moment recover guarantee challenging contrast approach incorporate generative discriminative feed fisher feature feed behind unsupervised find learn leave generative discriminative prescribe sample unlabele discriminative decomposition tensor
initial random subset zero otherwise ensure adaptation pick interested pick serious use dependence say disjoint subset intuitively dependent probable pick pick pick important negative chernoff independent general approximation base adaptation two negative weight unweighted modify bind material detail state guarantee relatively ideal optimal bound reasonable trivial regime cut bound edge need approximation cut intuition find directly linearity subset number weight unweighted let minimal cut edge edge let weight mc intersection empty prove reasonable graph cut small low cut generally cluster precise vary algorithm scenario spectral graph spectral towards graph consist matrix element assume imply assumption relatively weak connection cluster laplacian simplify angle way measure theorem approximate show obvious supplementary theoretical guarantee guarantee use notion nothing make partition cut separate small assumption basically inner relatively observe edge separate cluster cut proof chernoff bind cut supplementary material proof provide depth analog theorem structure weight constant bind think vertex clique connect sensible algorithm connect clique take try however sampling connect clique probability connect need scheme connect connect component component wrong right small probability connect sampling uniform intuitively distribution edge approximate structure original non motivated leading distribution weight estimate weight scale without probability replacement iteration moreover sample negligible modification suffice hold unfortunately find unstable bad clustering initially mix attempt weight pick unseen probability graph modify pick uniformly pick biased scenario toy early pick edge wish use desire discuss early mix appear initialize zero pick otherwise cluster pick connect ij aware algorithm somewhat nd drawback specifically design nd laplacian eigenvector extend cluster require partition suboptimal eigenvector decomposition costly impractical cluster single edge add eigenvector although couple another option several make test inferior spectral perform surprisingly inferior cluster cluster percent frequent weighted circle comprise four gaussian unnormalize weight half classic easy bad dataset dataset cluster image dataset suggest test unnormalized previous hoeffding chernoff dependence approximation change edge quite change proof paper pick equation change zero first second small unnormalized eigenvalue prove show two small unnormalized min spectral inner fact need main tool matrix chernoff without consider hermitian define replace requirement without proof function result without sample edge connect cluster probability laplacian matrix chernoff p en prove theorem use cut cut weight prove extension trivial round good minimal edge weight small n original graph finite n increase pick gx x bound drawback minimal unlike situation prove cut c chernoff hoeffding multiply set pc pc completing get simple sampling bind bit see weight observe cluster cut mc cut separate cut cut inside mc small great cut cut need show cut separate consider cut separate need dependent chernoff mc finish adaptive well theoretical uniform cut pick approximation accord pick pick accord pe view subtract suffice bound give cut define choose nonzero dependent ik get easily verify monotonically finish proof prove concentration output weight chernoff replace independence markov induction union finish show chernoff bind bad adversarial bad clustering graph pick least besides edge choose uniformly
contraction equal k implicit td td affect contrast implicit td implicit td benchmark td td evaluation compare size alpha method method alpha implicit alpha experiment perform library fouri final step may see alpha size stable cart td algorithm justify evidence implicit td td great td preliminary many ahead error exist result td implicit td surely td implicit td future extensive evaluation several adaptive compare proximal update successful td rl actor policy instability kk equal eigenvalue q eigenvalue b algebra solution ready td eigenvalue replace e discriminant lead contradiction thus corollary remark usa technology department statistics usa reinforcement method implementation td choice step empirically high instability cost td evaluation implicit typical td td state art wide applicability rl fundamental td linear make pair td successfully apply scale domain drawback td empirically study try solve stability stochastic sgd algorithms asymptotically significantly moderate sample stability implicit sgd motivate connection proximal inspire enjoy standard td explanation introduce maximal td suitably evaluation implicit td within td convergence implicit td markov finite probability underlie transition chain denote reward time weight value vx discount brevity sequel calculate sample state transition standard td note td introduce td discriminate standard implicit td implicit eq solve inner td td expectation take suitably td td thus implicit shall implicit
list notable tuning practice time sort diversity weight maximization basis item generate recommend item fa list recommend list objective function therefore treatment technical allow overhead optimality argue problem modular submodular diversity independence polytope problem dimension greedy solution item sort order weight section interpretation aspect finite topic movie diversity measure unique topic cover highest covered belong item belong item list another item must item item cover increase item objective diversity gain view cascade scan list top stop item due diversity gain second last equality interpret none early item expect recommendation list depend diversity consider diversity item result item return sort utility mathematically diverse useful recommendation maximum choosing idea diversity suitable user may redundancy recommendation interest assign q recommendation list characterize function diversity return recommendation item utility moreover length prove contradiction item choose list item contradiction topic among test value diversity number add allow controlling length irrelevant rule evaluate offline compare optimal solution evaluation list diversity time amazon mt separately recommendation mt worker movie match relevance movie study mt worker movie list finding offline fine assessment create preference profile evaluate diversity utility recommendation application frequently rate rating assign movie maximum utility I maximum include popular movie restrict movie reasonably movie evaluate mt worker intelligence ask worker interest generate list three list choose worker ask movie list match address contain worker movie ask movie recommendation choose address cover movie list mt show recommendation list movie contain movie movie recommendation list differ worker movie inherently recommendation adopt want number movie put disadvantage generate list either utility diverse table complete worker mt worker quality work ask complete average second second evaluate permutation highly hypothesis low quality worker evaluate result present compare percentage find movie match choose percentage movie movie match movie worker match movie percentage time match firstly percentage find match find matching list well perform method compare equally answer worker generate reject nan secondly time movie list perform significant time good list second perform compare hypothesis answer worker statistic less likely nan list generate ratio matching movie find imply movie likely recommendation due high popularity movie ground practically recommend difference real outperform movie movie match however insufficient cover five movie list movie assign diversity cover relatively movie cover likely user dark action dark dr dark c dark action action diverse movie cover parameterize ask worker go take prefer movie four recommendation three none mt preference recommendation time frequently rate movie rating assign setting generate movie diversity randomize suitable neither percentage identify recommend list complete worker ask guarantee complete worker second second list movie suboptimal answer baseline statistically permutation test recommend list suitable compare equally observe likely nan permutation interpret unlikely worker rate randomly reasonable evaluate dark action action action dark dark action action action action dark dark dark action life cover four popular assign insufficient diversity popular item happen movie recommendation movie similar strongly dominate movie assign diversity therefore movie movie less popular movie sum outperform topic utility low utility utility item item goal evaluation recommendation multiple profile million rating star user rating create profile recommend movie movie rate split randomly ratio factorization predict predict utility three report split use create profile whereas along utility profile movie list step belong create multinomial popularity movie rate rate create user preference recommendation proportional preference term diversity utility compare rating movie utility b rating list reason predict keep evaluation recommendation rating movie metric metric consider diversity choose first evaluate diversity first metric combination avoid compound metric diversity different metric distance recommend intra diversity list movie diversity exploit accumulate utility recommendation discount list item user user assign movie ndcg ndcg ndcg achievable item utility score compound metric utility diversity intra list discount avoid use c compute every recommendation list setting metric user figure utility rating movie utility movie recommendation step factorization trade mean diversity ndcg diversity situation diversity superior regardless score high ndcg high highlighted utility simultaneously also superiority compare figure observe improve hence recommendation various recommend utility operate diversity curve intersect diversity metric setting significantly utility start importance diversity take objective superiority balance diversity goal parameterization combination diversity system focus composition recommendation list diversity list pose utility diversity important propose diversity maximization modular different aspect user aspect cover study utility movie diversity movie cover study find baseline maximize diversity diversity result execute individually moreover diversity superiority objective combine modular submodular utility orthogonal maximize modular submodular conduct variant future item respect consumption user diversity apart diversity contribution item list investigation recommendation furthermore tolerance redundancy item agreement opinion redundancy item opinion future remark united com yahoo united yahoo com recommender diversity recommendation list replace diverse work method recommend diversity user optimal study offline incorporate evaluation superiority baseline popularity recommender system social network service item user typically recommender interest list top na I score item however sub recommendation instance recommend popular appear recommendation target topic recommend potentially diversity recommender diversity deal recommendation list cover single recommendation topic increase chance answer recognize recommender heterogeneous term interest music incorporate diverse member music diversity recommender system interpretation query desire query come unless topic reference aspect document formation virtual together maintain diversity come account decrease item relevant objective modular diversity find free maximize item subject diversity interest cast find modular list interest represent high utility primary concern maintain avoid redundancy interpretation suitable diversity conduct extensive crowdsourcing list baseline diversity superiority list superiority prominent variety diversity utility recommendation successfully overall analysis recommendation diversity priori tune two fold computationally aim improve list maintain evaluation online user offline demonstrate support propose element instead furthermore instead represent refer approximation relevance metric combination metric trade utility diversity give rank item create choose maximize relevance minimal movie also cover preference movie select recommend movie list substantially user movie recommend movie list great another movie utility ccccc utility x movie select first recommend movie recommend movie movie ground movie whose movie utility movie utility movie recommendation irrespective chance idea notion diverse recommendation contribution movie movie place diverse movie higher
particle practice instance infinitely solution matrix transform transformation mode behaviour parametrize general set er rao low parametrization major jump linear express sufficient eq parameter write use detail verify inner indicator compute smooth calculation tu indicate sa sufficient argument maximization last product particular lagrange multiplier maximum algorithm property rao matlab author identification jump markov mode n backward trajectory initial value mean run transfer n particle black blue particle use plot new rao significantly compare mode r low pass white noise initialization rao pick parameter particle particle figure run estimate plot figure multidimensional tb error mode type smoothing expectation solve leave smoothing key introduction filter maximization step could obtain term particle idea smoother great outside jump model something worth indeed possible turn contract contract department uk jump consist identify jump challenge derive expectation recent solve inherent conditionally discrete thought different variable hence variable furthermore zero measurement define via interested identification jump number mode formulate problem measurement input eq c unknown static throughout input challenge maximization type strategy separate first nonlinear state nonlinear solve monte particle mcmc particle mcmc systematic strength need mcmc estimator jump markov model inherent rao rao particle approximation recently switch exist jump approach stage segment approximate hybrid recent relationship smc approach mode dimension bayesian nonparametric iterative maximum em maximize maximize iterate solve maximization think select estimate likely involve nonlinear intermediate intractable force solution e step still intractable carlo smc particle general arbitrarily q view smc carlo algorithm space kalman expression filtering f tp relevant pdf normal mcmc kernel smoothing make ergodic markov define fold jump simulate limit initial ergodic allow smooth arbitrarily k particle important particle similar approximate accord draw probability proportional trajectory kernel map useful ergodic admit detail requirement fulfil rao fusion kalman filter particle rao somewhat process markovian handle rao rewrite adapt cholesky etc includes find rao discrete draw linear handle implicitly define algorithm estimate construction ta ia ji ni I ip
variable shrinkage summarize topic tuning penalty ridge value lasso mm penalty red square blue mark rectangle versus shape seem pure rectangular contour illustrate matrix lasso look equivalent derivation set solve present start lasso iv plus positive obviously decompose positive continuous mixed sign twice disadvantage case real disadvantage intend often two sign gradient interface solver tight obvious decomposition residual yield absolute solve derivation constrain equal derivation tucker lagrange multipli hold extensively section incorporate row analogous condition scale define solver quadratic model qr decomposition h green derivative sx contribute similar factor projection contour straight slope ellipsoid conjunction every lasso matrix penalize quadratic lasso penalty hold e huber aspect mention iterate compute lasso sign restrict square e g b calculate numerically qr solution lar coordinate descent model relevant incorporate advantage via qr svd interesting library qr svd algorithms quantitative qualitative path identify outer decomposition bridge regression ridge nlp augment penalty uniform b variable ol augment negative nlp minimize nlp minimizing solve nlp
similar computational aspect review resort non whose global difficult remove lasso square second hand notice bias contrast total denoise automatically remove early stop widely linearize bregman extension boost descent method solve problem linearize add linearize gradient ascent apply lagrange dual solution without setting early regularization necessary recovery basically nearly lasso stop without loss differentiable bregman linearize iterative soft widely different name literature q move shrinkage linearize generate path easy implement distribute store distribute choose load balance variable reduce input matter independent unit nearly parallel truly scalable implementation divide block row column recently unit communication link center long scheme multi party regression internet incur introduce denote complement submatrix generalize adjoint inner similarly omit reason discuss right throughout rest property bregman bregman generalization linearize bregman proof preliminary summarize section piece regularization iteratively ty kn ty go piece ensure existence uniqueness uniqueness bregman differentiable solution addition uniqueness define case uniqueness impose I involve must impossible argument hold f continuity nonzero remain either stay continuity kkt identify specify time strictly unique strictly strictly unique lastly column existence uniqueness linearize show follow existence linearize bregman right unique solution become lipschitz theorem ode close piece sign remain reader sign oracle least subject sign sequel question remain continuously differentiable continuous uniqueness path necessary noisy bregman restrict strong x restrict empirical linearly say one variety name exact recovery e covariate effectively check support alternatively mutual incoherence eq hold hold sharp translate kkt relation temporal bregman lie support incremental add drop distinct projection mean piecewise incremental sense bregman drop bregman omp fail bregman gets plug side ensure mean kkt split trick lasso part integration eq reasoning q incremental process tt q lose dropping happen lasso mean bregman statistical consistency path let false time moreover signal e reach sign sense statistically mean bregman path light incur path bregman path averaging cause bias tell always dynamic pick select incorrect reach bregman return path false strong eq probability unbiased strong consistency ensure square root assume subsection linearize section take omit linearize bregman linearize bregman give establishe consistency linearize path time magnitude eq c n sign consistency guarantee bregman rate monotonicity appendix x statistical linearize bregman linearize bregman big enough satisfy path enough x nk k linearize bregman ensure noiseless analyze differential associate restrict evolve potential fast potential study dynamic oracle continuity multiplying obtain dynamic induce oracle one restrict evolve outside subspace true strong decay reach goal decay nonlinear differential case gr bellman lead tight generalized right strictly ensure concern stop time reach equip generalize consistency say potential multidimensional dynamic drop ready sign discrete find support lemma depend noise depend noise thus datum leave stop q bregman stop minimum magnitude path along sign bregman must one stop bregman full addition rt comparable factor bind arbitrarily present proof theorem show probability bregman stop sign consistency algorithm stop bregman sign high residual x mean stop stop p x experimental relation lasso linearize experiment identity three lb lb big curve goodness roc receiver regularization lb level positive roc curve auc roc large auc indicate pick path repeat deviation noise reasonably become big lb decay h c lb lb noisy signal dynamic bregman linearize discretization lead widely linearize stop regularization achieve selection consistency unbiased estimation linearize bregman linearize stop bregman direction rule setting q operator mean false positivity consistency directly proof differential inclusion path matrix integration side obtain follow take directly union inequality end denote eq noticed mind continuous convexity generalize min min min min min dx min let notice sn n cs sx sx lemma false positivity lb monotonically upper suffice b negativity ensure cl big min min first version linearize bregman lyapunov subgradient side iteration present discrete q hold x recover continuous step size discrete end continue lb monotonically lemma admit monotonically
elsewhere establish completely analogous instance theorem computation must uniformity repeat seem intensive maximal since interested check contain ball intersect radius equal center contain exist sample center check uniformity rejected determine mr uniformity reject compute performance see addition value maximum cycle b r smoothing parameter convex estimation estimation mention table illustrative really estimation rs estimator theoretical convex hull estimation reference multiply table support estimator h r h support estimator last benchmark multiply size rs rs benchmark multiply rs mm mm estimation rs result rs present behavior hull provide result case estimation real estimation mainly see significance also discuss outlier provide risk satisfied select smoothing rs estimation error criterion large accord rs increase would closed ball subsequence converge subsequence converge subsequence denote contrary addition contradiction exists converge since necessary guarantee existence point boundary nonempty satisfie necessarily unique aa h pt pt aa aa relate existence coincide onto onto contrary condition line accord q pt let condition proposition without generality prove impose restriction verify accord auxiliary necessary first nonempty converge clear define ball accord prove take account notice arbitrarily give exist meet nonconvex nonempty let exist convex ensure exist open show contradiction satisfy straightforward convex nonempty accord loss generality replace ensure r bc imply nonconvex parameter converge r theorem consequence proposition ensure theorem hold converge notice proof criterion consider support project develop major science point drive propose select value hypothesis drive rate convex hull much flexible shape condition uniformity reconstruct practical reconstruct figure imaging make shape sort center see influential radius ball whereas large considerably ba ba shape respectively sophisticated priori instance convex contain analyze depth restrictive introduce flexible say center closely c literature produce reconstruction point approximately set although convergence see depend influence present however practically dot comment special emphasis especially give select aim drawback select available problem selector minimum span automatic see figure area ball sample calibrate maximal result uniformity opposite compatible uniformity back two usually set bound nonempty hausdorff hand borel measure hausdorff physical proximity measure distance completely useful shape hausdorff boundary evaluate ba section optimal establish drive achieve hull estimate practical analyze performance selector analyze proof defer reconstruct estimate definition define let nonempty compact nonconvex property verify obviously condition however take radius convex define gap maximal point lebesgue denote uniformity reject quantile instance applicable et select maximal region volume ball radius b dot show bad big value smoothing allow clearly uniformity hypothesis contain uniform must small idea technical section existence guarantee consistently compact nonconvex nonempty converge exposition infinity value reject behavior set prove guarantee go ensure rate hull compact nonconvex nonempty define define converge
consider determine asymptotic derive belong alternative assume condition normality determine weight maximize restrict test power asymptotically class estimator et unbiased pn tr tr tr tr show assume pn lemma adapt asymptotic asymptotic improved estimator derive expansion percentile addition percentile replace percentile compare sufficient allow high low local power asymptotic among test compare test test define significance level power valid normal hypothesis covariance replicate n point provide z consistent level table multivariate normality significance level approximate nominal reasonably tendency conservative power select replication size r ep summarize among show test test good small test always high close behave location asymptotic local asymptotic test stable statistic power conclusion recommend range author extensive discussion ms simulation numerical preliminary central form random diagonal express tr nz lyapunov lyapunov lyapunov lyapunov moment distribute diagonal tr tr tr tr tr tr tr al expand stochastically orthogonal eigenvalue independently statistic expand pn expand stochastically follow tr pn pa pn stochastically c pa tr tr monotonically find c derivative get tc power define pa pn h e h h tr moment relationship moment obtain coefficient relationship characteristic ct result use lemma therefore necessary test asymptotic test superior local asymptotic power pn pn pn result section ex plus location average statistic mathematical mail com school business loss exact approach du paper test comparative difficult local power new weighted statistic determine maximum local induce local asymptotic respect show statistic parameter power location subject dimensional hypothesis test traditionally n nan use determine important issue study maximum asymptotic power equivalence easy asymptotic size simultaneously go sample
challenge naive ignore presence set dramatically valid see reference therein involved focus target irrespective selector generalize crucial quantity confidence desire also quantity denote throughout selector standard explanatory precise eqs beta set explanatory obtain explanatory correspond regressor model call design dependent coverage target typically lead confidence irrespective design error cf explanatory optimality little relevance thus cover dependent target may next coverage target denote standard coverage infeasible cf remark particular target suffer discuss precede paragraph model interval cover coverage minimal nominal level interval valid irrespective selector representative situation extend confidence design active problematic select model component situation costly irrelevant component resolve variable maximize inactive width interval moderate extend seem consider discussion remark interpretation introduce model variable observe explanatory design confidence cover minimal coverage nominal computing interval report proof stress naive confidence interval selection correct target valid easily set confirm simulation model full identity matrix refer eq linear hence outline square observable chi square degree exist n p extra generality provide additional estimator possibly observable well full column retain relation use follow empty write cardinality write abuse square estimator inverse variance imply restrict square obviously estimator function depend allow procedure make allow case procedure restriction range power case consider universe quantity vector empty paper aspect force infeasible cf presence predictor post feasible infeasible target emphasize design apart cf infeasible discussion interpretation remark nominal confidence confidence interval design form depend depend interval zero reduce thus contain replace eq ci naive see numerical line relate quantity distinguish observation regardless post select observation component consideration observe follow general ci hand display neither p x consequently distribute cx collection ratio rx positive hold quantile universe subset maximum model rs consequence precede immediately interval general form ci proposition guarantee procedure post selection prop feasible compute select define replace hence via course depend want stress see arbitrary interval satisfy costly compute prohibitive sphere column convention eq distribution degree freedom represent expectation depend hence hold f quantile whenever obvious besides want stress full full full appear extension contain publish bound note counterpart reflect model possible interval event fact available confidence also cf section k immediately imply corollary arbitrary eq aim remark note coverage represent call full clear coefficient target inference depend dimension rise component mention amenable combination mx argue rather infeasible apply observe justification argue note applicable desirable problematic iii proposition post approach infeasible predictor sense nx draw otherwise optimality feasible typically force ii carry dependent provide universe ignore exploit expense complex sketch change degenerate x x x xx illustrative subset describe situation act regressor correct belong happen satisfy select overall interval presence want universe precede highlight require subsequent overall distribute possibility rise universe e present extend appropriate contain ol framework make become problematic model except cf framework continue otherwise continue proposition chi denote constant stress union bound conservative showing lem exist vector infeasible inspection also analogue replace random sphere column space close finally use k x bound improve marginally assume full column almost accord matrix moment denote maintain section distribute chi square distribute section conditionally section dependent conditionally shall estimate size furthermore result fix subset motivate target eq inverse interpret inverse justification infeasible infeasible call subsequent remark section design nothing post analogue predictor mx argue xu u additionally force straightforward computation prediction normally thus predictor depend square normally good predictor class depend shall remark interval section interval procedure mild typically procedure bic design sequence random weakly dependent moment also degree freedom course choose condition present recall interval model confidence asymptotically valid ci confidence arbitrary arbitrary sequence positive integer replace relation asymptotically target minimal choose theorem also continue replace condition uniform consistency property generally note satisfied post condition lemma consequence use interval remains valid allow also early infeasible term depend converge repeat result target continue hold independent ci continue consequence noted begin design dependent continue hold conditionally consequence turn drop instead continue cover rate ci generalize set employ provide statement concern probability converge difficult remain interpret statement outer establish presentation treat canonical coordinate compute proposition square estimator set show canonical decomposition section x replace precede analytically integration solve shall f freedom case calculate kx c probability obvious costly alg involve search solve search around one tractable one hour computer since alg solution monte replace involve range define decompose continuous analytically c approximated function suppose satisfie identically sphere calculate quantity choose hold solve uniquely always except hold uniquely r p beta observe except h furthermore r converge kk solution search negligible algorithm second equation run convention argument positive obtain version one integration negligible run much search compute universal bind constant negligible compare case uniquely determine uniquely determined exist unique exist unique positive case approximate equation convention constant remark modification square distribute freedom ii appropriately generalization alg tractable report quantile beta always reasonable numerically length confidence interval length confidence interval selection selection lasso matrix obtain exchangeable design obtain ci six replace either degree constant interval hold provide standardized contain ten explanatory term intercept square water square slope percent thousand surface index capacity water water hour period hour take except intercept explanatory peak choose length interval ease burden standardized length belong ten universe obtain approximate supremum monte follow monte step keep large evaluation monte algorithm second step monte standardized improve standardize constant combine increase standardized length indeed standardize increase standardize obtain decrease standardize almost standardized length say interval short interval standardize base standardized note costly especially length interval close confidence interval standardize length size confidence discussion confidence length yield sake brevity standardized confidence always increase respect evaluation minimal report costly would report sake brevity minimal estimate various aic bic procedure procedure explanatory intercept information constant universe intercept explanatory bic penalty equal aic bic aic minimize result intercept explanatory lar outline regressor regressor residual lar cross cv lars intercept model comprise column obtain standardize similar obtain exchangeable datum independently component nine row use target vector probability target investigation coverage estimate monte random gaussian column monte overall record currently confidence constant averaged small coverage first repeat carlo sample record small second confidence time estimate minimal coverage consideration coverage find obtain search upper coverage
multivariate say successively optimize hold solve successively achieve natural statistic netflix attention rating whose view coordinate recommend item user align every miss coordinate multiplied penalty introduce avoid typically quite impose fix decade technique attract much attention suppose penalty ridge convex recent year start mc hence detail scad readers original function interesting currently prefer penalize model coordinate coordinate solve fix successively lasso descent scad guarantee global make get inferior solution find get serious getting saddle introduce share perspective share perspective expand search slightly comes avoid possible thus proposal undesirable location search choice space upon traditional include space idea mc penalty stress motivate minimize point fixing thus see actually particular point able search would suffice main simple strategy undesirable observation follow run continue search sufficient strategy work process improve starting outline minimize key insight switch different undesirable location share perspective early alternate search component scaling adjust accordingly next optimize scaling begin somewhat helps scale view respective space interpret illustrate time search still much understand scale conduct improve well illustration imply equation suggest expand simply joint kind search view avoid full simultaneous illustration versus imply joint search denote fact would matrix solve switch saddle inferior solution minimize eq simultaneously bfgs search quite expect space effect expand scale type subspace propose restrict search factorization search item mathematically feasible item search important search notion informative describe two trying determine set vector large establish baseline space serve illustrate power search index incorporate shall report even sophisticated choice specific take still space observe improved context front highly would expect decrease lead depend negative selective correlate pre selective compute fit surface warm start penalty think fitting introduce challenge work fit concerned good pair surface solution sequentially surface warm start desirable solution inferior warm worse empirically common occurrence keep surface surface warm point surface coordinate use previous warm like obtain previous point warm keep actual surface surface strategy conceptually think easy surface triple computation experimental mc matrix demonstrate factorization million review dense subset million review rate item rate time restrict allow number item good performance wide test rating run absolute mae mse amazon baseline translate rmse mae baseline dense subset information item subset fair comparison cross show average show table optimal cccc scale subspace sec sec k factorization cccc sec k appear conduct greedy strategy fast factorization mae versus mc demonstrate mc predictor generate mean whose entry linear predictor space logarithmic spaced logarithmic small percent converge restrict indicate coordinate find relatively remain solution expand reduce small half percent decrease little large nonconvex average percent decrease point computed percent measure show strategy lead improved mc terminal regression comparison selection call conclusion article general upon nonconvex switch conduct different illustrate problem namely factorization mc penalty help search subspace carefully undesirable produce notable algorithm factorization regression contribution acknowledgment support explicit problem solve identify satisfy condition write hence
outer maximization f study extension show simple strategy efficiently output dependent however threshold achievable technique focus conditional consequence particularly set corollary derive general make empirical batch observation relate depend example explain prediction optimize micro macro false increase optimally thresholded characterization maximize relationship achievable score score discuss version result start valid lemma confusion tp ps ds fp ps ds tn ps maximize ambiguity rule conventional sensitive latter side replace later elaborate undesirable divide region except similarly besides add decision negative add simplification limit claim case output calibrate predict assign rule maximize decision calibration optimal assign eq simplifying calibrate half maximum assume intuition theorem save version result section label confirm length thresholded output define count gold standard total function ap achievable macro tp fp tp aa loop predict positive sort proceed stop number pick follow stop designing maximize batch observation uninformative potentially undesirable prediction depend distribution prediction exceed note exceed threshold never exceed predict instance batch assign great uninformative classifier calibrate uninformative thresholding maximize result uninformative seek maximize label denominator positive predict actually difference always positive simplification derivative whenever positive uninformative expect maximize optimally thresholded uninformative base close figure point context macro optimal batch micro macro micro rare additionally macro average uninformative imagine know base th label rare perfect actual positive rare nothing predict positive gold label thus single trivial system adjust constitute average calculate rare consider label trivial optimally thresholde trivial however improvement perfect common macro rare label macro uninformative classifier micro optimize predict macro optimize calibrate half positive classifier confident prediction micro macro thought threshold distribution assign low base predict maximize macro micro evidence study discuss thresholding macro undesirable real macro f consider assign tag vocabulary mesh article literature represent abstract bag vocabulary consist word tf preprocessing step word rare multiple square probabilistic overfitte approximate training value consequence rare lose problematic probabilistic classifier theory relate threshold plug rule micro macro prediction micro thresholded among mesh count human middle label base uninformative thresholding positive label macro system overall maximize experimentally rather ground maximize winner future phenomenon overfitte different threshold f empirical performance high threshold converge true higher demonstrate idea uninformative uninformative uninformative uncorrelated independent label accuracy convergence irrespective base threshold depend optimal threshold occur threshold close threshold consequence consider threshold experimentally sort order uninformative lie score include accordance maximize threshold f lower optimal far f optimize identify threshold whereas threshold identify problematic lead issue identify arise nonlinearity score treatment label choose label threshold maximize select plot predict threshold shift positive negative base low even number derive decision positive theoretical well achievable optimal make threshold classifier uninformative predicting maximize maximize macro uninformative contrast micro maximize micro potentially undesirable rare every desirable precision reporting alone sometimes practically optimize compete choose winner behavior edu insight maximize score binary context harmonic classifier micro average macro use value achieve optimum calibrate conditional uninformative classify example behavior undesirable surprising metric prediction commonly classification classification area supervise machine micro average average instance average use macro impact performance per experimental property performance minimization incorporates convert numerical classifier latter scenario alternative case belief produce thresholded
million point since keep gp use parallel computation achieve induce auxiliary involve al al adopt combine parallelization gpu acceleration experiment inference consideration construct likelihood induce covariance see practical application parallelization decomposable specifically write nm notice associate treat introduce gp maintain tight straightforwardly propose variational induce analysis unsupervised scenario namely w parallelism naturally distribute portion although computation recover computation collect node get locally distribute optimizer determine locally exist bfgs currently collect gradient global parameter optimizer rest computer write gpu write write scheme mention scale exploit gp quantity constitute computational bottleneck go something dataset make gpu acceleration hardware number small core efficient ideal advantage specialized architecture computation onto gpu making properly divide parallelization within gpu gpu synchronization memory expensive division difference try choice design assign block gpu computing assign block gpu intermediate share local write memory gradient depend division apply acceleration parallelism section integrate assigning subset portion gpu b gpu acceleration synthetic mapping radius rbf function recover representation gp reduction latent induce parallelization configuration processor directly run iteration time scale speed parallel implementation communication overhead negligible speed gpu core time show computer speed additionally algorithm gpu acceleration constitute counter gps applicable implementation plan model
section statement existence compact fix scalar determine c prove compact invoke existence establish iterate need classic fix theorem map iterate map every begin classic theory able proof arbitrarily converge mention concave whereby unique first necessary fix iterate fix show merely iterate repeat concavity unique yield rewrite introduce new scalar choice addition decrease large large respectively multiply numerator denominator first I p multiply definite substitute small rearrange write p multiplying span definite therefore equal let datum large zero small section one find imply converge large three decrease toward zero increase previous find large one case proof toward similar possible show ii end explain easy convergent calculate eigenvalue become eigenvalue small eigenvalue become multiply obtain small want calculate eigenvalue invoke proof convergence existence ml generality consequence ml solution exist still converge convergent possess structure subspace generalize present strong method highly outperform manifold technique experimental easy observe one choose become two variable first without fulfil firstly entropy namely expression knowledge surface sphere obtain hold expression average loss radial need radial follow square easy f radial expression average kl divergence expression follow log summing observe rotation axis kl increase kl become infinity fit substantially generalize modify fix parameter simply use counter em derive refined merge identify false division cloud cause minima practice iteratively find candidate merge split split merge merge one merge explain difference merge however entropy clearly k nk propose split solve need therein split improvement limit split lead improvement split threshold split stage find component reach perform step merge ml describe splitting small parameter merge typical optimization overfitte amount kn old measure bias test stop fine propose successfully alternative explain split recover cccc pt pt stage cccc different stage explain experiment show speed three bfgs trust also test like plot dataset size van dataset contains exclude etc extract image patch remain pixel intensity white standard evaluate mi mi bit pixel intuitive patch formally hx entropy pixel procedure mi dc component different observe mi change increase show simple capture layer reach parsimonious propose radial follow ica correspond deep boltzmann machine mi method explain emphasize difference mi mi improvement capture distribution claim effect baseline rate gauss spherical mi bit high different patch study elliptical existence uniqueness maximum modeling art remain direction study involve tool investigate non gaussian mixture gamma process complement process investigate behavior model rather gaussians hope basic outline encourage researcher rich author title author ac ir school college engineering institute institute biological de institute study elliptical parametrize tail peak rich ml nontrivial develop globally convergent merge expectation propose mixture modelling model analysis often capture express heavy tailed independence situation unsupervise mean usual gamma additional factor encode tail behavior gaussian pt leave display study stem broad mean successfully multivariate financial modeling pattern application survey application reference paper potential direction recovery multiple various field encourage nonlinearity easy implement work appeal concept brief mixture context paper make fix ml parameter handle numerically nonconvex despite optimality demonstrating gain obtain manifold variable dimensional random variate one variable kullback divergence zero gamma multivariate begin special scatter matrix exist
yield mini variant filter normalization convolution require modification incur negligible overhead different filter crucial version mini max pool convolutional two investigation imagenet dataset dataset contain million image image image assign category evaluate image train network supervise criterion work imagenet black box visual front involve possible benchmark involve digit involve imagenet deep mini network convolutional employ standard max pooling fair comparison case six convolutional follow connected way softmax throughout normalization connect layer type conv conv conv conv max dropout channel filter input x pooling baseline specify employ pooling accelerate neuron convolutional difference architecture specify neuron layer use mini max layer mini invariance layer accelerate layer conv conv conv conv weights texture report imagenet yield well find improved aspect ratio manuscript aware work report max pool neuron convolutional c max pool top quality propose imagenet feature purpose dimensional without fine weight new svm experiment preserve aspect per report different pool norm classify network follow set find beneficial convolution layer rate maxout art c maxout model maxout cifar mini mnist cifar comment property imagenet validation correspond et contrast max pool without quality somewhat improve paper representation deep neural successfully substitute consecutive convolution pooling empirically learn around well pool baseline challenge imagenet develop exploit amenable image reconstruction objective software framework get publicly source code file fully acknowledge convolutional neural recognition task propose replace consecutive convolution max model use mini small learn organize propagation learn assess classification benchmark imagenet convolutional network architecture pre imagenet mnist offer learn increasingly work demonstrate benchmark build like sift success interest learn computer refine aspect feature successfully employ deep successful image feed neural fashion propagation availability annotate dataset gpu partly build block deep image around convolutional layer field spatially share abstraction resolution max build around image translation aware representation position deep consecutive max pooling layer convolutional neural spaced patch across filter max pooling hand filter dictionary within input layer center alternative pooling employ mini mini enough desire position output mini maximum across mini use output per spaced within propose primarily train propagation mini achieve conventional pool convolutional network classification converge especially normalization accelerate convergence imagenet report excellent classification network build initially towards video modeling unsupervise explore mini variant sift learn similarly maxout critical difference layer hard extract value overlap share significantly reduce maxout conjunction max substitute moreover maxout perturbation create inactive contrary require regularization connect explore pooling avoid build paper unsupervise deep learn adjacent share area thus relate c mini max pool convolution filter pixel span channel represent element convolutional densely position also resolution map produce window position pixel apart map max pool convolution consist channel filter point attain propose convolution scheme replace spatial mini filter extract patch regular maximum position attain fix dual alternative filter max
different question date concern discovery approach practitioner expectation science brief end preliminary cluster achieve management maximize adopt specific extended pair base investigate assess potential learn continuous assessment sensor accurate use highlight really setup assume really hierarchical experiment highlight ball link different cluster learner identical imply learner previous behavior static fig south percentage cycle duration fig west mm involve duration maintain speed group goal increase stroke goal single second increase stroke receive arm arm arm suppose learner trial cycle period successive maximal equip angle frequency hz act individual relative define anti relationship series relative fig record cycle per trial cycle science view specific aim study assess behavior potential learner assess investigate possible existence behavior learner condition possible search group analysis point point entire cycle may highly beneficial discriminative cycle learner machine view tackle cycle describe feature continuous nevertheless preprocesse priori trial directly describe fix feature use cluster reduction stage cycle similar cluster observation represent observation label likelihood know observation fisher em principle mixture lie lower choose efficient medium example hold realization observation group indicate decide discriminative let complement dimension lie discriminative lie discriminative complement conditionally help deduce observation follow center ensure impose projected discriminative subspace generative discriminative parameter gaussian enforce gaussians combination iterative fisher discriminative fisher use observation computed probability projection maximize variance whole tn ik probability parameter ik computation efficiency nevertheless back observation original feature enforce fisher penalty criterion penalize algorithm cluster purpose informative sequence cycle clustering consist number bic level highlight result qualitative learn learner visit cluster across different lead pattern group analogy whereas use lead type practice gray bar level height bar nan corresponding latent cluster point movement fisher existence highlight transition cluster group high receive second cluster analysis allow highlight additional strategy preferred fisher highly correlate feature interestingly sparsity qualitative need term c
family r enyi entropies bound entropy shannon enyi nonnegative order shannon explore entropy min entropy enyi entropy min entropy relation easily extend class entropy end bound enyi relation section explore analogy analysis feature examine atom investigate dictionary topological overcomplete explore diversity measure zero measure diversity measure synthesis dictionary input study connection conduct illustrate learning gaussian process network diversity quantify overcomplete spirit recent degree receive ph optimisation security technology research associate system model laboratory technology interest include statistical processing wireless sensor process hyperspectral author paper learn processing past year review linear collect dictionary representation orthogonality condition yield overcomplete dictionary overcomplete elegant nonlinear define dictionary dictionary coherence overcomplete dictionary associate uniform spread entropy definition entropy map space generalize shannon sparse recognition gain increase popularity denoise representation atom formalism combination define basis predefine dictionary analytical wavelet dictionary widely dictionary deal orthogonality bi orthogonality deal analytical structure orthogonality year dictionary adapt call advanced statistic fall orthogonality increase overcomplete dictionary largely several dictionary problem counterpart overcomplete dictionaries relevant increase diversity measure quantify diversity simple diversity certainly measure examine either fashion thorough way characterize pairwise correlation correlation atom yield last pursuit dictionary also extensive compressed sense diversity overcomplete analysis framework pairwise distance atom neural indeed interpolation unit distant turn operate pair atom thorough measure atom combination machine gaussian online filter component aforementione consider formalism atom nonlinear give atom transformation conventional nonlinear study network nonlinear adaptive aforementioned diversity heterogeneity dictionary relate mechanic randomness generalize overcomplete diversity illustrate atom spread comprehensive kernel within entropy propose finally entropy derive depend diversity connect input space follow introduce nonlinear formalism present diversity measure quantify overcomplete dictionary throughout examine section space section estimator kernel investigate network particular l reduce complexity pruning contribute learning measure paper connection consider analysis entropy term coherence provide dictionary extensive space generalize diversity conventional linear model well conclude outline issue banach space theory give elementary estimating eq atom approximate span namely include transform cosine basis orthogonality overcomplete dictionary investigate representation coefficient representation sparse assume view seminal consider available drive iteratively alternate data posteriori probability dictionary correspond problem good dictionary subject method direction respectively detail therein worth difficult atom formalism elegant framework tackle feature kernel hilbert space product induce reproduce I divide category projective product radial kernel kernel radial norm paper restrict kernel conventional model nonlinear investigate rbf include take q deal easy linear consist residual rkh linear investigate dictionary determination difficult technique turn literature problem recently filter context element factorization j projective linear radial inverse j synthesis overcomplete dictionary grow atom heterogeneity atom require diversity pairwise thorough atom consider diversity measure paper connect information source alphabet measure quantify randomness need average store investigation aforementioned definition investigate r enyi entropy finally section extend rkhs studying quantify diversity dictionary diversity criterion dictionary diversity simple distance atom tight atom factor tighter say distant residual onto atom scaling take yield impose atom latter rely atom exhaustive analysis quantify capacity approximate atom combination dictionary dictionary say follow correspond residual project subspace cost coefficient q respectively study thus atom dictionary approximated characterize correspond large atom mutually atom two initially match pursuit union basis quality consider measure coherence give coherence correlation atom definition coherent norm atom I coherence construct dictionary enforce cosine candidate include coherence latter exceed dictionary basis coherence rely correlate thorough measure large atom way connect coherence investigate norm operator indeed gram atom explore connect latter sake simplicity construct dictionary include exceed threshold throughout rigorous overcomplete diversity attempt bridge follow theorem coherence dictionary quantifying dictionary vice versa atom exceed coherent also coherence coherence exceed proof fundamental analysis approximate exceed approximate dictionary nj ii coherence aforementioned measure theorem randomness give enyi entropy shannon entropies quadratic see deal random code definition yield correspond datum know entropy estimator overcomplete end initially generalize
efficiently well efficiently undirected dynamic mix correlation decay show underlying update dynamic runtime nearly process epidemic cascade infection epidemic observe broadly number chain include spirit show relatively generate literature noisy game author structure study infer observation rest learn section present give theoretic reconstruct ise graph assume binary variable gibbs partition normalize distribution edge vector simplicity suitable minor modification accommodate external field implication node allow chain natural time discrete version dynamic write configuration time start rate spin notably spin eq randomness later efficiently graphical plausible generating process check stationary stationarity family local chain include know slowly allow imagine access potentially information obtain spin purpose convenient heat version chain take update denote node identity first arise update sequence observe time expectation spin argument amount convenience set performance use minimax good triple tend infinity determine focus derive arbitrary assignment implicit dependence imply conditionally hand side sample claim decide suffice simple quantity justify follow combine give q generality replace emphasize turn possible sign depend allow contribution scenario select poisson determine two independent connection formula conditioning edge expectation event shown estimate eq q inequality estimate plug spin spin remain plug already side reasoning bernstein find implication adapt surely theorem denote bind imply kl k k observe z x ij ij bind take union see state inequality suppose kx kx bind suffice clique single remove I large removal difficult identical consist clique odd fix perfect clique cardinality edge value capture effect edge dramatically weak suppose section prove theorem use inequality kl divergence kl zero bound parameterized construction marginal clique divergence projection equal abuse project relevant clique measure keep update initial configuration draw index uniformly note configuration select symmetry construction clique consideration last symmetry bind probability likelihood later event give term mu mu l mu last equality expectation justify symmetry bound lemma eq times suffice uv uv show uv uv plug clique get second inequality take display main message paper quite setting underlie ise markov generalization observe acknowledgment grateful comment nsf grant office award nf david laboratory systems electrical science center school technology mit undirected graphical dynamic chain sequentially nod frequently additionally natural access direct low main work reconstruct binary pairwise theoretic might include stock financial user network markovian govern interaction interaction site dynamic underlie fit traditionally pose abstraction many natural generate graphical I sample hand represent focus low algorithm generate turn approach scale procedure observe paper node neighborhood search candidate node maximum prohibitive focused find
harmonic parameter secondary stepsize vary achieve stepsize secondary stepsize early result performance run quality cause slow necessary secondary stepsize run stepsize mdp stepsize stepsize outperform secondary suggest insensitive choice secondary stepsize line figure value range rule slightly performance value achieve secondary stepsize robust secondary figure rule worse slow horizon harmonic choice large consistently bad iteration good compete stepsize tune continue increase even slightly approximate iteration perform however become discount increase appear discount signal harmonic stepsize good perform poorly contrast insensitive parameter simple yield robust several stepsize mdp manner randomly pick let transition state state lead variety value stepsize upon state n accord next random briefly reasoning policy function make also implicitly stepsize stepsize affect affect visit affect ensure good visit practical algorithm quite problem outside scope stepsize policy state action run stepsize iteration evaluate follow find cs ns ss quantity iteration algorithm measure stepsize harmonic rule secondary stepsize secondary stepsize order good magnitude also order harmonic number visit see discount performance later half conclude yield harmonic tune sensitive visible secondary competitive horizon bias period behaviour run cs v ts n evaluate policy performance discount approximate horizon rule harmonic rule well easily horizon state rule competitive overall mid largely close harmonic finite horizon magnitude stepsize formula synthetic state transition stepsize identical last horizon apply early horizon stepsize observation propagate backward across time early period go exploration curve suitably calibrate last demonstrate conjunction present problem appear finance reservoir management abstract setting wish keep stepsize generic contain dimension amount resource currently price resource represent represent positive time increment action incur make decision resource minor change could demand continuous impossible furthermore programming post extensively state post pre obtain post state adaptively price discretize sec optimistic initial discretize continuous price discretized detail volatility spike pure choose process visit level generate random price stepsize architecture secondary stepsize secondary stepsize stepsize rule iteration decision cp ts horizon average path report maximize large require several policy consistently improvement figure harmonic horizon several iteration improvement harmonic experiment offer evidence new applicable complex post difficult expectation continuous advantage stepsize set encouraging sign propose mathematical stepsize single mdp derive new slowly application importance stepsize stepsize value approximation single state stepsize inherent prediction single considerably simplify estimate extended mdp horizon test lead rule stepsize rule harmonic tune sensitive stepsize test stepsize find rule conclude rule conclusion harmonic tune particular result strength adjust evolution function lack know converge equation stepsize bounding reach fast time differential approximated derivative step definition onto positive interpolation define recursion greatest less great writing weighted property omit increase derivative interpolation function eq dnn v bound bound increase across generalize define boundary condition differential equation solution order equation integrate dm cn dm plug equation imply require clearly similarly write n side observe q v complete inductive denominator show numerator inductive result n g n proposition rewrite subtract side side impossible contain denominator additional relation rule originally optimal refer alternate name adjust kalman filter processing sequence choose formula n smoothed stepsize minimize prediction term represent design general goal track scalar move signal general violate assume use rather bootstrap old approximation single construct reflect derivation improvement first explicitly incorporate dependence approximation updating handle bias would value use rise special c average period easy also secondary sensitive secondary stepsize conclude note may process observation construct inherently lead discussion fa fa nsf contract theorem definition prove wide care management algorithm dimension stepsize update operation research computationally intensive obtain popular parameter produce result tune stepsize prediction improve short insensitive new dynamic health care management management period planning period operation often require solution nonlinear integer quickly stepsize control merge horizon dynamic value state possibly action horizon describe take difficult solve curse bellman solve approximately name dynamic reinforcement however literature rule produce theoretical provide insight stepsize nonetheless stepsize slow construct single poorly remain part remain hundred poorly part stochastic adaptively estimate include rule stochastic bar variant additional challenge mdp work heavily tie whereas practical learning approximation arguably approximate view adjusted kalman selection study behaviour general dp stepsize bias tradeoff dependence model contribution computable demonstrate form easily stepsize stepsize account dependence tune general stepsize rule limit stepsize provably numerical comparison set sensitive rule insensitive importance allow focus strategy concern poor tune stepsize rule demonstrate action derive optimal stepsize set section new stepsize motivate stepsize commonly stepsize slow dynamic programming stepsize commonly refer refer inside observation albeit reinforcement gradient kalman filtering processing challenge close reason stepsize adopt adjust exhibit stage require extensive adopt different instead general close reduce equation independent identically prediction formulation simple general mdp tradeoff variance govern stepsize observation consider finite bias general issue exhibit complex behaviour subject insight issue stepsize capture able program allow high stand horizon mdp recall decision iteration optimal state steady illustration multiplicative bound free plot upper proof imply stepsize enough show apply proposition convergent subsequence subsequence find proposition know accumulation subsequence subsequence convergent subsequence follow accumulation immediately result observe vanish stepsize standard rule benefit rule design avoid stepsize valuable finance price management tight dynamic stepsize secondary stepsize choose stepsize become stepsize calculate optimal secondary stepsize require estimate period straightforwardly collect extend many action replace reward action depend visit reward however policy expect state system wide dp sufficient keep estimate similarly store dependent quantity dp example separate pair lead stepsize mdp generic complex problem store procedure initialize function solve let wide cs n stepsize x x x increment suggest stepsize avoid period mdp steady policy briefly mostly carry surely arbitrary
I irreducible two get call property ergodicity converge global balance gb condition stationary book mix strongly recommend book mathematically gb equal total employ gb special pairwise balance x py oppose gb db gb flow reversible markov usually numerically enforce mcmc gb db prior due metropolis mh generalized use algorithm first specify acceptance probability way new element total converge obeys db fulfil acceptance hasting mh ax mh process know converge section eventually interested average correlation observable spin transition equal ft ft ft ft fx go infinity converge stationary f xx omit dependence time subscript average dynamic autocorrelation describe correlation observable autocorrelation autocorrelation lag formula simplify ft f measure equilibrium exponential autocorrelation integrate autocorrelation inverse autocorrelation observable q autocorrelation relaxation observable place equilibrium inverse frequently eigenvalue lie disk large degenerate follow eq spectral define large yy expression tx fast converge obey db observe obey gb absolute part equilibrium call autocorrelation ft check substitution integrate autocorrelation control carlo average excellent another source monte algorithms statistical book mcmc detailed balance dynamic phase example lattice steady mh unbiased neighbor occur rate mh diffusion require step imagine sometimes beneficial momentum helps spread air mh slow suffer critical fluctuation sampling phase transition computer create implement convergence open discussion add cycle cycle steady practical way create stochastic impose skew detailed enforce adjust block transition state simplicity skew balance stationary turn essentially cycle start another west east west uniformly node turn need correlation inverse system walk visit site east west site random site vertical horizontal make autocorrelation see steady spectral site suppose ise vertex exhibit symmetry infinite vertex carry spin spin configuration run ground energy ground degenerate pointing point equally probable size temperature dominate tend align though still perturbation break mechanic transition energy solely eq e em q fm fm energy free minima degeneracy functional perturbation external functional q fluctuation proportional give vanish fluctuation explain transition give excellent introduction spin reversible e confirm numerically autocorrelation copy copy would result system make ise ultimately notice degeneracy external remain saddle vanishe represent ise blue represent mh red label mh online dots respective represent reconstruct fit large asymptotic reversible respectively slope pt initially system state ax new x periodic three converge reversible excellent explain analytically mcmc ise reversible skew detailed chain converge example control speed ise critical create change balance speed change size configuration gb convergence field model observable natural besides several similar propose mcmc significantly reversible variant detail balance violate db reader rigorous paper compare reversible reversible asymptotic observable increase markov chain notation mh chen improvement still review reversible bottleneck energy rare energy vast rely symmetry ise lead reduction equilibrium violate balance useful phase soft matter protein structure interesting explore convergence know complete center support thank discussion rgb tool explore property feasible physics reversible chain reversible physical relax detail yet balance certain case acceleration root improvement reversible introduce ise complete ise review applicability converge physics quantum field sum evaluate amount carlo square already carlo outperform point carlo extremely
value copy appeal weight connection change student reach algorithm extremely whose student critical high student sense many character teacher sure response teacher stop teacher teacher pass ambiguity information dark teacher second get seem within gd similar table gd sgd gd sgd systems also simulation non domain repeat spin rather case find identify quantify system lead support acknowledgement david suggestion simulation mnist acknowledge support gpu research institute ny city university york facebook ai york ny minima real function challenge whose agree spin prove tend teacher mnist phenomenon network finally descent descent level interest need surface question landscape challenge especially complex critical problem context seek find match learn describe form desire another come physics align equilibrium reach reasonably fluctuation machine spin mechanic connection attractive nice slightly exhibit stochastic critical minima spin interest critical point spherical spin establish name example spin high point absence contain minima level long ground matter slightly random random extreme external landscape polynomially count critical critical point surface minimum flat jump critical landscape exponentially lie clearly existence point regardless alternatively favorable performance context spin ask question address difference discover hope light future simple two spin spin consider interaction interaction total energy represent lattice geometry interaction strength weak field particle alignment configuration matter landscape easy achieve assume favor alignment favor picture drastically introduces simultaneously attain rather landscape spin discrete state sphere particle interest study detail critical use far find reason triplet move remark critical hamiltonian eq critical landscape explore continuous symmetric global point sign index critical find asymptotic horizontal analytic description function analytic minima show expect decrease beyond critical keep denote number increase random variable sum implication hamiltonian extensive scale give sphere low energy ground respectively level vertical expect low index probability finding confirm deep trying level consider centralized loss approximate I hinge entropy sample loss increase fluctuation life true approximate property expect converge necessarily test fluctuation paper first landscape really imply surface energy arbitrary drastically landscape starting point descent narrow moreover irrespective simulation spin clear qualitative dimensional surface surface practical aside reach gradient start fix variable start trial criterion previous section hold concentrate hope nevertheless asymptotic fully connect spin couple modify polynomial achieve version way machine view sphere point product hamiltonian instead iw normalize
maxima give whereas deviation considerable head table gps high small type gps gps gps cm type gps gps gps h cm gps gps gps gps gps gps plan methodology control establish rely base calculate take production dependent aggregate relevant operating characteristic numerically simulation work stage accuracy formula stage turn estimator well gps extension inspection allow investigation firstly remain aggregate extent stochastically lastly end many day arise measurement curve iv curve extension effort well beyond article acknowledgment thank sc read part grant b dependent auxiliary observation central limit theorem suppose variable detail proof expansion independent expansion obtain continuity drop virtue continuity ensure validity well recall follow argument q virtue asymptotically normal fourth moment see virtue first assertion estimator standardize available additional third fourth consequence p standardize remark g measurement module newly virtue er device ed array easy nb entail go along line independent may jointly calculate account namely conclude unknown correlation replace observe replace p thing arrive proposition corollary keyword acceptance dependence energy deal construction inspection produce item reject motivated output panel capture appropriately production acceptance represent produce distribution power non side usually tolerance randomness true mean replace decision thus critical account module item quantity control although away specification item module directly since inspection infer fraction acceptable probability acceptance fraction unknown production control module module customer production far pass later check requirement operation stage done avoid e acceptance second seem double differently acceptance statistic e say one control sample instant reject first lag stage fact inspection already sample propose item form necessary batch spatial carry general apply sampling well organize acceptance operate two two inspection additional construct valid cover expansion control normality operating control independent realistic dependent lastly simulation acceptance back seminal contribution overview study double robustness normality type likelihood estimate fraction short production compactly approximation normality powerful mind favor acceptance rejection discuss focused type normality sampling inspection quality continuous fourth employ theory sample distribution sample simplify find extend difference additional discussion production smooth estimator cross validate kernel density adaptively bernstein purpose estimation application extension side specification focus quality become area result certainly adopt production panel high throughput production process cell sophisticated regard stack pair ease movement cell amount cell optical filter ensure process let briefly work compound positively bind four adjacent fill meet energy contact move contact top cell internal due differently power load cell make physical material cell stack bottom top substantially chemical material process associated weather may heat absence cause serious physical chemical electrical degradation degradation module couple internal module micro arise module production site construction improper image characteristic serious impact long degradation failure several year micro experimentally heat test loss drive cell surface ground degradation surface circuit market degradation reliability consequence inspection combine production construction notion rigorously depend lot pair sample lot accept operate item specification plan two acceptance procedure lot examine production construction module take decide lot accept lot instant inspection agreement specification realistic variance measurement acceptance accept lot deviation take deviation replace j accept notice sum behind rule inspection pass quality control comprise favor respectively lot accept close inspection drop aggregate concrete impossible know underlie shall appropriate approximation operating characteristic allow optimal arithmetic average otherwise replace turn depend standardized assume regularity quantile measurement take estimator moment consider natural candidate order quantile acceptance concentrate degree define coefficient bernstein polynomial mse sense attain parametric degree choose control density closeness distribution result establish quantile inverting kernel unit variance bandwidth rescale nonlinear estimator result consistent consistent use function case quantile quantile stochastic process index interval quantile example bernstein attain density far refer decision rejection q operate explicitly valid particular cover generally accelerate module approximation expansion involve q expansion hold denote obtain overall approximate necessary quality inspection module inspection module rely panel design random draw panel analyze inspection module sequel establish take inspection paper aim aggregate inspection happen sample subsample item draw control item pair one already yield pair observation draw item lot stochastically dependent observation size satisfy subsection long valid standardized extension handle dependent share trivial stage satisfied expansion jointly stage bivariate normal variance say attain plan approximate stage apply assume satisfied satisfied plan without inspection acceptance methodology example lot rely likely stochastically production I put plan reformulate clarity quite spatial batch arrange spread module module site course observation affect wrong wind direction stress
kernel train statistic k n asymptotic retain normal limit define equation normal forest build randomization simply randomization subsample apart condition response implementation produce asymptotically prediction tree load select build randomization obtain statistic formulation resemble computationally mention appendix forest theorem establish parameter obvious determine straightforward share lead page equivalent select initial size training must subsample record mc carlo point final random mc mc calculate note identical simplify select fix subsample include subsample record prediction average select build subsample value situation iteration accurately depend factor course ideally estimation choose computationally feasible many sample necessary accurate tree correspondingly external estimation produce forest estimating parameter need inference begin parameter method outside could generate estimation initial subsample predict record average prediction add produce estimate mean estimate conduct building theorem use uniformly introduce procedure carry prediction provide distribution prediction produce prediction variance quantile formally bound quantile variance n recommend mention introduction interval statistic reject great quantile rate check within calculate confidence interval fail hypothesis otherwise expect prediction tree building consistent produce accurate prediction asymptotically valid occur rate build true underlie sufficient confidence however precision limit distribution way significance situation datum feature interested make prediction reduce feature mean full prediction utilize would whether determine hypothesis hypothesis reduce feature prediction test size take subsample give average tree build prediction tree finally difference function ensemble single test interest case vector q consistent estimator variance clarity obtain appendix prediction statistic reject nan hypothesis procedure though decide building contribute response repeat compare full dataset prediction commonly reduce feature trees us contribution prediction two additional prediction randomize reduce additional randomized feature final randomize unlikely due structure present illustrate limiting function regression visualization multivariate adaptive spline investigate response form limit comprise row bottom subsample histogram subsample title ensemble least splitting node parameter normally mean subsample build variance procedure interested distribution prediction estimate worth lead case figure row fit near alpha incorrectly nan hypothesis central build full reduce utilize hypothesis variance estimate setup none result nan alpha histogram confidence recall confidence capture though conservative repeat internal alpha level statistic internal shown build forest ensemble establish asymptotic forest histogram generate forest tree node random split terminal limit parameter take prediction estimate prediction new calculate histogram forest prediction dataset part science project report specie effort ask contain report characteristic united states national database predict specie abundance restrict specie far restrict little report either like abundance primary goal confidence abundance feature month month figure point obviously absence month report next month highly predict abundance report issue throughout abundance month categorical category miss value include calculate remove month consist size root build estimate build internal abundance show positive interesting high abundance observe observation report expect variance prediction positive nearly visual appear certain conduct test conduct month perform statistic internal estimate calculate statistic month highly abundance month statistic still ensure add prediction training statistic difference tree advantage generate month calculate statistic month significant predict abundance month feature significant month specie year significant predict training dataset consist sample perform method year mean significance month find follow manner randomize year prediction statistic month significant abundance add procedure supervise learner mathematically demonstrate ensemble statistic limit prediction allow formally additional computational cost traditional interpretation modern algorithmic primary prediction among concern formalize hope see something distribution focus bag forest learner supervise satisfies prediction procedure carry way reasoning modification subsample primarily ensure subsample extra computational small select subsample surprisingly remain repeat replacement statistic class also point subsample theory differently say different involve practice raise issue procedure prescribe fashion true building show consistent negligible careful hope address future beneficial select result could complex hypothesis interaction work grant nsf nsf dms science ny usa formal inference ensemble bootstrappe bag forest improve predictive aggregate bootstrap averaging build result statistic prediction allow prediction form incomplete result moreover internal develop procedure tool algorithmic combination variant bag base learner build demonstrate demonstrate allow regularity subsample provide consistently increase test supervise binary long predict oppose vote value additionally process inference result look illustrate consider contribute outcome statistical begin calculate statistic statistic interested field reject chance coin also seek correctly powerful clearly conduct prediction generate simple often enough reject order formalize plausible prediction course prediction combine hypothesis scientific probably approximately develop pac bind error hypothesis appeal estimate account uniformity minimize uniformity pac ensemble fit statistical modern frequently parametric test become increasingly subsample require extension tree bagging suggest recently may bag forest estimator employ confidence interval receive pointwise recently extend suggest determine relevance introduce allow interest largely devote study discuss partition chapter seminal book context prove certain bag individual consistent discuss general bag sample proper consistency forest behavior presence mathematically forest suggest prove forest investigate demonstrate prediction converge prediction distribute class building ensemble subsampling view consistent limit parameter carry limit section forest statistic explicit introduction thorough treatment eq generality symmetric argument size normal variance x subscript enough asymptotically normal subsample interested make build subsample write tree treat order pair estimator form independent thus
partition eq likewise recall write vector use transform quantity transform multiply eq similarly partition obtain q alone error multiply side leave obtain use collect follow primal dual dynamic connect primal evolve I I square examine behavior recursion compute side regularization ensure examine study stability derive error recursion already transform dual variable correspond correspond redundant transform n recursion orthogonal incidence simplify arrive interpretation diffusion seeks arrive different allow different metric square deviation desired extend evaluate argument verify step adaptation regime power argument conclusion result match actual small replace rewrite equivalent moreover kronecker derive analyze possess negative real relate concept stable let eigenvalue establish rely auxiliary stability follow possess full procedure correspond pg possible follow result regressor definite relative complement eigenvalue laplacian connectivity precede ready stability al write q lemma since conclude lemma stable conclude algorithm stable aggregate positive hand fact restrictive definite definite matrix enough exists regressor definite generally individually solve either small size sufficiently square partial similarly corollary restrictive positive I argument similar note stable non insight analyze eigenvalue assume definite matrix eigenvalue call upon demonstrate corollary u ml carry al although encourage performance al definite sufficiently simplifie q examine al large diffusion strategy recall regime proportional conclude diffusion topology laplacian case primal even al topology identical addition steady strategy close state slow bottom else less fig obtain steady convergence rate primal steady sort rate plot steady state algorithm steady hold constant variance choice convergence algorithm see scheme primal exhibit phase dependent second largely curve primal choose stability move observe step general guarantee guarantee furthermore assume converge guarantee large substitute al less primal diffusion network positive random converge consensus utilize doubly metropolis note design fig simulate bad furthermore far make match strategy important increase necessary find find network would enhance well examine primal particular analyze discover match solution stability limitation show match unfortunately increase fix al link modification al match strategy change stability restrictive illustrate unstable partial fully incidence furthermore covariance even incidence r straightforward spectrum let guarantee al guarantee see diffusion consensus converge simulate scenario converge scenario average surprising yet desire indeed experiment property kronecker observe size ignore power collect since q enough see define substitute theorem construction primal adaptation base discover performance conclusion exhibit necessarily steady consensus find unstable partial observation strategy step regularization show algorithm strategy less stable augment lagrangian diffusion primal lagrangian slowly solely adaptation rely update step become gradient network assess provide algorithm consensus diffusion strategy belong class strategy step parameter stability square reference therein broad literature second primal augment rely primal dual main deterministic ability ill conditioning solve constrain exist useful primal g shall examine adaptive static minimizer cost variant continuously dual determine explicitly long employ instantaneous approximate direction influence noise measure direction dynamic trivial surprising behavior primal comment finding version work useful assume exactly agent explicitly adaptive turn stability strategy al stream primal version construction explain anomaly al strategy exhibit update cause respective carry availability bridge regular network homogeneous capability variant multiplier steady node role lagrangian important conclusion refer agent aggregate entire discover fail recover become unstable allow arrive surprising illustrate analytically mean al strategy able range network able solve still connect consensus network disadvantage strategy range examine steady state adaptive discover processing employ agent achieve consensus al must step size value denote kronecker throughout column exception capital letter scalar letter aggregate across network formulation primal diffusion consensus later agent access random arise contexts application channel localization second process allow possibility matrix across definite correspond scenario agent node determine aggregate unique algorithm particularly stream prominent type latter superior mean size learn diffusion strategy sufficient enhance equation small coefficient combination satisfy stochastic use comparison consensus strategy important note state strategy source instability solution connect one agent self trust square error stability sufficiently small step deviation consensus strategy show doubly hold emphasize strongly connect stability expression doubly find manuscript focus compare doubly matrix across individual agent w notation quantity agree order conclusion strategy able estimate agent agreement solution furthermore guarantee long agent consensus implementation become unstable topology agent highlight diffusion execute presentation attribute present fact add encourage explicitly examine previously incidence exposition interestingly turn study deterministic optimization nontrivial necessary notable conclusion incorporate constraint limit primal strategy motivate incidence graph incidence loop exclude laplacian matrix whose agent hold incidence exist edge connect access column laplacian aware network connection node access incidence possible rewrite extended quantity constrain lagrange multiplier associate q regularization function minimize know convexity determined determine saddle lagrangian method rely gradient take alternate direction multiplier admm know result determine either consequently directly rely stochastic saddle variable vector evaluate eq amount give u ki I index saddle cost context reinforcement target reference consider employ decay adaptation step persistent end rely variable benefit iterate
note perform dimension remove independence feature em em actually posterior assign expert bayes em step rgb rgb lemma claim observation remark bold ff consider mixture model mixture distribution condition variational bad optima insight input moment tensor establish consistently recover mixture degeneracy assumption critical ingredient mixture mixture classifier score tensor hide employ model combine expressive latent predictive capability framework widely syntactic parse machine traditionally xu however optima slow rate approach guarantee moment pearson pearson involve observe recently highly successful unsupervised tensor community rank spectral decomposition tensor degeneracy tensor guarantee correctly recover method dataset effectively important assumption model suited rule high moment set high moment label information learning consider early challenge moment appropriate form amenable detailed ingredient ingredient employ feature accurate framework representation exploit superior purely discriminative many incorporate tensor high exploiting quantity differ high derivative density capture input establish label yield expect derivative input expect derivative nice unknown generalize glm e u establish employ tensor learn scale spurious framework classifier g xx thus weight decomposition cross moment input kernel yu complexity assume regime regime refer variance regime number classifier recovery method incoherent spectral compute whitening slice tensor assume value matrix recovery high guarantee method computational sample complexity learn weight technique maximization discriminative yu get optima employ learn construct feature classification framework svms learn weight construct discriminative guarantee input moment mixture specifically label consistently degeneracy mixture expert divide consider xu alternative carry usually hierarchical xu guarantee guarantee decomposition moment linear problem alternate convex restrict subspace gaussian hessian eigenvector order moment matrix subspace outer however fail moment vanish overcome transform result vanish line gaussian density stein handle bring feature use tensor individual weight vector employ tensor involve moment tensor gaussian class procedure drawback dimensionality eigen sir label project preserve eigen assume elliptical slice slice surrogate establish strongly monotone provably paper consider single glm monotonicity vanish derivative activation third utilize throughout denote order tensor I pi canonical rd say te ai cl set assume draw probability density incorporate generative mixture glm linear mixture activation although limitation classification glm employ glm u rr extend tensor specifically adjust follow stein identity weight method present result full vector tensor algorithm appendix follow recover vector scale bias estimate method handle violate constraint overcomplete dimensionality tensor recover incoherent detailed score compute j appendix gaussian scenario extend ingredient ability label derivative distribution result distribution use order derivative respectively th continuously gx mild learn general moment tensor cp component recovery mixture respect w guarantee provide section obtain activation know need fully propose far provide linear q similar framework nonlinear assume propose extend connection density ti equation z recovery svm function ingredient lie estimate estimation fit addition deep argue auto learn order score estimate multi versus analysis since tensor decomposition algorithm recovery need difference number require let u normalize guarantee first power line state satisfied guarantee input remove need power whiten orthogonal perturbation whiten gaussian regime appendix gx I discuss theorem appendix initialization normalize independent denote incoherent hold incoherent satisfie output satisfy since know function normalize bound output let almost surely bernstein I perturbation incoherent output employ learn mixture classifier expectation maximization discriminative svms yu discriminative since optima convergence local optima employ learn construct exploit direction variable work spectral general expert variable interest microsoft fellowship nsf award award award h award tensor multilinear form view multilinear form tm eq mi u mu multilinear combination mode multilinear combination slice stein lemma simplify first term give gx obtain obtain hand substitute complexity general empirical state take frequency perturbation translate bound see equation perturbation provide low order
mdp entropy sample visit region way always beneficial yet counter mode illustrate section admit action right source reward start middle know everything end probability terminate reward illustration belief ts end since coin take bayes htb thick thick might failure lack ts adapt dp optimal several episode explore problematic discount objective complicate material similar generate planning avoid integrate future planning deal inference rich probabilistic search adaptive planning guarantee lack performance sparse increase share ts combine less sampling filter forward need update thus horizon number root require belief reason choose forward supplementary material huge domain adopt strategy area bayesian permit carefully likely consider rich contextual bandit however solve planning say real likely motivate realistic dataset uci repository instance specie family attribute g color instance mdp ignore ignore incur illustrate initial observation indicate represent indicate component supervise learning ignore contextual bandit however unlike contextual bandit early reward valuable late one exploration dominate task parametrize context generate denote ss increment state update update cost key aspect joint matter uci unclear planning possible base inaccurate agent assume allow substantial underlie characterization evidence observe particularly parametric chinese restaurant pt crp assignment measure dirichlet assume hyperparameter inference scheme detail draw component crp correspond sample state infinite horizon generative label straightforwardly uniquely characterized context imply configuration stress really section somewhat realistic agent highly challenge particularly natural ignore lead neutral return run ts statistical result surprising result bayes adaptive agent obtain return despite abstract exploration performance investigate label free reduce uncertainty ts improve inferior adaptive et al regular ts outcome time integrate purpose simple contextual bandit apply ts ts bad crp base demonstrate large discount ts crp inference start step discount ts ucb agent control domain handle customer make decision different consider generalize share address actually draw plan mis key decision contain parameter generate dynamic denote either leave x b draw generative domain different assume know generative contextual model usual contextual arm option play contextual bandit exploit even extension include intra supplementary focus investigate performance sample reward material sensitive discount dependence exploration exploitation strategy ht concentration hyperparameter inference avoid maintain similar performance despite run simulation researcher powerful sequential exploration parametric emphasis planning factor mdps capture exist problem safe planning monte scheme depth branching factor limit benefit exploration deal mdps discount objective structure consider mdp combine bayes inference mdps specific domain unbounded state infer size state online search planning limited depth sized gps employ infer excellent however capture explicitly exploration planning address plan uncertainty reduction generally learn ultimately label concerned discount return fine labeling base attractive particularly domain carlo powerful optimistic planning severe avoid explicit planning domain benchmark uci bayesian exploration exploitation demonstrate feasibility advantage adaptive various planning roll interesting think function within tree open domain truly computation explore simple computation gain policy planning challenge model parametric amongst readily domain arm extension something collection measure expert solver figure mdp reader tuple sa discount component mdp mdp planning estimate line latent accord observe action tp uncertainty current inside augment possible tuple form bayes adaptive mdp solve obtain augment space readily action execute constitute action agent mdp equality degenerate support model agent go end hand aim right equally htb thick thick black c c b thick black action ts p v ts arbitrarily bad policy therefore choosing imply construct optimistic action across take present decide action result denote sample px showing add usually mdps decision put policy perfect first n c pn depend bad easily since policy action stress strong objective bayes severe pressure short horizon ts label free report agent fewer ignore show horizon sampler tractable concentration couple every simulation assignment sampler infer planning tree generate pool slowly htbp bs ar htbp restrict task contextual extension section mdp contextual informative motivated exploration site come know contextual e ignore site run type action type site model intermediate get site correspond except establish model reward binary variable ts bandit depend environmental ts act ignore conservative act large ts dynamic return sort execute explore cumulative ts solid line error theorem rgb rgb powerful bayes planning parametric simple planning thompson fully thompson leverage efficient adaptive parametric perform qualitatively conventional thompson rich inductive bias allow confident inference limited benefit control safe exploration balance act look plan partial exploration trade problem great unfortunately costly leaving might compare similarly computational cost treat tradeoff focus discover demonstrate despite sample base planning rich challenging planning provably optimistic thompson sampling fail risk perform highlight behavior uncertainty non way consider include material discuss reinforcement rl outline exist planning thompson introduce exploration motivate mdps finally adaptive
convex convex relaxation tensor connect well investigate technique develop address give utilize trace norm showed given improve latent regularization infimum convolution analyze rank tensor problem reduce efficiency expense address possess exist bayesian method study basically construct tensor decomposition decompose performance analyse collaborative filtering spatio gaussian decompose decay speaking obtain cp analysis favorable property without assume adjust rank priori significantly approach convexity convexity sparse tensor exist inner k rank rank tensor exist k u relation write cp paper investigate cp rank cp regression predictive q observe input observation completion denoise observational completing unobserve j j ia accurately tensor obtain sum standard task dimensional vector cp space show rank extension rich analogously gap envelope cp well envelope matrix norm np envelope present regularize assumption procedure compare provide u positive suppose gaussian k condition gaussian posterior mean square give rate define tensor pa key characterize convergence well concentrate around large bayes estimator truth location truth much specific wide possibility balance concentration dispersion trade normalize scale simplicity assumption assumption bind tensor technical convergence predictive suppose follow assumption constant appendix speed mass around eq outside show follow jensen strong state mean estimator assume symbol actual degree log number rate basically emphasize true placing estimate rank give assume convexity sparse lasso convexity require reason tensor practice turn accuracy accuracy error bernstein infinity mean tensor estimation avoid large tensor tensor completion recommendation apply large conditional accordingly sample proper corresponding truncate assumption bound front gap population simplified observe term analyze actual assume impossible derive focus weight tensor completion l px recover rate estimator respect max sample accept much well previous inside improved hand rejection recently analyse tensor work utilize tensor unfold unfold tensor th rank tucker general cp analysis empirical see bind achieve estimator k nice point automatically rank unfold minimum rank large bayes remark rate tucker cp suit apparent latent setting counter bayesian rank place prior author utilize strong convexity require bind utilize scheme concentration analysis course tensor bayes tensor investigate applie set element observational tensor randomly execute five repeat repetition varied addition actual manner figure scale ratio accuracy curve scale accuracy mean scale accuracy behave match predictive paper investigate convergence rank base predictive without convexity adapt rank describe behavior bayes negligible however experiment show behavior investigation thank discussion partially determine let eq k k eq combine definition unnormalized fix think random utilize originally convergence bayes parametric model technique denote da construct eq indicate test even eq check event fix hold q calculation rhs bound r yield prior therefore posterior rp rhs simultaneously pack packing unit ball bound
baseline add consist architecture probable viewpoint orientation fc bound baseline correspond tend table appendix ap clearly cnn one favor baseline fc valuable orientation stochastic momentum decay patch balanced patch discretize sec patch sec batch experiment select validation patch proportion patch selective iteration start avoid divergence divide stop decrease successive train discretization pose outperform state cnn explain training orientation increase interpretation train training imagenet annotation present increase considerably ap pt r car avg c c finding separate representation improvement detection orientation obtain pose explanation observe fine detection drop perform joint pose discrete detection treat continuous orientation detection joint orientation approach show cnn art acknowledgment work imagine des centre building partly support project c car r car avg search c k k h variant c c car train variant car v r c c car train v variant c c car avg variant car train c car c c c car rectangle fill gray center draw minimum height rgb study application detecting pose representation orient energy choice pose continuous variable object detection benchmark detection pose exist baseline benchmark performance cnn specialized vision optical availability increase cnn recently outperform less constrained vision seminal work imagenet apply explore potential cnn image namely annotation detect pose computer vision work contour match contour instance towards object category focus without take sift drawn section overview joint orientation real image rotation degree cope strong appearance object appearance intra illumination difficulty orientation tv rarely follow pre imagenet learn tune last idea pyramid manner output fine tune allow computation layer provide art classification adapt tune architecture orientation perform pose use vision attempt object category information representation orientation several feature pose handle pose pose patch benefit vision pose alignment alignment alignment recent one specialized pose handle intra category geometry simplify applicability real classic challenge dataset class average viewpoint standard metric evaluation viewpoint orientation use adaptation perform well cnn neural design learn successfully apply many specialized digit face attract vision advance improve vision problem object detection special easily box selective lead drawback candidate reason propose convolutional lead achieve slightly propose neural network available imagenet train imagenet art detection classification connect top imagenet unclear invariance encode could pose discriminative b b pose approach associate show subspace method network combination pose regression classification leave pose space state pose class roll angle slightly angle cnn predict pose pose set point define discrete feature point representation adapt develop pose predict probability orientation plus background contrary aspect associate patch circle angle see approach patch jointly class angle angle option function feature detail finally cnn comparable choose base pyramid pooling framework efficient testing good result pool similar imagenet pick selective image rescale layer tune function cover suppose training orientation dimension interpretation choice architecture pose bin k k category otherwise imagenet window patch extract extract map rescale follow softmax minimize log softmax sum minimal orientation framework appearance vary continuously discretize network suppose jump viewpoint vary appearance view one develop orientation unit enforce far circle positive local negative circle instead far avoid effect dimension circle live classification without softmax output respective negative natural loss follow property parameter indeed huge negative negative example clearly large radius circle probable orientation angle pose treat extend mutually exclusive distance category distance inspire consist divide orientation classification follow softmax pose
regime soon quantity dyadic multiplicative analysis single vary routine tuning issue quite surprising consequence loss value hold use adaptation hand well induction hand definition line sum combine rearrange achieve tune sum logarithmic working mention rate vary loss pick rule mixture k learning cumulative material give idea complete non sequence rate update tailor induction prove aim bound core update could handle inequality main notice work version quantify gap inequality precisely rate price vary also regret present indicate work analysis simpler elegant resemble dependency achieve order sequentially rate vector round wise nonnegative loss loss vector justify expert report refer expert differ round expert expert round weight k never empty regret account k expert confidence therefore depend bind also scale issue generic set immediate latter report detail essentially exist expert prediction reduction easily run modified round weight another strictly loss loss follow equal modify loss set q subtract side regret loss confidence second upper technique regret excess loss introduction plain expert confidence expert expert suitable lead expert leave prediction second bind already case key feature excess instead plain loss order eq cumulative introduction close loss gain ready symmetry nonnegative indeed regret satisfy loss translate scale canonical visible ta significant improvement worst realize nice affect loss even positive negative therefore expert substitute initial state substitution guarantee adapt non adversarial identically bound case least expect satisfy satisfy constant regret strategy form regret law number cumulative exceed cumulative order large bound theorem require bernstein basic cumulative deterministic factor let martingale v derivation study typical able consider variable sense play role vary learn instantaneous induction conditional inequality solve quadratic inequality yield claim bound q substitution conclude order loss fact body shown rely simple convex line consider value indicate low show induction hold round algorithm induction hypothesis inequality already equivalently trivial since expert proportional desire bound rearrange useful number eq first follow stem bind inequality term note hold inequality ratio particular square root apply apply substitute bound b alternatively exceed latter proof associate q part entail element diagonal fix eq entail instantaneous develop product equal hence lead inequality substitute finally impose increase consequence nonnegative q increase rewrite get fix rely variable pick q define bind apply far proceed induction side inequality application conclude show increase remain use bind put thing conclude specific section reduce loss trick unified reduction expert expert predict possible prediction determine function choose weight compete wish combination expert component forecast reduce confidence section reduce linear loss pseudo tx convexity inequality eq equality linearity regret imply compete set another paper expert selection selection expert converse couple expert like equivalent vector algorithm expert respect indicate although tune sequentially believe consider optimization tuning reduction suited expert report general entail therefore bound state
keeping allow split subsample cv frobenius norm adequate propose cv justification method memory dependence evaluate thresholding correlation functional connectivity datum multivariate covariance four px model n px norm ability recover sparsity positive define conduct sample range replication vary cv cv candidate range increment estimator correlation perform temporal overall perform ordinary ex hard soft r ex norm ht loss technical lemma second hand inequality tx nx ni eq nz ta constant sufficiently thus proof let plug yield unknown integer without q equality schwarz hold q r constant plug yield constant line dependent obtain thus hold convergence condition theorem constant polynomial constant key impose equation proof equation see complete proof proof follow line equation omit theorem inequality since eq q eq assumption mean norm frobenius similar exist yield inequality r corollaries general assumption set set thus acknowledgement wu principal van mh centers support center grant partly grant dms dms dimensional decay generalize thresholding convergence consistency impact temporal investigate intuitive thresholding parameter good method temporal implication fmri connectivity nx inconsistent overcome covariance regularization approach cholesky correlation researcher become particularly useful analyzing fmri assess connectivity temporal process traditionally impose overcome difficulty introduce recently hard interpret straight cross impose weak cross covariance extend weak fmri stationary memory autocorrelation rate much slow important example invertible integrated move generalize decay cross cross matrix simply dependence weak dependence decay temporal cover restrictive correlation aforementioned study brain connectivity moreover entry surely dependence condition pp extra care true replace sample mean unknown article estimation correlation consider series decay restrictive violate mainly focus rarely brain image intuitive show generalize thresholding keep originally develop matrix organize temporal dependence special temporal describe broad range result long thresholde cross validation method evaluate contain theoretical brief norm mx nc n xx x ij tm independent matrix column define product ij ij kt spatially average voxel within pass stationarity false fdr control approximated check mild rate dominate rate practice figure dependence seem fit least linear yield illustrate two brain clearly assumption homogeneous decaying time mild consider generalized estimator sample matrix correlation first temporal subsection detail memory temporal specify subsection matrix eq correlation matrix eq thresholde popular soft thresholding smoothly example generalize thresholding estimator thresholde c pm n replace respectively p I tend additionally probability tend replace without impose define correspond correlation consistency pf nk n q frobenius norm q r j eq would estimator like subsection temporal applicable long may
complementary representation constitute improve contribution entity augment semantic intuition entity play central heavily expression case identification implicit explore entity include role syntactic status despite solid linguistic foundation feature contribute word pair status entity distributional semantic throughout entity compositional begin phrase mean current semantic attribute distributional information linguistic structure distributional compositional supervision enable semantic include rnns rnns propagation rnns nod probabilistic entity propagate node entity semantic equation outside inside combine parent term style parse distributional semantic relation word shown utilize idea distributional word incorporate semantic text stanford argument parse semantic compositional representation entity semantic apply syntactic induce representation entity overall compositional annotation outperform previous relation semantic resolution annotation addition share parse edge edu relation small linguistic text automatically identify require argument subtle relation link level entity distributional representation syntactic work compositional representation compositional relation distributional entity system obtain substantial implicit relation characterize adjacent relation task annotate automatic identification art implicit parent one poor predict implicit relation fundamentally semantic may difficult implicit sentence seem surface unless annotate etc far address compositional semantic argument series syntactic predict bilinear compositional compositional operation classification implicit purely capture relation see make corpus sentence long appropriate preferred distributional almost unchanged syntactic representation span single capture relation entity role put meaning address issue compute entity capture play entire span account feed compositional combine pass structure parent tree combine bilinear resolve combine representation achieve classification outperform classification novel entity compositional model surface syntactic feature work prediction semantic relation indicate sentence justification sentence sentence refer sentence entity share entity relation sentence clear pair share entity relation totally change suggest relation sentence model capture semantic sentence semantic share entity relation without new composition capture entity sentence semantic combine distributional sentence composition part recursively combine distributional architecture illustrate start composition include share entity entity entity identify sentence implement stack composition show jointly composition composition composition work test focus outperform relation also discover formulate model sentence relation composition easily extend follow approach semantic clarity exposition extension section feed pass terminal syntactic distributional child rnn parent child element tangent composition matrix compositional find leave pre word representation sentence combine obtain feedforward little distinguish certainly relation one role semantic rather logical would parse logical representation role neighboring make pass compute composition recursive occur root procedure parent compositional maintain influence pass also feedforward influence node since pass feedforward node feedforward efficiently inside computing score fashion outside inside sum observe describe constrain tensor contraction involve parameter composition predict argument decision bilinear product parameter scalar entity share entity sentence root low apply reduce classification surface surface set representation advantageous experiment serious overfitte backpropagation present argument gold regularize square frobenius euclidean hold depend delta update similarly matrix composition every unified derivative form information also computation include compositional operator q composition word derivative objective compositional operator set topological convenient way equation graph illustrate start reverse edge trace review take train implementation use norm trick fix latent set latent dimensionality regularizer composition composition classification vector classification initialize composition initialize uniform train induced unit experiment pre give broadly syntactic binary stanford sentence syntactic sentence identify span branch automatic gold berkeley entity instance entity gold annotation intersection automatic gold line supplement use include four lexical feature dependency contextual mutual select feature three lexical level journal corpus annotate two argument identify challenge meaning focus challenge problem classify implicit temporal contingency fine relation specify contribution main evaluating relation explore relation binary build evaluate primarily multiclass however correct relation pair among second relation type exclude five relation annotate annotated instance ex cause account simply mean sum bilinear publish implicit relation feature lexical syntactic implement system enable comparison online method relation multiclass identification line outperform distributional improvement accuracy great distributional surface system individual prediction use sensitive significance semantic significantly outperform surface choose set figure accuracie narrow identification relation outperform prior distributional improvement choose development range latent entity line entity entity semantic without share therefore seem sensitive entity gold annotation intersection entity find inclusion entity
soft thresholding np np group index sparse gradient give build vector choose build closure operator value decomposable task positive reflect decomposable ode k resort two optimize semidefinite differentiable respect differentiable function algorithm reduce projected sdp lipschitz step pc onto cone semidefinite eigenvector non loss sdp eq numerical highlight novel framework ode finally ridge learn decomposable scalar summarize cross search trajectory model descent manually mechanic noisy equation spike potential neuron recovery assign prove numerous optima spaced add isotropic mean figure present learn figure top smoother estimate trajectory match true tend sharp curve represent truncate use space truncate depict smooth predict trajectory smoother learn peak reflect trajectory consist biological gold exploratory ode smooth well considerable uncertainty smooth true level approximately half automate time interested trajectory arbitrary true error model accurate ode non classic initial parametric well ode realistic comparison good highlight error ode ode parametric solver fail ode specify mse parametric parametric free ode matrix especially flexibility penalize rkh way address realistic ode learn nonsmooth help proximal also discuss approach presence issue theorem de division france universit de france france paris france dynamical view dynamic ode reproduce rkh matching approach ode smooth ode derivative nonparametric ridge ode dynamical physics attempt understanding eventually make prediction ordinary describe state account dynamic ode two choose ode noisy obvious favor choice rely test angle ode nonparametric issue principled parameter knowledge precisely govern differential equation dimensional dynamical ode value ode length additive point eq ode square subsequent parameter optimisation approach intensive suffer name gradient matching estimation iterate procedure iterate asymptotic enjoy consistency nearly work approach ode learn estimate learn differential play capture trajectory value assume value function want derivative reason contribution reproduce work subject rkh theory flexible definite scalar value svm rkhs value attract elegant structured supervise semi nonparametric scalar reproduce first endow inner property corresponding include sequence property rkh theorem follow empty kernel build let function hilbert want theorem ignore independently along act learn gradient p expansion coefficient matching stacking stack matrix literature sdp scalar kernel pairwise similarity kernel parametric use series come different non want nonparametric suppose propose nonparametric smoothness impose close estimate input multi strongly manifold regularization observe start condition learn describe minimized q stack value matrix ij ij rr elsewhere anneal g avoid average
metric vertical axis precision compare similarity instance similarity rbf change bandwidth score score similarity mle look incorrect similarity determine application least rbf dpp mkl rbf kernel exclude mle horizontal axes line mle mkl similarity deal appropriately similarity introduce dpp video dpp task cluster summary naturally want representative text understand conference testing time news article reference summary summary identify agree human summary oracle practice use algorithm evaluate human reference separately accuracy use package gram p additionally length character yield dpp task sentence similarity standard frequency frequency tf modeling similarity baseline enhance cosine method competition dpp decoding set real data subset achieve consensus consensus measure metric depend f actually flexibility dpp user infer output metric select necessarily diverse bias towards specific summary summary summary balance precision dpp summarie five summary one contribute gain package develop section describe frame stop negative summary achieve oracle summary able target independent oracle learn application specific summary selection result package summary number match view match visual color difference match frame vice versa develop number match experiment recall cf balance hamming cf text contrast neither mle able summary include hamming pair interpolation draw curve dpp generate high mle rise summary want turn dramatically conjecture u california science science california ex plus minus modeling application diverse subset dpp label make contribution dpp modeling flexibility propose novel trade error extensive contribution document video modeling kernel matrix balance imagine search retrieve retrieve image frequently cite need incorporate notion ground might contain many item ensure exact diversity point dpp technique diversity power set diversity likely diversity physics dpp find retrieval extension dpp dpp markov dpp dpp space dpp crucially square every pair ground reason quadratic impractical element necessary secondly document document annotation expert costly difficult many task dpp select summary estimation maximum typically underlie limit number reliably restrict precision two dpp label improve model dpp kernel domain fewer whole correct subset margin reflect desire measure closely selection error sample dpp principle novel superior video organize dpp work study conclude task large margin base approach discuss relate translate item unlikely occur subset arise matrix quantum nature determinant dpp select summarize rank search possess analytical margin analytical research restrict force dpp focus propose dpp individually structure model map resort algorithm extensive effort dpp explore surprisingly diversity dpp kernels mle approach note selection mle maximize joint margin discriminative dpp kernel limit successful large model explore outperform mle application handwritten character speech approach analytical margin bring modeling flexibility meet practical learn dpp testing stage recall review dpp excellent define symmetric semidefinite column proper call ensemble dpp subset matrix computable submatrix despite eq marginalization dpp marginal either case lead never item similar diversity subset diverse subset attain probability map l interested mode hard approximation investigate suppose annotate discover thus specify item unlikely represent share compute characterize optimize diverse subset attain rise estimate mle estimate dpp limitation multiple representation estimation function dpp apply large optimize reduce advantageous advantage optimize track component attain dpp semidefinite item ground right however application item retrieve image diverse sentence redundant also represent document decomposable relevant depend item encode contextual item feature sentence set descriptor represent item whether optimal adapt thus largely severe retain aspect gaussian rbf secondly base kernel annotate parameter estimation technique consist select frame impose dpp data dpp benchmark training subscript decompose quality ij measure reflect bag word visual appearance far compare mkl j turn mkl parameterize key synthetic question come learn parameter mle follow training maximum closely track error improve likelihood likelihood subset mle mode subset highly mode problematic dpp fall error approximate extract margin incorrect constraint loss measure discrepancy intuitively maintain probability incorrect explored structure exponential counting subset item unnecessary severe add trivial sentence type error ham function q item towards incorrect demonstrate real challenge deal constraint hard jensen inequality th diagonal detailed see undesirable contribute term hinge function tradeoff coefficient tune objective likelihood objective subgradient descent supplementary introduce parameter balance force fix descent project weighted hamming distance margin discriminative incorrect subset distance violate overfitte training yet careful reveal system people expectation document example reader may something summarie article usually piece online video summary mle merely focus fortunately large modify hamming hamming distance recall show marginal towards conversely make dpp model put effort item give rise high result course type discriminative meet dpp kernel form beyond quality diversity limited function kernel parameterization high summary learning maximize observe potential mle optimal test maximize enforce mode large py dominate tackle margin dpp minimize dpp researcher introduce dpp restrict dpp offer diversity adjacent structured dpp dpp generally hard guarantee alternative resort activity work explore popular estimator mle mis flexibility incorporate posterior minimize margin dpp make tractable additive large
forest goal depend furthermore write forest forest give large section compute quantity show term call primary tree forest bound paper partition almost show proportional leave say partition rf lead sake simplicity I compute lagrange c j x appear forest true c proposition precise bias infinite much tight low convergence smooth toy obtain piece formally uniform random forest key sx decrease section point make goal corollary get term appear toy let smooth show corollary inequality dx regular density framework surprising model regular difference regular randomly translate histogram regular randomly translate effect boundary highlight phenomenon forest suffer phenomenon e tree given compare notation define integrate therefore consider order constant leave tree precisely equal risk statistical risk build respectively x forest follow assume addition rate dx assume attain function whereas except constant infinite forest minimax next bias account involve rate reduce practical estimator need infinite forest independent random denote q far enough forest bias estimate appear decomposition true involve rate valid except avoid however conjecture toy issue scope paper small forest reach proposition consequence section corollary tree soon multidimensional purely forest partition piece piece make z split random step set tree whereas split model partition end average weight average infinite contrary xx plot compare p slow leave estimate monte appear bias hold corollary contrary toy effect approximation compare infinite forest term term q allow forest single approximation height infinity emphasize decrease slow cubic partition set indeed risk choose control control point forest bound proposition eq volume hence subsection risk assumption sx proposition leave point available l p infinite forest estimator follow infimum distinguish ii integer soon ii slightly apply imply p decrease tree infinite forest rate function model minimax split small split partition proportional function finally suffer lack next set choose would certainly would tight smooth following hold eq previous forest magnitude eq mathematical previous r section toy consider rf hold rf original tree extra consequence rf approximation tree take input function proportional range model choose toy remove realization estimation rate accord adapt significantly quite surprising investigate phenomenon systematic scope forest model forest improve regression forest equal forest compare toy forest instead toy tend leave stay leaf leaf contrary keep split leave forest precisely analyze latter partitioning set variable put node effect output weight infinite forest fast even quick study forest result fast smoothness regression combination analyse regression beyond research well rate tend balanced tree choose consequently reach large minimax compare forest approximate smoothness whether analysis suggest mechanism seem next justify reach minimax consist length choose uniformly practical forest get single tree build forest finally apply forest partition learn random appear forest particular mind rf define address research hold rf grateful discussion acknowledge grant detect prove conditionally classical q last term come convention sx integrating separately eq done separately hold definition appear prove schwarz conclude bind last toy ix proof assume occur x eq interval change eq conclude q k proposition quantity toy random variable uniform therefore follow eq quantity result gx yield integrate directly q gx gx p x p quantity appear gx gx get integrate variable define consequence v independence variable k conditionally since binomial every convention follow take resp x jx jx summarize k dimensional deduce deduce computation key appear follow since proves prove eq integrate integrate proposition key q prove proposition eq integrating eq taylor expansion direct integral section imply quantity b formulation distribution section accord px belong define every j hence x px l b px relative position point finish furthermore sub uniformly variable uniformly multiply p p j j every main summarize repeatedly proposition since I odd integer let integer sequence prove I additional q I apply prove take since sequence every jensen inequality get eq increase every proof j directly combination proposition proposition appear q every integrate yield every yield eq proposition proposition integrate partition induction clearly unchanged change one multiply uniform variable get end eq pi z p p z pp particular chebyshev every combine every eq take since function b fx follow straightforward upper integer u define definition remark pt forest forest order forest framework focus tree forest regularity bias infinite rate size tree single attain risk rate furthermore sufficient product purely forest forest rf henceforth machine remain deal rf bagging bagging posteriori rf rf neighbor towards theoretical rf purely henceforth simplify rf establish obtain partition independently first easy secondly mechanism obtain partition usually simple enough allow calculation theoretical describe try compare performance perfect ensemble rf randomize encourage good birth variant understand rf precisely model focus regression partitioning space base mechanism abuse recursive way leave tree element recursively belong decision tree classical tree partition denote obtain partition throughout leaf response forest sequence correspond aggregate important rf define recursive partitioning put repeat meet choose among find split variable split split crucial method forest heterogeneity quadratic put randomly choose split uniformly split uniformly put forest
onto area every crucial mean every solution k lebesgue assume eq integrating give equation derive point note cause nice involve multidimensional explicitly second critical development lebesgue calculate value interval marginal sum active analytically sample carlo carlo easily interest motivation spirit understand respect pp four figure define concrete rectangle integration emphasize tuning parameter estimate ignore regard sampling covariance n p na argument normal particular simply affine pa restrict bivariate lasso thresholding simple last obtain apply theorem estimate corresponding estimator example normal nonparametric method ni h h reduce estimation besides normality motivated perspective lasso draw use distribution assumption valid specify aspect explore introduce direct assume n draw draw residual routine form calculate expectation much alternative although method example high set reversible markov subspace dimension ordinary mh I new say p dominate measure standard mh involve mcmc distribution matching component reversible jump contrary assume move dimension reversible usually hard mh call sampler hold group accord follow proposal proposal j proposal symmetric mh simply efficiently especially remove add reverse mh analogously proposal one need ratio side efficiently dynamically detail proposal consume proposal consideration proposal let vector suppose ta mh mh mh input parameter numerical section distribution gibbs consume algorithm estimate interval approximate accurate monte conditional distribution rare contrary involve calculation tb proposal ease tf understand mh move design jacobian q coincide jacobian current jacobian account use view computational computationally tractable quite sampler conceptual direct linear transformation univariate size univariate numerical move analytically fundamentally relatively consume model greatly unimodal chain converge amount computing confirm numerically next routine initialization reach equilibrium totally remove detailed however routine numerical initial example coefficient give design matrix cccc b weight numerical lar dataset choose implement package determine estimate coefficient type error distribution correspondingly section routine lar package examine serve weight reasonable see routine notation estimate chance proposal proposal iteration normal plot sample illustrate subgradient away autocorrelation among decrease update proposal update proposal acceptance mcmc estimate conditional simulate sample sample quantity calculate accurate mse great mse ratio run around ratio mse estimate estimate estimate furthermore simulate sampler confirm serve direct simulate lasso routine parameter previous quantile standard compose practically use algorithm ground truth example accuracy variance across table model approximate estimate sd b c effort establish penalize high consequently positive eigenvector vector linearly independent augment estimator n augment row achieve constraint word lie augment restricted mapping denote unique satisfie constraint lie fix constraint determine jacobian simple constraint minimizer satisfied rank matrix n n dd accord n rd confirm continuous fix ready report estimate extremely order moderate around cccc multiple matrix kk coefficient true many relatively choose active coefficient along path summary well cover wide procedure therefore simply ht range value run value figure dataset confirm variation huge direct previous improvement ds tail bad variation bar give result dataset minimize section section augmentation publish extend definition proof q p eq immediate regard vector condition construct fix assume condition assumption lemma probability least assume borel depend give bind pr nr n satisfy sufficient least decay zero apply dimensional depend identical quantify difference let q satisfy satisfied fixing establish pr tend scale sufficient tend assumption beta comparable theorem one strong eigenvalue residual precise residual routine residual consistency spirit fix previous general explicit imply consistency recently number condition fundamental consistent stay result initial magnitude nonzero coefficient considerably generalize draw augmented become give set routine I reliable sufficient development mh design explicit draw importance expression unnecessary choice sample augment selection intuitive geometric respective active allow grow sign unique size active definition valid kkt via affine component consequently definition therefore consequently intuitive column regardless establish follow condition necessary bind example one c nd min replace vanish applicable posterior continuous nonzero bayesian lasso thus invertible assumption tn incur mahalanobis norm encourage sparsity loss e joint kkt kkt distribution n reasoning n discussion framework decision although improper confusion call estimator instead lead solve kkt condition therefore interpretation lasso vector therefore decision familiar correspondence worth loose sense interpretation kkt depend lasso monte estimator advantage direct show limitation augmentation relative augment randomness stress direct sampler similar way handle determine draw direct routine easily since proposal importance independently certain reach overall computational multiple draw sampler markov routine reach iteration computing assume access run initial draw sampler routine reach routine last assumption derivation routine decay autocorrelation always decay case component first decrease fluctuation empirically suggest direct need node direct sampling bootstrap density augment article mcmc method linear regression approach clearly method gain flexibility room idea augmentation use penalty study however difficulty uniqueness penalize augment mean manifold another future theoretically empirically finite coherent truncation pp p assume ii suffice q therefore since v thus complete proof let condition event equality e bb event construction consequently minimizer j direct kkt minimize give minimizer loss lead due lastly distribution event assumption least conditional choose see proposal index immediately proposal computation jj readily reject position proposal convert routine regression show determine numerically many distribution augment tractable low variance carlo draw sample augment respect concrete example offer regression obtain monte interval monte carlo regression design coefficient widely sparse estimate vector minimize penalize choose approximation however except special type complicate close covariance fail confidence develop orthogonal design limit bootstrap bootstrap algorithms lar homotopy time apply hundred time point circumstance modify justify develop linear several article significance asymptotic various lasso hand distribution useful selection penalization stability possible obstacle estimator sampling density interestingly joint normal regardless distribution simply marginal study sampling may accurately efficient another evaluate minimize numerically bootstrap furthermore mcmc multivariate locally remain article organize derive mcmc calculation inference provide theoretical estimate establishing include design selection interpretation sampling article conclude notation regard column set v b j
logical distributional two neighboring node pass parent si composition function bilinear product relation entity decision classification argument progress parse representation semantic compositional induce distribute argument entity jointly classification compositional operator base find font draw style edu linguistic element coherent text identify understand semantic link sentence sentence link element entity distributional tree key compositional distributional semantic compute representation entity novel compositional distributional also entity result obtain substantial improvement art predict level organization text adjacent relevant task sentiment coherence automatic identification implicit relation roughly predict implicit fundamentally semantic relevant may relation sentence appropriate indicate relationship however surface compositional sentence series compositional give argue purely expressive capture happens make change sentence original relation long hold despite syntactic address issue compute sentence entity mention play compute entity novel feed compositional combine pass syntactic combined help approach achieve accuracy identification syntactic automatically stanford ask whether might recurrent language preferable language resource whenever possible think unlikely key language recursive language processing semantic see strong evidence history natural syntactic capture accurate annotate language topic substantial differ substantially order language question whether leave recurrent extract language entity distributional semantic relation name distributional compositional entity augment distributional semantic pass composition distributional sentence entity representation sentence non syntactic distributional computed distributional child representation word
onto kkt condition w sufficient subgradient always hand side due hold condition screen al parameter plug define b bit screening overlap overlap bridge primal dual primal problem solution note path priori linearly spaced screening screen perform show screening screen base path omit proof theorem theorem leave hand valid tight screening feature goal optimization thus algorithm hand adopt minimization k group intersection kl determine compute set finally minimize overlap screening l minimize bind obtain iteration set process counter squared hand fix subgradient subsequently upper z k iterate side take root f overlapping include simple fast coefficient individual algorithm overlap lasso summarize mention fast lack algorithm similar various utilize appeal ratio lasso however necessarily small zero nontrivial compute couple simple denote technique summarize demonstrate speed rejection ratio discard coefficient zero overlap ol overlap screen image dataset genome image ad individual ad know predict variant code rna brain genetic information randomly color object image select dataset digit nine test induce locality output jointly image genome structure group consecutive group four root consist parent overlap group consecutive overlap overlap knowledge ad group nucleotide locate region start end site overlap solve overlap lasso screening likely except single report average performance solver screening implement matlab group structure size screen scenario rejection ratio structure figure rejection ratio represent dot diagonal rejection ratio ol except ad reject feature overlap hand dataset ol use ad size size ols increase rule ratio regardless size ratio single machine ol comparable measure speed gain rule dataset overlap give solver without solver ols solver illustrate portion run time speed without running screen ol dataset ols ols solver ols discard feature appeal portion hereafter observe experimental dataset structure group rejection ratio rejection ratio plot maintain rejection drop rejection ol group interestingly even tree group ol rejection ratio penalty htb ratio experiment use overlap group size change size ratio size increase group increase number test group left keep rejection size rejection decrease fixed window rejection start include make independently test group advantage test group ol screen verify screen tight develop various loss research overlap regression regularization determine sparsity primal q lagrangian optimization present author machine author author author author author author projection author illumination database author author author author author author proposition notation section theorem develop efficiently discard screen infeasible develop lasso arise group take
edge related maximization seed topic seed topic intuitively reasonable assume topic cccc seed overlap seed randomly mixture topic greedy select seed mixture percentage mixture intuition seed seed network seed topic contribute seed topic topic influence solve item mixed influence probability apply mean topic influence maximization inefficient choice well focus online mis compute marginal achieve competitive convenience budget algorithm value preprocesse idea minimize online seed mixture good influence call selection item pre seed seed preprocessing guarantee exactly deal issue spread use round method seed explain preprocesse value online enough maximization sophisticated rounding notation return neighboring basically round every influence spread round di output topic sub additive preprocesse sub every mean spread seed mixture spread topic verify network assumption could even layer edge odd layer topic structure believe real influence sub constant p p grain follow spread I p additive worst tight focus topic explore focus maintain reduce seed search seed seed topic preprocesse stage seed item seed topic mixture I greedy describe call stand small much spread seed find seed j ps derive seed pre seed select seed seed set motivate especially intuitively nod much whether mixed pure precise seed I separable topic case definition topic recall every seed let otherwise greedy also mi p advance sort mis round seed pre add marginal influence simply influence return seed I although mis original greedy algorithm topic mixture separable reasonable assume seed topic seed seed possible mixed denote final seed let fully set disjoint select topic seed set v j pi I uv conclude greedy suggest mis network verify well marginal sum individual thus mis verify real compare influence experiment preprocesse mis comparison aware greedy algorithm preprocesse aware greedy employ evaluation hundred greedy ic influence achieve optimize degradation spread paper degradation work ic ic use node seed output choose preprocesse ranking factor use degree high finally compare greedy preprocesse mis utilize seed experiment ghz cpu core server code write probability node topic scalability large www node author million topic practice medium influence practice usually mixture topic cover three dataset since case describe section maximize learn test greedy pre seed except algorithm equally distant pre core different seed test compare spread run carlo simulation influence seed bind solution influence spread greedy seed multiply theorem set influence spread influence influence minimum greedy seed total day day day n sec l sec ms ms sec influence show preprocesse greedy dataset seed table count cpu time need cumulative greedy suitable two preprocessing separate gap among algorithm small low average spread seed seed analysis first perform well aware observation ignore topic influential seed topic thus spread demonstrate mis fast seed three order report order magnitude aware sort baseline perform aware replace greedy mis spread indicate preprocesse reduce preprocesse table separate among utilizing greatly save online mis well small degradation seed spread small small topic overlap preprocesse still seed topic topic suffer mis online aware need least slow large indicate time mis graph run influence close competitive spread suitable large performance heuristic vary significantly minute long complete fast processing achieve time graph influence spread outperform heuristic furthermore greedy slow finish mis achieve spread mis use preprocesse first formulate cascade provide influence maximization large body influence study optimization number study machine g social influence topic influence topic aware influence et ic maximum aware computed seed complementary list introduction follow combine advantage guarantee world investigation topic wise preliminary bring insight influence guide acknowledgment provide observation lin seed seed node base certain diffusion aware model issue influence user dependent idea aware maximization topic preprocesse couple stand candidate spread preprocesse effort idea social connect maximization seed seed large people social network model direct represent relationship propagation seed influence maximization set seed network spread activate influence seed maximize wide text word decade improvement efficiency extension model information etc refer item influence friend influential influential influential topic aware cascade item topic mixture individual aware world task adopt propose topic influence maximization scheme topic maximization every diffusion preprocesse influence item mixture come seed study available aware preprocessing purpose analysis user relationship significant different topic property seed mixture come top seed topic influential influential category motivate explore preprocesse first minimize online seed ratio influence sort mis influence computation provide justification show mis conduct evaluation art result mis stand preprocesse spread either aware preprocessing comparing include world motivate theoretical seed mis novel complementary mis simple achieve competitive spread within need influence focus cascade ease presentation parameterize direct graph represent influence ic capture discrete activate activate chance inactive stay active activate stop activate define influence seed influence node diffusion computer mining like investigate topic overlap define topic p u I overlap topic fairly network apply among topic topic node network may research type movie cccc dirichlet seed overlap seed greedy come seed check source seed mixture seed topic mixture seed mixture seed separation even significant seed topic contribute seed network two present influence raw prior american site discover raw trace network represent user edge friend rating rate movie node direct topic maximum action trace topic influence probability due lack
problem especially organization hardness binary neuron unit binary infer pattern assignment constraint act constraint cause critical nonempty perceptron build complex learning memory weight weight hardware implementation perceptron thus rule structure mining code compression biology perceptron effort devoted solution case local density increase explain relate statistical past computation picture topic science machine physics recent landscape focus solution hamming distance element comprehensive description solution sample boltzmann equilibrium landscape reference potential state spin spin physical meaning cost temperature temperature work term entropy temperature physical insight understand organization demonstrate space perceptron isolate solution density minimal separate solution grow reveal hardness perceptron become exponential time require fixed finite learn binary potential replica conclude remark feed neuron perceptron association desire input pattern desire randomly actual nj pattern impose constraint denote perceptron configuration energy incorrectly e convention ensure order unity sake statistical limit perspective perceptron able extensive pattern capacity configuration quite nontrivial replica method analytic landscape perceptron densely connect learn idea select constrain equilibrium constrain angular partition constrain averaging quantity reference coincide typical current aa nm mx interaction replica replica mathematical identity average nr get constrain eq gaussian q saddle transform characterize lie apart hamming divide extract value concavity numerically point work move identify message pass word extensive barrier landscape always state solution problem sense make isolated solution separate picture report study moreover convergence solution replica symmetric introduce replica break weight zero internal apart result require analytic perceptron describe entropy landscape reference equilibrium independent reference solve saddle point equation concavity lead distance constraint minimal density imply compose one flip extensive refined picture organization perceptron understand connection study algorithm analytic analysis offer basis mathematical study hard information spike time neural classifier partially fellowship researcher grant core soft matter information current context equilibrium configuration temperature perturb constraint configuration satisfy free coupling configuration interest ground set equal substitute definition energy zero temperature evaluate value replica identity aa nm characterize average u q replica approximation symmetry overlap share average h v q deriving use gaussian parameterization structure pattern eq integral dominant saddle transformation saddle laplace respect keep consistent equation eqs characterize equation equilibrium affect system perturb depend get follow together saddle derive measure saddle equation way time perform integral retain covariance dp concavity saddle fixing coupling field method solution give derivative
may regularity fourier wavelet edge set translate penalization lead involve extensive inverse problem recent many mainly focus variation address optimization vertex incidence play intersection ball variation image follow display ability present significant decrease admm denoise matlab intel ghz core dual measurement original corresponding spin obtain solve j diagonal modelling transform ng redundant frame area background round subsampling strategy array acquisition maintain subsampling factor account reader therein reconstruction shown implement apparent subsampling matrix iteration require gradient effect absence slice snr wavelet primal learn audio management streaming network great computer vision dimensionality inherently noisy model aim explain usually interest relationship particular suitable methodology solving offer great dependency hard modular manner use powerful non hand discrete highly nonconvex np hard discrete unary encode order encode dual offer important advantage characteristic example provide problem theoretical rely schema art applicability mrf apply low level include view match tracking estimation image motion top bottom compare per iteration expansion form challenging task medical two domain grid cost project deviation system advantage complex divergence modality furthermore admit broad graph employ fig minimum surface matter mm result primal another application input right seek left problem measure difference patch spatially depth nonconvex exist obtain level convex relaxation discretize solve primal dual discrete optimization similarity pixel two image employ surface curvature dual principle fig especially np also minimization upper optimum approximation throughout assess primal extra practice correspond estimate almost review primal employ contribution perspective although prove effective extend also accelerate parameter speed parameter various technique fast method block distribute mention relaxation combinatorial np problem challenge e g high field label set appear main generally develop dual bridge kind problem blind deconvolution duality height em theorem corollary theorem theorem pt playing method problem drastically derive bring play idea important optimization nonsmooth emphasis present principle give method propose large discrete provide usefulness discrete duality computer optimization extremely paradigm constitute branch mathematic signal communication popularity approach stem characterize form uncertainty signal uncertainty noise often inherent perfect inexact application characteristic encounter area scale example computer solve low require least pixel image bad therefore exception similarly field due ease collected cope truly naturally situation arise application g network network constitute address tractable exploit problem possible take regard concern particular class method proceed task well formulation problem mention broad primal dual primarily problem duality nonlinear composition operator advantage method scheme iteratively compute subproblem involve scheme former gradient operator latter use proximity operator implicit implicit may easier exploit property flexible efficient dual achieve known proximity linear separately result expensive require significant last least becoming increasingly efficiently handle primal prominent vision labeling seek include task segmentation optical flow estimation matching mention image problem highly nonconvex optimize offer lead message easily handle regard estimate call principled derive powerful combinatorial bad aforementioned primal dual primal solution optimization background theory originally may appear topic allow broad class combinatorial problem relaxation certain discrete good solution encounter constitute source develop technique thorough intuitively principle behind continuous discrete advance place concern primal solving connection method alternate useful context signal notion duality duality duality programming notion subdifferential proximity section explain various devote admm deal schema well technique combinatorial dual base relaxation domain inverse field summary discussion definition use dual algorithm later hold space dimensional allow take modern discard search optimal problem component intensity must nonempty domain establish proper low f see nonempty fig htb subgradient proper differentiable subdifferential reduce singleton nonconvex extend definition subdifferential may reduce subdifferential growing introduce early work proximity define thus result minimization show variation p soft operation equal projection p operator view projection proximity projection view contraction engine banach proximity ensure algorithm proximity operator nonconvex guarantee uniquely define notion deal notion case graphical illustration conjugate fig particular subgradient necessarily proper convex conjugate express result always see subdifferential important characterization minimizer question exist subdifferential subdifferential provide property link proximity useful ss parallel draw multidimensional fourier transform tool process main kind nonsmooth encounter feasibility transform conjugate fourier notation define r dr denote dirac cm l invariant translation translation multiplication invertible jx nu nx j inf convolution element addition product offset define support nonempty positively norm norm compressive sensing variation area sharp contour convex set conjugate hypercube impose array express easy know solve dual bring information first basically solve provide primal precisely proper addition condition exist vanish gap equal zero sharing suppose composite distributed manner assign vertex reach consensus original rewrite space indicator orthogonal complement form u want utility saddle saddle relation actually prove tucker primal lp j nb mean formulation view property conjugate readily duality lp solution obtain lp lp convex sophisticated one wide q differentiable fidelity encounter respect introduce additional smooth may flexibility problem see possibly term inf convolution reduce trick section jointly instead focus find tucker mention quite simple algorithm admm direction multiplier view belong since possible saddle point augment lagrange lagrangian lagrange admm split gradient result perform potential advantage parameter norm nonetheless exploit generally comment formulation benefit split objective example operator quite operator introduce conceptual fact account role differentiable strongly convex I point amenable architecture successful problem dual method idea parallel dimensional mf mf projection separability proximity note consensus derive version simultaneous multiplier even onto simplicity instance involve formulation allow propose manner mention another image primal express l lx nk nb formulation model broad hereafter encounter lead principled approach find np call much extent reasonably obtain make former relaxation relaxation relaxation instance integer define relaxation much easy quantify hence approximation lp relaxation relaxation tighter interestingly tight lp relaxation involve expand feasible mention derive relaxation use maximum objective relaxation allow nonconvex relaxation discrete naturally primal lp powerful relaxation exist many number grow especially semidefinite second cone relaxation observation primal dual focus principle powerful heavy underlie lp aim lp fractional try much whereas schema technique optimization worth start exact program initially problem tight probably go back maximum max max essentially schema branch minimum span case schema drive fact initial iteratively total guarantee optimal since integral end notable primal often lp series combinatorial unweighted minimize complementary primal use instead algorithm np hard problem admit formulation applicable tool combinatorial cover network feedback location domain vision analysis schema broad np integral complementary since lp could schema turn relaxation need primal compute optimal due relax constraint give attempt solution primal rely explanation primal feasible assume show approximation optimal e dual principle rely fact sequence primal lp duality optimal exceed lp unknown guarantee dual feasible cost gap cost cost thus approximation principle heart impose directly dual convenient viewpoint generate program latter duality primal applicable duality gap exact relaxed variable check hold j relaxation give hold clear follow ny feasible satisfy complementary satisfie principle x base yield dual employ follow solution update apply primal schema strategy maintain primal complementary satisfied introduction measure degree minimize opt maintain relaxed complementary condition infeasible iteratively infeasible primal solution would improve feasibility dual ensure feasible matter strategie gradually bring primal close together complementary primal versa fact improvement lp lp compute problem relaxation costly primal schema purely combinatorial algorithm notice require primal schema np element disjoint subset assign cost subset cover minimum cost see determining include cover ensure least set obtain replace boolean relaxation mean number schema relaxation call
number uninformative voxel carry category include informative possibly redundant focus feature fmri voxel often redundant completeness stability feature discover possibly redundant accurately mainly aim uncorrelated reveal discriminative voxel cognitive credible three category filter embed score algorithm use score feature subset use embed typical embed method voxel fmri classification treat degree voxel proportional value component selection identification nonzero component consider challenge focus voxel motivate however inference plain sparse hard interpret voxel portion voxel result potential trust success plain column condition correlate noisy subspace thus plain incorporate structural brain imaging segregation achieve reliable interpretable hypothesis voxel discriminative group cluster correspondingly strongly use elastic net try use voxel regularization deal highly recently penalty add correlate besides penalization total penalization use simultaneously voxel tv make activation spatially voxel fusion successive certain smoothing plain regularize group structural correspondingly explicit segregation structure plain enforce structured solution voxel group either drive group agglomerative cluster grouping sparsity discriminative voxel completeness voxel result selection uninformative voxel year important method selection voxel bootstrappe behave disadvantage plain select even bad instability informative set information major advantage positive obtain expect addition stability recognition brain fmri plain regularize characteristic focus ensemble idea apply randomized tree discriminative voxel often spatially contiguous common clustering run subsample selection add help result rescale implement stability voxel fail clustered voxel paper via constrained subsampling voxel wise fmri voxel subsample classical sensitivity discriminative voxel stability achieve include control positive voxel new summation stability structural case structural rough voxel highly subsample scheme help shape numerical perform computationally organize section introduce new voxel selection real high specificity short future give denote fmri voxel idea follow difficulty effort detail voxel intercept voxel discriminative plain enforce structured correlation way grouping adopt grouping compare main belong sparsity feature prediction interpretability provide grouping allowed incorporate redundant therefore help voxel induce plain regularize due grouping reliable either voxel obtain incorporate discriminative make choose simultaneously like plain choose regularization parameter sample effective way reduce difficulty voxel connection control positive expect redundant voxel correlation explicitly consideration paper probably discover aim integrate stability selection common one sparsity subsample former likely yield latter whenever pure regularization subsampling easier extend subsample combine structural sparsity first explain subsample baseline returning feature special except stability sample subsample version drop important randomized structural sparsity incorporate structural consideration fmri structural sparsity general concept implementation depend fmri datum propose specific implementation name subsampling adopt replicate subsampling subsampling voxel group voxel lie cluster subsampling reduce negative size partition quite correspondingly solve group part select voxel subsample either drive select subsampling constrain prior selection able shape discriminative flexibility voxel belong area voxel though might aim voxel association response label part response would pay save lot correspondingly baseline subproblem prefer average idea apply variable positively correlate feature yield fit variance voxel pick block subsampling lie single voxel variable greatly result recovery much therefore greatly improve pose compatibility apply th large magnitude pick voxel lie cluster average simple boundary structural drive partition voxel patch algorithm mean spatially group cluster reduce fraction total large come constrain block stability stability subsample perform matrix subsampling select integer voxel notice run result small cluster expensive procedure input classification subsampling term score voxel brain perform voxel perform row block voxel calculate pick voxel pick voxel basically stability mainly observation point subsampling guarantee false positive adopt method plain finite bounding ratio expect false feature adopt sparsity base pointed term incorporate interpretability learn voxel functional relevant might bring bias intuitive explanation thorough study subsample reduce constitute cause stability rescale improve efficiency cluster large proportion paper algorithm univariate voxel voxel pattern include randomized logistic selection random feature implement logistic implement projection software logistic great software code otherwise might inherent allow block match easily positive negative block probable prior knowledge performance synthetic fmri experiment respectively chinese problem block repeat nine break block state block subject acquisition preprocessing mr west china china fmri image te ms flip angle head slice without gap centre uk www uk head motion institute imaging template brain three isotropic mr fmri high pass hz noise fmri cluster algorithm figure show brain base score different thresholded visualization meaning zero zero noisy localize brain identify visually recognize voxel balance control false false negative general identify control positive control strictly need threshold filter wrong confirm knowledge carefully threshold positive brain notice threshold via small positive achieve nearly apparent false control brain extra region likely brain science pattern successfully region indicate work cognitive default human task state stability selection method identify brain functional structural part randomize logistic common slightly concern make distinguish false positive compare advantage fmri size subproblem h public face object consist eight group ms ms inter stimulus interval brain fmri record volume stimulus volume fmri acquisition previously fmri house cat consist evenly spatially constrain software number evenly use sample select limit map first score thresholded visualization thresholded algorithms requirement fdr control method validation svm candidate code threshold correspond high except sensitive voxel potentially build logistic discriminative voxel select table extra voxel voxel could relevant voxel explain validate discover discriminative viewpoint rand tv rand maps phenomenon study area setting contour quite similar thresholded select stability maintain control false sensitivity compare voxel region unlikely positive cross validation voxel tv large discriminative voxel voxel positive even estimate discriminative voxel around positive test positive share common logistic conservative control false positive setting positive false estimate however reveal contrast voxel keep false positive result table algorithm rand tv select positive rand false look voxel pick randomly consider accuracy among combination achieve high randomized logistic reveal discriminative accuracy predictive specificity feature voxel degree prediction significantly high window matlab run intel eight processor processor base frequency ghz gb parallel involve master cognitive smooth lasso svm logistic take short notice write computationally master master test test spatially constrain take long time tv l directly software efficiency present algorithm rough rand logistic rand master voxel selection drive extraction implementation variate correlation multivariate mostly empirical understand limitation spatially well alternative method distinguish false positive address tv l integration thanks ga team provide randomize anonymous many constructive suggestion greatly support program cb cb project science foundation china specialized research education china z z slide volume voxel incorporate voxel stability effort fmri give brain brain proxy brain make brain fmri level bold grid know fmri compose series bold voxel high segregation area differ global integration consist voxel whose spatial voxel organization subsequent identification perform set allow discriminate brain localize discriminative due directly interpretable growth activity order multivariate argue determine sample distinct whether mean differ supervise analysis training svm training build set marked belong two category new common svm notation section regularize regression categorical variable regression necessarily score predict dependent intercept scalar regularization adopt intercept elastic hybrid elastic generate reduce value outperform follow intercept scalar logistic regression randomization stable brain imaging traditionally htbp htbp region accurately stability less contiguous block voxel positive positive sparse purpose receiver roc versus specificity different support roc selection ground voxel base positive specificity rate tn respectively practice use spc roc large false see voxel select two brain combination voxel select two test ground unlikely
parallelism simple effective big lda strong model topic million word parallel implementation face difficult challenge access explicitly access input categorical assignment topic lda token position token assignment topic topics lda reformulate collapse fast accord perform iterate analogous analogous lda schedule rotation scheduling mod return empty w p f b return update sufficient f c parallelism identify assignment statistic speed sampler worker one subsequent schedule amongst worker subset partitioning divide token denote worker worker subset schedule worker gibbs assignment worker word parallelism observe assignment choose sampling sample exactly fast gibbs sampler fast lda machine parallel gibbs sample unless conditionally disjoint document worker conditionally except dependency extra end worker denominator thus induce ij ij word proxy copy cm total token must lie plot error wikipedia refer topic machine processor core total throughout lda exhibit parallelization benefit consider collaborative predict preference give preference tend rank interested enable decomposition pose challenge purely datum sgd address divide across worker mf incomplete item preference discover product use miss user formally observe index column cm schedule row index disjoint index set compute analogously merely supplement motivation counter prevent dependency candidate next represent naive scheduling execute worker partition worker submatrix partition manner rule accordingly worker th iteration u illustrate h frame lasso schedule scheduling get new draw c safe u j j partial sum f frame z aggregate demonstrate implementation lda mf size baseline fast baseline partition baseline implementation distribute lasso rr alternate square implementation mf lasso rr random scheduling choose popular distribute mf two machine effectiveness parallelism hardware cluster lda cluster mf core cluster contain ghz cores gb interface cluster contain ghz cores gb ram via interface use size unless otherwise english wikipedia tokens token token large publish demonstrate topic large create extremely rating movie vary exceed paper feature world add noise otherwise reach run mf either handle converge quickly attribute fast parallelization lda reduce synchronization requirement mf nearly limited quickly dynamic figure schedule cause second mf achieve well parallelism fast trajectory machine fix confirm fast machine big parallelism scalability efficient memory partition invoke worker parallelization correct parallelism study enable scheduling aspect parallelism static partitioning parallel coordinate abstraction direction cost star topology eventually become bottleneck issue wish asynchronous parallelism parallelism also want machine logistic theorem computer fit take long natural processor naive ml make inefficient proportional develop parallelism explore schedule ml speed inference correctness demonstrate efficacy versus implementation model factorization digital storage medium improve massive scalable machine numerous heuristic principled big million deep problem attention world challenge efficiently worker access large k furthermore parallelism important demonstrate suited big subset allow partition memory allow variable parallel tackle big exist framework show variety crucially framework automatically decide next choose next user automatic scheduling convenient offer grain subtle graph structure moreover allow user criterion schedule lda rotation scheduling collapse gibbs topic scheduling descent dynamic scheduling coordinate sample framework parallelism schedule execute aware way dynamic parallelism schedule specifie specify individual partial variable specify aggregated full primitive ensure distribute automatically execute user utility implement schedule popular ml implementation enable solve programming modeling vocabulary matrix application j send worker update j frame u u worker return frame update worker schedule basic signature primitive parallelism refer parallelization model high change quantity ml iteratively iteratively use strategy like parallelism parallelism allow problem massive machine memory advantage model memory machine machine algorithm whereas strictly constrain small practical consequence pair enable topic enable user systematically parallelism framework map paradigm user ml repeatedly create iterative figure schedule primitive vs less
n j paper one notational ease efficient preferred paper get difficult large effect deal due adjust root parent hypothesis continue child adjust factor denote cardinality incur fine effect effect specific marker partition region marker form elastic allow non zero coefficient final operator individual marker score apart involve genomic kernel parameter relate net post process standard performing typically hold explore detail hyper parameter choose accuracy improve informed opinion reflect resource aim depend number marker marker hyperparameter estimator allow control value model kernel flexibility detail locally test importance genome toolbox seven trait date md trial year line evaluate model display genome wide use replication split trait trait available divided rest line evaluate fashion display level accuracy association diverse genetic diversity target also lin locally trait whole genome length trait display trait genomic trait individual individual select angle angle width data marker national day accuracy region lin score display figure width angle propose although seem population single kernel additional advantage utilize score use marker occur context advantage solve matrix matter dimension involve matrix memory problem marker loading marker genome miss outlier matrix separate region marker genome divide linkage proximity calculate code sequence grouping trait allele frequency marker absence marker guide incorporate membership marker subsequent set code file acknowledgment award section example property ph genetic additive genetic lose argue line genetic map additive effect parametric mixed genomic relationship design lasso performance explanatory genetic semi parametric mix marker successfully predict study genome marker always prediction value bi population express design effect eq kernel hilbert connection recognize model implicit input dimensional space refer number symmetric k k exp taylor reveal kernel regression extend additive though option marker genetic incorporate additive marker implicitly incorporate study empirical prediction gaussian however increase lose issue overall effect additive potential argue kernel estimate genetic article argue marker effect marker locality model additive marker since incorporate view semi incorporate contribution article genetic whole genome local effect genomic snp marker easily reach million hypothesis focusing argument segment genome building block schema predict complex evolutionary mechanism fitness tend well structure building block example fitness genome lose segregation would build block parsimonious genomic utilize counterpart genomic product remain section briefly literature notion similarity come source literature include method commonly component calculate component perhaps genetic gene interaction kernel kernel learn building stage subset divide marker subset genetic process local fit training step obtain marker nest subset marker conclusion subset annotation marker however capture genetic possibly nest genomic define marker linkage genomic region inform illustrate root whole genome genome width hierarchical allow region coarse divide detail availability nice coarse fine hypothesis tested error keep scheme
calibration approach structure sense calibration estimation phase compress sample sparse shannon number measurement denote transpose compressive recovery retrieval imaging optical imaging vector fourier reconstruct measurement retrieval recover essentially vector definite convex pl trace low among one acknowledge retrieval study theoretically large measurement therein iterative describe propose measurement compressive quadratic pursuit addition sparse magnitude need provide isometry condition reconstruction signal pursuit solve class compressive define contaminate cross hermitian semi definite signal recover program become identical though determine sparsity structure constraint joint induce objective induce improve constraint know norm suitable therefore ambiguity recovery problem value system investigate perfect generality compressive room improvement numerically pursuit problem basis convex empirically pursuit provide bound good recovery performance insight sparse determine perfect show eigen operating element eq projection onto equal represent order imply strict assume bc show constant choose ensure eq phase w convexity x lead convexity space definite local convexity optimization imply eq solution q equivalently suggest since satisfy consequently satisfied provide combine satisfied similarly condition true set p tight transform non nature constraint requirement simplify criterion tighter guarantee perfect straightforward sparse whether perfect recovery respect non entry measurement varied level complete perfect signal lowest highest select among figure increase input signal mainly recovery broad majority upper bind feasible perfect recovery result tight estimate recovery display phase diagram compressive different phase demonstrate accurately report observe match simulation evaluate performance tight perfect http advantage simulation evaluate method determine possibility successful
information reconstruct coverage map allow robust resource allocation combine mobile user balance wireless resource performance scheme far enhance addition predict promise like discussion corollary institute department electrical address reconstruct map evaluate adaptive alternative offline algorithm tailor iterative subgradient user quality estimation measurement process simulation realistic algorithm world application mobile estimation coverage attract attention challenge enable network wireless service reconstruct importance device device channel extract provide device reliable decision make crucial allocation decision resource allocation present future propagation quality service mobile information user utilize resource allocation media streaming buffer mobile reach area avoid failure decision traffic traffic load transmission cover problem estimate two loss ingredient reconstruction coverage development resource scheme application maintain coverage prediction base user measurement path quickly adaptation coverage reconstruction need requirement important arrive continuously online nature ensures take account continuously quality addition exploit enhance information understand information concrete location exploit incorporate measurement input due various wireless user usually position measurement control network non uniform sampling significantly handle situation prediction overall despite area kernel aspect paper enhance provide coverage aware base adaptive algorithm incorporate simulation attention realistic publicly real trace spatial short work online loss user kernel estimate map side algorithm time enhance weight underlie wireless network challenge researcher wireless empirical path decade measurement measurement mainly machine neural ann aside technique recently krige apply track map capture spatial correlation scheme spatial available prediction scheme model measurement wireless author fit correction real wireless university building type subsequently update corresponding path refinement measurement refine iteratively study propose mechanism map readily without need environment rely project subgradient iteratively generalize subgradient easily tool machine affine successfully network image recovery real respectively letter thereby element element product trace frobenius inner possibly dimensional notational convenience notation respective space context infimum nonempty closed set uniquely point singleton reproduce rkh property ff reproduce calculation carry trick high calculate f particularly suit design minimizer specify model outline important basic side scenario problem pose reconstruct path map use availability wireless wide low cost device physical method input information trajectory technical notational clutter section deterministic good performance mobile sequence receive signal th measurement relation path measurement unknown measurement arrive give operator finite quickly estimate function computational online algorithm keep improve measurement arrive easily investigate highly base subgradient technique cope operator technique feasible computational limitation practical algorithm grow measurement relevant relevant sequence call estimate projection proximal gradient smooth function notational dictionary treat computer interested reconstruct path location devise computer define construct path future location expect visit attention outline section model selection scheme correspond specific application belong contain account unlikely contain enough provide reliable precisely set contain expect point reasonable obtain problem feasibility possible example able store whole sequence n practice recall able soon come back estimate finitely many use set however mild assumption estimate ii approximation I propose variation project subgradient detail set approach choose continuous constant solution produce solution note average averaged follow fix adaptive hope obtain tracking time comprise differentiable function compute easily forward modify end comprise structure iterative proximal step base lipschitz mapping time vary two measurement add column euclidean close remove irrelevant discard refer algorithm weighting row enforce cost employ weighting compressive first stability order keep weight row enforce inspire minimization algorithm connection aim minimize induce enforce convex absolute vector one fix obtain follow incorporate omit constant sum address intractable minimization detail convex concave g additive form q variable become find monotone try many norm alternative formal approximation need scope approach element completeness literature reweighte enforce pointed numerical suggest minimization recover report good frank wolfe type closely wolfe construct move correlation scheme regard increase roughly factor numerically propose estimate noisy uniformly parameter variable error mse communication loss realistic city measurement subsection devote numerical except kernel common applicable kernel free l projection weight ci width l reconstruct planning network wireless channel assess real evaluate realistic city language create meet realistic precise city loss datum ray base momentum appear provide path valuable mainly capability algorithm reconstruct path loss long sufficient specific resolution underlie location low server assignment cell base estimate define base assume poisson far randomly end realistic movement trace conjunction simulator tool simulator generate movement trace realistic limit intersection mobile trace enable processor interest area parameter bs pixel run capability propose figure simulate respective show sake similarity original path estimate loss user system estimation location dash dot solid green path trace area mean time instant compare evolution predict note one frequently available measurement completeness account pixel belong define sufficiently evolution outperform speed value report location incorrect measurement variable absolute offset e pixel usually gps device relevant error sensitive path inaccurate achieve drastically small dictionary figure simulation show evolution size observe sparsity algorithm tailor raise experiment select concern impact factorial choose indicate subscript value table l result accuracy default figure choose small gain achieve bad versus steady capability set dataset project receive strength measurement collect university area trace measure measurement take particular find remain performance gives compare low due
signature element generative employ foreground histogram foreground image contain essence appearance foreground complementary body symmetry complementary aspect person appearance extract salient feature weight exploit principle geometry capable provide meaningful solution distance video separate automatically give person large diverse camera view comprise extraction descriptor riemannian stein stage detail subsection reduce vary pixel image generative use advanced pixel locate xy xy colour channel xy colour g r indicate magnitude channel rgb colour select relatively certainly thorough beyond several noisy sample iii correlate symmetric definite interpret riemannian manifold space handle manifold non due largely challenge manifold take function even riemannian structure consider riemannian somewhat size define logarithm demand essentially eigen furthermore prevent conventional riemannian embed tangent distance distance give start discriminant find intra distance scatter similarity map classifier refer relational method person dataset image person cover aspect represent improvement caused evaluate stein conjunction classifier ie manifold without create stein capture real arrival video image normalise height overlap illumination person probe stein method outperform also conjunction preferable stein direct base capture move camera contain variation randomly select image rest commonly set propose considerably outperform pls direct stein par create apply cluster ensure signature close stein pls appearance person comprise represent covariance matrix foreground treat riemannian manifold similarity class aid stein divergence iv discriminative vector final classification traditional latter result inaccurate modelling manifold person identification dataset recent histogram accumulation communication centre shot person relational box university school identification particularly challenge due view people signature effectively illumination pose camera appearance space application model non end represent covariance matrix interpret riemannian manifold manifold similarity bregman divergence stein similarity classification similarity vector manifold tangent space suffer comparative evaluation identification obtain technique histogram partial accumulation feature surveillance identification person match overlap camera diverse location within context surveillance large candidate pose body illumination variation make human person approach typically use entire challenge person separate camera view challenge change body illumination variation appearance person generally argue geometry end recent covariance interpret riemannian manifold embed manifold pairwise tangent shot base identification manifold tangent space require adapt recently riemannian similarity similarity aid form stein divergence task manifold convert vector discriminant analysis method separate continue follow person
primarily web message phone comment movie precise ratio aim rnn hour list epoch newly pass learn ensemble hide apply normalize training example total consistent spaced filter compute window ms audio file training set divide global input describe train phrase character phrase alphabet word keep token speech benchmark situation create web dl speech benchmark range hundred benchmark hour speech rare utilize expressive recurrent synthesis approach vision find convenient speech properly present base scenario environment gpu synthesis strategy build system noise combine speech rely processing stage believe continue increase dataset size future acknowledgment grateful dl speech forward project also ng ai art speech recognition architecture significantly system rely process traditional tend poorly environment hand background learn concept key well optimize synthesis obtain varied data speech test speech also widely art system rely compose stage paper speech call speech stage language achieve network rnn learn directly specialized setting corpus system speech heavily process stage input acoustic hide expert deal model introduction algorithm improve speech acoustic deep play speech noisy robustness deep recurrent advantage capacity learn performance sufficient power however pose build must train enough effectively utilize challenge handle text speech address enable neural meanwhile rapid et demonstrate speed gpu aim vision large scalable rnn complicated vision partly lee al techniques map novel scheme parallelization quantity speech collect learn robustness take idea suffice build end traditional task achieve new noisy speech recognition system error rate remainder recognition begin describe recurrent follow gpu synthesis experimental deep conclusion core recurrent rnn english sample vector audio denote th audio frame sequence character hide unit recurrent frame frame non layer compute eq relu activation bias fourth layer hide recurrence backward recurrence note must sequentially must sequentially recurrent take backward softmax character probability slice denote column weight bias respect output character point back nesterov accelerate rnn considerably simpler relate model literature long term memory lstm circuit one disadvantage lstm cell store multiple neuron forward backward bottleneck homogeneous model computation recurrent activation relu output involve highly gpu wise nonlinearity length expand fitting reduce several dropout feed layer recurrent employ computer vision feed version beneficial translate audio ms half bank propagate use rnns output speech character character rnn external plausible english example occur never appear set speech language know impractical integrate gram language train huge corpora language section corpus phrase support vocabulary train n experiment rnn weather right weather rnn character probable language character word aim objective validation n gram maximize objective use highly search network amenable speed execution homogeneous network implement optimize call fully connection experiment feasible multi gpu accelerate two parallelism multiplication prefer many example prefer gpu wish support parallelism across separate minibatch iteration parallelism parallelism implement multiplication resolve problem sort similarly sized inspire format accelerate rnn parallelism train minibatch face example update improve yield also inefficient minibatch spread example minibatch gpu scale partitioning parallelism due sequential layer layer comprise backward perform unfortunately split go depend less model half dimension layer trivially decompose along time series assign gpu another gpu first gpu begin forward activation begin backward activation mid intermediate activation swap gpu gpu forward work run recurrent rnn recurrent layer library deep require abundance record correspond english public extensive consist hour comparison dataset read fisher read corpora publish linguistic expand potential training even synthesis context primarily environment capture speech environment way audio generate superposition noisy speech audio track audio track noisy speech track audio necessary form together realistic audio risk hour clean create hour speech track span roughly say repeat since become recurrent single hour shorter public video source separate noise end ensure match synthetic datum datum reject noisy effect encounter recognition environment effect overcome collect ensure induce play background person record induce thus allow capture experiment system use dataset character level fed word yield highly researcher split report new alone full rate
aggregate name bootstrap dag overall entire ensemble structural hamming dag node fitting thus positive implementation hill regime package keyword skeleton hill positive consist edge direct node interpretation dag dags relationship intelligence genetic learn quantitative research field body topic topology graphical structure learn indirect system nucleotide gene practitioner expression examine conditional would conditionally impose constraint parameter learn future instance dag sample huge exponential lead challenge commonly dag tackle challenge utilize search acyclic check sample gb learning procedure highly drastically perturbation tackle use achieve reduction consequently ensemble dag base aggregated minimize family metric aggregation perturbation approach inspire name study procedure false positive bag initially build recent year successfully perturbation aggregation previously dag learn propose measure confidence graph learn stable log perturbation locally average learn though graph acyclic aggregated dag propose rest brief overview model hill structure aggregation strategy dag aggregation section brief overview direct acyclic reader direct acyclic edge node dag graph figure give I x pa set set show graph obtain discard parent seven skeleton middle edge due connect highlight red node sequence node meet head neither meet head node figure separate next relate dag model say admit probability formally dag admit recursive factorization specify local conditional namely distribution dag df dag respect admit obeys set I say map obvious independence vice versa say perfect map hand compatible dags dag say I encode follow skeleton edge structure equivalent dag converse true relation partition dag equivalence bottom right since separation separate clear equivalent x x ccc impossible dag solely equivalence formally dag structure distribution dag structure p brief overview method dag score dag commonly score early exponentially nod exhaustive search therefore greedy dag include search dag dag view independence hybrid method structure test restriction focus implementation hill search performance dag structure procedure score discuss decomposable hill dag propose hill search make dag structure identically map goal learn map denote moreover dag define score define dag space depend node parent hereafter update rest subsection include matrix property negative negative estimator decomposable residual sum dag reasonable model candidate bic decomposable score aic I set score pa mean node hold minimize strictly score add edge imply ii eliminate hill update distribution term dag want find among dag mention due space moderate heuristic employ algorithm initial subsequent operation operation maximum operation able decrease score exist operation result cycle acyclic check operation apply major hill acyclic check potential cycle calculate score note operation one operation decomposable score change neighborhood score update score neighborhood involve operation graph lead change operation nevertheless score costly operation step moreover take stop e greatly high efficient hill search information facilitate score acyclic current illustrate idea operation current hold operation involve result score cycle cycle cycle operation operation dag search dag dag aggregate dag hill subsection metric hamming conduct much set dag learn dag aggregate crucial aspect dag dag aggregated subsection metric score theory vector need convert dag hamming lead valid hill search edge dag among specifically edge always operation j distance unit lead generalized ij define j j I j correspond correspond compare dag true study penalize skeleton edge reasonable orient support aggregation dag tend edge well sf dag ensemble p additive function individual edge indeed hill simplify input ensemble dag sequentially sf sf initial graph current sf pass acyclic check edge add proceed stop set cyclic hill decrease moreover addition hill conduct cyclic edge ensemble dag node dag minimum proposition dag generalize direct dag aggregation q constant hill simplify proposition sf search depend sf orient edge penalize extent frequently edge appear possible selection sequentially stop empty pass edge acyclic add stop cyclic edges hill score selection frequency reverse large lead hill reach proposition conduct extensive simulation examine several dag learn sample independent mechanism variance deviation part consider namely standardize score replicate dags size http snr graph dense graph extra large skeleton v edge table reason report skeleton structure table skeleton edge edge learn number skeleton edge identify correctly identify report average standard less fair quite identifie besides facilitate search skeleton skeleton edge graph total versus across end result stop package package independence test drive run series draw trajectory curve one method structure aggregation effectiveness empty method number positive graph omit graph node size figure skeleton point indicate stop presentation stop number skeleton structure skeleton learn figure aggregation color non black curve skeleton demonstrate superior moreover aggregation implicit model result aggregation aggregation reduce positive edge inferior well curve stop early demonstrate penalization discrepancy efficiency figure sample aggregation false aggregation able performance main list detailed learn curve table snr aggregation curve aggregate skeleton detection detection snr graph curve poorly lack curve range stop early aggregation induce positive skeleton curve difference snr term difference lie stop study procedure snr vs much aggregation aggregation low false mainly edge bic method graph infinity conduct demonstrate skeleton curve size size false give effect topology study topology focus ease figure
f il n r compact z trivially restrict neighbourhood z b neighbourhood z x numerically calculate rr desire b x software integral describe expectation decomposition boundary define restriction interior cube map parameter sufficient claim boundary parameter closure hull polytope polytope therefore cell relatively open cube cube give note parameter note cube translate map equal convex vertex properly separate meaning lie space generic face lastly follow closure use space induce decomposition boundary space figure face highly behaviour right also begin show boundary cube duality relationship euclidean eq similarly say image become despite dimension enough give enough kf sn kn sf sg sf kf ss sn sn f topological relatively face dimension association relate closure closure closure hence face lie induction hypothesis kf tt sg sf f g essentially g sn k n prove induction ss rs sn establish theorem say face boundary say image approximately approximate large relative parameter generic theorem function continuous ideal one prove connect recall generic lastly concentrated intersection hyperplane numerically volume every show jump large reflect degeneracy let degeneracy subset row row generic jump z n endow sum square open define converse volume generic always degeneracy enough rr dominate generic matrix compare q approximately see since rx denote hand approximately generic element argue b ib z jj combine degree degeneracy let reasoning last equality perhaps logistic novel finite jeffreys volume simple elegant version give model therefore parameter though arise principle boundary space logistic map cause implication behaviour exponential deep duality lastly acknowledgement author would thank interesting discussion remark prove generalization classical finite volume interpret volume generic matrix less nearby way tend prefer arise general principle lastly topological boundary draw bernoulli canonical riemannian transformation geometry behaviour concentrate geometric volume prove consequence regularity jeffreys interpret theoretic measure maximize natural datum minimum description length parametric model previous logistic study prior place principle hyper code adapt logistic remarkable geometric matrix complex nearby tend choose matrix behaviour though fit equal fit derive mild full covariate approximate volume compete volume choose small maximize meaning surely go principle giving volume lastly linearly raise possibility generic relationship decomposition boundary open closure boundary approximate radius divide spherical hyperplane decomposition dimensional boundary dimensional become behaviour interesting right implication computation rest model cube embed calculate metric unchanged rescale cube show apply image processing generic topological section generic conclude remark though consider component realization model q odd odd odd odd interpret column family sufficient correspond family riemannian likelihood hessian expectation assume regularity distinguish fisher independent map sense natural odd q eq riemannian manifolds riemannian manifold turn either euclidean cube space model isometry open cube light euclidean identity dimensional respect matrix prove lemma euclidean odd domain partly establish logistic regression volume invariant real unique odd consider matrix real substitute family natural fisher know prove publish author space function differentiable function jacobian back natural odd recall dimensional riemannian metric tensor local notion integrate jeffreys prior determinant volume density strictly everywhere rank everywhere invariant recall subspace column de theorems model let parameter model row real follow design natural euclidean isometry onto euclidean volume hausdorff inside embed around circle monotonic use volume logistic say onto co matrix sum subset generalization de binary cube unique obviously identity complete proof bound sharp degenerate particular volume upper q show generic close interesting trivially identity follow column range x horizontal integral continuous would would close compact effectively restrict fix recall derive theoretic statistical behave logistic regression model space suppose countable set model logistic design choosing minimize turn choose model large case interest principle likelihood x moderately instead use eq valid note observation regularity condition valid exist exist derive begin section generic rank mean full converse unless covariate lebesgue e component row probability generic fisher iid random number region integral region arbitrarily error assume hence give say limited computer experiment hand lastly minimum degenerate assume generic approximation fact row e row row approximation approximation selection give criterion choose datum result show criterion almost goes note n consider difference true enough sparse reduce information however bic approximate volume application processing application partly problem particularly process black white pixel black picture signal noise pixel pixel follow interpret white black subset pixel column represent segment pixel contain approximately lasso implement regression fit parameter validation
node skip infect cascade always zero whether network cascade cascade regularizer la successfully network estimate yes estimator cascade network survival hazard concave hessian sum outer matrix table hazard hessian capture co cascade check impose restrict identification hence definite diffusion cascade answer question crucially incoherence diffusion sampling cascade way child relation co compare commonly naturally satisfy specifically population take cascade hessian city index parent complement index ss min connect co reasonably cascade incoherence exist ss jk node neighbor infect cascade neighbor cascade hazard norm n appendix cascade lipschitz condition boundedness feasible evaluate absolute remark state strictly condition exist remark depend observation study income node easy incoherence condition ba ty node source cascade cascade incoherence direct condition consider star edge transmission like incoming leave long cascade incoherence incoherence condition remark pairwise satisfy analysis cascade need polynomially node ga mi million parent compare network individual edge cascade q exist constant follow regularize unique uniquely specify incoming node incoming edge sample improve remain remark co parent union bind provide co child remain largely large parent node primal previously ise good knowledge technique optimal share pattern far unique prove primal subgradient vector moreover hessian strictly unique next primal dual furthermore construct kkt pattern deduce incoming primal ne tucker kkt regularize primal solution condition substituting construct state kkt hessian lift hessian satisfy eq kkt condition hold construction step provide use expansion remainder entry eq algebraic cs ss small help lemma incoherence converge incoherence condition holds apply lemma condition hold cascade pn min lift dependency incoherence impose population regularize implementation network satisfy natural incoherence exponentially decrease cascade cascade record future cascade confidence infer edge network exist window value incoherence condition satisfied finally activation allow missing activation nsf gm fellowship song survival concave hazard pairwise hazard express sum semidefinite concavity survival concavity hessian positive definite semidefinite cascade order cascade within cascade hazard cascade index cascade matrix triangular sort continue sort cascade source position infect across cascade order break tie cascade infected never cascade infected index node cascade order similarly column correspond cascade node infect early finally remain assign cascade order lead desire spectral problem constrain constant convex constant across primal primal kkt lagrange negativity solution primal gradient since primal since optimal strictly strictly convex respect sg deduce segment strictly radius bb value segment inside entirely end detail function regularization neighborhood ci series separately bind reverse bound remain challenging introduce quantity q proceed n k b next lower set kk c convexity z j kkt respectively hoeffding start hessian proceed condition lipschitz continuity difficult q first condition continuity boundedness cauchy term inequality condition eq select regularization difference nuclear cascade jk express fisher cascade ss ss ss reasoning start incoherence score vs cascade hold prove infinity jk jk z jk prove confidence proceed final bind incoherence easily apply ss conclude eqs ss cs cs c success incoming canonical fig super cascade use cascade one super node window lead line well probability infer incoming kronecker neighborhood cascade transmission window lead edge cascade transmission method outperform cascade polynomial cascade need success exponential number cascade information across network structure trace cascade kind cascade cascade despite increase availability cascade infer ti un question literature continuous regularize mi cascade na tu condition framework structure ty cascade maximum parent soft consequence alternative pt behavior disease temporal trace give rise information neighbor influence observe person get infect cascade model infer unobserved underlie attract attention predict path spread sale stop focus evaluate experimentally synthetic real network go analysis enable answer condition run recover network cascade cascade ba li ty direction view identify relate interaction cascade make depend network structure sampling cascade recover occurrence parent nod cascade use li cascade rate cascade finite sample guarantee cascade theoretical solve especially suit scalable find sparse soft thresholding demonstrating outperform state art inference cascade general net recover network cascade cascade par discrete cascade instead diffusion validate author network continuous independent cascade source uniformly cascade network cascade additionally study bound degree source decrease polynomially cascade incoherence network cascade infect cascade generative cascade introduce transmission parameterize contrast associate differ discrete sense cascade iteratively round transmission continuous
practice learning term popular iterative nonlinear aspect fit directly term via hyperparameter penalty neural network help variance hand fitting optimization performance hold use optimization proceed loop iterative outer hyperparameter perform rigorously application search computationally train light effective systematically principled attempt good evaluation optimization train quality rapidly useful even loop complete explore hyperparameter effectively fit goal iterative procedure bayesian framework hyperparameter favor start partially complete old maintain train theoretic one continue loss roughly decay towards nonparametric decaying characterize temporal gaussian able partially train hyperparameter many different ordinary bayesian optimization space definite process prior gp gram applying form form gps become ability characterize come computational grow invert gram implicit hyperparameter characterize behaviour ern infer constant prior integrate slice domain generality hypercube optimization receive develop tailor hyperparameter sensor optimization proxy determine location choice determine utility trade explore uncertain exploiting region yield result acquisition ei eq q density correspond minimum far posterior variance probabilistic ei minimum choose input location minimum represent minimum acquisition location yield entropy expect minimum simulation ei sample natural curve develop curve specifically develop decay fix integrate allow region mix basis function use model factor procedure row draw independent gp prior learn curve partially train curve gp colored training color surrogate require computation I model training gp would expensive computational incorporate independence curve draw global generalization curve specify column vector generative setting gp another constant mean mat ern gp prior illustration restrictive machine block omit derivation require gp marginalization property gaussian give eq size use lemma efficient distribution likelihood gaussian new curve omit space absence gp pre time total practice keep increment monte simulation observation use condition observation carlo equation select run gp develop create system decide user fully train one low asymptotic discover reflect hyperparameter setting curve maintain represent model entire ei use standard bayesian ei task become choose ei favor one iteration ei similar acquisition pick location minimum method search outcome unseen input consider subsequent information common hyperparameter report low epoch empirically validate art method logistic task allow report epoch run epoch training procedure run five method stationarity specific provide optimize train descent popular hyperparameter include weight minibatch dropout optimize hyperparameter latent lda wikipedia topic implementation optimize rating penalty optimization procedure distinct hyperparameter curve epoch color bayesian optimization promise promise start show specific epoch clearly experiment significantly art due dynamically prominent online predict observe negligible small additional incur explicitly gain rapidly reach visualization optimization pmf empirical analysis
greedy greedy require priori hierarchical prior sparsity priori analogue et factorize user author estimate column induce low induce generalize low reconstruction induce low e leave singular left definite determinant precision diagonal wishart generalization gamma naturally log q relevance iteratively update maximize r give estimate resemble weight one problem unbalanced balance eq balancing remove limit computation parameter numerical p leave vector equal right kronecker precision computation thus far reduce corresponding want complement decompose variational bayesian become I likelihood iteratively since convex iterative converge objective accelerate computation uncorrelated reduce block iterate block time point operation iteration reduce numerical measurement matrix vb block know toolbox nuclear completion matrix variational bayes vb nuclear norm varied db vb varied method figure accelerate less dimension sense take combination draw normalize vector accelerate low nuclear nuclear small nuclear see figure sparse kernel machine singular vector prior construct call vector machine iterative design priori complexity accelerate outperform outperform relaxed develop bayesian reconstruction relevance singular machine accelerate computation reconstruction problem numerically relevance low reconstruction sample attract considerable generalization compressed sense greedy method hand develop heuristic provide refer development non
attribute field color map represent proportion market relative distribution job statistical perspective job besides advantage discrete plausible form encoding form quantization embed highly summarize job naturally lead public policy important job attain job eigenvalue find coordinate figure sub along directional gender plot strong correlated flow sum plot explain due job augment represent economic latter use intervention depict panel suggest tend region seem specialized gender observe diffusion gender slightly worker corner characterize proportion corner embed specific corner economic force towards end show situation outside flow corner blue right panel overall seem theoretical view direct vector field plausible general take make manifold moreover result node spectral graph knowledge extend generative possible set value component result coordinate recover recover albeit theoretical embed framework check immediate currently since manifold education force core ball integrate tangent integration change approximate reproduce explain origin locally flat normal along orthonormal tangent orthogonal laplacian coordinate finally denote polynomial remain arise derivative along use radius meanwhile differential come volume drop term take interpretation variable ball around form expansion important global field substitute integrate vanish meanwhile manifold change term ultimately come pick come amount divergence field implicitly xt asymmetric em height em consider graph directional direct diffusion endow kind direct graph estimate laplacian type construct highlight strength advance visualization foreground laplacian element infinity put study embed foundation lead elegant lead development manifold map laplacian undirected graph social alignment international citation naturally asymmetric type affinity work propose contain popular cut principled way similarity asymmetric result cluster successful adopt purely concerned actor affinity assume statistical step work explicitly manifold field account relation direct sample link determine overall connectivity laplace limit recover manifold local intrinsic pay attention tell extension also help previous method depth denote node generative compact make undirected graph node asymmetric similarity kernel define vector assign top leave corner aa ss embedding preserve generative process recover distribution practical process increase laplacian diffusion map answer infer aim undirecte direct laplace eigenvector eigenvector eigenfunction thing laplacian principle directional geometry know scaling increase embed converge original manifold preserving embed detail start observe asymmetric affinity part unique em symmetric diffusion essential add correspondence skew part good family interpret something retain radial smooth field orientation worth note domain locally appropriate field pointing though seem rich describe form omit define density transition represent reader recognize operator give normalize denote correspondingly eigenfunction limit definition study limit operator like one limit convenience operator manifold field combination operator eigenvector counterpart next core follow exploit obtain algorithm method graph transition close operator asymmetric integral asymptotic expansion expansion origin help come vector limit remain curvature tangent interaction follow mention assumption obtain use form interesting operator briefly present mean x derivation apply differential calculus drop specifically general operator complex eigenvector nevertheless play role extract meanwhile instead generalized q discuss add theory become author procedure part cut well cut criterion eigenvector symmetric play precisely represent affinity diagonal dense small natural equal normalize graph laplacian derive limit immediate symmetric kernel correspond pde describe right act source worth point absence pde field diffusion cause source flow recognize flow motivation theoretically separate manifold whether embed direct thing generative manifold locally laplace diffusion describe step discretized version eigenvector constant embed coordinate coordinate simplify every task reconstruct gradient plane exploit simple serve coordinate unit use component figure component run replicate step operation affinity n ss q ss order right eigenvalue embed n aa tr
cc robust cubic spline knot cell cubic knot cell variation discard gradually decrease shape objective converge variation middle datum upon amazon actual partition curve figure show solution rough similar fitting regression indeed spline linear peak transition curve present narrow cluster differ peak middle left contain curve less narrow peak large middle look flat cluster spline regression help addition unsupervise hand profile present attribute scheme pre map som apply piecewise notice two fold indeed som som raw step infer observe behavior narrow narrow peak contain increase follow slow decrease cc spline knot cluster knot algorithm converge variation show gradually variation objective partition middle mixture optimize penalize likelihood world application curve degree always cell degree good solution retain datum clearly spline cubic cubic smooth kind continuous approximation take second fitting number model I mixture fully regression conditional mix proportion expert proportion softmax direction fit expert hierarchical expert number expert sequence knot respectively th spline eq basis ij l b mx ij spline consider find w mix proportion lagrange multipli function multiply summing finally update proportion figure toy toy figure universit france universit france widely however crucial em perform also mixture criterion pre fully carry em spline regression mixture approach unsupervised proceed criterion accurate initialization apply curve propose robust result real confirm propose practical one successful estimation discriminant focus model density density component estimate e estimate vector model mixture density achieve datum functional concern paradigm analysis curve dimension observation curve series mixture spline modeling arise assume generate spline regression widely study analysis algorithm initialization number cluster case criterion choose estimated candidate provide gaussian multivariate focus cluster unsupervised consist likelihood expectation maximization polynomial spline spline regression mixture propose proceed rather standard brief maximize penalize criterion model formula propose approach conclude cluster generative mixture formulation assume multidimensional model set mixture gaussian matrix estimation likelihood base curve compose density way regression mixture approach overview model em mixture assume draw suppose realization polynomial corrupt mean noise polynomial degree tp ik represent correspond denote conditional density polynomial regression parameter vector iteratively via em expectation likelihood log complete z nz value otherwise em regression start initial two current update maximize mix maximize lagrange multiplier consist square solution know solution cluster estimate component fuzzy represent fuzzy observe assign cluster high summarize standard mixture htbp randomly partition mean initialize equation output ik extension previously polynomial mixture spline spline constrain mp derivative spline continuous piecewise write spline spline matrix version nearly singular spline thank matrix spline spline finite support everywhere else b spline curve regression polynomial use polynomial em notice standard em regression sensitive might estimation mixture initialize drawback em start em cluster know criterion issue mixture mixture separately among regard automatically estimate em algorithm algorithm number attempt indeed call minimum message mml penalize negative observe penalization control start cluster penalization proportion discard simultaneously estimate initialization still become serious dataset number concern datum observation assume reduce adapt individual structure rely analysis lead problem analysis model limitation functional mixture attempt limitation mixture curve cluster propose regard proceed polynomial spline spline derive robust em fit present regression spirit extend functional spline fitting mixture furthermore adaptive similarly mixture adapt proceed integer maximization r consist maximize analytic update proportion estimate cluster represent hard partition compute maximize denote estimate initialize regression model parameter middle stop code summarize algorithm curve regression mixture mixture curve converge nk use equation compute em equation discard cluster proportion q ik may simple data linearity even spline adapt choose spline knot spline use cubic spline spline twice kind piecewise spline adapt order piecewise constant concern knot knot space range technique location cross knot place knot determine either sufficient knot easily type selection knot much fix knot paper sensitive location knot knot sufficient dedicated propose approach simulate world implement develop code available request em spline spline perform estimate accuracy misclassification simulate real world course simulate arbitrary curve curve simulate nonlinear curve cubic spline spline regression estimate interval actual correctly classify cubic mixture mixture knot quasi identical accurately estimate middle table normalize mean one classify actual cluster function one actual em simulate number iteration cluster rapidly majority discard iteration see penalize likelihood value objective algorithm middle iteration iteration number precisely iteration obtain regression also accurate result rapidly provide elsewhere consist curve h h mean unit temporal interval generative cluster spline correctly spline slightly well polynomial em robust cubic spline three misclassification absolute cluster retrieve misclassification slightly regression spline em objective model highlight evaluate datum original one describe namely construct five five dark iy dark water retain frequency consider discrimination
max game adversarial net conditional model generator extra label data modality perform conditioning noise combine adversarial compose mlp player minimax illustrate conditional condition encode vector net dimensionality within hypercube relu layer hide relu generate mnist maxout piece piece layer maxout layer piece fed sigmoid architecture critical maxout unit typically batch exponentially momentum dropout generator likelihood validation show log mnist draw sample detail adversarial approach include adversarial net efficacy exploration hyper architecture match exceed non generate condition one label generate mnist adversarial adversarial site vocabulary adversarial model work experiment tag tag image tag repeat tag generate top cosine similarity vector vocabulary annotation tag generator receive relu layer vector relu hide map vector relu hidden image layer piece join finally sigmoid unit mini batch initial decrease also momentum generator hyper mix manual albeit limited annotation people tree house water net interesting thorough analysis tag multiple generative hope achieve obvious construct scheme suit specific acknowledgment would helpful author acknowledge vision production frank yahoo train model adversarial net simply condition digit condition label illustrate modal preliminary example adversarial intractable probabilistic adversarial net wide incorporated produce realistic generative control conditioning conditioning could construct adversarial net demonstrate experiment digit one modal despite network challenge accommodate predict category issue date focus mapping many instance label tag human use different describe help address additional natural language corpora label geometric make g predict fact make predictive generalization
frequently model apply mathematic mining twitter tweet analysis pattern collection tweet extract twitter start combine tweet relate cluster topic tweet algorithm give nmf prove explore result tool mine world group relative text cluster analysis collection text extract twitter contain game start begin tweet tweet english keep tweet contain user political news twitter security able search could country certain range size research algorithms factorization mean twitter center choose space assign centroid assign close minimize close centroid continue centroid metric cosine magnitude distance contain give distance however magnitude sentence contain world contain cosine purpose research cosine deal matrix distance tweet thus tweet might consider short tweet word cosine tf document result range non value range initialization initialization random mean different disadvantage order algorithm world sometimes difficult many algorithm dataset create together consensus running mean times parameter case b nmf topic non non topic text pointing direction vector multiplicative alternate constrain aim multiplicative rule converge local greatly one multiplicative costly depend initialized sparsity element long mine remove start path least speed initialized matrix initialization flexible create multiplicative disadvantage lack replace negative replace square advantage alternate add matrix sparse fast since nmf base spatial form mark noise removal h input dataset dense radius point dense radius remove parameter drastically vary variation tweet tweet twitter call much thought tweet decide identical tweet preliminary exploration tweet tweet tweet remove eliminate original tweet much vocabulary stop look collection tweet want tweet tweet closely figure keep removal create run varied could tweet cluster remove drop consensus tweet consensus matrix drop row sum another tolerance entry consensus average remove noise decide tweet noise distance use cosine tweet cosine distance dense sure dependent create tweet run tweet point also create tweet decide keep cluster apart decide distance keep tweet create classification decide tweet mark point consider noise unique remove cluster frequently strength combine look represent least mark tweet remove keep tweet decide diagonal entry row consensus identify gap eigenvalue topic topic broad choose gap eigenvalue figure cluster major text mean widely algorithm highly consensus give tweet tweet text file tweet word cloud visualize overall throughout use mean tweet know tweet help cluster tweet cluster visualization tool discover word
infeasible instead take approach imagine represent individual entity datum record categorical advantage variational posterior model data vocabulary challenge particularly inherent clustering model process dirichlet non proportion change clearly entity resolution cluster record constant behavior record record record complete every miss record database make database field lda noisy record record database regard record latent capture let th kf non trivial assume th categorical putting assume individual record latent individual length dirichlet assume vector draw dirichlet encourage split merge record take special record hybrid assignment record split merge health care database record hour database database million hybrid take wish million multiple census contain million record database country million record model show entity approximate generative show full appendix generative amenable lda word model vary latent contrast part allow proportion topic key regard mixture model entity cluster size would order simple apply via strictly datum may finitely finite infinitely dirichlet case cluster without grow theory cluster cluster natural assume record individual rather traditional must ask regularity inference draw cluster infinitely dimensional grow generative proportional solve derive follow ascent variational q new much fast since make simplify assumption incorporation realistic assumption record moreover remain individual grow construct wish solution address entity broadly domain grow bound proportion tb berkeley fellowship nsf grant dms grant gm independent conditionally independent give independent within notation database record field categorical field th record index define dr model record database latent separate whether full value aggregate likewise record divergence maximize concentrate field parameter parameter approximate recall ascent sometimes variational first eq zero proper together database minimal represent latent bayesian allow generative share across principled quantification query final resolution mcmc modern database contain
pc motion human acquire marker system fig pose position gp metric see latent periodic pattern motion riemannian geodesic show geodesic straight pose comparison pose see geodesic straight reconstruction stay reconstruction straight drastically length geodesic match truth h pc train dot green denote dash straight line pc pc euclidean original different difficult operation metric latent distribution smoothly point local metric short straight interpolation new gp expect uncertainty long force avoid latent desire behaviour worth riemannian reasonably track riemannian kalman classification potentially metric entire although low less understand g almost surely metric manifold lead curvature worth investigate influence acknowledgement author great european project project ref foundation education rgb rgb rgb rgb section figure figure box intuition appendix chapter computer university university uk structure dimensionality riemannian geometry tensor treat variable riemannian expectation distance expect lead represent dimensional potentially represent nonlinear mapping metric metric uncertainty space metric capture thereby provide low useful illustrative display representation underlie rotation want analyse datum insufficient space go datum raise choice euclidean still meaningful question approach generative reflect intrinsic recover mapping metric observation space riemannian computing distance short path riemannian metric natural trend concept riemannian paper overview state art reduction introduce extend probabilistic metric finally path experimental discussion latent embedding surface manifold learn interpret manifold underlie basic geometry surface surface surface dimensional surface machine terminology correspond chart straight intuitively chart illustration q jacobian surface chart integrate riemannian riemannian symmetric smoothly product point riemannian metric smoothly change suffice curve geodesic q show imply unique starting reduction provide probabilistic dimensionality define unobserved joint variable dominate mapping associate feature account model typically choose advantage therefore dimension principal component analysis probability datum write independence nonlinear basis tensor compute short path differentiable embed section explicitly compute riemannian metric jacobian eq define conditional jacobian follow naturally row central wishart number degree freedom centrality equal central wishart distribution joint vector function possible lead formulation gp latent notation differential derivative gaussian long derivative jacobian mapping compute every partial latent jointly gp model observe define support observe embed differentiable jacobian mapping metric jacobian independent jacobian form tensor compute metric tensor imply uncertainty curve uncertain metric define gp exploration way dimension generative map explicit example latent colour proportional space endow riemannian compute short path short problem space result grow exponentially latent quickly infeasible geodesic differential equation geodesic independently dimension nd ode st ode matlab gives solve repeat derivative illustrative provide specific square eq
g equality covariate relation accord j p set sum identifiable since pair identifiability complete q transpose q furthermore take mm mm mm parsimonious family cluster account attain eigen impose component sufficient condition identifiability compare show accounting dependency regression offer use last decade arguably methodology parsimonious cluster package typically variable insight gain accounting dependency cluster incorporate perform response whereas weight logistic often utilize represent investigate statistical paper discuss mode e distinct linear various parsimonious univariate eigen covariance response response currently correlate recently model decompose covariance investigate deal response eigen parsimonious recently response parsimonious eigen impose parsimonious schwarz family hereafter organize basic summarize section recall identifiability likelihood issue assessment base conclusion idea response framework covariate disjoint multivariate density normally matrix normally rewrite density variate variate single parameter necessary decomposition eigen yield entry sort constraint eigenvector accord geometrically determine orientation group split model align orientation belong volume free spherical spherical align align axis align align g g covariance decomposition lead refer sequel usual asymptotic theory estimation cf multivariate mixture generally class finite function q identifiable sufficient general follow identifiable variate parameter via unconstrained nn incomplete note incomplete ig complete ig maximization expectation update closed form detail q equation update family choose fit among family model criterion bic asymptotic development well practice extensively incomplete commonly additionally mean component equal occur note convergence maxima unbounded surface comparison package make work package user initialization facilitate initialize section analyse dataset hereafter covariate correspond component group respectively lastly matrix htb family result bic choose fit result bic choose estimate model contain mixture analyze concentration body fold percentage composition variable response summarize htb ari vi vi comprise vi ei yield ari estimate second bic quite freedom pick result ari well grouping package provide width variable ari difference bic select freedom ari together bic choose ari model ari result ari two family ari assign essence choose ari implement pick component ari pool htb contain blue highly width measurement take variable colour know algorithm run select ari bic pick ari htb ari parameter model ari ari table estimate model lead good membership agreement estimate ari ari basically put component pick ari model utilize also note surprising multinomial logit multinomial condition covariate density assume give result ari respectively bf om account multivariate correlate response covariate allow eigen decompose response covariate component structure identification parsimonious unconstraine
carefully transform various naive would dimensionality complement nuclear convergence epoch stop epoch line prove convergence worse careful lead outline objective log loose weak fully apply former latter dimensional batch well loss within I n loss locally since gradient bound derive somewhat weak lipschitz provide guarantee additional denote log determinant let j limit need guarantee bind contribute directly property however condition utilize result note epoch improve factor compare lipschitz replace satisfie intuitively condition initialize consider network observe zero mean multivariate gaussian distribution vector column diagonal allow limit cm sep fill circle mm h observe name name name h joint rank among triplet parent child term presence additional edge convert introduce parent direct graph remove undirected case loss fm I bind singular bounded method previous sufficient walk efficient guarantee online slightly reason scenario ball good initialization ensure subsequent involve section approximate applicable somewhat weak local limit bind contribute radius log reason q hold local depend factor reason intel gb ram see design nevertheless accuracy within addition epoch high fluctuation therefore projection good method st e e st admm reason st ccc run inexact alm design epoch work epoch specific epoch delay cause compare reason art inexact alm direct eliminate inexact alm reason reach useful error reveal projection far either reach projection expensive svd projection alm inexact alm thus omit high multi multiple regularizer optimization problem reach factor component propose modify multi admm algorithm sparse optimization matrix decomposition outperform particular accuracy consider decomposition provide address nonconvex additional addition descent acknowledge detailed discussion thank valuable recovery author discussion point regard microsoft fellowship nsf award award number minimax decomposition component provide guarantee p match minimax low respect latent admm annealing consist projection ball sparse reach high accuracy rank regime stochastic extensively uncertain involve scalable scale contrast traditional technique far operation work alternate direction multiplier admm scale employ many g locally via augment apply update admm solve globally regularize since natural encourage optimal g sparsity employ admm problem regularizer regularizer illustrative set result simple modification inexact anneal huge implication rate dimension instance scale projection certain ball anneal introduce dual constrain obtain begin epoch average pass estimate projection also decomposition admm update sparse project nuclear norm admm dimensional problem problem scale minimax linear guarantee minimax rank guarantee matrix literature comparison framework convergence dimension noise well compare admm decomposition inexact alm method recover admm improve expectation per rate table contrast admm admm rate require function regularizer contraction whereas sparse study anneal achieve one derive capable variable incorporate online dimensional admm matrix poor low impose batch fairly convergence low model note weak condition matrix suffer e noiseless set rate rate however worse fix however epoch follow establish derive modify epoch size batch combine epoch vary epoch additional ensure estimate trivial different careful enable first online decomposition set match guarantee many interesting model change stop c method st st admm condition admm sc e batch df reason r p q generalize result setting optimum solution regularization nuclear later optimize inexact admm employ base set estimate epoch constrain close go expect provide admm extend involve block admm access epoch length prox radius rate impose high efficiently implement discuss update carry set consider matrix model access update estimate impose desire propose detail algorithm reason recall assume update linearization use inexact proximal follow project epoch constrain matrix impose rank impose nuclear initialization nuclear encourage rank entry note impose assume discussion projection perform efficiently appendix auxiliary eq efficiently projection step approximate efficiency projection stand compute initialization epoch epoch prox shrinkage initialize I provide reason efficiently dimension efficiently need follow addition fm fm intuitively constraint control low separate type sparse low f require jointly nuclear dual update step total guarantee least appendix improve vary epoch max scenario lower bind provide bind convergence rate match scaling factor attain discuss otherwise intuitively bound term need two batch average sample efficient concentration observation conjecture set provide incoherence constraint incur even noiseless identifiability error decay online list ki k set well mean gaussian entry another independence diagonal obtain q batch entry e matrix assumption connection variable efficient set match conditionally variable long since compose inverse precision express
stationarity ridge convenient could stationarity worth linearity nonlinearity linearity linear work nonlinear work linear nonlinear datum nonlinear indirect causality causality extent without extent compare side transfer extent side spurious causality good causality yes transfer yes causality test augment lag somewhat sensitive promise somewhat sensitive somewhat sensitive lag lag histogram kernel parameter lag crucially financial measure causality dependency aware first research decomposition decomposition asymmetric causality causality third suggest building intervention causality describe prediction becoming successfully build u obtain gram intensive perform cross calculation kernel validation validation point whole select appropriate use calculate testing datum recall still significance window experiment measure perform believe strength measure reason parameter optimal nonlinear optimal employ cross choose parameter learn embed split subset th belong create range kernel scale dual dual weight calculate particular average correspond calculate prediction optimal mention undesirable college uk discussion valuable feedback comment max device social centre es study collection university college bt centre economics political science series concentrate causality causality causality schmidt transfer examine attention ability nonlinear causality theoretical benefit dependence set generate nonlinear dependence bivariate highlight month sp rate circumstance research series causality financial beneficial causality distinction intervention causality answer patient survival answer operate intervention involve tool analysis causality direction knowledge financial model intervention use causality past expand must lie joint observational need distinguish capable discover cause causality describe causality method independence useful causality characteristic know well case financial stationarity exhibit nonlinearity become broad provide review practical aspect methodology synthetic financial application contain supplementary material causality appear wiener predict use past one concept introduce winner economic autoregressive cause occur unique information contain variable include causality mean concept deterministic say signal time signal simultaneous instantaneous coupling expand publish feedback instantaneous causality instantaneous coupling causality instantaneous side quantify paper measure therein crucial place strength causality one causality measure definition series subscript understand random variable accordingly causality cause natural stand cause couple bivariate case way instantaneous coupling alternative optimal instantaneous coupling definition assess formulation square many introduce process generalise multivariate model way linear allow everywhere restriction quantify usefulness causality instantaneous eq q x later linear quantifying machine causal machine perspective initially independence causality last become method become search meaningful requirement individually pairwise kernel function interpret also suggest causality ridge surprising introduce clear property way alternative causality square schmidt space permutation quantify kernel concept hilbert schmidt normalise independence explore method causality please hilbert create definite kernel positive definite name semi henceforth kernel define provide feature identity fact product space replace dot section trick causality nonlinear show reproduce causality standard univariate good linear infer causality alternative particular four model past new lag represent reasonable assume lag typically causality early involve look value depend simple univariate well drawback poor sample size address least square solution write primal q notation w tw px x kernel form inner explain combination representation result dual trick allow kernel element gram denote linear allow operator prediction way square whole denote fit analogously define index causality instantaneous coupling use way causality covariance use analyse pointed assess independence use measurable finite correlation appropriate importantly equal even maximum attain nevertheless machine completeness schmidt product element covariance analogous cross operator covariance symbol tensor product definition operator two covariance follow denote reproduce hilbert rkhs k topological field expectation ensure random expectation covariance normalise conditional operator normalise operator rkh dense supremum norm independence independence appendix normalise conditional independence denote hilbert schmidt operator use normalise marginal normalise operator information schmidt normalise square schmidt normalise conditional q hilbert schmidt operator straightforward good behaviour define use cross normalise cross covariance inverting next construct construct estimator schmidt necessary inverting introduce alternative information theoretic measure provide comparison method measure sense transfer entropy transfer popular among transfer develop entropy bivariate nonlinear causality improvement max review causality present perspective design state bivariate omit side causality transfer prove variable causality decompose shannon well mutual information mutual assume random probability uv shannon independent mutual information mutual lack mutual direction natural extension directional entropy previously stand transfer entropy generalise define hx calculation already joint impractical serve comparison couple deviation generate dependence consequently need assess measure assess achieve permutation test permutation test significance causality time since causality rely create keep permutation causality surrogate small causality quantify depend permutation hypothesis causality level significance set overlap window latter useful stationary case believe kind subsample beneficial concern world eight relatively lag instantaneous try lag eight subsequently shift relation lag lag lag lag network show lag causality occur data correlation testing dependence cause variable ts ts ts ts ts ts ts ts c ts ts ts purpose source matlab transfer matlab toolbox causality open access matlab measure negligible comparable range code cross grid histogram performing incorporate write accommodate implementation permutation test permutation median inter lag causality occur causality lag four lag allow analyse effect range lag different lag present conclusion measure similarly dependency range causal direction short range well lag two direction detect causal coupling permutation acceptance fail spurious analyse lag inherently reason inefficient range lag need couple mutual lag entropy report zero value relevant causal direction fail direction lag instantaneous coupling correctly impractical range lag range handle higher correctly causality correctly report lag instantaneous value lag bottom measure transfer te lag retrieve c c ts ts ts ts ts ts ts ts ts ts ts ts ts ts te ts ts ts ts ts ts ts ts ts ts ts ts c ts ts ts ts ts ts ts ts conclude gaussian lag misspecification seem transfer direct direct causality introduce refer causality test four degree acceptance rate direction particularly linear detect exception causality detect spurious case effect different lag three possibly turn test eight describe size play crucial causality however kernel work type example present causality demonstrate distinguish indirect cause gaussian zero know effect indirect calculate assess causality repeat experiment time lag causality measure obtain linear expect causality side take consideration indirect causality pick kernel dependence h equivalent causality z blue face calculate gaussian face calculate hilbert schmidt normalise independence transfer allow achieve face year economic around scientific causality formulation field methodology successfully field characteristic distribution generally finance economic tool devote mostly information reduce dimensionality g subset factor causality structure help relevant forecasting causal parent future financial characteristic biology physics finance long biology though dependencie many researcher stationarity usually kind clearly direction rate considerable concern real interest country causality analyse namely index economic country reflect contrast set interest reflect ask whether economic indicator statistical sense run similar gaussian kernel median investigate windows significant interpretable observe long window clear direction short window considerably often dependence window lag month lag report value lag show scale causality series month assess causality direction accept reject level causality translate roughly explanatory somewhat pattern separate interest fall consecutive explanatory interpretation causality h hypothesis red lag chart scatter set causality price index one month model lag scatter plot causality set month lag scatter hypothesis causality month red lag scatter causality lag long clear lag rate time direction causality linear result direction strong separation lag lag reason perform well interpretation causality conclusion aspect detect causality transfer causality u month transfer entropy significant direction lag often stress might please refer direction one blue line way red lag scatter causality set u month lag scatter causality measure carry trade six exchange index sp daily period sp exchange volatility information logarithmic return measure window length day window method case employ result separation direction especially sp causal consistently sp indicate period period sp effect cause sp red obtain regime purely lack causality series test specific lag explanatory lag permutation causality autocorrelation introduce bias also high datum correspondingly instantaneous causality transfer appear bias similar side causal sp trade causal effect include pattern exchange rate lose explanatory sp distinction direction h hypothesis exchange cause sp volatility blue relation ask method quantify causality context well develop firstly often field science economic lack context secondly question management question cause loss tool quantify causality currently develop quantify causal inference understand result understand enable reader set part describe comment testing direction conclude nonlinearity causality assess causality efficient causality good dependence financial normally exhibit arguably analyse requirement causality causality bring causality distinguish indirect consequence reduction repeat indirect causality introduce notion indirect whole variable concept indirect causality measure distinguish indirect follow compare conditional causality indirect causality explicitly build condition cause intend notice sensitive call partial transfer indirect cause spurious causality cover cause spurious causality wider indicate causality infer relation introduce add spurious exhibit dependency none dependency instantaneous coupling nonlinear causality reader causality domain numerical mention high sensitive causality bivariate good significance expensive regression layer calculate expensive unless
svm find margin label justify consider voting behavior statistic proportion york randomly instance individual proportion sample bag form instance population chernoff proportion population proportion enough bag sample bag generalization b conditionally bag title group training instance spaced scale large far formal learn recover real datum predict census label cover subset census instance binary education status week bag people feasibility form bag divide testing bag label propose solver bag proportion validation report first simulate case draw instance assume instance bag instance sampling bag relatively test feasibility training bag bag likely solution simulate attribute group assign simply base perform random replacement time select group figure l country education bag error world application bag predefine rather bag new instance grouping attribute large predict predict elementary education negative proportion elementary education assigning bag education individual education grouping performance baseline bag form bag bag bag redundant novel analysis individual label proportion affect bag proportion bag mild bag proportion b c proceed conclude denote use standard concentration inequality assumption easy verify latter negligible obviously complete contradiction put assumption equivalent prediction come label classifier misclassifie ne p several define involve differential access small database substantially satisfy differential database q coin function adjust differ eq laplace database database draw differential privacy differential privacy preserve extra conducted application mechanism privacy differential notion different proportion private mechanism serve paradigm construct overall private reduce step partial publish guarantee often proportion decision tree leaf label item access explicitly time structure different attack privacy often item feed label proportion later publish differentially way laplacian standard construct differentially structure know private algorithm point give privacy guarantee make count projection loss generality assume object instance label differentially private mechanism count sensitivity count consideration disjoint parameter one output differentially private mechanism n nx thus proportion close version explain differential scalable tool label proportion instance use differentially publish training bag task predict instance political answer fundamental match proportion analysis vc bag bag sensitive bag show mild together formal guarantee label set application proportion paradigm privacy guarantee demonstrate feasibility base real world predict census information individual proportion proportion proportion disease public predict recent setting call bag work available individual show combine capable correctly predict acquire census lead promise privacy sensitive attribute label proportion namely optimize match proportion formal mild assumption recover learn bound generalization proportion bag sensitive bag word bag possible proportion unseen bag conclusion get good proportion bag predict disease certain predict rate department mild control bag imply point good proportion concern increase aim private preserve world demonstrate census seminal work estimate label proportion conditional exponential restrictive instance conditionally label learn generate consistent label proportion show know proportion provides learn related bag boolean bag positive draw single easy sided label world arbitrary sample tool generalization good predictor bag consist attribute bag bag inside bag bag bag bag proportion proportion learner bag proportion instance unobserved th bag denote instance low prediction erm instance however label therefore try proportion define h select bag set compute framework immediate label bag proportion show sample bag proportion sensitive bag mild bag give bag proportion generalization bag possible unseen bag basically bag hypothesis smooth loss proportion term proportion measure utilize relate proportion main intuition bag instance hypothesis formally adapt class number generalization bag bag bag give appendix bag sample size grow bag bag create fortunately grow sensitive result bag proportion bag proportion predict bag proportion discuss provide insight proportion already proportion section bag bag bound inequality either instance error control bag proportion learn justify bag least summarize assume bag least correctly bag nr fraction bag learner bag extreme bag label proportion instance learn provide fail bag study insight way bag conditionally bag lot sample individual individual bag bag bag consider draw bag firstly pick note utilize bag generate proof straightforward bind close hx probability bag proportion small match monotonically generative strong result mild express consider generative assume instance assumption restrictive proportion prior approximately match adjust bias prior cdf want curve monotonically control
approach process implicitly match gradient observation requirement look remain since uncertainty know integral generally form expansion avoid explicit case process see eq xt linear gp covariance observation time boundary gp whose covariance gaussian formulae yy yy globally give cubic evaluate much small ode solver smooth approach approximation gp derivative gp interpret assume subsequent outline practical approach subsequently retain measurement form gp fx value subsequently compute continue gp right indistinguishable line standard gp retrieve integrate densely complexity gaussian much solver use solve ode solution experiment use squared covariance procedure substantially bad accuracy reason function namely similarly predict experience extend drawback technique alternative many direction solver outline three solver plot derivative stepsize plot indistinguishable small follow obtain eq simplicity deterministic ode forward future requirement ode derivative derivative requirement ode subsequently f nx repeat formally procedure differ significantly firstly ode requirement also condition complexity cubic say limit complexity approach bring complexity demonstrate despite excellent comparative computationally x n draw iteration xt perform compare solve number iteration gp calculate curve requirement ode derivative variable condition final variance term independent equivalent problem suitably constant optimisation derivative rapidly converge principle carry optimisation solution solution therefore move pass experience perform implicit plot approach window stepsize indistinguishable solution benefit conditioning derivative jacobian order derivative gp conditioning set option order novel solver ode sample correct derivative estimate gaussian ode operator direct solution xt joint define euler method approach recursion x ft k accumulate generally method weighted combination carefully define future value require algebraic red x red red x x derivative infer tx ft add observe derivative collection obtain integrate draw may curve extension version note drawback unclear make practically suggest collection ode solver implicit date carry limited believe promise exist gp approach explicit gp method analogous ode solver experience forward evaluation sophisticated implicit gp numerical van outperform explicit ode solver implicit explicit comparable implicit though require
draw independent py exponential ratio use kullback divergence drawing substitution interpret monotonic decreasing tell rgb evaluate
large arise privacy concern researcher decentralize social since design implement focused movement gradually fact date accumulate million user establish discovery constraint observe whole whole motivation service towards literature decentralize privacy preserve discovery matching problem attribute interest social activity way profile common primitive speaking hold two protocol compute either party raw scheme hardware generic construction common protocol protocol adversary security efficiency drawback social connection topology heuristic precision discover result span towards extend discovery community topology privacy preserve set party possess social translate preserve protocol circuit construction impractical world tradeoff privacy efficiency contribution propose community detection largely improve recall topology discovery decentralize transform walk preliminary result community preserve privacy extension variation end widely discovery topology graph mine term community closely community mainly scenario contrary system one decentralize execution work exchange much exchange adjacency privacy formulate omit interested reader survey survey intra dense linkage review classical centralized decentralize formulate privacy make formulate problem graph partition vertex detection try maximize minimize depend remove get overlap artificial surrogate community necessarily tractable via modularity classical truth decentralize scenario observer view whole partitioning tractable ask observer formulation observer stack community restrict overlap encodes outcome application privacy adversary passive adversary node execute protocol single adversary capture connection otherwise begin knowledge community one guess connection procee even without preserve detection even scenario researcher use tool incorporate specific heuristic perform heavy tuning amenable community base graph model encoding connection community number matrix illustrate protocol main involve truncate live initially rw record accumulate reach intersection answer intersection pairwise privacy intersection scheme reveal reveal reveal intersection size adapting follow existence primitive extra decentralize preserve truncate community enough enough intersection come proper ensure protocol adversary exclude community priori protocol limited assume adversary use average challenge omit summarize follow protocol adversary successful base example strategy information end advantage successful rate strategy problem community community edge protocol rw negative false advantage preserve privacy proper heuristic optimize repeat protocol design formulate privacy paper multi protocol truncate walk thorough protocol meet objective protocol suppose protocol exchange cell adversary know infer intra community generation different guess link measurement network size set represent node rw infer think hash define w community identity prevent adversary guess protocol version know nothing indicator weak widely variation adversary know intersection set sequence community adversary potentially exploit protocol problem preserve define security privacy preserving scheme topology new connection find community update protocol cause normal since walk preserve requirement decision minimum party propose community
cancer reach paper resp easy discrimination vs cancer dataset performance vs good evaluate discrimination systematically roughly evaluate correlation plot cancer positive normalize model also note numerically roughly report mrf discrimination cancer benchmark discrimination sketch namely discrimination involve spectra yield list site code mass spectra two respective frequency event within focus site site highest low select fit distribution average percentage correct discrimination decision set accelerate vs reach signature identify ratio table retain involve clique list statistic table margin interval asymptotic normality evaluation margin sm quality datum evaluate yield respective quantile signature discover discrimination either score score coordinate list clique involve correspond discrimination score signature clique mass spectrum activate score belong cancer otherwise promise algorithmic acquire select strongly link specific group use automate far incorporate protocol interpretable signature cancer stage tree artificial mass spectra acquire cancer patient efficient discriminate generate black box biological rigorously fit parameterized spectra dataset acquire efficient signature interpretable signature group variation observable homogeneous acquire systematically field classical spectra several hundred thousand strong peak potential spectrum binary status distribution study dependency realizations automatic spectra spectra reduce two point systematic discover explicit signature enable code peak fit achieve quality signature discovery experimental spectra acquire three stage cancer publish acquire cancer patient final computed leave one performance good performance report concrete advantage signature interpretable key ratio thank sciences university cancer thank institute spectra acquire mathematic publication none state literature mrfs configuration space result implicitly element proof proof space length parameterize vector give generate maximize fast concavity pseudo likelihood strictly function reach empirical pseudo concave surely notation resp hence quadratic vector positive since form take configuration condition binary strictly obviously reach prove configuration derivative let conditional specification random index identity index conclude normalize asymptotically positive determined equation see section outline technical concave sure vector tw formula expression sample hessian q conclude hessian large formula situation normalize variance normalize vector computable via hence hence gaussian zero covariance conclude particular asymptotically vector z conclude probability recall percentage must true hence systematically precede pf joint inequality norm use bound similarly imply nd imply resp compute correct resp achievable discrimination arbitrary small find computed formula large q explicitly elementary eq result prove eigenvalue become variable last impose eq percentile proved immediately combine immediately apply prove equation spectra acquire cancer patient technique automate discrimination stage implement new signature lead interpretable signature model homogeneous spectra parameterize markov random present detailed theoretical successful discrimination acquire cancer well cancer patient broadly technology study protein present biological identification cancer stage mass enhance mass mass intensity quantify ratio acquire acquisition modality range specialized software machine artificial forest box discrimination level group variation develop software tool spectra automate discovery signature power easily interpretable automatic combination quantify impact presence paper interpretable signature spectra co markov mrfs recall mrfs dependency interact discrimination achieved study successfully mrf discovery acquire patient spectra acquire process clinical http home cancer group spectra spectra spectra reference range z ratio newly publish cancer patient provide science mass acquire research institute usa include spectra early cancer group spectra cancer spectra call accuracy ratio signature spectra mrf implementation performance mrf benchmark discrimination describe acquire cancer cancer vs cancer processing mass spectra remove acquisition affect intensity peak could relative acquisition hardware raw processing normalization extraction baseline removal peak detection outline spectrum step peak detect peak could potentially stage peak list ratio spectra site binary index activate detect peak spectrum code vector site activate mass generate observation binary set systematically unknown identify site site call field mrf q describe system clique recall pair site clique gibbs clique naturally parameterize space denote coordinate see binary code spectra generate binary vector length cancer fit reduction achieve site specific site correlation potential clique order often still respect seek enforce moderate clique clique discovery set clique seek impose whenever parameter force precise achieve fitting classical benchmark example fit introduce play spatial intensive configuration fast site clique zero coordinate pseudo likelihood brevity restrict theoretical impose coordinate constrain specification q binary configuration coordinate eq principle seek vector estimate observe configuration denote hessian gradient pseudo hence non linear q existence follow likelihood concave strictly concave proof supplementary concavity standard stop inferior benchmark discrimination roughly around order site clique automatically select among ghz computing spectra small second group full focus gibbs asymptotic proof precise asymptotic supplementary material configuration asymptotically observation normalize error computable supplementary proof material provide tool decide coordinate approximate appendix n explicit whenever interval estimate descent implement iterate estimate benchmark accuracy coordinate display desirable acquisition phase study involve model quick show automatic discrimination main criterion goal signature enable discrimination principle quite strongly nonzero patient mass spectra simultaneously reference large cancer processing yield fitting spectra one want achieve statistically high set must select presence absence provide set spectra resp frequency ensure significant select one two number site benchmark study site easily justify site weakly small number equal focus site fix positive integer low site select binary systematically restriction generate binary datum binary site compute frequency four contingency table quantify stochastic dependency significance dependent retain event within typically quite study mass spectra acquire dataset size moderate estimate dependency achieve reasonably clique hence include clique successively inferior optimize far retain precisely clique high pair fix clique clique integer moderate integer set select site site determine clique parameterize set technique explain section error margin different force mass spectra formula partition sum heavy task approach justify law partially prefer implement function subset q robust plane separate estimation affine quickly compute discrimination discrimination repeat regression discrimination ideal unknown classical leave cross spectrum eliminate outline correctly classification evaluate range leave costly fortunately unnecessary discrimination maximization immediate access base reference involve constitute explicit indeed compute restriction site classify mass absence signature mass add present jointly sum yield actually equal classified sign performance develop margin frequency decision frequency provide rough obtain maximization provide rough good compute leave quite subset verify constraint discovery reduce hour maximize properly normalize leibl kl well know iff thus recall fx view formula expect typically statement distance account weakly normalize mrf separate dimension conjecture easy compute maximization fast discovery
aspect bound identify true describe conclude column also pm tr ix pi covariance group group overall correspond eigenvector normalization uniqueness population classification define mahalanobis project discrepancy optimal classification come distribution observation within group tn correspond show canonical express orthogonal transformation directly decomposition unique express interpretation group exist base population proposition rather penalty choose suitable function deviation respect transformation define discriminant correspond exactly common context although penalty feature eliminate canonical word vector individually element preserve imply overcome orthogonal row penalty refer overview choice fact preserve suggest sample make arbitrarily instead regularization quite encourage accord substitution preserve result convex bound eigenvalue analysis literature take otherwise equivalence method set belong view correspond choice value group problem equivalent proposal affect optimization coordinate descent interior reader overview sparsity induce choose coordinate advantage range fast convexity solution kkt row lead matrix subgradient far lead jj block ij optimization subproblem solve block algorithm proposal selection serve selection denote support aa bound v motivation rely normality normality possible supplement establish variable coincide population rule p evaluate alternative refer report concern sample structure take definite estimate dataset dataset bernoulli base scenario consider structure range error value structure setting perform well knowledge terminology programming discriminant within sample need simulation requirement affect misclassification misclassification replication report component population canonical truly perform bernoulli comparable feature rate misclassification difference small bernoulli reveal suggest significantly unclear optimal positive identity autoregressive consider group structure structure group though popular versus versus discriminant vector ambiguity consider discriminant group case find canonical nonconvex mean replication table row matrix zero truly scenario comparable feature tends select good tradeoff oracle identity autoregressive bernoulli positive autoregressive package value precision level treat canonical restrictive produce package adaptively final respective fold run detail follow method program solver much global biological genetic integrated genome environment direct analytical advance possible hundred parallel perturbation researcher platform compound study demonstrate identity investigate dr seek lead compound identify throughput response patient replicate follow patient misclassifie replication split training test report term misclassification misclassifie lead perfect significant group achieve selecting whereas less substantial replication sample far canonical use illustrate figure perfect group choose one tuning replication validation tried project though variation replication variation compare publicly analyze author patient gene free discriminant follow independent split contain sample use misclassifie number split report well select comparable misclassification group addition tractable perform feature selection canonical penalty vector possible propose canonical vector regression context propose size perform enhance interpretability addition result effectively low modification direction research extension manuscript aware consistency sparse proposal case however future appendix follow n x nx first group nx tx tc g unbalanced denote row h tx c n r r dd h r analogous population analogous define classified group last matrix eigenvector statement q similarly ab note follow definition r since group n furthermore expression far simplify remain constant rewrite eq cat aa proof directly eq constant rotation tw aa triangle constant follow c hoeffding proof corollary mapping imply gp text corner
autoencoder purpose use ht onto pool product response mapping linearity sigmoid mapping activation first reconstruct transformation encode likewise second obtain reconstruction turn shift span class component absolute quite relational look relational kind hide partial view subsequent dynamical sequence assume unit explicitly keep video motion seed able video first multidimensional dynamical subsequent view derivative high order second relational pyramid first relational analog partial derivative experiment layer support view frame video describe relational autoencoder modular order construct module relate mapping refer mapping mapping accord module transformation rotation sum response yield detector angle delta angular acceleration frame directly way reconstruction adjacent mapping see compute infer amount prediction transformation assume train minimize minimize reconstruction supervise guide mapping representation image help ht transformation violate transformation look ahead iterate ahead prediction compute compute amount relational step prediction make order feature describe follow relational describe prediction multiple ahead repeat inference prediction e one compute next low activation activation seed frame gradient ahead training dynamic video vary complexity synthetic accelerate transformation transformation whiten work descent momentum reconstruct subsequent mapping evaluate yield transformation reconstruction video transform berkeley video shift rotation set uniformly pixel divide sized bin contain filter unit choose search good performance filter unit model epoch learn momentum mapping input logistic classifier experiment set accuracy reconstruction objective generate explicit transformation content shift train training sequence sequence shift rotation image berkeley initial angular angular scalar angular sample angular angular sized accelerate velocity acceleration pixel acceleration shift mapping perform grid train rate epoch epoch layer infer first frame accuracy use mapping descriptor layer mapping mapping predictive accelerate shift concatenation mapping descriptor show bottom prediction predictive bottom predictive layer achieve significantly high predictive concatenation mapping angular base simple shift acceleration transformation improve increase concatenation relational show improve capability evolution explicit state generate three seed frame filter pair datum figure set figure introduce change author object instance instance frame video frame long train performance stop improve prediction reflect well frame datum dimensionality overfitte localize dynamic number number mapping show model capture ball prediction frame major sequence model deal range correlation deep address well input learn representation temporal evolution input aspect input allow prediction future interesting predictive analogy take frame new analogy task target may relate model datum relationship play crucial acknowledgment work support education grant google award bi feature way frame reconstruction frame previous encode transformation inherent thereby encode motion bi introduce encode frame transformation encode structure bi show natural way commonly force evolution input future achieve
anomalie nan period expert together acceptable exploring range numerous hundred indicator indicator link score easy presentation source least presence absence anomaly class desirable discriminate anomaly source term however one binary indicator also state interpretable cart hundred indicator coverage parameter expert maximize anomaly seem obvious redundancy choose feature reduction limit information operator redundancy excellent high difficult early sign tend remove sign could detail record early sign anomaly close huge datum goal validate methodology justify label methodology describe case assume therefore distribution observe model notation accord signal anomaly therefore change slope observation increase point choose uniformly th balanced corresponding anomaly anomaly htbp slow deterministic randomly period signal amplitude shorter difficult model anomaly degree distribution choose randomly anomaly type anomaly anomaly explain expert position present window test conduct shift signal occur window u shift two test variance parametric window define sample signal test detection complex binary way build classifier one indicator fraction test observation two consecutive window minus indicator consecutive change take consecutive window window recommendation original use simple average configuration lead indicator subset indicator vector signal shift classification et compose keep three divide estimate put report random acceptable see performance confirm reliable fit forest satisfactory generally difficult interpret indicator allow perform indicator rough global decision fitting argument reduce mutual base estimation forward approach indicator acceptable performance indicator take account redundancy indicator test summarize median white bag accuracy quite indicator performance expect around indicator accuracy circle subset summarize dot inside white accuracy bag estimate result general expect difficult expert particular distribution achieve performance satisfactory aggregated forest table indicator average ccccc window ks indicator correspond good classification indicator length window ks selection induce aggregation indicator understand operator methodology engine health monitoring build expert parameter hundred cover introduce turn diagnostic indicator understand decision automatic choice indicator interesting methodology decision model illustrated working methodology sound reach predictive performance selection behave instance fulfil univariate set complex anomaly extremely important note forest easy cart simple indicator majority voting probably simple indicator health firstly health monitoring involve class imbalance secondly cost asymmetric fr paris paris de paris france fr paris fr engine collect large engine help optimize cost article study build sign anomaly detection final idea generate indicator anomaly score expert scheme lead reduce indicator tune illustrate method contain sign reliable thus operational event produce jointly per hour rate take availability nearly engine monitor external monitoring among typical message status overview engine send anomaly early sign degradation anomaly automatically expert anomaly confirm recommendation send company operate consequence measurement sign degradation delay despite case inspection prevent availability avoid general build automated decision algorithm build hundred sign sign health monitoring indicator interpretability base human decision operator health detail propose methodology dedicate monitoring base acquisition equip sensor physical quantity pressure temperature mention etc engine good health engine potential detect diagnostic make diagnostic send company detect change sign overview consist recurrent obtain analyze operator methodology article kind traditional engine monitor expert survey sign drastically operational partly indeed currently design integrate nature partially via logical probabilistic monitoring help decision present health monitoring engine produce multivariate engine addition critical time turn know engine phase behavior anomaly detect anomaly major informative long indicator capture failure partially indicator approach couple experience coverage transformation monitor standard classification indicator decision decide whether anomaly engine type anomaly responsible potential describe indicator early sign anomaly display numerical real world shift slope instant roughly center expert typical situation source
wave wave intervention passive quantum upon depict system measurement update accord adopt version measurement correlate cause trick splitting causal connection distinction two distinction scheme intervention passive observation show problem completely passive scheme performing determine restrict specify draw scheme system experimental scheme passive observation quantum equivalently understand wherein restrict experiment implement circuit fig structure vary create h denote vertical measurement pair swap gate mode detect representation circuit fig take one produce preserve cb hilbert must circuit fall channel formulation proposal motivate least goal describe causal circuit formalism multi formalism formalism object describe suitably generalized understand causal quantum latter field causal present causal bipartite purely describe cause describe state bottom probabilistic cause direct give even general cause cause contribution act c pdf reconstruction scheme observation three causal show cause maps cb array represent positive red negative cb b identity reconstruct state tr passive cb tr slightly suggest experimentally quite match reveal causal finally expect find fidelity scheme appendix real hilbert measurement span output complete set allow conventional process achieve causal describe causal constitute scheme include scheme obtain implement scheme result average fidelity reconstruct map constitute description grain instance probabilistic common mixing causal show extract probe common direct passive probability common colour logarithm least narrow correct thereby causal square deviation passive unlike scheme passive map find therefore although span operator span nonetheless signature demonstrate turn one distinguish unitary pure maximally bipartite state probabilistic nature process bipartite thereby inference choice implement mixture remove aforementioned ambiguity obtain display reconstruction causal map passive average fidelity result well passive implement context alone pattern explanation cause observable measure perfectly positively correlation correlation pattern constitute universal gate three identity correlation sometimes row explain correlation signature causal discussion example make coherent channel common cause separable state direct cause mechanism implement break channel measure correlation causal case conclude causal appendix purely common cause relation scheme extensive markovian act cause system help correlation lead process several future infer observation mechanism measurement passive intervention produce embed centre nm component match nm wave place horizontal upon exact set wave maximally separate light mirror pass quantum fidelity consist h h half wave wave setting wave completely would extract desire leave gate directly swap fig distinct phase difference input implement shift implement swap probabilistic switching control three wave path implement path pick gate implement phase gate pick effect gate swap switch hz swap choose proceed mode pass another desire light send gate output gate mode detect detect ensure produce detector rate hz acknowledgment valuable research part innovation institute innovation develop project experiment numerical calculations author show completely specify describe circuit box main article limit purely cause show generic reduce bipartite process respectively value onto dt dd article transpose normalization express assume convenient essence type act maximally mixed scenario outcome measurement imply hilbert schmidt certain operator reconstruct component provide complete basis refer component wherein zero correlation system subtle express hence wherein expectation outcome subtle trace calculate schmidt inner reconstruct eq scheme reconstruct causal bipartite process case purely relation neither influence causal cause consequently consequently cause reconstruction yield eq purely cause distribution state meanwhile tr consider possible causal causal reflect underlying observation projective outcome measurement setting measurement outcome causal case relation common causal bipartite follow quantum note marginal expect produce serve measurement bipartite operator call ref emphasize causal encode infer certain sort affine ref analogue law like state semi completely composition map cast alone direct purely cause map output tr b therefore determine entirely facilitate comparison common scenario operator ref conditional trace preserving q positive transpose ultimately specifically cast operator mechanism indicate measure cause summarize statistic positive admit purely cause form admit explanation direct cause common explanation note ref cause observe passive scheme contain apply describe system interesting common cause respective two connect cause cause case set fact direct allow trivially conversely wish study restrict namely maximally eq q condition possibility set observe give rise ultimately restrict maximally mixed otherwise common cause direct cause relation marginal direct assume maximally prevent marginal bipartite maximally common cause summarize inference problem eqs possibility statistic nontrivial problem wherein signature passive quantum cause unitary common pure maximally previous causal rely cause express term state define bipartite rise cause maximally constraint maximally sufficient maximally describe unitary achieve causal case determine refers apply euler formulate representation effect sphere correspond scale maximally representation easily sphere ellipsoid code colour ellipsoid instance image anti several analytical basis encode offset centre ellipsoid channel maximally unitary channel maximally state vector index encode ellipsoid show axis length context channel ellipsoid coincide know imply channel implement pure unitary meanwhile whose ellipsoid maximally simple rotation sphere colour green sphere describe distribution improper rotation follow conditional change multiply main article ellipsoid image input denote anti unitary green identity leave maximally state mixture two cause height orthogonal realize experiment point mix channel produce point direction radius plane input lie connect centre span probability turn state effect origin angle decompose image alone rotation image lie remain meanwhile coincide offset plane along associate fig transformation therefore semi along degenerate semi axis root direction oppose ellipsoid disk dominate axis image lie plane magnitude q ellipsoid combination straightforward extract angle semi axis oppose still read normally follow scale plane similarly cause contribution ambiguity rotation take convention whether point axis ellipsoid generate possible solution rotation ambiguity implement unitary process maximally bipartite mixing angle previous show process perfectly passive probabilistic object causal without observation causal channel state separable ambiguity identify ellipsoid al separable channel break ellipsoid define fit sphere order condition mechanism channel common cause mechanism coherence conversely coherence causal purely separable partial transpose imply explanation process break rule purely cause explanation identify scheme reconstruct observe passive observation statistic statistical fluctuation causal square determine causal close passive analysis section frequency outcome value run model causal eq causal probabilistic mixture direct cause ie trace satisfie predict simplify rather appropriate normalization subsequently parameter operator unnormalize seek parametrize frequency express infinitely unique close passive common cause cause recall scheme state channel combine unnormalized operator write count passive seek minimize represent mixture pure maximally bipartite unitary aim type remove ambiguity consequently fit close realize impose maximal
instead optimization current bit depend bit well alternatively eigenvalue loose lead inferior modular formulation inference search large group block optimize block time let cost block denote optimization modular block hashing leverage similarity easily block meet modular sub modular word block modular z prove sub modularity method need variable optimize h code bit train hash bit usually one loss surrogate loss exponential adaboost boost coefficient tree train threshold minimize feature time speed summarize highly recent fast conventional implementation feature quantization largely consume linearly feature apply weight iv apply splitting summarize hash alternate iteratively code learn binary training encoding test retrieval train precision c cifar n comprehensive experiment image method retrieval specify tree depth hashing way retrieve example denote area curve dataset cifar contain scene category cifar truth tag annotation image annotate identical keyword portion allocate image training retrieval aside cifar split randomly select test query employ codebook soft thresholding patch test codebook feature code step much less perform much objective relation inference time outperform set compare different binary tree function spectral codebook retrieval fig type hash decision tree able code rbf data rbf svm training compare hash low codebook extract cifar resolution dimension dataset include hash semi hash codebook consistently feature result train dramatically increase solve eigenvalue expensive compare train vector sample large codebook order magnitude retrieval codebook feature plot retrieval c cifar cca pca cca dimension reduction codebook compare combine dimensional reduction train whole except cifar slow improve decision tree hash time performance hashing unsupervise hash lsh spherical hashing poorly preserve margin bit high feature large length linearly l increase high run bit high bit length training increase train outperform margin retrieval c train time precision map lsh outperform challenge scene codebook set bit subset training almost intractable short bit challenge dataset bit whole example weight example splitting may training due less margin usage contrast tree method involve comparison easily advantage retrieval significance many image cm van david hashing aim map original ham hash function demonstrate advantage encourage price achieve linearity suitable hashing modular hashing binary code inference solve hash decision precision especially order hash compact code hash fast search table hamming ranking code extremely data storage retrieval preserve distance hamming supervise try similarity locality hash lsh randomly hash cosine hashing learn affinity iterative approximate euclidean hamming space hashing manifold take intrinsic supervise hashing preserve take hashing increasingly hash kernel step embedding world usually hash despite hash interest may demonstrate dimensional example codebook remarkable thousand exploit advance feature desirable able deal efficiently sophisticated high hashing leverage efficiently incorporate hash however could feature supervise contribution ensemble decision hash hash number high thousand map general hashing decision efficiently binary code inference decision binary modular formulation significantly outperform retrieval high order training employ function inferior loose dimensionality decision propose method inference
test stochastic block network vertex define bethe block result negative sign curve recursion isolate bar black axis leave bethe hessian eigenvalue decay towards axis informative reach decay top finally information interestingly decay bottom informative eigenvalue choice bethe hessian generate informative one informative eigenvalue straightforward eigenvector backtrack must relevant inside number argue regularizer backtrack claim numerically compute building efficiently solve bethe straightforwardly carry vertex fact bethe certain weight unity reduce argument generalize immediately relationship backtrack non backtrack ise spin connection along spectrum bethe hessian backtracking operator index remarkable efficiency backtrack sbm uninformative disk lie outside real precisely real community eigenvalue bethe hessian notice eigenvalue correspond bethe definite prove e circle theorem eigenvalue course phenomenon takes translate negative eigenvalue adopt q outside circle radius come close eigenvalue spectrum stress backtrack matrix correlation small setting refine guess world choice informative eigenvalue infer membership standard bethe theoretical ise distribution parameter control strength analogous statistical physics approach machine bethe moment belief restrict bethe goal justify independently uninformative eigenvalue eigenvalue bethe spectral delta peak remove vertex degree belief propagation recursion cavity formula marginalization graph solve lead density locally limit dynamic iterate excess pool updating justify analytically eigenvalue bethe linearly around exist introduce equation rewrite jacobian backtracking operator identity square contain derivative invertible around implicit function contain exist around show jacobian eigenvalue modulus strictly long continuity respect exist recursion real prove reach far regularizer fail cluster region backtrack bethe propagation bethe systematically backtrack operator illustrate spectral propagation optimal measured overlap true maximize bethe systematically complicate run community bethe count number eigenvalue feed real graph illustrate show block bethe identify several detection large well backtracking consider identifiable particular eigenvalue backtrack case word matlab reproduce result bethe synthetic bethe bethe gave combine real non backtrack oracle answer tractable parametric perform optimally reader file matlab impact wide clustering expect impact spectral value similarity oppose backtrack promise arise use generalized maximize e modularity else eigenvalue carefully choose could solution give relaxation np discussion european grant agreement grant triangle universit et paris sup paris france paris de france approach node low adjacency recently argue symmetric operator detect simpler know bethe hessian combine performance backtrack theoretical limit computational advantage symmetric cluster community range biology benchmark sbm create matrix infer concentrate algorithmic case concentrate equally size refer connect group important conjecture rigorously prove also detect community meet perform optimally stochastic block cluster transition far detect transition pass propagation fed well limitation spectral adjacency matrix remarkably version cluster suboptimal detect backtracking
accuracy bp vary bp theoretically capture solution principle design formulation express rely linear corresponding indicator totally underlie constraint matrix combinatorial use technique acknowledgement european grant tu group otherwise tu text convex feasible start leave look sum simply child leave ball tu text convex envelope envelope use linearization trick tu let envelope eq definition structure sparsity tu structure relaxation polynomial programming framework sparsity introduce argument important reduce familiar p pn typical absence impossible reliably learn nontrivial must exploit knowledge impose structure sparse theoretical broad generalization rely convex establish sample algorithmic obtain describe fortunately convex encode within effort description inherently restriction coefficient structured review find tractable surrogate capture combinatorial end combinatorial issue arise quite simple summarize encode identify whether totally tu verify investigate notion derive combinatorial convex relaxation illustrate tu description popular norm hierarchical show tu description support relaxation fact tu lemma specific induce induce complement submodular modeling go beyond tight norm novel theoretical group group sparse root lead description provably totally exclusive lasso overlap scalar letter letter entry yx I pp identity context introduce definition sequel submodular f g totally tu every square proof omit see supplementary material encoding combinatorial support hence task find surrogate determining envelope low condition computation f pt p ball completeness conjugate otherwise last tractable noting without necessarily lemma restrict sequel unless general conjugate hard numerically generative submodular quite popular know checking approximate light three allow tractable convex relaxation regularizer fact equivalent submodular minimization minimum empirically run recent solve may non zero magnitude make sense combination continuous envelope dual p seek combinatorial approach satisfy sparsity encourage simple via inequality structure tu admit relaxation tu us simple template tu tu penalty support feasible modeling arbitrary tu extension envelope tu envelope tu tu lp still despite non note simplicity need tu interaction tu penalty weak hold penalty definition besides tu capture study tu relaxation structured naturally therein group together collection support figure represent group set iff graph iff pe intersection two iff structure group cyclic black fill thick sep black thick pt rectangle draw white thick inner sep transform auto label label leave label v v v g v shape fill label g node node node group typically seek express non decrease submodular sum penalty express ig sum weight force group select tu entry zeros condition envelope intersection penalty induce application union seek minimal bipartite representation correspond minimum cover problem propose potential tu penalty admit convex envelope worth note latent homogeneous envelope g lead tu relaxation structure matrix acyclic shown induce acyclic tight far induce level variable enforce two lead non surrogate give tight group tu penalty surrogate q x propose group surrogate group otherwise provide formulation enforce sparsity signal signal bipartite actual minimal f small cover seek signal tu tu group lead tu envelope eq result convex program lasso case material hierarchical organize rooted subtree model wavelet tu model circle draw inner sep inner sep pt shape mm child scale mm mm mm child selection norm hierarchical model incidence iff tu give group hierarchical norm envelope far encourage implicit within speak opposite within sparse opposite form model exclusive prove tu structure tu partition exclusive lasso actually relax tu necessarily group tu matrix tu penalty tu acyclic tu trivially tu dimensional see period two eq tu exactly exclusive constraint exclusive actually version relax desirable
miss occur arise perform principal retrieve spectra spanning hereafter l principal well describe remain hereafter subset retrieve maximized computing component first surprising iteration since write dominant dominant eigenvalue uniqueness depend obey dominant eigenvalue eigenvector failure check satisfactory iteration similar retrieval ghz present principal idea intuitive flexible generalization algorithm exist give behaviour usage assess already author science policy office national foundation office science web site http www iii laboratory group de berkeley national laboratory physics new university york university de universit b li straightforward analysis weighted method retrieve principal amongst meaningful weighted principal retrieve component illustrate usefulness digital spectra measure shorter benefit fast component pca design huge datum set principal arithmetic mean corresponding variance coefficient coefficient principal interested reader deeply wide variety digital spectra point describe hereafter spectra cover problem require pca decomposition limit tool assess long nevertheless limitation weight inherent classical pca difference variance come come limitation focus problem miss case factorize deal none account orthogonal optimize describe explanation give maximization include observation well individually datum come fact finding well individually take em order current alternative algorithm simulate extension use bold I th denote element product hadamard contain retrieve reference matrix I within ii potentially sense mathematically minimize regard generality like retrieve component number ie decomposition orthogonal variance minimize covariance differently explain order clarity pca become solely drop orthogonal transformation maximize vector purpose choose practically already deal pca component equivalent minimize cf rely latent hide iterative procedure optimize fast pca fulfil follow expect pca algorithm converge still relative change give design general approach dimensionality spectra specifically attempt equation part smoothing strength non negativity constraint reflect interpretation spectra drop negativity constraint restrict concern optimize comparison deal ignore constitute major drawback implementation result go approximation solution solved step clarity matrix orthogonal straightforward gram schmidt retrieve secondly principal eq state begin regard retrieval matrix optimize equation consider hold equivalent preferred step retrieval decompose observation combination principal q moreover single iteration regard huge lead good insight component fit individually hypothesis reasonable since component account result implementation one apart order cross suppose retrieve principal q retrieval whose last retrieve orthogonal nevertheless manually check finally end though find orthogonal decomposition suitable set none component reconstruct idea weighted result principal variance relevant identifying necessarily variance variance weight convention straightforwardly definition write fulfil accordingly constitute implementation observation variance fig unable suppose purpose observation point explain maximize variable consequently zero retrieve dominant covariance eigenvalue dominant q maximize sake clarity equation reference may exhaustive proof hereafter implementation fast iteration recognize nonzero direction dominant eigenvector unity round inherent number eigenvector associate algorithm subtract find section method nevertheless design algorithm scope fact understand associate normalized unit converge regard point minimize principal retrieve real distribute happen correspond small observation problematic principal regularization factor expression equation result weighted allow strength regularization go behaviour rare conversely assess regard namely come fact fairly competitive test observational consist basis take shift period schmidt evenly interval provide amplitude uniformly value discard contiguous observation latter assess performance weight retrieval simulate perform observation realization various retrieve five principal compute follow follow chi quality use completeness decision estimate really strongly depend section deal quickly study account preliminary need describe set twice number give refinement miss top middle average fit various fairly algorithm somewhat high dispersion set show mean increase miss datum average difference presence miss still parameter simulation moderate increase miss average first latter dominate show reach unable converge maximum detail average clarity plot remove solve explain already suggest choose retrieve component noisy maximize variance release either uncertainty determination insufficient spectra frame spectra variance inspection show fit variance spectra mainly attribute spectra thought cause signal retrieve previously algorithm assess various assess variance amongst amount presence large perform retrieve keep assess quality fig ccc
imply equality family follow dot express definite assumption symmetric definite reproduce hilbert norm pe dx dx imply aforementioned fact statement kernel condition compact suppose result respect exist sufficiently detail requirement theorem inequality kullback hellinger distances bound positive bound hellinger depend guarantee moreover part note imply also enough full correspond bind admit disjoint credible set source base small simple subset alternative view approximation random measure approximation median combine small size credible performance rate cover hellinger exist particular typical theorem yield moreover hellinger bound k condition hold satisfied recall geometric f k let hold discuss method respect misspecification distribution primary probably p pp conjugate course sensitive achieve concentration k ks nz sn moreover besides proceed mean conjugate l correspond chebyshev concentration event subset result comparison way particular posterior exclude simulation evidence favor thresholding rate acceleration viewpoint modern see discrete w tb w ji two previously comment choice theoretical guarantee many interval acceptable size g available yield suggest pick among posterior computational achieve run distribution goal compare run parallel tm approximate degree tm n refine way advanced optimization reference two improvement achieve magnitude outli posterior simple univariate gaussian outli linearly increase index replication flat jeffreys prior q I repeat replication datum representative posterior fix compare consensus posterior credible calculate replication empirical contrary consensus overall posterior across length identical length wide interval posterior wide absence outlier ht value hereafter mean robustness grid gp standard convention standard obtain equally grid location algorithm draw across case subset correspond employ posterior median location represent band correspond quantile posterior replication extremely sensitive shift truth outlier coverage band location gp produce true unstable instability matrix avoid work approximation massive subset posterior subset gp computationally stable contrary great subset gp computationally carefully depend computational regression promise massive datum general social survey capital consist contamination small survey answer question incorrectly use process dp multinomial probabilistic response detailed description generative include appendix divide sampler account remove atom weight associate posterior posterior mode case tend subset accommodate density density estimator around mode slowly however heuristic approach explain remark follow space occur imply cardinality part proceed event occur triangle inequality part complement wasserstein take hellinger hellinger follow eq every satisfying exist note chebyshev finally recall note q let proceed support conclude numerator hence large put bound numerator denominator together c generative generate stick break construction represent response latent latent generate stick break hyperparameter shape fix latent sampler obtain analytic conjecture ex author support grant es institute environmental health sciences support grant dms provably computational technique evaluate measure propose measure equip distance quickly efficiently practice evidence improvement datum pose general challenge cluster storage contaminate outlier identify remove place statistical literature progress point method understand main propose provably scalable big allowing implement parallel split part implement markov chain carlo another draw probability properly section overview exist literature explain goal aim introduce main proof remark constitute study robustness indicate research robust study sensitivity uncertain uncertain typically heavy tailed likelihood usual assumption e g contamination large place outlier removal also robustness misspecification progress scalable design distribute subset machine local communication machine optimization approach distribute limitation dominate subset posterior sequel knowledge rigorously justify combine posterior major evaluate return likelihood master appropriately combine conditional repeat among sa successively learn batch sa hamiltonian langevin dynamic method parameter learn variational posterior see excellent well substantially uncertainty fall avoid communication independent chain posterior combine variety simply draw alternative density call limitation applicability justification model unlike method provably inspire median technique apply framework key fact throughout distance pp pa trace totally space packing number structure ball hilbert k form see application median discrete characteristic sign borel integrable definite transform characteristic compactly support almost always question favorable upper corollary wasserstein well hellinger absolutely lebesgue density explain theoretical index absolutely hellinger metric eq assume let value vector define inference borel algebra observation borel assumption towards mean surely address happen arbitrary usual concentrate corruption description propose construct distribution integer divide typical prior subset depend disjoint median evaluate geometric identically define fix choose grid mesh principle prior discuss possibility practical application weight properly rescale overcome approximation subsample unstable metric measure improve set coverage often numerical note
apply meet iterate one end process construct formulae smooth es es imply guarantee one exponentially satisfy gain tend fast formula quickly standard similar back increase circumstance norm view asymptotic formula sense incorporate thus hereafter idea final normal ensemble square produce associated root exactly normal square root filter kalman jacobian matrix eq gain computationally expensive take purpose residual implementation evaluation reduce computational cost element may magnitude less evaluate preferable improve take hybrid enkf give instant minimizing ensemble way enkf enkf perturb assimilation ensemble smooth ensemble complex user therefore analytic jacobian end stochastic example relatively real accurate expensive correspond inverse calculate factor consider jacobian relatively relatively requirement may deterministic inverse rule residual let inverting factor spirit one experiment adopt method adopt otherwise transition trajectory discard rest respectively truth assimilation measure odd element fx observation error assimilation observation integration normal initial background ensemble way previously whose residual norm reach may experiment extra investigate localization experience suggest default experimental beginning member however presence process equip covariance localization norm descent purpose conduct investigate relative normal randomize lm iteration fix constant aim cost essentially choose minimize instead necessity circumstance conduct comparison worth lm lm normally construct enkf though toward reduce illustrated test cubic section vary factor width ends suggest enkf nonlinear report norm panel lm cubic figure background call hereafter solid line together dash line residual background assimilation indistinguishable opinion lm iterated average member criterion update estimate continue eventually negligible background almost reduction apply odd assimilation error run ensemble member step nonlinear apply error observation two observed scenario odd half four ensemble ensemble observation variance error take even increment average size frequency panel ensemble rmse monotonically increase relatively appear increase large tend tendency may b observation nonlinear observation figure case clear rmse exhibit shape achieve low possibly observation observation c observation panel suggest nonlinearity hand panel b half observation panel nonlinear incoming pose consequently freedom constructing tend state toward scenario pose solution well observation ill pose small low show panel instance linear covariance background introduce localization circumstance rmse however certain value half localization even guarantee residual effort impact localization investigate also assimilation window consist step nonlinear investigate ensemble setting show figs variance panel size frequency case indicate variance relatively seem opposite variance error affect conjecture occur combination variance background ensemble step variance necessarily effect rmse decrease examine mis term experiment term true variance observation test variance report function sensitive mis specification variance possibly cubic may increase linear estimation mis variance might achieve certain find certain error improve assimilation overall uncertainty work concept assimilation derive handle observation residual implement iterative ensemble numerical handling achieve reasonable term mean assimilation realistic conduct resource future iterative filter thank anonymous constructive comment suggestion project realistic financial eps mm time residual rmse lm cubic operator upper background solid cubic panel assimilation solid reduction example reduce toward upper final analysis assimilation window exponential function time step one half scenario p panel nonlinear variance calculate panel rmse legend panel assimilation mm mean observation panel magnitude iteration scale horizontal observation visualization scale international gate kalman enkf assimilation circumstance enkf author improve stability enkf adjust able accuracy extend nonlinear operator modification iterative suitable assimilation illustrate iterative filter enkf nonlinear operator various kalman filter enkf variant implementation kalman finite ability scale assimilation problem enkf receive assimilation influence enkf enkf relatively estimation covariance adopt auxiliary call localization improve enkf covariance increase increase robustness enkf various covariance hand localization localization suffer certain circumstance especially error assimilation assimilation residual residual norm one show circumstance assimilation equip stable well operator suitable nonlinearity observation operator main gap adopt filter assimilation cycle use residual suitable convenience refer linear nonlinear cause confusion organize introduce residual observation section aforementioned method extend modify conduct stability conclude work observation state instant instant project space respectively zero discussion drop end suppose observation certain instant difference space set residual find euclidean hereafter reader behind prevent combine combine call inversion original state work residual proper explicitly extended reader formulae ensemble consider slight generalization kalman positive scalar analysis ensemble objective general suffice conventional gain resemble kalman gain enkf analogous multiplicative covariance residual satisfie denote satisfie ease inverse transpose give follow b number directly alternative obtain computational cost omit brevity reader refer restriction need relate certain discuss kalman kalman covariance kalman variant variant put emphasis robustness estimation kalman reader therein nonlinear complicate since explicit analysis residual may long operator generally satisfy large continuity state assimilation relatively readily iterative framework aim long enough low process solution least square problem aim solve remark residual intuitive term hereafter minimize pose inverse uniqueness inverse theory assimilation introduce e b aforementioned avoid presence derive behind bayesian state observation gaussian interpretation situation may dynamical reality often evaluate e state scale fix estimate purpose value combine iterative enkf construct enkf optimize work state assimilation
good standard get less grows adapt life bias estimation really assumption static need window static variance window big bias contrary lead bootstrap simplify run experiment static yahoo take portion experiment ground obviously news compute average ucb evaluate ucb ground truth use acceleration interpretation htbp close please tend batch enough recommendation evaluation realistic focus offline estimate reasonable prove asymptotically convergence counter intuitive introduce fast static accurate publicly make yahoo server present acceleration highlight acceleration issue evaluation desirable property context risk estimation bandwidth extensively study kde safe controller company recommendation behave certain collect tight replace ascent policy notation apart notation modification experiment non contextual exhibit importance sake algorithm history triplet efficiency learn set triplet hx ax rt pp recommendation news recommendation recommend come challenge recommender set seem solution purpose evaluation rs live avoid offline evaluation option thus trust method evaluating literature nonetheless satisfactory mainly fraction limitation bootstrappe estimation latter risk online proof superiority compare various name become common activity web think movie recommendation netflix amazon news job recommendation application yet profile order attractive serve item recommendation piece software rs item interaction click recommendation train predictor click user past recommendation implicit paper recommendation recommendation recommendation continuously replace new characteristic sometimes dramatically web news find yahoo tv example item recommend item movie contextual nonetheless recommendation recommendation predict offline argue idea require continuous effort item picture static greatly political movie star rs portion engineering effort able offline rs recommendation computed accepted community nevertheless give dynamic offline fairly yet use argue web issue evaluate explain previously bootstrappe theory empirical clearly term bootstrapping allow estimate sense especially evaluation decide algorithm also synthetic detail publicly motivated order deal dynamic precisely contextual bandit framework recommendation problem also arm variation thompson variation contextual tuple reward pair reward user action recommendation game round player choose round game reward reveal whose score player important game reward action offline evaluation typical learn try exploit therefore player face exploration exploitation either uncertain improve explore perform believe armed bandit study ucb deal upper bind contextual problem study additional without although basically normality action estimate linear bind contextual triplet choose action say maximize reward convenience per click output simplify systematically drop bandit live fact period understand impossible concern region different potentially different thing evolve equivalent playing go reality likely acceleration contextual look lot challenge evaluate yahoo news chapter work contextual record acquire yet deal problem evaluate dynamic model line understand protocol suffer stem bootstrappe thus dataset draw bootstrap underlie yield converge bias concentration speed recall bias want map history policy map action appear efficient implementation would also contextual triplet kx bx bt b protocol bootstrapping bootstrap dataset subsampling allow classical purely policy obvious datum formal reflect together interaction estimator estimate deviation expand last contain expectation expand dataset prevent amount context neural avoid overfitte practice network online bootstrapping smooth smoothed bootstrap kde sampling kde bandwidth get smoothed bootstrap kde core loop henceforth analysis evaluation denote recommendation algorithm generate recommend asymptotic series independent moment explain admit realization actually adaptation bootstrap convergence result introduce produce dataset evaluate expectation mean estimator convergence allow evaluated sketch respect consist guarantee gap subsampling order index estimate dataset policy chernoff denote inequality obtain probability admit expansion recall q admit thus
model share representation consider work criterion salient model tie salient share component framework improve propose e feature salient use length document message dirichlet optimize work concept model sparsity proportion specific one two subset salient follow occurrence component machine specialize derive na penalty term interpretable effective paper organize lda topic parsimonious section derive bic joint bic objective corpora image dataset conclude remarks corpus dictionary unique word topic follow document document indicate whether specific word originally extract generative process topic topic variable algorithm briefly review variational log family change dependency dirichlet value document determine leibl kl low document log next variational parameter probability meet proportion control corpus optimize proportion document lda model document essentially estimate topic hard assign maximum document lda develop new fundamentally differ treat deterministic rather maximum bayesian hyperparameter hyperparameter parameter topic document topic develop treat model deterministic bayesian set alternative approximate approach overfitte corpus select function pmf pmf indicate present possess topic generate aforementioned together specify structure full model constitute double product switch th word constrain pmf nu likewise topic satisfy pmf assume derivation sequel seek bic moreover dependence bic respect equivalent accordingly parameter maximization em element treat incomplete increase iteration consist complete step hide current parameter origin add normalization estimate maximization topic proportion derivative satisfy normalization constraint respect achieve multipli multiply side sum word e initial estimate assess iteration next meet estimate share principle take respect optimize estimate initialization hold show quite work count jointly optimize bic fix bic alternate locally learn computed sizes taylor integration derivative thus taylor negative taylor mean evaluate topic document specific share fix irrespective approximate usually uninformative description treat deterministic minimize tradeoff negative proportion number q hessian information approach bic block q become penaltie different type log derivation parameter interpretation equation particular sum estimate total lead generalization cost penalty configuration topic principle invoke uniformity across topic generally topic shannon accordingly mn topic across define configuration select uniform algorithm jointly determine switch global bic generalize step guarantee minimum note applicable bic apply monotonically likelihood toward substitute log complete taking term bic complete datum datum incomplete term iteratively step iterate expect datum form incomplete datum term complete fix term bic dependence minimization complete log step via visit current change respect bic ensure descent bic repeat occur predefine reduce current plausible remove mass repeatedly reach corpus respect class include whose probability compare implementation com approximation log fitness divide hold keep part disjoint expect proportion topic proportion correspond also compare extract topic exhibit coherent covering concept agreement expert occurrence probable contain similarly document percent specific topic top coherence lda coherence initialization validation topic accuracy create document topic document mesh document label document word removal four model topic massive show bic curve bic std show lda train complexity trade label dataset distribution count ground proportion respectively label label class proportion assign label high criterion discover divide ground number correctly label assign criterion different threshold report area auc parameter topic document unsupervise lda auc lda well entire auc curve coherence lda compare occur least topic average specific total topic word occur document corpus suggest well class also number average indicate great overlap topic salient topic std c topic topic std document standard removal corpus initialize topic two show lda std lda likelihood parsimonious performance hold topic single label compute label document consistency fig good class compare topic coherence lda plot lda table small topic although add occur corpus label unique specific share table separately share word topic write first topic cm model patient bank health gold year share year plan patient treatment bank gold lda come law ga price share stock month thing price disk window hard record write post program offer unit problem file record compare include unique removal elimination hold curve hold set compare consistency classification across minimum bic curve class order high coherence report measure lda small compare top sample topic lda report section report comparison class unique extract sift slide grid collection give learn center sift descriptor nearest cluster perform mean pruning cluster represent length text corpus accordingly equally model reduce topic lda std hold lda order classification corpora fig consistency consistency lda sparsity proportion however word probability fashion four hold order achieve hold bic minimum lda increase much topic moreover average per label per document curve fig possible achieve class recognize unsupervise salient choose huge lda label topic word interpretability hold class consistency model process core processor execution smaller comparable lda typical document time lda running time lda step em improve nevertheless run nsf consist parsimonious salient give word derive bic objective penalization jointly topic performance log agreement ground email department university university pa parsimonious topic corpus model even salient explain universal share document document bayesian goodness interestingly identify size minimize specific text corpora test ground design reduce number free covariance cluster grow number parameter across parsimonious less overfitting rich various document bag introduce topic thousand word probability topic expect topic model dirichlet vocabulary lda mixture corpus proportion model topic lda intuitively many model universal principle every proportion cover allow nonzero proportion document method identify word may
sequence stack v instead comprised frame frame vector notation express multiple proceed error assume real subsection experiment relative error second experiment compare propose algorithm true sparsity algorithm sparsity change estimate sparsity give rough initial sparsity level five evaluation sparse index nonzero gap simulation carry xlabel nonzero row ylabel style draw fill legend align leave color solid color solid width row crcr color width pt crcr color solid row crcr show sparsity root square fewer non zero estimate twice true estimate support height xlabel iteration ylabel draw black fill legend cell line width blue width pt crcr determine total time configuration ensure carry number measurement experimental threshold mean less term varied actual support plus level experiment repeat time experiment measurement vector measurement vector achieve row zero increase decrease row vector row axis xlabel ylabel recovery rate fill legend color mark option sep crcr color solid mark table crcr color solid line width x option sep crcr green solid width option row sep crcr different calculate twice success experimental repeat three multiple recovery rate multiple determine rate height xlabel ylabel legend fill white align red width pt mark option crcr blue solid mark solid crcr red width mark mark mark option solid row solid mark mark option solid sep crcr mark mark row sep crcr blue solid pt mark mark row crcr choose recognition comprise audio people sentence person contain head audio sequence contain view short sentence office video camera video image resolution contain face loo evaluation person compute recognition lower large cccc norm heuristic experimentally sufficiently actual experimentally find value twice sparsity sufficiently simulation multiplication converge art use propose technique matlab website email www ac iteratively randomized exponentially weight solve measurement measurement solve common modify model face recognition project gradient synthetic confirm project system speed application various area process art row randomize reconstruction band prove converge algorithm solve equation term square solution interpret I observation explanatory useful ideally know observation overcome solution lasso try sum error solution solution well greedy approach outcome address algorithm solution similar experimentally fast almost equal multiplication imaging solution e measurement stack column sparse requirement sparse multiple measurement propose modification problem propose algorithm recognition video compare sequential almost sure estimate max j cx I ts element show support sparsity initial index heuristic follow optimization follow call number multiple measurement decompose single problem initialize row large choose support vector index lx x change
bandit pre process collaborative completion solve relate observe mixture learn sample produce set reveal rating recommendation reveal two set rather item classify efficiently support nsf grant office award support fellowship derivation clear context indexing writing reproduce ease presentation lem en step appear begin ensure user user chernoff explore user bad explore thus jointly item exploration z z iv change constant decay potentially assume bad event lemma occur hold high suppose item rate least user neighbor lemma verify bad neighbor enough jointly explore rate item pair user proof four less bad happen tell provide suffice require lemma tell neighbor suffice since satisfy inequality preliminary upper probability user suppose rate item jointly rate user neighbor vi yield guarantee inequality subset cardinality shorthand user suffice good neighbor arrive rate neighbor bound item note exploit n finish show comparable recommendation popularity amongst friend dm friend popular rating user friend dm beyond preprocesse compute online dm item simulate recommendation system movie rating netflix movie rate collaborative real recommendation rating simulate item address issue dense vs top rate receive rating dense rating initial reasonably explain source movie rating effectively user structure user item could reasonably movie experimental rate star star less look subset rate corresponding simulation upon mark thus item long recommend netflix dataset top result nonzero netflix nonzero item average reward recommend item reach I movie movie simulate recommendation item feature vector dataset reveal provide thresholded rating thresholded rating dm estimate datum next user rank number movie rating simplicity search choose set achieve high reward netflix dm wide dm expect roughly around time mostly item fact mit despite recommendation little work especially recommend address introduce recommendation cast item recommendation learning analyze cosine user either common probability item user distinction bandit item goal maximize recommend time establishe collaborative filter know exploitation step type step explore explore user recommendation vast tailor prominent amazon netflix movie collaborative decade news amazon recommendation netflix win song million song dataset challenge recommend recommend already movie movie separate case item recommend user good recommend different item success development justify effectiveness gap paper online bandit cluster bandit impose user cosine similarity collaborative filter key inclusion two type exploration space different type problem setup near nearly logarithmic exploration recommendation system collaborative give guarantee section overview consider item recommend simplicity immediately reward system step recommend formally item recommend time rating indicate rating yet objective q item item maximize clearly user recommend focus maximize aim recommend random whether maximize former user music prefer music like user low find diversity experience merely item maximize equivalent challenge preference item preference user rating broadly inference possible preference relate paper preference type user identical item preference type heterogeneity ease exposition assume belong user user correspond latent user movie model clustering relate version bandit armed infinite solution armed bandit determine keep apply cluster seek capture bandit available adversarial combine collaborative aspect bandit dynamic availability impose strict greedy bandit explore item far vote greedy exploitation use similarity exploration item ask rating item randomly choose let fill user rate recommend item rate recommend user rate maximize threshold either ask explore recommend user choose describe exploration space decay exploration decay user score indicate give yet reveal restrict jointly precisely cosine cosine overlap support user neighbor cosine propose collaborative respect state reasonable necessary condition establish item noise user preference u uv item classify incoherence cosine example suggest incoherence reasonable allow rather generality divide recommendation independently pool performance pre input define latent proportion recommend initial step algorithm recommendation oracle item recommend recommender would item meanwhile give period scale step simple incoherence consider user rating type produce probability random variable probability inner product rademacher standard incoherence previous choosing source vi vi scale suffice example user assumption event proof focus neighborhood event neighborhood event enable argue initial rating bad part exploration sufficient neighborhood event hold user think enough user type decay thought yet correctly neighborhood accurately item user hold exponentially decay term thought classify last two decay thought cost explore proof combine appropriate constraint user lemma specify user combine lemma corollary lemma generality meet bind ask great depend statement detail appendix simulate online system rating netflix rating rating consider recommendation reveal rating rate issue simply item rating latter top vs item user rate item receive rating dense rest rating rating interactive online beyond reasonable dense look behavior across dense top movie
know cauchy conclude definition institute ac optimisation gain popularity parameter machine model hyper optimisation reasonable advanced fail optima surprisingly little bayesian optimisation apply acquisition hyper optimisation become development machine recent medium interactive environmental monitoring combinatorial optimisation automatic one quantify uncertainty illustrate parameter fall range integrate use monte carlo advantages sophisticated treatment knowledge introduce estimate hyper symmetric goal unknown exploitation exploration process optimisation natural function mathematical function evaluate massive bayesian whereby introduce encode smoothness rule derive carry location process flexible place reader detail process covariance input jointly eq convenience type kernel mat ern theoretical general type bernoulli presentation focus gaussian noise refer reader introduction evaluation point marginally mechanism update datum turn crucial must computable optimize every exploitation exploration acquisition ei default popular optimisation package ei write closed form density case improvement member member cause stochastic mean seem alternative choice ei scale enable gaussian distribution despite statement discuss must smooth assume reproduce intuitive review rkh proof rkh property f rkhs theorem consider inner non positive clearly f k cauchy cauchy precede rkh converse theorem condition eigenvector eigenfunction eigenvalue expansion rkh therefore rkhs suit present consideration proof appendix material discuss could restrict kernel condition without regret obey analogous convergence ucb agnostic ucb detail rate convex kernel power decay mat kernel dd dd proof idea sketch consider instantaneous challenge gp quantify way cauchy dedicate inequality separate reproduce hilbert concentration combine aforementioned challenge relate ei quantitie easy analyse via build bound turn instantaneous regret variance sum subsequently regret maximal say gp accommodate
optimisation problem good problem use discussion approach close form eq cross avoid either inverse treat unknown qr speed become multiplication discuss enable practice avoid calculation principle matlab qr inverse pseudo inverse inverse package exploit cpu core modern inverse significantly magnitude hide become multiplication require time acceleration describe incremental streaming offer insight offer output large potentially training double gb ram typically modern pc problematic identify follow key activation sum activation similarly simplify key introduce column write way keep memory solution ram expand runtime rather face memory limitation batch subset implement iterative output describe typically uniform binary small make possible unless typically product albeit dot zero row occur orthogonal aim weight match statistic ideally datum could sense focus primarily selection several introduce layer bias dot product weight publish al call constrain third neuron rectangular visual call rf although machine aim limit visual convolutional rf superior pass rf considered follow backpropagation adjust weight simultaneously train weight maintain backpropagation backpropagation follow repeat iteratively convergence rf mnist ht symbol vector symbol multiply vector weight difference field rf neuron random rectangular field image motivate backpropagation operate weight sum weight define term argue reason strength bias conventional backpropagation mm normalize subtract dimension divide deviation layer neuron block train sample class size block random sign multiply transpose product block normalize row unity input vector sample sample different space addition propose overlap shorter near sample largely unnecessary implement elimination select mm distinct difference weight difference sum randomly normalize row input weight difference select solve blind percentage advantageous storage rf resemble tuned preferred region tend contiguous visual frequency aspect biological image create rectangular influence generating start input integer coordinate mask mask small discard repeat two zero vector correspond matrix hadamard term multiplication weight unity find beneficial exclude mask mnist database ensure exclude first small note convolutional beneficial diversity layer pixel towards class sparse provide unit specific enhanced performance combine shaped weight either rf rf follow obtain weight follow hadamard normalize input unity length rf bias difference rather weight use rf c mnist benchmark combination combine output albeit neuron ten output ten first label quick hidden neuron note consists therefore combine multiplication increase network combine effectively middle layer development report learn unsupervised train error solve capacity weight capacity able suggest capacity however return layer specific backpropagation tune hide introduce possibility well understand mode backpropagation neuron solve weight equation backpropagation mm whole indicate matrix weight continue desire illustrate solution test maintain six method describe row input unity seven randomness inherent input train hide plot marker rate figure actual training result occur rate training improvement fit verify cross first classify point point training train layer marker actual combination three part total therefore rf outperform total number unit rf produce second illustrate hide unit overfitte error continue note train seven method iteration backpropagation using infer still relatively give converge hand improvement surprising backpropagation backpropagation iteration backpropagation describe mnist achieve backpropagation hide rf cp former backpropagation describe unit time report slow shaped weight outcome input backpropagation outperform backpropagation time percentage apply mnist backpropagation marker error improve method little impact runtime matlab ghz intel core os gb ram plot exclude load memory file matlab exploit four cpu core negligible consume formation solution square large none depend conclusion runtime individually minute runtime mnist comparison backpropagation minute report second unit second hide rate benefit train various size train testing exclude load mnist file backpropagation apply scale linearly backpropagation trace iteration classify handwritten digit improve rate preprocesse apply affine elastic sensible way datum comparison problem nevertheless train able rotation scale elastic date training increase runtime reason point large significantly multiplication systematically mnist enhance expense neural network benchmark network define hidden simply find one combine accuracy comparable publish without augmentation preprocesse denoise network effort belief network rf input input sparse close weight implementation part pc require little highlight avoid calculation moreover possible set iteratively compute streaming application every inversion principle hide engineering framework utilize large model potentially boost course hard argue entirely mnist cifar accuracy cnns
empirical different order suboptimal satisfy slowly propose implement specific implicitly guarantee reject graph g center matrix xx update xy lb b lb l ji l j j qp l sf lb run purpose get convergent instead terminate usually network modern come make fortunately structure early complex individually idea appear concentration decomposable refer dynamical reliably capture topology see whole getting detect exact block contamination possible treat identify robust refer effective rely estimate may identification different purely estimate perfectly decomposable noisy datum tend remove deal robust decomposable write conduct fine fashion screening learn smoothly drop case reveal alternatively enforce sparsity set take stage constraint drop instance screen reduce fortunately fall framework f show search ny popular graphical similar screening handle coordinate worst come stage complete scalable lie fine implement matrix separate package topology consist example sized measure ij ij ij ij accuracy model I xx matrix scheme I multimodal run comparison whole screening parameter throughout spectral existence minimize validation evaluate screen tune quantile show I I infeasible suffer simplify fail frequently conditional dependence sufficiently seem try correlation topology low true connection sign shrinkage surprisingly degenerate job rate quite infeasible take minute run intractable design nonconvex accurate efficient stage remarkable exception comparable validate unnecessary successfully examine network decomposition index obtain membership come cluster true define true negative auto sized element network estimate decomposition consider screen nature figure setting show decomposition rather great ease practice network experiment pattern comprehensive cost consist equally sized generate manner computation computation without graph screening slow path sparsity report decomposable large computational conduct resource infeasible analyze stock keep record stock stock take transformation decomposition cluster place interestingly obtain varied systematically base stock category comparison clustering decomposition quite seem category reflect design decompose network nine isolated reduce tune package huge graph thresholding mask truly exist inaccurate correspondingly course lot fact transition matrix conventional mse window past e use estimate repeat end define n h synthetic bic b category relatively even offer comparable forecasting ccccc category category consist large stock market collect price stock remove consideration consist sample segment consist get idea cardinality graph connection whole take particularly node com intuitive three large service technology technology group product share connection negative condition differ although cause purely causality fortunately link screen perform examine detail scheme investigate forecasting include intractable horizon transition segment suggest existence dependence beneficial take correlation forecast stein work reduce search fine finding cccc segment dynamical sparse structure direct transition second order undirected dependence topology stage synthetic current state contaminate multivariate include stock describe evolution translate matrix order translate notion association joint edge screen small manner refer screening reduce problem fine stage world identification dynamical shrinkage dynamical study stock market stock interact evolve dynamical resemble random infer topology ax finance evolves translate causal observation conventional fail identification perspective shrinkage sparsity prefer produce network influence graph perspective nevertheless totally network even observation node ideally structure capture gaussian attract lot desirable unfortunately directly challenging substitute mean comprehensive picture necessary experience big infeasible ordinary pc make reliably network topology accurately energy stock motivation graph short correlation isolate helpful graphical statistically speak similarity exist joint regularization isolate index also notice brain connectivity network network detect much possibly course decomposition connection propose jointly undirected dependence topology identification dynamic association screen decomposition scale identify remove unnecessary link search problem reduce enhance successful develop decomposition estimation mask screening decomposition describe call section algorithm graphical screening screen network exist dynamical network behavior point component previous multivariate gaussian characterize node translate cause node translate undirected conditionally nod second topological dynamical topology perform exhibit dd decompose completely mutually regularize problem extremely inefficient moderate computational performance boost propose framework consist stage identify structure group induce group element form stage maintain sparsity package operation possible popular parallelism bb sparse another covariance bb impose feasible motivation screen computational algorithm nonsmooth unknown result depend appear algorithm efficient asynchronous line odd unbounded soft thresholding throughout sign define thresholde define general general nonconvex discrete rule guarantee universal convergent refer point equation satisfy rule practically nonconvex cover important role follow form choose group
qp alternate direction multiplier qp otherwise write lagrangian lagrange estimate constraint modify admm nonnegative entry nonnegative part finally admm apply apply multipli inequality consist success crucially rely minimum apply qp augment lagrangian step quadratic unique separate minimize whose see fig order apply argument auxiliary consensus move involve search compute positive step fast stop time form cholesky compute operator qp originally need high possible constrain qp costly solve efficiently cholesky cholesky factor solve linear cholesky factor solve linear permutation feasible relatively dense cholesky add iterative solver gradient warm start inexact update problem cholesky add much thus first slowly admm problem dense take pc subproblem large warm start outer otherwise stop fast use matlab follow implement use cholesky program qp inefficient runtime converge code qp separately implementation b r rt rt break end lemma proposition observation example example
evaluation aforementione easily full system first write fashion pick row position q multiplying give parallel orthogonal j tx q proof substitute present take expectation randomness st iterate randomize I I tr tr invertible differ essentially consideration column mention imply substitute set converge square verify upper bound try proof wrong hyperplane hyperplane project hyperplane alternate span confirm hold optimality x tx term optimality iterate randomized modification full early easy imply norm claim behaviour start span orthogonal span converge important since update never early go consistent however span component orthogonal unfortunately opposite iterate converge mathematically convergence proof carry hence convergence residual getting convergence iterate preferred reason prefer view suppose positive want pose update basically lipschitz update randomize psd treat similarly psd treat follow rule pick proportional normalize uniform distribution normalize column must norm rule exhibit ridge style randomize descent psd system update look pick contrast randomize descent xx take update lastly count evaluation kernel ridge maintain function rkhs different parameterization make ridge ridge subroutine machine application nonparametric major issue involve scale formation gram matrix style get issue never form avoid form great algorithm receive instance view stochastic easier understand perspective sgd viewpoint opposite unique extremely direct however system consistent converge preferable inconsistent prefer unfortunately preferred update exploited avoid explicitly form invert potentially randomization technique help scalability thank point class correction version manuscript lemma conjecture iterative convergence row prove linear randomized work direct relationship often stochastic examine discuss ever store form recognize encountered topic limited randomized algorithm involved work column represent dimensional one want minimize residual consistent sparse ridge extension represent coefficient regression depend stepsize coordinate update like stepsize depend randomized descent lot like perceptron know gradient descent algorithm difference descent stochastic derivation bring traditionally present manner kernel ridge perhaps subtle manner connection explicit reader build understanding first aforementione minimum assuming row use solve stand minimum return situation section understand situation deal two focus specific thorough relationship analyse setting proof direct inconsistent square solution iterate preferable hard mathematically solution explain linearly iterate
manuscript base reconstruction interesting hope drawback alternate tumor tumor copy allele frequency incorporate maximize single estimate make thousand simulated tumor population reliably sequence use correctly simulate reconstruction alone base patient tumor previously manual deep finally state art breast tumor advantage correction tumor reconstruction possible enable automate reconstruction medium depth sequencing read throughput variant read variant allele position allele reference population depend sampling allele population affect fraction cell variant position dp prior dp generate frequency pa I infer group occur furthermore nonparametric evolutionary structure rooted stick break height unique frequency multiple node constraint tree infer monte frequency non great enforce rewrite observation explicitly result use auxiliary root design frequency via child construction ensure population appear posterior distribution auxiliary variable sample frequency new copy reconstruction allele reference allele copy proportion population change available one possible absence able region site determine relationship allele frequency tumor population allele frequency half model population pseudo represent binary mutation read uncertainty frequency reference read support allow region relationship cell expect copy lie proportion read mutation copy allele allele sequence reading contain allele vice versa proportion read contain reference allele looking population un look population population copy potentially number copy evolutionary relationship infinite contribution first find population five population contain contain population occur occur population contain cn occur ii rule occur cn iv reference occur cn copy copy occur branch calculate observation cf circumstance place nearby genome occur genome easily multiple tumor regard applicable simultaneously structure share main model lie sample evolutionary satisfied tumor metropolis hasting move mcmc burn fix hasting factor package convergence complete trace autocorrelation increasingly sequence genome result sequence error different sequencing mutation define tumor allele frequency frequency introduce principled copy improve diverse population expansion reconstruct insight present population frequency increasingly whole sequence tumor automate method reliably reconstruction attempt heterogeneous base frequency solely nucleotide know read genomic explain copy variation read depth current proportion cell mutation magnitude typical preliminary evidence number decrease read size region reconstruction reconstruction need population new copy copy status population share resolve often reliably resolve unclear automated open question overlap reconstruction population impact allele overlap make mutation neither place important reconstruction genome sequencing unlike method appropriately region overlap enough five thousand previous method probe read depth absence still automate less copy overview evolve tumor result process tumor variant allele frequency inference iii show evolution tumor time grey blue tumor tumor circle reference genome mutation indicate low case letter mutation also mutation contain mutation include cell increase mutation even division define rapid expansion large acquire selective population drive indistinguishable frequency noise sequence panel tumor mutation analyze tumor copy case population present point tumor evolution tumor b tumor exist always every attempt cluster mutation identify without reconstruction overlap introduce mutation tumor prevent overfitte balance fit versus parametric iii cluster recover appropriate cluster mutation set still define tumor panel ambiguity one evolution powerful site evolutionary tumor evolution perfect persistent subtree tumor rare compare genome nearly valid incorrect reconstruction alone permit tumor principle require many actual application resolve ambiguity small therefore select maximize tumor ambiguity figure validity establish condition branching occur false either assign mutation mutation weak guarantee whenever identify handle multiple tumor n site multiple report reconstruction violate strict sample carlo mcmc posterior consistent rule sample area determine major read mapping quantification change low applicable region genome overlap occur affect region infer value negative integer infer average copy always allele upon resolve attempt high also know tumor cell tumor population attempt affect computing know independently occur affect copy number computing require know copy figure illustrate information would interpret two cause region ignore allele method change relationship allele frequency infinite associate describe automate properly account comprehensive first provide brief explanation incorporate pseudo perform illustrative permit tumor effort quantify relationship accurately recover apply simulated read next application real sample patient single tumor assume already sequence estimate two first population read imply simulated variant reference count contain evolutionary ignore incorrectly assign population infer reference count run run copy produce identical integrate tumor sequence order structure answer population count read depth per tumor read read depth number complete relationship runtime log plot core intel mcmc hasting runtime decrease implication three single intel k complete complete remove identify result show read depth recover true population population subtract first read estimate relationship characterize population increase decrease number intuitive sometimes demonstrate stick breaking eliminate ad removal cluster leave read depth experiment correct x need resolve six accurately systematic account imbalance precision recall curve matrix cluster construct co cluster matrix sample burn co average well predict co compute chosen presence imbalance plot result relationship cluster population line per infer read provide qualitative user example infer matrix co population row correspond co cluster probability tumor normal free depth time read depth apply importance incorporate compare see relationship precision curve plot result overlap variant examine consist various proportion publicly file variation bic seq find result read read collapse read two take intersection previous verification tumor achieve run bic seq output require see varied return nearly composition decide rely copy simply remove seq identify leave despite still able change composition content run infer benchmark patient extract supplementary tumor treatment equally collect simultaneously read examine mutation gene proportion variant read variant cell copy location proportion cell imply expert manual nearly exception assign child leave expert generate deep sequencing allele five analyze datum coverage tumor analyze analysis genomic status genome region affect copy run normal copy normal perform correction manual identify assign performance look panel continue
necessary formula np suffice univariate proof dependency et al base matching literature notion distribution equivalent moment match polynomial support behind program match dual equivalent likewise distinguish concave match string moment ask consider project moment hypothesis generality unit expectation distance concavity convert cumulative concentration distribution explain yield upper univariate technique upper bind moment canonical density moment moment integrate observe factor exponent concave factor moment attempt theorem depend side yield vanish vanish actually fail vanish classical polynomial dense weight kind sequence make little bernstein excellent polynomial apply prove dense immediately result dense assertion continuous approximate marginal polynomial since turn concave arbitrarily confirm conjecture even arbitrarily polynomial normalize formally exist follow roughly say derivative sign try lemma markov survey exist exist polynomial force therefore idea let must case enough yield show value polynomial approximate impossible product density specify polynomial approximate arbitrarily polynomial univariate polynomial latter let w exponential law uniformly specify get dominate factor grow law dominate generalization power dominate question tail bound distribution reduce namely discrete amount binomial truncate apply weighted approximation know origin degree grow imply must introduction find follow program formulation question fourier character tail tail program program upper shift univariate question polynomial upper appropriate optimal capture prove barrier may weight binomial distribution upon distribution degree key move polynomial evenly space sufficiently degree rearrange anti well entropy eq proof eq integer interval whole interval complete infinite determine interval origin near origin th chebyshev kind property rescale prove formulation enable sake q applying give q contradiction learnable distribution rule task polynomial nothing work show combination hope non polynomial obtain boolean work least hypercube polynomial suitable concentration inequality focus distribution et give sophisticated generator nearly seed thank comment know completeness lemma immediately let require start let wish bind moment pick term without increase change odd rearrange prove moment bound tail theorem uniform degree substitute r question theorem conjecture polynomial approximation generator limit technique prove sign agnostic model fact concave approximated polynomial ask distribution show polynomial concave real strong limitation secondly chernoff chernoff sum schmidt et establish variable independence tight factor study well various distribution area classical area approximate simple computer science application capture quantum query algorithm polynomial approximation measure agnostic polynomial et polynomial distance show idea yield bound computer science establish al tight characterization amount wise chernoff fx otherwise back building include pac perceptron difficult problem agnostic concept agnostic even restricted class uniform hypercube sphere information theoretic hardness result learner hypothesis np hard moreover arbitrary pac open problem high dimensional form linear compute program subsequent certain approximate learnable approximation much hope circumstance distributional assumption technique address approximation exist namely absolutely distribution laplace distribution threshold arbitrarily possibility classic approach reference therein give threshold give log good exist extend coordinate establishing various study restriction learner capture besides gaussian elimination regression fact agnostic hypercube limitation agnostic algorithm different hypercube time leave determine learn fix constructive seed give short concentration fundamental objective replicate study question namely linear generator generator seed use suffice wise r small new seed work whether tight strong tail independent independence hoeffde like independent denote show essentially previous due low support independent theory good knowledge indirect imply existence wise independent tail bound idea force wise linear maximize kn
pass pr source match additional source resolution improve matching source source imply movie likely movie informative netflix greedy similar pass close similarity movie fundamentally similar greedy design follow real datum already experiment next weight output weight match calculate pair movie movie movie netflix high comparative multi six source bipartite display message greedy weight approach six source source pass get weight low suggest matching movie match far maximum matching far publication show match vs compare improve create noise vary primary greedy truly competitive message pass movie represent generate represent dataset point marker movie noisy experience approach message pass operate far greedy finally perform equally value along approach vary message dash gap figure even gap small gap severe least message perform message pass pass depend need converge examine empirically efficiency message pass approach entitie message total change increment candidate threshold figure iteration message threshold message graph entity pass approach converge message cc total conduct empirically compare efficiency message pass much slow pass source increase computer intel gb fix number per fix number greedy message pass increase acceptable hour movie source around increase increase integration pass greedy motivated source ratio latter conduct movie recall leverage message slow message direction area formation connection surrogate precision zhang ad team twitter microsoft facebook ph dr publish numerous retrieval database pc major computer science berkeley database security major area work microsoft google yahoo research production system entity degree dr world life author wide area life video database microsoft product include false definition usa mail twitter com zhang principled explore optimization graph arise entity resolution integration perform structure typically proceed record block record match similarity often match member statistical record linkage bipartite appeal natural global improvement unfortunately bipartite max matching inference algorithm theoretical latter world literature publication result quantify exploit matching discover complementary recognize explicitly entity replicate entity would copy site rely netflix rating split multiple product business page amazon page maintain uniqueness database record linkage community er system employ leverage natural lack initially benefit take significant quantify reason score poor er however impose one one local kind bipartite matching combine community widely applicable integration source little multi er community np community maximization successfully er requirement pass combinatorial optimization approximation lead statistical record linkage extend greedy bipartite sophisticated passing enjoy easily bad competitive ability leverage perform economic service drive customer world services state unconstraine experimental recent cite magnitude measure benchmark typical paper crowd conduct publication generality vary main contribution principled factor pass constrain greedy approach sharp bad example sequential matching sufficient matching demonstrate precision second real world publication datum message enjoy superior discuss paper source entity section setup entity record linkage investigate resolution web negative linkage approach base uniqueness attribute value many constraint entity actor name attribute census record however resolution use entity systematic record linkage source match source instance exponential prevent principled pass well entity resolution problem million entity approach pass principled tractable bipartite maximum solve polynomial come weight extremely special imply fast run principled weight presence endowed competitive entity pass bipartite graph prove desirable finding bipartite loop design message match bipartite graph differ max programming weight tight compare pay attention application entity tune experiment finite number size score entity etc database appear mapping involve source represent mapping entity source entity together one tuple show experimentally leverage one source yield recall linkage match netflix movie netflix focus exploit global particular global source simultaneously source individually equivalently real naturally due argue significant netflix crowd site copy site rely suffer recommendation make attribute alternate past quantify raw largely publication source rather source perform source match heterogeneous characteristic individually variability source overall source sequential entity previously sequential resolve poor poor propagate possibly fashion reduce global demonstrate weight source true b b c truly bipartite preserving achieve global local optima quality contrast look ahead develop strategy approximately maximize co iteratively match entity locally greedy discuss exact np yet principled loss generality illustrative max algorithm reduce include message eq minimizer combination combination observation replace similarly derive source show message keep update iteration reach optimizer similarity reduce gibbs optimizer optimize entity one entity matter optimizer keep update optimum optimum always computation need converge always getting begin optimum entity entity choose combination among optimum among output round less max follow formula configuration final message therefore q employ omit general entity need least message decompose source optimum candidate require time sort candidate favorable pass message leave explore natural multi pair discard order far pair match resolution already entitie clique examine entity clique entity singleton entity derive different clique merge clique merge clique include resolution clique merge clique would sort selection extremely implement max weight generalize competitive greedy duality weight matching least max matching primal note enforce ensure value lagrangian introduce bring thereby form unconstrained optimization ij uv uv appropriately maximize lagrangian g ij uv variable constraint lp original dropping dual match weight lp lp greedy duality feasible primal demonstrate fact sharp max behave practice usually achieve much well experimental movie aim application publication add data synthetic stress multi main data movie meta movie vertical note order production entity source entity source netflix attribute netflix comprehensive list strict one thousand movie netflix score exact normalize discount well among release year release year cast count cast member five name cast match divide short feature score tf perform inexact understanding matter focus entity accuracy regularize score score pair train logistic model evaluate entity matching label truth movie source hundred movie source ask human matching exist match movie assign matching movie source pair share protocol publication source detailed title author publication pass pair
learn dictionary polynomial store centralized task acknowledgement van de provide quadratic objective function set particular row row stack vector column objective constraint express affine inequality vector definition desirable ability specific implementation dictionary signal challenge incorporate intrinsic datum structured particular signal combination overlap graph pattern dataset dictionary learn competitive learning algorithm dictionary localize manner processing task compression classification dictionary laplacian suitable modeling structure live domain signal network simple example sensor traffic city illustrative interested meaningful representation capture signal weight graph overcomplete class combination atom additional challenge design dictionary graph geometric characteristic euclidean domain dictionary often adapt implementation dictionary direction mod therein realization signal given learn costly apply processing task dictionary wavelet transform overview structure dictionary realization represent accuracy numerically train impose structure dictionary structure generally desirable implementation list reference generally day b day c day benefit analytical dictionary incorporate signal combination pattern describe localize evolution similar incorporate underlie graph encode atom concatenation parametric adapt signal representation graph graph necessary understand learn describe synthetic real signal discuss overcomplete dictionary past restrict design signal approach mod signal learn neither fast structure meanwhile dictionary signal overview reference signal transform wavelet wavelet sample frame vertex feature pre implement generally hand two diffusion wavelet try bridge transform numerical dictionary learning algorithm propose learn structured dictionary graph topology close necessarily lead take consideration explicitly dictionary author laplacian code smoothly along dictionary none able provide class exactly compose laplacian overview definition graph find vertex edge weight laplacian diagonal degree define throughout eigenvector avoid large power laplacian matrix nonnegative signal function vertex characteristic eigenvector signal live notion laplacian fourier vertex frequency inverse transform besides harmonic transform translation convolution center vertex allow interpret operator act spectral localization around vertex smoothness localize around translate localize atom diagonal note power localize topology atom center graph learn dictionary signal combination overlap pattern learn dictionary capable use definition translation learn generate form main directly support detail dictionary pattern translate give vertex localization atom representation spectral impose semi upper signal consideration component cover impose eq constant particular prior behavior prior incorporate problem modify frequency certain spectrum choose flexibility derive dictionary c ny generalization leave schwarz combine would tight atom learn summarize parametric graph represent signal atom equivalent cast signal objective ii stability optimization solve alternate code fix parameter q orthogonal pursuit dictionary omp dictionary omp atom dictionary sparse code remain method pursuit soft thresholding fix coefficient dictionary parameter step dictionary learning dictionary converge local dominate computation laplacian enforce constraint line split well example interior method htb target sparse attribute kernel learn performance svd depend size blind unable pattern particular significantly slightly training signal dictionary show much stable well atom neighborhood tend poor contain localize signal respect svd fig localize area course signal translate pattern training dictionary translate version learn dictionary instance translate pattern appear intermediate solution svd form rather continuous evaluate contain pattern graph since necessarily complex implementation discuss structured term generate c study signal polynomial even though signal preserve true fig combination atom different level omp graph polynomial noiseless testing scenario performance polynomial divide spectrum four band concentrate four particular atom band generate uniformly entry index band zero atom vertex localize randomly atom generate match generate spectral band generate fig kernel localize notice approximate band generate dictionary similarly behavior fig atom atom topology smoother particular fig atom spectral kernel atom concentrated frequency laplacian associate smooth generate exactly performance improve polynomial attribute atom localize neighborhood reason explain generating dictionary flexibility smooth generating achieve nonetheless learn dictionary efficient observe dictionary learn svd graph structure generate polynomial behavior entire concentrated band generate two band construct generate atom atom accord signal linearly order polynomial learn support frequency band illustrate exist training since concentrated part generate signal support particular spectrum eigenvalue examine synthetic localize graph world take represent graph construct assign two location distance shorter threshold daily california data dataset throughout major area california connect distance euclidean gps weights proportional bottleneck persistent drop length bottleneck active maximum learn use testing normalize respect energy fmri acquire five brain contiguous subject measure state completely movie treat brain euclidean coordinate centroid determine short edge dictionary atom use dictionary signal validate learn normalize norm dataset synthetic section adapt clearly atom dictionary signal six omp decomposition brain dictionary learn polynomial note polynomial dictionary consist localize poor localization clearly ability localize
segment rating segment netflix dataset million movie integer ordinal log user mf vector ask whether movie typical per figure indicate quality possible enough store main handle exchangeable cause subtle likewise similar mapping rank respect focus produce individual contribution inference explore exponential space demonstrate empirical suggest art collaborative university department edu ranking arise group item movie properly subset combinatorial approach procedure explore discover variable large collaborative filter considerable machine community rank datum cast generate document arrange decrease relevance compatible object object likely group contain complementary movie beneficial recommend compatibility group pose group subset situation need stock store rank rating example quality package rating somewhat relate learning produce label input inherently subset introduce situation pattern object partition scheme partition partitioning order former exploration metropolis hasting iteratively possible way partition order consecutive two consecutive merged propose term order latent fashion machine rbms posterior hide visible use collaborative filtering g group rank list unseen public dataset competitive art rest section together main introduce extend collaborative filtering review follow conclusion merge middle b conversely split represent preference order causality modelling depict group subset direction look impose distribution difficulty partition thus al careful partitioning allow care property group order impose capture relation compatible order linear potential account parameterization allow flexible possible take split position subset remain unchanged reverse e appropriate guarantee entire armed sampling procedure stochastic give object denote rank belong furthermore collection object partition subset usual partitioning among subset denote notation write order subset wherein element grouping complete govern recall divide partition pair perform partitioning consider give know super fast log size standard permutation encode compatibility among encode property potential effect relative hereafter refer propose mcmc inference evaluate mh mh proposal sample move accept define proposal ratio intuition local partition cost change walk slowly move singleton split take subset guarantee possible configuration illustration singleton distinct chance drawing subset merge back ratio compute potential depend order rank operator merge subset merge subset recover merge operator ratio metropolis hasting present initial l sub iii acceptance probability iv accept move consecutive subset merge keep subset unchanged evaluate acceptance eqs accept procedure introduce temperature uniform anneal repeat computed z configuration linear f ax bx ax need unfortunately inherently chain follow iteratively e b far introduce latent serve purpose collaborative choose ranking reflect discover partition g cluster datum conjunction activate thus boltzmann potential admit x capture relative hidden model figure hereafter refer posterior indeed shorthand ph ph representation configuration generation involved need explore order alternate straightforward remain px potential jx kx eqs product product xx ix x j except rao straightforward alternating usual simplicity similar modify assume share specific ax bx ax bx statistics statistic ax equip gradient trick parameter respect application application preference item rating star item user often rating thousand million create sparse discover latent user factor individual limited partitioning order unseen item rank want reconstruct complete ph ph rank let approximation j jx resemble ph due rank task think intractable approximate treating completion fast simple assign worth grouping factor contribution item compatibility first compatible worth order item
section volume direction chance practice visual process volume double check euler surface triangle close surface sphere adjacent triangle share face total number characteristic check mesh satisfy binary volume surface exception topology heat construction lb sphere find compare iterate diffusion surface study representative z treat measurement surface smoothed comparison smoothing root rmse error surface heat eigenfunction correspondingly bandwidth effective sufficiently size iterate reach figure b square iterated kernel mainly localize iterated kernel heat kernel comparison limitation iterate heat kernel smoothing converge heat diffusion heat give iterate diffusion smoothing heat gave sufficiently iterate give original one visualization actual replace coordinate kernel discretization error converge heat discretization converge increase smoothing solve discretization smoothing heat bandwidth know image perform small snr shape surface almost region surface differently signal take ground variance mesh vertex black group variance measurement mesh vertex add region measurement group detect region heat sensitive I simulate substantially snr iteration necessary smooth empirically heat eigenfunction determine performance test threshold detect iterate heat addition heat diffusion incorrectly visually figure discretization scheme approximation step higher simulate functional substantially group figure smoothing bandwidth heat eigenfunction detect however heat smoothing diffusion smoothing sensitive snr raw iterate smoothing due heat negligible region minimal well substantially snr diffusion iterate perform well analyze growth ct human divide group iii main biological localize growth acquisition previous surface alignment surface surface subject affine f template map construct choose old identify template template remain template remove subject remove difference surface perform template metric framework metric construct transformation differential ode vector constrain sufficiently integrable generate lie field reproduce satisfy template connect define application employ approach template template simply initial subject template template template individual subject initial template ii ii row growth direction mean difference color top row significant bottom row group level correct bottom iterate use growth ii iii corrected determine significance length much easy length assume smoothing measurement make smooth heat eigenfunction less heat response testing group ii show degree freedom comparison iii statistic map black finding finding simultaneous growth also perform iterated diffusion result diffusion size bandwidth split smoothing figure iterate snr perform mesh vertex kernel diffusion number significant present novel heat regression framework analytically weight laplace weight expansion relate isotropic heat diffusion validate discretization regression parametric establish equivalence diffusion wavelet growth growth identify quantify first decade life overall growth small currently ct grant dc ct study develop institute grant center child health development clinical award science associate thank university wang comment edu present scalar eigenfunction heat formulate new bivariate expansion heat kernel isotropic heat wavelet validate characterize localize surface ct surface template kernel laplace eigenfunction diffusion medical surface represent triangular surface process likely noise mesh widely reduction technique surface heat surface brain subsequent involve random isotropic isotropic iterate widely solve surface surface smoothing brain smoothing spatially heat discrete tangent manifold heat linearly heat bandwidth process heat analytically eigenfunction lb avoiding use although introduce heat vision heat kernel descriptor manifold heat machine however heat never framework machine kernel wavelet surface wavelet map sphere local wavelet however wavelet surface serious metric subsequent less parsimonious surface intrinsic lb expansion spherical transform graph primary unified wavelet coherent mathematical define manifold apparent provide theoretical extend heat kernel surface diffusion wavelet explore transform mathematical equivalence explain growth surface identify show significant localize growth surface snr sensitivity surface continuous fashion assume unknown signal estimate square integrable area image eeg filter boost surface filter isotropic form laplace diffusion control amount green cauchy follow dirac differential initial isotropic various numerical smoothing need discretize discretized fp mesh triangle share neighboring mesh angle opposite contain adjacent otherwise triangle diagonal matrix adjacent diagonal ordinary discretize estimate laplacian row euler scheme need iteratively weight heat surface mesh heat break iterate small expansion heat bandwidth concentrate vertex sufficient iteration numerical use eq neighbor truncate localization circular angle north outside band laplace operator spherical laplacian spherical degree order use spherical harmonic expansion spherical least sphere use heat bandwidth harmonic expansion severe localization wavelet phenomenon close eigenfunction laplace surface numerically solve among many medical formulation global individual implicitly consume amount memory surface thresholded visualization eigenfunction numerically parameter estimation basis minimize residual q square method coefficient harmonic mesh mesh eigenfunction condition numerical mapping template surface use study statistic mesh vertex comparison heat datum smoother enhance snr integrate inference level signal normalize fashion bandwidth I e previously smoothness quantity along mesh surface bias often smoothed sufficiently small motivation develop heat interested determine significance note consider continuously index underlie level heat kernel subsequently often comparison correct degree manifold functional euler characteristic ec pf cumulative freedom second bandwidth incorporate propose ct model
evidence toy test equally spaced measure toy htbp reconstruction handwritten goal lie center split example digit dataset root average scale lie color give outer test image tune unimodal bag context training solution impractical large human dataset apply neighbor software motion capture frame average mod true pose view consist pose camera use half frame test pose marker coordinate frame pose pose pose vector theoretically affect affect step optimizer step prove change power theoretically motivate role cross converge dataset suggest purpose role purpose cross cross range practice cover set initialize prediction regard table parameter share choose justify result sm sm l notice toy improvement toy figure tuning perform well pose report give toy refer toy toy conclusion outperform argue b htbp sec sec sec another toy example slightly perform least dataset bias towards input however learn towards output justify sm powerful divergence optimize however member tune conclude result report knn knn knn five comparison toy dataset knn neighborhood see compare outperform knn knn knn propose structured sm divergence analysis understand sm part argue could finding kl understand perspective cover analyze instead cover sm base maximize correlation test highlight computationally sm complexity reduce complexity equivalent sm major contribution practically achieve structured tuning sm perform task experimentally observe generalization experiment outperform toy example dataset tune validation would indicate hour dataset hour grid significantly decrease time enough validation like save instead cross validation experiment prediction name depend interesting future measure efficient form sm divergence main section number operation compute sm operation operation practically achieve structure perform intensive experimentally adopt work propose divergence report new cubic time simplification straightforward gradient prediction theoretical parameter generalize name propose yet computation quadratic cubic structured causality extensive validate finding pose two nsf award appendix relate etc b sm number calculation calculus follow q invertible matrix multiplication hard final dd simple expression sm gaussians ignore multiply e xy xx k yy xy xy factor sm gaussian lemma cholesky operation ignore operation ignore cubic contrast require decomposition compute multiplication need appendix require could efficiently sm accordingly time sm sm write x xt dt xt dt xt dt proof analysis function start denote xt notation sm equation px py px py z compare xt k k indicate proportional k theoretically title journal volume page number year book title page year proposition example ex ex minus nj usa computer present generalize divergence divergence mutual measure enyi leibler entropy divergence kl insight divergence experimentally framework result offer big since lot probabilistic shannon powerful mathematically development communication lot physics science reliable divergence kullback leibler divergence use machine texture negative pose lot connect turn view measure information r expectation jensen gap equivalent uncertainty lot relationship alpha divergence investigate consider machine entropy sm later entropy parameter converge shannon entropy al suggest sm equilibrium sm harmonic similarly sm mutual enyi kl domain sm close form sm motivated set utilize sm kl particular sm process structure structured gaussian divergence sm sm divergence study context probabilistic specifically presentation community generalized base sm divergence subsection simplification divergence variate cost evaluation subsection theoretical sm subsection experimental sm toy example rest sm multivariate gaussians gradient present theoretical perspective discuss simplification sm framework sm could associate subsection sm toy thank understand concern review perhaps look kl sm community address valuable prediction similarly start adopt share regression perspective cover miss theory analyze kl cost cover kl claim theoretical sec material write line line paper sec proof b valuable also present computationally derive require determinant require matrix computation simplification sm function computation cubic however simplification sec straightforward new illustrate agree comment complexity extensively evaluate various method cover proof show correlation test argue another entropy sm generalization rd ref sm reference refer sm distribution sm divergence sm divergence enyi kl sm enyi divergence enyi divergence sm originally recently alpha enyi relate divergence boltzmann shannon information sm divergence reason generalize suitable structured possible consideration work entropy sm divergence close expression main entropy affect structured analogy motivate study physic enyi entropies generalize extensive present non extensive linear enyi interpret quasi arithmetic sm generalize linear mean enyi limit trade quadratic expression introduce lead sm gaussians write vanish form expression apply sm expression divergence new cost xy xx yy xy x multiplicative ignore ignore contrast py px determinant stability minimize factor depend constant yy xy xy xy xy xy quadratic improved closed form sm cubic complexity gradient complexity expression context compute cholesky decomposition decrease significantly quadratic cubic expression time fast need operation need proof determinant efficiently sm conclude equation identity multiplication ignore need time need less need compute appendix indicate sm speak prediction unknown discuss detailed subsection property subsection interpretation subsection size marginalization extension marginalization term matrix determinant distribution elliptical eigen eigen hence determinant notion volume elliptical orient eigen one could interpret scale look closely decrease new close point eigen eigen value term maximize small I e produce could think make input however discuss uncertainty extension subsection detail sm straightforward think space equation kernel lemma xt xt dt sm sm compare since xt dt xt x p xt yx prediction xt dt k x maximize k px x py x xt px py px claim achieve x xt k py p xt px
variable lag z solution order autoregressive multivariate time similarity correlation time assume derive solve approximate marginal transform sample linear piece wise fit piece wise case piece divide domain interval coefficient pa k I doubly bivariate order moment moment get approximate piece transform invertible give piece wise invertible monotonicity cdf thus piece wise process accuracy call naive z z jx iterative naive r difference r j r h x iterative require require solution start coefficient close desire case transform bivariate increment gaussian correlation amplitude moreover guarantee monotonicity monotonicity interval large threshold close safe practical purpose monotonicity check interval number search accuracy closed expression know feasible marginal bivariate feasible maximum feasible correlation deviation matrix difficult condition feasibility procedure check feasibility feasible solution correspond correlation semi definite gaussian checking gaussian positive lag modify slightly positive correlation comprise correlation change eigenvalue identical differ solution structure eigenvalue slightly positive value component second repeat entry replace entry back cause definite repeat work step coefficient var equation var express uncorrelated determine stationarity root reverse lie unit plane equivalently modulus positive stationary process possess typical piece approximation former marginal possibly correlation piece marginal display thick clearly due correlation var marginal bias pass correlation monotonic occur sample differ theoretical correlation monotonic marginal transform match example make var monotonic vector match even show monotonic transform var black line grey realization black line denote fisher confidence row cubic thick report computational efficiency single correlation non correlation correction derive close marginal density monotonic three pair uniform legend indicate correlation relative match evenly around mostly concentrate large somewhat rmse matching attain approximate succeed slight output figure fast pair three correspond line power relatively bar standard shown respectively indicate generate continuous transform correlation respective e skew number purpose converge reach approximation marginal always obtain unless e time moreover make doubly could expression time iterative transform demonstrate piece integration investigate insight close practical derivation need consume dimensional numerical obtain proper matrix definite directly correlation contain introduce iterative turn match marginal process auto correlation lag serie moderately say auto cross however study system system var practical generate proper time use randomization marginal transform encounter matrix eventually extreme positively skew strong time series randomization nonlinearity truly comparison multivariate nonlinear dynamical system future generation realization multivariate extension univariate process transform autocorrelation piece autocorrelation transform determine vector autoregressive demonstrate marginal autocorrelation gaussian randomization u series consider fit generation marginal simulation randomization particularly model dependency among constitute multivariate give marginal internet temperature randomization testing dependencie nan nonlinear surrogate test distribution compute series preserve distribution surrogate nonlinearity field investigation dynamic linear structure surrogate series arbitrary call autoregressive modify series rely numerically double product two marginal computation simplify approximate marginal pareto surrogate nonlinearity approach developed match marginal randomization marginal transform refined transform autoregressive process call transformed form autocorrelation autocorrelation use numerical parametric originally piece series simulate multivariate briefly univariate multivariate simulation start univariate autocorrelation lag equivalently spectrum problem q sufficiently spectrum frequency without series solution solution transform transform eq variable standard density cdf objective random though constrain realization test surrogate nonlinearity another onto generate amplitude iterate realization attempt identify generate give realization two approach decompose step marginal
foreground incorrectly predict foreground background speed get intuition incorrect encourage accuracy stop policy prediction obtain evaluate compare ap alg let exclude location since evaluate part method evaluate class vary ap decrease incorrect grow evaluate part due report positive speed require mistake policy evaluation experiment table ap grid ratio significant accuracy negligible versus log versus training car cat person tv ap car cat person tv ap ap car cat tv cascade speedup ap ap car cat person tv speedup cache pca cache full cache full cache pe cache cache pe cascade versus baseline cascade cascade ap part relative speedup experiment dataset publicly cascade process label additional part decrease visually score agree intuition correct location terminate part posterior reflect score ap demonstrate reduce irrespective feature sec show negligible recall via ap second cascade feature location location projection low discard filter fair adopt similar selection policy schedule filter pass select filter make slow summarize discrepancy cascade second second individual stage filter evaluation full significantly slow combine fast cascade slow dimensional high blue bottom visualization part pixel color color leave car heavily selection accuracy unlike pre specify threshold schedule obtain potential include image position detect simultaneously assumption problem edu object optimize describe response formalize schedule classification look base cascade detection optimize negligible model powerful appearance accuracy demand cascade branch scheme response location future introduce art pyramid part decision confidence approach optimize apply part part quantifie false mistake threshold utilize idea object name part phase learn scheduling inference policy image image probability update sequentially response suggest terminate evaluated part contain original score proceed foreground maintain location sequentially response low learn filter round color policy stop location confidence foreground evaluate versus approach art evaluation time cascade make detection filter evaluation use detector achieve without additive use sift point refer detection optimize representation identical inspire acceleration detector cascade however evaluate cascade select next maintain foreground ensemble classifier stopping optimize branch bind bb search location easily search test yet give location early bb constrain slide window transform sequence object test orientation minimize total coarse classic cascade adaboost study introduce cross cascade response classifier exploit prediction cascade optimize pose refinement cascade structure pose resolution filtering pose state emphasis structure scoring pose recover general feed maximize use close filter evaluation accuracy optimize filter cascade still part e parametric representation pdfs annotate emphasize hold simplify likelihood conditional part stop assumption expect learn joint pdfs fidelity indicate g car cat person train tv example place score bounding filter record place score positive obtain negative example pyramid collection smooth fig likelihood discuss order detection proceed round apply root part topological ordering next part past take location ts zero everywhere part uninformative independence score bayes seek choose run plan sp repetition admissible termination formalize make error hidden stopping error label small infeasible relax lagrange multipli lagrange interpret incorrect elaborate cost choose incorrectly cost incorrectly introduce mistake solve use compute proceed part apply force terminate bin term stop choose alg summarize step sec bins store
function spherical j mm lee normal skew discriminant pattern recognition york york independence distribution normal development foundation www mm skew instrumental gaussian direct unobserved b discriminant function pt pt theorem question discriminant mm mm normality time practical situation index modelling consequently discrimination method skew elliptical study skew normal quadratic family simulation word skew elliptical skew normal unobserved variable goal rule describe van several researcher normality study flexible discriminant discriminant normality quadratic discover skew linear discriminant discriminant robust estimate generalised rule van adjust skewness skew distribution skew elliptical lee al linear regression longitudinal method sense al recently perturb distribution ml interesting application skew elliptical distribution case classify give skewness skew classification generalise extended skew elliptical see skew skew normal accommodate skewness heavy tail model mle place class unify skew elliptical distribution start extend skew elliptical section explore extended discriminant obtain find know proper selection pdf al x x consider mechanism disjoint classify belong fall actually assign misclassification group assign classify decision classify know assigning equivalent classification q assign large extension rule elliptical distribution elliptical elliptical distribution al elliptical introduce skewness perturb screen unity specifically elliptical variate generator word elliptical h h k fx h q univariate induced generator write random skew elliptical se denote two distribution hence group conclude set equivalent large selection classify se elliptical otherwise generator convenient popular correspond normal multivariate normal scale elliptical density generator extension skew al variant al dimensional location shape variate cumulative unlike linear rule ax assign minimize linear rule normal classification whenever multivariate skew mahalanobis distance two normal index variate population namely complete depend assume rule population u eq jointly yield cn ab replace consider propose assign cn maximum discriminant em well complex comprehensive know ht iteration assume miss k ij ik eq ml depend th depend propose work parameter correspond location scale al conclude likelihood appear satisfactory simultaneously remove discussion case classify discriminant simulate discriminant accord carlo consider proceed auxiliary representation multivariate em obtain discriminant rule indicator parameter generate individual use ht ml bias ht allocate total group simulate development team cc indicate fact overall accuracy classification value versus data classification derive skew classical property distribution focuses skew potential skew elliptical well research reference mm pt mm normal j mm r g unified selection mm b shannon mutual multivariate skew distribution extend skew unified skew elliptical skew normal family skew related york multivariate skew mm
help day c c test arm arm boost c upper body detector order width detection window scale factor train scale detection window percentage correct pose tool calculate correct body correct ground truth present low head part set achieve part get bad result whole mis subset predict bound box achieve arm run code full per denote get training contain increase cost day need model compare arm factor ground predict position th show figure accuracy large accuracy bad strict suggest estimate pose exact location full improvement rl effect multi weight table poorly weight detection significant increase task low error gradient dominate case greatly guide enhance generalization detector datum feature among seem feature detection test convolutional operate reflect neuron sensitive expect detector rd mid find neuron instead neuron middle input neuron receive backtrack filter feature neuron map local one optimization patch contribute test rd body detector maximal activation map occur visualize nd rd convolutional head mid feature detector head arm fig localize body right level body fig c two band horizontal window frame could useful identify context location filter cc b cc paper heterogeneous task pose consist task pose slide jointly learn generalize testing visualize mid maximally find neuron selective pattern localize pose combine unsupervised pre would like extend video object china acknowledgement city li edu liu media city computer science city university edu pose neural regressor slide window part architecture good empirically localize body pose computer vision application video retrieval pose depth format mobile device equip camera estimation general classify two part method regression part graphical structure find configuration match pose estimation globally expensive two definition part part avoid orientation part appearance capable capturing complicated slide window pose hand allow multimodal rapidly pose view regression pose good information currently approach training calculate prediction expensive deep success computer task network popular computer vision fully model convolution large capacity hard train network generalize task pose pose window detector heterogeneous detection train benefit greatly local activation selective localize multi training task review regression relate heterogeneous encouraging regression share joint find feature pattern force heterogeneous task good convolutional scene labeling define category classification define slide window window contain detection task window train sensitive neighbourhood neighbor feature pose location regression stage predict prior region increase refinement improve performance network task could length convert stick annotation indicate absence window eq body portion upper annotation indicator map body several window minimize truth probability combination detection part image weight base consideration share task motivate learn task helpful second sharing parameter generalize body part predict position translation position preserved context sometimes difficult bound box part long arm lower distinguish look include neighboring part help detector part detector box image human network rgb task layer pool activation information consist weight share neuron previous pool linearity integrate neuron convolutional layer regression layer neuron neuron receive neuron activation neuron relu show task train jointly train regression global back propagation update image task calculate gradient
call formulae kt represent number know empirical characterization natural invariant parameter always alternative type adequate currently describe limit deviation efficiency analyze asymptotic new regard theory calculation test kolmogorov nan hypothesis therefore applicable coincide supplement research power efficiency integral asymptotically degenerate statistic degenerate calculate nan x x k px ps side px remain calculate term j x summing follow calculation degenerate degenerate statistic considerably simple calculation family test describe attain level exact follow hold p always leibler nan quantity alternative kernel non kernel see asymptotic large nan hypothesis deviation degenerate see state moreover accord law also hx present alternative eq gamma eq negative see alternative exact leibl distance similar omit q variance q regard asymptotic analytic moreover eq slope sequence state satisfie alternative table use alternative well alternative gamma power simulation replicate htbp alternative kolmogorov type projection f distribution variance complicate case equal elementary besides limit use gaussian find simulation type obtain c kernel sense large supremum family degenerate result equal deduce hx gx get tf se te e satisfie calculation therefore considerably exception probably relate kolmogorov statistic get te e figure degenerate describe use case impossible find limit statistic family supremum degenerate sufficiently moreover alternative alternative projection section te e te te slope see previous table power four alternative htbp gamma maximal nevertheless favorable sequence sense describe local efficiency relation hold give density hold easy consequently local bi e schwarz constant constitute simple density alternative gx e kolmogorov hold schwarz inequality constant alternative class facilitate presentation e gx gx x two family integral many favorable asymptotically power test closely order correspondence criterion recommend try statistical test test local efficiency turn rather common alternative power optimistic change regard probably closely intrinsic however virtue especially favorable deep send
behavior sequential involve hide generic mapping condition relaxation follow two condition algorithm associate automatically attain prediction write likewise second monotonically admissible relaxation relaxation enjoy bind claim version enjoy paper admissible forecast relaxation enjoy regret offset rademacher schema derive non eq prediction derive schema enjoy eq estimator derive simple admissible relaxation enjoy problem online notion alternatively play standard know schema design admissible enjoy regret study work value minimax hence expression fashion back minimax supremum supremum denote since linearity expectation write eq arrive supremum prof first expectation denote jensen eq upper random conditioning ensure proceed proved except end worst along exist easy leave subtree contradiction depth tree sign choose splitting block last stay close need require mean q low may change examine illustrate let choose delta optimal clearly view discussion initial trivially check note check expand loss rearrange eq jensen eq supremum admissible far ty enjoy exactly notice closely notion derive since offset relaxation derive relaxation initial condition hence apply condition expert rademacher soft arrive inside hence give obtain round write eq hence dependence tree may rewritten conjugacy relaxation relaxation view inequality arise use conjugacy relaxation ty relaxation square notice forecaster final bounded online regression introduce optimal transition analogous frequently sequential match rate generic forecaster establish design computationally derive exist expert online arrive stream forecast square datum leverage law learn aim develop prediction formulate regret family notably upper require past progress include study rich nonparametric class function regression partly motivated remark appear obtain approach forecasting aggregate exponential interestingly respective property view online since algorithmic main algorithmic aggregating mention aggregate beyond supremum underlying remark arise pac bound day require notably long recognize cover potentially entropy growth empirical characterize loss cover combinatorial rademacher regret correct behavior minimax logarithmic loss partition critical radius minimax employ ball aggregate integral main difficulty precisely sequential minimax decay low logarithmic real mild behave noticed regret completely many situation relaxation framework provide develop characterize rate relaxation admissible constructive generally feasible relaxation dimensional make slightly regret notion suppose converse minimax regret extract get handle minimax regret inside round describe range range minimax notion mention scenario thank online scenario subtle binary entity capture end definition depth complete root binary root label child th level level word cover one complexity tree form function sense polynomially parametric behave necessarily main technical statement normalize constant sequential cnn cn parametric logarithmic uniformly growth uniformly modify entropy growth cover priori additionally optimistic cn np regret bound optimistic slow obtain rate subgaussian tail upper factor state play style control parametric coincide cover evaluate growth scale choose extra yield particular finite use small normalize rate lower optimistic bind obtain entropy infimum evaluate extra infimum evaluate take optimistic follow recall rademacher eq rademacher minimax fail rate regression give complexity extra fluctuation complexity give critical parametric irrelevant g purely curvature loss help square phenomenon occur risk observe enough sided concentration statement rademacher complexity side minimax complexity minimax response range tree range tree complexity upper bound entropy analogue bind rademacher crucially choose scale value depth optimistic offset arise depth argument lemma value eq also compare approach compact subset
decompose pc ij w ij de rapidly increase also confirm error pc pc reality work see fig issue pc w cross element trial error choose asymptotically adjust bias acknowledgment helpful domain representation vector search matrix minimize datum regard embed multi include multivariate canonical vision paper propose code domain single version augment domain cross discuss illustrative domain get image datum vector image typically hundred like retrieve alternatively retrieve image strength theory association association image unlabele color classify remain match true association otherwise ij project single transformation transpose trace diagonal block domain eq minimize weight supervise let unobserved weight transform look common perform across domain formulation matching graph multi domain similar popular recently vision formulation multivariate connect code indicator call canonical discriminant correspondence cca let paper code mention similar code vector domain vector domain represent get solution embed graph embed minimization classical model illustrative example parameter reduce matrix easily compute say cholesky us error subject minimization work regularization properly data domain vector ed matrix objective domain solve spectral graph express consist simply graph would match final code element diagonal nonzero code weight maximization kind pca code idea find whole vector input e association cross matching reduce assume matrix specify coefficient cross matching connection regularize show case array de simplicity correspond matrix factor simple generation repeatedly define element generate vector standardize variance grid
network ignore machine support generalization I train consequence due numerous example towards address dependency form study machine effort general relaxed elegant classic concentration example improve early weight well naive ignore illustrate law classic structure example learn derive variance concept theory several weighting section task improve conclude summary contribution future work introduce intuition however share get sharing make introduce formalize problem fundamental learning hypergraph hypergraph set zero hypergraph vertex hypergraph vertex group number abuse example tuple tuple special case example member network contain member share binary relationship object friend network prediction vertex two type hypergraph hypergraph partition disjoint vertex example tuple movie rating person illustrate setup denote object suitable usual supervised label member contain set may wikipedia want active target whether friend feature interest education friend movie rating movie actor etc movie describe contain amongst gender rating give movie hypergraph hypergraph label hypergraph tuple hypergraph vertex alphabet alphabet label labeling labeling object assign value form assumption dependence strong perfectly make I assumption still believe several way explicitly dependencie dependency detail work dependency information hypergraph label draw assign vertex target value sample possibly identical choose training vertex draw assumption long possible training test movie example may may hold rating preference movie experiment movie participant member ask movie would hypergraph partition kf iy assign independently relational logical direct relational dependency template
interesting color relate transformation model affect layer template layer configuration variation seem effect segmentation million million foreground augmentation augmentation simple sample heart single neuron neuron kind create layer feature map layer final position filter edge channel color tailor neuron activation allow neuron forward propagate correspond zero produce neuron keep viewpoint input look like much visible vs achievable neuron row neuron fc viewpoint transformation representation viewpoint much realistic semantic activation layer fc row activation actual generate single neuron neuron edge like result hence must fine spatially neuron c segmentation stream fc fc class fc fc stream scale correct size normally image analyze layer look modify observe approach already material level neuron smoothly activation extent neuron feature gradually go almost level outcome clearly sharp observation layer correspond empty rd filter explanation pattern frequency fix size regular figure happen map bring pair c source previously unseen learn previously unseen view cost separate source remain number transfer train knowledge miss angle transfer visible interpolation fine preserve start view pair transfer produce satisfactory interpolation work reasonably view bottom row fine lose euclidean transfer dramatically suppose bottom row simply view find similar neighbor match source view miss view try distance rgb descriptor although figure performance suggest learn linearly remarkably object meaningful representative show bottom naturally row intermediate look like however intermediate supplementary cnn appearance optical consist apply optical flow concatenation optical flow connect optical flow refine flow numerically test ground ask people manually mark first total pair ground truth average manually annotate validation set tune compare sift pyramid baseline intermediate error pixel spatial pyramid sift human performance training network standard task generate image task merely smoothly meaningful relatively fashion parameter additional state manner show powerful applicable modern expect elaborate involve kind might beneficial probably extension apply predict depth future variety different people add network realistic generative object viewpoint network merely heart find meaningful allow similarity network different approach cnn successful vision task predict depth task supervise problem cnn label network mapping input output object position work stick supervise cnn image train capable give color etc neural network image backpropagation enough network perfectly learn heart reconstruction behave input observe namely capable transfer object interpolation describe detail internal network unseen track accurate model typically datum representation priori infer procedure prominent rbms boltzmann rbms build encoding solve direct wide range consist attempt forward perform generative image latent second try discriminate contrast approach image generative rather give reach approach label image problematic uniquely identify significant noise unsupervised incorporate label form semi unsupervised approach variation include rbms rbms rbms digit condition autoencoder restrict structure generative formally refer convolutional turn composition layer fc build share three fed layer neuron stream process connect layer fourth fully fc connect split stream fc generate object mask share fully fed layer filter layer follow relu nonlinearity layer see convolutional layer pool conventional cnns span oppose shrink replace entry block corner elsewhere width convolution apply deconvolution use parameter train euclidean error reconstruct segmentation mask omit weight
kernel adjust scale individually gaussian ard optimization exact become use expansion kernel mixture gaussians describe marginal rotation invariance novel piecewise radial scalable gaussian individually location heavily gm model gm equally describe learn section initialization initialization pick minimum marginal continue signal deviation dimension scale pick random optimization run ard standard initialize initialize ard close parameterized controls origin ard initialization rbf distances uniform rbf width radial follow ard initialize cluster basis expansion rbf ard kernels htb compare method accuracy leave score large reciprocal performance across htb also ard rbf kernel small medium hence gm take kernel place r datasets rbf ard gm cancer hardware auto stock energy road song compare htb time basis gm function component flexibility gm become although sensible combination parameter tune give allow large parametrization allow setup wish emphasize gm adversarial order gm ard gm method hyperparameter commonly rbf ard gm ard rbf entirely expansion gm expressive ard rbf popular vary gm continue increase function gm fitting basis extent mean true gm seven method normalised predictive training consumption score low time despite require similar expressive ard good memory show clarity plot log supplement gm gm greatly outperform accuracy valuable family parametrize function expansion minimal diverse runtime consumption additional gain group short simultaneously expressive hope work flexibility method sense flexibility scalability problem expressive rgb rgb rgb dark medium blue great representation compare scalability introduce flexible learning basis expansion mechanism learn spectral expansion class speed consumption entirely control represent product typically flexibility whereas expressive large rich representation spectral arbitrarily basis conversely recent offer expansion kernel priori hand issue might indeed expressive scalable novel radial expansion mechanism automatically likelihood optimisation adjust learn parameter basis computational frequency computationally allow great group require interval four evaluate advantage accuracy consumption describe background kernel include basic tool kernel contain evaluation fouri flexibility expansion expansion fix recently incorporate scalable parameter enable parallelization jointly proceed descent achieve instance optimize frequency expansion spectrum formalism gain scalability expansion expansion computation dimension mixture gaussian require hyperparameter kernel generalise incomplete grid learn large naturally enable statistic flexibility make ideally suit dataset datum partial efficiency gain consider expansion weight expansion expansion property inductive purpose novel radial flexible require denote domain label joint hilbert schmidt speak semidefinite key kernel allow product high implicitly mapping desirable might solve combination beneficial amount create approximate manner exploit translation invariant ensure integral product kx suggest carlo approximation fourier transform popular normal distribution use rbf refine approximate admit expansion random randomness allow efficient uniformly zero expectation define recursion dense subsequent hadamard reservoir sampling uncorrelated diagonal draw result iid gaussian encodes draw define length via straightforward change adjust computational remain adjust rather generic additional la keeping introduce flexible learning allow translation accomplish additional sample moreover learn translation basis individually optimize frequency still computationally expensive fitting undesirable optima particular want enforce also scale location frequency result parametrize section describe assume expansion next piecewise radial scaling use formalism introduction process next derive kernel general purpose particularly expressive choice objective assume gaussian drawn feature parametrize involve infer mixture model access component parametrize mixture weight integrate away solely denote design parametrize covariance index minimize negative cross training inspection simplify expression immediate storing provide predictive accomplish via formula require rank approximation projection computation kernel formalism preferable kernel yet rotation violate rotation leave follow term fourier choose frequency symmetry rotation invariance linearity transform expansion translation approximate gaussian density estimation fourier approximation fourier amount directly amenable fast expansion provide kernel form insight shift fourier accomplished multiplication x inner operation multiplication multiplication original accomplish preserve translation otherwise preprocesse random group product feature dispersion likelihood describe individually less fitting local efficiency retain rotation able adjust spectrum piecewise radial recall instance rbf inputs rbf design integral radial analytic remain two transform efficient parametrization provide explicit piecewise range parametrize eq basis piecewise piecewise via explicitly invert considerable cumulative pick particle substantially mm automatic relevance use radial radial component piecewise require basis partly approximate expressive efficient algorithm optimize likelihood matrix rbf adjust vary radial procedure obtain marginal kernel hadamard retain preserve modify main multiplication uci flexible scalable vary intractable propose competitive intel operate ghz gb ram marginal objective group rbf ard gm test grouping datum rbf ard partitions every average rmse partition fewer tractable rbf ard kernel hyperparameter ard kernel relevance determination scale individually large expansion gaussian mixture gaussian describe regard marginal likelihood rotation radial sparse process individually optimize location frequency heavily case model gm section optimisation section sort quantile initialization minimum optimize signal noise initialize pick well multiply ard draw uniformly standard gaussian covariance assume scale initialize ard function parameterize control ard technique rbf random bandwidth rbf kernel kernel radial ard specifically variable fix expansion ard htb accuracy comparable compute take reciprocal score large accuracy ard rbf five dataset gm basis medium take typically intractable ard gm cancer hardware k protein ct slice road song assess respect compare function htb time leave gm gm become valuable many although sensible combination fine parametrization wish gm model adversarial order gm comparison rbf ard gm hyperparameter commonly implement ard gm large exact ard expansion gm expressive ard popular figure rmse gm gm indeed optimize gm model investigate compare average normalise testing lower correspond gm alternative runtime expressive model ard third testing runtime time efficient consider log transformation supplement gm gm scalable expansion parametrization collection accuracy runtime memory consumption gain group short scalable expressive help scalability flexibility flexibility expressive ex example conjecture axiom rgb dark medium great rich statistical dataset neural network scalability introduce fast flexible parametrize expansion mechanism expansion wide alternative consumption entirely represent tradeoff speed flexibility function class fast adaptive expressive need rich flexible require arbitrarily basis lead restriction conversely offer expansion kernel priori address priori appropriate form expressive purpose radial function frequency expansion mechanism automatically optimisation individually adjust frequency flexibility free lead fitting limitation spread location method computationally number control many basis function furthermore special four kernel range advantage speed describe relate background basic approximation evaluation discussion approximation kernel carlo great flexibility consider kernel learn sum learn parameter separate enable parallelization jointly stochastic achieve promise acoustic instance deep optimize location frequency expansion formalism gain scalability expansion expansion basis flexible kernel mixture arbitrarily basis combine modify generalise kronecker product incomplete grid statistical representation kernel enable ideally suit domain partial gain mixture weight expansion expansion several learn spectral inductive also novel piecewise radial kernel overall perform learn regression likelihood additional innovation denote domain finally schmidt speak semidefinite key represent theorem might infinite instead kernel eq beneficial amount create burden expansion explicit map translation invariance yet invariance violate rotation representation fourier choose typical pick frequency symmetry linearity transform inverse component universal expansion translation approximate density density fouri theorem domain hence directly amenable expansion provide shift however modification allow efficiently form key insight fourier accomplish cost multiplication induce accomplished order preserve translation
volume construct hard rank decomposition achieve k ok high overview approach explain start tool connect opt obtain sample row adaptively row sample sample approach ok far approximate svd choice costly method bind carefully matrix adaptive lemma additionally sample turn opt section column near immediately play orthonormal obvious orthonormal desirable would c ok ok row desirable row step next seem would really want apply ok inequality require need additional construct ok opt k sampling argument lemma derivation relative error call design matrix relative opt ok span opt ok sampling column sampling pick condition lead desirable address primitive know include bss e primitive find employ sample primitive address primitive row summarize algorithm error deterministic issue ok ok primitive primitive primitive column primitive row primitive sample primitive algorithm proportional implement time implement sparsity already section know rest step develop tool design version combine idea input lemma combine idea algorithm possible within subspace restrict see matrix construction product matrix embedding view implement deterministic tool version obtain adaptive body computation provide kernel square low approximation area discuss mention construct problem frobenius relative method select cx theoretical column build theorem error error subspace sampling construct subspace sampling probability proportional leverage row constant see discuss decomposition trace norm relative error near method along zhang eq summarize table ok ok ok k ok ok c ok ok nk nk ok ok ok ok ok ok k k row use lemma twice state article validate careful comparison step score unknown particularly helpful algorithm limitation rank wang zhang et bss column orthonormal ok ok step within numerical skeleton rank svd bound summarize know provide perturbation rank rank opt decomposition k pseudo n rank costly speedup make svd matrix considerably sparsity time omit se instead approximation running describe deterministic relative sub arithmetic deterministic arithmetic construct obtain column assume k result correspond exist randomized compute arithmetic operation plus nk construct summarize exist literature bss ingredient dual ns nr ns equivalently introduce row n jj rr write k r q residual approximate sample adaptive improve column approximation theorem n ip arithmetic operation r np ip q operation connection transpose section nk nk algorithm qr nk simple corollary nc prove matrix construct well want proportional address good good within inside specifically requirement denote detail discuss subspace embed preserve well geometry sparse call I entry choose subspace dimension choose sparse embed opt opt theorem dimension remove review method preserve element b r problem opt opt kk prove combine subset selection tool om next sparsity version c first probability satisfy omit probability proof continue repeat idea although take construct careful projection routine operation ok r ok ok qr r equivalent algorithm run convert truly decomposition un rescale introduce carry version unchanged goal fast simplicity potentially randomize algorithm fast though difference algorithm closely detailed finally theorem analyze precisely input c ok ok algorithm detail choice time three row iii analyze describe lemma next operation om kn k om om om r k follow prove eq q intermediate matrix approximation precise lemma satisfy least suffice notation algorithm k recall argue offer column low argue strong norm follow norm value failure lemma probability argue satisfie write notice fail guarantee eq immediate lemma follow lemma similar satisfy proof satisfy ready implie expectation take expectation failure bind probability sparsity convert truly keep un rescale scale factor carry unchanged closely complexity ok ok qr step input rank input matrix ok ok k algorithm closely sparsity run optimal row intersection construct detail section lemma pointing replace implement detail use next analysis arithmetic need nk k sparse subspace om km main quality theorem implement via column make algorithm argue offer factor rank column indeed choice term offer probability prove combining relative relative follow prove identical construct opt randomly subspace equivalent opt immediate combine satisfy opt opt opt opt opt opt opt eqn write lemma failure eqn follow matrix drop optimality furthermore follow orthonormal failure probability q deterministic convert truly un rescale version introduce give complexity algorithm input ok r ok closely specific implement run iii third construct refer lemma ok ok ok qr r r k formula arithmetic om need find om asymptotic algorithm present prove might independent implement via preserve necessary low first argue satisfied prove term norm connection spectral follow hence offer column base combine relative eq immediate lemma result lemma proof ready prove construction bind follow fact overall unless column outline k ok symmetry corollary approximation observation consider factorization statement theorem three assumption contradiction error ok ok ok ok ok continue give basis look k lemma lemma ok ok integer look like matrix rank n later small error occur approximate cc result possible describe diagonal precisely copy row select ready q column sufficiently q least symmetry immediate corollary symmetric ok ok q extend symmetric mention symmetric appear choose intersection david institute berkeley work like acknowledge advanced project air laboratory contract
furthermore equation impulse still write gaussian depend vector system estimate compute remainder effectively joint impulse parameterize hyperparameter introduce represent flat improper prior accounting positivity factor accord variance choose map solve hyperparameter state assumption recall hard reason scheme end obtain iteratively iteration hyperparameter indicate iteration latent ease notation define iterate step compute employ guarantee subsection scheme key iteration em hyperparameter use accordingly e predictor time instant accounting impulse vector posteriori diagonal element define impulse accounting hyperparameter vector iteration method employ estimate rule student hyperparameter element remarkable establishe solve sequence scalar crucially depend posteriori impulse update admit close hyperparameter principle admit objective function available robust identification obtain experimental em attain output repeat posteriori total residual impulse response update compute operation posteriori differential impulse energy relate posterior recall impulse coefficient parameter flat become accordance conversely must student outlier parameter detect outlier desirable automatic scheme estimation treat aim integrate precisely follow include scheme assume iteration selection criterion satisfactory e consist fast process could reduce integer integer hyperparameter adopt update rule remain average pay principle choice group accordingly reasonable expect force converge estimate nominal behave section numerical ht estimator outlier perform performance monte carlo run generate order gaussians eq way outlier variance generate probability equal noiseless ease trajectory impulse response true impulse response carlo estimator new kernel adopt laplacian hyperparameter except noise degree section opt student model introduce take fit operation impulse thus practice ss ml impulse model see outlier accuracy robust ml presence outlier estimator em opt slight degradation price pay parameter suggest anonymous employ situation scenario find misspecification end perform experiment student noise variance show monte carlo simulation em offer estimator ss capture feature student r heavy around box estimator section equal noiseless carlo runs estimator ss estimator laplacian instead estimation value variance value section different share note case correspond force white plot fit variance degradation em accordance ml method identification laplacian description via gaussians monte see detail impulse gibb noiseless report score time burden lower fit avg em gs average motivate quality score absence em ss regularize identification nonparametric method constitute particular kernel stable spline kernel limitation paper heavy tailed laplacian exploit joint noise hyperparameter efficiently kernel base rely closed show effectiveness outlier compute introduce assume correspond compute expectation respect make proof compose convenient rewrite rearrange position since recall follow first proceed adopt laplacian minimized deal plugging obtain kernel note eq rewrite due hyperparameter depend minimize partitioning adopt two form se recent development system attention type compare classic respect novel method output heavy tailed probability density focus laplacian gaussians cast identification require hyperparameter overcome difficulty posteriori hyperparameter solve maximization method outlier experiment currently history series regularization strategy impulse response compare parametric approach selection usually require method establish aic validation set square estimate machine tc autocorrelation process depend context usually available interpretation impulse process whose hyperparameter marginal integrate dependence impulse retrieve rely measure experimental outlier output describe impulse feed white first signal measure estimate impulse right panel situation outlier output much impulse outlier novel denote identification impulse namely adopt paper allow theoretically impulse implication sufficiently define toeplitz matrix problem make persistent require ease assume laplacian
perform rank curve illustrate regard skeleton principal curve curve fail consistent connect besides fail order curve strict present strict monotonicity function dimensional piece monotone ranking rank like monotone list tangent another pair follow attack attribute performance account always produce acceptable object rank list work rule meta assess ranking list second serve rank perform skeleton principal produce list embed principal cubic five meta existence principal highlight contribution design assessment meta rule unfortunately rank observation object rank link rule theoretically attribute object reasonable list rank approach improve construction et monotonicity consideration improve ordinal domain also capable assess manifold able unsupervised observation object curve serve rank molecular surface bring preserve principal explicitly one cubic strictly monotone corner box avoid confusion end end point control cubic red shape rest formalize next rule namely principal prove five meta function cubic carry rank indicator vector ranking point I I rank list totally require ordinal thing partial proper x easy verify subset eq ranking vary task prefer help rank rank score order order monotone preserve requirement partially totally point state rule indicator life quality one country numerical eq order monotone mapping strictly order converse readily monotonicity respect define differentiable eq strictly monotone monotonicity equal monotonicity monotone decrease versa monotone monotone mapping monotonicity origin exist theorem assume smoothness output value rank score point list sort verify list rank five reasonable list level level rank five feature serve rank able rule rule namely translation example thousand range translation order strictly meta rule rank ordinal problem monotonicity object classify class require strict monotonicity ranking hold indicate high otherwise example refer linearity relationship nonlinearity take rank task score case rank task meanwhile nonlinear smooth smooth yet continuous derivative guarantee rank two line rank approach interpretable parameter allocation ranking characteristic design biased propose perform principal cubic rule summarize dimensional seek direction explain maximal cloud project regard project point order rank skeleton order smooth express skeleton skeleton produce discriminate project parallel horizontal line pca extensive application comprehensive recall principal extension pca summarize indicator appendix principal assume principal cloud project curve ranking unsupervise still rank skeleton instead curve pca score noise measure error influence exclusive indicator score remove correspondence curve inverse take numerical five rule correspondingly minimize optimization determine residual reconstruct obviously achieve means associate rewrite q eq pseudo computation substitute always ill condition optimal intermediate would thereby employ column converge maximum respectively solution rarely root method respectively consider root design curve datum learn vector adopt summarize perform rank task numerical minimum vector unchanged scaling perform end point procedure automatically b curve occur begin therefore find infimum decay stop rank produce along summary unsupervised rule weight ranking cost little assignment weight expert proportion indicator whole rank completely rank five meta rank meta capable evaluate hand rank guide dataset ordinal information determine relation include object influential select rest still observation nothing formulate error adopt observation object object order aggregation b denote keep aggregate well rank ranking suffer monotonicity ranking list information model meta detect ordinal illustrate object dimensional show order respectively since rank object rank list remain table curve give table ordinal candidate unsupervise attribute object significant journal illustration index reflect aspect list evaluate comprehensive propose attack provide skeleton task evaluation open source software gb memory list list cccc cc united life year per control et people indicator indicator visualization fig illustrate shape include linearity nonlinearity indicator begin bring exceed person increase little decrease matter hard evolution control list bottom space point table three curve depict skeleton fig al center assign country country take understand principle parameter size understanding present five meta explain good life quality interpretable easy carry four cc c inf uk mis en learn web journal citation report science social sciences report citation article score remove try science artificial intelligence application illustrate rank fig journal high indicator linear like take frequency transaction rank high transaction bring therefore get comprehensive place mean one whole list account indicator different behavior human positively ranking activity challenge greatly rational unsupervised critical truth unsupervise rank candidate observation multiple domain rank motivate five rule invariance monotonicity linear nonlinear smoothness assess principal formulate cubic b restrict control interior hypercube computer science model rank indicator rank feature principal summarize pca ds ds originally define manifold distance denote wide e variety principal curve perform try smooth meet expression interpret curve formulate bring make interpretation even hard curve rank white box monotone regard totally hold one monotonicity set subsequence converge uniformly sequence converging assume converge eq
economics management university long range bivariate strongly mean expansion full reduction limit process available process stable move average memory application goodness goodness kolmogorov random variable rv cumulative strictly event nx know ep non either weakly dependent rv range constant paper study properly stable orthogonal polynomial discuss cdf rv cdf framework analytic expression testing purpose generate stable rv rv paper approach provide reliable method generation framework key ep discuss reader excellent review general specific relevant discuss bivariate non bivariate expansion average study devote non worth follow provide discussion show essential moving obtain sequence ep concern goodness test organize follow argument ep rv final presents need rv expansion ep provide define sequence stable rv random copy unit lk stable rv characteristic consideration analytic see connect equation rv detail possible notation let g proposition rv transformation specifically finally admissible parameter discussion stable rv well know box transformation integrable density polynomial denote du measurable polynomial v converge explicitly dependence refer transformation otherwise technique cdf r rank polynomial indicate nz mn mn md md expansion ep exhibit bivariate principle weak result construction multiple wiener integral weak equip sup multiple wiener integral copy gaussian integration indicate exclude integration gaussian normalizing introduction non would result z z simple bivariate consider reason satisfy exploit section properly normalize case discuss exploit outline formulae derive formula formulae formula function formulae derive formula compute formula induce discuss detail prove recall need highlight write one follow g z ax need denote integral concerned detail g last obtain concerned bx cx note expectation definition x need integral q transformation turn reasoning reasoning yield z cx proof z parallel reasoning bring follow expectation transform recovered reasoning one far coefficient computation transform recovered reasoning result hypothesis stable previous rv satisfy rv obtain worth appeal ks
marginal learn inductive useful unlabeled identically name optimization globally analyse well art mnist rest paper review multi analyze equivalence equivalence later next input stand confidence rate soft negative class equally approximation definite pairwise hypothesis sign theory define underlie available distribution k k equivalence geometric confident accurately body alone introduce associated orthonormal geometrically origin center ellipsoid angle hypothesis principal crucial iv conversely lie small equivalence multi fit unknown subset pick reveal label unlabeled unobserved account modification suffice set set generality label multi knowledge former study latter study fully partially label label datum possible set label set measure cardinality classification unique work volume soft concern em possess possess th class jt negative label predict label form stacking pairwise provide positive matrix due kronecker positive equivalently e g consequence geometrically origin ellipsoid sign nc nc nk geometric stack soft argument frobenius equivalence class conversely lie constrain nc np identity necessarily domain originally nan propose volume method develop analyze define entry depend whether ever appear label otherwise label loss measuring volume like eliminate scale region become although carry axis subsequently dimensional multi class counterpart fourth fourth loss identically constant undesirable issue discuss thus version rewrite use stacked origin fortunately could fundamental kronecker qp ij ci qp objective lagrange stationary eq locally theorem feasible globally plug sort z nc globally minimize globally proof root eigen summarize fix find case stationary v eigen algorithm dominate computation sort find third problem multi recover cost like comment firstly employ loss latter loss optimization eq would compute eigen complexity secondly eigen decomposition sort eigenvalue hyperparameter setting finally complexity improve fix certain stability bound matrix bound ground truth guarantee fact meet assumption note unconstraine bind train label optimization unconstraine ground soft firstly ensure implement large secondly already correlate correspondence label unlabeled position close unlabele completely uninformative recover notice need even multiple totally number possible globally solution unconstrained optimization theorem appendix instability bound constrain section numerically directly prior latter see impose detail class regularization belong cluster belong specify influence class closely unconstraine motivated propagation viewpoint constrain use generate ratio ground uniform distribution result mean classification rate plot task performance decade cut regularization state art call report similarly standard repeatedly consider another cosine define neighbor involve select cosine fix result cosine similarity local similarity singular exception often state volume approximation setting convex theoretically justify analysis derivative increase interval
ball goal square square assign whenever ball four square shot assign assign point take shot otherwise goal player guess goal square matrix vector encode belief game develop indicate independently distribute neither knowledge guess complex subject belief ball position likewise non likewise type reward perfect perfect correlation mutual deviation lead vector accordance weak symmetry singular present procedure recover ard reward ard result evaluation term column recover original experiment perform combine mean construction player leave infer benchmark reward show red corresponding mean case htb whether algorithm goal receive square receive square stem infer ball exchange affect change calculate result figure conclusion actual reward goal recover player toward consequently observe substantially player shot position distance use quality response situation notion learn game environmental consider discuss typical want policy reward sum minimax policy simulate c rate rational minimax reward minimax reward recall recover reward assume reward first c develop reward rest game comparison episode outcome column ht base c vs vs vs vs vs vs mm vs vs sm vs sm sc vs term win draw outperform tie reasonable b know truly ard ard small win notable winning ard become drop implication crucial infer setting game seem challenge demonstrate difficulty person ill good know observation fortunately many problem additional prior important inverse observe player observe deterministic infer finitely observation action strategy often infer many take game reward likely appropriate generative include perhaps person challenge primary equilibrium possibly nature equilibrium strategy general equilibrium player specify assume equilibrium player player uniformly equilibria reward another player drive toward equilibria player inverse acknowledgement international proposition theorem conjecture axiom term inverse formalize game markov optimality mdps mean nash uniqueness equilibria reasonable inversion equally problem establish foundation competitive person sum reward play follow context abstract extent quality reinforcement aim reward agent behavior environment subject extensive expert demonstrate preference graphic motion controller human decision action understanding experiment move quantitative evidence inverse planning predict goal inference solve agent jointly make rational significantly lead multi propose conceptually rl simplify former treat system environment agent game involve agent propose agent payoff depend action adopt concept agent choice concept weak condition observe agent reward know weight distribute note converge multi planning involve exchange consider include payoff play action take formalize game mdp game bring mdps equilibrium nash uniqueness equilibrium give equally sensible paper study appear present reinforcement multiple compete key consider simultaneous game solve non system decentralize active player want infer third party observer system person sum broad deeply help examine prefer yield tractable computing apply dynamic numerical relationship quality space game play agent equilibrium learn win game terminology definition basic later work reward reward reward play success conclude remark work two player play begin finitely select finitely hence reward sometimes player make transition dependent jointly select action repeat horizon discount reward specify person r kk provide rule player follow loss generality mass refer play state follow player player accord know formulate optimization adopt assign learner make complete observable point reward reward know observe give effort focus determine function optimization generate function formally model let denote denote observe maximize observe selection select minimax reward minimax select minimax mass eq minimax countable normalize reward q minimax posteriori maximize equivalently maximize provide detail use focus gaussian prior reasonable represent nominal reward add leading covariance probability nonzero minimax reward function function otherwise compute map computing map computing maximize prior reward solve devote feasible static minimax person sum payoff theorem direct inverse special goal minimax constraint stage theorem value minimax must constraint relate function relationship game player reward select action expression let express formulate concave equivalent problem solve person sum class reward depend depend action stochastic reward satisfie note expression second policy take action policy inequality finally develop demonstrate
especially sgd significant improve performance speed convergence combine sgd quasi newton rate hoc even work implicit sgd definition statistical comparison furthermore would require solution multiple exploit efficiently implicit efficiently root finding find narrow exploit monotonicity transfer nb r estimate identical implicit optimally tune stability covariate continuous glm n nk n limit sequence lyapunov inverse operator map sgd define decrease sgd iterate typical sequence satisfy assumption form asymptotic optimality averaging scheme impose weak constraint appropriately correct rate satisfie assumption consider matrix n assumption show asymptotically leverage summarize suppose asymptotic explicit estimator n n asymptotically explicit sgd converge sufficiently become implicit stable small variance subtle sample show asymptotic variance definite asymptotic satisfie variance I way compare assumption need prove normality central exploit regularity n satisfy sgd loose compare mle dataset variance n quantify achieve leverage experiment order achieve comparable efficiency gain equivalent minimize eigenvalue need recently extensively tune sgd estimator bias simplify remainder I initial decay explicit limit method however explicit procedure inspection magnitude eigenvalue dominate stability setting convergence stability price convergence implicit j critical sgd eigenvalue less one finding implicit specific consideration avoid etc misspecification learn implicit sgd asymptotic extend introduce assume write parameterization extend sgd sgd eq multiply generality notation statistic generalize result hold regularity expect scale explicit implicit method asymptotically rigorously several notably refer geometry argument prove aforementione impractical consistently inversion sgd transformation sgd new rate complete sgd parameterization asymptotically optimality appropriate estimate optimally invariance parameterization efficiently parameterization correspond parameterization inverse transformation thus jacobian thus eq mle demonstrate particular perform follow experiment conduct running core ram gb technology implicit poisson poisson log sgd unstable rate result implicit preferred equation concern compare function mle procedure mse compare package mle much r package develop elastic net aforementioned implicit sgd vector sgd counterpart machine study additive air public health design fit scale demonstrate leverage theory develop explicit computational devise strategy work uniformly illustrate variance analytically n variance particularly contrast sgd unstable order implicit quantile deviation sgd procedure modify avoid explicit understand determine learn take substantial effort implicit iteratively glm optimal estimation experiment implicit dataset normal binary row determine experiment generate sampling replacement second generate nh slightly algebra eigenvalue sp bs thus use derive theoretically optimal compute numerically range estimate implicit run regression approximate runtime mse well size memory model row observe come loss average mse implicit glm simulate implicit sgd thus almost sample mean square roughly se se implicit sgd fitting computing task view project project update incremental qr decomposition requirement simulated roughly efficiency sgd satisfy run package method hoc file report computation exclude technical unlikely e optimization component utilize regularization efficiently iteration achieve component experiment experiment package first pp control experiment outcome generate tuned experiment generation time average replicate implicit sgd consistently fast sublinear section affect update sgd method regression normal model replicate sgd method maintain scale linearly contrast affect covariate slow sgd significantly slow numerically covariate cross sample report second line median ccccc cross averaged repetition ccccc mse c direct comparison net parameter produce aforementione elastic implicit use therefore situation indicate trend big high sgd outside family procedure compare standard sgd typical benchmark author variation regularization complete observe remarkably misspecification sgd small affect mean rate however maintain stable implicit log sgd air aim risk public health daily roughly g concentration health variable separately city due dataset procedure qr decomposition gb construct datum identical observation implicit sgd fit entire second almost fast home estimate random mb implicit mle replication aforementione reveal suggest limit moderate key efficiently property reveal implicit subtle fitting implicit obtain implicit iteration thus explicit factor fisher information sgd procedure incorporate involve implicit sgd combine efficiency extend implicit available least quickly equation computationally sgd generally n either expect available another sgd estimator easy situation general exponential sample assume parameterization similar f b n unbiased monte implicit monte approximate implicit sgd apply carlo accord suppose satisfie usually normalize pg g triangle normalize generally summation graph use fix independently normalize constant mle assume edge sgd edge binomial edge iii variance obtain p iteration various rate confirm high bias rate slowly bias estimating parameter depict line replication explicit trivial would modify sgd draw show quantify variance theorem invoke asymptotic normality create testing normality justify theoretically consider bootstrappe normality plausible construct consider chebyshev paper algorithm massive term asymptotic bias theoretical principled common achieve theory exponential family suggest term sgd model procedure suggest explicit stability concern procedure outlier misspecification shrinkage combine computational efficiency first experiment suggest implicit repository contain code implementation repository formalize implicit defining procedure adapt clarity unique start define approximation furthermore implicit counter intuitive observation current effect e possible implicit depend future implicit suppose implicit reference definition procedure q furthermore substitute term equation term carry completeness let term converge construct series construction identical definition find requirement since fulfil monotonicity requirement glm notational convenience subscript let fy give fy part multiply get side become solve finally implicit correct h u n straight wish point monotonicity furthermore nr algebra n search successive map recursion recursion n b see therefore collect hand side b e ai ai e n finally substitute second part recursion identical definition recursion n identity obtain line b lyapunov apply n ni ni eq side n side use q explicit method asymptotically implicit n establish explicit notational convenience variance side simplify rewrite remainder mapping q variance since side n mapping application corollary also proof q obvious even formula stability sgd covariate n positive definite variance n asymptotic carlo parameterization inverse ij n corollary efficient descent popularity task approximation contrast implicit version iterate amount shrinkage depend fisher latter implicit hyper gradient affect asymptotic contrast rate agree fisher information bias estimation efficiency maximum avoid careful parameterization estimation stochastic demonstrate real stochastic descent superior efficiently efficient procedure popularity estimation stochastic term next iterate context formula estimation show relate implicit explicit shrinkage observe fisher information never benefit scalar hyper guarantee contrast require agree efficiency suggest careful parameterization imply observation methodology exponential class method compute underlie theory extensive involve compare maximum posteriori however widely fisher iteratively scalable modern hundred million hundred thousand root insight develop naturally extend generalize produce form log assumption er rao traditional running range newton compute mle procedure mle inversion per method scoring replace positive definite fisher similar two algorithm actually identical quasi newton calculate mle quasi hessian update iteration algorithm favorable estimation iteratively expensive estimation massive time complexity roughly sublinear optimization notable sgd simple carefully define stability sgd sgd appeal expensive inversion single furthermore single observation essentially convex parameter I keep simple naturally regression variance regularity theory j sgd procedure write motivate true n procedure derive close explicit sgd converging implicit determine act exactly procedure stable misspecification indeed cause explicit sgd insight stability procedure regard efficiency n n optimal generalize concrete optimally sequence implicit sgd derive aforementioned
jeffreys perform considerably believe interval paper test focus test hypothesis type composite e invert behave interval theorem brownian bridge scale bridge generalization brownian bridge scaling determine force estimator improve substantially bias simulation brownian brownian motion brownian bridge wiener bridge scale brownian alternative brownian brownian interest brownian bridge brownian brownian absence transaction follow stock tend option example phenomenon however price market start early brownian bridge brownian bridge stock bridge suitable order stock show phenomenon stop bridge possible include brownian brownian brownian attract considerable community brownian study focus mle study finally proof well bridge self self extend brownian suffice quite substantial particular mean market market bias correction inverting expectation correct observed posterior prior proportional fisher jeffreys tailed median seem unlikely prior information might preferable informative bound support estimate realization rectangle estimator compare qualitatively nearly unbiased mse thereby upon considerably jeffreys biased correct bayesian estimator towards nearly reliably market heavily correct mse recommend one reason bias preferable jeffreys jeffreys bayesian market situation frequentist financial decision
experiment synthetic answer template scene dataset multi human st question template category test experiment human remove turn demand incorrect answer wrong association difficulty unseen category train uncertain semantic segmentation create world severe drop automatic segmentation class part serious bottleneck world prefer human last generate fact observe trend high counting language substantial incorrect attribute segment g question answer world visual despite uncertain program indicate bring idea scene parse symbolic reasoning combine multi computer vision language seem bring open challenge h answer accuracy c accuracy human baseline de pt minus method automatically answer advance combine discrete uncertain prediction represent human realistic count list answer benchmark modern attempt vision technique like object increase full scene often understand correct labeling term annotation prediction inherent visual propose learn question fact world infer interpretation correctness fact core address answer real world formulation different interpretation scene date substantial serve answer test chain relate ai build test vision early day ai argue answer task step direction combine real world symbolic reasoning framework answer dataset question produce human modern visual benchmark advantage multi approach challenge ahead source different inspire answer natural language supervision rely manual annotation logical form contrast never language goal connect sentence image physical world constrain domain object instance object block build diverse world scene relationship moreover beyond scope reasoning impose challenge answer scene representation deal language efficient spatial reasoning alternative learn language bind unclear integrated execute restrict execute restrictive scenario system color green table probabilistic database entity recognition segmentation apply visual answer visual scene interpretation visual scene account picture confident possible denote although confident opposite benefit wise spatial object coordinate build answer part latent variable formally world perform semantic tree logical logical world pa recursive evaluation relationship subtree template template string predicate search tree model parameter engine answer logical form linguistic phenomenon engine detailed exposition refer reader operate world correspond world overview fact automatic semantic image purpose art semantic recognize position color position represent max define minimal axis object coordinate schema release purpose predefine relation table association class answer consider fact segment different probabilistic database possible interpretation visual scene segmentation answer question semantic segmentation logical semantic segment assignment category segment bind set tuple practice draw every show parallel need synchronization cost sum parallelism question answer end answer pair contain induction consider fact million predicate bad image induce fact batch fact space batch boolean variant tf measure training test dataset website tb description image many color room depict room object image object image color object bag room object answer dataset v annotate consider canonical view depth define spatial uncertainty visual technique prop spatial content collect question answer dataset work annotation synthetic answer pair template template fact answer ask house participant provide answer answer color number set answer impose question question believe robust human jj mean
pde account model physical stochastic pde extend gp vector develop pde physical make adjoint regression application process obtain estimate state bilinear gaussian process regression standard gp finite base pair method incorporate albeit relating combine greatly system nonparametric method functional noise nonparametric bayesian inference gp regression different view think functional define functional inference functional inference theoretically interpret nonparametric bayesian avoid construction present introduce pde demonstrate remark nonparametric bayesian appropriate space pde model state v functional pde describe physical practice knowledge pde find paper shall drop indistinguishable measurement differ output additive distribute acceptable validate good close trust behavior many good due source geometry good produce process proceed review assume training vector vector aggregate output add perspective encode belief account observation mathematically eigenfunction joint conditional joint distribution predictive combination covariance explicitly function predictive implicitly depend impact likelihood see choose gaussian provide among family choose hyperparameter different term model characterize us function regression maximum must describe output linear functional regression knowledge carry advantage model able far disadvantage standard mathematical I differ knowledge characterize output model capture various source model namely first operator operator characterize account observation play operator functional process stochastic pde pde accounting formulate functional adjoint q functional bilinear pde output relate characterize relationship characterize introduce training wish given emphasize collection adjoint determine collection joint observe function give q notice describe share similarity important way difference regression gaussian process ex ex ex n n nonparametric bayesian framework gaussian bayesian framework linear avoid operator introduce family bilinear hyperparameter covariance crucial ingredient bilinear operator form symmetric specify fortunately process allow determine hyperparameter use depend convenient hyperparameter maximizer determine operator compute function subsection functional sense light combine determine describe compute pde pde model basis knowledge pde choose arrive system j follow gaussian spatial ij note term evaluate functional matrix showing grow figure ex compute compute ex compute posterior least square q adjoint coefficient follow mean sum adjoint mean pde light output observe covariance bilinear adjoint equation recall delta adjoint equation complete differ exactly whenever expect state good show optimality state assume bilinear introduce lagrange multiplier constraint iv ex ex partial two adjoint give adjoint knowledge adjoint complete covariance bilinear square also posteriori functional discuss know least advantage whereas mechanism pde heat dirichlet boundary satisfy eq source easy serve everything boundary space observe knowledge replace counterpart problem element element specify functional iv vx chebyshev interval functional element mesh grid free remain observation model furthermore bilinear operator likelihood utilize scale likelihood c xu u mean prediction posterior gp knowledge state figure show considerably prediction heat gp regression square root length exponential see table small gp use good model functional whereas posterior albeit slow regression indicate standard rigorous estimation show region panel confidence region mean measurement state error remarkably rapidly increase confidence see figure poor prediction rigorous summary simple functional gp f statistical improve incorporation prediction state superior propose conclude point nonlinear pde adjoint depend pde preserve due nonlinearity extend oriented infer output rather management mit discussion fa mit framework describe delta functional eq functional target noise I output formalism need without generality diagonal diagonal always diagonal work obtain normalize substituting eq capture weight prediction functional posterior give predictive however fortunately equivalent recall formula inversion obtain need attractive observe need l follow
block imagenet cnn precede operation follow protocol image reflect patch time net imagenet epoch top ht testing building net cifar imagenet imagenet mini share layer parameter model mb base cnn ce cnn ce pc imagenet hold category cnn fine tune decrease mini batch build nevertheless top wise cnn mistake make build net fig collect building net fail receive mass fine layer category protocol edge perform average refer detail imagenet imagenet build category hierarchy overlap gpu exploiting parallelism fine k decrease memory fine classifier directly compression cnn obtain imagenet imagenet cm layer dense imagenet dense net network multi testing net achieve evaluation layer net cnn error net cnn slightly outperform prediction net cnn architecture net building net future extend cnn architecture theoretical image separability object highly demand dedicated convolutional network cnn flat effort leverage structure deep cnns embed deep cnns hierarchy cnn separate easy class coarse category distinguish coarse term category cnn scalable visual cifar imagenet benchmark experiment cnns cnns ht cnns visual scalable algorithm cache batch huge volume training well recent year dataset become object category becomes come along separability category hard cifar belong coarse category cifar nonetheless deep model make flat structure adequate intuitive alternative attempt deep large model impose hierarchy inference dedicate category classifier category cnn classifier cnn slow consume principled hierarchical architecture image task step coarse easy another class fine category adopt principle cnn build upon build currently rank cnns cnns benefit progress cnn fine paradigm fine category block correspond cnn increase contribution novel classification organization coarse fine cnn fine tune logistic category consistency term cnn category cifar imagenet class art cnn integrate hierarchy main cnn cnn image detection parse considerable interest nonlinear either improve expand capacity building building hierarchical deep cnn recognition vast category structure build hierarchy class predefine learn imagenet utilize trade specificity leaf hierarchical achieve speedup certain attempt insufficient various label relation encode hierarchy cnns category mainly scalability constraint exploit hierarchy novel cnns cnn follow notation image respectively category hierarchy fine category end end illustrate comprise namely coarse multiple category left side layer receive pixel low share layer precede block layer component configuration building cnn produce coarse coarse aggregation layer coarse fine coarse one coarse purpose prediction category thresholde enable fine category component coarse probability bottom layer classifier category prediction fine component category fine prediction category fine category layer build final fine common reason first precede agnostic level corner face coarse reduce execution significance last decrease success cnn side receive fine category coarse produce weight coarse image coarse category fine stress layer configuration flexible modular design hand building category grouping category dedicate fine category employ hierarchy hold set evaluating net matrix diagonal set entry discriminate category spectral cluster fine category coarse result level coarse coarse coarse category coarse classifier mistake correct ground label implicitly separability coarse add fine coarse category fine prefer add fine coarse category hold aggregate category branch full set category hierarchy coarse prediction update coarse classifier category linearly coarse amount train training hand training mini batch fine mini gradient fine category large mini decompose multiple training complete sequentially component fine building block cnn precede coarse category cnn coarse fine independently classify category use coarse precede layer initialize keep layer except last cnn coarse category fine tune cnn tune cnn hierarchy category focus classify fix fine fine tuning semantic category predict coarse category keep category category conventional learn mapping coarse category image coarse within mini batch use fine show size mini batch regularization cnn memory execution linearly visual develop execution parameter technique necessary weight negligible conditional weighted accelerate use parametric fine classifier evaluate layer category grow coarse category memory product quantization row store cluster center precision achieve compression hyperparameter benchmark imagenet implement software test cifar natural e whitening size stack cifar cnn building block fine share precede account independent category hierarchy hold coarse category visually layer rate iteration fine tuning mini batch factor corner cifar cnn testing improve net hierarchy coarse either disjoint investigate fig occurrence coarse dataset affect category coarse cifar dataset category occurrence overlap category consistency cm average net double cifar cnn ccc share layer memory sublinear fine building cnn category cifar build net without compression memory execution
know shot shot strategy utilize advanced human annotation stage base eliminate achieve comparable four cifar class exist object recognition zero learn image classification base video surveillance et al humans able class object detector million likely come shot learning paradigm abstract unseen available training recognize unseen object intermediate cascade enable recognize example use semantic relationship reference unseen approach extensive human supervision attribute tight unseen class training propose replace supervision relate loose relationship hierarchy start bag image herein codebook employ probabilistic latent base signature topic unseen class object visual object perform signature representation publicly cifar effectiveness rest show description representation bridge transfer shot issue class attribute categorization attribute indirect approach image subject subject rich language supervision description commonly though attribute shoot intra belong relationship replace order human supervision need shoot attribute topic attribute free jointly semi latent space modal attribute motivation attribute focus object learn topic unseen eliminate consume human annotation replace topic latent model latent dirichlet allocation model topic representation object class shoot conventional image tag insufficient shot attribute redundant many evaluation effective tag codebook concept utilize stage concept common cluster deduce specifically integrate image namely coarse hierarchy topic among approach attribute inter inter similar shot et model wikipedia represent object object million document wikipedia visual embed devise al concept class unseen classify object relationship computational knowledge instead relate unseen class class learn shot attribute direct use shot concept inference random rf decision recursively split training shannon probability codebook codebook occurrence collection codebook joint document infer unseen zero shot paradigm handle concept discuss infer class shot learn nested figure broad coarse narrow visual concept coarse concept share conceptual fine coarse relationship concept device water house codebook example forest example tree tree build tree difference codebook codebook similarity codebook fine coarse fine substitute shannon utilize codebook codebook drastically belong therefore j fine codebook rf total tree similarity associated histogram codebook bin joint coarse fine c codebook shannon yes yes yes summarize property still rf tree splitting codebook handle tree utilize rf splitting split zero see unseen collect associate class belong conceptual describe introduce novel namely associate create relationship unseen class unseen similarity could relate signature set infer shot predict concept unseen unseen rf codebook use either codebook j codebook codebook build calculate signature pick relate test employ public cifar pose visual challenge term illumination classification perform unless cifar extract pyramid histogram pyramid bin instead put codebook rf therefore descriptor globally nature codebook shape well patch rf codebook public face face random extract similar attribute feature identical combination find nearest unseen class choose relationship propose accuracy compare al increase system histogram expect unseen c feature relative number number unseen category perform test test comparison consistency handle intra variation annotation c propose beyond unseen concept cifar class unseen use testing major exist people dataset resolution pyramid feature learn method compare outperform unseen achieve unseen computational low employ also see class drop class indicate robust capable cifar dataset collect codebook learning strategy utilize rf c unseen electrical
play move eventually move player however since belief condition b probability hypothesis weather sum hypothesis subject player make move move possess draw action prescribe odd underlie correct low correction identify transition state state low subject assign every transition lead remain show reader mathematically conceptual motivation arise account device player correction causal subject belief transition save hypothesis weather htbp term belief illustrate nest vertical axis axis carry conclude calculate belief unchanged observation indistinguishable action read subject weather analogous calculation purely observational virtue action conclude weather weather model intend replacement framework provide common abstract contain minimal causal detailed limited countable axiom causal event causal causal dependency dynamically cause come force suitable cause gene restrict causal le stress causal seem view boundary boundary maintain identify removal tag event attribute subject self subject unity life identity classical cut distinction similarly control artificial intelligence distinction indeed intelligence describe construct plus develop whenever intervention intervention belief subject probability fix induction equal intervention belief along plus measure possess original measure contain trace intervention form action causal intervention suppose intervention basic treat event observation learn gain distinguish world classify classical theory hypothesis explanation light translate somewhat statistically generate external world subject term regard play word numerous reason choose status efficacy study conceptual framework intelligence great modern idea otherwise prohibitive literature stress mathematical agnostic reader indeed compatible interpretation idea forward causality writing wish anonymous suggestion manuscript abstract causality thesis biological principle grant office research modern theory use way provide outcome experiment mechanism bring quantity choice probability phenomena measurement reveal discard generative subject aim observe choose distinction crucial identical stage either stage randomly decide exclude experiment odd right odd last colour swap yes pick colour white figure set protocol player name flow htbp carry auxiliary device fair select prescribe previous odd ball give stage probability space eight outcome measure figure eight probability cccc pick colour black right right white yes white yes yes contain space plan outcome swap colour construct generate outcome colour swap perfect accordance law experiment obtain plausibility black pick belief probability outcome instance first paradigm instead let observe determine take turn randomly leave stage ball protocol swap colour make probability subject belief last stage behaviour stage interaction familiar analytical case sequential order swap change calculation previous exception carry stage experiment whether hence statement swap yes swap attempt first regime information specification possible drawback belief axiom equation tuple attempt add auxiliary whether swap form treat another however fundamentally extend swap extend swap swap consequently would variable indicate whether infinite indicator conceptually swap experiment situation specifically pick right independent follow law odd pick albeit probability outcome new law list experiment swap pick black leave yes yes white yes black yes thought reveal requirement change probability reach consequence knowledge familiar probabilistic correct concrete recall pick infer plausibility would choice first experiment conclusion probabilistic context learn conclude derive game brevity exposition connection game player variety mechanic move chance move game root leave player illustrate internal take move terminal label pay notice strategy walk away empty assume white make move pay bring replace player move strategy choose suboptimal move fig drop pay description white previous know time move game game player von information belong player semantic reach correspond extensive specify player decision partition htbp loop around set omit brevity fig result contrast strategy knowledge see good hard von restrict game result thought precede part causal dag htbp dedicated hold status abstract empty iff member iff neither follow say axiom see set tree nest subset root limit reasoning impossible order partial endow correspond intuitive depend determined nest possible event partition recursively branch reach leave termination class namely union potentially say collection subset consider two turn think possibly exclusive base representation exist axiom induction respectively v n cv n must choose false member prove equality intersection member member axiom place unconditional probability probability connection measure view henceforth drop disjoint q causal provide support skeleton construction visually fig figure htbp measure completely exception introduce observe subset take aforementioned uniqueness two measure stable intersection either either member next lemma measure agree agree prove previous ready define object space serve dependency event causal tuple contain information represent probability say immediate consequence essentially compatible equal importantly unique causal say give causal rise crucially differ causal additional causal serve relation functional dependency outcome direction interval define q q interval previous close closed context two sub process initial different define former instant instant determine causal course notice drop subscript close start instance consider interval unique point discriminant unique discriminant lead contradiction since must true similarly partition prove part part always interval capture split exclusive causal branch give identify respectively process generate representation iff figure member instance appear htbp member algebra generate theorem countable countable repeat construct representation respectively must member similarly imply countable arbitrary countable follow analogous countable ready define space definition intervention define event mass intervention algebra say iff intervention think remove transition root uniqueness member intervention nan intervention check definition axiom contain intervention intervention member algebra compatible intervention eq intervention contain moment branch lead remain intervention branch generate iff gain critical formal measurable value member collection algebra measurable learn far specify understand causal one expect necessary space respectively say picture illustrate also converse representation x measurable converse ss next define endowed causal intervention algebra intervention abstract intervention variable algebra intervention causal interval contain define share perfect accordance far instance space induce member root leave value assignment tree path necessarily measure variable illustrate possibility model dynamically causal succeed causal dependency control obviously causal critical highlight intervention pick direction interested bring intervention critical mutually exclusive circumstance causal pick intervention change theorem successful framework reasoning define notion theory must theoretic theoretic inside extensive game player example induction bayesian causality modern development political shift collective notably axiom put think everything else proposition fundamental subject operate heart economic system education subject sciences inter experience scientific recent advance massive capacity internet improve process scale history medium stock user aim system know hold responsible progress pose novel old one range investigate basis learn make understand question address adequate mathematical subject enable discussion program argue implicit basic concept counterpart study conceptual second I forward probability need causal measure theoretic causal intelligence economic back early discussion idea follow trend fundamental concept seem several dominant see subject study discussion firstly theory secondly abstract nature subject entity unity vary different account speak acquire separation inside rest early belong instance distinction term crucially subject divide belief belief know distinction latter constitute interpretation terminology aforementione describe subject self word responsible reality image symbolic pre structure language entity linguistic material language thought detect possibly cascade crucially association establish related logic computer symbolic subject experience namely perturbation subject pick symbolic thereby belief finally question incoherence consequence point chain randomness entail post detect pattern mathematical prominent theory formal synthesis lebesgue kolmogorov modern start degree hand observe action reflect belief probability logic limitation account subject statistic world belief update govern bayes theory capacity recall theory broad include thought subject throughout life involve randomness form algebra subset operation comprise complement assume pick nature device something phrase pick conceptually algebra universe question proposition aspect extract via symbol complex collection potential hope furthermore three typically next play conceptual constitute associate ground symbolic reality however require symbol boundary model symbolic flow occur particular respectively causal intervention causal relation relation static symbolic analogue theory systematic appear idea causal intervention ccc symbolic space flow intervention establish economic term hope aforementioned suffice summary fully mathematical draw logical inference subject shape necessary formal thorough requirement defer synthesis theoretic interactive subject summary main idea therein always central aspect explanation discussion year receive attention partly express prominent figure logic recent decade computer causal rigorous thorough exist arguably causal draw informally intervention hold choose acyclic influence mention tree capture rich causal although aforementioned definition scope degree ultimately mathematically much later
speed feature drive involved contour simulation construct aim connection cell horizontal preference stimulus intra connectivity pattern find across specie include difference specificity axis orientation affinity matrix fundamentally orientation prominent visual stimulus velocity seem play orientation selective prefer orientation nearby motion direction horizontal strictly order segment align environment segment dataset depict assign instantaneous embed r velocity describe motion affinity understand extension provide improvement velocity change unit velocity way fix maximal velocity assign belong velocity vary random bottom row space r velocity producing contain stimulus magnitude orthogonal velocity apply section use fig affinity velocity evaluate grouping compute number stimulus number point incorrectly noise incorrectly recognize partition correctly obtain repetition change stimulus calculate kernel grouping region proportional percentage repetition stimulus kernel vary stimulus velocity assignment performance orientation velocity separate correspond stimulus compose element low grouping curvature contour stimulus path minimum approximately error dominating indicate width fan stochastic connectivity contour separate worth coefficient mainly length element distant element induce recognize distant give kernel potentially distant contour grouping correlation stimulus contour grouping mostly contour generally contour grouping tend together element order contour connectivity diffusion contour successful property image discrete add curvature scale numerical connectivity argue curvature scale concept deep aspect address analyze non velocity inspection contour curvature orientation however significantly completely eliminate random influence correlation detailed parameter length arc assign explain velocity choose contour diffusion constant grouping capability preserve result comparison obtain spatio view rely affinity temporal assigning point spatial operate instantaneous high indeed eigenvalue level level partition result row element bar stimulus geometrically affinity integration quality angular diffusion kernel coefficient kernel stay value subsection kernel perform connectivity consideration effect velocity contour visual study visual grouping contour trajectory set motion contour level trajectory describe two parameter sensitivity change velocity reflect connectivity high composite spatio surface relative bar level separate level random element total grouping result random noting retrieve fail contour thus partition though inspire carry grouping understand mechanic segment temporal position orientation connectivity propagation direction movement shape v mt measure stimulus orientation solely tune stage subject movement direction speed refer continuously example orientation selective position contour velocity contour tangent paper model construct grouping spatio capability geometric low vision mechanism already indicate spatio play important suggest concrete allow primarily spectral clustering spatio feature space velocity present geometrically isotropic belong contour form spatio surface visual grouping affinity stimulus position detect capability stimulus analysis show velocity affect low percentage spatially extend grouping time extend previous affinity instantaneous affinity position activation detect cope causality evolution work asymmetric one connectivity segmentation spatio realistic assumption behavior like model neural aspect visual task address mean delay dynamic certainly matter aspect understand physical observe implement spatio plausibility accordance framework generalize extension depend extension lead modality discuss tune visual behavior mechanism apply rgb rgb definition well understand property stimulus execution global visual segmentation concept visual task underlie visual experiment present orient patch align along continuous study phenomena stimulus patch recognize co linearity co compatible functional primary detector range link orientation directional operator specialized organization naturally model integral curve algebra plane within seminal therein global movement velocity spatial stimulus motion path level specialized cell spatio temporal spatio visual neuron experimental motion integration lead extension stimulus provide geometric contact position time detect feature structure purely association mechanism extension capability trajectory task spatio temporal completion accordance aim capability spatio temporal grouping address refine structure properly describe dimensional spatio weight whose affinity connectivity grouping laplacian simple deal hypothesis algorithmic focus geometry temporal connectivity stimulus capability visual system implementation principle apart study address problem association motion grouping grouping co circular natural basic actually neural dynamic visual role indicate strong motivation segmentation property could grouping plan arise spatio architecture detail connectivity graph basic principle adopt affinity spatio geometry different constitute spectral artificial feature mechanism spatio temporal segmentation contour shape also relation intra european fellowship prop european community architecture visual contact diffusion admissible take compare behavior layer pathway visual stimulus make reconstruct filtering cell spatio selective cell basically local directional stimulus uncertainty hand cell dimensional eq spatio frequency spatio simultaneous worth capture tune depict subset optimize orientation fundamental stimulus consider fixed spatio temporal image space filter v interpret spatio directional stimulus derivation maximal direction smooth level always orthogonal vector field coordinate q text wise organization primary spatio temporal image hyperplane two represent orientation velocity present induce surface tangent complement hyperplane call contact whole name contact contact several contact among geometrically field concrete along aim contour propagation along force diffusion frame streaming differential eq brownian motion diffusion field mechanism aim move contour spatio spatio segmentation apparent propagation force diffusion process density eq process assign spatio stimulus diffusion worth static consist along associate provide stochastic connectivity appropriately connectivity normalize interpretation process increment see decay represent propagation replace transform another identically consider reach evolution intermediate depend evolution uniform keep track length path notion connectivity worth diffusion visual curvature simplicity treat address outline numerical compute connectivity lack notable obtain several approach flexibility differential carlo discretization use loss discrete covering subset path eq value number pass divide multiplicative result connectivity compute deep numerical reference vary track connectivity kernel stand diffusion diffusion parameter projection connectivity kernel together e vary counterpart contact horizontal curves orientation association extension association spatio orientation velocity see relate fan curve motivate role spatio temporal neural path leave kernel projection variable projection horizontal curve spectral graph previously geometric task broadly problem locality literature huge address reader therein cognitive spatio temporal visual grouping interpret spatial visual three dense embed throughout system normally object lie environment dash embed field segment random orientation stimulus rise line quantify formalize set task rest consider know easily purely cluster branch devote development spectral technique address issue property symmetric affinity construct locality preserve embedding set project affinity matrix segregation step input datum algorithm group geometric construction associate spatio visual vertex originally real vector basic grouping partitioning recursively separate foreground argument improve minimal essentially affinity see symmetric affinity reversible row normalization matrix normalize give eigenvector node edge connect result diagonal eigenvector piece constant function affinity matrix version possess binary spectra purely make pose ideal case affinity point weakly connect several normalize affinity relax cut nice probabilistic real choosing possess relevant cluster look maximum particular cost g adopt fix cluster eigenvector block spectrum order decrease smoothly fig ill pose case facilitate suggest sufficiently close threshold spectrum see transition walk eigenvector row thresholding parameter dynamically assign background table h build affinity upon affinity order decreasing define integer belong fix join less remaining partition fig endow isotropic affinity choose cluster result decay gaussian intuitively describe cluster kind similarity visual second affinity fig perform assign remain noise worth many mostly element orientation position orientation together affinity try separate boundary contour additional suggest consider point contact besides purely presentation whether plausible responsible visual discussion connectivity constitute step motivate connection neuron income heavily incoming discussion possible implementation field symmetry break sufficiently eigenvalue input eigenvector connectivity model eigenvector locally unstable hence generate activity aim reproduce combinatorial principle unit stimulus eigenvector since relative weakly stimulus concrete step already step almost neural eigenvalue magnitude population gamma aspect mark substantial computation proportional activity artificial correspond spatial spatio temporal stimulus different specialized neuron kind nan feature stronger weak locally consider value simplification capability deal feasible represent synthetic generate feature indeed seem connectivity one stimulus affinity connectivity compute compatibility contain one indeed kernel markov generator adapt theoretical setting couple geometry carry reasonable neural model cell spatially feature connection selective connect kernel hermitian mean fundamental fundamental solution angular rotation transform kolmogorov symmetric hand angle degree turn angle process properly reason deal modify cell detect rather geometric connect along angle orientation apply differ affinity depend role connectivity spectral stimulus stimulus segment position center
study lattice state early consider size design complete lattice miss miss disk mcmc lattice disk report mean deviation average conditional contain true parameter latter size lattice average depend increase lattice complete similar design lattice iteration incomplete disk iteration incomplete quantify regression base fit model increase like root roughly lattice iteration mcmc lattice lattice lattice lattice time consider em efficiency profile conditional ts independently maximize note expectation dataset monte ti generate use substitute function maximize generate complete lattice miss disk three parameter close density carlo method prediction also e spectral implement give random disk disk disk multiply table em estimation method replicate note definition difference approximate mle mle comparison mean square define design parameter inaccurate largely negative histogram unimodal fairly notably strong compare posterior deviation ml error maximum note substantially method km km relate fine scale variation vs estimate standard agree quite estimate identical method mle exact mle mcmc error bayesian method likelihood method algorithm find incomplete cholesky composite stein conceptually require updating challenge periodic embed lattice well c plan publication multivariate spatio report reason sampler unobserved embed lattice require infeasible simulation simultaneous updating plan matrix block fourier namely multiplication quadratic summarize inverse multiplication respectively vector multiplication involve vector multiplication require algorithm solve symmetric positive solve equivalent relative tolerance matrix inverse imply block likelihood q nm zero one define block conditional form j j j approximate ignore constant fully lattice domain store unique write sum pt google propose miss likelihood miss surface composite approximation augmentation markov spatial lattice environmental science realization stationary process extremely value make likelihood exact lattice spatial widely lattice value composite method spaced approach maximum spaced field krige process need base recently stochastic function increase feasible solution e one converge paper propose new lattice view periodic value periodic augmentation realization periodic efficiently markov monte maximum carlo em simulate practice compare complete lattice miss approach full probabilistic composite recover surface method application introduce process lattice embed likelihood illustrate extensive stationary isotropic z goal n spaced likelihood computationally require rectangular lattice miss toeplitz toeplitz block incomplete lattice rectangular likelihood require lattice datum incomplete domain lattice consider periodic length embed block augmentation unobserved location embed compute embed simulate lattice lattice highly assume evaluate lattice periodic rectangular matrix fourier simulate random variance eigenvalue random field embed number positive problem require large prohibitive embedding cutoff limit use isotropic modify cutoff compact ensure periodic provide certain covariance definite figure extend scheme lattice miss lattice disk shape panel minimal scheme embed lattice illustrate embed cutoff size miss disk embed lattice lattice periodic complete point inverse matrix advantage multiplication compute excellent stationary grid efficiently embed covariance unconditional fast fourier generate periodic covariance ignore transform product multiplication operation observe datum observe unobserved location denote lattice observe z nn conditional observe infeasible store conditional matrix cholesky substitution direct simulation avoid cholesky proceed two simulate complete field unconditional approach note solve require iterative solve compute exploit form u u conjugate system system system original appendix stop iteration tolerance tolerance multiplication former partition compute multiply also element solve multiplication two fourier transform multiplication cost ideally criterion low include block block incomplete cholesky decomposition generate part imputation maximum expectation consist elsewhere infeasible approach conditional complete parameter avoid computing covariance matrix determinant therefore initial algorithm proceed maximize complete use newton conduct lattice generate isotropic z exponential ratio contain squared covariance still effect variant cutoff describe cutoff quadratic zero approach definite embed allow parameter choose trial em algorithm change size variable reversible use simplify throughout trial simulation lattice function lattice cutoff behavior become dense lattice
quadrature loo handle integrate loo integration make deterministic equivalent importance parameter quadrature also importance loo propose quadrature approach robust tail original maximum easy case numerator close bias difficult towards truncate truncation raw idea limit avoid tail capture level deviation far already show usefulness truncation quadrature loo loo loo well loo asymptotically bayesian loo provide refer originally mean density density mean training density optimistic interpret interpretation correction gibbs change describe criterion posterior quadrature integration series prove loo examine loo apply around expansion leave predictive density expansion expansion match expansion loo expansion loo negligible contribution show gp low observation posterior close log predictive thus clear happens depend accuracy marginal instead could series loo desire accuracy cauchy concave high limit monte quadrature eventually loo numerical drop dependency loo cv approximations handle importance weighting level full style posterior review instead reason experiment property review loo list four datum one available internet classification likely skew often approximation classification set affect difficult loo cv select often analyse survival result report probit probit probit probit logistic censor square function separate scale except probit use censor toolbox tp loo loo loo fact fact la loo loo classification datum performance approximation take length gps flexibility difference density interpret fit relative length get small full loo marginal loo happen locate corner ep loo use cavity marginal cavity distribution look loo estimate large quick la loo ep loo start fail vertical tp vertical dash line flexibility vertical flexibility show combine loo loo weighted hyperparameter work unweighted map importance tp loo loo la loo tp l loo loo show la loo loo provide fast distribution prediction point distribution global quadrature loo give accurate ep loo ep combine accurate full ep loo ep loo fail relative likelihood group multi class low grouping loo loo acknowledgment thank acknowledge resource project response loo derive leave express mode loo solution computational difference treat taylor expansion give two derivation classical example remove change remain variable quadratic linearity likelihood define collect contribution give recognize approximate coincide introduce approximation second account explicit removal likelihood leave simply divide square obtain response equation log likelihood non define obtain remove term mode ie ii side indirect due last contribution linearize laplace get mode derive linear use article model laplace propagation validation forming accurate generic predictive leave laplace propagation cross assess bayesian include bayesian leave loo validation laplace review approximation leave computed cost form explanatory variable variable notation use denote focus scale latent joint prior covariance applicable posterior generalise gaussian clarity interested application expert simplicity logarithmic application specific logarithmic bayesian cross future q approximated validation shift estimate density interesting may reveal influential straightforward validation posterior leave replace fold cv approximation computational form full comparison already show likelihood want usual practical map ep integrate latent change substantially integrate whether improve predictive base continue well predictive additional expectation propagation paper good property linear review notation discussion prior mean covariance characterize correlation zero q variance multivariate gaussian pf analytically need la form latent unnormalized approximation propagation approximate non normalize gaussian site pseudo normalization ep likelihood ep update site site first remove cavity marginal analytically cavity form leibl distribution moment moment use dimensional site approximation match single sequential ep site approximation update parallel ep laplace construct taylor mode posterior marginal laplace write leave marginal brevity drop exact likelihood approximate marginal represent except local locally different marginal approximation cm explanation la la py ic global py ic approximate ep ep use marginal expectation ep denote variance pseudo site simply simple improvement ep ep laplace write site order way cavity response ep la new gaussian predictive account also term approximate laplace ep marginal consume global marginal correspond marginal intensive find taylor method refer la cm correction take taylor mode la discuss value interpolation model fit marginal density ep ep cm factorize term improvement la practically global slow ep approximations small approximation previously posterior integration commonly method marginal posterior approximate posterior narrow may negligible marginal density loo section review loo list base loo importance weight quadrature integration truncate quadrature integration la loo loo loo ep obtain match term loo taylor loo drop bayes add correspondingly remove integrate loo analytic monte version affect result loo unlike leave predictive density often easy sometimes leave alternative remove impact furthermore produce integration gaussian equation leave joint use matrix result predictive integrating loo ep explicitly cavity form site observation loo posterior loo obtain product ep ep loo likelihood analytically generic quadrature method usually use converge cavity cavity cavity accurate loo visually marginal marginal improvement shape moment loo accuracy response theory loo also consistency derivation cavity approximation write unnormalized cavity compute loo numerical loo response theory compare loo equation restrict obtain carlo posterior approximating drop importance proposal loo importance form importance weight explicitly leave loo weight reduce presentation tail obtain loo mass loo towards full loo cause harmonic unstable harmonic marginal harmonic sampling loo effective sample q normalize furthermore detect variance weight loo truncate truncate mean towards provide use importance
lasso paper receive claim topic focus topic explanation centrality natural choice merely report area detection discuss undirected community analyze b undirecte connect think union subset call stand nothing across community community belong I community network cluster approach profile pseudo key modularity lead eigenvector idea first recursive approach method chen profile thousand slow al profile aim improve speed price ignore make tractable score spectral ratio community adjacency associate simple matrix community remark degree remove ratio propose undirected analyze type extend score score network citation remark different measure adjust rand ari information vi large ari vi predict vector component figure plot suggest run record vector label largely inconsistent vi ari vi focus vi ari ari method moderately inconsistent size identify community dot white circle represent four community north community researcher north north researcher parametric parametric compare major lie fan score fan community north explanation fan tie suggest instead assume method inconsistent reason score north follow include branch north cluster connect regard result score meaningful differ several small branch two branch fan htb htb component large dimension reduction node sophisticated theory yu research li help meaningful evolve university bridge connect de berkeley ph berkeley group ph department university start li large west stanford include david etc quantile experimental group ex ex di david ex ex ex j ex ex ex ex de ex ex david ex zhang ex ex l frank ex b also hard node consist call primarily detection present apply assume finding compare correspond ari vi somewhat surprisingly inconsistent ari vi show substantial reasonably ari label ari predict agree three community identify interpret arrange size objective bayes researcher range size variability community triangle north include ann university dimensional size three range include researcher variety area high g bioinformatics community r community identify worth note mostly university ann behave differently either counterpart score identify size community compare panel top community fall community score second fan think belong community branch north cluster north community member b group bayes b score citation direct network way additional network detection citation network htb direct cite citation usually focus component connect citation network edge cite citation community network spectral modularity undirected network modularity however properly direction represent therein method undirected adjacency network think split disjoint community bernoulli heterogeneity degree heterogeneity motivate community network adjacency let leave singular define node edge node cite common two cite least network respectively note separately restrict cluster assume community community partition group th community citation node assign large sophisticated citation n illustrate citation panel cluster suggest community section lk citation axis axis column index show blue bar dot identify multiple citation node weakly account restrict attention associate respectively detection present plot inconsistent score briefly identify scale testing node researcher include bayes berkeley stanford groups david david fan lin li zhang taylor lin yu lin spatial nonparametric short discussion figure consist hard interpret end restrict network I ignore obtain component parametric statistic david li nodes stein parametric statistic david wang communities present htb order understand community network column connect semi parametric spatial c dim exp var selection test among citation semi parametric sort wang chen lin li zhang lin david group north tie tie group score section score assign citation network remain network theoretical stanford strong evenly three community citation former weakly connect citation latter focus component comparison first community two part part researcher close tie lee selection community network seem many node chen li consist total lin yu lin j wu david zhang reasonable yu lin david zhang community part include wang david high taylor lin third node nod david david david cox subset non testing community citation wang second part large testing additional insight testing part consist subset community second close another significant researcher bioinformatic htb bayes citation parametric hard researcher model functional etc sort lin david rao dimensional high sort fan lin community researcher nonparametric david mac stein vi vector community moreover community score observation properly detect investigate centrality community exploratory tool sophisticated tool network present array interesting paper spatial objective machine also set collect limited paper publish year period recognize core limited science science economics finance science recognize one david period bias present serve home serve network space focus paper discussion brief set provide array centrality score detection network underlie method analysis also sometimes detection inconsistent happen light theoretical framework strength interesting issue mixed relationship national influential work popular future citation information study research trend community informative abstract studying report finding pattern trend author suggest per year total number paper publish average number paper result largely year paper paper decrease drop ten collaborative area people enter area increasingly difficult view top wider make substantial also present paper author author way count divide approach way cause insufficient author contribute axis paper author approximately look straight distribution tail htb approach panel suggest contribute top coefficient dispersion paper physics community seem publish evenly another span year set four statistic period usa range fan david high degree author degree suggest investigate time panel year see network community mathematic community physics usual moderately author citation per largely cite neither cite receive highly skewed highly cite paper receive citation coefficient suggest highly skewed observe pattern return favor citation early among distant period proportion decrease roughly proportion distant slowly probably communication increasingly easy blue cross leave probably effect publish cite publish include datum delay citation later overall proportion self distant respectively confirm appear online website department early publish overall delay e publication paper mean distant year quickly overcome focus publish object journal number cite raw set paper remove item book review correction etc title remove leave paper information citation directly every collect author keyword abstract journal etc challenge online strict eventually overcome science little source find good paper could serve format resource paper paper substantial successfully identify web paper combine citation relationship information one mention paper effort uniquely author paper interest consume publish name middle author cause wang wang wang wang second name list consistently example list li three california ann li internal none user also people try hard author spirit introduce however use program use author name g may manually author additional e email address file reader www publication file review correction remove review correction author author rule name cluster manually define cluster name author txt list author txt bipartite adjacency txt adjacency citation txt citation jj david fan li partially grant network paper published analyze focus centrality community pattern trend cite meaningful community group statistic well one machine find author distant suggest increasingly collaborative competitive finding topological ground relate social frequently interest area scientific topological researcher understand useful range researcher research topic citation network convenient address question hand resource e google convenient collect citation network provide community many aspect also study help assess study recognize people tie researcher effort pay collect aspect network know community structure community truth analyze effort collect citation set base publish half statistical journal american association journal provide social truth set theory understand topological structure last project collect cover long period analyze network centrality area collaborative
small consist document article simple document layer multinomial visible see minibatch set th h x derivation bind h h expressive belief inference none approximate propose non sampling inference jointly maximize estimator apply variance reduction sigmoid belief network show outperform mnist achieve powerful globally normalize deep fairly counterpart system highly latent challenging difficulty pose tend suffer simplest difficult state observation update provide alternative method efficient approximate fully factor variational posterior expectation analytically however highly expressive one expectation simple variational difficulty variational sigmoid belief net optimize log fit often derivation propose combine feedforward implement inference maximize gradient although network apply technique result feedforward compare many exact highly independent much store observation training discard handle discrete continuous latent variational complex dependency employ sophisticated variational bind primary naive practical range applicability show train sigmoid well capable effectiveness scalability state intractable option variational standard variational serve exact simple easy kullback leibler divergence distribution maximize variational distribution well variational observation variational approach feedforward distribution architecture inference architecture deal architecture approximation locally maximize scalability estimating gradient variational simplify notation parameter involve involve intractable special carlo objective anneal schedule convergence however heavily section gradient behave pose scale inside potentially result slow gradient next practical estimate useful practice eq inference effectively seem want distribution turn distribution affect see depend affect equivalence evaluate price pay estimate suggest subtract simple make adapt systematic subtract observation affect expect depend elaborate implement though account magnitude baseline improve incorporate baseline contrast elaborate baseline distribution easier center trivial magnitude dramatically variability fix center run normalization rate stop signal great computing update provide supplementary material make assumption however advantage property noisy instead global signal involve remove term signal layer latent posterior naturally denote layer learn variational use law iterate rewrite use drop without signal signal simply layer signal expect within structure apply inference network whether factorial case far yield leave explore idea training approximate variational go back derive autoencoder encoder inference respectively however gradient model infeasible feedforward perform initialization boltzmann machine recognition match marginal involve limited inference call stochastic bayes feedforward model perform approximate train optimize considerably use model handle inference dependency benefit treatment value converge fast relate sigmoid belief framework feedforward approximate concentrate deterministic handle latent thresholding unit ignore thresholde variational absence generation feedforward year optimize use unlike traditional learning network fully factorize field share enjoy scalability applicability wide algorithm introduce machine augment analogous updating phase update eq recognition generate cause model distribution optimize optimize bound recognition easy estimate training network high estimate naive section improve see reinforcement learning depend output update output reduce thus serve considerable baseline reduction rl likely contain intended demonstrate handle generative randomly use early base configuration see input dependent hide unit superior anneal report performance preliminary experiment rate report dependency within signal instead report variational bind considerably evaluation perform benchmark generative handwritten test training model center subtract work layer baseline normalization dependent baseline appear compare two gap combination exclude reduction effect
run current modeling approach extend naturally work datum accurate look infer simulator produce tractable detail free set eq vector modeling simulator generate indicate simulator pseudo parameterize control approximately infer delta repeatedly act slack however prefer improve unfortunately trade approximation large large rejection sampler marginal sampler iteration mcmc simulator accept parameter denominator carry marginal unbiased interestingly view lead sample sampler suffer eqn attain mix away sometimes denominator numerator procedure convergence mix lf marginal interpret fluctuation propose approximate uncertainty acceptance fluctuation repeatedly produce distribution clearly delta confidence allow local discuss pseudo sequential monte next discuss introduce pseudo parameter e replace simulator order also eq analytically give satisfying simulation imply bias use similarity motivation extremely sampler accept result see analyze decide sufficiently confident accept study general case eqn explicit sampler accept probability simulation would similarly significant replace normal expression randomize error either condition integrate unconditional error cdf accept mh probability actually probability accept error unconditional monte carlo analytic cdf hand tool adaptive start mh fine draw user threshold accuracy mcmc simulation higher uncertainty around mh confident usual actual another close nevertheless remain serious expensive simulation mh section simulation improve consequence eventually eliminate input c eqn mention introduction simulation extremely expensive mcmc unless store accept perform provide gaussian purpose simulation conduct simulated able away simulation marginal us conduct confident accept decision synthetic likelihood frequentist favor nonparametric literature surrogate surface surface directly independent process well model single joint co although may assume robust full gps experiment gps well enough bivariate distribution covariance input kernel evaluation th acceptance evaluation eqn gps abc mh eqn bivariate step expectation take carlo mh input force current step analogous acquisition implicit goal mcmc simulation training training output hyperparameter may key gps frequency simulation surrogate confident region gps uncertainty introduction consider reduce ingredient gps abc differences procedure aspect gps abc approximate gps abc proof approximate step chain stationary distribution mild bound abc fit proof approximation stationary add two adaptation ergodicity gps abc acquire decrease adaptation resemble ergodicity gps abc latter experiment toy stochastic biological system experiment correctness synthetic henceforth gps gps abc demonstrate illustrate output independence demonstrate along parameter spirit simulation run marginal abc unconditional mh additional vary mean different gps run one simulation great involve gps illustrative gps statistic generating rate parameter exponential vector draw distribution observation statistic simulator process exponential e generating experiment seed experiment show row abc histogram abc abc magnitude simulation bias bottom right gp figure algorithms kernel abc abc likelihood gaussian posterior dash line discard run sampler gps point run sampler algorithm use kernel abc toy pseudo sampler rate abc slowly note smooth measure may per adaptive simulation gps significant kernel abc require simulation gp model mode shown dash interior training red circle uncertain axis mode indicate line uncertainty indicate population compete population use simulation produce setting model simulator replicate series along generate series generate q acceptance total significance maximal maximum replicate base log series degenerate etc simulator ht last require simulation call gps call compare use gps abc abc abc twice gps control step mean gaussian deviation dimension scatter plot abc interesting relationship inspection mode abc due gps abc run estimator force gps full interpret approximation difficult determine estimator influence enforce likelihood lack abc call abc million expensive versus heavily gps abc gps abc top posterior similar covariance posterior potential gps apply cell cell terminal begin initial update statistic log consideration show simply arbitrarily broad figure experiment allow number change bottom miss value solid curve distribution draw three predictive whereas shift towards value row row case predictive large compare distribution remain demonstrate checking algorithm reason choice check pre part observation simulator simulator b n
c profile equilibrium eq q player nash player game finite player player type player transition resolve along infer opponent adopt odd game game similar characteristic corollary game various economic field importance political biology describe solve rational interested model opponent player behavior infer move player motivation present introduce whose game try know repeat incomplete describe game state know actual nature player actual past payoff action know markov stochastic player actual state transition game recurrence game lack bad formally repeat player player probability state type introduce game payoff know know type action transition know solve player hide compare way stochastic process exactly observable state observable know idea get weather help office office day weather weather possibility depend worker observe people come weather know weather state consist state transition store state store observation state produce array store state htb transition label describe game inspire player hide payoff opponent player player state case player know opponent opponent markov game player transition sequence behavior opponent due tool game aim game chain play accordance game bayesian inference accomplish markov observable equilibrium infer player array storing finite array store state nash store opponent playing game observable great else play frame versus situation strategy server try open receiver action server type choose strategy player interested receiver need opponent reduce loose obtain game hmm opponent choice use opponent payoff c open center open profile scenario original observation every observation game behavior hmm stay hmm e picture odd bayesian hmm stay increase odd train hmm close c bayesian h two take transition hide information think type know opponent game use infer type markov use method quite work present solution special lack
analysis resort construct graph adapt use rate movie movie rating movie rate rating rate movie take star share distance become natural weight w ij distance fast one nearly identical around weight fast distance increase star transfer star distance find well nn explanation user positively recommendation essential emphasize paper access movie business cross validation evaluate vary test result plot level clearly outperform observation seem nuclear regularization alone green dense nuclear reach combine observation sparsity nuclear message improve row optimization matrix offer recommendation combine collaborative solved pose iterative admm scheme system real validate recovery suggest usually people completion construction furthermore completion propose algorithm way uniformity matrix partially special weight nuclear non movie influence quality correct improve firstly scheme scalability deal matrix big dataset qwertyu qwertyu b qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu I qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu n qwertyu qwertyu qwertyu p qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu u qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu qwertyu ed universit ed de miss recent although low hard exactly matrix completion proximity community make recommender receive encode graph manifold constrain order force pose iterative study propose recovery completion namely sense recovery actually possible reconstruct appear recovery problem low sparse important world cast matrix completion problem include recommendation completion indeed become movie recommendation netflix amazon netflix see collaborative filtering widely use infer recommendation pattern miss entry recovery show perfect refine reference therein study consider propose column assume completely netflix age education etc movie release year actor origin country advantage movie rate movie recovery new literature induce factorize low good knowledge graph cluster along methodology collaborative content user idea force solution manifold movie standard graph user movie uniform achieve laplacian heat diffusion manifold non convexity challenge optimization problem recently exist completion collaborative recommender netflix popular rank denote element replace surrogate semidefinite incoherent sufficiently minimizer coincide minimizer observation contaminate depend type square frobenius mask notable representation column smoothness row close linear eigenvector row column laplacian overall simultaneous outer product wise laplacian efficiently success choice introduce follow handle constraint constitute success fact require sub rather augment lagrangian x close strong duality primal saddle augment lagrangian e find saddle iterative admm prove approach fast sub x k v singular c g h h k condition ba symmetric kronecker product conjugate gradient complexity dominate nuclear cg nn evaluation recovery synthetic netflix like model dataset two assumption column graph movie recommendation integer netflix rating movie graph group connect within neighbor edge belong likewise movie group type graph cluster fig movie often noiseless performance compare particular reconstruct report observe green line poorly nuclear line nuclear obtain alone discretized laplacian tendency try reconstruct nuclear fig quality green nuclear norm smoothness green line dash add noise connectivity green dash wrong benefit low regularization solid therefore graph graph observation observation
n final equality homogeneity exponent location totally always challenge work spectral closed form exponent equation density combinatorial dimension increase rapid respect situation expression lack close block chapter compute one prior define function true datum generate produce likelihood generalise need available importance sampling produce appropriately use observed weight algorithms order importance algorithm w useful computationally prohibitive procedure evaluate procedure develop bayesian operate parameter quickly general precise credible dataset retain conversely unlikely discard draw repeat exact yet approximation intractable principle heuristic make precise generate generate reweighte generate metric mahalanobi discard correspond receive weight retain sufficiently close reject sample draw posterior auxiliary sense quality kernel get dy dy otherwise accept candidate parameter exactly reproduce unless discrete unlikely practically typically great extreme dy dy show weight abc approximation distribution give clear fidelity lie closeness inferential low aside approximation practice commonly great quality abc univariate taylor expansion substitution h dy du function form estimation likelihood interpretation limitation abc know suffer curse dimensionality arguably impractical appropriate univariate measure unlikely reproduce observe dataset dimension simple statistic manner image summary summary possible posterior give p ds l due consideration give summary sufficient information sufficient loss precise abc precise reduction kernel offset information summary input vector simulate statistic h generate work avoid draw come abc pls pg intractable illustration block year central notable extreme cause historical flow world human economic previously daily mm maximum daily record open univariate approximately model generalise specification easily obtain posterior markov chain carlo gold specification dataset three parameter analysis choice practice abc curse summary model dimension prior implement algorithm distribution improper situation explicit set exploit know scale correlation estimate need approximately correct identify area statistic convenience smoothing kernel scale summary note definition line strictly first however procedure draw determine abc practice e posterior left centre dataset solid four centre abc decrease apparent generate independent small decrease course sufficient even close quantile million abc approximations dot line however beginning improve quantile obvious although achieve match likely occur due dependence likely element summary appear overhead vector line broad even quantile enough use slightly well posterior information seem quite increase decrease far statistic practice regression adjustment improve abc determine true unknown see abc intuitive procedure evaluate summary highly parameter price pay discuss suffer curse purpose number still sample great posterior explicitly regression correct simplest least adjustment variation regression adjustment ridge adjustment qualitatively different adjustment improve conjunction adjustment describe approximation estimate panel solid statistic abc line regression grey figure illustrate distribution focus vector statistic approximation grey line adjustment indistinguishable use analysis adjustment effectively initial relatively top bottom occur effectively unchanged varied large obtaining adjustment rather identification huge research provide comparative classified good indirect approach include abc reliably single well good currently list sampling generate monte circumstance variety importance importance sampling construct markov augment auxiliary target ls proposal l result always section tractable term detail hasting g monte alternative abc density estimate production clean term production consideration size important inclusion two dimensional inclusion great measurement threshold inferential unobserved cross slice focus extreme problem mathematical process inclusion mutually conditional approximated generalise distribution scale shape standard accordingly spherical inclusion inclusion associate unobserve inclusion diameter observe inclusion great chance term large inclusion adapting problem numerical difficulty function treat unobserve latent assumption inclusion independence plausible family addition poisson consider specify generalise define large diameter uniform ambiguity measurement principal diameter inclusion spherical analytic difficulty extend family inclusion approximate specify implement total million summary denote quantile interpolation distance either identity analyse uniform scale sample approximate dataset bottom row correspond inclusion solid model approximation posterior dash indicate identity mahalanobis grey version subsequent adjusted marginal bottom inclusion dash dot grey adjustment solid black dot marginal spherical clear approximate dot fairly identity dash high variability namely likely highly happen correlation take closeness substantial improvement clearly adjustment inclusion marginally show suggest require initial available qualitatively conclusion good accounting correlation statistic agree posterior difference reflect ellipsoid principal diameter ellipsoid intersection inclusion predict inclusion inclusion analysis suggest depend strongly likely strongly assumption computation factor datum slice inclusion choice minimum weather weather weather report pre weather contract undesirable weather occur low weather reason weather variable payment event interest threshold payment payment excess show mathematically weather positively correlate financial outcome weather derivative positively consider payment derivative recognition essential max process stable generalization extreme hold subset process stable stationary poisson intensity replicate max fr margins stable convenient interpretation center independent take process extend stationary intensity measure eq fr margin bivariate underlying correlation family correlation ern realization mat ern correlation range summarize simulate realization bivariate sample term location get location bound coefficient complete dependence think independent location consideration pair eq observe bivariate show write coefficient explicitly transform fr coefficient eq take coefficient grow rapidly dimensional statistic prefer add detail triplet value draw fr margins location average produce absolute deviation measure parameter quantile maxima daily take location united record locate west fr spatial parameter aim candidate process location credible parameter space shape shaped shape function incorporate parameter solid posterior line either evaluated numerically write form various model complexity intractable require simulate suitable summary former trivial subtle accurately simulate continue
force eq besides also effective return pareto behaviour true normalization enough correct behaviour return return normalization guarantee cover critical area difficult objective case parametrization force pass limit show without concentrate center distance area normalization correct behaviour slightly seem try convex combination obtain normalization without try produce behaviour lead dominate converge correct completely choice figure among return pareto accuracy pareto reservoir model water store reservoir control release transition depend reservoir objective problem reader consider objective immediate reservoir threshold min due water discount objective different phase episode use policy objective radial convergent parametrization force pareto capability decide simple indicator start parametrization report start pareto covering approximation produce important policy release reservoir phase modification unnecessary penalty try add return mixed insight use thereby set precise pareto cover indicator penalization term monotonic previous take section go metric propose idea dominate solution enough figure behave mean use dominate solution dominate short pareto solution mean increase use wide achieve cover circle circular fill draw thick document material multi pareto markov decision policy pareto differently algorithm optimization perform gradient ascent run step improve idea exploit optimize get close pareto besides metric assess quality pareto finally propose empirically evaluate pareto publish conference artificial supplement complete provide smooth map volume r bb directly volume direct indirect derivative expand kronecker bb permutation last term expand I differential give hessian reservoir study paper addition present normalization take pareto area integral unitary eq combination area function idea obtain dominate area cover pass iteration trend objective study discrete quadratic follow dynamic column semidefinite couple policy extended objective particular
bs electrical engineering university institute fellowship frank fellowship paper award conference electrical college engineering college interest recognition datum storage book technical apply associate security technology award filter object track traditionally use fourier dft correlation exist account fact circular previously design truly optimal criterion eliminate ensure optimization circular benefit diverse challenge associate version http net project filter correlation filter localization transform signal processing recognition tracking possess attractive make task localization classify scene shift invariant object recognize peak location object attractive stage template refer spatial domain image concept temporal template design sharp origin image peak peak scene may large template peak object detect peak peak peak traditionally typically metric minimize correlation actual correlation frequency element fourier dft image algorithm multiplication multiplication circular correlation circular fig circular part add influence localization cf circular avoid template resolve ignore carry correlation cf use exist energy cf design design present localization obtain example circular correlate appropriately zero linear output accomplish dft circular correlation output generate dft signal part correlation effect affect design mostly ignore paper force template correlation rather circular zero consistent criterion design correlation capability offer cf span decade introduce filter extend subject extension scalar efficient descent base numerically comparison demonstrate superior main convolutional convolutional neural potentially thin thin green cm inner sep pt rectangle fill corner height fill red distance height em dft dft dft node zero p image cf template dft dft cf cf cf c p zero correlation cf template circular guarantee correlation original criterion organize summarize relate literature circular issue tradeoff discriminant filter filter margin applicable discuss design application face recognition automatic localization object detection detection detection action couple recognize image recently interest bind face track use detector image face use appearance localization object prove detect string video improve wise search recently base correlation pose video detect action motion combine activity apply vector optical histogram orient transform sift ignore problem filter spatial circular domain handle circular recently circular image circular originally cosine effect great detail undesirable fundamentally discuss fall eliminate circular effect practice image take dft although true stage train design eliminate must stage template coupling constraint optimization correlation result get cf next correlation design circular design filter illustrative notational ease expression rest cf design aim gray scale formulation accommodate g sift bold transform operator act channel independently complex transpose typically filter signal design interpret mse signal template cf express mse desire channel template origin peak center dot signal template column whose result solution filter minimize correlation equivalent closed form solution filter output sharp however term precisely multiply together circular correlation spatial circular attempt circular obtain dft result circular force zero eliminate extension explain intuition benefit template generalize template domain mention expensive begin mit long expert main peak signal class template mention zero dft conventional filter design index template effect eliminate formulation demonstrate train template length template template element template signal compute formulation term template energy circular imply peak fig filter note conventional decrease simply adequate considerably become equivalently template ensure template template whereas conventional differ significantly bridge zero circular filter filter three image person database face database face computational reason dimension template output template ht plot template conventional template outside template size template display value template ht filter correlation filter mathematical detail incorporate cf perspective group cf dot image specify unconstrained shape actual inequality constrain idea well derivation template eliminate effect cause circular template dft set note aggregate channel modify formulation minimize enforce peak force template filter notice improve requirement inspire early experiment see notice rapidly evaluate although implement impractical perspective algorithm solve descent unconstraine filter equality filter idea optimize impose constraint spatial impose spatial template project onto signal respectively describe design f iteratively allow satisfying impose unconstrained template tail I transform template outer width height thick blue drop minimum cm thick node xshift yshift xshift yshift amplitude xshift xshift yshift xshift pt yshift black xshift yshift amplitude xshift yshift sep cm height thick fill drop minimum rectangle thick cm minimum rectangle node pt xshift xshift yshift amplitude xshift yshift xshift yshift amplitude xshift yshift xshift pt pt mirror xshift yshift node xshift yshift amplitude mirror xshift yshift node xshift yshift amplitude xshift yshift yshift amplitude xshift yshift black xshift pt yshift spatial constrain step update extract portion template close template desire peak contain signal p loss descent hinge solve closely accelerate along gradient objective newton filter conventional cf conv conv prox accelerate proximal descent conventional prox conv prox prox accelerate descent filter prox conv prox conv prox prox proximal peak spatial form constrain advantageous save memory resource need system complexity fast efficient scalable demonstrate benefit descent comparison close form zero proximal method design terminate use face image platform matlab window intel core ghz ram experiment compute close solution near limit fast show perspective amount require large prefer proximal memory times filter close horizontal image recognition formulation observe counterpart original design account circular fundamentally performance realize cf design performance apply task namely automatic recognition localization object baseline cm cm cm apply database face face database face subject image cf subject result image one correlation repeat calculate energy measure compute solution accelerate proximal descent version comparison demonstrate effectiveness close extremely high computer therefore available filter delta function desire close form also true remove less vary subject select build cf subject rate rank proximal simplicity output achieve performance video database video frame eight video range note make circle long allow frame fig general one template template classify variation select manually background select frame testing frame template plane assign image class classify localization peak e half truth horizontal correct correct cf template training add false positive help filter observation traditional localization recognition performance conventional help development image base base face important accurately determine location bound face face detector experimental setup localization face detector transform translation factor rotation image people partition rest testing evaluate distance ground truth leave respectively human patch suitable great design result design descent learn version localization cf template
intuitively make additive easy false future simulation indicate irrelevant positive dc zero independence coordinate concentration replacement argue probability control false negative hold restrict regression let boundedness strong additive full proof follow theorem hold ambient attain reflect scale relevant smoothness achievable convexity demonstrate condition draw additive symmetric vary ambient diagonal combination set use ac dc mark ac dc plot exact recovery variable reader two minute ghz intel cpu gb vary extent b diagonal case design dc generate figure dc even true additive third use correlate see design moderate correlation ac dc covariate detail repository dc regularization range indicate lar ac dc consistent finding fourth variable select ac dc component ac stage zero dc capture shape agreement spam validation describe ac dc slightly select fold dc figure ac ac dc find similar ac spam optimization optimum cccc additive additive cc ac dc paths mse estimate ac dc framework estimate convexity suffice carry variable additive model convex coordinate quadratic programming establish result scale ambient analysis work building model convexity concavity program concavity grant grant amazon education learn grant convenience impose solution restrict kx coordinate ki optimization omit boundedness constraint verify item conclusion construct variable satisfy complementary c exp nc claim begin define gx support likewise minimizer follow concave analyze likewise version dc index convex ac dc argument risk error small reach risk risk th enough theorem theorem restrict cb proceed empirical risk bind entropy say empirical plugging thus lemma remove bind w f corollary least h lx verify uniformly h l gaussian vector least put c number furthermore union multiplier plug another union substitute concave step bn nk theorem step term proceed identically concave least condition np statement convergence plug fourth likewise term take dimension complete constant possibly dependent set absolute verify hoeffding union bind probability likewise plug equation norm balance choose n n decompose bound take union twice dependent kx fx second bound away bound function shorthand prove term derivative additive bounded suppose sake contradiction x positive imply gaussian scale inequality least enough least list proof take exist constant result absolutely define line px px extend convex component clear f l uk lk l direct union hoeffding inequality make expression easier suppose fx cc additive distribution verify guess verify interested constant need satisfy section lemma example machine department pa department university il sparse subset quadratic convex follow concave sparse additive smoothness appropriate setting yield false negative efficiently advantage effective screening shape restriction monotonicity convexity concavity natural estimation constrain understand minimax multivariate rigorously problem regression sparse dimensional significant statistical contrast general fit convex sparsity penalty concave residual procedure population mean result negative density second simulation analysis estimation arise reconstruction theory interest assumption natural tractable recently increase activity shape analyze convex surprisingly estimation mle latter estimation lower common convexity estimation knowledge variable variable selection adjust local estimator minimax isolated advance provably scale hold penalization theory technique hashing scheme problem show adaptively scale achievable smooth intrinsic give showing selection study perhaps working induce penalty formulate convex smoothing hilbert space tune additive smoothing regression naturally use finite convex additive convenience actually selection scale dimension intrinsic polynomially exponentially summary technical include give program reformulate efficient scalable grow full appendix provide high description technical detail precise main identifying variable convex false negative population quadratic program throughout vector denote one restrict denote shorthand convexity finite quadratic program equivalent quadratic support hyperplane dimensional quadratic formulation curse subgradient sparsity effective reason appear constraint regularization group small additive additive component appear convexity selection play approximation error component approximation additive additive approximation assumption x support satisfie discuss condition population set convex boundary property twice differentiable fixing prove show zero maintain show screen sufficient additive additive fit series concave notation additive component boundary convex continuous set respective univariate depend result naturally suggest two stage stage convex additive shorthand ac dc ac fit additive function residual nature dc stage process optimization convex function analogous multivariate represent hyperplane subgradient point equation impose support hyperplane suffice univariate characterize subgradient monotonically observation constraint solver descent sparse involve package establish variable consider proof technique manner separate rate irrelevant ac negative primal theorem differentiable involve signal make additive procedure make stage regression assume future boundedness new hold ac dc np n achievable ambient estimation reflect respect smoothness mark irrelevant mistake inherent population example convexity assumption begin additive mild integrable eq stationarity solution result use result build stationarity state e strictly convexity guarantee turn fix eq uniquely optimality convex therefore kx x fx distribution let fx particular fx example uniform mark therefore expect intercept slope pt additive understand capture call intuitive function differentiable depend suppose eq say imply population yield general support weak satisfied property boundary density follow support twice differentiable present fix slice show slice convex together still maintain convexity fx f shorthand likewise integral boundary zero integral necessarily hessian must zero column gradient respect play variable function additive twice expect condition find direction imply depend zero additive case additive component parametric function similar next certain convex dimensional appendix additive close easy additive violate concave function numerical arbitrarily mixture gaussian uniform distribution square square boundary closely approximate distribution additive integration true nonzero zero approximate htp gaussian importance boundary although condition additive define reason convex prefer free approximate abuse notation use represent function variable fit x satisfies continuous function let differentiable depend kx h f identical replace x let concave center close k kx twice differentiable away large second must eq center kx first xx conclude imply uniqueness constraint objective strongly uniqueness screen population concave population mark irrelevant fit separately second procedure ac straightforward construct screen encourage sparsity penalty estimate penalty component appear dimensional equivalent show reformulate ac additive describe estimate scalar offset support hyperplane I impose constraint suffice since univariate characterize subgradient scalar increase monotonically lead nk sorted order explicitly kkt condition additive constraint qp relatively notice couple error term optimization vector nr ik r ni k introduce impose penalty subproblem package use www com cycle covariate convergence cycle admm qp input dc modification modify inequality reformulate
brownian motion k x mm specie group specie record level tend recorded aim remain variable improve group mass motion conclusion refine well explanation study body mass brownian motion parameter check brownian motion size size analyze remain front end relationship df log aic aic bm ht remain front end notice value bm aic homogeneous unknown body suggest trait homogeneous currently available parametrize describe laplace motion comparative relax homogeneity present investigation analytical formal couple process type evolutionary author centre f foundation scientific research education mathematic author comparative multivariate adaptation date page work article author title model road journal page article author title comparative volume page author origin species article author author author title journal date page author author title comparative identify shift character journal evolution date page article author title comparative journal date page title quantitative character date author title journal volume page article title selection comparative adaptation journal volume page f title assess trait comparison journal evolution date article author author author h comparative adaptation randomly evolve environment journal evolution volume date page title evolutionary science page author title detect character journal date page author author title generalize journal page title adaptation journal date page author author author title continuous space journal date page article author author process option pricing journal date page author title g return journal volume date page article author title testing rate trait evolution volume date page author title account comparative adaptation journal biological statistical volume date page article author author title accounting journal page author author competition journal date page book author team title foundation statistical laplace majority assume evolutionary process homogeneous offer rather expensive way preliminary whether offer establish species specie take point specie come sample due history due specie trait notice birth availability genomic allow comparative challenge comparative run observe trait currently branching mean way increment trait label motion xt trait specie mean trait brownian major drawback variance estimate discuss correspond however consider complicated brownian motion comparative vast comparative limited allow sequence well something never majority laplace application consider laplace motion variance hold non interest idea laplace relate put formal mathematical evy parameter em exploit laplace treat length variance miss comparative equivalent change branch
bandwidth trick batch performance achieve sdca propose perform except sdca dataset prefer rather sdca good feature sdca sc sc preferable sophisticated decision utilize could computation requirement huge costly computational cost efficiently sdca operation prohibitive ccc mnist cifar imagenet block compare convolution filter layer specifically filter dimension apart predicate bandwidth median trick batch dimension random mae neural nets chemical predict molecular database trick batch million contribute method scale artificial randomness doubly stochastic machine successfully meanwhile achieve compare several dataset comparable sophisticated support nsf grant microsoft fellowship fellowship nsf gm nsf edu school institute technology edu science cs edu general scalable learning try hard novel concept rely express solve gradient number function income reproduce hilbert readily regime net achieve competitive net million energy million handwritten digits feature pt pt pt ex general scalable scale theoretical remain incomplete enough neural net exploit sophisticated architecture virtual deal invariance bottleneck scale storage kernel usually store attempt effort algebra kernel require operation basis nystr om incomplete decomposition follow strategy observe percentage loss performance regularity kernel typically nearly kernel good ability impractical dataset preprocessing memory feature directly approach point compute operate generalization need approach often feature well procedure solution obtain straightforward issue kernel coordinate coordinate iteration incur strategy thus computation memory requirement however serious drawback keep test big classification summary want inspire novel call rely rkh descent make functional behind property long unbiased random generator fact initial exploit memory method interestingly possess ridge logistic different type kernel adapt generator optimize flexibility method stream random computation key generation seed point complexity memory doubly allow prohibitive streaming keep program sample pre seed need guarantee nontrivial analysis involve newly recurrence relation might outside rkhs optimal introduce contribute rate kind independent interest regime net net handwritten digits mnist material million imagenet suggest method replace neural net large scale world nonparametric remainder support kernel kernel positive definite pd process play design pd x relate process kernel shift characterize fourier invariant pd kx b kernel distribution ball explicit sphere random feature design dot additive homogeneous hellinger kernel reverse kx another reproduce hilbert rkhs rkhs exist kx exist pd dominate inspire work concept issue potentially subgradient respect rkhs try linear direction kx stochastic point inspire process additional doubly doubly outside since outside b associate functional source randomness development scalable meanwhile also create analysis deal carefully mm seed mm seed key intuition behind randomness another artificial kernel intuition feature doubly functional determined seed suffice task proceed functional gradient algorithm summarize perform feature maintain collection feature align obtain need well modify convergence analysis respectively iteration simple form doubly functional size convergence mini random empirical covariance potentially expectation high rkh generalization compare gradient estimator rkh inequality estimator ahead exist optimal solution loss generate x mm present theorem due sketch proof appendix q lm eq probability fashion technical difficulty rkh construct intermediate difference f term due apply rkh randomness recursion slightly error bind lipschitz continuity jensen rate classical strongly convex classical speak convex feature contribute therefore still able log achieve adopt classical refined sophisticated reduce sgd batch tuning trade desire error quantity number memory achieve prescribe require rank functional sdca sdca sdca om method run dual similarly combine stochastic mirror nystr r htb cc preprocesse doubly sgd sdca sdca sdca r one method r sdca dependency factor interested random procedure sdca clear refined term iteration memory requirement c doubly sdca r compare medium net yes last virtual mm c cccc dim ridge yes forest svm yes yes imagenet compare seven method kernel adopt criterion purpose stop entire sc stopping criterion design motivation bottleneck ability advantage within time sc preprocesse nystr om
extract show approach efficient flow chart figure htb publicly dataset http roc auc disease relatively compare technique rest organize processing component system present detail experimental methodology section extraction classify specific reliable decision serve segmentation propose field tangent field enhance connectivity structure classify disjoint simple texture descriptor train classify image information representation reflect geometry structure signal technique extract filter representation supervise sign appear red dot efficiently preprocesse primary sign occur shape also detection combine preprocesse well figure related image htb refer vision automatic dr roughly center location define incorporate circular shaped structure area center correctly reference e system aspect appearance certain position indicate advanced rare serious like detection computer reliability separate dr detectors section basic concept literature concept formalize base dr definition vector correspond respectively function majority voting weight assign weighted majority voting classifier follow classifiers dr know classifier test select dr search classifier select far end formal dd member remove description ensemble one search classifier member ensemble contain good experimental study publicly database compressed image pixel range r clinical patient stand serious appearance proportion r r http fr feature extract also table assessment number image severe lack part screening describe detection find confidence represent normalize divide center regard feature diameter fm negative scalar confidence dr large dr description feature result dr severe dr confidence level pixel confidence euclidean fm dr ensemble alternate decision knn perceptron naive forest svm selection q false false negative classification apply realization classifier one give respectively energy ensemble fold cross energy function database sensitivity specificity fit receiver evaluate strategy investigate sign challenge minor visually sign dr advanced r r table specificity fusion dr table accurate bold c vs c forward htb dr dr regard perform ensemble specificity use fusion dr dr sensitivity specificity achieve search see table energy vs dr r similarly ensemble dr three energy function fact r bias balanced dataset vs specificity accuracy dr specificity backward search similar specificity backward strategy htb c vs sensitivity specificity vs r sensitivity specificity accuracy forward fusion effective aggregate confirm however suggest strategy sensitivity specificity c fusion strategy specificity ensemble search method recommend automatic screening group evaluate proportion image dr completely different meaningful area provide regard comparison lack sensitivity specificity auc dataset comparative r dr also base solely confirm necessity component htb htb vs system sensitivity specificity auc htb htb dr dr specificity single propose ensemble dr opposite art level create extensively publicly area outperform system specificity close recommendation association dr european european grant nk develop computer system office research technology contract om european european inf ensemble method screening extract assessment pre fm
paper challenge vision perspective camera view overlap helpful appearance camera angle illumination camera miss low individual camera camera view view pair assign structured bipartite simultaneously potential match fig camera view probe set entity red matching require similarity entity view learn instance manually formulate pose text encode word priori text learn way test weight word visual similar word text ambiguity issue word significant variation appearance due illumination handle distortion base occurrence visual word weight different occurrence statistic motivate co visual behave similarly view occurrence see observe image large view co pattern instance statistically speak white color camera camera light camera negative pair novel occurrence capture important first encode sufficiently word result embed kernel incorporate appearance locality sensitive co occurrence interpret transfer illumination common comparison change camera visual camera method appearance across learn visual occurrence statistically camera identity co occurrence robust co occurrence contribution structure match deal shot shoot unified appearance visual word co outperform art benchmark efficiency testing receive seek probe broadly focus focus metric learning aim spatial video sequence discriminative representation viewpoint attempt match attempt learn appearance ambiguity distortion aim transfer template matching contrast metric learning attempt positively appearance base shot shoot entity image literature et al good discriminative mid describe descriptor entity redundant shoot wu et locality regularize image boundary al representation overall content deal entity entity probe meanwhile handle shoot multi shot accounting appearance adaptive function fundamentally base learn entity impose semidefinite enforce consistency person camera person view person match camera pairwise camera contrast testing approach bipartite priori structure liu utilize structure integrate metric color ensemble individual classifier feature pair feasible unlike approach appearance express basis feasible appearance change occurrence decision function co structural induce ground truth enforce globally assignment occurrence partially people recall paper align identification benchmark level deal issue person track association world object tracking association temporal lead totally goal aim correct entity test appearance information associate appearance adjacent locally prediction detail list report camera common sequel overview location let camera match depict scenario entity image shoot multi shot entity probe green exist reason pairwise structure matching intuition binary match probe entity bipartite graph account ij probe probe correct among scenario unknown arbitrary similarity model function similarity goal build intuition collection document probe tuple probe namely w training instance similarity obtain v v match bipartite truth bipartite constrain rather need encode visual typically match challenge appearance camera similarity co occurrence pattern insight appearance way static pairwise occurrence visual ambiguity motivate entity camera view view along analogously likelihood co visual match instance function empirically estimate visual contribution location visual occur visual simultaneously parameter along truth approach let bipartite matching entity represent two probe insight structure narrow space structure graph knowledge node probe entity match entity probe correct match among however nod test enforce structural probe entity probe match likelihood constrain help avoid entity probe entity graph test probe node accordingly degree set degree degree calculate degree bound narrow match structure bipartite probe view predefine probe operation spatially close distance map smoothly co probe multiply location guarantee descriptor insensitive entry indexing occurrence descriptor match probe image sparse appearance descriptor simply p u otherwise compare computed make however multi shot paper visual multi shot three spatial q denote image patch shot entity location explain accordingly base person spatial control locality image utilize represent ignore relation turn outperform art spatial quite truncated gaussian filter definition predefine illustrated euclidean much make question probe view bipartite formulate structured weight predict bipartite denote basis function ground truth predict vector constraint enforce match structure ground substitute eq construct utilize knowledge chance subset slack svms cutting training alg violate feasible set add current resolve search adopt ij indeed efficiently thresholde trick test speed alternatively sample trick widely demonstrate without notable implement strategy extract vector patch encode pixel centroid patch map visual model descriptor probe descriptor extract dim color sift feature pixel patch visual randomly sift per camera sift word euclidean pixel encode weight pair view person pair black contribute employ pixel cross standard cumulative curve recognition match solve far follow rank entity probe ask solver three single shot multi shot comparative try figure comparative either necessary know report trial consist view follow describe randomly training capture camera per pair form probe camera entity image dataset overlap camera view camera scenario single shot people shoot pair sequence vary people test camera view form probe ss ms shot multi except typical camera pair always negative indicate camera view compare white weight unlikely black camera compare within region white pair occur contribute black contribute mid filter mid level semantic super super fusion color semantic single fusion metric colour ss discuss shot list comparison dataset rank curve comparative non fusion method aim combine metric overall fusion well always comparable lack interpretability fusion mid filter utilize discriminative mid filter powerful utilize foreground comparable outperform method perform compare utilize occurrence integration explore previous structured reduce rate ms colour ms multi shot learn definition image person list result art rank shoot also improvement shoot latent kernel shot shot entity shoot visual co shot improvement robustness miss scenario different purpose utilize different metric I learn similarity image apply sm short utilize similarity simulate match comparison structure help improve performance general among summarize rank sm fig since sm general rate however much compare result demonstrate probe set respectively probe sm display structured matching match incorrect match structured matching match simulate probe set occur time randomly match randomly probe positive match negative non divided entity matching help improve two issue world descriptor calculate three descriptor entity matching similarity structure color since record storage probe roughly speak visual
r implementation two crucial vast array collection correlate principal know factor component capture successive remain precede commonly core dimensional largely small cca compute projection maximally principal random variable mutually order amount explain cca modality ability modality available training keeping application cca extraction finance biology processing speech computer pca cca reveal relationship variable study limitation extension cca autoencoder neural network tend rather high complexity cubic theoretical separate research help reveal basic regression incur loss achieve complexity cubic feature amenable contribution variant cca sum dot analysis rely extend effectiveness randomize world state deep canonical information scalable autoencoder neural numerical validate lastly implement stream research dot kernel development hadamard randomized component hadamard transform appear twice statistic canonical projection nystr map cca achieve learn presentation recall key nonlinear sampling nonlinear version fit class seek approximate minimize risk true convenient nature insight independent obey ultimately nonconvex optimization remarkably theorem claim let function argument take fit operation computation yield interest help connect kernel feature normalize invariant transform kernel approximate fourier may kernel matrix scalable method exploit extend straight feature nystr om superiority om aforementioned experience section random store orthogonal variable uncorrelated computing transformation nonlinear variant know em pt reproduce computation principal operation autoencoder artificial train representation transformation autoencoder bottleneck principal propose nonlinear randomized pca may equip kernel nonlinear load pca pca loading approximations nonlinear belong function consequence theorem approximate tend infinity feature dot converge evaluation study operator spectral grow analyze henceforth let exact counterpart express define shift approximation q counterpart define bound triangle inequality note eq invoke thus upper bound error divide please aspect spectral clustering use spectrum applying mean therefore easily variant cluster analysis cca multidimensional cca basis stand correlation refer canonical vector act cca speech audio pair transformation particularly view available use kernel trick nonparametric nonlinear cca exact variable deep transformation correlation mapping autoencoder used autoencoder use nonlinear randomized equip shift invariant understand cca pair randomize mapping computational complexity cca perform pca interested characterize counterpart grow avoid spurious characterize approximation random shift kernel approximation analyze first analogously individual eq q recall hence deviation expectation q deviation follow apply old twice turn issue variance term take follow norm jensen eq argument show definition yield worst bernstein inequality analogously bound k take maxima concluding extension linear discriminant analysis seek separable paired therefore nonlinear canonical copula matrix apply section introduce tool train autoencoder set describe paragraph section compare nystr om set form evaluation first projection form cm value norm equation vary agree effect feature close modality cca cca fourier nystr om unable run cubic show inferior replicate accumulate canonical canonical unseen mnist representation mnist width height cca validate simultaneous acoustic speech frame yield measurement vector use validate cm cm fouri nystr om minute cm
suit recover perform source direct state orthonormal wavelet domain explain couple db invariance wavelet synthesis db natural source reconstruction nmf confirm effect nmf peak also adapt capture structure wavelet local peak smooth enforce sparsity sharp peak structure peak width smooth conversely extremely synthesis interestingly experiment visible perfectly case parallel domain wavelet width several kernel wavelet conversely version outperform wavelet domain appear structural provide source complex correctly especially source regularization preserve peak correctly reconstruct large scale smooth component lastly handle sparsity base synthesis scope blind separation already inverse generally contrast synthesis regularization retrieve decompose limitation possible validate use regularization bss regularization superior separation separation performance active entry model approximately source take example wavelet effective noisy display namely gaussian show ground mix matrix observe part head visible picture sparse exhibit similar perform retrieve correct separation special treatment take overcome produce author introduce author propose iteratively soft thresholding opt call reweighte consists choose amplitude transform less amplitude usually source update source use source estimator base absolute mixture image reweighte display sake regularization square estimate reweighte square snr sir reweighte provide denoise snr decrease sir greatly regularization reweighte sir snr apply simulated spectra introduce synthesis formulation wavelet yield improvement db figure sake synthesis constrain benefit see improvement high regime may wavelet mainly paragraph synthesis reweighte imaging set standard coarse coefficient relate coarse account wavelet mixing wavelet use b source reconstruction noisy analysis synthesis redundant wavelet reweighte retrieve clearly direct source provide analysis exhibit lastly reconstruction low finding turn enforce synthesis lead significant article cope blind source transform end novel enforce synthesis formulation enforce transform negativity direct recently calculus blind separation propose enhanced separation performance neither direct correctly finally synthesis formulation arise imaging generally sparse improvement define non lagrangian optimization tucker eq consequently projection negativity follow projection ball aim compute eq column proper eq von min min finding relationship value rearrange term straightforwardly projection matrix positive everything analytic implement synthesis converge w proximal proper continuous relationship computation synthesis analytical reformulate notice variable straightforwardly proximal projection constraint blind separation bss refer factorization different retrieval nmf domain negativity together sparsity transform simultaneously deal domain article impose negativity along sparsity transform domain formulation present reweighte version blind separation encounter scientific sense usually spectra mixture elementary specific physical entity source recover source underlie still way blind source separation negative nmf spectra notation number source bold capital matrix call measurement spectrum mix measurement account instrumental pp multiplication hadamard n orthonormal wavelet domain mark subscript source transform provide several separate symbol bss linear mixture notation form negative matrix nmf source arise mining audio hyperspectral intensity spectra instance necessarily relative physical entity I nmf update least square hard minima beneficial minima desire display feature exploit nmf recover therefore carry multiplicative sample smooth tuning toolbox implement smoothness difference signal energy non large signal nmf sparsity descent update update efficient recent automatically handle available datum fidelity non negative analysis author use technique order automate describe implementation online explore enforce direct favor constrain go perfectly algorithm continuity none aforementioned nmf domain depend see basis capture representation signal enforce transform domain problem therein nmf enforce sparsity deal domain impose transform limited impose orthonormal clearly performance nmf enforce introduce preliminary detail show yield provide however tackle transform either redundant tackle sparse synthesis well never nmf comparison mixture spectra type yield estimation bias generally dramatically major present reweighte prior carry show propose yield enhance normalize design extension tackle formulate subproblem solve regularization separation transform different synthesis formulation synthesis unknown minimization denote aim reconstruct linear possible dictionary call synthesis build atom synthesis domain directly carry multiply synthesis aim synthesis formulation indeed invertible redundant dictionary advantage redundant dictionary come atom available redundant wavelet translation redundant transform formulation show behavior minimization space analysis latter w orthonormal transform extend synthesis begin k indeed backward apply generalize forward two semi proximal proximal operator close orthonormal redundant improve reconstruction particular tight operator still analytic proximal generalize proximal could redundant pseudo synthesis algorithm synthesis admit convenient analysis transform use transform carry synthesis proximal yet proximal analytic need compute subroutine computation need subroutine transform code experiment redundant transform iteration multiplication forward projection negative multiplication transform length order mostly redundant wavelet transform sample multiplication note indeed update multiplication physical identify make mixture mix spectral acquisition spike laplacian measurement naturally however benefit wavelet polynomial wavelet redundant wavelet
solve minimization iteratively square new dictionary sparse metric metric datum matrix nmf address et online example sequentially svd offer flexibility simultaneously dictionary aim minimized problem iteratively metric fidelity preserve svd atom entry simultaneously svd convergence learn tailor member complete mean symbol stack cost noise adjust sparsity result versus fidelity solve adopt alternate wherein guess coefficient current start guess typically begin iteration update n x n denote entry diagonal entry denominator stability choose result package available threshold sparse application specific propose interpret follow prune ccccc iteration iterate scale u u c c black red laplacian input report average five realization update atom atom absence denote seek z whose index ambiguity restrict key idea respect set eq solve description algorithm expensive train validate conduct experiment metric rate recover atom consider recover exceed unit near atom compute recover atom measure closeness indicate recover closely truth create generate zero subsequently contaminate noise initialize training example iteration repeat optimum find variation average trial facilitate fair omp svd k svd one superior k term one robustness laplacian attribute simultaneous dictionary execution respectively execute platform system gb ghz processor decrease plot recover atom suggest conduct denoise laplacian similarity output image svd adaptively noisy noisy direction noisy extract dictionary initialize iteration repeat sparse svd form experimentally noisy follow hard patch fraction experimentally clean patch generate gaussian report svd comparable high svd especially noise denoise svd job preserve structure range comparison image contaminated laplacian indicate improvement ccccc error apply motivation metric fidelity achieve robustness texture edge image svd update dictionary flexibility superiority counterpart svd ground superiority example limit denoise k input svd algorithm turn
specific several moment processing deconvolution convex application communication often intractable sometimes devise tractable nearly equivalent formulation differ across principled convex notion simplicity atomic norm application application choice simplicity constitute hull atom atomic norm operation atomic ball induce serve regularizer atomic norm well effective regularizer atomic hull often atomic follow constrain norm euclidean ball lasso overlap processing show atomic application communication sensor atomic atom sequence support methodology wavelet graph biological network application atom define subset hierarchical modeling fmri atom group atom regularize use penalty correlate variable interest decomposable rank atom consist unit tensor deconvolution component atom typical include atomic set sparse basis sparse signal loss representation observation subject constraint impose besides generality aspect improve fidelity dramatically existence ambient superposition small atom origin additional cardinality ambient operator use often sometimes slightly iteration refer column induce eq convex hull collection equivalently representation atomic atomic dual amount argument achieve norm know atom atomic rigorous remark algorithm optimization interior impractical difficult formulate scale instance often scheme popular scalability find machine find atom optimize objective atom basis perform include gradient iteration prox method fista accelerate method completion prox part computation lead singular structural cg latent lasso extend signal employ amount overlap increase prox intensive cg scalable solve problem oppose quadratic program projection prox solved retain guarantee interested solution sparsity usefulness scheme atom may ultimately contribute acceptable quality atom remove recent iteration truncation simply discard alternative remove current least nature atom limit seek completely basis atom backward opposite possible linearize direction new improve property contribute sparsity backward greedy consider previously omp extend set basis seek norm type square method warm perform simplex equal maintain without organized specify parsimonious atom section describe compare apply scale algorithm deal deconvolution problem major element backward truncation discuss construct alternative characterization acceptance threshold output step f forward step linearization current specifically solve argument minimizer attain atom perform efficiently shrinkage prox method search exactly discuss option backward truncation amount sufficient decrease criterion modify close frequent removal expense seeks expand quadratic prediction removal atom approach removal iterate frank wolfe subproblem relative objective accuracy lower check similar spirit inexact approximate linearize fraction solve signal problem test eq atom sign reduce randomly construct I measurement check performance run value count logarithm vs cg step well convergence conditional cg figure quality cg pursuit hamming trial trial corrupt cg choose pursuit variant group require atom step amount accelerate pl approach consider size index take add htp prox overlap observation element first top singular area cg practical approach rank approximation range multidimensional latent tensor tensor fold product tensor reveal wherein ft atomic form approximate efficiently implement basis via toy various rank tensor always component tensor stop plot entry h c c sample give observation find frequency infinite step operation formulate semidefinite program since well high limited implementation form initial frequency refine add pair select control accuracy discretization simply select negative implementation backward step algorithm frequency heuristic nearby frequency multiple adjacent spike cg sample signal length recover indicate small spike recover role play result report cg right spurious frequency cg grid blue spike circle solution circle solve comparison include signal combination arise physics kind atom define triple atomic infinite atom choice superposition limited sample wavelet gaussian characterize much learn key ingredient atomic mention solution via discretization shrinkage atomic norm formulation define atom standard deviation vary perform succeed recover form sense express mention section adopt optimization drive outline arrive describe informally start choose nearly respect step respect backward step proceed unless backward beyond repeat termination rank type consider graph consider simple undirected graph adjacency superposition interest sample wish graph note neither edge class cyclic corresponding information cyclic permutation yield grid atomic adjacency set cyclic permutation cyclic order weight fix canonical atomic permutation observe full deconvolution norm
nonconvex expect algorithm globally structure possible algorithm global formulation optimize subject constraint formulation program convenient suffer behavior geometric optimizer orthonormal guarantee plant exist follow plant eq orthonormal produce optimizer quite nonconvex next heuristic near stationary round technique recover sparse blind separation however develop solve slight variable penalty huber make optimize close soft k easy recover nonconvex unlikely produce dimension extremely ambient initialization purpose normalize program plant g g z n I bias probabilistic gram schmidt condition bias biased global optimizer biased direction optimizer suppose global optimizer matrix global optimizer invariant row optimizer output prove algorithm fall radius recover equivalently linear prove recover describe succeeds plant orthonormal q pi provide constant suboptimal optimality theorem demand aside guarantee low mostly lp round succeed rather iteration illustrate main detailed deferred invariant generality work go sketch nearly proof simply argument note vector favorable analysis view fix numerator independent q study process round general q inequality significant portion sphere move direction observation one bind numerical imply iteration allow feed scheme programming rounding whenever input magnitude claim enough round return produce synthetic plant learn planted generate sparse basis gram schmidt operator regularization use initialization p repeat simulation five dictionary observation row pair nonzero construct u repeat plant sparse present successful phase transition seem beyond linear sparsity regime whenever plant sparse gap future direction extend vision appearance approximate low pixel image person illumination dimensional subspace subspace select row orthogonal sparse continue subspace figure different illumination htbp manually experiment show interestingly concentrate differ etc first experiment reader think discovery cast vector concrete handling meaningful sparse adopt subspace extend believe paradigm work preliminary gap result likely bind place cover potential structure despite mention start hope provable nonconvex approach estimate various structure setting dominant dictionary geometric algorithms initialization plant sparse thank foundation partially grant nsf compact notation index denote scope always context probable dominate safe rounding theorem theorem theorem subsection edu problem dictionary machine convex target exceed nonconvex alternate direction provably succeed assume plant target embed challenge contain arbitrary efficiently numerical algebra control structure spectral dictionary graphical manifold contrast relaxation optimally understand np hard computational surrogate nontrivial
delay message delay delay system identity respectively certain adversary observe infected subgraph source adversary likely metric protocol spread reach scale fast achieve perfect infect immediately solid message trivially center infect subgraph contact tree infection even randomness node center infect subgraph attack infection right illustrate break combine warm spread infection insight spread maintain center virtual infected novel protocol diffusion provable guarantee protocol inherently distribute message adaptive diffusion reach scheme pass neighbor contact source perfectly user infected source among infected class graph cycle numerical showing protocol nearly warm protocol tree combine insight approach analyze empirically discuss limitation discuss contact warm protocol protocol message source protocol fail broad contact network line protocol develop contact reveal high large two develop contact source high novel call contact network protocol source spread neighbor scheme trivially identify center infection add randomness random time infected protocol study estimator scale give source detection vanish appropriate perfect insight infect node likely origin infection figure equation adaptively choose away infection source likely spread infected node infect boundary summarize goal anonymous rate contact start spread line protocol location expect infect bound expect fast deterministic line deterministic delay detection source source estimate size infection achieve illustrate protocol simple diffusion choose random spread accord protocol detect computation appendix example distribution message send source source line protocol provide detection many path center I match tree size infection contact analogous fast protocol neighbor fast trivially identify infected subtree infect subtree balanced leave diffusion infect infected source hide node present protocol infect keep infected subtree infect subtree leave infect subtree illustrate protocol regular message one refer virtual virtual infected subtree infect balanced neighbor virtual source message make tree notice equally follow form protocol source infect subtree significant infect node contact start spread message protocol tree protocol adversary location use property number infect least estimator expect source proof explain protocol protocol perfect large contact network line two line independent infect subgraph approach node neighbor node combine show infection adversary protocol provable line graph tree line tree protocol source away tree protocol infect balanced source close leave protocol perfect regular tree line diffusion protocol infection function example partially illustrate contact illustrate ensure infected depth call virtual true regular subtree another illustrate infection diffusion source start infection pass virtual token I virtual spread infection balanced depth root pass virtual token virtual virtual take spread infection balance root node consistent depth root infection virtual symmetry underlie contact virtual neighbor virtual chain virtual h v figure show leave whether virtual token construct appropriate subscript contact example virtual right pass virtual leave fully let transition represent virtual source pass one virtual symmetry equally likely infected except virtual source origin statement precise together ensure choice show except virtual equally contact regular start message protocol adversary estimate use likelihood infect source protocol diffusion step source neighbor virtual source pass virtual source virtual source pass along choose virtual source leaf infect subtree leaf virtual infect subtree infection subtree pass happen message spread choose remain virtual pass virtual token randomly exclude previous virtual thus virtual receive virtual message get pass virtual cause infection one subtree leave subsequent virtual source message asymmetric infect symmetric virtual panel infinite regular contact cycle degree realistic contact ht contact source infect select tv uv infection tv study underlie finite network degree still apply diffusion graph challenge first immediate degree maximum adaptive diffusion sensitive long depend discussion approach odd adaptive diffusion virtual computing virtual path since virtual source compute virtual constraint diffusion pg v tv v p v virtual introduce always choose virtual virtual token virtual specify pass virtual token trajectory wish compute summation valid path exposition use contact spread snapshot infect infect subtree path understand spread compute give node assume give large source close leave subtree leaf node two regular virtual identical regular long infect subtree virtual source infect subtree virtual candidate v g ta v infected equation still equal infected virtual v ta pg tb leaf tp g v g protocol efficient ml naturally computed message pass algorithm gets pass virtual leave every message degree virtual source start pass continue node turn child divide reach leave discuss depend leaf regular give leaf adaptive tree degree fix illustrate random degree trial number infect represent underlying value tree expect infection source likely boundary infection successive illustrate probability infection underlying legend indicate choose degree average size whereas degree suggest perfect average infection adaptive cycle connectivity network facebook facebook eliminate three friend guess friend spread diffusion pass source estimation could preserve possible constrain infect node also adversary undirected infection explicitly identify pair spread infection subtree contact adversary source message expensive find trajectory depend whether virtual one ml describe graph cycle denote time loop cause varying candidate ml leave likelihood long exponential problem virtual source loop virtual percent also distance small say informative connectivity graph induce adversarial adversarial example like adversary protocol network activity adversarial attack chance source design anonymous protocol extent adversary exploitation difficult anonymous protocol ensure contact infect subgraph contact infinite infect boundary infection message current naturally contact infect subgraph select neighboring message virtual source eventually infect subtree protocol procedure infect virtual create virtual decrease stop infection ensure true source infect virtual choose virtual state contact regular tree keep infect subtree start set node set message infect keep state virtual source infect subtree center integer subtree example leaf height subtree depth node dot move stay claim prove therefore true assume px px px px k infect subgraph chain chain leave write line protocol hence infect sum random accord probability adversary infect subgraph moreover know bad performance assume pg maximum node source whenever could reader verify whenever put distribution contact finite put prior location fix large remark posterior infect normalize constant ensure probability remark infect infected subgraph g g kt k formula uniform first protocol exception root even root whenever odd follow immediately protocol start claim leaf leaf evolve ensure equally leaf prof indicate exception depth even therefore statement tree exception child depth whenever odd probability root eq lower infect virtual since pass token virtual iff virtual exist virtual evolve protocol path path virtual candidate claim observe regularity symmetry pg pg tv tt design equation combine infected virtual get pg pg virtual virtual stay case
bold response shape shift remain total description inter stimulus activation pattern epoch random dataset bold voxel plausible course precisely use subject variation process process simulation instead use mean corresponding canonical outline allow inter stimulus repeat epoch set description simulate construct outline except relate design stimulus randomize trial description simulation dimension voxel wise standard related spike stimulus spike second stimulus single randomization scheme outline upper corner fig correctly specify glm profile various degree order direct standard ol ordinary multi subject fmri stage begin coefficient subject across subject component subject coefficient variance method estimate population inference determine significantly comparison fdr control inform university drug pre reliably apply rating fmri device ii fmri compatible mm diameter calibrate warm moderately tolerance heat period trial participant ask stimulus point analogue fmri compatible weight bold image tr voxel acquisition functional minute correct slice acquisition delay adjust head http www ac high image voxel tr ms collected run manual check adjustment ensure alignment institute template avg image pass filter cosine deviation voxel slice z cover temperature high heat basis fmri subject combine run fix voxel alternative outline control control fig simulation map thresholded glm hierarchical glm ol delay within square upper left corner dramatically duration point et left corner handle large deviation see misspecification rise autocorrelation though white simulation clearly irrespective shape improve improved sensitivity specificity simulation activation however voxel activation method particular previous datum difficult detect use glm canonical derivative impulse logit il show comparison new activation estimation multi fmri model determine forward second researcher use therefore potentially ignore contain vary subject population benefit inferential perform test voxel also canonical canonical field diagnostic purpose canonical situation population activation flexibility latter present activation detect canonical temporal dispersion impulse use date manner common voxel amplitude currently extend datum consist spatially neighborhood brain resort exist g drive see decide discussion present include canonical temporal canonical temporal derivative impulse set version il change bold substantial among term bias derivative shift shift il amount bias il amount examine handle large suggest purpose encourage balance specificity none addition il activation extremely signal reason propose towards effectively multi fmri matlab implementation propose methodology available request time fmri datum moderate parallel voxel fmri estimation solve matrix multiplication definition slow voxel small voxel programming problem solve arguably necessity inverse algebra regressor acknowledgement anonymous remark grant omit index notation identically n n lag minus twice augment n term r get optimize entry l r compactly take element voxel aggregate estimate voxel carry justification sampling estimator estimator rely whose property consistent increase number reduce estimation sampling effect pilot shape nan pilot step derive least average except assume apply v kn n negligible argument n tt convert paper simultaneous estimation shape response fmri allow vary across subject provide activation inferential allow activation test shape validate application fmri researcher fmri activation certain date accurately assume many fmri attempt magnitude example glm arguably towards time combination component test whether typically assume priori focus analysis obtain across assume constant subject rise assumption relax several function within glm stimulus canonical type function brain use set response impulse response consist parameter cognitive arbitrary brain canonical temporal shift width choice basis cosine radial functions logit critical glm subject analysis analysis fmri glm provide assess group predictor status behavioral problematic truly self contrast compare difference number issue notably basis entail corresponding derivative nuisance canonical fall apart shape begin differ coefficient derivative create subject enable voxel population magnitude offer flexibility basis simplicity inferential allow test activation canonical idea towards impulse constrain gibbs later number suggest spline random draw level voxel across parameter include canonical model simulated set study flexible outperform glm derivative smooth model logit approach propose simultaneous group suffice estimation inference assume acquire bold voxel scan right nuisance sum stimulus whenever nuisance typically cosine basis head heart nuisance subject specific strength across subject represent suggest knot say basis desirable spline I possibly reason immediately interpretable inference greatly computational shape stimulus determine shape determine amplitude stimulus orientation considerably reasonably maintain one activation voxel across previous autocorrelation kronecker delta specify structure partition domain suitable ar variance e te j spatially characterize smoothness function subject voxel express convolution k v amplitude j j v variation variation summarize notation subject index voxel voxel subject fmri population voxel voxel deviation subject voxel voxel matrix nuisance nuisance voxel effect voxel subject effect voxel matrix outline voxel mathematically generalize guarantee select good start pilot objective final describe subsequent pilot residual estimate obtain voxel pool suitable median voxel voxel generalize square pilot voxel first square pls q nuisance signal canonical temporal whose contain determine closeness cross validation randomly rely j v j consistent smoothing v latter voxel activate experimental step voxel residual consistency j j voxel take voxel temporal estimate temporal efficiency estimate number voxel aggregate suitable median voxel usage first iteration good start value likelihood section voxel write multiply q fix concerned likelihood voxel assess step optimization detail linear voxel variance variance residual subject quadratic
formally indicate function specifie belong worth region restrict define wise act model w capture region linearity fundamentally important convex loss regression wise model formulation linear optimization region sparsity induce structured function space two region principle prefer representation motivation utilize interpretability although rule hyperplane region empirical wise model well appropriately optimize special global employ wise linear active rewrite integration global determine affect globally always return regardless f residual although convex wise propose let vector zero formally introduce induce constraint effective partition enforce natural practical preferred interpretability optimum indicator make region complicated standard constraint constraint penalty follow equivalently rewrite term decrease modular index pair confirm envelope modular envelope become convex respectively derive proximal tool problem iteratively convergence gradient acceleration iterative shrinkage fista incorporate th decrease empirical evaluate step regularization frobenius backtrack avoid calculate step employ convergence fista proximal matrix follow solution confirm wise depend step improve efficiency norm computational determine node large size alternatively paper employ idea decompose proximal first map map perform decompose proximal discussion q subdifferential subdifferential norm subdifferential hull max value max derive soft operator decompose feature derive include condition regularization group wise problem projection efficiently proximal describe map single utilize gradient gradient partition practice stage step dominate operation dominant sort ordering become computational derive value width previous step consecutive terminate derive proximal initial considerably affect initialize initialize among random initialization initialization etc initialization empirical local present generalization model discuss sample relate overlap lasso rademacher condition rademacher expectation value rademacher almost surely partition wise reformulate basis special assumption third norm apply give residual straightforwardly discussion uniform uniform global residual model satisfy function within eq partition candidate term practice summary able candidate fit wise linear comparison linear classification binary uniformly dimensional space rule first feature add noise e sign feature nearly output candidate logistic iteration residual illustrate learn red line weight apply line piece yet exist structure make capture benchmark art candidate calculate partition value categorical active feature yes value several regression dataset census internet energy community breast nh bank fm summarize specification cccc cl census cl twitter cl cl breast cl internet cl energy energy nh bank local global local sp discrimination svm kernels rbf kernel note region linear compare respect stop early low cc sp tree svm census breast cancer internet table error global consistently achieve rate dataset sp census breast significantly bad census twitter internet partly partly initialization svm case obtain census internet stop regression compare cart rbf performance depth validation experimental setting use summarize rbf well global many c cccc nh fm comparison warm start warm rate warm calculation become bottleneck technique promise respect incremental incremental minimization despite stochastically approximate gradient direction parallelization parallelization calculation importance direction greedy solver matching work reasonably advanced partition take locally analogy know anchor considerably advance propose add piece train applicable model boost approach partition generation concept treat candidate sparse sparsity induce notable define hierarchical partition structure structure understand structure regularization directly technique optimize sparsity induce structure optimization thank dependency derive acknowledgment majority central school science technology application highly propose model partition key assigning region linear combination region globally formulation make proximal map sparsity inducing demonstrate model well competitive art divide sub assign well linear understand interpretability advance specific prediction challenge convexity inter optimize region arise bad local lack generalization analysis propose model distinguish help convexity propose partition wise divide possess weight apply input
integer write hold theorem degenerate triangle let consecutive supremum improve subtract multiple affect digit base independent uniformly distribute case diameter discrepancy triangle suffice study boundary boundary segment area centroid triangle place centroid segment segment neither subset great discrepancy eq attain pass centroid row triangle case discrepancy pass centroid row centroid pass leave right else discrepancy increase two disjoint outside band case sign line contribute discrepancy horizontal passing centroid left triangle result discrepancy area similarly portion sign discrepancy contribution fact record line summarize contain sign pass centroid centroid discrepancy table triangle horizontal play important role analysis sign discrepancy contribution line centroid pass centroid triangle hold triangle line base include exclude centroid figure line meet centroid sign triangle line include centroid right low relevant show discrepancy column show sign discrepancy discrepancy line line meet meet leave sign sign case contribute contribution discrepancy irrespective sign column discrepancy empty triangle sign sign discrepancy exceed second discrepancy intersect great discrepancy arise triangle configuration exceed fourth band intersect discrepancy triangle one line great discrepancy suitably copy lattice angle cube begin say natural integer perfect repeat representation set angle subset side angle horizontal axis say exist list angle give satisfy take lattice angle remove point arrive inside right angle triangle eq side horizontal always convex side angle axis angle choice write integer denominator perfect finite angle integer lattice angle lie list constant hypothesis point yield set point point linear linearly triangle attain different angle scale discrepancy either lattice run sample satisfy lemma map subset contain end rotation rotation point lie step onto latter leave relatively triangular triangle integer respectively satisfy figure angle already range discrepancy roughly parallel cite parallel discrepancy triangular lattice angle solid may add point exactly use average dividing sum randomization usual find book triangle triangle let integrable integrable k modify integrable bound everywhere sign respectively parallel side subset sign set sign next panel set either integrable triangle proof attain discrepancy construction van construction continuously randomization produce root square kronecker result construction smoothness triangle digital proper corner average project back arc acknowledgment foundation dms restrict band band outside another attain end band result search triangle carry quasi focus graphic quadrature triangle paper present vanish discrepancy van sequence integer angle tangent attain discrepancy indicate discrepancy construction available construction accuracy construction also integrable triangle require quadrature problem numerical triangular quasi monte integral arise quadrature method correctly survey classical rule poorly choose attractive integration weight inequality star discrepancy point sense mapping cube equal hand use approach difficulty composite suited version simplex variation simplex simplex vanish discrepancy also integration simplex mention discrepancy function variation discrepancy measure bind factor cube simplex point neither simplex vanish present construction digital construction van exploit partitioning resemble point kronecker construction rectangular grid intersect retain combine vanish well digital amenable construction vanish also vanish believe construction triangle along describe twice former whenever first triangular van copy choose keep lebesgue measure volume lebesgue low exclude list distinct counting discrepancy measurable sign discrepancy absolute shift consider bound finitely extend take ns understand simplify omit box star value sense inequality continuous variation sum integral face simplex indicator corner extend corner vary spatially simplex inequality discrepancy triangle corner real value let illustrated vertex triangle discrepancy study list
task work focus prove nlp ideal lexical lexical across induce lexical agnostic set lexical tailor explore formulation induce task lexical embedding predictor lexical learn compatibility lexical linguistic refer predict probability noun sentence like capture compatible complexity lexical embed obtain employ agnostic embed retain lexical along line confirm task vocabulary denote noun relation relation paper semi fashion exist unlabeled simple distributional contextual word skip model access pair e able query unseen relation word lexical essential word minimize use regularize constant controlling regularizer interpret query highly predictive interpret original tailor relation matrix relaxation penalty common setting identity inner common evaluate semantic projection noun noun conduct initial lexical configuration experiment six syntactic noun query query always unseen pair report active need query candidate vocabulary distributional word dimensional skip embedding window thus main skip embedding compression window prediction non dimension embedding test unsupervise noun relation speed regularizer similar low start bag skip embedding three former relation latter conclude agnostic useful necessary retain nuclear norm representation embedding table present scheme rank relation relation l city membership membership law law code law show set
reason sentence time reliable quite sentence argue help optimal solution notice training put aside backpropagation analysis bottleneck layer ccc bottleneck autoencoder refer reconstruction ability highly size bottleneck bottleneck autoencoder project near adequate bottleneck estimation bottleneck layer comprehensive autoencoder model differently focus training text metric allow identification critical metric autoencoder fine back orient autoencoder characterization linguistic phenomenon carry ei fp texts c project multimodal es research star edu sg autoencoder model differently explore construct autoencoder level novel text reconstruction capability autoencoder critical bottleneck dimensionality language lose text space deal usually document thousand meaningful association try relevance explicit reduction dimensionality reduction inherent latent semantic probabilistic semantic allocation topic document reduction prominent similarity language association document language broadly linear technique compare principal component large dimensionality projection multidimensional representation language project unseen projection matrix reduction autoencoder da autoencoder structural autoencoder reduce document deep biological study task issue give find deep reduction technique nlp entity unlike retrieval similarity representation text question reduction qualitative assessment reliability reliability capability space comprise metric autoencoder capability distortion reconstruct autoencoder analysis explain provide adequate critical dimension carry dimensionality level assess autoencoder finding rest autoencoder autoencoder discussion critical bottleneck dimensionality estimate future datum brief autoencoder autoencoder sub describe autoencoder detail model differ way autoencoder presence document softmax deep autoencoder count similar input approximate reduction map hide variant autoencoder multiple remain add pca stack multiple deep architecture truly powerful space stacking restrict boltzmann machines rbm bipartite visible usually unit vice versa primarily bottom rbm rbm layer rbm base visible layer sigmoid non visible document hide unit state visible unit bias weight eq sigmoid rbm softmax vocabulary variable define count term visible layer softmax multinomial visible unit word count recover distinction bias way document visible unit autoencoder fine tuning cd rbms train one rbms take rbm autoencoder train rbm epoch cd cd rbms autoencoder show activity replace value entire show word input vector length layer stack rbms upper rbms take lower create autoencoder tune propose use subsequently comparative analysis projection reconstruction unfortunately refer poor neither detail reconstruction preserve justify bottleneck autoencoder dimensionality text quality task modeling estimate poor decide whether metric intend aspect autoencoder capability distortion semantic metric datum compute cosine similarity data eq q distortion attempt capture reconstruct measure reconstruct strength normalise accumulation dimensionality reduction play role document literature autoencoder
asymptotically optimal change point generalize kullback leibler yet communication sensor stream observation present point represent sensor system presence fact although place physical case monitor traffic opposite may correlation environmental wind appearance interference dependent happen sensor curvature field sensor region produce matrix literature include stream observation correlation time treat general correlate stream partial post change stop character alarm partially know stochastic matrix stream post drift rule delay examine upper similar methodology efficient behavior derivation sharp dimension handle non markovian methodology alternative dimension although prove formulate exist result introduce stop rule establish detection delay rule section paper complete correlated remark section omit denote filter p stochastic differential th process treat however sign change latter case assume know positive hold singular correlation brownian cover instantaneous correlation arrival pass sensor place subject yet formulation even capture realistic scenario observation high ti stationary white arise full detail subsection facilitate canonical n derivative comment dimensional brownian motion trade delay alarm ultimately follow performance bad bad detection delay take rule detect regime delay detection delay impose follow problem false alarm describe acceptable alarm stop detect concern know change brownian observation optimality criterion page process stop incur negativity bad delay occur process optimality stop rule know drift type stop optimality process rule stop rule h chart stop rule semi respect small independently subsequently alarm central fusion namely center instance sensor easily see difficult devise rule achieve rule detect say second optimality examine stop rule present detection delay upper delay stop ever detection system subsection detection threshold upper dominate delay introduce detection imply q uniquely instead choose able computable bad detection delay q last stop law tuple essential side definition strong function monotonicity give square integrable decrease measure fact integral set conditional expectation measure g threshold result rule alarm robust respect drift treat general case assume monotonicity slight abuse derive mean false suppose threshold stop alarm proof integrable decrease together expectation equality side get singular stochastic instantaneous correlation choose threshold delay rule bound big hence proceed rigorously heuristic argument process satisfy mean alarm kk argument one similarly sum side complete result stochastic instantaneous equation rule imply optimal delay detection delay ik false alarm define similar q see imply sum immediately result proposition drift correlation threshold imply delay stop rule strong rule optimal detection accomplish derivative tuple brownian tuple exponential define brownian driving find stop threshold clearly assertion similar formula argument stop tt exponential martingale drive ex integrate expectation support delay alarm argument proposition theorem non py py inequality yield optimality either hold hold examine detection delay alarm increase proposition delay respectively determine either optimality assume drift stochastic choose asymptotically detection optimal detection q optimality result drift singular instantaneous stopping equivalent stop asymptotically finally know partial ik assume ik singular instantaneous rule define asymptotically delay stop upper stop asymptotically optimality theorem give optimality proof asymptotic know decentralized communication suppose sequentially employ asynchronous fusion fusion want signal stop sensor change dynamic distinct point suggest health monitoring fusion sensor occur sensor alarm implication center receive take decentralized setup word distinct rule asymptotically detection easy device central fusion center processing efficiency valuable problem instantaneous exhibit optimality tradeoff bad independence optimal delay fold design stop stop trivial exist robust unified rather analytical treat change multiple especially acknowledgment grateful anonymous comment local martingale ready
status influence tweet logarithm status tweet cross datum show solely twitter acc table percentile partly situation considerably expand pool ccccc svm c svm rbf dynamic mm project city political characterize micro establish differently depend active frequent attract project act project match new project extremely important common failure work website input project recommend list twitter extent study focus mm google european fellowship social computing thank say valuable comment yahoo bring market often option name internet support project post project medium site mainly look project way set propose way twitter project drive analysis finding recommendation good accuracy list potential twitter account key insight behavioral sciences g internet crowd project successfully community fail project cite propose automatic matching mm description period recommend tweet project behave differently depend site project one support project tend interest pay less attention aspect project upon quantitative potential twitter percentile baseline order list predict twitter derive random conclude discuss practical implication finding start site successfully million project usa goal reward offer sign cd exchange success video visually connect dedicated account small traditional capital flow attract researcher economic computer science example economic yet tend come friend focus predict whether project successfully category existence video capital friend project project accuracy base predict indeed find powerful success phrase mainly principle evolve series tweet duration conduct behind contribute fail potential fail project cite leverage online conclude automatic project carry three step hypothesis behavior collect twitter hypothesis finding match project another family individual project extent distinction behavior depend formulate convenience active behave frequent good mistake look fact good segment frequent pay management project translate frequent update make e project reach goal receive tend realistic goal reasonable project goal video tend frequent since friend tend expect project project dispersion project reveal location city convert city coordinate location dispersion live close dispersion live frequent familiar site able quickly project look project interest concern tend support interest keep track project topic classify topic twitter build run project page art technology check project page change project eliminate outside project project category project proportion mm duration final number tweet mm period publicly twitter tweet project tweet project title project page tweet project report report project project project successfully meet goal success published retrieve rd oppose successfully financial goal goal dataset day however take previous representative mm mm mm mm project project match frequent span extent project span find largely span project frequent high span coefficient support recommender system project match confirm hypothesis find tend high growth contrast project happen majority project limited activity project growth hypothesis twitter run lda tweet description project represent twitter interest cosine project project interest interest project project variety tend stick project correlation activity cosine similarity well frequent frequent project goal grow interest instead decision thus infer considerable act appear friend member happen facebook successful project characterize considerable friend probability facebook friend project facebook friend moderate partly previous recommend specific project support basis situation recommend potential twitter twitter user project twitter project twitter project matching initially formulate predict predict whether prediction project twitter likely project include project interest twitter dynamic growth project dispersion comment datum add project pair construction evaluate regression fold project subset project project repeat result resort acc recall characteristic auc train pearson correlation coefficient update lr include feature c ccccc lr dynamic linear static static static rbf suggest point separate static good acc dynamic slightly type slightly feature individually g matching ts exclude number comment accuracy individually behavior category similarity improvement inspection give category red bar project active bar project goal case balance set create unbalanced find obtain yet acc binary classification problem project return rank evaluation resort reciprocal reciprocal ranking order predict project project definition formulate percentile project list list rank project score highest rank rank rank c lr static dynamic svm static cc rbf mm mm activity
illustrated reveal current currently possess stop second state immediately start immediately progress degenerate allow write make path piece connect utilize conditioning expand macro window component annotation enable iterate alternate draw macro observe consequence parametric form respectively monte ease motivate independence simplify leverage expansion express evaluate explicitly categorical constrain evolution obtain four pass similarly shoot attempt shot yield far thus pass shot discuss context difficult require integration trajectory evolution time path characterize secondly play second detail independence marginally denote markov process generalize time semi markov denote visit record term transition combine transition markov embed state rewrite system deriving actually homogeneous ultimately much smoothly evolve useful making time section hold prevent marginal estimate enforce interpretable checking computing meaningful computationally tractable full integrate ball acceptable describe evolution process depend rich homogeneous interpretability model level high resolution simplification resolution coherent require resolution distant shift intuition shift approximate begin describe simple assumption specification computation encounter model parameter specification model full discuss repeatedly draw macro sample formally stop move call shot attempt make pt employ state shoot treat red pass receiver state intend future represent possibility pass annotation induce draw favorable decision specify possess possible option attempt without generality correspond pass event attempt event begin complement implicitly express term aspect decomposition type location intermediate process critical connect basic draw blue draw player location resolution black loop degenerate event pass player correspond pass shot attempt nontrivial shot shot resolution prior shoot parametric lastly factor essential computational formalism let simplicity restrict consider optical snapshot tracking exactly high include coordinate player ball game situation time etc annotation occur pass shot intuitive path provide available tracking second value time expectation take end evaluate amount integrate use lebesgue annotation component dependence quantity imagine different achieve potential resolution player meet consistent aside datum value model guarantee consistency interpretation stochastically curve derive stochastic evolution consistency require sensitive grain without spatial configuration trading potentially methodology current process choose resolution expectation contrast estimating combine level player movement chain exact portion require map markovian plausible summary transition represent meaningful event column style anchor east column fill xshift pass c pass player player shoot shoot pt p east player north east east player south north east north east player north north north north player north shoot bend shoot north shoot south south south north edge loop pt north leave north west west south east ne west player east se se ne se ne cycle corresponding group together player discretize player position gray transition represent player bottom figure three associated value whenever player possess define possesse live represent colored diagram reveal discretization indicate annotated resolution currently transition air pass shot pass progress progress list transition pass air shot attempt pass design notable discriminate pass lose ball marginally semi embed markov chain associate matrix specify full conditioning define additional possesse attempt shoot state pass leave transition history label
gap exceed threshold detail c let number choose indicate tt word mod hold optimality gap wireless use aware detailed estimate appendix confidence performance particularly receiver time action incur choose arm within arm translate overall appendix theorem overall cumulative overall acquire discuss mod strategy reward high confidence theorem duration explain worst well predict environment theorem communication environment receiver pair scenario code base code encode message combination linearly successfully message message recover scenario high help decide successfully wireless receiver achieve indicate prevent exchange application video transmission consider length symbol say need affect least achievable sufficient duration knowledge choose difficult slow present ucb inside instead ucb evaluate reward explore arm associate mb detail later benefit numerical along ucb scenario receiver vary study adversarial reward entire action however pick adversarial strategy fashion interference wireless channel understand fashion strategy wireless channel un fashion avoid interference randomized strategy regard continue irrespective strategy learn communication irrespective strategy level wireless widely allow optimize action regard employ arm environment certain might assume employ random unknown choose manner strategy I mention predefine level include section select strategy learn repeatedly interact regret scenario r indicate cost take employ appendix incur incur case derive case due lack adapting strategy duration typically wireless system employ track discuss discuss receiver employ static compare symbol receiver scheme receiver receiver feedback learn strategy choosing enable previously via know parameter optimization optimal strategy contrast strategy expect try db db use various discretization show fig fair initially assume discuss receiver db agreement use learn db use phase offset learn performance factor instantaneous algorithm achieve due result due exploration phase discretization discretization initial take discretization greedy explores try exploit e use unless optimal strategy strategy perform significantly bad novel algorithm see discretization achieve satisfactory close scenario algorithm sub incorrectly know performance behave coherent unknown derive indicate performance consider wireless environment observe receiver performance receiver optimal strategy db remain h ensure maximize objective predict theorem optimal extensive ucb loop inner loop discretization arm thereby learn scheme converge step simulation send instant employ choose uniformly every instant level range employ employ scenario assume algorithm db strategy track change may important history achieve slide environment use adapt strategy round step step term passive slot term slot frame take active passive slot slot frame slot per frame however passive frame update ucb index frame start every mean reward action passive slot frame estimate used slot frame exploit exploration phase splitting horizon quickly strategy please window randomly across employ conjunction slide user adapt fig fig level track strategy vice versa successfully capability subsection scenario strategy depend factor wireless ignore consider error feedback different user fig learn signal vice versa agree complete set strategy user level compare mechanism reach maximum overcome interference cycle window capable track satisfactory mean receive several mu case allow receive rather spread improved expense etc worth applicability wide cognitive type without knowledge novel algorithm bandit optimally receiver pair capable coherent signal either asynchronous commonly algorithm capable strategy pair confidence successful theorem department electrical engineering electrical email edu adapt adaptively optimally receiver armed scheme power duration present learn efficacy receiver term rate prove optimal fast particularly dynamically change wireless characterize static pair inherent wireless medium make wireless largely adversary passive adversary wireless try infer attack active adversary order transmission hybrid attack adversary transmission agent attack receiver traditionally theoretic theoretic principle major disadvantage pair gain practical match strategy strategy receiver pair contrast develop learn learn interact receiver pair act approach environment canonical example reinforcement rl agent success feedback transmission action specifically learn optimal repeatedly interact wireless channel receive feedback action bad take meaning depend specific consideration throughput cost address anti multi channel go guarantee critical severe consequence opponent none work environment exist mix guarantee action novel arm mab algorithm communication offset paper know choose level duration scheme propose receiver wireless unknown exposition theorem scheme power alternate send enable error average instantaneous energy characterize fraction reward formulate mab mab action power duration action propose bandit learn interact receiver receive observe receiver pair estimate symbol
source audio sound meta like influence attempt modeling geometry take modal heterogeneous modality audio generate combine symmetric dominate challenging song infer music audio tag modal music infer modal evaluate retrieval similarity similarity relation index word w similarity latent dirichlet incorporate distribution modality multinomial explain object outline object strong feature modality model inspire improvement section estimate take expectation respective dirichlet define hyper optimize point topic asymmetric parameter respective modality symmetric already similarity document focus solely document measure kullback distribution product cosine candidate introduce dissimilarity proportion document log document similarity hold topic proportion fold style parametric similarity nan statistic permutation matrix exclude diagonal column maintain examine similarity subset song track compose last fm tag tag audio modality audio naturally occur count word audio approach total pilot topic combination modality list style similarity set fold cross split combination hold fold calculated figure correlation permutation nan complexity issue similarity approximate randomly model gain insight result similarity seem provide inter significantly positively possess label increase topic model link describe modality model conclusion modal predictive extend direct evaluation correspondence work
gaussian much solution boundary correct boundary minor case one never result resample make mistake assumption hold dataset uci machine repository characteristic name object source cross fold nine nine unlabele rest unlabeled semi supervise curve unlabele datum repeat determined curve figure semi outperform lda measure test significance four semi l dataset outperform significance semi minimize could hope empirical improvement loss applying relate goal unclear margin negative offer lda step principle semi supervise open question extent converge unique global proof objective lda canonical assignment self self weight control influence hard parameter help bring perform supervise counterpart semi supervise seem learner rather supervised learner partly public research laboratory university technology department computer university department molecular medical supervise pattern variant work expectation discriminant analysis new principled supervise linear discriminant implicit sense expect improvement misspecification unseen real recognition task expensive obtain document image unlabele object easily web unlabele semi classifier decade improve learn add additional datum effect unlabeled lead unlabele offer robust lda implicitly constraint unlabele compare expectation study supervise principled semi supervise implicitly lda comparison supervise version discriminant explore expect offer improvement variant term organize discuss several discriminant illustrative toy benchmark work semi supervise later refer closely relate maximization generative unknown likelihood maximize relate object assumption usually manifold smoothly manifold low separation support incorporate unlabeled leverage rely unlabeled object dimensionality normalize lda propose accurate subsequent semi adaptation theoretically practically successful introduce discriminant supervise semi procedure additionally design distribution function biased object find new q employ assign high semi straightforward adaptation supervise known bootstrapping classifier train classifier new classifier done predict underlie treat possible add integrate unknown hard supervise objective expression log em em sum jensen maximize update obtain practice sum label unlabeled object e bind effect self instead self learning suffer label update approach idea certain constraint parameter use feature alone lda instance matrix link covariance covariance latter accurately rely label point mean accordingly hoc update estimate unlabele label alternatively force objective numerically hoc ideally implicitly constrain intuition train true unknown would outperform classifier two problem one label safe know
good validation probability make available follow procedure unit preliminary regularization need single layer interval epoch fine epoch hide layer hide choose test log l rbm mask mask mask mask mask short refer table layer h ensemble achieve performance comparable belief hide layer mask confirm auxiliary mask input improve epoch suggest long variational help mask model train b outperform corresponding increase bad network consider effectively train start decrease fig input investigate test training case seem show performance drop fig continue seem phenomenon show digit digit argue task restrict variance mask c mask clearly filter random decode evaluate sample compare mask mask optimize hyper mask experiment hide regularization mask without rbm rbm mask mask h h see table addition rbm outperform rbms result well iterative neural extend conventional neural maintain original intractable boltzmann machine belief network extension inspire probabilistic boltzmann rbm deep boltzmann like multiple hidden infer perform number compare utilize model able generative come sophisticated future instance theoretically empirically well see testing fully sequentially efficiency sampling ensemble confidence corner miss confident reconstruct digit correct digit acknowledgement would acknowledge universit universit cifar estimator miss learn improve reconstruction reconstruct step combine compute use engine boltzmann machines competitive art two test machine potentially mcmc estimator mode autoencoder simple corruption paper distribution neural deep training selection train backpropagation observe room miss imputation criterion three recent training pseudo autoencoder corruption function generative generalize disadvantage sampling train resampling imputation inference cope cope vector reconstruction next previous paper inference inspire evaluate agnostic start define factorial conditional denote observe component conditional give permutation propagation parameter draw variable x one train deep experiment reconstruction replace equation agnostic consider special see sharing additionally mask initialize miss see one use mask probabilistic task imputation probabilistic variable get conditional often p factorial qp one posterior parameterized eqs rbm boltzmann
minute estimate viewpoint mala consider proposal merely point study insight method note calculation list question insight reader note matrix add computational overhead require step covariance geometrically correspond curvature could understand inherently property global geometry author relate research simplify manifold mala term large calculation correspond manifold place make appropriate lack cause geometric ii subscript tangent assign tangent set vector along thing along basis directional tangent case derivative consider along onto derivative linear derivative basis drop basis satisfy product direction argument partial directional derivative surface vector normal c know th riemannian q repeat usual manifold chart symbol q geometric employ monte wider diffusion euclidean connection carlo commonly physics idea geometry highlight method introduce manifold adjust langevin hamiltonian since article idea consider detailed unnecessary hamiltonian scenario interested reader review necessary markov chain riemannian geometry minimal measure inform reader prefer skip section provide derivation langevin diffusion riemannian manifold intuition langevin challenge geometric manifold discuss literature instead question research practitioner distribution density measure measurable appropriately construct chain invariant posterior briefly concept overview markov define x aa pm call admit equivalently write stationarity invariant chain certain see useful sufficient relation eq integrate respect reversible stationarity relation primary invariant ergodic markov chain analogue law element estimator estimator reversible provide first chain first efficient clearly arise intuition sort chain assess chain far need measure choice markov literature q informally difference event distribution admit density write proportional imply typically unbounded distance often inequality geometrically grow provide qualitative central limit assess stationarity average suitable mcmc visual simulate devise chain distribution suitably relative perform iteration also little practical exist chain choice hasting accept reject case remain focus step qx x behaviour move change zero many reject remain place autocorrelation see challenge balance proposal acceptance transition result markov chain way invariant reversible propose move reversible limit simple consideration broad researcher proposal mix large discuss extremely simple choice denote acceptance simplify intuition move accept typical structure conduct property rwm show proposal target propose former autocorrelation move walk sometimes proposal result acceptance autocorrelation ht show rwm markov chain choice efficiency increase rwm exponentially light tail necessity geometric ergodicity mean fast require demonstrate tailed pose far markov almost proposal remainder discuss choose superior rwm proposal sufficient benefit property specify continuous markov provide introduction langevin dynamic govern differential brownian motion informally imply change define part e often description evolve differential drift later smoothly drift typically form whether class side find volatility diffusion user basis metropolis langevin describe dynamic molecular eq suitably regular exist construct mean commonly encounter langevin way metropolis adjust langevin mala whereby euler used candidate tuning offer langevin proposal deterministic shift towards relative part part dominate vice versa opposite though hasting propose large enough accept optimal acceptance form differ factor efficiency result highlight mala geometrically ergodic typically result tail tail converse true offer away rwm also fix quickly far away neighbourhood reject strong mala also condition rwm proposal tune invariant idea information geometry successfully widely geometric insight common differential method diffusion turning discuss geometry make property evolve tuple number additional chain mala advantageous impose metric draw current position occur reduce reach efficiently attractive constructive dynamic calculus riemannian understand define notion euclidean purpose dimensional riemannian way chart exist invertible available sphere challenge coordinate sense say differentiable inner product aid intuition think curve think pass define give straight line agree geodesic space always give orthogonality thought flat since straight manifold level think always define product define curve velocity lie denote think local define dx riemannian purpose define coordinate restrict still use convention manifold object euclidean sphere lie ambient nash embed rx x seek idea concrete example graph embed coordinate euclidean linearly canonical partial use curve r although start object riemannian distance coordinate also explicit knowledge nash essence manifold define suitable high euclidean induce define map vector trivially volume riemannian coordinate follow area jacobian case manifold riemannian measure manifold local actually mean diffusion desire diffusion sphere produce brownian motion draw surface sphere piece flat brownian image brownian define euclidean langevin diffusion object appropriately technical reader wish motion manifold increment move along manifold tangent generator euclidean local deduce stochastic denote laplace operator trivially laplacian value brownian motion operator generalise unit neighbourhood also derivative provide directional onto tangent vector lie therefore field manifold seem coordinate ambient tangent define e generator brownian motion diffusion familiar formula drift integral rule ordinary calculus mapping typically drift simply ensure correct lebesgue q putting become upon simplification diffusion require map onto distance map hasting mala eq tuning parameter drift turn sometimes switch direct posterior distribution choice define distance parameter key explore rao year although measure often fisher tailor fisher combine negative log style understand mcmc proposal conditioning method match structure locally hessian long match discuss previously unless hessian globally definite et may scenario efficient procedure set metric problem ensure likelihood contribution positive globally provide log mcmc geometric start viewpoint appropriate absolute hessian way computable name negative decompose remain act value close fisher expectation tractable effort mala proposal long way possible one give diagonal fall difficulty numerically positive definite hessian accord metric well induce simple style difficulty tail take long variant mala hessian may avoid since major take proposal tail mala choice need
leave cardinality uk technology framework sequential via sequential distribution space auxiliary use able unbiased monte order construct algorithm involve capable efficiently tool intuitive underlie area application construct utilize carlo last year unbiased partition normalization method importance estimate partition decade first nonparametric propagation message send monte carlo method propose provide normalization constant another branch model build sampler tree subsequent add accord anneal particle field discrete hold somewhat inspired produce hand particle underlie clique clique node circle right node distance scale rectangle right edge edge subset graph consist random factor simple toy undirected graphical corresponding graph decompose approximate sequence probability recursively update weight typical model sequence observation limit applicable sequence target marginal iteration intermediate target arbitrarily intermediate sampler decomposition amount simply distribution iterate factor add construct artificial albeit use sampler probabilistic graph clique define emphasize need factor anneal order practice affect order decomposition auxiliary target function sequence intermediate modify target set sure establish unnormalized corresponding normalize subgraph circle style main style circle draw right cm style scale rectangle leave edge edge b distance cm main node scale rectangle leave left edge node style rectangle edge edge edge scale node draw distance scale right cm scale factor right edge edge edge edge target collection particle empirical iteration conditionally independently smc adapt resample target increment density particle index particle assign smc w r resample propagation skip importance apply comprehensive collection result limit sampler adjustment multiplier choose sampler say interesting likelihood statistical mechanic energy capacity channel partition discrete simple function contrary provide normalizing eq obvious unbiased due also offer sampler besides asymptotically theoretic capacity channel inspire art solve p k evolve reference propose play construct exact construction change asymptotic fit straightforwardly make kernel index leave joint detail implementation instead setup illustration enable useful general autocorrelation however context ideally sampler simulate systematic random amount simulate subset gibbs get directly conditional make facilitate gibbs sampler let construct depend throughout sample applicable sampler useful reason set decomposition subgraph partition subgraph sampler three illustrate additional result available reproduce mechanic xy sequential resample confusion choose second scoring toy markov mrf potential decrease xy model mechanic ising spin periodic order normalize constant xy adaptation proposal von distribution node right add towards site update linearly start geometric importance design fairly evaluate performance order box sampler match cost spend give r algorithm order depend temperature option easily implementation anneal step node cm main minimum circle width fill gray topic latent corpus often conduct hold w learn challenge since procedure sequential decomposition graphical decomposition include node order rao variable reduce exactly propose sufficient case original learn simulated simulated compute keep particle mean demand perform plot real held particle sample run bar estimate bootstrappe logarithm show see simulated could degeneracy long tailor toy illustrate incorporate deviation gibbs tree sampler moderate particle admit correct among result hold model gibbs sampler gain simulate moderate particle scope beyond model fully drop zero fast surprising latent jointly reduce autocorrelation strongly note iteration new propose see improve acknowledgment thank kind provide lda project contract contract research additional main direct state note result new provide proof state member mechanic ise spin describe angle lattice periodic condition individual site spin temperature hamiltonian describe size target
truly mix task example unlabeled set accommodate different penalty annotation intuition entity confidence explicitly label example indicator total training high replacement entity tb wikipedia point distribution label wikipedia regardless choose improvement stable language maximum language wikipedia create bias entity canonical mention usually throughout remainder belong belong single bias reflect name entity tag link inside entity link tb stage annotation word entity article annotate tag tag tag link exact string matching mention entity appear link exclude frequent vocabulary improvement add improvement especially improvement recall tag p english heat ba pe en v pe name change france want european de twitter star win di circumstance result chinese news public security east bring death throughout english build house david thompson cross en subsequently team record home operate pour le successful l sa rl da short da arrival city ia di di york york chinese party become head china security france unite several organization denote error label acquire google translate label translate annotation analyze produce qualitative efficiency solution wikipedia annotate example language correctly annotate mistake show system name language example system identify entity scenario robustness stem vocabulary language sufficient contextual error group category word appear hard system tag error occur include confusion tag entity chinese tag company name lc english corpora dataset english wikipedia evaluate alone competitive outperform english apply rule appear wikipedia evaluation contextual include english dataset name map vocabulary embedding rely tailor preprocesse step difference approach scalability notational rely tailor preprocesse wikipedia suit human annotate training domain train wikipedia well annotate scope evaluation far seek translation tool preserve count annotation language aggregate sentence emphasize indirect annotation quality translate name entity language preserve name entity hold language set entity phrase belong category category belong source language measure entity entity sentence translation pair language english wikipedia annotated sentence pick sentence detect translate sentence translate language calculate entity positive language large wikipedia article english vary category english origin source annotation benchmark highlight language evaluation skew metric general find translate translate language translation english affect count translation original english investigate effect count wikipedia wikipedia attribute coverage wikipedia average versus wikipedia article negative hand wikipedia entity versus wikipedia entity versus wikipedia article language feature language annotate comparative analysis translation language wikipedia languages author subject keyword nlp extraction massive diversity language introduce ir write language family build massive intervention build name wikipedia language resource parallel corpora therein agnostic competitive language automatically wikipedia finally linguistic performance annotate gold community content process nlp number language english internet correspondingly text surface form address aspect work build name system entity know text phrase pre name essential processing nlp retrieval ir systems extraction base address rely supervise drawback human second design linguistic require language building work address language technique wikipedia automatically language c preprocesse stage yshift language language dictionary tag corpora annotation dependency language yet comparable encode syntactic language internal article name entity link entity include anchor link address propose matching avoid annotate standard language machine tag constraint appearance indicate base mechanism semantic syntactic unsupervised successfully develop embedding acquire huge amount raw language capture co syntactic semantic abundance embedding specifically language embed range feature language investigation train wikipedia without label frequent representation consist embedding objective word cluster name proper word w I f tag neural hide unit hide tag
knowledge guess mix community dendrogram partition dendrogram enable systematically remove community root level dendrogram tree subgraph statistic iteratively select subgraph high statistic value since remove likely member community step nod remain mixture statistic location mix idea dendrogram similar removal summarize mix mix follow dendrogram set amongst si es complexities complexity complexity give network complexity exhaustive virtue mix method outline probability derive procedure tail field differ major aspect rely variable normal distribution depend complicated eq second derivative variable expectation respect special upper es mix community case relatively false community set contrast false raise alarm calculation http www edu method delay es mix sequential exhaustive es hierarchical mixture mix sequential ratio es complex quick community drawback community mix incorporate dendrogram decomposition theoretical mixture demonstrate focus paper community accomplish subgraph high mixture extend outline bernoulli variable large central theorem unit variance use equation tail function variable recover expression keep keep integral desirable combine equal use since replace sum adapt series detect model graph form es mix run length detection polynomially exploit active community h mix dendrogram decomposition analytical mixture threshold approximation determine mix detect community quickly es detection variety cancer often consist differ characteristic detail often clique detection realization true observation divide dynamic shot observation static concern sequential increasingly network either structure change latter community due real processing approach many datum previously stream usually base heuristic recognize theoretical community therefore cast framework community network count inside detect graph adopt differ sequential statistically property formulation three exhaustive es h mix es method perform exponentially complex size know polynomially fact raise active community mix address impose dendrogram detection procedure method numerically verify numerical explain mixture contain numerical matrix community community illustrate figure represent adjacency red form problem nan th alternative subset node baseline case either know define run minimized expectation define member interact typically represent anomaly replace setting replace community assume test likelihood possible possible window window limit grow exceed exhaustive search size recursive statistic u statistic formula calculate
around fails skew normal skew well fit assume show covariate small skew numerical normal method skew et omit probit skew coincide poor lack dependence correction confirm little situation logistic show bias reduce apply covariate probit regression appear widely r r regression skew logistic skew carry assess effect type consider normal assess degree skewness assess normal latter derive bivariate skewness type respectively usual fit omit unit skewness scalar covariate one omit solely df et al regression parameter log variable covariate addition treatment throughout approximation provide close covariate covariate apply apart binary treatment indicator clinical variable category include predictor randomization covariate treatment covariate analysis adapt randomized treatment continue e partition factor denominator extend arbitrary number rather restrictive correct treatment arise allocation statistical model would perfectly lead trial calculation odd ratio course practice importance element zero penalty asymptotic however issue relate account logistic expense investigation nevertheless help inform analysis probit analyse logistic probit correction essentially correction estimate treatment less probit prefer logistic false approximation equation false probit maximum probit presence factor e equation skew expectation logistic expectation give denominator author trial randomization take assess summary bias arise omit randomization accurate asymptotic normally covariate omit compare convenient form insight apply additional asymptotic logistic probit trial use linear important baseline omit randomization ensure relevant omit treatment necessarily carry identify also important logistic asymptotically covariate author randomize approximation treatment omit general scalar exposition article taylor restrict whether fit varie article skew logistic obtain false covariate omit taylor approximation excellent form apply usually take treatment arbitrary number result wider compare skew extension allow additional covariate require assumption conclusion draw suppose respectively e assume tend false expectation take distribution extend skew scalar dispersion mean distribution multivariate normal principle could dispersion change analytic case partition outline detail derivation even I variation fit repeat probit trial fit omit expansion imply scalar covariate fit normality make author series expansion omit fit unconditional expand exact analytic normal increase close change correction slightly reduce solid line dash line dot adapt fit include addition give still th limited alternative multiple probit regression distinguish analyse natural probit tx estimator essentially denominator place although result slightly see appendix
understand immediate thus minimize use q form original definition thank fx x fx fx fx finally induction conclude thus conclude fx come second hypothesis show nesterov accelerate descent vary sequence initial smooth nesterov accelerate unconstrained obtain q multiplying obtain remark put get u induction show constraint behave euclidean ambient meet gradient technique dimension rate instance euclidean fx mirror minimum ball mirror descent devote mirror alternative presentation inspire chapter intuition situation forget dimension observe project arbitrary hilbert situation banach make formally dual update point point space lie outside need way project back mirror map chapter compact convex fx fx bregman useful closure mirror boundary mirror mirror point lie notion precisely map take ii write result primal point outside one project mirror projection bregman associate precisely uniqueness projection follow lemma bregman essentially imply immediate mirror let xt illustration thick side inner token label token label token token token control map strongly satisfie claim one lead mirror conclude observe mirror mirror view mirror try linearization move far away measure bregman mirror simple mirror descent mirror furthermore mirror project descent recover early project descent write equivalently bregman map kullback divergence bregman simplex result know one mirror achieve case von gradient update write equivalently exponential logarithm projection trace non show easy word simplex consider descent true advantageous situation mirror descent extension mirror averaging replace ask mirror descent average w x x x proposition x putting display schwarz obtain would computation latter hypothesis mirror chapter mirror attain mirror prox mirror prox smooth mirror descent mirror prox make mirror instead rectangle side inner token right token right token token token token label node control mirror fy fy x separately mirror obtain term schwarz sum straightforward computation mirror mirror satisfie come learn see mirror descent saddle calculation setting satisfy prox arbitrary eq dramatically already always function optimize globally specific structure smoothness smoothness overcome chapter description interior rather oracle know simplicity clear description inspire recall want minimize assume quite natural locally approximate n gx fx iterative shrinkage come elementary computation fista accelerate fista easy fista fista metric euclidean composite mirror obtain detail quite smooth see structural attain nesterov smoothing find subsection saddle computation proceed descent chapter warm powerful mirror prox next compact convex yx explore algorithm produce candidate solution duality gap x key similarly eq z g x view mirror mirror map later field sp saddle easily md imply norm thus use mirror prox mirror algorithm sp saddle point mirror describe follow tw z mp satisfie light suffice field r introduce sp md mp minimize mx mf il x l sp order mirror obtain much box iteration non smooth let denote zero correspond obtain attain furthermore step sp mp dominate multiplication get nash sp mp separate look generality replace column find write sp solve iteration vector multiplication fundamentally far type q objective self empty interior fix enforce live subspace span modification algorithmic consequence newton stay word equality lagrange multiplier refer chapter explore optimization randomness key go quite computed progress optimum small gradient correct vanish picture case long step order oracle take point output possibly assume need smooth stochastic oracle follow interpret loss convex find minimize stochastic draw distribution report second describe want minimize report stochastic quite situation access want oracle pass indeed expectation use point contrary pass mirror md thanks ft x mirror map strongly md satisfy generalize md size q let strongly oracle optimization basically exact bring acceleration acceleration square sharp exact descent thank next use smooth mirror map convex r furthermore assume fx cauchy schwarz strong obtain thus yield sum conclude modification mini batch sgd conditionally query oracle convergence mini modify stochastic return thus obtain call mini mini batch sgd call mini batch two situation iteration sgd multiple processor central unit processor gradient independently serial mini particular calculation several estimate examine detail unconstraine context compare basic independently everything else gradient computation improve sgd computation low reasonably gradient positively sag sdca coordinate ascent require gradient computation descent sag natural one nesterov question sdca rate sgd typically center oracle precisely center also convex f ix ix ig ix particular ix ix write respect observe yield claim center rarely idea lead epoch initial everything else convex clearly simplify follow dependency upper thus one obtain fx fx fy x fx fy note fx finally section simple optimize denote everything else draw md lipschitz accuracy next greatly differentiable maximal bounded see smoothness independently preprocesse context basic descent attain improve function attain potentially iteration obtain I fx fx fx putting obtain computation potential acceleration la directional assume attain let strongly follow elementary convexity elementary calculation fy fx proof theorem hand randomness computation true sp sp md stochastic saddle point assume x r r satisfy md section column quite oracle step sp md index take overall nash equilibrium sp mp dependency instead subgradient modify note ij unfortunately turn free specific situation briefly relaxation randomization study let entry dissimilarity two maximize total weighted adjacency rewrite follow turn optimization problem existence combinatorial difficulty stem replace efficiently eigenvalue power randomization hypercube clearly furthermore immediate implie sample uniformly hypercube arbitrarily approximation even display know sdp relaxation sdp find solution point follow state solution sdp relaxation lemma v j v quick I euclidean remark use fact one l l x positive expense constant interested sdp relaxation sdp show b semi latter taylor denote matrix entry one conclude gram randomization naturally section assume could prove generalization cut convex draw uniformly particular isotropic set isotropic one near isotropic position replace method isotropic one obtain chebyshev nn ensure randomize center progress constant fraction progress would unnecessary isotropic x n least isotropic explain convex picture direction run enough distribution total uniform randomize good distribution correctly like ellipsoid informally center need random either step walk overall need call oracle walk chapter chapter corollary chapter address financial engineering edu chapter chapter study convex segment algorithm write compactly precise constraint machine problem interpretation cost simplicity I mf iw detail origin one take hinge obtain svm row lasso capital letter observation unknown matrix complete way result sense result extensively separation guarantee immediately support hyperplane support hyperplane notion set result essentially admit existence fx fx fx recall obvious g f fy g rescale interior build subgradient hyperplane exist let infinity enough imply conclude differentiable fy interest empty yield interior affine generate notion particular function exclude sometimes extension notion algorithmic around subdifferential information another instance global minima global minima minimum happen enough algorithm alone justify optimize surprisingly admit excellent describe aspect argument already many learn logistic regression formulate conclude extension constrain sake simplicity optimality trivial decrease black assume resource objective input oracle input subgradient interest many sufficient minima reasoning need box allow derive theory matching limit resource course pay box early reference recent year algorithm popularity quite high explore detail chapter chapter chapter dedicate noisy discuss cover free black section seem extremely objective globally structure address ultimately optimization extremely interior technique fista mirror able efficiently program direction lp consist problem semi frobenius sdp section sdp quick specific hard summarize emphasis present expense make practical lipschitz know also tune reader adapt potentially reference text consideration denote define depend whether subscript positive rate iteration dim non ellipsoid separation lipschitz one smooth nesterov smooth fw opt convex lipschitz one smooth nesterov fista gradient prox sp mp md md newton empty interior chapter box solve let let stop query center side comment optimal center query well need fast rate follow require attain digit double comment concern turn computationally carry general section center randomize method turn theorem geometry center prove point otherwise would proof clearly obtain e nr x rx convexity fx ellipsoid convex geometrically semi length eigenvalue simple lemma heart ellipsoid method x furthermore one ellipsoid computation derive show ellipsoid ellipsoid norm quick picture sense ellipsoid would one inverse semi axis principal axis look half look ask write eq latter quite naturally minimize ellipsoid show maximum attain elementary computation ellipsoid ellipsoid c h c x ellipsoid ellipsoid formula conclude ellipsoid access separate hyperplane separate tw x h ellipsoid stop observe remove ellipsoid ellipsoid much bad indeed need call call situation case derive basically always intractable instance context oracle overall interesting property ellipsoid give radius back simple minimize differentiable function point fix behind make minimize local descent see complexity feature attractive section chapter operator follow study rectangle token label token token otherwise chapter simplification contain euclidean note subgradient inequality schwarz make modification gradient exist importantly make update onto iterate prove token token leave token label node project subgradient satisfy elementary identity x plug value directly recall convexity reach need call oracle sense complexity ambient dimension quite center ellipsoid put differently could hope ellipsoid dimension explore restrictive optimize complexity computational bottleneck often case admit analytical think combinatorial algorithm operate function finally recommend depend iteration practice undesirable vary one next indeed go auto say continuously differentiable explore improvement theorem iterate previous smooth gradient descent eq represent integral cauchy smooth give improvement next lemma inequality smoothness smooth fy denote show display imply thus together constrain come constrain q gradient iterate fx gradient smooth optimization gradient started decrease see case expect may cut lemma progress constrain x still prove result convex give show decrease see g describe compact convex introduce perform direction illustration perspective key replace case token right token side turn gradient descent former precisely fy x follow r norm true easily arbitrary convexity induction inequality produce iterate finite set definition know iterate vertex dimension regime see vertex representation interesting property together rate prove simplex minimizer clearly function schwarz one norm iterate efficient example inspire problem open signal dictionary way q dictionary instead constrained situation nonetheless reasonable want assumption polynomial problem meet combinatorial dictionary span lin lem finally correspond seem take save admit minimizer discuss exploit descent study compute needs derive smoothness fx fy mean respect effort descent significantly first follow inequality immediate verify strong convexity curvature instance lead rate optimum step optimum course smooth careful tune
anti nominal interested na b quantity compare asymptotic otherwise always write easy integration determine I must q figure error nominal range asymptotically conservative variance choose depend large line point conservative rate increasingly formula solid line anti conservative nominal rate conservative small error feature suggest critical value hybrid bootstrap critical asymptotic examine detail penalize ridge statistic let I nn smooth ridge interpret assume write write q penalize testing effect penalize whether fix solving ridge penalty know regression lasso pattern standardize group understand appropriate penalty penalize implement package acknowledgment thank provide valuable insight state prove additional lemma need proposition mutually ij sub gaussian follow sub jt kt union hold otherwise depend convexity ball expand enough write lemma hold satisfie without p b tend minimizer argue show hold add subtract c tending condition write multiplying give recall central q distribute dominate dominate corollary interested association assess constant variance formal method likelihood ratio set test low case q condition identify value interval formal test remain year technique formula bootstrappe expensive involve variance suffer zero recently produce feature become parameter produce test nan hypothesis variable lasso however give individual conditioning framework permit regression interpretation datum place alternatively inference dimension invert stationary lasso regression gauss markov unlike knot lasso start decision make need diabetes measure diabetes patient list remove covariance sake regression outcome feature propose penalize section choose validation htbp l lin tc age refer refer code author test refer table ease display order decrease stop observe stop upon reach covariate produces variable interpret presence model association accounting trend interpret discrepancy answer question broadly however propose regression feature separately lasso decision association outcome result related lasso rest method penalize score general penalty consistent special test extension induce penalty bold font matrix font k procedure notation way also hypothesis assume score statistic large appropriate reference apply set multiple regression variable feature wise select penalize serve penalize non statistically hypothesis discuss appropriate since interestingly pattern lasso tucker lasso choose mixed effect model effect assertion precise connection explore suppose log l regression bias incur inference penaltie classical vs however treatment depend throughout scenario regime define parameter stationary association effect note concrete parameter include associate extent concept fix choice theory special regression come briefly greater penalize penalty vector proposition appendix depend may distribute p need central limit large quickly require c constant condition zero distinguish particular require inactive residual ensure grow correlation inactive grow quickly boundedness allow quickly convenience tail slow zero detect condition condition hold approximately hand approximately n variance use test estimate residual option fan rely replace appeal interpretation light test adjust turn pattern regression apply conservative nan surrogate classical unbiased relationship unbiased related condition test consider connection variable selection penalize score variable penalize score interpret shift classical score score take eq order give differ statistic center variable penalize consider case omit ease exposition statistic testing know advance order support hold condition among thing connection recovery support penalize statistic close statistic reject correctly test meaningful result score depend se expense freedom spend explore numerically score proxy test small behave generate matrix outcome rest zero size average replication dimension penalize turn sake regression result use residual simple obtain test truly associate positive unbiased nominal rate give function choose control examine produce non consequently behave multiple hand include feature demonstrate test penalize test penalize test respect type measure ph formula penalize multiple plot summarize replication simple regression marginally correlate whereas relationship nearly surprising identical simple decrease dimension power decrease enter correlated typically high htbp indicate six correspond penalize bottom panel line true average bottom panel close nominal rate beneficial assertion support show
quickly whole segment consumption orient purpose ideal good entail coherence clarity summary compose part song may issue produce summary summarize music automatic human redundant performance rely portion audio compare music contiguous truncate begin middle song music comparison purpose present review review similarity describe detail introduce classifier report conclude remark music propose music extract person automatic consumption coherence requirement summarie centrality similarity sentence page text another social focus improve speech sentence sentence previously select another technique segment select include summary segmentation aim extract meaningful segment segmentation change detect repeat segmentation boundary build contain every apply cluster output first similarity change cluster produce final strategy label belong call extract long piece filter building lag embed find position summary term lot segmentation use simply allow look use important part aim meaningful segment song later include summary lead combine generic review algorithm choose approach segment duration song later song aggregate whole song second keep distance pairwise centre sound select build similarity value feature calculate sum column row since similarity accord desire whole song length piece index start maximize evaluation whether summary song evaluation average summary clarity diversity summary produce summary take select sentence maximize model metric sentence previously select query sentence relevance diversity sentence centrality rely pair centrality google web page list rank one first cosine step create accord weighted calculation convergence successive vertex guarantee vertex unweighted edge vertex summary length reach determine score sentence mathematical text reduce dimensionality element times occur sentence global sentence singular diagonal sort use correspond calculate column extract sentence iteratively singular equal increase sentence never summary sentence introduce half ij music impact classifier classifier song music solely classical used song concatenation song energy base hz hz range low frequency hz hz frequency matrix frequency along min peak peak max distance peak frequency contain string content distinguish half dataset encode hz microsoft file request cross calculate first begin middle end summary algorithm extraction extract also operation frame overlap vector generic need additional music audio frame piece k vocabulary song frame vocabulary segment allow sentence depend sentence cosine sentence apply summary tf calculate iterative picking length reach type implementation operation segmentation concatenation sentence singular small score explain rank weighting combination frame overlap word size widely music present try interpret overlap g
complex extract together represent adopt code sequence site follow biological great challenge area system biology main focus theory biology protein study individual application biology p tumor important study individually measure system compose however despite great theory limitation lack classify redundant measurement mine theory important bioinformatic characterize gene extraction relatively information succeeds even consider belong category extraction genomic sequence complex network use among sequence highlight difficulty recognize predict present genomic adopt genomic graph theory connect formalism real property sciences physics thus complex connection internet group topology represent biological characterize term possible complex list shortest give length minimal determination characterize short cluster agglomerative network two friend common define kind centrality pattern find maximization start site dataset various specie work contain sequence extract genome remove code segment merge reverse reverse genome select code region genome database network information global relationship genomic sequence total maximum entropy sec networks accordance approach occurrence propose consider next consider genomic consider complex nucleotide network network measure standard deviation node genomic measure evaluate extraction dna coding method machine adopt ten fold cross change radial feature task perceptron bayes notice indicate genomic figure show roc possible effectiveness classifier propose genomic network occurrence identical possible identification sequence correctly correct genomic pattern combine composition genomic sequence potential methodology achieve classification addition obtain complex network
hyperparameter compete decomposition one sparse representation optimize hyperparameter automate also effect desire micro min macro min low representation document combine present advantage supervise eq expression depend concavity term inequality relaxed update rewrite matrix table table summarize matrix division one indicator il il corpus tag readily lie utilize variational factorization coefficient discriminative tf representation label yield pattern inter represent convenience inter categorization latent supervise essential machine cover research principal analysis factorization well upon formulate entirely application available appeal use dimensional preserve discriminative incorporate fisher discriminant purpose lie mathematical offer graphical family along semantic baseline mining modification probabilistic know robust prove denoise missing property formulate labeling relate syntactic word simple intermediate word term result frequency term approach specific term document measure tf normalize rare entire corpus document document corpus document contain approach mine know space pattern frequency relate rather learn perhaps good formulation assume inherent nmf decomposition worth nmf kl minimize objective term intermediate nmf decomposition decomposition general extent model divergence impose paper categorization regard collection represent tf organize semantic component word model superposition model assume gamma noise exponentially distribute indicator formulate mixture discrete eq indicator convenience notation variable represent expectation prior tail around prior impose additional indicator large activation constrain notation px il il joint p gamma additionally formulate sparsity decomposition sparsity share tag variable hyperparameter unobserve general bayesian introduce instrumental kullback rise bind density factorize q show update low conjugate factorize approximation expression analytical specify computational convenience include outline treatment b rather generalization potential classification representation test sort date multiple computational load reduce tf consequently classification place consideration poisson unsupervise tf learn dimensionality test optimize component learn gamma decomposition fix shape gamma distribute coefficient run hyperparameter optimize maximization directly bayesian varied hyperparameter optimize follow initialization fix burn optimize predictive k square root cardinality latent varied method nmf minima initializations metric macro average document belong denote belong split label accuracy macro average imbalance measure l originally context penalization take purpose column treat document component expect document belong model pattern support denote sum coefficient accumulate label document motivation behind exclusive pattern sparsity relaxed definition paper micro accuracies initialization vary semantic constraint improvement regardless penalization explanation though representation natural differ greatly label representation well boost result present decomposition experiment beneficial label sparse micro accuracy matching produce
prediction interaction glm glm interaction impulse response linear impulse response direct interaction roc supplementary material detailed outperform matrix predictive baseline normalize ten graph presence absence interaction experiment roc identify glm hold outperform spike homogeneous process standard glm power supplementary discover interpretable world interval sep price price event process yield trading course day daily variation incorporate periodic supplementary scale compare prior train outperform interpretable structure std ex bottom model stock nearby stock interact describe embedding plot stock list com notably energy tend together indicate interaction stock broadly suggest stock influence slowly vary may diagram bottom top interaction top notably suggest activity cascade fourth eigenvector drug perhaps encouraging report weighted interaction period cluster break offset share day per related underlie network occur mutually occur frame training cox rate baseline process uniform process community consider community separate group community cluster material process network latent interaction identity improve south allow area due overfitte insufficient potential interaction prior help community predictive high predictive log likelihood come cluster model distance suggest discover look figure safe gold buffer neighborhood activity might report west show increase rate consistent trend point wise add periodic capture interest previous poisson intensity maximization fully bayesian consider special case infinite exchangeable discover interaction process gmm identity prior suffers recently leverage conjugacy prior promise discover network perhaps inference discover tendency similar interact recent development nonparametric linear interaction consist purely outperform discover background unobserved process identity uncertainty fully bayesian noisy prior interpretable variety real generalizing beyond promising wish thank discussion fellowship text parent first induce process spike weight recall impulse response spike proportional gamma distribution rate impulse response dataset allow scale cox equally spaced background rate grid equally spaced calculate integral elliptical empirically period day year well know trend scale offset log set homogeneous event training able distance place log characteristic scale weak gamma place uninformative gamma shape typically scale know scale scale invariant prior root information generate event spike glm test second run chain simple cross correlation w thresholded point process model external external impulse response likelihood concave easily allow link direct roc homogeneous top component logistic impulse sample glm last second glm figure likelihood os model interaction glm compete average across collect week positive negative change price activity would generate stock ignore outside trading price short choose parent interaction high vice versa brief jump stock markov interaction last sample trading vary day peak market cox periodic posterior main sbm stock interact interact stock latent distance belief explicitly block model connectivity vector probability place prior sbm comparable infer correlated com h std net sbm year predictive consider iteration likelihood expectation illustrate main dataset intensity spatial normalize community introduce location prior belong encourage spatially cluster cluster os discover cluster prior localize model fail dataset intensity figure prediction latent model spatial likely suffer complete prior spatial gmm process name play central role modern analysis enable study part analysis edge many enable probabilistic combine random poisson superposition enable elegant empirically modern characterize via vertex infer critical pathway trade disease associate identify low representation e know entry literature concern directly perform implicit infer noisy dynamic structure financial stock market execute time stock relate infer interaction wide throughout discover interpretable pattern insight explore stock infer trading example edge attribute induce cascade vertex unobserved financial possible dynamic infer case identity spike activity mutually interact recently result markov augmentation efficient parallelism process set poisson canonical govern nonnegative intensity moreover subset set event poisson poisson poisson event interaction event known point specify history event add nonnegative impulse generative impulse draw number induce draw time I enable computationally intuitively background rate event attribute constant fluctuation share variation trading trend fluctuation log cox background periodic capture daily fluctuation offset intensity process recover constant identity infer alternatively occur cluster
lyapunov condition variance define lyapunov condition specifie demonstrate lyapunov note limit term membership hence lyapunov lyapunov convergence rearrange term show stack observation k define ab ab process state mutually linear term give kalman linearity filter estimate estimation enough linearization vector mean optimal vector membership item item address space alternate item search hill estimate sbm factor probability previous plug section recall form probability exist occur state initialize ml estimate initialize search cluster vector assignment estimate assignment compute predict phase two make mistake cause test real dynamic facebook analysis day step people less leave people tp initialization singular value snapshot adjacency actually community diagonal block snapshot enter quite regardless occur variation histogram edge facebook sbm sbm network step sbm fit histogram sbm figure assumption majority show fraction sbm propose provide fit well forecast notice network appear replicate effect add dynamic simulate edge depend whether step would forecast number create future author thank provide access facebook great recent model network inspire well stochastic sbm absence influence future sbm derive procedure reproduce analysis statistical network static consist modeling appear representation allow refer evolve target discrete network assume snapshot particular greatly simplify flexible replicate observation network dynamic static extension utilize absence time edge would version adjacency advantage property combination extended kalman dynamic ability accurately replicate dedicated modeling dynamic mostly several excellent survey extension static exponential model continuous extension stochastic relate membership sbm propose dynamic extension sbm model discuss markovian dynamic snapshot tractable realistic interaction interact interact facebook interact month hide markov influence occur influence weak incorporate future current currently propose satisfy latent static represent edge node square adjacency matrix denote edge vector static adapt adjacency static membership vector membership class identically I adjacency assume estimate interest focus difficult posteriori set evolve edge snapshot mapping index mapping correspondence invert remainder mapping time follow denote adjacency let dynamic stochastic dynamic sbm dynamic extension past sbm sbm parameterize specifie author simulate annealing place evolution govern dynamic apply kalman procedure magnitude fast relate conditionally previous edge tie form edge density particular undesirable propose edge long density time iid likely appear edge class respectively sbm iid edge accord two matrix denote probability accommodate node membership formally model generate accord stochastic transition adjacency static sbm node statistically independent time choose satisfy node scale probability adjacency matrix follow sbm transition valid finally provide connection sbm derive satisfie three must valid next present change step becomes scale property inequality meanwhile substitute combine arrive q low bound function accomplish choose argument current substituting value satisfie property derivation
loss classifier excess newton method yield expectation contrast excess learn learner access advance additional exponential loss excess function utilize complexity tailor exp sharp convergence setting sequential notion loss margin condition focus concavity derive study concave risk learn classifier follow optimization concerned optimization classifier step parameter receive n newton difference online smoothed version covariance receive newton gradient classifier online covariance examine several also vary focused excess classifier first batch introduce key note unlike hold respect ii make verify strictly convex property concave ii first monotonically therefore sufficient surrogate calibrate binary excess batch learn eq provide risk fashion final intermediate return indicate excess online general modulus dd function unclear excess exponential question batch achieve batch advantageous aspect batch second important determine step proof technical result defer variant improve also eq rooted concentration use key follow deal concentration e fix bernstein eq domain next proper net bind probability moment return theorem overall prove excess loss state fact combine imply plug excess complete plug turn prove base notion rademacher work consider rademacher small hypothesis set certain divide variable sample domain item side p rademacher rademacher complexity bind use next g together union obtain complete combine bound turn prove online exponentially theorem ii add bind separately start indicate inequality martingale lemma condition lemma complete proof address online show address remove excess concave careful open question investigate future improve dependence sparse accord excess risk plan sparse analyze generalization exponential concavity combine rearrange obtain I I conclude take optimal proof bernstein e bernstein inequality martingale denote variances martingale eq martingale kt kt kt kt kt r kt I bernstein desire inequality goal excess exp
tried achieve demonstrate base heuristic vs comparison multi task average test small specifically outperform plain advantage compare robust feature solve nesterov employ accelerate result superior perform convex formulation rule intuition refine estimate threshold intermediate alternative gradually improve synthetic effectiveness theoretical acknowledgement china cb natural science china like constructive suggestion task aim feature alternative present might performance adaptive additional adaptively determine expected selection algorithm embed come empirical world effectiveness variant limitation common incur potential unlike structure successfully stock exist relevant feature share task commonly share learn well common share convex restrictive suboptimal regularize correspond task feature algorithm achieve convex computationally prohibitive notice regularize formulation employ prescribe row unknown value adaptively determine achieve structure intermediate current stage threshold stage solution scheme iterative wang et jump rule threshold though method could apply prescribe feature obtain first conclusion scalar denote denote capital letter euclidean frobenius denote scalar brief introduction feature share learning row represent scenario share learn quality measure function throughout magnitude reflect assume establish situation commonly case unconstraine occur standard impose constraint obtain reflect example aspect really natural choice parameter feature base regularization I iw unknown unconstraine formulation balance loss sparsity individually make almost relevant another prohibitive computational burden make joint sparsity norm regularization th share relevant share certain share task regularizer type convex loose moreover employ may convex formulation base task feature learn task performance model prefer regularization th problem denote row penalty feature formulation multi algorithm regularize feature theoretically obtain proceed circumstance interpretation l initialize w jj indicator like multi limitation fix prescribe may optimal difficult well aim threshold refine adaptively last rule appropriate adaptively determine value follow q still function balance threshold dependent assume prescribe significant jump significant review correspondingly modify task feature threshold detail threshold accord keep update proceed eventually reach empirically j jj end though lead achieve refinement model regularize well reweighted algorithm compressive regularize non convex formulation gradually iteration proceed necessarily inherent error intuitive explanation could undesirable local correct decrease rigorous addition exist intuitively critical constitute original prove solution one return unique rule sort small amount jump set rule capable detect false fast decay heuristic kind matrix adopt method formula could try true many quality though case tuning key typically large gradually improve result intuitively jump partly magnitude one magnitude spread false one cluster jump compressive observe pattern importance rigorous value challenge effectiveness jump multi task row nonzero generate data distribution noise sample response subgraphs subgraph clear
euclidean empirically vary dimension perform validation neighbor six near synthetic weight weight synthetic fig point manifold euclidean base dot right fig geodesic distance compute function learn ground distance remove effect mr achieve poor preserve section world database experiment face vary leave image second datum contain category image set label query retrieval query precision recall evaluate result different algorithm relevant query score rank position ap ap query c mr le map average scope scope rank scope curve precision table map provide recall indicate reliable ranking list comprehensive improvement indicate manifold embed study field perspective provide characterize geodesic novel heat field learn experimental result demonstrate develop machine designing algorithm use heat differential show heat accord appropriate boundary dx dx derivative derivative self adjoint v derivation positive manifold give heat xt xt eq essentially heat field next analyze behavior heat equation primarily along connection heat geodesic dx p please represent dp dx lie line segment connect unique parallel dx field flow field around heat field field gradient heat view specifically vector field uniformly riemannian riemannian manifold desire function measure intrinsic distance geodesic distance simply call euclidean tool study manifold tangent smooth let point linear derivation vector totally abstract space tangent define derivative direction derivation write directional derivative denote use local chart verify coordinate tangent dimension manifold embed sphere dimensional plane denote union vector usually property field nothing point tangent derivation worth note tangent manifold field summation convention imply summation index might write next assign manifold inner tangent smoothly mean smooth subscript omit context write metric define close curve dt axiom positivity difficult give function riemannian field coordinate define direction measuring coefficient rely coordinate constant vector constant fundamental field connection vector field manifold compute coordinate use hold rule connection equality due since manifold symbol symbol curve chart field curve vector along curve tangent vector manifold parallel three curve might worth parallel field curve field along gd v along angle parallel field curve field suppose vector parallel along three general parallel along curve call geodesic geodesic recall derivative result also dt second partial equation geodesic geodesic constant parallel curve geodesic tell short path must geodesic geodesic short length measure minimal study geodesic tangent tangent exist complete every geodesic connect hold please fig example map minimal geodesic connect opposite long geodesic connect geodesic unit measure opposite everywhere except laplacian operator field field smooth field smooth tensor q tt lt x laplacian adjoint connection orthonormal notation function denote hilbert schmidt show derivative space open subset say represent appropriate evaluation differentiable define origin define therefore exactly derivative functional equality self adjoint due since manifold geodesic complete actually unique minimal geodesic inequality ds rp equality geodesic hold whenever geodesic curve integral exist curve pass curve pass first prove neighborhood point vector since gauss pass curve geodesic vice versa curve pass pass manifold complete geodesic connect geodesic uniqueness geodesic prove function connect note geodesic speed finally connect cut unique cut limit rx dp next compare hessian evident g r hold due fourth distance theorem section section ji center evolutionary state college university china manifold great learning euclidean learn preserve function manifold euclidean well paper directly characterization gradient field theoretical propose learn field whole heat gradient experimental datum demonstrate effectiveness learning pattern recognition desire retrieval depend label classify category unsupervise supervised point label unsupervise learn view space lower map euclidean preserve principal mahalanobis typical le diffusion try preserve original manifold reflect connectivity diffusion preserve preserve manifold learn distance original may euclidean preserve sphere embed geodesic geodesic distance variation definition property intuitive geodesic short short distance consume handle manifold geodesic pde grid ambient impractical ambient note tangent equal ambient field tangent ambient inspire field heat propose geodesic characterization heat flow field geodesic field unit learn learn initial local heat flow asymptotic approximately gradient geodesic obtain close normalize optimization linear system complete synthetic demonstrate riemannian goal desire distance provide natural measure fundamental follow briefly concept detailed introduction tensor inner relationship let neighbor approximate estimate tangent space neighborhood mean true mean component vector recall tangent abuse vector j jt tx jx please function block diagonal th block x ix j ic tx ij n selection tangent parallel worth note connection laplacian semi matrix normalize derivative via field normalize solve plug remove th side varie column field summarize dominate part search near tangent space
context dynamical manner interact consist whereby become replace step parameter gp perform dynamical ability state high use likelihood apply variational dimensional nonlinear system distribution gp mat ern particle lag smooth particle lag compare gp take particle ern ard identification offer substantial available test performance obtain batch time cccc rmse tr train time gp min var gp min gp gp min neuron ec region potential rapid little overfitte learn space crucially tractable oppose length series believe future work variational eliminate smoothing well gp prior equilibrium limit etc material side noiseless q transition independent induce interpret deterministic time radial deterministic lead opposed parameterized perspective analogous regressor undesirable variance away induce input exactly opposite behaviour would process transition emission augment variable emission straightforward latent matrix log emission induce trajectory optimize hyperparameter ascent compute uk successfully sparse process system parametric straightforwardly trade avoid overfitte main hybrid bayes fast widely success diverse finance popular series auto bayesian sparse gps encode continuity variational complex risk overfitte lead posterior prediction linear use smooth state dynamical inference accelerate learn scheme work uncertain prediction enough nonlinear create principled arise engineering finance instance adapt characteristic nonlinear dynamic new adapt control situation quantifie avoid risk flexible later extend parameterized process view tailor develop nonlinear find map latent learn issue apply much small could mapping state recently manner employ particle smoothing prediction article state lead representation space eq noise future another noise state dynamical useful time deterministic signal omit function mechanic application specify ability thank specify straightforward way e light place process process severe strong correlation variational formulation apply double supplementary prior convention return group hyperparameter note restrict particular distribution gp convention gp induce matrix argument equation doubly transition rich nonlinear hyperparameter qualitatively instance dynamic panel tractable approximation dynamical tb prior generate qualitatively linear panel appear limit cycle nonparametric necessary find distribution integration expensive since trajectory alternative aim transition series technique induce latent density induce induce omit conditioning variational popular making lead tractable maximize inference methodology induce distribution relate inside denote analytically variational ability location induce location posterior tight trade without risk distribution evidence eq mean optimal depend x smoothing likelihood augment tr smoothing markovian modeling strategy characteristic particular system severe likelihood heavily multimodal smc present auxiliary present alternatively propose hybrid variational whereby sequential carlo smooth discuss section characteristic dynamical appropriate smoothing particular respect instance covariance assume respect variational use optimal
nn sc nn mat ern experiment sequence video pixel ability dynamical variational vast allow video learn give frame pixel near achieve firstly benchmark processing video challenge create scene frame periodic contain video mat ern functions compound kernel periodic reconstruct frame original video outperformed nn demonstrate visually video material l reconstruction dynamical nn frames e reconstruct video variational gp aforementione recover smooth nn video dynamical var reconstruct dimensional gp smoothly whereas problem file ht successfully ard video training frame generate frame future frame amongst video experiment demonstrate ability reconstruct without frame smoothly generate gp divide gp latent variable allow digit bayesian approximate low bound describe classify determine label class comparison vs approach c misclassifie logistic kind dimensionality scenario seek find low gp temporal propagate uncertainty treat unobserved uncertainty full spectrum range unobserve fully unknown rise auto gp model extensively great success setting denote unknown however world application uncertain come gp extend aforementioned also model input show explicitly model input input free induce typically give pass subsequently generate form make variational variational probability parameter implicitly uncertainty prediction autoregressive manner uncertainty output denote subscript input output shift uncertain aforementioned way inference particularly consider around center iterative ahead future approach predictive make iterative step beginning include predictive point prediction immediately uncertainty happen predictive input framework consider benchmark represent see something dataset challenge train dataset create point future comparison firstly give model output give mention refer make ahead predict add straight uncertainty autoregressive propagate autoregressive gp robust handle something low notice method expect prediction part input obviously uncertain general row latent supplementary derive mathematical formulae expression low derivation complete square recognize definition hand detail quantity jensen use square equation distribution equation complete recognize derivation complete final put quantity exponent get algebraic use replace eq replace well replace choice equation computation kernel ard statistic analytically ard quadratic eq j variational ard function ard integral tractable derivative variational variational transform vice versa scalar diagonal single element software datum therefore derivative obviously rule write prior term involve involve turn depend demonstrate forget dependency mean calculation account result equation replace eigenvalue numerically task explain bind aforementioned illustrate certain standard formulae variational observation gp far expand term equation test second exactly augment observation version latent variational could approximation correlate set experiment optimisation base output explain q exactly phase product easy appear accord form project sparse distinguish calculation identity gaussian find fully denote row partially miss point replace distribution delta observation area area mean accord subsequently along parameter gp whose behave partially define initialize fix variational u train distribution prediction figure obtain experiment datum depict ard experiment predictive angle space mat ern kernel gp scale switch correspond initialize add part original line dynamical variational gp use body mat encode run separate subspace space learn train investigate information subspace consider employ mat ern covariance constrain ard project interact dimension separate walk regime value belong multiple latent vary dominant output row dimension belong region vary motion motion clearly different latent generative model ability produce projection dimension blue dot dimension point output background predictive variance latent position generate output ht rgb rgb rgb rgb center box equation chapter chapter sep font draw fill sep distance sep distance corners distance institute university k gr department university economics business ac institute university uk flexible widely maximize method introduce inference framework subsequently marginal method gp iid observation learn non dynamical show robustness overfitte automatically dimensionality nonlinear generic flexible extend purpose process synthetic machine benchmark dynamical dimensionality seed rand seed class recover simple pass present problem mu alpha mu mu axes position exp alpha pi alpha axis xlabel center axis p xlabel axes position text center option option diagram multimodal normalize appear use autoregressive uncertain uncertain computer dimensionality several dimensionality state physical manner nonlinear result density extend dynamically lead resample suggest nonlinear kalman transform representation space variational input gaussian combine process represent process dimensionality perhaps function low datum subspace map linear principal analysis component tractable place datum mu alpha mu mu mu mu exp mu axis x hold axis xlabel center ylabel plot end end axis pos pos pos pos pos pos position axis axis center text option false height diagram dimensionality reduction suggest spectral locally attract attention closely multivariate interpretation explicitly underlie mapping dimensionality reduction seek low precisely additional dynamic modelling incorporate algorithmic paper prescribe kalman step rather specific datum entire function restrict filter basis cast gaussian specify distribution gaussian dimensionality linear challenging extension rely posteriori principle desirable automatic dimensionality formulate variational framework propagate process marginal follow jensen infeasible inference build variational gp induce result define product distribution variable generic easily extend measurement input partially consider dynamical significantly extend kalman filter nonlinear markov place trivially spatial derive string graph deal model demonstrated conduct experimentally extension input fully unobserved variant prediction gp finally theoretical current variable specify review dynamic datum discuss map estimation characteristic introduce process determine property map traditional covariance rbf infinitely use covariance linear latent space appear covariance denote give equation input kernel hyperparameter form construction density term p come space discuss detail latent pass challenging fixing treat omit notation give straightforward latent suggest subsequently author notice gp structure simple observation fully factorize latent space prior incorporate discriminative literature seek constrain system temporal auto model whereas track temporal employ smooth path shall dynamical dynamic define component draw via datum follow covariance evaluate covariance fully factorize row function use rise markovian choice experiment find rely procedure optimize hyperparameter several drawback firstly fact could overfitte provide insight optimal add require slow consequence employ typically complex help latent automatic determination ard unnecessary almost benefit use occur bayesian provide rigorous limitation map variational demonstrate avoid overfitte automatic selection dimensionality detail simply approach immediately tractable auxiliary tractable variant concern outline extension method enable application modelling dynamical trick vast field treatment marginal associate joint latent integral prior nonlinear nested write complex infeasible progress invoke variational variational aim approximate jensen obtain low log eq mean field problematic intractable intractable observe integration appear analytically tractable bind close jensen expand auxiliary induce variable originally induce speed extra gp expand extra induce constitute evaluation augment joint take q marginal prior induce expression evaluate covariance induce induce illustrate augment ht node latent latent node latent node latent latent u variational gp node level augment value model hyperparameter induce drop expression variational approximate distribution form however proceed variational allow analytically key reason term derivation follow eq shorthand expectation easily gaussian explicit expression j see numerator explicit value trick explain optimally tighter long depend I q notice appear refer require tractable straightforward analytically worth decomposable appear pair respectively particular term associate one point speed computation parallelization computation suggest average separately constitute ard quadratic form aforementione appendix lower write q sum side essence follow maximize standard main optimize variance parameter gp investigate carefully observe term column closely resemble low contain variational finally restrict set additionally place parameter subsequently integrate apply gp property separate kl divergence easily accommodate form rise incorporate prior require suitable compute variant section resemble gp across variational prior variational appear result simply form variational bayesian dimensionality dimensionality resemble fully term optimisation many free reduce stationary diagonal denote dimensional parameter transformation newly inspire iterative optimisation scheme depend actual newly lead treat case jointly appeal improve optimisation indeed equation confirms couple according treat one optimisation readily dependencie different nature kind dimensional simply replace non use section observe take non gaussian extend let sequence dynamical capture underlie priori block sequence block sequence use induce approximation variational cubic gaussian reduce select induce gp handle variational thereby induce computational gradient substitute svd cholesky practically million video kind interest calculation density estimator secondly reconstruct unobserved overlap index problem miss reconstruct covariance discuss task dynamical task predict discuss write likelihood marginal variational numerator integrate newly construct ratio low appear denominator specify numerator approximate q exactly analogous impose equation mean well need decompose average extra minima sensible initialization associate discuss wish reconstruct thus totally previously instead approximate moment achieve introduce underlying free latent variable form obtain methodology marginal low quantity detail project sparse compute calculate describe standard solve dynamical specifically predictive input optimisation distribution challenge shall forecast empty fully gp optimisation find gaussian mean close save diagram clear rbf bias else load end fr height eps system eps fr eps eps load else height eps fr width gray eps fr eps fr height eps part eps fr width eps eps clear generation bias rbf load eps system eps eps eps fr height eps system eps gray eps fr frame eps eps else height gray eps fr height eps system eps eps width eps eps rbf bias else rbf body eps eps white else eps system eps eps eps eps reproduce mat matlab variational performance gp dynamical ard determine subspace construction present evaluate reconstruction class source video structure covariance dynamical world capture dataset able handle work raw video dynamical benchmark framework evidence enable bayesian proceed actual review mapping output nonlinear thus selection method infinitely dynamical mat ern covariance
algebraic copula generalise copula dimension towards nonempty subset function volume lie follow central copula nothing interested application consider cumulative otherwise uniquely determined conversely cumulative briefly bivariate copula empirical ranking object rank retrieve associate dependence give respective equivalent pearson rank value substitute rearrange seem consequence rank constant copula cdf empirical indicator allow express form integral unit unbiased monotonically increase nonempty real function box domain multivariate margin probabilistic reader statistical background real unit continuous cdf distribution trick marginal analogous bivariate tend theorem express copulas two vector generalize product configuration proof positive similarly inductive product similarly product positive inductive assumption way product observe configuration positive product half relate integral copula integral copula form change reduce change similarly induction multivariate vector marginal copula give argument j j possible way product possibility difference probability integral form function copula copula possible multivariate bivariate multivariate bivariate fr hoeffding minimum possible copula correspondingly rank object provide rank put access discuss far require rank recall rank range copula express rank convenient normalised expression copula interested ranking plug expression integrate product position derive correlation rank geometric capture notion rank generator rx rx generator idea generate rank order element rank rank rank generator show possible showing imputation reverse ranking imputation ranks particularly consider list belief discussion imputation scope manuscript task aggregation extreme core extend bottom rank common partially label generator close rank rank label weight ranking label rank like weight see weight rank list appear twice calculation interpretability dimensional optimisation decompose bivariate notation product observe rank correlation bivariate express term square rank normalise rank use algorithm extend rank convert rank learn weight transformation naturally scale consensus rank give order though bias offset least interpret rank interesting closely consensus close among st supplement test method challenge aggregate pre fold provide leave expert expert identify bad benchmark bottom potentially label tie tie evaluation website implement discount cumulative ndcg well stagewise recall consider fashion approach perform apart rank weight imputation assume ranking perform quite rank suffer arbitrary demonstrate importance st good fold fold aggregation relevance unclear form intersection retrieve contain order label list dataset maintain exactly cross split report note outperform confirm justify well importance choose ccccc rank aggregation rank solve estimation normalise imputation complete aggregate list surprisingly art dataset without computation simplicity model expert learn mining bioinformatics department communications centre program result may show express tie two denominator express express appear match term numerator expression numerator denominator together soon aggregation want worst rank extreme main parametric aggregation multivariate extension copula normalise multivariate propose geometric square list miss allow apply aggregation benchmark rank retrieval recommender bioinformatics one major normalise hence combine many learn lead formulation task rank rank aggregation permutation object instead association combination pairwise multivariate measure association informally value observation inequality concept capture ideal multivariate copula corresponding copula review mathematical construction familiar expert turn learn expert scale rank denote normalise interval square contrast involve complex sampling rest theoretically square expert geometric mean normalised multivariate target extreme rank
prove suit tailor dimensionality operator encode wise constraint respective alternate modification variant consider follow augment quadratic initialize follow optimize optimization step lead involve measure feasibility convergence know special aspect wise negativity constraint constraint property concern state converge optimizer expensive involve eigen decomposition limit applicability large modify call exploit low rank alternatively express allow k k see bottleneck store derive solver evidence experimental meaning rank idea invoke encode throughout iterative us memory problem dense efficiently whose efficiency nm nm turn complexity initialize k k theoretically ensure modified address iteratively lr lr rather converge state reveal lr optimal small iterative file marker gm cpu gap inf cpu inf cpu gap inf inf e e e e na na na mul update na evaluate lr benchmark solver mix k gm gm gm dense number instance run lr experimental evaluation three benchmark see table new consistent camera comprise representative category evaluation surface gm chinese character gm mrf gm match gm category gm gm character gm map gm category relaxation tight comprise category evaluate lr exist sdp admm present cg conjugate nonconvex interior relaxation serve heuristic propose lr cutting method incremental fashion small add mul sdp solver consider category algorithm applicable align gm label contain expand lf star na na align gm na gm na gm na na gm match assess program duality nc different report solution duality run dual turn remarkably scale compute point method provable scalable report nc large nc solve optimization local turn inefficient solve mul update lot perform expansion make max cut iv lf vi tree reweighte problem assess evaluate objective result assignment category report necessarily high percentage summarize art map category lr superior gm lr either poor advantage map em method exhibit specific potential explain programming relaxation moderate structural huge space get lr outperform dense high obtaining art gm problem tight gm comparable sense energy attain arise highly optimize design lr optimum issue result gm occur gm match gm combinatorial star find lr round preliminary matlab less gm label medium align gm run lr minute hour scale gm hour huge room improvement alternative accelerate eigen future propose admm solver sdp solver result confirm sdp various enable relaxation algorithm benchmark direction combinatorial formulate estimation light combinatorial acknowledgment nsf grant grant fa n award examine kkt begin involve detailed various orientation estimation construct kronecker condition exist uniquely characterize dual block rise recovery produce problem submodular constraint otherwise relaxation example optimal introduction lf object align lf gm na gm gm na posteriori inference discrete fundamental wide world semidefinite map develop accelerate direction lr exploit comprise hundred thousand compare lr remarkably commonly hold relaxation mrf evaluate quality demonstrate outperform art produce computing span scope scenario range gene estimate approximate various formulation surrogate semidefinite usually dominate formulation programming programming despite superiority obtain applicability paradigm purpose propose relaxation refer map pairwise undirected key marginalization programming semidefinite accelerate variant multiplier admm scalable problem pc state per node practically remarkably variety lr collection inference consist experimental computational well concern model scope reader refer topic convex relevant structure linear method convex formulation nm quadratic objective exhibit property non relaxed replace complement replace constraint sake relax constraint crucial sdp scale negativity necessary loose submodular state property
thm proposition red test accelerate address sparse regression upon idea worth computational dictionary preprocessing stage small believe even great iteration one may advantage computation iteration henceforth dynamically formalize screen principle inside number first order assess show screen thresholde focus numerical optimization problem consist fit induce dictionary fidelity regularization non smooth induce rely information gradient base suited method demand iteration fit application transpose application extend algorithm challenge fast multiplication bottleneck govern fast associate dictionary screen understand cost inducing entail optimum many aim zero screen test screen draw detect zero relation equivalence remove zero locate screen remove matrix atom column index imply solution zero depict upon screen cost take aforementioned improve exist computation overhead consequently dynamic use small update em illustration dedicated reader approach mathematic figure particular combine version safe screening screen effect illustrate evolution atom draw independently unit sphere dimension actual begin iteration dictionary dynamically iteration get second screening particular screening iteration equivalent state static screening use tt start atom new formulation improve algorithmic scheme problem adapt screen give screen detail discuss significantly reduce cost optimization definition integer indexing th sub element notation extend select respectively primal vector dual ar b dynamic accelerate term notation refer might formalize update compose primal extract update convex duality scalar characterize order accelerate family screening screen dual design screening begin line screening update show line enable inactive atom dictionary variable screen transpose thank successive share variable acceleration impact assess experimentally section np screening ti iterate screen order state preserve convergence minimum convex n dynamic screening inclusion solution sequence exist update general algorithm may various regularization associate describe may accelerate order specifie namely define primal aspect first algorithm algorithm formulate way computational optimization l backtrack fista backtracking make proximal beyond subsequent proximal description may embed correspond norm sparse norm eq respectively necessarily link eq define solution trivial simple screening may feasible maximize focus initially quantity base concept geometrically contain upper admit sphere safe et al formulation rely three screening generalizing thereby use present screening efficiency proof screen boolean value screen lemma safe screening st exactly except aim reduce screen atom form inclusion screen overhead mainly multiplication already perform algorithm acceleration dynamic screening implement screen screen stage evaluate screen I determine inactive atom fortunately begin computation already overhead zero finally algorithm may small first order experimentally dedicate assess principle deal measure screen dynamic screening acceleration extent gain datum screen static dynamic screening evaluate focus static dynamic run reflect acceleration screen dynamic strategy base computing quantity sparsity current iterate algorithm iteration compute operator operation respectively dynamic screening screen compute operation static separate screen number operation c n sub group dictionary implementation screening screen represent time equivalent check synthetic type dictionary I realization introduce draw I dictionary experiment build randomly atom coefficient active group draw I inactive observation sum ratio audio cosine dictionary adapt audio music image digit split testing compose digit observation taken address fista tf strategy algorithm dynamic screening several st pdf normalize time dynamic screening circle value account fast plot fista typical strategy acceleration parameter run range ability really normalization conclusion fast observe time trend fair group median run represent dynamic screening wide group grow inactive group confirm intuition screen inactive group trend discrepancy discrepancy computation screening test previous fista dictionary gaussian fista pdf fista screening screen acceleration improve static audio bring important safe static strategy safe counterpart atom improve safe dynamic allow acceleration show practically algorithm use induce strong acceleration make mean screen principle apply conversely apply modify preserve convergence first order answer anchor screen theoretical present whole study screen combine adapt solution successive computation high give exact examine dynamically iterative nice behavior orthogonal pursuit
impossible achieve edge label isolate node resort objective correctly strictly error least think always contain component label node thus enyi contain connect show fundamentally impossible two impossible attribute fundamentally correctly whether indicate need exploit encode graph work infer distribution community case positively positively correlate positively infeasible sharp simulate embed pick large eigenvalue nearly useful embed large eigenvalue plane show fig well ten color attribute coincides find simulate pick normalize square label square ij fig divide three correspondence spectrum algorithm satisfie surely surely eigenfunction get existence eigenfunction decay slowly bad bad display decay zero yx far eq independent conditional r therefore introduce notation matrix use norm norm root sum norm use usual denote sort norm overview scale font thick right theorem surely orthonormal eigenfunction ki sharp control spectral symmetric entry independent universal lemma perturb symmetric denoting exist eigenvalue absolute proof next proposition apply observe inequality ok n hand orthonormal eigenfunction v ki r main distance define fraction eq entail note constant hence right hand side successively go technique adapt label attribute root attribute couple two attribute agree agree coupling agree child child check agree grow branching branching well branching eventually bayes question remark objective microsoft centre community whose general randomly unknown latent infer partially observe inference conversely show well without provide spectrum eigenvalue law community network attention find numerous across various include physics biology statistic exposition reference assume network consider graph blockmodel sbm k plant simple node partition extensively provable guarantee real display cluster observe interaction rating question accurately interaction absence cluster answer sbm assumption allow carry come accord attribute label graph label collaborative user movie label one view label person person relationship either bipartite gene label edge label expression predict blockmodel formally seven endowed finite denote measure attribute manner independently draw finally retain infer inference shall statistically indistinguishable combine emphasize label inference without sparse propose computationally efficient spectral assign construct adjacency label spectral finite lead adjacency use empirical neighborhood space underlie true isolate neighbor edge isolated provide impossible make meaningful fraction impossible trivial threshold asymptotic specify appropriately prior work main theorem sbm spectral attribute space reduce spectral use underlie sbm rely edge rank establish compact clustering base attribute analyze assume attribute finite latent node attribute simplex endowed dirichlet sbm reduce sbm study community fit exactly exchangeable edge point approximate spectral eigenfunction exchangeable focus component phase identify sharp cluster degree rigorous community multiple positively correlate impossible threshold sharp tv throughout almost surely tend notation adjacency goal time spectrum suitably detailed description step weight label top integer step node spectrum construct label exploit labeling encode know priori irrespective indicator weight adjacency decomposition extract eigenvector unit let estimation give small define guarantee spectrum operator define operator act function zero see g eigenfunction decrease eigenfunction performance guarantee continuity appear exchangeable random graph uniformly modulus continuity fix integer characterize estimation positive satisfie give satisfy least second vanish simplify go go strictly successively sufficiently small measure endow stochastic block small theorem interesting fix adjacency precisely fashion derive expression operator lebesgue fourier series
series section series performance detect cs format mm p noun noun noun noun proper noun noun noun noun noun noun well detect fail detect significant window l table format mm game book amazon review tweet series word column value usage table current demonstrating change sharp frequency could attribute rise popularity movement popularity political false positive intuition illustrate google useful visualize detect linguistic shift last detect tag noun noun indicate meaning suffer high false negative rate linguistic however linguistic annotate domain false negative linguistic resource detect attention demonstrate detect seek point introduce understand acquire previous detect word detect second word detect well change continue production pre music mid significant word yet introduction flexibility popularity book start book change life consumption table start noun dominate common computer word company mid shift mean operating decrease corpus method require intersection field discuss linguistic point detection language evolution detect political frequent period study quantify linguistic tracking shift fine track language still able period entity rely parse construction method linguistic resource enable compare sequential embedding moreover fact web word learn mapping symbolic continuous embedding language outperform effort propose computation big word embedding capture fine structure prove range natural task embedding change area describe change bootstrappe establish significance outline excellent survey series internet language medium influence internet online form medium like medium focus usage implication mail instant I language online excellent internet provide include linguistic analysis medium twitter google different series approach statistically significant linguistic shift represent medium analyze google able historical tweet amazon review game book capability detect shift ambiguity system language implication semantic internet real change acknowledgment provide access stream gray rgb pt significant linguistic shift mean usage shift especially rapid exchange quickly meta construct word usage statistically change linguistic consider analyze complexity generate property use distributional infer word occurrence language word time unified coordinate construct distributional time series word linguistic demonstrate scalable linguistic change micro twitter decade review movie amazon google book pattern language medium information storage artificial language inherently evolve accommodate internet rapid idea linguistic shift media micro review book semantic word throughout mean propose tracking shift model temporal series per investigate series extract second construct word pos tag contextual co construct significance word compatible illustrate semantic semantic shift last observe semantic year construction type regard word usage difference frequency two google trend sharp united word acquire mean distributional observe scalable investigate linguistic across year micro decade review corpus movie review amazon books books corpus introduction product book aware web understand request detect semantic word contribution statistical speech tag construct series investigation first sound linguistic change score book review corpora time several change rest structured define language word evolution describe significant change language qualitatively limitation blue change mean quantify linguistic shift mean temporal create corpora dictionary e usage appear vanish corpus corpus snapshot reflect several calculate construct quantify significance usage shift usage question shift happen first quantify significance semantic syntactic usage syntactic distributional aspect evolution significantly influence type detect phenomenon immediate change frequency google trend google book analysis index corpora linguistic frequency word method track appear snapshot corpus language construct occurrence word snapshot information time jump tb metric calculate bias popularity entity could significant meaning detect significant usage involve syntactic serve word evolve syntactic category noun acquire speech proper noun describe company figure leverage syntactic corpus pos calculate distribution tag snapshot quantify corpora specific pos shannon divergence dark line tag noun dramatically popular company stack chart noun tag dramatically increase dark blue shift restrict speech acquire sense device categorization subtle semantic change deeply use distributional appearing context semantic space space representation appear recent variation track learn snapshot corpus track embed shift discuss embed construct time change snapshot goal begin vector representation context vector equation occurrence snapshot word word appear within equation calculate probability classification reduce cost normalization factor use stochastic propagation training epoch normalize force word experiment implementation model unless snapshot align embedding change complicate train alignment change distributional word snapshot aid alignment simplify assume structure assumption alignment model fail align properly linguistic near word tw normalize set shift normalize step shift observe determine series track shift snapshot relative occur across period capture linguistic distributional word embedding semantic introduction popularity constructed discuss determine word score threshold change preprocesse normalize series bootstrapping draw w b ip describe exhibit normalizing attempt detect shift variant shift analysis outline method normalize transform series score snapshot w
solve sum task column row stand traffic anomaly detection range source case pattern regularization weight certain feature task feature problem regularization nonsmooth gradient among trace extraction flow second address address rate capture mb mb anomaly field privacy concern use maintain trace repository traffic trace capture link united traffic start traffic comprehensive provide could achieve well derive header computer use ip ip protocol complete flow account variety select characterize flow include information deal time simple count content server number arrival arrival inter arrival inter arrival arrival lack truth traffic difficult label step detector analyze traffic report similarity similarity stand traffic strong community community measure reference traffic traffic anomalous anomalous stand traffic traffic anomalous efficient detector label reject none detector traffic method dataset process effectiveness task raw traffic characterize flow dataset table cc experiment multi feature package package svm matlab apply task top turn vary build classification svm evaluate data estimate overall metric selecting achieve high result multi observe improve classifier work selection lasso top multi improve detect anomaly employ real network generate anomaly detection generate evaluate multi plan extend evaluate selection network simultaneously wide h wang pt research area accurately detect anomaly traffic multi consist flow different period task different period perform detect anomaly anomaly area detect anomaly simultaneously particular task know base selection show accurate utilize simultaneously united state anomaly outperform norm anomaly behavior user become fast accurately classify behavior traffic sensitive prevent effective traffic anomaly inspection flow fourth learn state art traffic anomaly effectively traffic al traffic modification conduct compare seven commonly use feature accuracy anomaly mention traffic meet traffic task character disease diagnosis computer process traffic consist flow period task learnt simultaneously extract utilize multiple anomaly apply selection area employ effective preprocesse extraction step deal raw datum generating anomaly task potential
decay slowly polynomially kind tail treat unbounded regardless impossible general class surely underlying compact rather belong framework optimal rather natural framework outcome interest sparse compressed sign symmetric isotropic euclidean n rt x rt r section scale incorrectly accurately show erm absolute outcome incorrectly notably error become noise persistence explore great later poor outcome rather merely get picture suboptimal boundedness wise boundedness invoke tool contraction function behave interval argument put supremum concentration play essential exception supremum empirical rademacher average g reference role rademacher contraction careful realistic hope bound introduce totally barrier reflect nature insensitive level describe depend variance would distance problem almost tend among phase exact possible insensitive level totally cm entire contraction address handle must scale correctly view concentration argument tail suggest arise statistical article need erm title substantial sided state high low may concrete integer straightforward copy event variable well behave estimate obvious assume fix constant binomial obtain estimate immediate sided heavy tailed class heavy heavy tailed two sided take centre actually side make eventually lead sided concentration impossible consider different regime difficulty explain difference expect interaction complexity intrinsic regime another control intrinsic depend exact perspective localize scale deal significance obvious property agree mistake away mistake mistake erm business identify appropriate formulation immediate outcome function function proportion distance control interaction mistake happen turn interaction sign measure endow nature scale supremum measure maximal random f f definition maximal generic nothing happen bound contraction every dominate bound moreover insensitive contraction affect close right sense multiplier small description role reasonable splitting capture two concentration contraction mechanism every l l loss shift minimizer satisfie belong rather interaction quadratic solely side obtain upper highly quadratic early side type restrictive assumption lead term generic small minimal choose appropriate absolute notion ball concentration concentrate behave trivially satisfied highly contrast sort weak nontrivial ball see formulate convex set particular introduction way persistence reasonable target persistence probability error study fit outcome lead minimax whether technical scope fact class note characterize regime diameter estimate functional persistence endow turn diameter version lower indeed arbitrary target space diameter base show problem procedure exist therefore regime exception example typical version describe interaction target optimality gaussian result mild canonical without go subgaussian canonical gaussian index parameter regime estimation subgaussian subgaussian coincide optimality unfortunately extend somewhat subgaussian beyond scope turn heavy class member useful nevertheless class every constant immediate outcome see f constant cm pass smoothly sense let mean variance vector nc lead apply distribute put ff either erm eq fall scope rapidly decay illustrate difference compute independent accord mean playing assume surely r proof q probability erm big contrast abuse exist constant depend put q yield much diameter capture exact far rather iid relaxed tailed measure may technical reader begin class star shape around shape around eq sometimes also shape around example component let close give follow empirical ball sphere hx np function every inequality every vanish contraction process eq least eq star shape shaped consider q star shape imply star shaped end distance characterization metric projection hilbert assertion q consider fix star shape q claim present claim section prove strong statement th isotropic vector unconditional coordinate tail belong fix mean despite claim p functional possible symmetric variance coordinate may hold unconditional extend follow isotropic unconditional function attain ball set isotropic imply unconditional every depend isotropic unconditional z l follow subgaussian variable independent mean absolute cm first intersection ball radius equivalent convex increase z constant proof tail en jensen proof follow independent copy independent copy distribute accord subgaussian moreover constant range verify suitable fit member scale show high event claim proof use evident depend thus depend next multipli subgaussian standard lemma recall choice concentration inequality bound see relative endowed least
cumulative priori posteriori indicator put mass probability dp address issue dp name bb prove bb frequentist bb diverse priori view event previously report value large bb also remark imply strong reasonable side dp inference example inference spread answer characteristic choice issue worth explain model information observe specify concern unknown prior lack prior propose generate event reflect concern class beta span posterior inference reader one parameter call inference expectation prior calculate specify behave define dp class obtain let normalize vary inference satisfie calculate proof appendix respectively prior since obtain degenerate degenerate bound one let place e posterior expectation finite explain mean infinitely drawback view inference distribution unobserved consider posteriori greater insensitive tail finite measurable case statistic recently interest bayesian nonparametric dirichlet coupling may follow example population population one well response traditionally rank test population equal come population fact eq computed stress use limitation estimator nan although weak see detailed overcome issue common goal versus interpret therefore limited nan drawback test meaning value approach making practical much row e letter probability dp besides limitation extremely weak instead inference obtain lower accord inequality satisfied decide great evidence large value decision figure probability result posterior evident great say distinguish appropriate sided great prove derive presence draw hereafter simplify px dirichlet priori upper prior satisfied furthermore reasoning satisfied measurement posteriori I I n z ji ji bound posterior extreme low distribution bayesian test population compute dirichlet remain correspondence prior suggest decision additional information know posterior would hypothesis dp bb dp information analyst collect additional eliminate start bb dp test loss monte loss dp response evident bb practically coincide noticed case bb dp bb dp response run bb dp clearly accuracy bb useful analyst well available data situation bb dp issue well turn coin although bb dp precisely percentage bb dp large easy instance bb dp loss action significance accord criterion decision reject bayesian accept principled determine lose put decision format believe equal significance level adopt power figure evident performance test practically coincide verify random answer maximum run random lead conclusion coincide well correspond random value choose show test c also return response repeat minus increase test trend decrease together result see guarantee observe difference test gaussian distribution htb propose nonparametric extend set develop variable strength normalize vary prove predictive dp strength dp make computing inference exist thus avoid demand stick break base conservative nonparametric develop test numerical compare test prior strength go robust almost work plan statistical test infimum definition degenerate class exploit ds degenerate proof dp break dp write px f f px f equal always kind prior base equation exploit equality linearity result thus property ta te trace te te ij ij il correspondence posterior give give realization computation theorem goal convergence dp iw normality dp prior vanish state sequence limit low converge variance first rewrite numerator ta note equation tend u ij ij j var dp covariance acknowledgement work support grant em pc pc pt pc pt dirichlet prior information require consist dp set bayesian frequentist test commonly particular robust instance aforementione dp dp nonparametric etc consider prior statistic naturally arise extended function censor data dependence bivariate dirichlet dp justification classic nice promise result development testing package nonparametric relate dp well fact characterize prior parameter prior strength scalar parameter lack information bootstrap bb prior go bb quite since actually viewpoint base drawback bb generalize idea robustness review lack family prior carry consider order natural candidate turn set learn prior statistically useful suggest alternative behave inference e prior credible inference bayesian model already extend near post assumption straightforwardly aim dp dp class strength let normalize vary probability behave priori inference statistical show nonparametric test sum sum health status trial speak near present several advantage formulate problem evidence favor near allow prior view inference expectation close exist monte distribution specific dp come free natural expectation nonparametric near prior efficient effective practical share similarity significant come rank produce test translate minimize
accurate ng high I large amount none kind mistake ic kind ng difficulty ic get summary accord simulation attempt method ol basic choosing method elastic net always predict low elastic net ic ic mcp ic often accurately intensive kind ng get national foundation china remark technology systems chinese sciences china square simultaneous high scad simulation term present still helpful scad mcp elastic field business people demand variable square successfully decade popular lasso elastic net study popular concern practitioner paper numerical micro array fan propose sparse least focus traditional study reader rest organize method end paper response vector variance intercept expression ol lasso lm compute ols penalty covariate correlate ridge explicit tuning select generalize shrinkage ol multiply ng u solution optimization ng directly aic bic accordingly aic highly ridge estimator ng lasso derive fold validation put validation fit expensive mcp nonconvex scad mcp simulation implement scad mcp developed algorithm bic convexity choose appropriate mcp vector datum mean I estimation incorrect ic kind incorrect ic ic ic via intel ghz three configuration ii case iii htp ability selective perform well make show ic training ridge elastic behave reduce ic ic htp htp next sparse vary
unit logistic output network stochastic mini batch increase linearly final epoch entire million feature take machine intel core gb memory graphic processor software package use burn bayesian space possible six decay momentum momentum six unit scale initial momentum log epoch eight layer seven momentum mini performance parameter deeply explore good single decay factor momentum network deviation last root dropout algorithm stochastic activation difficult primary avoid overfitte david comment help optimization support acknowledge nsf grant google award think provide interaction power barrier pair tune training result layer improve performance even improvement discovery equivalent increase accumulate lead discovery collect come back direct mode mode branching ratio measurement consistent claim physics physics improve analysis small scientific learning area software package rely artificial result advance vision speech linear efficiently generalize difficult vanish advance deep network significance high physics operate time million parameter overfitte thus challenge select architecture detail letter deep worth tune parameter carefully maximize statistical systematically heavy form unstable decay successively particle decay decay mode involve large momentum direction visible particle intermediate observable generate particle difficult show process identical particle distinguish particle momentum direction visible particle examine perfect measurement mass state detector impossible calculate mass momentum sophisticated program produce simulation distinguish possible show model classifier detector high predict example describe network similar network train two feature type hyperparameter minimize expect bayesian optimization algorithm combination hyperparameter million example generalization rate momentum momentum epoch layer per constrain neuron space eight hyperparameter random train random weight deep achieve area receiver significance performance boost create classifier auc show network bayesian include hide number large network architecture bad deep ensemble subset level optimize network put practical boost discovery dnn measure need significance nn dramatically operate complete nn dnn dnn dnn dnn ensemble expect derive capture poorly alone train low level deep complete though perform complete deep alone high regard include dnn
work face straight manually build output representation become choice work build feature process unsupervise screening feature generate candidate direction difference cnn black box pair utilize stage cnn image representation map closely identity irrelevant minimize correspond especially keep discriminative power encode abstract another minimize design ideal close within uniform representation eq contain complexity express complex computation preserve information unsupervised method aim modeling unsupervise pattern relate task influence include illumination expression consideration supervise impose preserve requirement deep neural raw capable non abstract property image cnn face pair person numerous match pair also adopt face produce whether pair belong indicate whether image computation network stand encourage person learn preserve specific recognition intra person convolutional compose layer convolution operator q deeply large powerful grow pyramid cnn accelerate multi figure pyramid pyramid cnn cnns divided compose part share network scheme large region share first view pyramid cnn supervise layer share level network f relatively input filter image level train process image become greedy continue level train purpose coverage unsupervised target reflect share another pyramid cnn scale architecture pyramid naturally patch face fed pyramid multi region level computation verification complete dimensional detect improve enable fair pyramid scale feature position system acquire web belong outside pyramid take training representation outside face thousand benchmark slowly fairly system furthermore face system make mistake attribute face detector pair case human argue rely knowledge people comparative effect pyramid cnn pyramid networks pyramid pyramid amount low level improve expense slow room verification increasingly protocol reflect world application especially pair significantly match pair though case receive actually person attack protocol successful attack within attempt pt improvement recognition suffice million face match social rarely significant gap accuracy similar effort suggest pyramid apply area classification scale typically pyramid cnn significant challenge face image part object object believe extend face crucial face optimal discriminative numerous room face pyramid pyramid cnn operation enable pyramid naturally share face result achieve recognition accuracy extended pyramid system performance benchmark face network validate face recognition system image vector learn task verification search
tailor adapt ei couple scope much beyond facilitate handle multiple complicated constraint allocate meet policy requirement knowledge case despite region treat ignore infeasible utilize base lagrangian tool mathematical programming problem novel ei tailor constrain al utilize burden statistical optimizer subroutine mathematical condition derive ei guide subproblems numerical carlo alternatives importantly carlo unlike stochastic method utilize global convergence experience herein scheme number specificity paper problem constraint statistical throughout remainder describe synthetic statistical introduce framework handling combine statistical surrogate toy potential test give tb figure local optima global minimizer solution strictly hold bind bind local may present challenge design boundary set toy problem characteristic common notably objective highly response lagrangian implementation defer date since flexible help one option surface true deterministic objective since gps accurate conditionally distribution surrogate focus extension accommodate constraint often restrict valid region uncertainty capture change normal surrogate function variance together exploitation toward global statistic variable likely show ei analytically case standard cdf reveal exploration global branch later ei improvement yx yx yx ix ix weak algorithm ei optimum gp surrogate ei see one wide family radial surrogate local viewpoint computer rarely ignore loop fit distribution search slow local solution practitioner prefer global ei search nonlinear favorable property find local device method augment lagrangian serve lagrangian define stationarity problem reduce approach constrain unless considerable penalization introduce ill subproblem sequence parameter lagrange multiplier approximately solve subproblem lagrange multiplier inner approximately termination set optimization termination primarily computational section involve evaluation e cumulative could stop approximated lagrangian example threshold one stop determine inner solver dependent solver motivate convergence inner accommodate constrain leverage available section benchmark solver briefly al output obtain determine next direct generate spatial mesh size determine mesh limit fail find parameter take software recommend maximum mesh fine subproblem outer iteration progress spirit statistical propose employ local radial trust region method solver build subproblem approximately solve method design minima ultimately converge minima statistical surrogate offer potential simple directly separately model via predictive mean ei form expression special surrogate order denote number obtain inner outer f ny l ax ic approximate without maximal ei attractive option modular tradeoff modular software ei exploration al iterations inner search region several drawback apparent consider nature exhibit primarily boundary valid gp stationarity boundary region regime behavior gp accommodate schmidt change latter option pair public partitioning divide accommodate limited stationarity challenge change roughly align modeling ei treat motivate example unless inefficient address composite stationarity likely violate simplify many extension improvement surrogate provide c nx nx serve trivially surrogate convert composite calculate swap deterministic way choose composite random predictive ei ei yx include three modification asymmetric entropy uniformly multiple variation comparative discussion perform none poorly easily dominate sensible counterpart require al separate modeling part section involve via analog analytical alternative indistinguishable ei case ei total report al without ei ei separate gps initialize ten pair I start fast mle augment algorithm inner inefficient objective improve candidate easy rejection nice fix densely improve progress two consequence convergence candidate like ei address inner ten search time however find exploratory early search inferior local great explain salient consideration please implement package work toy summarize carlo experiment toy ht cm ei ei model ei ei ei ei monte carlo repetition track iteration distributional repetition valid value plot middle quantile variation eventually global minima five case ei fail instead brief near ignore method ei dominate ei mark behavior drop ignore seem help early sa bad evaluation sa eventually converge reach evaluation compare ei ideal simulation experiment asymmetric perform ten subsequently consistently motivate contaminate environmental increase attention year water prevent disease place water back spread site locate lead contamination site provide illustration expansion identify city website boundary include pose treat version rate operate subject let well objective operate solution element treat never boundary challenge treating via figure valid leave e g abstract stochastic may surprise simple sampling new replicate suggest implement varied search randomly early stage observe half trial minima global slow converge success illustrate clear exploitation early experiment repetition initialize randomly cm ei ei ei ei ei ei al surrogate worst good monte outperform run achieve behavior particularly regard moreover substantial proportion repetition valid comparison initialize contrast surrogate difference final solution well section expense likelihood convergence could thousand iteration attempt sa competitive ei progress rapid classical never quite one sa result amenable approach process method computer attractive augment unconstrained method optima conservative programming ei composite objective show sensible variation scheme leverage traditional usually still room toward success ei provable local global let direct solver toward front constraint leave potential cross idea prove format treat composite al one keep pareto strategy promise offer mc calculation important ei like loop eliminate yield entirely deterministic sometimes clearly usefulness tight nice setting good value rough way monotone case drop al attractive yield improvement inside region outside another slack variable return might guide lee gray lee extra perspective ei straightforward allow exist statistical software directly attractive relative mathematical programming statistical optimization practitioner package box hard imagine matching engineering capability constrain acknowledgment thank american institute
article represent advance piece inference hope piece ultimately activity fall article rare enter statistical nan lasso difficult tie essentially heuristic abuse framework account think sort formula quantity distribution comparison keep grow remarkable particular forward document naive routine nonzero first select assume statistic statistic nan predictor maximize explanatory proper illustrate stochastically nan way effect outline conditionally far test lasso guarantee arbitrary remain already include predictor version stochastically importantly never enable inference predictor proceed test nan predictor simulate exist value correctly account whose properly estimate first correctness adjust significance else naive multiplying approach conservative predictor method coefficient adjust treat obviously easy obtain correction test predictor obvious significant perform stage distribution one simultaneous combination inclusion stop indicate combination predictor select stand magnitude statistic remain statistic distribution statistically significant outcome indicate need power implication statistic although difference half sample eight ph free conclusion endow poor lasso guide curve connection test statistically jointly one remove remove model turn lead example issue test notion lasso issue hypothesis procedure hypothesis truly start arise sequential provide cancer significance carry add carry inclusion previous conditional six carry rejection article guide nan become tight orthogonal predictor assume value step show figure multiply value oppose fact kind behave try control fdr somewhat suggest aggregate series smooth repeatedly inferential fair author actually recommendation methodology read test range article discusse rule fdr emphasis present considerably author work arise meet practice insight comment response issue limit ahead observation assume situation anti conservative day experience rarely sparse stick like need make come exist test sequence test ultimately home message may help form nan distribution may live trade procedure random predictor comment post inference fdr would decide fdr sense mutually error predictor orthogonal error meaningful concept adjust statistical inference article necessary path large underlie test assumption misspecification misspecification place nonlinearity become yet effect inference due test lasso test test ultimately augment sequentially however validity whereby selection nothing realistic situation analyst try forward device datum heart mind face meta favor practice benefit selection meta accord subsequent big graphical exploratory
entry eq exactly place invoke theoretic result ij provide exploit semi triangular strictly follow nonempty nonempty lemma stay away influence eventually two positive respectively difference oppose lemma three every extend remove start near remain satisfy q proof meta proof triangle r cauchy note subset coordinate update algorithm convexity optimal convex strictly follow argument e e state lemma minimization uniquely uniquely minimize thresholding define eq note solution strictly turn minimizer define value infinity belong trivial problem use provide argument similar establish ultimately follow every assumption need deal oppose argument taylor fix arbitrarily convexity negative eq q follow obtain follow parallel version exactly suppose repeat q bound least limit point note arbitrarily q every show iterate appendix follow solution system large depend result approach et al provide constant definite minimize get follow hence hold definition iterate converge problem special applying stack precisely permutation z z p note I view kk verify convexity none independent logistic regression number tackle situation respectively define follow note typical necessarily cyclic variant appropriate choose coordinate minimizer method cyclic application convergence global minimizer proof cyclic matrix iterate cyclic satisfied every logit appendix automatically follow hence establish contradiction lemma obtain r e non non negative induction every eq definition j hold every definition prove require part proceed claim go every exist hence result part every contain absolute obtain contain absolute exist large occur observation hence induction establish non q x contain absolute large entry assumption exist conclude argument sum strictly subset norm appear modern associate optimize employ effective consequently crucial iterate cyclic regularize likelihood variable function establish ii establish convergence establish inexact iterate produce variant usefulness application employ twice strictly domain empty interior suppose also everywhere boundary let give column follow extensively particularly example typically log pseudo correspond like however modern obtain sparse solution exactly inclusion objective challenge convex hundred penalty finding problem scalable theoretical convergence form cyclic minimization consist minimize often offer computationally respective form involve dimensional numerically achieve high step hence understand cyclic rigorous cyclic minimization optimization iterate cyclic minimization stationary point thought descent function differentiable subset contain effective every follow inexact ensure produce consider way choose quadratic alternatively choose clear result variety descent hessian non entry many minimization cyclic substantially different block coordinate minimized iteration speak possible ordering coordinate cyclic suitable situation time amenable important convergence rate establish reason motivate random descent cyclic former allow easy actually non cyclic establishing iterate objective quite although function converge cyclic solve separable descent type trust detail propose extensively outline rigorous iterate cyclic algorithm build extend incorporate cyclic cyclic shall smooth challenge non trivial question provide summary assumption algorithm cyclic respectively convergence problem cyclic c establish later I provide domain empty twice curvature everywhere minimization empty h tolerance q r return stop claim contradiction x follow x attained f r contradict case lead contradiction theorem formally establish convergence produce theorem generate cyclic value detail successive iterate go establish square difference zero sufficient cauchy combinatorial iterate cyclic cauchy limit follow zero coordinate bound coordinate stay away influence eventually zero therefore establish immediately distance iterate boundary zero solution consider problem recall make application arbitrarily fix however separate statement two algorithm iterate set initial tolerance go minimization unique closed form establish produce cyclic second paper cyclic descent minimize converge compare method prove start minimization follow argument therefore omit eq definite open I differential version kkt imply alternative provide iterate descent appropriate argument euclidean matrix depend vector also continuously uniformly follow say say empty cyclic contradiction hold since bound subsequence follow minimize along coordinate contradiction establish successive iterate cyclic arbitrarily let subsequence around follow every every fix exist constant r bound satisfy sd since argument trivially establishes follow every consider second coordinate
learn create often overfitte interpretability expensive actually less classifier svm mean curse might also incorporate genetic cause might end participant gene assume gene cause end possible fit perfectly discard variable attain gene might actually produce cancer might need take low dimensionality well learner might become infeasible article broad introduction selection ten year passed state art feature cancer patient conclude responsible human pose reduction present overview modern justification take validate continue author identify strength build accord selection accurately overfitte might well treat valuable selection author gene tumor preliminary selection sophisticated procedure select top call implicitly feature uncorrelated ground absolutely select perfectly high would probably discard tumor denote tumor grow separately would ignore connection subset try reduce variance boost filter necessarily see figure correlate get tb approach proxy look break superior heuristic train depend atomic feature anneal branch subset exhaustive search either add way replace least treat necessary simple way classifier probable similar p take optimize distinction could step reason motivated selection small compare compress store express bottleneck recover low concept create place create intuition efficient code occurs often give representation much body model dimensional hide gaussian low dimensional map body track body shape neither fix training never approach independent identically distribute split identically take historical book pixel pixel perform well page split author author instead modify datum modify add variable chance ratio subset feature examine expect add feature call examine objective function evaluate necessary linear predictor variable variable model take noise eq pca case assume iterate natural show optimally score drop fix globally optimal subset find wang classify expression determine tumor think vector huge dimension chance require interpretable discuss feature embed describe understand separation direction maximal discuss information criterion descriptor minimize exponentially criterion n ignore influence one length algorithmic store descriptor give good cause variable paper start rank discard repeatedly respective collect seem unnecessary even overfitte discriminative autoencoder neural find fitting bottleneck transformation tb minima stack machine optimisation stack extract distribution somewhat sift enough autoencoder learn concept cat face supervision simple logistic feature concept sift bag human create variety discriminative possibility neuron single selection extreme benefit advantageous necessarily deal dimension computer actual pixel image document image incorporate heterogeneous evaluation useful instead thing whether prior useful example tumor gene could unfortunately segmentation causal proxy pixel smoothed limit big advantageous one variable rank bad classifier fast due allow area many still apply justification application develop drive advance high autoencoder motivate integrate usage principal goal avoid interpretability opinion integrating learn expect embed approach fit ensure treatment support vector machine extend incorporate expect usage
parallelism amount neuron activity neuron activity data parallelism per weight parallelism arbitrarily synchronization batch batch hundred thousand example consist layer property layer connect computation representation ask whether data parallelism appear attractive layer parallelism attractive connect layer propose explain mention nice convolutional net rely heavily parallelism parallelism fully worker reference pass like worker example worker convolutional batch activity worker switch parallelism worker activity activity connect worker convolutional activity gradient example stage layer activitie worker worker activity worth consequence big worker batch undesirable device big batch worker layer activity consequence much fully far advantage scheme worker scheme utilize major usual way forward pass stage convolutional worker pass continue usual worker compute layer gradient gradient responsible worker fully connect backward propagate convolutional computed example must operation I reason backward worker parallelism three stack parallelism connect layer pass replace six pass convolutional batch process fully layer pass backward backward pass worker layer weight gradient simplest worker accumulate implement modification backward propagation nonetheless run batch size notice backward pass wish backward pass extra layer kind batch algorithm pure parallelization long update consistent convolutional turn batch size fully minima question somewhat complicated question answer benefit batch size large widely imagenet million fall scale iterate many training minor winning consist map dimension equivalent minor arise convolutional another softmax layer multinomial final unit minimize perform easier require normalization multiply progress use w momentum decay learn gradient batch size hyperparameter plausible momentum big batch batch multiply multiply expectation constant adjust weight decay batch size like decay weight application batch weight batch size w w learning rate decay coefficient w net approximation neural net aside heuristic multiply multiply practice multiply instead decay patch compute error patch machine eight intel cpu amongst simultaneously gb express incur penalty big batch size greatly parallelization scheme well scale dense multiplication output multiplication inefficient size gb spend product gb scale size dense connectivity kind show gpu communication simultaneous batch connect parallelism convolutional parallelism connect hour table publish alternative parallelism parallelism parallelism speedup relative gpu implementation hour train epoch sgd gpu parallelism parallelism speedup relative gpu train accuracy gpu gpu cluster neural network worker spatially across neuron activation edge could potentially convolutional net
argument goal topic mixture vote resort approximation analyze utility supplementary vary proportion support maximize utility support brief vote bottom sometimes bad utility function present perspective support vote know side optimally write brief ip odd vote favorable marker actual simplify ii consideration writing style choice etc fully brief treat share author author despite tool generation research ip roll infer position vote ip multidimensional multidimensional study especially influence present interest study decision encode text text evidence behavior relate maximize look extensive structural decision ip researcher familiar algorithm distinction matter maximize utility domain improve vote prediction capture simple text secondly importantly interesting question write fact differently acknowledgement part fellowship sim nsf google award resource kkt th issue q first vote topic equal marginal value highlight ii large whose vote receive iii controlling care utility expect utility sample turn hasting public code national vast majority range year public dc management collective north collective action member preliminary matter title vi act discrimination old worker life reasonable impact claim party construction plain mean dc circuit year relation master interest rt act reservoir video circuit circuit fourth circuit instant fourth fourth probable agreement national bank rule act public international law international law convention united challenge school strict education discrimination school interest local political law final removal proceed proceed death death trial death direct new environmental water clean water master public material water school private school belief organization free circuit art matter act act drug act service high public law party private corpus award state trial ideal depend various reading law notable expect unless attribute california law survey rank vote ip ip influence suggest present ranking note encourage hypothesis l curve vary topic vote hence model favor prior interior proportion experiment inactive school university pa usa institute university usa idea piece maximize one utility make concrete decision unite past work quantitative political science framework empirically model decision incorporate friend separate decision benefit improve piece huge array behave take incorporate united states american far reach nine public organize group friend hereafter author know reveal explicit attempt vote language build establish political vote although incorporate analysis draw rational agent argument toward favorable outcome derive inference substantial gain importantly answer would change brief characterize brief different fact applicable goal behavioral response review commonly put argument party file brief response argument recommendation side necessarily conclude vote relate vote political science etc ideal ip ip time simplicity interpret spectrum ip model vote favor case popularity favor capture otherwise recover maximize opinion additional right vote embedding incorporate text evidence infer dimension build latent dirichlet popular corpora issue lda vote vote outcome determine mixture proportion issue ip infer mixture proportion although text serve incorporate label supervision infer address label multidimensional g multidimensional due preference issue fact fact influence outcome argue public group unite goal present argument u recently ct neither consistently attempt position strongly ic proportion ip influence form text embed rescale discrimination parameter generate relative vote hereafter b proportion infer support support share influence ip equally influence vote capture collective focus position vote suppose e vote favor outcome indicator policy cost increase fact control notion text carefully frame costly match unnecessary role outcome uncertain brief expectation incorporate ip maximize impose utility checking expectation make difficult constrain imposing prior negative negative write brief discrete choice relax precision assume topic expect consider likelihood utility resemble vote maximize assign principled manner rational expert variable ideal diagonal direction dimension case ip dash leave ip blue issue ip diagram ip non use text associate hyperparameter token text share similar dirichlet right fig diagram importantly note serve structure topic ip argue conceptual define rotation multidimensional preliminary issue joint vs stage latter relevant stage mcmc latent gaussian diagonal covariance target likewise univariate gaussian proposal utility utility hyperparameter supplementary topic case fact vote cast brief label manually label label support take content support g phrase give interpretable extraction preprocessing brief material ability vote probability vote find side multipli expected ignore non utility due specification ip vote vote imply voting towards vote distinguish identify evaluate actual vote validation vote regression topic ip ip na I vote use proportion baseline exhibit perform well adjusting suggest model furthermore believe insufficient slightly well paired accuracy qualitative
understand accord clearly preserve increase due step result td easily result text rank td exceed corollary assumption minus height supplement analyze omit zero j jj nj kn nr nj kn jj row product therefore add either conclude independent subset vector ready x full column column lemma main iteration q matrix definition utilize operation involve step show output td
surrogate multiclass surrogate define surrogate risk learn prediction learn make fx several minimize multiclass space tf condition surrogate minimize example raise natural loss amount surrogate loss square multiclass rectangular ignore generalize surrogate loss w imply w proof calibrate follow calibration surrogate mostly concern note calibration follow result straightforward loss f surrogate calibrate cx goal surrogate k end certain multiclass multiclass relate multiclass multiclass surrogate necessary probability call diagram diagram probability calculation l easily calculation n empty optimal extension note converge positive r set section give necessary calibrate r normal surrogate normal calibrate start derive happen positive intersection calibrate imply contradiction necessary calibration behave loss discussion strong positive individual contradiction sequence l tt l calibrate exist q empty contradiction include special direct surrogate look positive figure clear zhang give sufficient calibration helpful surrogate contain hold finite normal nr z calibrate calibrate normal surrogate surrogate calibrate ordinal regression apply surrogate calibrate additional provide necessary also convex calibration section necessary calibration involve normal order calibrate compute characterize set computing surrogate result compute operate number applicable u w u w attain minimum subdifferential subdifferential u u n computation lemma two surrogate operate calibrate another calculation surrogate positive surrogate u satisfie note satisfy similar p positive compare u normal insensitive insensitive u z r consider compute condition lemma p computation z figure theorem calibrate detail calibration surrogate calibrate ordinal sufficient surrogate loss calibrate w multiclass surrogate raise support calibrate multiclass lead otherwise cc result multiclass small let appear surprising surrogate surrogate calibrate loss however multiclass accurately accord probability show composite task helpful algorithm operate cc depend tight nd parallel hull column correspond vector u u dt u u r u calibrate definition normal probability q corollary l follow immediately hamming hamming loss bit representation r r therefore dimension loss illustrate section obtain existence calibrate surrogate surrogate space convex denote subspace p bind make feasible dimension p p v v subspace point us dimension let denote one identity matrix ordinal loss know theorem n class show theorem tight p therefore immediately immediately dimension rank loss framework rank together simplicity fix predict permutation document popular loss subset discount cumulative ndcg disagreement pd loss view multiclass ndcg loss relevance acyclic cc ndcg section ndcg result show dimensional calibrate surrogate ndcg pd et surrogate map ndcg set relevance say level permutation non ndcg view multiclass calibrate surrogate ndcg direct acyclic associate instance th document document prefer object permutation label g I pd multiclass term sum simply subtract column minimizer loss loss result loss therefore one show exactly et certain popular calibrate pd calibrate surrogate exist et exist surrogate calibrate pd allow go surrogate pd prediction dimension predictor permutation object label q bind result et al show surrogate predictor calibrate map fact one surrogate unify surrogate multiclass define multiclass cc multiclass possible calibrate respect analyze loss multiclass loss example loss surrogate must surrogate lee value learn implication disagreement pd average loss rank rank problem exist surrogate pd loss admit calibrate surrogate convex surrogate loss operate surrogate learn value value scoring cc tight class loose characterization cc dimension develop design calibrate surrogate operate according given design calibrate surrogate surrogate space form always possibility surrogate space certain loss issue contribute understand calibrate surrogate multiclass proof nr z z z claim banach theorem corollary hyperplane strictly separate w w calibration fix e n z jj p contradiction calibrate j give q l claim pt u tm tm p require relate feasible intersection property lemma contain get contain row affine nan p recall convex eq clearly condition far satisfy q let clearly therefore orthonormal f verify eq subsequence converge point orthonormal n satisfy condition theorem take limit thus lemma equation get z n therefore l b l calibrate eq claim follow l graph row linearly sake contradiction direct coefficient permutation b apply two eq definition verify column two give b j I show q since r true vector form removing dividing establish make denote rr rr pt linearly independent exist cccc exclude diagonal entry moreover vector intersect trivially column permutation together trivially expand recursion establish fellowship thank technology fellowship technology k z calibrate pt conversely see u z p u u p n p satisfy multiclass lemma calibrate p z z tm suppose calibrate h convergent still call say p tm contradict conversely calibrate particular consider mass f mx complete eq symmetry loss p p p p p u z set z z algebra q q lemma thm study function general multiclass problem multiclass notion calibration multiclass calibrate multiclass loss measure size space matrix upper quantity various loss dimension tool calibrate surrogate result et al certain surrogate rank surrogate loss interest surrogate binary multiclass classification problem finite prediction multiclass structure understand loss consistency unified study minimize surrogate loss surrogate respect target give sufficient multiclass matrix fundamental surrogate calibrate quantity difficulty loss class calibration consistency practically class value give quantity term framework tool arise rank discount cumulative gain ndcg pd practice subset relevance query learn loss sort document pd loss admit calibrate surrogate together sort operation convex lower bound surrogate work year consistency calibration give brief overview body risk focus largely classification example consistency universal show result boost zhang calibration binary particular seminal classification calibration surrogate yielding give necessary sufficient surrogate calibrate calibration enhance surrogate require strong early calibration theorem convex theorem tight pd additional ndcg example throughout minor improvement emphasis notation terminology example throughout formalize
result circuit bound monotone circuit size work function computational theory respect influential boolean monotone uniform learn consider various generalization arise monotone circuit obvious monotonicity boolean circuit small circuit access compute denote boolean generalization start alternate characterization boolean monotone chain increase monotonicity flip terminology position write denote alternate maximum tight quantitative inversion complexity ii motivate circuit circuit theory power circuit show circuit contain circuit circuit non circuit work circuit give result structural circuit establish theorem boolean monotone conversely yield every express either well know consequence shall circuit significant possible upper boolean must investigation circuit extension markov fourier learning circuit tn give circuit run essentially match monotone slight though membership learn monotone fact membership bind membership query accuracy thus give fairly answer circuit low match wide learn unknown membership strong make arbitrary query tool hardness strong task balance monotone hardness strong boolean function moderate hardness lift circuit moderate ingredient crucial final low detail extension theorem write denote write f I jt f simple observation thm fix x conversely boolean boolean immediate inductive express get converse observe induce possible monotonicity immediately follow corollary circuit express monotone every exactly circuit goal boolean fraction significantly suffice ff inversion complexity boolean approximated circuit bind boolean boolean th coordinate influence fraction boolean hypercube easy together prop lb must inversion theoretic showing learn circuit significantly shall cn sketch learning start concentration high estimating degree coefficient refer learn fourier monotone monotone armed straightforward extend fouri finish even boolean function immediate give answer subsection theoretic low limited number start establish membership bound bound example defer exist balanced variable gap monotone hardness learning alternating exist balanced boolean ii tradeoff range analysis query k require require function eq overview alternate get hardness moderate repeat moderate detail idea take base monotone hard monotone sensitive hardness must take constraint monotone alternate possible useful recall notion play approach draw obtain quantity coordinate left build sensitive give stability minor detail exist constant infinite g upper hardness uniform build hardness theorem deal run learn algorithm inspection proof query class boolean distribution membership query uniform learn accuracy hardness accuracy claim lower give completeness appendix hardness class balanced universal range hardness noise low stability bias proof sufficiently function variable membership contradiction learn membership query infinitely many membership contradiction mr error achieve indeed balanced trivially recall accuracy q improve give monotone family f bind hardness monotone hardness establish hardness hereafter monotone every bias zero recall boolean use function could use variable instead hardness function need scale odd rx middle hypercube chernoff layer sake proof define balanced sufficiently n membership query infinitely membership impose constraint impose lemma need ultimately show first constraint meet setting remain check constraint inequality
estimate cost machine memory factor since computational besides obvious handling incur centralized method point centralize adversarial achieve contrast naive division algorithmic machine significantly robust appeal division though division averaging helps preserve robustness bring prohibitive might practical moreover offer computation error instance machine many individually take less precise concrete distribute roughly require center aggregate suffer build framework us sample parallel take simple average robust corruption one robust partially reduce machine linearly sample finite target matrix underlie additive another problem covariate parameterize lr explain iid well many learning corrupt particular contaminate challenging fraction outli fraction briefly concept develop robust median call detail median importantly let hilbert close inner norm geometric define practice version admit atom later estimation machine median point eq median exist rather calculate median employ limitation collection strong presence robustness geometric median set median long least particular geometric median skew significantly away implementation example concrete division strategy core definition fusion step aggregate suppose ready evenly subset onto specific estimation denote estimation previous division propose separate estimation reduce average aggregation estimation many outlier machine lead aggregated estimation base sample communication error differ bad concentrate single point estimation lemma base provide distribute input guarantee characterize corrupted estimation present ground truth k basically even machine break either base communication guarantee final high monotonically monotonically monotonically account geometric concrete specific rapidly failure machine trade pca failure need trade machine real world series outli outlier machine potentially final distribute learning algorithm way exactly expense potential introduce robust point robustness point corrupt algorithm provide concrete centralized averaging counterpart classical component analysis pca propose reference therein cost prevent big first robust pca enhance efficiency ij product product product remove remain small select index obtain estimation covariance eigenvector decomposition produce estimation md perform eigen decomposition large sample n aggregate plugging distribute robust outlier simply arbitrarily span subspace eigenvector arbitrarily due limitation proof supplementary material divide onto large least eigenvalue denote onto suppose outli projection matrix small basically say fraction outlier machine corrupt integrate underlie detail covariate n p h pair kx k aggregate output reader detail similar proof supplementary depend fraction outlier guarantee design probability least straightforward constant outli linear regression problem whose iy p sample outli generate estimate conduct outli degree uniformly thus outli fraction outlier distribute instead outli machine similarly adversarial case design repeat simulation implement pc cpu ram take centralized second distribute cost parallel procedure negligible improvement performance fig division centralize non robust outlier avg pca offer high efficiency hold fig begin centralized fraction increase centralize blue division average green line still robustness well indeed compute break favorable aggregation computing besides result machine error machine user machine quality aggregated performance machine finish randomly estimation sign estimation aggregate averaging offer strong report avg far solve recently large scale around tag provided employ tag tag deep cnn large impossible gb store training implementation divide division robustness well provide lr achieve compare division average geometric median actually negligible avg lr robust different subset memory preserve learn addition bring node adversarial example xu department corollary remark distribute traditional robust learning order robustness property show robust precisely adversarial outli break contrast naive averaging node framework component efficiency advantage tag big challenge scale current
possibility non factorize partition assign consensus distribution modularity extent statistically structure converge neither factorize stable spin replica symmetry break word exponential bp jump obtain current marginal long bp spin retrieval modularity consensus many factorize compute derivative factorize magnitude like factorize fix random know stability census reconstruction show average excess number statistically phase spin converge factorize retrieval factorize bp converge thus statistically note fall retrieval heuristic necessary scan linear perturbation oppose respect backtrack bp eigenvalue network sbm assume correspond community bp factorize long disk complex inside structure isotropic community retrieval eigenvalue spin study heat mcmc hamiltonian cut measure modularity analytical transition fix spin spin sparse problem sensitivity bp whether converge fix spin bp expect happen phase retrieval free fix exponentially bp start message fail energy spin beyond replica bp replica symmetry phase quite narrow c grateful mark draw software http skew de work grant modularity community however maximize modularity modularity poorly produce address modularity hamiltonian finite temperature belief propagation partition modularity partition numerically work transition network generate work claim show recursively statistically detect hierarchical network method address physics try partition network maximize modularity seek modularity partition message treat modularity hamiltonian apply cavity analytically perform network method determine community biology connect many spectral adjacency statistical stochastic block variety review partition modularity partition label group node modularity network edge edge number neighbor kronecker delta thus edge community fix give modularity random community modularity even modularity exhibit amount degeneracy poorly small perturbation notion statistical hypothesis nan enyi however community true partition distribution block right regime correlate modularity right approach hypothesis appear work hamiltonian spin gibbs modularity search look modularity partition single one analogy marginal gibbs assign tie achieve call modularity claim modularity language marginal posteriori prediction informally efficient propagation bp marginals cavity algorithm scalable number group sense way provide community tend statistically significant validate work claim find obtain community easily model plant sbm popular ensemble community group plant connect commonly group entry ratio community become er call transition bp transition establish rigorously number group complicated mark hard time succeed easily behavior er sbm regime two phase bp converge equally replica symmetry break transition spin modularity return bp bp jump little bp assume replica phase sbm spin bp statistically significant retrieval enter increase retrieval modularity er convergence transition spin sg transition sbm r find retrieval modularity indicate statistically analyze factorize stability correlate perturbation cross analytic fig leave diagram retrieval phase retrieval excellent agreement find retrieval define plant I e emphasize optimal subgraph apply indicate subgraph network sbm er find state subgraph stop network large world algorithm repeatedly retrieval subgraph suggest political find two ground split eventually find hierarchy total leave fig level modularity explain split community explore report give level split community stop indicate remain leaf denote degree color group final division order partition algorithm popular statistically modularity group network find modularity algorithm find modularity community emphasize modularity find community show network apply sbm normalize rather overlap group plant show correctly choose appendix heavy degree distribution drop transition er find average base statistically community graphical modularity attempt marginal gibbs bp algorithm cavity next likely marginal essence look consensus partition modularity indicate opposed fluctuation retrieval spin correct bp correct modularity however em block variant clear optimize work sbm barrier community community difficult appendix give evidence barrier namely clique clique proposal determine group eigenvalue backtrack spectrum network political large deeply detailed generalization modularity weight internal interesting normalize cut consider run bp marginal fix bp marginal reinforcement external toward leave graph modularity partition nearly uncorrelated language material landscape optimal modularity hamming distance replica symmetry breaking jump optima focus optima contrast
different sign saddle eq intersection look like corner rectangular confidence confidence co fluctuation repeat previous confidence region unbounded geometric linear relationship comparison correspond panel interpret throughout subsection stationary remark co increase give increase actually conclude responsible european combination keep useful combination ga conditions equation equation ga li develop identity acceptable positivity yield co trade unit e co level formula show develop quadratic interaction compare relative relevance variable table complete variable co instead relation specifically arrive relation variation direct derive application thm proposition thm order finding optimal second linear second study dimensional region risk part model co various include mixed interact produce ranking start develop region comparison indicate part per million collect co emission datum list united member exclude present individual period former respectively cm co goal response summarize canonical respective effect decrease neutral type region region elliptical parameter recommendation optimal management emission trade policy order requirement country restriction second interaction determine eigenvector equation kronecker define computing eigenvalue arrive software precision overall factor eigenvector find represent bring form constant non use become conclude find degenerate point normal quadratic write find case cm width positive negative define interior right panel width
data absence train unknown noisy estimation know disjoint life non possible sample differentiable training ensure absence datum attempt system function available effectively point value comprise datum measurement uncertainty learn value system point within difficult task paradigm unknown function word bin bin value define try impose exercise translate density course identify unless construct subsequently model hard learn wavelet fundamental spline wavelets capture amongst difficulty particularly invert learn high arise variate uncertainty vector capable datum could meaningful big within practical determine paradigm lastly paper discuss hand section briefly model methodology subsection inference synthetic round density space available practitioner estimation parameter west observation current modelling error interest parameter another equation system high well uncertainty space treatment replace state observation embed variable partially thus include static addition address methodology present inference mass simulate galaxy mass project useful dark learn vector give comprise th light suggest f term however deal sub must project invoke embed accomplish rhs generate since functional relationship attempt upon impose place size equation variable cell equation cell unobserve vector general give compute rhs impose edge cell call identification mapping detail consider use incorporate measurement refine density advanced inference likelihood equation give hand discuss advance drive suitable version implement credible region learn value ask treatment estimation unknown measurement pursuit express inversion function bayesian method variate learn computation unknown test absence available datum e paradigm model functional possible yield connect unknown therefore alternatively learn learn learn instance becoming value address result share relation paragraph nature dark science dark matter dm individual observe system exercise fundamental quantification dark direct observational evidence dm light quantification measurable physical field I dark physical property temperature live play light field act extraction learn total mass matter subtract latter self matter matter reliable mass physical include pattern perform galaxy attention galaxy learn measurement particle particle refer particle star signature mark include old cluster state refer galaxy coordinate particle system velocity galaxy vector playing rise mean stationary aim physical velocity align line observer e particle come towards observer similarly particle plane x x datum k allow galaxy high addition measurement noisy typically datum typically incorporate measurement data matter dark well galaxy unknown mass proportional product function observable conditionally likelihood write application explore situation thereby achieve evolution space discuss tell evolution time td however particle inside star scale age galaxy tf boltzmann q state attain stationarity constant time bt correctly suggest stationarity inside write another steady equation invariant lie proof respective connect vector dt dt attain stationarity space variable dependence rotation cross product simple system along location r evident high circular circular path bt ready methodology section aim computing domain range place grid state application particle galaxy live inside galaxy galaxy particle attain e rr attain normalise maximal low light keep discuss bin bin range cover normalise bin make physics could available recall aforementione less less thus clear normalised extend bin width learn inference equation place centre learn mass lead choose discrete fix radial entirely radial learn express r v unobserved grid mapping coordinate turn function cell depend integral grid inside space consider rd grid cell cv circular lying provide particle bin follow ks lie bin semi minor lie lie implication equation give observe energy bin lie within circular extend elliptical extend semi minor axis cell writing volume area region integrate recover rhs multiple way region distinct overlap know overlap excess identifying allow numerical computation area irrespective allow bound area plane area overlap plane equation express integrate th grid cell low value recall discuss vector parallel q root equation attain value hereafter triple give contribute towards equation dependent within implementation metropolis hasting ensure value enhance integrated max equation conditionally measurement particle project plane galaxy uncertainty particle velocity denote galaxy hand usually implementation simulate define uncertainty measurement likelihood nothing suggest angular opt inference mass bt monotonically decrease numerically motivated community refer purpose mid hyperparameter prior experimentally probability density constant define integrate interval within generate metropolis hasting write joint next make along constraint perform metropolis maintain inference let scheme current let achieve variable zero mass radial variance choose empirical I ease proposal acceptance criterion metropolis discuss accept update line inference directly let current empirical th update element nf proposal interval inference rhs equation proposal density sample ratio posterior I propose vector accept tn lf numerical synthetic display explore trace posterior part learn synthetic black solid high credible represent bar line learn plot parameter dot line synthetic histogram size sample mark solid mass state space learn galaxy exercise relatively run isotropic achieve fix bin dimensional isotropic model isotropic result chain relaxed incorporation learn plot value recover modal learn bin middle parameter estimate chain bin long estimate state learn use isotropic black figure estimate I panel green focus measurable physics connect comprise training relationship either wavelet estimation bin write could missing project onto term domain projection identification prior perform galaxy galaxy kind particle density r dr function compute learn plot infer mass turning result absence prior isotropic state galaxy really mass galaxy mass distribute space crucially exercise possible space vector vector state remark mathematics associate research department statistic theoretical physics university department statistic university al physics unknown model parameter relationship function method modelling
construct contain follow connect besides vertex connect group distant object operator minimization eq sort matrix namely motivation intuitive graph similarity reduction method perform reduction kernel bring gain computational discrimination power reduction space empirically stay classification accuracy crucial classifier experiment conduct instead part classification serve relevant algorithm adopt new query transform basis residual discriminative show algorithm corruption design dictionary lagrangian formulate penalty representation vector lagrange multipli alm lagrange multiplier monotonically increase implement iteratively q shrinkage come transformed kernel project classification apply normalize regularize normalize unit mu mu identity locality use measurement term training sample fine small increasingly sample nn similarity near dictionary locality locality dictionary adopt straightforward way constrain locality reduce computational cost locality category biological criterion recognize speak locality constrain discriminative distance discriminative curse kernel idea locality mathematically experimentally atom example formulation mu first obtain via different atom fact locate near efficient enough database gd simply several perform measure perform distance conjunction flexible review subsection several pixel measure simple effective situation city pixel grid another assume make move move move count horizontal lot pixel distance present texture texture similarity main idea appearance characterize distance difference texture sum response histogram filter metric measure histogram texture initially surface category surface crucial tangent distance affine stroke descriptor relative descriptor iteratively match shape final match shape transformation shape descriptor define gray shape descriptor map sift descriptor descriptor method combine texture unified locality construct enforce locality via texture similarity three size set diagram atom dictionary combination similarity construct combination similarity e lc lc lc texture unify use suppose equal use unified similarity measure recommend raw use dictionary namely sample sample per generate set gd gd b slight classification gd stay global dictionary projection use consist face individual face experimental setting evaluate namely dictionary sample person mostly dictionary informative adaptive important dictionary degenerate gd become locality become well less enough performance dictionary sample gd size global note pyramid similarity database contain object background train rest use global sample comparison spatial pyramid feature use classification competitive especially classification improve high lack extra discriminative gd obviously feature category difficult categorization database experiment category constrain svd classification dictionary contain scene office select rest testing global dictionary setting lc superiority htb pt c c gd gd gd conduct run time evaluate public mnist experimental table use compare approach locality fast gd gd circumstance htb category pt gd gd gd mnist metric greatly enhance classification superiority discriminative compare database euclidean mail image fairly rate width setting detail local histogram divide face area histogram histogram database handwritten digit randomly size raw result pyramid experimental properly select enhance power distance metric bad extend mnist since work bias fortunately metric adopt framework show flexibility allow flexibility unified unified still close basically automatically select reason unify complementary complement diverse enough use complementary tangent believe could bring htb pt extend euclidean n technique smoothly combine discrimination reasonable additionally locality locality exploit enhance mathematically help database also reduce discrimination ability approach discriminative coarse strategy intuition appeal comprehensive superiority public database efficiency validate moreover construction experiment public database simulation discrimination ability idea discriminative greatly create toy experiment unified effective kernel predict become thm thm via collaborative locality propose collaborative representation similarity query atom locality dictionary addition measure unified superiority validate appeal aspect incorporate similarity theoretically perfectly classify conventional scalability state kernel regularize near neighbor locality constrain dictionary recent year great bring robustness regularizer ill condition represent basis basis call overcomplete atom call indeed impose sparsity return help recover despite fact nonconvex norm study et dictionary report representation promise approximate input atom predict select despite fact use still representation classification zhang far unnecessary enforce feature enough work representation cr cr improve reduce collaborative nature make poorly distribute function kernel principle component svm overcome sparse classification particular nonlinear infinite kernel group become besides major propose locality unify give gain cost gd cr database gd gd enable database classifier enforce locality serve locality appeal motivated finding list similarity important mathematically incorporate theoretically training obtain operate feature dictionary typically informative often bring gain globally consider concept link query atom play role support recommendation report gain fact atom yet well performance atom regular dictionary near categorization demonstrate classification fundamental tag advance since far able human infer abstract remain motivate discriminative efficient image outline discuss relate locality discuss follow remark technique al promising zhang al far mathematical conduct comprehensive evaluate overcome handling liu et address kernel representation incorporate model practical recognition discuss image however kernel formulate hyperspectral image method classification difference exist incorporate consider hyperspectral contrary strategy discriminant reduction generalize extend improve formulation detail reduction kernel notice idea locally dictionary pruning apply extension combine locality dictionary link conjunction bring scalability extend feature dimensionality propose briefly code whole essence class dictionary th denote I ji error formulate combination determine residual key reasonably regularize least signal prove collaborative important machine whose member task component classification raw explicitly specify similarity function via kernel easily kernel enable operate implicit computing often sequence reproduce rkh kernel include perceptron many transform non handle smoothly nonlinear mechanism map high via transformation mapping empirical mu mu transform
numerical nature superiority make skew viewpoint numerical outcome little want classical perform slightly numerical commonly triplet experiment figure would regard point examine comparison aspect opinion closure interpretability viewpoint plausible great possibility factorization independent lead preference skew statement lm subsequently adopt close skew variant prefer finite modelling limitation examine statement adequate correction avoid reader acknowledgement grateful discussion various aspect relate material theorem p skew continue popularity effective include formulation model literature property various claim well datum year dedicate class distribution special normal univariate elliptical study alternative elliptical impose symmetry simultaneously component closely meet symmetry elliptical skewness explain skew already quite stream skew symmetric elliptical similar recent account generate nearly compete alternative preferable specific considering examine far although parameterization early one distribution construction skew normal skew shall skew normal coincide dimension differ arise difference formulation viewpoint carry arise underlie parent normal wider elliptical call elliptical family examine classical skew counterpart skew question form skew reason great flexibility shape arise freedom traditional via substantial involve mix parametric th family role elliptical parsimonious possibly interpretable special distribution view quite inaccurate potentially impact community brevity shall lm paper appear adopt framework overall formal apply lee role elaborate skew distribution skew normal positive definite playing role parameter representation follow normal mean ct obtain detailed present chapter skew minor symbol retain parameterization write symmetric independent satisfied wise moment two analogue originally use skew match display scatter ht institute sn panel similar mass contour curve panel contour mode somewhat curve corner extensively illustrative literature become shall come qualitative include completeness parameter coincide family subset moment become factor handle burden rapidly purpose fitting common bayesian lm skew statement consider equivalent diagonal matrix allow independent skew component factor non vanish involve require classical incorporate logical frame describe subject selective expression skewness skewness excess numerical maximal appear coincide form case among thing hold remark formulation emphasis depend even wide limitation introduction skew possibly factorization consideration lack closure negligible skew advantage factor take interpretability remark relevance qualitative matter illustrate refer resemble regular student follow normal independent skew representation skew thesis range coefficient skewness marginally instead skew source point flexibility version appeal explore aspect lee early lm relate formal analogous concern defer section skew skew refer name incorrect constitute section lm skew view extension latent replace analogue stand multivariate skew near end place component add level highly restrict adopt terminology broad generality see early wide generality furthermore skew lm skewness sect study extension skew et limited form skewness expression skewness true classical skew marginally globally case coincide univariate skew expression distribution employ purpose potential additional variant tail prove considerable numerical diverse area employ context focus application lm place statement therein motivation present statement theoretical discuss lm contain claim superiority fm extra fm restrict besides general superiority formulation believe purpose systematic due selective reporting arise section note skew formulation lin lin lin amount skewed analysis herein skew herein conduct analysis correct rand ari perfect agreement zero classical software within package start lie skew classical skew component figure mixture performance give mixture ari h five datum package r base various focus size rw gender skew fit classical skew ari give poor produce well classification ari misclassifie present accordingly comprehensive wherein skew distribution triplet note consistent ari
overhead global gp modelling present derive perform guarantee achieve induce parameter communication embedding experimentally study suggest inference resource time power implementation demonstrate process show gp improve amount million test implement map multi architecture package derivation additional robustness failure drop software package extensively explanation recent propose variational train use mini batch successfully learn undesirable variational marginal variational induce target analytically derive additionally need analytic gradient induce advance strong correlation parameter heuristic work difficulty review variable aim function location input precision convenience analytically marginal prohibitive approximation aim term induce complexity input correspond infer posterior induce equation relationship induce link conditional overall make tractable optimisation modify fitting alternative greatly reduce fitting computational approximation distribute supplementary material latent aim mapping prior technique see derive finding next distribute model derivation inducing distribute allow easily model explanation derivation regression latent variable modelling identify marginal introduce induce expression multiply induce jensen low brevity integrate use obtain use derivation identical derivation break induce represent individual integrate triangular distribution calculus variation analytically plug q input obtain mi I sum communication supplementary using hyper induce calculate set global induce additionally px calculation supplementary material node central back global local posterior map central follow sum point material contain partial optimisation local range criterion well increase scaling compare scale distribute system square exponential ard automatically add conjugate inference give improvement running time transform linearly core space total run spend iteration show improvement close achieve core little sign interesting overhead inversion carry global core show assess increase resource available computation run per iteration effectively extra total take compare inference see sequential significantly computational resource inference spend step core research parametric sometimes state requirement practical equal load reduce computation worker maximum execution node core difference node suggest load describe series experiment demonstrate gaussian big task gps perform often regression point use present modelling perform imputation test digit far aware mnist na na k delay record distance datum test experiment setup different stationary nature performance dataset c c baseline depth estimator good different resource induce optimisation root rmse point core minute baseline take several gps big advantage uncertainty principle increase train large converge optimum bfgs mode optimisation likelihood mnist example large train use test take likelihood py e induce choose marginal converge get
notable feature build recorded ns performance sp stand quantization pyramid respectively projection convolution sp total highlight inference time pixel image matlab dual gb ram framework soft convolution pooling sp bottom manner adaptive map classifier table multiple additionally feed forward kernel table detailed comparison method one one sift descriptor extract sift descriptor generation adopt locality sort descriptor amenable parallelization gpu expect gender svm lc evaluate gender age net act baseline ar database code online include lc lc training testing code setup first project dimension matrix fed face recognition gender learn mid partition spatial neuron two comparison neuron display n compare furthermore neuron ns layer capture characteristic intuitively demonstrate propose ns work classification net include use appearance codebook spatial list demonstrate well year old deal perform slow resemble mid model reveal age face mae mae image nine learn original image upper panel image map sift acc infer lee zhang al l acc wang al lc popular database training testing resolution preserve aspect recent unsupervised learn sift feature psd method mid codebook code dimension model learn neuron category pyramid partition public categorization marginal reason foreground appearance sophisticated neuron alternatively sgd fix basis normalize pose negative newly adaptively class response n appropriate rewrite q sgd derivative calculate couple newly simple w hadamard initialization convergence decoder pre train merely pool activation specifically calculate pre initialization lack allocation category pre also initialize variable matrix purpose symmetry breaking inferior architecture ns layer random initialization work well descent initialize eq appendix due paper mid level soft convolution significant preserve orientation map illumination illustrate trivial filter illustrate two highlight full figure illumination soft convolution illumination require operation convolution soft thresholding put several feature produce convolutional map tensor normalize third scaling comparative third map threshold map produce normalization similar benefit local descriptor illumination invariance face individual challenge vary illumination illumination select image person original normalize soft map max maps illumination illumination evaluate framework database face face performance ever background soft convolution pool weak response remove soft operation keep three database randomly category face consist descriptor patch sift consist sift descriptor center pixel image generate display gray sift map sift mainly focus mid lemma mid feature greatly enhance automatically manner paper efficient mid level operation pooling quantization simple method need much time boost neuron ns layer mid ns neuron inference top extensive database achieve performance gender categorization recently code image sift convolutional network improve classification level far mid hierarchical neuron layer feature high fed panel descriptor window salient mid layer despite mainly nonlinearity pyramid apply sparse pooling nonlinearity focus nonlinearity sparse psd introduce absolute however point et al moreover complex simple carefully factor size density therefore instead design complicate mid efficient mid feature consist operation convolution quantization shown sift produce desirable might descriptor consider mid boost accord build neuron ns demonstrate neuron signal specific bottom inference ns improve notably summary mid feature give approach generate argue might ns mid classification ns support inference appeal achieve art describe level section mid mean build without structure via code sift despite promise accuracy extract low descriptor amount knowledge system autoencoder empirical confirm rule nonlinearity mid performance well include densely descriptor suitable difference simple study mid learn call mid sift adaptive mid generate filter decomposition map see rd tensor panel worth convolution adaptively feature several step convolution normalization advantage first convolutional behavior descriptor preserve information illumination filter thresholding sift pooling robustness map pair within macro size capture neighborhood pool nonlinearity map code feed forward descriptor result demonstrate splitting map map densely extract sift descriptor patch descriptor dictionary pooling code predefine partition different task detail pool code representation usually dimensionality reduction normalize mid involve oppose project discrimination reduce produce guarantee person illumination image display three filter map thresholded descriptor sift within unsupervised despite share similarity hard example eight capture subtle descriptor resembles derive forward pathway max build layer architecture produce complicate deep architecture comparison say use descriptor statistical incorporate along max also map convolutional map interpret consider contrary propose soft convolution negative interpretable mid manner build mid feature boost neuron principle neural signal category stay therefore call layer fed denote class build layer ns mathematically specific mid ns generate activation turn logistic activation encoder inference process activation pattern present produce structured activation top decoder activation successful field decoder weight decoder back neuron reflect appropriate level encoder bound generality consider decoder mid level length reflect neuron hence specific signal stay property structure impose eliminate besides activation denote minimize mean take column activation time activation activation h h h penalty several decay neuron automatically break separate part behavior rigorously three intra
circle specify etc loop generate diag library diagram option distance option useful want diagram persistence diagram bottleneck wasserstein package r bottleneck wasserstein two persistence diagram circle diag diag diagram bottleneck nd wasserstein diagrams code specify diagram loop diag diag summarize information persistence diagram briefly landscape persistence landscape sequence piecewise encode diagram landscape create obtain graph persistence landscape persistence landscape function max max middle diagram treat diagram persistent low persistence conversely persistent feature see value half life blue landscape right blue landscape function function persistence landscape specify interested st landscape length kk return band scenario landscape build compute prohibitive instead draw landscape subsample yield identically distribute approximation construct band valid illustrate sample circle xx subsample diagram subsample seq store store subsample diagram feature diagram kk construct landscape alpha plot code main landscape col col tb problem topological kde suggest describe work width band scale describe measure kde kind tradeoff maximize illustrate follow circle plus clutter specify limit evaluate xx xx xx limit among number bootstrap band progress bar alpha alpha display value criterion call kde f alpha parallel persistence maximize persistence kde example let density l dimensional subset maximal simultaneously tree tree tree branch tree particularly whose organization difficult cloud three well separate x xx x nearest knn alternatively use kde density algorithm connect density contain tree object middle lambda lambda knn access homology library package implementation persistent homology become comprehensive topological keep library new topological information analysis like thank ed discussion author develop package available algorithm package present package provide tool topological implementation given provide topological distance salient topological quantify persistent homology provide interface library include persistent homology persistent homology grid result diagram recently package include implementation allow visualize dendrogram persistent homology cluster recent advance actually take cloud topological lose collection topological persistent homology homology multiple simultaneously quantify nest persistent homology topology change diagram death devote presentation interface algorithm library topological information underlie estimator devote computation persistence diagram persistent homology set function grid diagram build cloud challenge persistent homology represent topological topological persistent homology exact persistence infeasible often confidence diagram allow topological compute draw x q prove validity kernel persistent homology implement package figure band kde grid grid show surface surface provide persistent homology reader basic concept persistent homology grid construct persistent homology arbitrary choose compute persistence library code compute persistent homology cloud object evaluate smoothing bar
exist partition core core regular degree particular remark hardness long dependence correlation hardness suffice proof literature express subgraph entry describe prove define constant leave universal fix black box give backward mapping model possible produce approximation trick probability mapping make call least take proposition get proposition marginal desire marginal know backward amount gradient optimization project reason project onto project marginal standard project np nevertheless projection address difficulty inside lemma consider operating amount lemma omit condition approximate project converge purpose completeness minimizer oracle error gx gx gx tx cs l gx translate approximate approximate equivalence convexity strong implication appendix x p difficulty black approximate mapping obvious closely polytope even onto np project onto goal optimization thresholding project project translate project simple per suppose consider section devoted procedure sx update tx sx p x I v prove subsection property keep iterate inside notation independent collection shorthand write lemma specifie want imply rearrange read fs fs fs fs simplify close polytope corresponding require show note b new check I contradict spirit next allow sketch give section supplement sx follow hyperplane hyperplane definition imply coordinate h h inactive require desire gradient h h argument require care prevent despite negative already complete paper address hardness backward within hardness even weak exist constant acknowledgment thank helpful discussion manuscript office award nf material amount node marginal subgraph claim obtain remove label induction base trivial formula summation inductive order approximation give approximation maximum independent choice suffice argue follow set map neighbor set removal lemma convex modification projection onto convex contraction prop inequality definition gx precede define divide convexity apply jensen give right show recall change triangle direct calculation value p p see chapter bind derive measure I I subsection follow restrict fs fs fs fs finally f sake argument contradict goal sx initial hyperplane except negativity write vector fact together zero observe h similarly critical active first inactive constraint require gradient constraint close x qx inactive consider fact critical bad rough coordinate increase sufficiently iterate argument might prevent projection start increase coordinate consist rearrange give follow prove ready prove coordinate know pp pp affect increment hence coordinate limited amount satisfy additionally eq together count move away hyperplane complete proposition corollary theorem david laboratory department school management mit edu specify undirected graphical uniquely mean parameter principle feasibility parameter learn canonical unless rp polynomial reduction approximate core know hard parameter reduction entail show polytope optimization procedure ellipsoid powerful high core intelligence variety finance communication biology undirected model rich applicability canonical consist wise marginal graphical computation marginal learn parameter polytope backward capture study subject interested task basic compute efficiently well approximate computational core simple pairwise define sharp compute core model tractable exhibit property exist despite hardness previously obtain undirected et show hardness show hardness require know section hardness mapping thus backward vector v physics literature activity eq serve normalize note play major role define eq core polytope equal hull vector polytope structure need large depend polytope condition notion entry next notion bound rp np difficult
slowly rescale treat saddle argue saddle instead become attractive newton minima entire justification quasi method e rapidly bottom less optimization previous report order rapidly saddle curvature fundamentally different method also outperform involve suggest minima provide come nature domain review particular derive point distribute negative critical attain plane critical concentrate monotonically imply error much exponentially saddle necessarily chance small r unless critical stand close one positive show eigenvalue shift global shift shift negative well eigenvalue typical slow yield perspective geometry surface saddle event pick become pick positive negative saddle qualitatively similar derive function error surface generic chosen minima exponentially saddle point negative result applicable analyze surface perceptron mlp linear layer surface show saddle indeed analyze saddle scale space mlp dynamic deep follow transition performance aspect soft explore network randomly choose teacher importantly unit within cause permutation among associated symmetric teacher interestingly curvature saddle curvature hessian measure relevant property surface develop generalize mlp matrix critical small mlp train sample version mnist newton fig setup detail test qualitatively critical point concentrate monotonically increase plane saddle seem accord leave critical give understand behave near us saddle point analyze appendix supplementary material detail new parameter step point sgd eigenvalue step along restriction drawback direction eigenvalue absolute saddle structure dot stand free rescale gradient eigenvalue approach newton descent move towards move along direction eigenvalue saddle point newton order curvature add rescale gradient modify eigenvalue every eigen direction increase drawback potentially eigen incur approach ignore regardless strategy bfgs saddle ignore curvature follow natural relie curvature behaviour take similar newton fisher matrix argue descent saddle effectively resolve arise however descent suffer curvature issue fisher gauss direction distant converge point matrix saddle exhibit negative mean landscape mean rescale descent use rescale vanish critical much near straightforward trust region taylor instead rely taylor trust constraint taylor region describe arise special saddle family near saddle sec simple eigen rescale newton preserve turn saddle hessian suggest example aware justification dimensionality saddle alg empirically theory suggest saddle validate saddle function network near order minimize approximation scale mnist direction cifar minibatch descent newton saddle free select coefficient small update saddle likely sgd near saddle point fig clearly small size saddle outperform large margin close behavior algorithm fig error see saddle get near saddle sgd epoch observe saddle newton rapidly fig shift toward shift suggest successfully error large recurrent saddle seven layer neural deep autoencoder benchmark descent newton rapidly sgd even confirm shift saddle newton follow free get art hessian free method well feedforward want saddle free method avoid saddle recurrent hide sgd saddle free newton method see trend feedforward sgd quickly suggest around soon drop find newton negative eigenvalue saddle method see sgd physics matrix theory neural saddle dimensional intuition minima exponentially dimension provide first neural surface test application confirm qualitative index critical positively generalize region saddle curvature fundamentally different define trust order saddle free method theoretically recurrent network saddle rapid descent newton trust show sensible saddle improve neural first beyond hessian second critical training neural far generally deep property surface guide design non impact engineering acknowledgment thank cifar research compute computational google fellowship thank vanish characterize symmetric number scenario eigenvalue minimum eigenvalue local eigenvalue zero critical saddle restrict span saddle maximum example saddle restrict direction move correspond pick sign point presence structure maximum saddle shape look along equal almost eigenvalue large structure structure circle shape max direction shape taylor neighbourhood critical reliable vanish eigenvalue span eq plot discover nearby saddle newton saddle seed select run among saddle free amplitude pick weight cube layer activation network different randomly train newton allow different critical point along trajectory absolute corresponding subspace slightly note find useful direction v beneficial subspace hessian w feedforward network strategy rate minibatch momentum sample maximize among protocol classical recurrent weight orthogonal rnn hyperparameter gradient
briefly introduce formally loss unified introduce misclassification denote label set penalize class hard soft formulate case hard little abuse predict rule margin combine obtain commonly hard classification margin approximate correctness non hard sign may margin direct non infeasible surrogate surrogate margin loss loss prevent fit commonly exist loss soft providing spectrum soft classification naturally formulate base describe option formulate form notation option pre specify express mention fit weighted specify class simplicity predict account commonly constrain interval generality weight margin surrogate consistency statistical surrogate classifier fundamental expectation theoretical g respectively margin function classification bayes weight calibrate theoretical loss monotone pc rejection option rejection option hard platform gap formally introduce underlying population shown p separate c soft show span entire soft formulate target unify margin insight order interval obtain splitting belong framework three dense correspond discuss later illustrate spectrum framework generalize collection classification bayes boundary simultaneously boundary formulate weight throughout loss class show theoretical usual multiplicative precisely figure function figure horizontal axis correspond giving except case smooth panel boundarie observation boundary incorporate step soft become loss indeed correspond learn task precisely theoretical loss hard option rule classification theoretical loss limit theoretical bayes correspond soft find describe section optimization use surrogate generalizing rejection option classification prediction common negative formulation theoretical differ along vertical f margin formulation surrogate first surrogate consistent class piecewise surrogate include propose empirical surrogate include piecewise surrogate consist segment boundary become dense tend denote convex measurable provide necessary sufficient surrogate consistent naturally surrogate loss condition exist possible intuition justify soft loss soft loss meet loss corollary next surrogate svm satisfy surrogate boundary contrast consistency surrogate surrogate build option circle theoretical loss panel appropriately first loss panel boundary consist non respectively consistent control segment surrogate loss segment surrogate loss observation denote intercept segment express intercept loss linear consistent piecewise surrogate piecewise specify b denote location along loss piecewise intercept slope decrease hinge eq segment order degenerate align hinge guarantee importantly next obtain satisfy logistic dot use dash piecewise tangent construct rejection state piecewise satisfy let loss construct line logistic piecewise surrogate satisfying figure dot surrogate vertical denote equal tangent logistic correspond differentiable highlight appear negligible notably piecewise suggest may additionally spectrum explore issue simulation surrogate loss show risk respect risk risk rate minimizer bayes optimal derive rejection far suggest separation function bound error piecewise class surrogate piecewise non differentiable svm quadratic qp also formulate qp complexity intensive moderately propose project similar rewrite surrogate define space rkh norm formulate intercept review kernel boundary write denote estimate iteration iteration let mi mb b b b iy iteration project illustrate illustrate achieve consistent piecewise loss piecewise constant derive boundary grid tune logistic loss time piecewise setting panel along black rotation variation respect space three piecewise bayes optimal setting linear loss boundary illustrate minimize tuning along decrease intuitive converge improvement classifier theoretical confirm classifier piecewise furthermore boundary black comparison piecewise logistic vary panel median loss standard replication minimal spaced sample rotation setting heavy consider asymmetric optimal boundary linear classifiers simulation piecewise great converge logistic nc institute drug private private md california effort many co date pc nc subject color estimate ad vertical consist feature nc ad process logistic validation determine principal pc subject subject interestingly appear ad subject include subject mild cognitive stable subject depend whether ad consider nc ad distribution ad correspond boundary vertical group appear peak appropriately divide subject disease discrete class several hard rejection option learn conditional framework hard soft classification provide perspective hard family unified margin surrogate previous behavior problem class problem part national health grant share u institute imaging association discovery foundation sciences company company company research company health provide clinical site private contribution foundation national health www organization institute education california imaging california research grant material option rejection option loss loss soft deriving limit q note bayes classification generally similarly recover let option rewrite equivalence option loss rejection option although traditionally limit eq minimize choose appropriately boundary boundary separately condition consistency necessary sufficient condition boundary define wish satisfied equivalently pair r r exclude boundary wish satisfied non class loss note equality satisfied derivation three convexity satisfied rule rule rule suppose express pg pg cf suffice pf pf g convexity eq rule margin theorem additionally inequality therefore combine choose letting fraction right surrogate boundary denote k k desire bind define f I exist bernstein tail h f I result since pf pf pf b pf rewrite eq p b similarly measurable uniformly class without f q net bernstein note necessary plus disease abstract learning covariate class literature distinction soft model extensively propose spectrum span reveal novel classifier convex surrogate descent disease keyword excess problem supervise similar regression describe generalization denote covariate probably covariate task correspond briefly target prediction hard soft classifier example include rule predict commonly hard probability classifier traditionally soft question classifier differ recently unify relationship classifier connect several discrimination far extend category base
ij topic cluster find mean partition compute th l high approximation dominant topic succeed probability find complexity notably document dominant document need coordinate note get long lemma essence proving identify partition call center correctly almost document prove theory thresholded document document able dominant topic assumption fraction topic topic help pure find complicated induce conditioning thresholded available author empirical synthetic real life coherence know advance test multiple give empirically well initialization project svd mean report dataset step standard select vocabulary term frequency remove less word paper vocabulary consist york dataset document dataset ng document vocabulary length sampling state burn iteration posterior show document corpus minimum topic fraction document pure also indicate justified fraction topic assign document high cluster table topic ij per column intuition local recall analyze synthetic corpus plot separately fix show plot monotonically hand unimodal c dominant pure topic mean topic topic ng semi corpus ensure corpus retain gibbs run final topic generate synthetic draw topic gibbs topic summary evaluated rigorously evaluation l reconstruction algorithm good dataset multiple real justify conclusion real corpora document present svd able thresholded svd svd massive corpora apart recovery thresholded broadly similar minute game team team sale book school book game game drug find patient drug medical music band company web www site computer software mail look room show look home house room look water house trade death car car com com author player team book author play character goal play team award million school teacher child plan plan million stock market percent team home room school shoot home company law company software company window million million company stock percent company million business com company com site quick product percent home com shot team team team political com www room million movie music movie character percent stock market percent price share wind weather water weather air article ball country percent right study student plane pilot company company medium company business customer million worker company pay shot shot game team team black primary blue blue lemma proposition exercise em height width microsoft topic latent word inference problem np strong give provable algorithm widely lda give provable vector aim develop intuitive svd provably solve lda co occur specific occur strictly topic individually major realistic corpora value svd step recover dominant sample empirical evidence corpus propose assume distribution word convex pick multinomial combination topic recover provably gibbs provably topic topic collection document sometimes topic pure word topic dirichlet development give provable corpus assume exist word topic single topic learn try occurrence keep group call occur topic frequency weak separability weight significantly motivate assume corpus paper document collection document dominant topic high topic every nearly purely contribution provably topic grow dictionary unlike grow semi synthetic corpus several topic document let k l give topic column pick document pick topic weighted topic I trial trial pick wise provable excellent provable start successful document recover collection document base primary indeed method numerical like satisfied anchor topic word polynomial note linearly dictionary every seem realistic ask like run assumption informally dominant document corpus reasonable assumption base propose inspire introduce individually occur topic much matrix subsequent mean dominant corpus negative dominant document dominant sl sense identifiability vector model unique group likely co occur assumption try word pick document could think pick pick multinomial dominant whereas technical weak sense justification generally word plot plot expect curve frequency reach unimodal empirically size close asymptotic think infinity large thought intuitively document intuitively need mainly need refer different svd
high fit spam response comparable spam lasso show estimate interval set spam estimate np loss convex j discuss loss penalty generalize descent solve choose eq special case omit brevity case logistic amount q form generalize penalty solution logistic piecewise figure set value set display expectation average replicate combination two model interpretable fit adaptive knot limit knot fit jump knot knot basis knot constant flexibility knot accommodate trend large fit suited trend previously future package make develop interactive demonstrating note fact u degree differentiable thus stein ts around block solve yield freedom add omit take p objective pz ji pz union n v u ip plug p n plugging inspection form begin lagrangian z note partial respect b v writing lagrangian obtain ball solve ex minus plus ex ex ex section lemma department variable observation interpretable desirable propose piecewise adaptively knot optimum provide show degree keyword predict response feature observation task offer interpretability piecewise pre knot flexibility instance non interpretability flexibility select feature include knot th element element reference contain review implement examine analysis set close section generalize restrict attention function might spline additive dimension propose extension spam induce estimate spam solve j td order datum fit capture complex relationship j td cubic predictor contain inner cubic spline also recently partially linearly basis choose adaptive interpretability many cubic spline clear change knot knot proposal knot adaptively begin seek piecewise constant knot parameter derivative adaptively choose jx permutation exclude solution value tend therefore optimization provide encouraging piecewise induce purpose much descent cycle repeatedly hold fix partial operation iteration leveraging solver solution directly optimum made initialize compute r p repeat fuse filtering show th trend filtering equivalent regression spline interpret variable locally adaptive spline illustrate portion spam however impractical v u u center triangular remove difference order equivalent solve lasso allow fit fitting modified ridge ensure convexity prediction zero compare predict one coefficient finite assume p provide smoothing package spam cubic knot space quantile ij four scenario display spam area area scenario scenario scenario scenario validation training freedom spline spam range degree freedom degree multiply freedom covariate intercept freedom estimate display mse versus three achieve low test mse spam outperform need preferred noise exception scenario scenario scenario scenario indicate replicate additionally summarize mse optimal j freedom except dimensional setting scenario achieve comparable explain performance dimensional cccc proportion mse freedom spam spam spam spam spam spam spam spam spam strength local constant qualitatively examine truly replicate fit constant spam spam impose encourage entire control flexibility variable fit fit varied greatly domain fit function purpose spam correspond simulated consider estimate country national country approximately country publicly unite
value namely reveal non step optimum decide round rather threshold truly semantic fold evaluate rank iteration top illustrate curve high compute mean deviation list optimal rank dataset gradually continue reduce explanation much overfitte likely htb reproduce previous program outperform sparsity degree table range b word truly effective semantic keep constant extent explain model extraction perspective criterion experiment feature exploit classification overcome significant improvement open relation extraction reconstruct item plan improve capable acknowledgment basic grant cb national science foundation china grant laboratory division innovation development laboratory technology university china china advance china com essence extraction label tackle sparsity noise problem completion factorize minimize classification complete label matrix underlie leveraging solve widely baseline knowledge text traditional hand label corpus achieve precision recall corpora satisfy increase demand large text distant supervision improve effectiveness paradigm intuition automatically text wikipedia york corpora heuristic account basic entity involve come occur appear text current diverse relation combine label relation paradigm corpus automatically come sparse kind nlp stanford extract variety name tag speech tag path unfortunately leading noisy instance relation explicitly noisy case extraction incomplete mention incomplete instance therefore distant incomplete corpora essence incomplete multi perspective use knowledge technique relation extraction supervision specifically sparse pair column relation relation classification transform complete unknown item item incomplete factorization de label contribute supervision extraction completion recover simultaneously logistic function influence incomplete suitable binary modify find global optimum widely discuss compare degree distant supervision firstly bioinformatic discover entity article maintain expert date web text build corpus without label adopt scale crowdsource knowledge online relation name wikipedia entities mention relation variety regression inspire relaxed replace sentence al entity multi approach label relation extraction jointly text basis address entity tag relevant pca collaborative distant supervision e perfectly promising apply area computer vision recommender controlling model classification robustness relation basic vector test matrix complete entry observable entry rank observable feature observable impractical entry thus label z optimization weight nuclear employ w w another call relation generate sigmoid entry derive function complete sigmoid calculate entity n relation np hard es suggest relaxation solving modify optima formulae modify solving contain infer follow gradually minima minimize singular value svd cut assign matrix parameter pt accelerate
input residual split suitable regression procedure apply lemma empirical taking arrive desire ready score either satisfy additive test non characteristic regression procedure additive population bivariate c bivariate satisfy test noise bayesian model input closely closely relate penalize likelihood minimize respect yield formula identity formula derive obvious penalize identical negative rewrite around value relationship score random standard sort white noise rule integral follow noise characteristic gp input add scenario without scenario scenario scenario gaussian simulated setting cause four scenario benchmark form competition benchmark consist cause pair consist pair statistically variable know cause task sample publicly available set select agreement truth pair consist weather obvious cause though due hide selection cause process relationship system generate whether intervention distribution perform practice original generating long available unfortunately available clearly cause effect set criterion cause relationship set cause follow subsection describe pair detail motivate decision scatter plot horizontal cause axis include overview benchmark dataset ground hour pair age diameter age hour consumption consumption weight consumption age stock age pair duration temperature pair age age age heart compressive compressive pair compressive pair water compressive compressive coarse compressive aggregate compressive strength compressive strength consumption consumption consumption pair consumption body pair age pressure concentration day temperature pressure pressure pressure relative day temperature pair concentration temperature concentration consumption acceleration dim dim life capital pair capital capital life capital life capital life capital life life capital pair water bank stock stock return stock return stock send http pair dim disease global temperature co energy life per pair growth temperature net pair temperature population protein pair temperature age merge weather weather datum six value year duration notation causal temperature temperature pair hour temperature ccc pair pair hour pair temperature elementary place tend level roughly think intervention temperature high hand perhaps location happen air let statistical south also place high lie south temperature empirically direct temperature dominate since occur air force rise air water due indirect via relation less influence temperature main direction wind relevant intervention allow pair temperature detect relation intervention west even unlikely east therefore duration positively high weather sometimes day sensor increase duration whereas causal influence dependence early temperature move change necessarily north south movement obvious somewhat weather west east expect relationship east west change adjust cloud uci repository concern nine height whole directly age year six cause relationship diameter pair age weight pair pair age diameter age height pair whole weight pair age weight height obvious intervention since possibility change observe consider intervention provide condition clearly whereas change change difficulty define agreement cause height age count change length natural good proxy age census census repository study age hour stock pair pair age per hour pair stock age hour instance per age already argue age difficult problematic intervention background life job experience year later old work job however sometimes long job experience intervention change hour intervention easy imagine would certainly age stock instance stock age vs hour intervention theoretically stock hand age influence stock thereby stock indirect age stock less hour uci city consumption car several attribute like weight acceleration come american association thereby cccc pair consumption weight consumption acceleration acceleration consumption air engine draw cycle engine consumption change weight change air consumption measure power engine consumption add engine car change consumption change consumption change consumption car powerful consumption causal consumption weight vice pair acceleration air design able certain maximum give car indeed engine big selection acceleration acceleration two combination variable multivariate three consider cause comprised consumption concentration chemical compound child year package language pair plot pair pair age concentration pair age concentration cause old set package contain subsequent old national usa consist collect pair scatter current duration interval interval duration repository patient use causal patient length h ccc scatter age pressure certain partly affect temperature storage daily air day air day time two drive scale weather causal day pair pressure pressure pressure surface mostly weather large scale weather pressure gradient hence pressure stem day pair pressure pair relative air day air movement place occur stay affect place reason scatter pair reason temperature surface pressure level pressure influence variable dataset website contain various national road connection around south scatter variable day indicate working day day cause introduce political amount traffic large number day certainly book pair temperature temperature strong impact adjust air little much heat deal relationship daily air produce presence surface give go detail complex chemical mention chemical instance apart air may influence temperature traffic weather occurrence path phenomenon lower high three temperature scatter plot temperature temperature concentration pair temperature temperature concentration daily concentration daily mean daily temperature daily place day concentration complex wind air global affect height formation influence contrast drive g place environmental wind temperature consist daily time pair temperature wind direction speed air wind air vertical source mix air different wind database united division cccc pair scatter pair life capital life pair life per pair capital life consist life year birth country capital various china correspond period respectively pair capital life pair life reasoning life water access percentage access water change people access clean water particularly disease feedback country development aid towards increase access clean water contain emission together sample mix world energy co amount across source change co use country term decrease change per life collect life birth country general rich thus care believe life human impact country vice versa collect live reasoning influence system determine minor disease per reverse causal yahoo database stock stock bank prop subsequently follow adjusted price yahoo finance base day use interpolation stock price calculate ccc pair scatter bank return stock stock prop stock bank return bank stock whereas pair stock stock reasoning stock prop prop major stock file http server institute internal website request send interval interval minute pair scatter internet internet website raise transfer create additional website access transfer website fact make inside outside datum room outside every minute day locate explain large fluctuation collect pair scatter outside inside outside causal expect outside temperature temperature heat capacity inside house reasoning pair take face face interpolation component face image answer scatter plot pair certainly intervention set repository order suffer disease choose decision temperature patient occurrence yes yes yes group six dimensional pair disease think disease create uci say create expert diagnosis disease represent patient consist temperature unit east conjunction centre office express mean anomaly description website count ten average ten small pair temperature phenomena surface cause entirely temperature anomaly correlation less proxy activity believe temperature organization un cover area period one consumption population describe consumption day c pair growth consumption growth consumption regard cause mainly people availability advance increase market economic short scale probably might consumption population could imagine fed reproduce response datum filter light response obtain consist three measure total net define release light intensity nm nm visible light time measure measure several forest available ccc pair scatter pair net direct temperature collect aggregated day site exchange set quality mean credible nan pair scatter temperature temperature exchange approximate release largely mostly consider truth pair site growth population nine file eight logarithm people eight scatter plot pair seem reasonable total versa people people believe rather employ might status child trial protein allocate mixed measurement drop stop value datum week discard drop week organize set see relationship protein protein produce c pair scatter pair cause vice website demand student interest size room month remove scatter plot pair daily daily originally concern historical daily bc temperature h pair scatter pair temperature tell cause effect whether dataset change consecutive age average take pair scatter relative age cause european union institute university statistics max institute institute kn institute systems circle fill black minimum pt circle black size black black thick discovery relationship purely observational science elementary discovery cause alternatively cause joint consider impossible causal different cause pair various evaluate performance bivariate causal discovery benchmark cause purely observational noise causal causal relationship rather association effect gold identify relationship control expensive impossible identify purely observational constitute influence cause influence common cause condition acquisition discovery attempt distinguish require condition call causal study purely copy task impossible could causal approach distinguish cause attract recently cause supervised shift variety able argue marginal distribution cause low factorization intuitively appeal precisely measure contribution extensive family bivariate discovery original benchmark collect year discovery definition causal review idea discovery review cause effect free appendix benchmark various joint observational distribution intervention consideration aspect particular perfect intervention explicitly force leave system notation intervention force lead consider label graph causal cause illustrate marginal infer inequality feedback relationship variable e present intervention costly still relationship I common effect implicitly feedback relationship infer decide upon causal say direction xshift node var z var z cm xshift var bend bend leave yshift node var xshift yshift causal relationship two observe variable latent cause causes explain condition hide variable explain dependence valid although except article review discovery exploit bivariate detail extensive literature assume effect field although linearity mathematically convenient generally np model possibly function cause latent effect cause cause intuitively reasonable model relationship nonlinear cause lebesgue assumption common upon feedback latent cause unobserved cumulative variable easy distribute construction direction another interpret q gaussian prevent draw introduce show noise allow distinguish recently variance lead high variable relationship identify causal influence precisely consider class consist function say interested introduce satisfy identifiable contour joint scatter sample distribution contour mean different model additive typically identifiable non gaussian distribution identifiable fall imply something might expect intuitively additive multivariate identifiability provide identifiability transformation call post know either additive I cause regard rigorous rather exactly quantify conclusion cause possibility happen satisfy identifiable would special unlikely discuss way helpful bivariate bivariate additive finite induce fx induce sample model residual estimate datum testing residual consider scenario split independent typically split big two identical datum different couple suggest come additive noise estimate residual one noise test parametric residual dependence care threshold ensure tight far choice lead way compare need algorithm scheme identify simply regression dependence decide additive I method measure estimate residual calculate measure principle subsection hilbert schmidt base alternatively statistic consistent differential entropy residual score originally finally briefly message score idea minimize possibility originally hilbert schmidt residual input definition propose score eq indicate independence possibility value option certain technical infer joint either splitting datum scenario kernel definition consistent additive dependency cause show weak vanishe method explicitly residual input entropy score use reproduce joint shannon proof application differential entropy identifiable additive exist e one order causal shannon advantage entropy marginal entropy mutual certainly disadvantage rely effect differential term identity identifiable noise show suitable assumption standard generative gaussian role calculate marginal evidence infer causal case dependence typically instead reason implicitly distinguish splitting use decide noise bayesian mml score function fit measure combination mml construct score trade infinity complexity mml direction identifiable mml refer mml like conditional mml identical mml base length mixture gaussian px optimization problem nonzero score difference former gaussian combine single bayesian score generally measure residual minimize respect respect challenge multiple local guarantee find addition strongly automatic prove consistency challenging residual section discovery cause effect cause base effect build mechanism formalize case toy strong assumption causal sufficiently provide introduce deterministic relation via translate contain intuitive case equality q interpret variable side distribution equality positively tends region contain illustration intuition behind cause correlated density high employ expression side zero section variable introduce rather define scope perspective base also priori range may choice let cause deterministic density exist interpret expression alone density special reference instead substitution side kullback therein amount infer cause density quite gaussians infer rescale specification essential implementation choice shift map preprocesse slope assume increase give entropy denote equivalence slope whether deterministic I normalization scale deviation discuss sense twice fine preliminary conceptually solution order remove occurrence original ignore repetition occur describe implementation criterion present section source code reproduce make available open platform library gp parallelization process gp implementation use exponential constant reduce spaced computation scale introduce error therefore try call order behave asymptotically remove way deal discretization comparison previous implementation entropy estimator toolbox entropy sp shannon sp shannon sp sp sp shannon shannon shannon shannon also make toolbox release near expansion detail toolbox gaussian take kernel asymptotically see also permutation gamma mean nan hypothesis estimate description benchmark set cause consist extension eight form competition publicly appendix cause justification ground scatter plot scatter plot cause effect collect benchmark ground decide ground straightforward method process ground simulate way realistic plot simulate look world idea structural want similarly standard normal causal process measurement default level expect finally gaussian approximately nonlinear cause expect scatter scatter scatter cause scatter scatter effect scenario standardize affine transformation empirical perturbation perturbation perturbation variable discretization discretization repeatedly cause merge add small add ideally robust perturbation estimate direction affect real effect weight pair come weight correlate age whole age curve whether accuracy increase evaluate force visually interpret significance experiment carry plot indicate weighted confidence interval confidence indicate evaluate bandwidth bandwidth datum splitting entropy reporting show simulate benchmark variant estimator different perturbation benchmark variant show obtain accuracy way additive measurement effect result turn perturbation additive misspecification originally size represent decision value direction use value suffer identical perturbation discretization slightly show performance base noise depend detail combine generally little differently consistent standard accuracy set right figure entropy six variant perform result nonparametric simulated exception varie discretization effect entropy treat occur multiple example occur lead chance level majority quite nonparametric well datum seem perturbation mml score score score chance probably typically violate accuracy satisfy scenario evident perform scenario employ measure setting practice perform scenario probably mml simple measure benchmark perturbation parametric well accuracy simulate behaviour understand match actual distribution datum simulation setting report variant show variant accuracy perturbation base accuracy accuracy perturbation bottom gaussian measure base accuracy around chance gaussian base low chance high chance require reference measure scenario
already process increment operational environment arrive fix half entire training feature human regression subset performance pearson coefficient comparable regardless train smoothed average model sample random replacement random well baseline change increase informative margin baseline small show percent fisher transform level test unimodal relatively quite interpretability percent calculate pearson mean anomalous achieve typically select select see htbp average perform prominent enough illustrate notable deviation great bad figure robust human well human validation production reach noticed entire perform optimal cause entire poorly also poor misspecification misspecification semantic human primarily avoid misspecification evaluate uninformative report generalize task promise score less answer answer employ tree forest model know use consequently row matrix row relatively active sampling agree subsequent supervise thus short human supervise forest train human maximized minimize perspective primarily human quantity enable safe solve large allow computer human effectively human score automate writing effective quickly adopt large scale context human requirement costly evaluate ensure consistently informative training maximize thereby reduce cost discussion integrate automated writing language statistical create automate scoring e english even wider automated evaluation recently length answer help platform massive contain automate writing train trait batch language input learn vector previously process require least requirement system human cause system allow yield perform adopt technology per paper barrier context example enable system choose human effectively integrate method literature choose sample old space optimal clinical study optimal design sample vector variance begin label unlabele paper active design choose prior learn good assumption match regression ask really active stick mostly terminology literature score automate assessment foundation set summarize ask write computer letter set read text evidence grow must tend experience important element target score range target predict set performance quite suggest student focus algorithm mechanic lexical style range b source automate student narrow determine inherent variable design number allow since feature eight ordinary square solution would suboptimal regularize ordinary square ordinary augment usually ridge penalty irrelevant towards since linear integer map back onto value score determine adjacent reasonable threshold use candidate wish give predictive score formally lack row predictive regression result consequence choice illustrate value great perform concept feature algorithm choice constitute maximally distant uniformly distant implementation implementation implementation exchange algorithm purpose design criterion optimality maximize set difference tend somewhat exposition limit optimality scoring randomly replace row row optimize algorithm optimum one maximally distant centroid see choose vector index q subsequent add initialization exist mahalanobis feature design distant distant another cluster extra final
c ic cost happen one select worker worker involve select set bid tc apply transformation post post transformation ex monotone allocation randomize post separate lemma constraint cost uniform probability round give number round round round round hence turn exploration e tt ucb difference ucb regret situation expect combinatorial solve optimization problem naive implementation ns lead computational complexity underlie combinatorial due able approximation algorithm monotonicity rule solve minimum problem np approximation problem ensure select general quality worker combinatorial eliminate worker eliminate elimination incorporate approximate return monotone give rule essential compatibility monotone allocation ic ic algorithm arm explore worker probability worker target decrease exploration select emphasize se identify worker worker cost require choose arbitrarily adopt ucb target check target ensure extra worker rest cost worker bind check run none violate average greedy true sample ns reduce iteration worker figure greedy number worker private worker aggregate select worker attain target novel setting develop constrain confidence bind post individually rational mechanism exploration depend inherently exist approximate monotone ir interesting research convergence attribute exploration separate solve optimization strategy example generalization possible require assumption soft form future thm identical labeling worker outcome aggregate problem challenge even develop accuracy mab constrain non upper algorithm worker cost ns adaptive call post allocation compatible individually give also select bind upper insight illustrative efficacy financial security company pool opinion aggregate majority probability increase company business provide minimum threshold set private report service threshold company learn assume learn give homogeneous abstraction homogeneous another incur design aggregate certain provide value level right answer call agent cost therefore sensitivity goal select subset give agent absence play know reduce learn worker minimize cost though worker worker significant cost thus face versus choose optimally learn natural multi mab work crowdsource challenge ensure unknown address try cost mechanism ensure theoretic learning cost simultaneously need game learn mechanism refer mab mechanism induce achieve require accuracy highlight solve problem select worker target optimal solve first version learn paper propose framework mab call ns make sure probability suboptimal worker set level true achieve match may modify ns separated confidence prove ex post cost post adopt technique ex mechanism separate exploit simulation knowledge learn agent information mechanism crowdsource reverse summary next stage discuss section mechanism extension incorporate worker avoid high cost exploration section variant mab address version crowdsource learn worker answer crowdsource provide natural review mechanism crowdsource quality satisfied hold worker certain meet micro aggregate answer worker assume crowd general number worker arm heterogeneous select go micro assign task task quality et predict analytic guarantee predict formulate oppose subset worker none address challenge also cost mechanism crowdsource literature crowdsourcing involve price online worker mab determine pricing crowdsource homogeneous know assume cost private information propose price mechanism mechanism maintain online consider consider heterogeneous et involve people market crowdsource winner worker period adopt mechanism preference task theory crowdsource either homogeneous assuming set worker learn mab rich body literature available mab problem far moreover satisfied oppose satisfied round probably pac closely pac subtle obtain approximately arm round provide approximation optimal set high satisfied respect moreover exploration round mab chen wang relevant pure combinatorial subset feasible learn mab discuss arm reward round exceed instead armed mechanism set combine area mab mechanism phase regret round arm slot develop adopt make click multi oppose traditional armed arm constraint consider forward mechanism procedure monotone rule randomize input allocation allocation rule exactly mab post transformation post monotone allocation rule reverse bandit setting propose translate accuracy worker preliminary appear ensure accuracy current improvement crowdsource worker work homogeneous crowdsource worker associate labeling worker quality task assume service quality worker incur target accuracy determine label l ex round confidence require worker aggregate worker right quality task ucb task bind task worker worker incur input depend rule aggregate abstract let capture select profile seek requirement monotonicity say profile I decrease bound smoothness satisfie increase continuous profile difference error profile continuous error probability next monotonicity majority aggregation monotonicity smoothness player task report label vector label label majority voting rule aggregation likely outcome lead worker mistake satisfy constraint respect give monotonicity focus assume verify smoothness simplify monotonicity describe recall fashion thus goal worker threshold make follow worker priori learn repeatedly also solve accuracy bandit mab typically measure achieve later way satisfied thus algorithm satisfied probability expected incur satisfy large would involve regret start important property profile follow separate sf framework suffer time worker algorithm reward function quality quality profile worker profile respect come profile change event markov eq worker profile worker get version strong law number fact suboptimal optimal arm number various mab ucb et mab arm maintain exploration number regret increase ucb reward arm work monotonicity satisfied reward confidence arm similar ucb high algorithm function know make algorithm error monotonicity bound smoothness assumption interesting assume label observe complete motivate trade company satisfy learn algorithm select complete enough worker worker assume publicly know learn since worker accord true satisfies smoothness algorithm black box aggregate aggregate label opinion aggregate aggregate voting aggregated ns ns ns worker confidence ucb select worker f explore worker observe true label k ts tt observe subroutine minimal return minimal present work ucb constraint input worker level predict decide worker set initially estimate report next observe assign similar maintain bound bound hoeffding prove lie worker since constant worker lie key worker effective meet add subset subroutine minimal accuracy meet even low error target round find upper confidence even use stop task minimal accuracy simply ns satisfie hoeffding worker I q monotonicity ns equation round assumption satisfied ns algorithm set set rest unique though easily return p round ns stop explore ns solve optimization eq aim non say round say round set optimal exploration algorithm set task bind overall regret round e get round f unknown algorithm require ns adaptive let h hoeffding n n select worker ns exploitation optimal whenever tu tu far exploration round total eq incur satisfied ns bind assume proper cost section call monotonicity theoretic worker agent allocate worker satisfy smoothness denote incur constraint equation eq linear every heavy incur wrong thus compatible say compatible report dominant worker rational worker always utility characterization mechanism provide player generic transformation take output compatible design mechanism set set monotonicity allocation
sampling motivation observation count sparse count discuss later applicable inference fast start dense maintain nonzero fractional term advantage construction linear procedure benefit contain mass one topic via synchronization scheme datum introduce literature fail handle huge inference mention big exceed ram share parallel inference correctness performance update heavily condition evident synchronization share variable statistical huge copy load place help memory worker share disjoint motivated addition specifically topic word outcome straightforward scalability allow share burden replacement flexibility inference rather model lda realize algorithm block block assign correspond worker worker token sample block worker another iteration worker block sample process synchronization share mainly asynchronous incorporate worker effort synchronization worker construct incorrect become even evident cluster cloud service parallelization proceed receive task request block carefully worker store server purpose distribute memory partitioning frequent background asynchronous simple hash implementation suffice dynamic partition strategy demand communication key store round key store task thereby global accelerate communication block block synchronization communication synchronization combine dynamic demand dependency frequently fast token fact method much complexity another count impossible term value denominator change final worker round highly receive vector key store worker aware change sense similar relax requirement major maintain show compare actual empirically negligible combine demand communication avoid parallelization protocol separable dependency magnitude time implement design illustrate partition worker generate assign coordinate worker maintain special key value store distribute synchronization requirement job worker scheduling constraint consumption acceptable since omit hardware equip specifically high network interface equip ghz gb ghz ram machine corpus wikipedia word token original word token phrase occurrence phrase vocabulary demonstrate scalability size topic extremely case surrogate measure lda sampler optima topic progress rise measure gibbs unlikely local reach might ask employ surrogate practitioner improper evaluate goodness model evaluate alternative generalize new system generalization issue learn introduce factor optima sampler optima control external measuring inference end parallel reach figure log trend similar dynamic partitioning suffer begin copy progress iteration almost synchronization worker leave round relax requirement minor huge count affect overall proxy copy worker round token local copy lie collect observe drop stay close procedure demonstrate parallelization error fast ability big table yahoo lda vocabulary copy long memory model big able perform indicate ability yahoo lda indicate parallelism big sized demonstrate effectiveness partitioning strategy c yahoo cluster memory ideal memory consumption observe parallel nearly ideal scalability start drop much indicate storage unnecessary yahoo usage parallel word machine machine big also machine different yahoo node traffic increase bandwidth contrast speedup closely inference effectively utilize resource significant overhead demand strategy parallel inference traffic guarantee ability inference handle end paper present parallelism implement parallelism efficiency parallelism process bring capability big metropolis speed already significantly block broad attempt parallelism investigation interested parallelism challenge dirichlet hdp regularize big cs application topic conceptual high become next especially grain online usually million conventional approach topic inefficient heavy centralized pose small ram address another parallelism namely parallelism enable integrate parallelism parallelism distribute element ability tackle collapse gibbs algorithm experimental computational ml advance technology big massive various ml resort parallelism task partition pose mild synchronization assumption valid huge file database suit parallelism due element share variable entity model clear persistent converging need convenient machine program model large basis topic basis couple normality negativity estimator must procedure treatment dependency parallel explore sub decompose strategy evident synchronization logical correctness cycle scalability fine grain variable early engine grain mechanism parallelization case prevent asynchronous update art lda yahoo lda advance server extent little correctness study theory support model update parallelization improve parallelization pose accommodate handle unlike convention application modeling instance online beyond extract topic visualization scale vocabulary topic readily available copy vocabulary require real feature augmentation word large conceptual raw model storage issue type parallelization parallelism parallelism parallelism update dependency among specifically make sampling share small computation find subset completely assumption block exactly result serial inference requirement handle modeling end model parallelism program parallelism parallelization
gradient algorithm smooth partly fulfil manifold identification light small bregman divergence linear book overview sharp extended prove term functional see literature state I condition lasso author prove though similar establish author nuclear norm note invertible restrict condition operator cover total fuse lasso generalize partly operator cover dimensional variation equal also performance family decomposable norm nuclear show high random operator noiseless ensure completion noisy signal level minimizer sensitivity non seek ensure usually hence assess stable sense stay notion partly smooth existence manifold partly behave identifiable move manifold behaviour smooth sum sufficiently locally partly first equality fact cone vanish u manifold class deduce p p derivative recalling manifold use constant manifold show together arrive accord lemma state normalization homogeneity sub simplicity slight abuse approach since continuity continuous applying lead contradict fact classical fourth moment finite scale partial relative enough partial smoothness partial partial particular smoothness mapping partial subdifferential continuity property smoothness solution particular define continuity subdifferential contradiction unify performance partly problem function popular regularizer use feature generalize guarantee acknowledgement author er european sigma vision least square partly convex class function solution notion force solution problem dim manifold make low generalized tuned level regularize tend regularize correct dimensional manifold generalize statistic operator science recover inverse impose solution consider value prior control amount follow canonical without stand pseudo inverse goal e understand close noise stability identifiability associate body literature include regularization turn special general theory partly smooth subspace hull set interior interior affine hull affine contain guarantee partly smooth originally hereafter contain continuity continuous say partly neighbourhood partly unique regularity proper discussion convex continuous subdifferential everywhere automatically verify continuity converge converge characterization popular smooth regularizer use imaging check partly smooth literature basis pursuit literature name capture sparsity overlap block group group typically therein detail partly value impose understand rectangular completion nuclear generally consider function partly smooth smooth proved matrix absolutely invariant define piecewise constant image sparsity enforce partly partly manifold start design impose sparsity use see partly partly force convexity state main introduce stability enough linearize pre deterministic show robustness perturbation design locally partly smooth constant deterministic vs typical one statistic machine consider regime row show hypothesis close sharp characterize solution
engineering examine split nonconvex direction multiplier alternate method multiplier sufficiently stationary nonconvex additional algebraic furthermore condition guarantee boundedness satisfy wide include identity gradient efficiently apply optimization problem twice value well element minimizer indicator engineering regularizer norm regularizer use nuclear therein proximal mapping stochastic linear block discuss map splitting apply proximal generate convergent globally choose modulus nonconvex cluster proximal apply one feasible alternate apply admm nonconvex particular author measure show square successive change iterate case nonconvex sum euclidean show iterate assumption motivate admm identity contribution characterize cluster generate admm replace admm approximation variant subproblem involve solve quadratic furthermore boundedness generate condition satisfy algebraic cluster actually semi verify recognize cover concrete admm show nonconvex statement preliminary material next devote proximal numerical concluding remark inner denote induce denote identity map adjoint product map semidefinite nonzero resp real value equal proper fx limit subdifferential immediately subdifferential subdifferential subdifferential z continuously subdifferential variable subdifferential resp respect general subdifferential enjoy base principle particular solution optimality always throughout continuously bregman continuously word union finitely strict inequality semi algebraic semi semi algebraic nice structural property proper continuous differentiable hold satisfying property proper close algebraic function study direction multiplier nonsmooth follow continuously termination criterion subproblem call subproblem proximal hence popular remark study suitable hx x say would chosen least continuity modulus pick interest update second subproblem pick hence iterate hence convergent subsequence stationary definition pass follow global conclusion point sequence admm produce quadratic indicator euclidean convergence admm proximal establish work idea admm note subsequently change primal like modification directly subproblem introduction establish lagrangian existence hand boundedness sequence general enough literature study scenario suitably initialize get strict improvement suitably initialize stationary stationary strict improvement initialize sequence decrease proximal admm stationary choose initialization approximate close obtain relaxation stationary point relaxed initialize observe take norm I obtain assumption make relation definition establish q operation preserve semidefinite point modulus minimizer summing eq subsequence put pass make use conclude desire minimizer see py py discussion precede stationary suppose initialized chosen end proceed consequently since must lagrangian combine recall conclusion inequality convergent modulus one take third ii choose picking suppose vector adapt apply algebraic suppose sequence admm converge stationary stationary point establish subdifferential py together constant decrease subsequence l notational minimizer relation continuity subsequence together lower imply combine existence claim furthermore decrease x z k kk meaning terminate finitely conclusion terminate finitely next function definition pick hard I dl fact next concavity eq divide take rearrange hold induction proof inequality consider q inequality monotonicity induction moreover hence claim relation complete comment inspection show theorem continue augment lagrangian property read z property case least interesting assume suggest experiment preliminary admm solve concrete convergence nonconvex set admm map problem reformulate admm let denote multiplier constraint iterate take ambiguity update nonconvex iterate initialization routine show cycle x k convergent successive change look nonsmooth mapping proximal backward update q hard stationary convergence flexible size exist continuously moreover indicate restriction small modulus allow continuity modulus definition subdifferential apply descent rearrange convergent subsequence along convergent x px px px px take limit along convergent subsequence conclusion concern hold example continuously function modulus concrete hold semidefinite open step lipschitz know lie clear lipschitz continuous modulus step choose h hx hx tw px n conclude cluster exist incorporate difference initial backtrack search perform numerical code matlab experiment bit intel cpu ghz ram matlab close present full count proximal admm admm guarantee latter solve admm obtain convex condition successive change hand generate instance relaxation report distance report cpu initialize origin allow violate always close obtain close initialization cccc ccc admm cpu e e e e e signal indicator slow heuristic generate initialize terminate occur benchmark solve use generate piecewise specifically matlab r consider computational cpu second cardinality error original always correct piece always original noiseless cm ccc ccc cpu next present visualize recover signal via
ray voxel row positive voxel neighborhood system twice continuously preserve edge choose class due quadratic linear standard come smoothed different determination ard originally ard likelihood diagonal treat evidence refer posterior interestingly many concentrated zero readily show concentrate concentrated mechanism specifically tailor likelihood type originally expectation em base step oppose mode ard variance ard computationally demand method mainly task extension ard approximation despite ill significant advantage ard avoid nuisance trade variance experimental design scope ard conjugacy ard poisson law conjugacy step equal hessian map result ct high inverse hyperparameter irrespective laplace principled lack objective conceptual difficulty reveal ard contribution extend determination ard convergence reveal ard preserve similar gain previous likelihood transmission adapt reduce considerably simplify parallel search pixel voxel analysis scalable guarantee global imply guarantee world ray rest ard poisson likelihood surrogate feasible sec analysis property algorithm connection method present ard model sec real parallel present sec transmission directly basis transform newly hyperparameter choose scalable discrete fourier wavelet transform ray ct explore fig usually row voxel average pixel e boundary neighbor outside domain boundary condition neighbor voxel agree many zero voxel piecewise smooth difficulty trying extend ard gaussian evidence q variational evidence low kullback kl divergence free maximize summarize solution zero kl minimizing kl nonnegative evidence increase likelihood e expression direct calculation accord integral prohibitive force posterior long evidence kl either decrease increase update evidence local except consider ard previous ard gaussian immediately clear extend ard poisson preserve gain ard surprisingly variational discuss still principle ard despite evidence bind call ard mention step perhaps common distribution mean variational conditional restrict addition restrict distribution univariate propose modify em two step repeat rewrite omit backward step reduce long distinguish similarly step provide estimate negativity add define form repeat reduce iteration increase importantly highly make sec drop forward operator deriving assume invertible square provide sec remove recall objective feasible practice ard evaluate prohibitive approximated straightforward verify fix jointly imply minimum jointly minima guarantee evidence understand somewhat estimation sparse explain kl transformation invertible exploit duality monotonically side obtain sharp envelope ard illustrate fig latter level kl shrink illustrate much mean concentrated posterior require responsible noise approach energy apply even expression evidence unknown gaussian noise force work al consider original ard backward level curve prior curve concentrate chosen kl student dash black form envelope mean prior posterior around fig domain lie simultaneous task compute scalability separable surrogate q zero th entry iff readily jensen jx j proceed definition simplify derivation define variance quantity associate th call type projection posterior type iteration back projection respectively jointly make separable denote set involve way modification requirement em instead equality obtain since need repeat separable lemma substitute estimate pixel voxel difference max deriving combine q surrogate step iteration step parallel describe initialize projection g mean forward projection require variance compute iteration line search component line search voxel variance parallelization provide mean share algorithm iteration algorithm number considerable advantage compute computer dedicate hardware gate array constrain obtain unconstraine problem sub solution measurement voxel complexity neighboring pixel voxel typically detail implementation search line give present convergence sec refer tucker kkt optimality solution give kkt kkt minimizer subject replace replace definition plug substitute remark update end iteration necessary kkt necessary kkt condition solution follow kkt plug solution let negativity solution introduce variance shall em necessary kkt substituting point reduce condition monotonically iteration par iteration divide done state assume ti term side sequence limit iterate zero imply par solution kkt iterate contain guess theorem positivity compact satisfied divergence kkt convenience proposition continuous map nonempty continuous existence follow line set several also constitute mapping therefore closed point composition close mapping close convergence present view sparsity minimum obtain formulation differential std domain note term without one global combination term serve avoid separable ard reweighte likelihood consists follow tune provide equation fidelity respect approximate difference variance serve pixel voxel remove substitute minima penalty easily avoid correct critical facilitate comparison reweighted sec correspond surrogate likelihood penalty reweighte consider iteration replace execute f vector estimation mle scope reduce mle surrogate close update surrogate decomposition execute parallel j shall huber huber pixel q th map q approximation posterior poisson b factorize emphasize principle ard alternative et inversion voxel feasible approximate since super super gaussian building sec extend sparse overcomplete square considerable update function take pixel voxel prohibitive problem objective rectangular objective long interpret due replace since independent minimizing minimizing kl interpret kl formulation although interpretation concatenation locate horizontal image horizontal pixel difference replace difference original neighbor pixel original correspond index respectively variance vertical axis neighbor ray ct medical ray avoid level intensity typical clinical par posteriori huber reweighte notation slightly par choice complete complete representation implement execute intel e cpu core run platform implementation search parallel core objective source intensity clinical angle full object detector measurement square map tuning reweighte one tuning run trial rmse comparable reconstruction table observe high much rmse properly choose important object rmse consume merely reweighte reweighted reweighted horizontal std realization negligible std reweighte reweighte estimator tune low rmse image visually map reweighte find fig occur tune significant fig reconstruct show observe region recall domain pixel std careful std posterior objective method image display include enhance frame mle use penalty vertical iteration recommend variance run pixel initialize reweighte initialize fit first objective iteration objective predict theory use assess practice reconstruction acquire view angle detector use setup section specification water inside air reconstruction different image reweighte show reconstruction produce reweighted repeat reconstruct correct pixel difference fig letter object mid air experimental letter horizontal one neighbor recommend figure objective monotonically water angular grid view reconstruction case fig include fig reconstruction filter reconstruction fourier reconstruction prominent view reconstruction fig display range enhance std fig reconstruction defer publication lastly fan geometry geometry search parallel line search lead whenever regime huber use trust appendix execute newton trust appendix newton work dominate region take converge minute trial reweighte take store transpose region memory back projection could take portion search lead comparable back iteration resource limit slow map reweighte twice trial low std view also determination ard call ard ard use transmission mean reveal mechanism establish previous ard important avoid tune good
important tuning likewise nuclear minimization enhance code original algorithm access code detail general tune noiseless result additionally emphasize superior clutter presentation limit variational limited completion publicly focus nonetheless conduct experiment thresholde inferior avoid presentation separately comparison fr begin reproduce hide uniformly trial percentage across varied capable limit beyond number freedom represent candidate pool tune author although parameterization entirely benefit strong theoretical always match art display motivate reproduce completion design superior generate varied evaluate reconstruction combination value reproduce challenging case superior reconstruction difficulty fr fr define blind rank result previously mean failure achieve next arbitrary constraint nuclear minimization type mapping ii latter condition operator display somewhat ill condition display include comparison case vary consistently able general explore range rarely always theoretical boundary explore actually np recovery failure certainly circumstance possible scenario probe carefully condition difficulty reduce measurement measurement even fix reduce exactly equal degree examine uncorrelated far examine failure singular case notice classified error state threshold almost correct hence nonetheless feasible theoretical maximally size feasible spectral indistinguishable importantly test fail much apparent motivate success account respect relative percentage trial whereby find denote trial new criterion failure become involve actual rank solution completion involve reduce fr break result perform besides metric achieve fr adopt limit reveal failure specifically dct coefficient process linear information figure replace purpose thing stand metric failure mostly rank secondly dct outperform report avg summarize demonstrate capable theoretical limit process even feasible nearly suggest failure failure tend near display test situation reveal nonetheless promising noise reproduce design observe observe although heuristic four adjusted value report exhibit superior class updating generalization performance limit special circumstance comparison true rank knowledge specifically algorithm test introduce priori nuclear rank match especially bad decaying phenomenon multiply th large decay drop finally regard complexity scale completion exploit difficult recovery increase though limit highly overcomplete show effect relatively versus world formulate consider collaborative recommender former latter observe basic idea order taylor approximation around estimate jacobian transformation accomplish project side feasible result original nuclear norm term simplify comparison result small successful significant transformation estimate collaborative technique recommender user entry item task collaborative knowledge estimation recommender system per appear strict validity assumption remain entirely unclear globally low observation computable necessarily lead fact report tend around almost imply provide discriminative type compare heuristic modification underlie algorithmic estimate completeness adopt offset weak image red transformation derive show comparison wide strict dataset million rating assess test ability rate item select user strong generalization three performance metric minimum result include algorithms gm generalization well course apparent fall narrow make optimally necessarily translate truly practical collaborative argue explore conceptually matrix affine capable broad nuclear norm break adopt principled justification entirely theoretical local empirical exponentially regard nuclear appendix brief lemma address compressive presentation basic aspect adaptation minimizer must infinity infinity span else objective drive infinity constraint constant ultimately maintain unbounded statement objective rank behind collection secondly ensure assume r scale minimize rank complete sketch column indistinguishable achieve theorem become note minimum positive infinity I drive construction involve first likewise take I kk great exceed consequently negative reduce except moreover consider shrink generality translation condition great preferred unless display display minima reveal counter iterative bound tailor symmetric likewise via practice sufficient obtain good application require recover minimal affine constraint notable special replace nuclear act convenient elegant theoretical replacement restrictive fail ambient high constraint poorly non alternative carefully tune locally failure like wide empirical theoretical measurement unknown rank surprisingly possible affine ill prove nonetheless condition whereby point locate optimum exist involve completion recovery subject involve mapping commonly apply collaborative problem low np rank non smooth consequently alternative denote concave nearly special retrieve surrogate prefer reduce norm quantify condition heavily matrix nuclear coincide minimal restrictive convex broad art operate constraint follow derivation technique adapt pca describe connection norm issue special whereby stationary discuss algorithmic performance contain efficacy image collaborative filter proof rule appendix proceeding take offer little substantial gain tailor modification probabilistic lead systematic analytical empirical insight justification consideration merely qualitative underlie convex avoid solution inspire require balance minimal truth include direct head design code original carefully tune even algorithm never demonstrate previously rank develop evaluate attempt locally derive maintain rank behave scale albeit well perform minimize homotopy merge replace function minima progress reasonably spurious avoid procedure pre reduce schedule specific solution ever derive apply function unlike suggest tune different class unclear choice substantially norm seem slightly quadratic less stage optimization trajectory minimization limit equal degree recover hence good boundary practice ill achievable nuclear truncate convex image via contain setting poorly somewhat non derive alternate low matrix solve approach require parameterize contrast emphasis regard experimental aware embed low rank typically feasible tune previous rank minimization affine constraint build summation element penalty apply iteration analysis affine apply similar model completion competitive state mention intrinsic focus challenge consequently general estimate refer particularly solve problem first close intuitively discuss desirable global minima conclude brief convergence minimize minimization reveal function nothing suppose apply term equivalently solve demonstrate optimal simplify nuclear arrive limit become conclude distinction nuclear intrinsic section convex substitution probabilistic function technical duality compressive please penalty approximately view still minimum constraint become possess attractive invariance mean rescale rescale optimum optimization much surrogate local defer small feasible block invertible block iff likewise correspondence rank theoretically restrictive require require likewise essentially guarantee possesse optimum rank limit indicator via argument certainly adopt function minimize rare largely process provably specialized quantify suppose diagonal restrict nonetheless case include generalized element instead block furthermore satisfy always global minima great imply cost function minima condition intersection merely rank simultaneously measurement still highly ill condition minimizer globally standard typical rank additionally unique one solution crucially minimal unlike true underlie importantly
emphasize complement either numerical quadrature mcmc number algorithm g riemannian sample correlation well mcmc posterior hx hx rx give rao estimator exploration update rank achieve effective argue involve introduce since replace reduction particularly space mcmc subspace offer additional advantage apply full low scheme full method root stochastic newton handle full much handle carlo product function either multiplicative sum expectation subspace numerical particularly useful example analytical reduced variance reduce reduce reduce estimate result constructing require much ensure capture variation choose solve markov mode project follow adaptively dimensional last state construction basis checking terminate evaluation fall return incremental distance inform heavily basis consist compute distance sample adaptively exploration might ignore direction inform update complement would construct numerical describe good construct posterior give essentially course algorithm sample mcmc pde inverse demonstrate construction mesh mean vary observational pressure pressure govern pose boundary boundary impose superposition width center corner bilinear endow log normal prior I prior length true pressure synthetic h pressure field collect black dot figure observation operator correspond mask operation deviation prescribe ratio show draw prior prior run order slow spectrum truncate frequency unless retain construction refinement mala simulate threshold dimensionality versus refinement carry coarse grid iteration diagnostic discretization h sample use black blue marker grid grid show show subspace h level discretization column order rapidly reflect log observe grid slightly grid effect discretization grid weight adjacent inform diagnostic order magnitude begin rate convergence diagnostic three level suggest local variation course explore refinement grid show figure subspace grid refinement refinement mode close hand reduction describe pde mala hessian full space hereafter result discretization langevin sde inverse sde empirical posterior mala dimension setup examine project onto vector result figure discard burn row benchmark mcmc produce decay autocorrelation use run mcmc iteration cost full difference second benchmark cpu immediately observe autocorrelation per cpu time reduce course construct roughly mcmc step cost h subspace result distinguish field central measurement sensor variance right region carry affect structure demonstrate likelihood sake computational mesh impulse system section evenly center place domain distribute evenly domain inter refine four pressure sensor algorithm spectrum note eigenvalue decay amount reflect impact lead direction area domain inform subspace however frequency might differ basis corresponding eigenvalue share similar eigenvalue different pattern carry basis realistic use stand year lose different intensity spectra inversion infer ill spectra minor totally inform small briefly theory setup reference transmission spectrum measure ray model height call cross section laboratory measurement discretize inversion resemble density assume within spherical height inverse layer fix layer approximate integral choose n geometry contain length line layer cb stack top know variance note linearize measurement synthetic solve profile discretize dimension simulated profile profile denote matrix value prior choose profile rough magnitude density know well totally construct diagnostic threshold compute hessian mala figure profile standard complement column show horizontal axis apply addition contribution cs entirely determined contribution low mean result avoid accurate illustrate plot six basis mainly inform expect mix space space mcmc compare mcmc test computational mcmc simulate subspace mcmc second cpu second algorithm iterations cpu cpu approach inverse approach divide subspace inform posterior distribution complement dominate explore project onto gaussian problem chain treat complement estimate expectation rao randomization greatly reduce particularly solution heavily dimension majority handle analytically approach show update generalize vary inform subspace global adaptive meet first pde flow parameter remain explore analytically treat produce via full computational dramatically problem infer chemical star dimension full specie appear offer exploit inference algorithm curse reduce order applicable acknowledge provide code sense support office advance scientific de sc sc example department usa intrinsic inverse affect relatively identify inform characterize influence support identification efficient bayesian chain monte sampling low dimension monte expectation variance pde sense monitoring carlo arise indirect parameter high moment quantile event parameter dependent prediction quantify posterior carlo mcmc affected dimension degradation increase higher posterior recent share scaling argue randomization estimate explain propose inverse identify subspace notion nonlinear approximation develop case reduction combine likelihood inform wherein independent datum particular approximate dimensional marginalization complement benefit evaluation expectation likelihood inform enable great efficiency step allow complement inform avoid analytically condition expectation previously way construct truncate expansion likelihood hessian log inverse problem nonlinear construct stochastic approximation mode either stochastic tradeoff hessian posterior proposal proceed low dimensional projection enable rao mcmc posterior amenable integration present mode choose orthonormal precision matrix preserve important seek form determine inform thus inform embed manifold aim global majority nonlinear inform forward linearization forward provide sensitivity observable inspire linearize newton approximation hx jx jx
describe descent practitioner decade inherent conceptual simplicity ease work code largely ignore recently report remarkable come understand cyclic greedy next coordinate increasingly clear complexity analysis link random describe subset coordinate select iteration capture smoothness useful function admit would allow would column identity denote property unit coordinate context descent focus extend reader inequality eq hadamard term hand side latter hessian importance descent perform design nontrivial deal design search influence complexity see table updating coordinate update lead perhaps resource whether understand study complexity dependence vector soon satisfy lead natural coordinate descent variant coordinate accelerate study study deal algorithmic aspect aspect mention employ serial sampling serial appear bound directly year nonsmooth primal n alpha method arbitrary apply serious problem function lipschitz regularizer nonsmooth arbitrary thing accelerate enjoy slow uniformity variety basic overlapping overlap nice serial parallel non serial sampling assign distinct necessarily lead bound speedup product associate intuitively speak sample linearly sample far almost inequality recover general give large submatrix briefly terminology matrix parameter inequality describe paper section review elementary satisfie however often entry eq functions require help assumption randomize coordinate descent method assume pick jx h jx inequality identity matrix element proposition matrix separable function coordinate gradient appear formulation th coordinate eq eq q form role design accelerate function appear dual apply set value value terminology shall never never coordinate descent key elementary elementary elementary associate intersect refer reader condition necessarily every sampling uniform additional uniformity property name doubly prove notable doubly pick uniformly give refer standard mini nice sampling arises distribute coordinate nice q nice sampling define nice pick subset uniformly basic combination intersection nonnegative scalar sum accord elementary doubly arise nice let sampling doubly nice statement definition intersection eq sampling necessarily eq collection constant add I via eq consider hadamard formalize independent ij ij restriction several alternative writing keep distribute sampling definition note belong partition nice sampling nice finally doubly uniform q nice note semidefinite diagonal denote normalize eigenvalue recall semidefinite resp resp quantity later useful compute I since elementary simple see consequence es cauchy matrix since add give sharp bind identity simplicity identity elementary whenever result combine upper study quantity statement maximal I bind elementary tight view normalize eigenvalue associate family rough sampling doubly mention upper apply proceed fix nonempty intersection eq apply q cauchy plug j obtain nice sampling let nice nice sampling apply calculation e give large doubly doubly develop hadamard product semidefinite study bound hadamard hadamard eq substitute expectation next direct consequence regard reasoning lemma sampling statement sufficient view hence pick similar class partially separable function arbitrary degree separability correspond see normalize consuming pass prohibitive next follow one issue avoid decompose different vector individually recall set coupling th j completeness equality finally theorem proposition conclude matrix large eigenvalue upper proposition see illustrate precede eigenvalue computable sampling way distribute sampling doubly sampling sampling serial direct vertex remark part improvement quality involve part compare small well admissible effort lead nice admissible dedicate formulae appear admissible computing size pass return approximate apply semidefinite number number notation pass n parameter strongly setup time epoch pass pass convexity formulae report big preprocessing compute formulae normalize use product partition method multiply value processing formula formula also magnitude take pass formula enough convexity
spectral previous scheme could boost spectral approximation show leverage let set leverage estimate follow iterative immediately enough allow sample matrix row cut recursive give clean argument prove version slightly theorem technique believe leverage score actually sufficient obtaining row powerful coherence row specifically coherence intuitively give describe exactly leverage row coherence spectral reweighte thus reweighte score uniformly sample never great sum score reweighte score row reweighte trivially leverage score leverage need obtain row score bound ensure spectral frequently lemma simple randomize numerous multiplication linear helpful survey runtime gain alternative algebra tool gain pattern require linear algebra process algebraic roughly multiplication approximate select projection challenge correct focus reduction step approximate randomized scheme combine require projection subspace go back recent progress significantly iterative specifically spectral incidence commonly primitive graph graph potentially row leverage spectral ensure reweighte vertex incidence matrix preserve row level accelerate edge incidence evaluate li fairly ultimately projection row preserve converge step r diagonal let singular spectral approximation imply preserve multiplication consequently singular leverage row also define leverage score leverage orthogonal row would change leverage row coherence removal affect composition row characterization help intuition optimal entry must constraint ix j furthermore score compute leverage pointing row remove could simplify approximation generalize score multiplicative leverage spectral lemma score fact exact score suffice I let denote otherwise completeness result argument diagonal valid leverage score indicator prove trivially formula fact bound break select always process select return score leverage score low fundamental proving prove study conjecture otherwise bound satisfy set exist total incoherence show reduce prove prove score evolve reweighte decrease leverage row leverage score rank update diagonal next claim row allows arise continuous ki place ready main require consider weight see decrease give lemma score non construction u ii increase leverage row remove row enough variety approximation however slight bound correctness sampling score intuitively coherence portion loose leverage score estimate score leverage leverage upper set bind bind come require sampling give probability statement match leverage leverage show hence choose accordingly clear gives start rate score cut leverage restrict leverage converge expect keep cut sum far correspond differ early g algorithm maintain row iteratively leverage enough row score summing cut row eliminate consider zero score obtain row c reduce sampling actually sample row rate computing return match introduction theorem yield extremely simple clarity initially present version first art solely preserve improve usage system recursive estimating row course compute leverage input matrix output spectral rescale make show set leverage quality leverage round obtain row output rescale reasonable fact solve time exponent emphasize trade rescale think fast primitive showing leverage efficiently compute generalize leverage rescale di obtain idea within multiply height score solver explicitly slight need example primitive analyze runtime simplicity spectral furthermore leverage constant row runtime refinement induction cut uniformly cut instead leverage increase generalized score increase another constant row runtime score termination terminate iteration factor decrease technique idea leverage use rough score respect row row finally leverage take come refine entry dd term hide tradeoff algorithm summarize use note matrix process recursive call output rescale note incur create result generic simply call sufficiently recursive modification head algorithm head recursive giving tail recursive w step error factor sample w give situation likely still give thank helpful discussion support nsf fellowship grant grant fa advanced project probability score random matrix independently concentration corollary semidefinite ip desire eq consider u trace cyclic semidefinite q equation directly ic lemma random concentration probability nonzero leverage change update diagonal eq
biased toward predictor ranking rank rank highly desirable find structural different ranking discover create aggregate link pair criterion whole consider ranking pair label fact output select ranking go name rank simultaneously ranking goal true value indicate contribution merge ranking predict pair link step exactly high true prediction come purpose link link window highest select throughout slide window one contain process iterate summarize cm x cm x cm x x step ranking gray window I counter prediction I l method step window represent gray initially pair exclude randomly join accord learn training practical test high already ranking namely consider rank step c phase pair predict give test learn top item etc tr n benefit ranking moreover store window array update complexity go ranking yield memory due ranking space complexity preliminary part aggregation numerically q first consider slightly aim q decreasing add recursion argument high eq I contradiction previous practical related ig life aggregation hypothesis classify rank increase condition nearly fulfil process mean problem get order process order efficiency first compare classic restrict near tree technique recall experiment explore involve view social crucial depend connection say link access phone order simulate divide three link phase define link link guess assign contain link link derive situation performance obtain predict link evolution vary one slowly reach maximum smoothly slowly aggregation improve precision rank exploit difference profile difficulty task recall impossible distant increase dramatically learn leave versus curve description discover merge link scale adapt learn ranking argue aggregate ranking information ranking redundant addition additional ignore performance plot curve obtain aggregate classifier apply well consequently well method explore aggregation area recall quantify performance quantity increase concern supervise comparable performance magnitude range poorly region minus average prediction retrieval benchmark leave prediction versus table comprehensive matter long ranking restrict example merge fluctuation rank source prediction suppose bring ignore process critical link consensus performance dramatically poor choice experimental encourage x x x x dependency indicate performance possibility phase intermediate small ranking c differ belong link link guess link guess method phase guess phase large miss ratio much average degree expensive costly consider large make ranking focus ranking ranking ranking experiment factor plot show begin closely follow curve performance aggregation mostly aggregated initially come rank soon allocation take good complementary aggregation always choose pair rank notice ranking poorly dramatically comparison supervise poorly limited highlight limit prediction largely learn result five link ratio area generate efficient unsupervised test unsupervised vanish grow stem dominate merge tend stick perform curve miss link ratio leave practical briefly magnitude experiment throughout cpu unsupervise ranking production index merge shorter framework combine ranking straightforward suited tuned need design prediction rank significantly improve purpose structural ranking node additional classifier option also consider user interact short connection theoretical mechanism link identify quality apply difficult detect application include security detect existence connection engineering combination active acknowledgement office scientific acknowledge european rgb rgb networks link relationship simple supervised rank aim various illustrate social improve performance rank also standard selection area relevance prediction link key mining practical go recommendation link view identification behind evolve example closure core link drive force link seminal snapshot network predict link attribute involve aim typical node irrelevant handle typically shall challenge computational indeed specific link role different type environment misclassification combine support biological scientific prediction desirable social network address establish accord scalar interaction predict top rank property example gender profile interaction last framework combine link method chain supervise retrieval document spam recommendation author approach pointwise feature fit undesirable rank link connect link popular account version direct spirit final adequate goal engine provide highly field measure discount however link whether quality work resp top quantity score effect imbalance
study population participant also requirement control design input p p compare decision trial requirement specify design sc power expect trial table file table report figure report create package center input panel main drop different show batch mode must results panel main panel output describe summary trial trial z z plot stage bar sake design h center default input top display power sample expect panel loading input save input contain trial automatically file parameter panel design bar web figure sake performance advanced parameter interactive change interactive mode automatically user batch top interactive advanced save click basic select save file input regardless load e already available real give match one must indicator contain treatment arm must binary outcome file header adjust setting give observe parameter basic proportion prior estimate design item successful control arm expect design outcome consider possible advanced per participant participant k stage include k participant alpha requirement design item alpha allocate use efficacy boundary applicable delta efficacy boundary simulation trial power trial duration time stop user reach cpu limit either time extend second item total item stage comparable efficacy boundary sc identical efficacy sc ss k constant ss final combine plot performance display performance three expect duration treatment treatment basic treatment advanced parameter specify metric plot metric page three metric table denote different ad reject h reject reject sc reject ss h effect participant ss duration necessarily total stop sample necessarily duration stop come plan trial goal iii trial aim stroke outcome plan iii phase trial little pressure monitor participant yield treatment effect ci think assess evidence efficacy scenario special treatment effect difference small participant treatment participant zero participant follow item furthermore control participant outcome participant p participant p true project design ad goal iii fully corresponding design goal achieve goal generally expect return goal achieve goal iii ss achieve recall treatment design default mean treatment scenario remain ad base design design goal achieve c f ad ad default achieve goal power power reject reject scenario reject scenario performance although specify alpha rate alpha package application criterion give explanation application output current limitation also outcome delay requirement acknowledgement acknowledgement research support institute ns drug comparative science contract institute environmental health sciences es publication solely color interactive designing trial criterion date design plan benefit trial goal specifically criteria duration user require programming experience application table sequential randomize trial occur strong evidence early trial benefit treatment study restrict example stop evidence benefit focus introduce combine feature design design define sequential design change trial except entire stop efficacy introduce package user type package densely application ideal user little experience core input allow standard several oppose automatically full use comparison report software planning phase treatment stroke new plus rt pa describe participant refer participant treatment participant baseline phase trial combine small determine small prior inefficient simultaneously answer combine focus trial throughout many formally design software currently available pe package computer locally web discuss interpretation demonstrate adaptive two refer certain likely trial example refer adaptive standard include rule stop trial early consist k stage assume newly participant correspond population pi participant stage entirely stop adaptive design restrict describe stage maximum end maximum cumulative participant end stage k k sample successful I stage treatment I outcome outcome available outcome c p average comparing versus give overview understand discussion efficacy hypothese nan treatment analogous simultaneous nan hypothesis h p compare hypothesis adaptive nan ad compare standard design standard sc design denote ss test stage stop early stage trial differ change switch participant discuss simultaneously implement standard area research global p p mean treatment cumulative statistic base participant participant standardize compare k population sum k right sc st st difference mean control combine stop analogous restrict formally z statistic follow subject k sum n I sum right left I I replace stop boundary statistic end study wide error strongly probability nan c asymptotically go ss control design ss nan single rule criterion standard efficacy statistic c stage efficacy design sc stop trial boundary stage stage complete k sc stop trial reject make simplification k efficacy boundary efficacy stage delta total range sc alpha cumulative sample sc sc boundary set delta efficacy sc z covariance sc c k delta stage nan define section ss ad bind boundary boundary ignore motivation prefer bind boundary control despite boundary trial duration assume sc user default negative stop although trial set equal efficacy boundary ensure efficacy boundary decision boundary ss analogously design except make simplification ss user efficacy boundary delta ensure alpha delta final efficacy boundary consider adaptive ad specify stage combine regardless stop end stage reduce turn option paragraph stage trial ad type two decision ad boundary cn delta ad ad ad nan hypothese alpha boundary control type rate level alpha ad c e ad certain ad boundary relative z delta ad delta k ad ad stage inf indicate reflect ad decide continue population trial stop entirely rule describe rule carry assess efficacy k reject c stop trial assess stop reject else stop future stage following iterate reject stage trial reject stop else continue participant ignore continue continue pi pi motivation evidence stop trial stop modification incorporate testing hypothesis consequence stage l remainder compute constant ad define efficacy k control alpha type suffice error global nan hypothesis follow size wide alpha interval type alpha initially allocate describe algorithm compute hypothesis define compute constant hypothesis nan hypothesis alpha ad leave k ad k kn right limit software comparable pe conversely pe pe tool implement many
angle might obtain simulated anneal cost anneal permutation convergent short annealing period physical six machine universit institute employ three tc enable forward enable enable ignore current outside force substitute part impact force passive coupling yield passive consist front front middle part connect enable around axis mainly inspire robot six light dependent sensor arrange front body sensor front body sensor pattern detect front light generate inside signal digital interface pc neural controller power sensor implement hz robot front wide experiment r period I right robot robot implement learn suitable period pass deviation period robot deviation period learn period show experimental diagram show experimental supplementary website directly transfer result mechanism flexible configuration robot possible c scenario r l work multiple complex behavior exclude three manual tuning robot introduction complete paper elsewhere implement control require precise calculate lot resource inspire already experiment control independent additionally modular independent biological perform loop sensor require benefit system loop instead implement robot carefully replace loop regardless robot four successfully present active generate pattern continuous self add selector artificial anomaly method realize change mechanism planning automatically adjust tolerance realize emphasize mechanism multiple generation adaptation orientation sensor converge easy controller control especially center maintain ground able body weight load result way configuration effective responsible robot body support robot robot overcome control balance investigate beyond work one two start exceed threshold learn algorithm period controller modular flexible six modularity offer possibility drive behavior obstacle module thereby focus neuron inspire generator demonstrate behavior self main advantage control precise mathematical robot multiple system frequency robot cause setup deal line learn mechanism simulate anneal technique automatically thereby considerably deviation original movement cause base learn converge acceptable getting combination period demonstrate effectiveness learn investigate might mechanism advance suitable furthermore cognitive goal addition employ controller reliably stable periodic unstable periodic controller pattern education grant bernstein g national foundation project natural science foundation project innovation foundation european fp specific communication agreement thank frank technical simulator implementation check language universit mc institute control implement generator pattern behavior controller deal present movement desire trajectory extend single simulated synchronization dynamic automatically resemble well first robot six result approach generation part independent multi robot pattern generator neural humans movement show level adapt create elegant vary report demonstrate achieve central pattern apply type kind control control review inspire way movement many loop control model robot integrate adjust several base sophisticated deal task control problem controller identical failure immediately adjust frequency individually change frequency independently also appear cat cat movement stable complicated happen proper contact procedure intensive traditional develop multiple deal find system controller output independently suitable automatically demonstrate propose real allow perform adapt rely multiple learn mechanism structure also state introduce platform effectiveness verify conclusion controller controller extend multiple also become indicate neuron generate input weight bias generate like pattern simultaneously add e signal period neuron detect control p period adjust control circuit sigmoid activation neuron weight w dynamic obtain period calculate every adaptively use period change pass post process module produce robot walk slow period slow wave fast wave blue area ground white one stop ground unstable pattern period generation useful neural circuit dynamic important principle mainly stable unstable period without dynamic stable period usually pass module neural diagram complete circuit since neural control study discuss briefly shape subsequently module feed forward phase shift simple feed network tc act capability increase tc even reverse tc turn finally neuron delay determined phase side delay see setup motivated perform period controller thereby lead drive single situation robot robot word robot contrast real control trajectory effective pattern inspire module processing output send line describe figure front neuron master neural gray module consist depict circle dark indicate module indicate synchronization mechanism delay output neuron signal neuron send last I mean period store store range increase time close much probable return combination period trial conversely probable period loop period straight robot automatically combination find parameter create tradeoff acceptance slow vice additionally observe learn often work empirically balance angle period simulate employ six controller controller frequency hz initially work master master period frequency achieve wave slow wave movement set consequence affect normally robot stay simulation orientation sensor angle orientation detect six lose synchronization angle subtract window deviation current period test different combination find robot maintain straight important period change period affect period keep illustrate column leave front six trial degree column decision period return one ignore initially robot right front robot turn balance body front randomly change angle period decrease right change deviation period trial return rather however keep combination period show advantage anneal provide change leave possible cope r l angle l body contact force environment figure fully phase air phase similar relate input example r body especially neighboring fix depict fig trial diagram start case state six one six scenario nine exclude counterpart situation depict r resulting
letter refer set blue mutation analyze detect tumor copy region proportion cell mutation mutation allele mapping contain genome panel histogram heterogeneous tumor sample large error dna even cell division mutation distribution central particular cluster tumor example mutation tumor sample attempt detect historical sample evolutionary tumor node currently tumor connect assign assign frequency infer frequency sum frequency child index object use cluster concentration dirichlet place assignment prior draw component distribution finite extend dp prior result unlike estimate thereby fix restaurant crp dirichlet assign object exclude generative infinite chinese restaurant associate object assign word unique chinese restaurant describe cluster produce rooted tree structure tree object proportional exclude child node exist read read matching allele variant position total read let allele population two copy recover frequency equation read count likelihood frequency infer assignment tree multiply identity equation complete pass asymmetric dirichlet burn acceptance discard burn gibbs allow result paper merge case infer natural ordering sum natural merge unlikely accept merge likely accept select tree merge parent child become parent child tree select select node e split leaf mix split node frequency uniformly population frequency parent decrease show leaf merge move construct either split merge move split merge move type gibbs simulation population population allele read drawn population read depth table quickly prior strategy mix fewer likely remain accuracy sample first divide actual take divide calculate package efficiency consistently imbalance population precision co simulation correction correspond difference suggest difference finally situation cpu great effort experiment heterogeneous run read runtime runtime ii variant adjust remain slow time ability dataset apply sequence patient five reconstruct sample simultaneously examine recover cluster together structure publication recover nearly difference frequency estimate direct parent child bottom substantial population occur cancer find patient middle bottom simulate runtime per per great accuracy five test amount cpu fold run furthermore decrease large number suit genome sequencing number expert dataset remain question decrease flexibility two acknowledgment national engineering award support science statistical increasingly popular infer chinese restaurant crp represent propose merge tailor improve time comparison stick prior superior sample cell
update solution trick also redundant identity admm iterate simplify simplify convergent split converge denote radius however really rate find split redundant convergent apply transition note determined penalty parameter split admm solve equivalent penalty transition surprisingly achieve asymptotic case lead small beginning proceed small section consider selection algorithm invertible high filter frequency band component band extremely huge algorithm case hence admm eq admm estimate method appear rate determine algorithm estimate sum two additional converge seem inner square minimization least square happen asymptotic solve still rate sometimes verify parameter discuss image show instance middle converge reconstruction penalty regularizer well resolution tradeoff note horizontal vertical solve efficiently instead inexact significantly thank curve error setting image setting solution convergence split immediately rate come inexact exhibit slow estimate admm might fastest split suffer paper regularize method convergent alternate multiplier admm regularizer nice admm lagrangian inexact method image reconstruction deep understanding admm analyze split admm method edge preserve regularizer insight tune region work interested convergence inexact inexact square proximal mapping complicated affect rate acknowledgement part intel split splitting image problem yield subproblem separable proof proof impose subproblem impractical many mr ray ct reconstruction inner least square usually shift image problem alternate admm inexact special admm augment concrete admm term measurement square smooth huber finite regularizer one propose lagrangian iterate least proximal efficiently soft thresholding method direction dual show convergence hold edge preserve assumption differently inner update impose convergent solve unfortunately inexact mr ray ct reconstruction lack equivalence method convergent inexact update inexact satisfie condition application ray ct pass necessarily shift difference laplacian high pass non vice versa nan ray reconstruction convergent admm inexact update ray inner iterate optimum absolutely look combine initialize iterate admm find
clearly stationary mab instance mab problem stationary formulate mab formulation reward budget grow time set reward evolve study mab formulate adversarial arm may study suggest finite regret regret change allow necessarily regret describe approach identify proof relative good action benchmark fully adversarial environment draw horizon batch batch perhaps accord arm expect change follow fix fix arm condition arm arm arm respective assume binary reward expectation k recall batch one addition j sum arm hold j sequence reward batch arm distribution one k tv last establish conclude batch single batch relative adversarial describe tune correspond associate regret j kk v kt dominate arm whole contradict e hold batch establish conclude proof sequence batch analyze action regret exp compare sequence compose part follow possibly parameter regret exp tuning difference policy throughout decision batch eq sum jx hold exp batch j take tuning parameter use exp subroutine increase one regret incur kt pc arm armed mab problem arm characterize reward arm play simultaneously exploitation due extensively assumption sharp characterization regret range reward maintain fully mab establish extent reward variation achievable analysis connection rather adversarial framework exploration exploitation stationary minimax presence feedback face collect trying optimize future fundamental variety internet web site seek recommendation priori price maximize select large preference customer select internet efficiently datum user know delay well instance decision daily future effectively acquisition future instantaneous base paradigm armed bandit originally context drug testing place general reward realization reward characterize maximize possibly discount reward receive mab modification extensively statistic economic dynamic clinical trial price innovation name reference programming formulation cover machine mab one typical benchmark instant select growth regret horizon converge oracle order characteristic sharp characterization regret traditional reward mab design growth domain several reward ignore mab formulation origin arm change rise term associate arm accord stochastic line lead various relaxation reference therein stationarity dominate mab raise fundamental uncertainty realization game see significant review adversarial reward realization single benchmark action reason static perform sequence time limitation adversarial regret static regard lie characterize regret mab problem non establish extent achievable bad four formulate stationary quite phenomenon remain mathematically constraint impose reward bound adversarial reward pick maximally policy within treatment mab focus reference adversarial treated evolve accord brownian explain second non bound reward policy sense variation arm sublinear adversarial stationary number time regret relative oracle treat broad temporal establish optimality regret horizon result order minimax regret range order grow linearly well achievable performance exploitation trade set compare stationary forget algorithms mab literature weight past stochastic associate time risk bias interesting draw adversarial mab stationary environment adversarial suitably calibrate optimally set establish introduce basic provide regret admissible lower contain brief discussion proof let decision epoch decision maker epoch maker arm random variable expect decision epoch reward addition assume beyond different collect capture second tradeoff stationary environment past reward reward old potentially relevant stem reward change reward old turn encourage enhanced exploration achievable build adapt set deferred idea admissible must regret order partition horizon batch except batch exactly good arm thus batch numerical realization reward batch realization expect correspond variation budget probable realization expect reward nature prevent batch since history sake simplicity discussion budget proof admissible identify arm batch epoch policy horizon select variation satisfy low develop variation budget policy number batch repeat initialization arm receive arm update begin allocation subroutine batch regret well adversarial regret dynamic oracle budget policy policy include weight chapter tend well numerically guarantee oracle study class algorithm regret order action adversarial subroutine two tradeoff versus exist capture subroutine good compare exploitation incur gain expect second tradeoff versus capture exp old discard characterize regret multiplicative logarithmic quantify impact extent environment achievable broad achievable upper experiment environment two arm arm epoch evolution select arm pointwise incur pointwise summing change reward approximate oracle regret incur instance display second depict third horizon fix tt describe change environment instance spend throughout whole spend horizon depict upper plot growth figure policy identify reward select policy reward selecting receive quickly arm keep trajectory reach policy tradeoff occur tradeoff subroutine exp exp explore epoch batch batch epoch exploration rate reward
seem latter new truncation contrast motivation aim loading rotation step loading rotation loading since pca loading approximate pca loading truncation tends produce vector equal less greedy globally block contribution devise truncation unify sparse pca together unified view find relation version drawback imbalance loading rest organize introduce present truncation performance give unified series relation new conclude c pca st decade ad hoc loading process interpretability rotation loading thresholding zero explicit objective maximize impose facilitate elastic net technique spc loading impose one suffer getting transform semidefinite guarantee unfortunately computational high expensive elimination order make large author greedy instance complexity lead solution solution cardinality range improve far full review generalizing recently power relate call propose augment lagrangian orthogonality correlation among loading global computational complexity summarize first give type behind understood loading constitute orthogonal span basis instead sparse approximate loading rotation solved alternatively optimize subproblem basis type soft thresholding percentage truncation sparsity linear diagonal express orthonormal thin mean loading obtain clearly eigen idea rotation sum hard rotation approximate loading confusion eq version approximation key sparsity penalty simultaneously fix subproblem close become form eq z tx z entry thresholding z far decompose entry otherwise express thresholding normalization compare add practically sparsity feasible evenly otherwise determine rotation orthonormal orthogonal basis basic table pca loading linearly loading truncation loading arrange wise pca initialize rotation truncation switch sort ji ix normalize discuss truncation type hard thresholding truncation sp truncation hard truncation operation nothing else devise truncation irrespective whose energy take percentage sort sp objective type correspond hoc st systematically seminal rotation simple discuss orthonormal four truncation study much orthogonality variance explain bound explicitly orthogonality angle define angle dimension span intuitively threshold sparsity truncate original bad side deviation sparse loading expect cumulative percentage variance maintain similar control appendix surface axis angle determine sum small orthogonality entry deviation generally well q hard usually guarantee moderately discrete truncate moderately en deviation direct orthogonality explain en preferable moderately en nice eq advantage lie want finally two tailor let svd loading close explain loading possible conversely much loading tends variance guarantee less loading loading another estimation diagonal sparse loading sparse loading pca loading explain loading energy loading guarantee en en sum project onto likely achieve proposition finally usually percentage subspace basis project span near proportional independently originally spc approximation variable alternate two subproblem substitute solution loading substitute two independent sparse net problem tx spc artificial original deal objective follow fundamental one appendix objective mirror exist penaltie ty unit combine original one unit loading via loading penalty constraint spc finally serve length small spc searching difference eigenvector rather key success drawback block ensure orthogonality also tend length length lead unbalanced among loading loading goal sparse mode loading less set weight appropriately align besides may loading truncation output loading normalize contain loading truncation loading arrange wise matrix compute svd major thresholding length type sp insensitive length improve algorithm block version matrix loading moderate comprehensive evaluation gene dimension random purpose speed test toolbox implement version similar code mainly five criterion sp loading std standard loading explain loading denote loading loading among loading instead std imbalance loading comparison direct sp let number loading original fall solution pca distinct set ensure avoid termination change loading exceed code common computer ghz load sp sp gp sp en sp en whether gp loading consider hidden word mutually correlation particular mainly common mainly dimension generate correlation latter feed accept datum reasonable share loading find support focus whether acceptable pattern detail nz loading std sp ta optimistic sp sparse accept artificial make test loading nz sp algorithm test criterion although suffer unbalanced std mainly lead maximal lead pca improvement tradeoff std sp focus bad rank patch vision pattern make comprehensive range gray patch dimension remove stability truncate energy loading variance explain evident simple provide section guarantee evaluation level absolute bound assume especially situation see upper cause dimension find specific well universal bind sparsity section comparable criterion plotted verify significant block orthogonality uniform unbalanced criterion get bad orthogonality orthogonality criterion perform generally good cost increase unstable sensitive consume increase sp en plot variance insensitive satisfactory across loading loading globally evident perfectly recover globally similar loading pca loading loading dataset motivate thousand gene determine show figure run involve mean report en fair sp grow much slowly already figure loading sparse pca call gp comparison exist conduct accord outperform trade sparsity explain orthogonality balance sensitive sp capable deal high sparse loading type work hard thresholding overall soft get good sp sparsity sparsity energy future effort objective en many prove bound achievable part deviation absolute value support overlap q soft thresholding operation combine z upper case pp element p k p p sx svd
five high follow one digit store visit calculate probability line store q k mean constant weight coefficient acceptable accuracy turn record path evenly mutually exclusive calculate validation initialize svm calculate eq classify number misclassification sample accuracy converge follow path mapping produce hybrid processor digital processors gate array parallel device time number one reliable node article use l visit global visit subset fx sign optimization convergence accuracy many genetic ga handwritten recognition particle optimization article graphic unit article improve algorithm large svm text bioinformatics slack determinant c algorithm optimization huge role simulate mix algorithm svm ability mean number sample svm means mean functional margin classify credible scale functional scaling b maximize slack basis
tensor index entire cross matlab standard three apply add operation multiply scalar operation use circular convolution circular convolution orient l di illustrate multiplication tensor size product develop fouri matlab notation tensor see proof transform rd fourier inverse rd scalars scalar refer form equip invertible module think generalization correspond scalar analog subspace rely property orient matrix set tensor size product framework multilinear form identity orient form module closed free multiply product illustrate generate linearly dimension htbp give transpose slice slice mathematically rigorous sum exception independence elsewhere product scalar algebraic definition see scalar orient article consider I independent zero union zero side orient orient come union derive section product write include help th use focus orient illustrative purpose position nonzero select matrix orient scalar shift original oriented matrix subspace contain permutation shift copy combination combination signal combination filter pattern argue matrix datum set adequate copy cluster believe suit many variation move subject camera shift usually collection application video pixel recognition provide useful framework capture natural circular family replace image step version algorithm algorithm arrange representation affinity n spectral clustering develop give analogous subspace clustering indicate ability traditional setting theorem strict must linearly merely special contain sum performance introduce notion cosine individually define describe algebraic proof find supplementary result speed sized synthetic datum lie mnist handwritten digit dataset synthetic image varied large database image total test display portion ssc entry penalize clear could perform tool face recognition know condition approximately subspace cluster basis face pose face via use successfully ssc segment person another training choice four nine ssc narrow succeed useful sign furthermore precede paragraph ssc parameter ssc unable face test original individual subject expression configuration w reduce pick first subject illustrate ssc ssc htbp experiment base image factor along normalize intensity lie take number pick person various pose different vs ssc ssc believe invariant preprocesse handwritten character digits instance digit curve show turn ssc competitive shift respect pixel good method preserve way aspect plan carry american sign video processor dataset take minute test effectively project fast ssc work state present computer engineering mathematics university combine recent field ssc come subspace respective subspace introduce multiplication ssc affinity build representation unlike ssc self flexibility ssc special array matrix whose element scalar take multiplication module retain property leverage vector preprocesse mnist handwritten digit database object assume embed option graph strong take approach come disjoint spectral theory final belong variation diverse array ssc employ resolve even reject outlier subspace two potentially exploit outside many reduction find approximate reference therein exploit multi present algebraic subspace column slice use tensor strategy incorporate cluster achieve less preprocessing necessary background summarize generative order call characterize performance add construct bad however generalization framework obvious key paper make present manuscript conduct face handwritten digits ssc
copula unique u dy eq margins cdf copula regression discrete explanatory copula copula univariate parametric th univariate margin margin response discuss simulate likelihood ce sequel since dependence longitudinal pmf response pmf rectangle dominant monte carlo reduction transform quasi efficient pass package advance probability high copula model discrete response simulate hereafter sl maximize univariate copula newton four place evaluation work poorly numerical log variable rectangle probability sl initially longitudinal reader copula hence numerical calculation simple approximate probability copula density dt dy univariate close q identity multidimensional copula asymptotic study asymptotic surrogate dt asymptotic sl estimate maximum sl copula exchangeable structure dimensional integral integral integral integral exchangeable pattern sl already take bernoulli binomial parametrization ease margin covariate distinct limit sl limit limit pass mle dimensional limit compute variety likelihood good limit comparison quickly vary effect finite truncation exceed probability ccccc limit copula margin bernoulli margin omit place sl lead regard surrogate asymptotic univariate latent correlation asymptotic individual count asymptotic count response univariate asymptotic decrease ccc ccc limit nb margins truncation point exceed sum dt discretization individual mean discrete decrease surrogate likelihood simple error se root divide margin place cccc se c dt error limit mle poisson truncation exceed small study estimation datum dependence concentrate model spatially spatial adjacency degree row label adjacent car copula construct inversion univariate variance car proper conditionally autoregressive margin consider nb parametrization binomial comprehensive chose covariate j count binary mean take poisson logit probit link poisson coordinate lattice size truly small per lattice car dt sl cc sl dt sl dt sd rmse sl dt sl sl bias sd rmse c sl dt sl sl dt sl individual dt decrease section spatially aggregated cancer second subsection incidence large section efficiency surrogate dt efficiency dt criteria aic bic aic include penalty aic nb spatial criteria dt sl parameter small aic fitting cancer incidence incidence period provide supplementary economic determined institute also interest cancer count assume poisson nb economic status fit ignore independence depict independence probability suggest likelihood reliable cc c size assume datum car copula perform via dt sl give estimate parameter along fit likelihood nb aic nb regression far car margin improvement one h cc nb sl dt sl dt sl estimate aic sl data copula margin horizontal line estimate profile profile significantly excess incidence status count response level gender observation whether different illustrate method analysis year well size discrete cancer count assume poisson nb preliminary ignore spatial dependence assume independence depict reveal individual probability suggest dt c cccc cccc regression dt est se est aic est est se dt est est se aic estimate se aic sl car perform dt sl parameters se sl obtain hessian usual theory bivariate margin test penalize copula nb margin copula poisson marginally nb cccc se se aic regression est est se value aic se value se aic standard se sl discretization car margin count surrogate application interesting glm reference dt poorly gender gender interaction gender significant conclusion latter analysis dt result true interaction good aic discussion paper study margin surrogate binomial car substantial estimate correlation parameter precision car discretization probability sl highly response although burden increase rapidly compute dt sl rectangle calculation replace simple however since cdf histogram reduce burden worth health interest field occurrence site occurrence threshold extreme area dt latent structure mat ern isotropic structure dt replace rectangle calculation hence discrete response structure pt sciences east uk pt distributional transform dt amongst computational multivariate normal copula model analysis normal univariate margin dt lead biased dimensional discrete simulated multidimensional randomized calculation maximum dimensional multivariate illustrate aggregated datum show via datum distributional generalize quantile rectangle spatially aggregate continuous straightforward distribution transform thorough categorical name hard limit marginal concerned generalize linear initially analysis correlate parametric family come continuous multivariate choice seem appropriate spatial flexible choice paper copula interval dependence modelling dependence copula study explore copulas experience discrete margin copula pmf copula cdf generally statement dimension negative pmf
section mixture experimentally acquire signal enable fouri conventional quantify chemical chemical place strong spin produce imagine bar bar field induce current frequency chemical mixture current call free local molecular permit conventional assume generate channel exactly perfectly model frequency intensity transform principle spike delta frequency relative magnitude spike adjust intensity relative due explicitly mixture conventional fourier general conventional procedure outline give intensity within frequency chemical group differ discrepancy know intensity calibration experiment measure slice imaging adjust theoretical intensity accordingly list experiment differ give composition mixture peak chemical section chemical show fourier transform dft fill take fouri transform peak dft width delta spike look like cauchy noise peak practice shift see figure together peak peak belong chemical index adjust conventional chemical q along peak chemical spectral conventional calculate concentration chemical credible ratio snr experiment definition eq peak snr peak simulation real conventional domain exponential fouri correction perform fourier spectrum national institute technology chemical frequency weight calibrate propose conventional ultimately wish induction two phase e signal white chemical specie mixture chemical intensity decay shift intensity frequency procedure section synthetic physical interpretation arise rf take sensitivity current stop period typical fourier delay correct fourier transformation raw signal delay relaxation homogeneity place effect amplitude ideal signal exponential perhaps range decay describe illustrate behaviour parameter generative model frequency intensity table panel dash line simulation multimodal phase explore surface frequency accurate optima gradient optimizer choose improve function every converge undesirable likewise popular metropolis hasting reject also optima frequency sensitive decay explore unimodal show performance compose component miss less initial decay information away thus increase surface natural decay decay decay find anneal simplex optimization reliable estimate chemical follow procedure uninformative analytically estimate nuisance parameter p variety operation channel chemical specie operation cholesky decomposition term motivate briefly selection provide review quadrature local shift taylor expansion gaussian however highly approximation multimodal break quadrature specification unclear test frequency metropolis hasting sample parameter quadrature time conventional mh explore multimodal surface become undesirable optima extremely unlikely find move gibbs involve parameter conditioning gibbs mix poorly gibb many focus frequencies monte carlo experimentally spectrum spectrum develop water similar coefficient water mcmc determine whether generally applicable quadrature focus snr spectrum description scheme construct domain modelling phase correct baseline correction leverage positivity requirement promise conventional discrete fourier dft promise statistical frequency quadrature develop heuristic quadrature robustness synthetic well transform focus aspect quadrature modelling develop novel chemical quantification quadrature solution thorough comparison conventional transform experimentally acquire behaviour section fourier ft decay signal automatically correction simulate signal intensity table presence use thus challenge ground truth synthetic resemble experimentally stress response ratio define uncertainty concentration gray bar credible interval standard datum show snr true concentration decrease snr figure snr around true identifiable std credible interval empirical percentage reconstruction marginal snr snr increase increasingly converge truth show snr bayesian credible true behaviour chemical credible bar confirm consistently accurate credible always conversely prediction ft approach vary interval broad bar ft unable concentration ft particularly sensitive become apparent study bayesian consistently accurate uncertainty investigate ft absence chemical chemical chemical hard snr whole ft red species concentrate specie peak concentration moreover duration acquisition frequency truncation peak make peak frequency window frequency nearby principled reduce difficulty absolute error spectrum indicate box peak peak truncation peak distinguished concentration bayesian show truncation concentration give ft systematically bias region bar experiment reliably distinguish concentration x axis experimentally ability level range increment create per snr ft figure expect mixture error bar horizontal sample ft consistently concentration bias ft scatter prediction always bind uncertainty conventional intensity peak peak almost quantify
classification carry linear range space input ik observation alone information side clear independently misclassification low regime feature condition overlap sufficient pass source upper misclassification extraction zero without unique misclassification diversity tend infinity signal condition information class distribution joint index ik ik ik ik ik ik ik j j ik ik otherwise ik condition classification expand fix obtain note particular ik specific value verify imply upper decay ik ji extract verify moreover achieve decay depend space span side intersect signal concatenation intersect span project intersect class interested reconstruct determine condition guarantee perfect generalize characterization also region simplify e r conditional moreover express sufficient guarantee regime approach condition upper observation alone e regime consider reconstruction directly perform diagonal property straightforward hand necessary decoder noisy noise vector stem coincide mean r side represent possible reliably r happen overall span project signal input dimension span moreover need span dimension alone sense projection capture characteristic share mean fig dot source possible conditional sufficient leveraging result provide regime correspond decoder operate decoder estimate associate classifier associate immediately gm immediately misclassification characterize noise transition draw condition distribution see show extract great large space component enough component information trivially consider side sufficient transition intersection region decoder reliably regime source obtain g signal correspond obtain formula inequality optimality input information derivation number model input condition satisfied phase input feature side reconstruction provide distribute union class characterization misclassification draw give misclassification expand report step consider expression integral expansion pair wise diversity leverage lemma asymptotic expansion bad indice diversity classification classifying whether associate reflect side entirely determined mean diversity possible classification leverage theorem determine misclassification hence accord condition ik ki ik misclassification classification ik ik j upper corollary number extract discrimination guarantee pair characterization noise expansion distribute nonzero appendix provide expansion case difference lie bind decay thus necessary proof expansion characterization source code counterpart entropy counterpart joint condition different immediately signal basis sparse innovation decoder result theorem gmm sufficient necessary class class condition briefly outline theorem transition associate derive draw use wiener filter bind carry leverage phase upper misclassification gaussian input accord class condition probability base estimator derive condition gaussian input meet condition scenario case size common innovation component mapped innovation additional reconstruct pattern report synthetic real cast aim theory approximate present diversity characterization behavior dimension respectively ik ik ik unit span class r ik ik ik ik ik different share sum space span signal different condition diversity yield simulation error fig misclassification well term transition diversity impact side representing value presence information obtain transition input linear span projection signal moreover increase feature increase analytically characterization base diversity align report need image correspond nm single snapshot accuracy reference image acquire hyperspectral image image perfectly measurement fig reconstruct hyperspectral information furthermore though still reconstruction correspondence block reconstruction six channel table notice fundamental classification art image extraction decoder carry classification also consider joint side correlation marginal likewise marginal misclassification construct asymptotic characterization quantity sharp sufficient misclassification necessary phase source numerical principled integrate reconstruction compressive hyperspectral imaging presence information direction one information source source possible model point appendix scenario design lead phase decoder side scenario side gain interest follow generalize translate signal modality recall misclassification give k k c lower simply follow derive induction upper differ multiplicative diversity integral also upper misclassification j ik ik noise diversity index expansion side computation diversity computation rank ik ik ik ik j characterization rank compact drop result ease notation straightforward space dimensional observe represent remain row tight proving hold consider span nan consider rank two subspace space conclude pick independent r n last observe force equal identity case leverage fact similar use r r finally lemma conclude pass generalization considerably complex misclassification transition occur semidefinite verify ik regardless assume ik separately follow r ik ik ik r simply leverage ik ik ik ik ik ik sufficient measurement ik ik r ik ik ik r ik r ik r r r ik ik r combine previous guarantee ik r ik r j ik ik r j ik ik r ik finally combine ik ik ik ik characterization expansion upper misclassification start expression leverage class otherwise expand expand index verify necessary verify define ik ik ik similar separately ik ik leverage error observation alone respectively ik ik ik ik ik ik ik ik r ik r ik r follow ik ik ik verify ik ik ik verify since eq previous expression condition ik ik ik ik state ik r ik necessary sufficient ik ik ik ik ik ik j order regime condition fact reconstruction observation consider bind incurred recover equivalently write regime approach zero introduce inversion moreover note immediately order leverage separately summarize state proof union necessary expectation hand simply complete positive semidefinite generalize positive probability consider apply recall rotation generalize substitute condition conclude necessity proof report ik p k c verify theorem ik prove leverage imply ik r respectively ik ik ik misclassification step proof misclassification gaussian class measurable converge ik separately state regime prove probability guarantee noise total argument bound c right use reflect gaussian provide input space denote ik ik ik j ik ik ik ik prove use write ik ik ik ik step proof theorem able follow cyclic independent identically division generalize interference interference interference multiple transform quadrature amplitude ratio identically input division additive cumulative tucker power profile quadrature shift compressive sensing matching pursuit side distribute reconstruction distribute pass isometry discriminant ratio code snapshot ci de da e mail fc university college email ac uk edu fundamental limit classification dimensional access linear interest feature information signal side assume signal interest draw correlate component specific gmm misclassification associate reconstruction associate interest transition quantity regime condition extract framework offer principled integrate high imaging art compressive image digital mmse misclassification concern extract salient signal method feature extraction dimensionality unsupervise dimensionality various dimensionality theoretic mutual information mutual information criterion linearly reduce lead art divergence express unified poisson channel enable signal reconstruction become acquisition paradigm offer simultaneously sense seeks extract low show reconstruct dimensional signal sparse projection greedy pursuit compressive paradigm processing compressive detection popular attempt aid dimensionality reduction prominent high include union wavelet tree manifold union lie collection leverage reconstruction conjunction root connect need reduce manifold linearly volume regularity decoder know beyond exhibit correlation concern high signal aspect attribute signal often live affine space side correlate connect feature dimensional relate compression classification distribute whereas side namely characterize two decoder side surprisingly rate compression associate joint compression discrete presence code compression encoding optimum compression distortion decoder contrast encoder suffer general information loss small case relate problem compressive sense side compressive sense compressive compressive entail desire signal leverage support decoder use decoder associate previous certain dynamic imaging minimization account distance image snapshot number image reliable recovery high mixed reconstruction compressive reconstruction necessary perfect innovation specific multi terminal spatially couple minimization matrix show ambient consider signal domain derive sufficient well algorithm multi compressive sensing description infer extract datum demonstrate various experimental datum signal unlike signal underlie represent adopt conjunction formalism represent counterpart signal signal lie affine translation rank prior approximate mild provide result dictionary classification video compression inversion noisy feature within moderate reliably extract relevance wireless communication metric gain measurement asymptotic carry decoder generalize joint condition low interest side also side use real resolution compressive hyperspectral imaging side traditional constitute work characterization compressive allow basis allow provide signal processing extract input classification presence remainder throughout section contain misclassification misclassification side notably sufficient classification proof notation case matrix low letter symbol identity drop dimension represent transpose rank respectively denote subspace expectation covariance denote symbol side feature associate desire projection extraction system decoder signal projection kernel side additive variance write eq draw distribution n rotation invariant special rotation entry fix cm cm draw east inner width mod sep west east inner pt height cm anchor west east west circle east sep height width anchor west east inner anchor west east near si anchor west index aim estimate input underlying purpose perfectly index component minimum probability classify classifier class reconstruction side objective decoder signal conditional mean observation minimize emphasize distinction previously multi task compressive recover compressive consider recovery addition objective jointly latter side information map aspect relate signal correlation adopt I ik k ik ik ik ik ik motivation fact accommodate joint pdf incorporate note c c ik p ik ip kp state reconstruction hyperspectral digit also common component generalize condition pair ik ik ik ik ik ik ik ik n ik ik ik ik factor subspace vector characterize share subspace dictionary signal respective perspective generalize previous scenario ik ik ik ik ik ik irrespective ik satisfy require prove contain leverage clear interaction connection ik ik obtain combination atom dictionary underlie phenomenon condition class hand ik ik statistically thus phenomena condition sensor describe innovation characterize formulation respect pick common exactly innovation basis range ik ik ik hand signal therefore express rank appear rank appear span gaussian span signal represent sum span input signal index correspond index span input side ik ik ik ik ik define represent span projection represent span projection draw index subspace span projection signal information component provide source
eq thus contradict complete ex approximation inequality desire contradiction ready put everything together complete proof corollary dropout effective setting sound theoretical need dropout exploration regularizer misclassification loss logistic minimize regularize remain regularization weight regularization last particularly surprising proxy contrast formalize compatible regularizer exhibit provably insight bias regularization provide result prominent dropout study inductive dropout dropout shape search strength co adaptation concern try function inductive dropout understand classifier remain popular dimensional third thorough understanding dropout inductive deep architecture preference learn decompose system node clean artificial classification dropout independently replace replace q note obviously dropout probability converge broad variety condition abstract dropout view stochastic view goal inductive variance case rather dropout may decompose eq negative marginal lead view style like rhs motivation proxy classifier assign weight preference strong get penalty show dropout inductive penalty surprising regularization monotonic absolute incur infinity prefer never reach extreme remain convex penalty matter detailed infinity constant convex shorthand drop play example remain dropout effect informally say dropout parameter dropout inductive bias source align inductive compare regularizer help illustrative handle use ensure control difference respective regularizers style useful tool study inductive work dropout family evaluate point case dropout regularizer regularizer discuss regularizers experimentally dropout view ensemble adapt require dropout variance study variant dropout compatible source preserved complement focus original dropout characterize unique minimizer regularizer separate dropout provide section feature optimizer drop keep paper introduction independently random analysis easier randomly one loss consequence optimizer well follow either attention feature r tie assume contradiction perfect tie xy decrease loss assumption minimizer remain therefore bind p r pr x e strictly minimum p summary vector weight dropout keep regularization additive dropout dropout dropout optimize vary criterion generalization regularization present specialize decrease exponentially prediction dropout second inaccurate experimentally evaluate dropout accord suggest penalty increase linearly proposition e w penalty new next show dropout regularizer regularization dropout regularization penalty remain single vector infinity figure dropout index substitution complete line use I function neither range expectation different character go remain dropout instead dropout vector initial remain go small consequence already penalty theorem dropout penalty zero surprisingly hold regularization proposition write q make penalty product prediction dropout identical us dropout monotonic fact dropout fix arbitrary locally behavior prediction like prove support dropout probability sign penalty infinity straightforwardly infinity penalty immediately lead nonzero support infinity together approximated dependence allow show whereas infinity sign show remain compatible alignment go infinity together complete multiple go infinity dropout discussion theorem proposition suggest dropout regularizer indicate grow suggest rare less weight frequent rare feature perceptron base approximation empirical dropout discriminative use suggest limit sign hand proposition indicate encourage help share weight correlate turn dropout pair strongly qualitatively prefer regularizer w separate dropout exist c family separate use consider weight classify perfectly regularize optimize criterion distribution vector classify encourage drop weight enough minimizer dropout criterion expect plot left dropout regularizer low dropout right green region mark compatible regularization recall contrast criterion p consider pressure pressure prevent grow correctly hand nearly wrong mean light proposition give advantage regularization multiple simultaneously go penalty remain bound weight increase base regularizers go infinity individual infinity characterize cause remain put sign suggest regularizer grow linearly dropout provide dropout work definition regularization regularizer regularization complicated dropout pair dropout particular dropout generalization natural well devoted prove separate pair dropout separate strong separation find criterion wish separation range dropout amenable exact difficulty separation dropout sign moderate analysis plot illustration dropout regularizer preference case single multi layer neural network deal open separation gain regularizer setting first show assume without generality multiply multiply change q continue sign numerator numerator since numerator locally note may slight modification nonzero sign negative move fix arbitrary support dropout sign regularization go go depend go discrete limit w w analyze inside third expectation I go infinity also goes drop drop non drop precise notation proof scale obtain regularize criterion simplify expression lemma repeatedly use closed separate start w decrease derivative increase proof continuous derivative term negative term positive complete correctly much assume contraction rhs negative contradiction suffice partial positive go infinity evaluate eq decrease increase negative desired show classify throughout proof subsection dropout criterion component independent bernoulli simplify minimize equation optimizer correctly one prove prove partial drop matter open negative proof multiply whenever proof misclassifie complete proof ex keep might classify prove misclassifie scale let simply implicitly minimize derivative eq rhs rhs show large contrary give bind prove lemma convexity either show complete scale criterion independent scale objective partial correctly suffice decrease even recall equation go infinity ex show term dominate assume fact give ex ready combining imply eq imply ready ex e complete piece succeed lemma classify complete ex perfect p minimize frequent partial majority vote notice existence node fill inner negative infinity positive hand increase infinity partial magnitude magnitude reach meet hold complete optimize dropout make difficult make jensen dropout convexity form last drop whenever become optimize fail want contradiction begin consequence assume contrary jensen inner w w w optimality w bn drop drop plus give bind bn n w dominate majority vote lemma ex loose large conjecture criterion fail produce bayes theorem minimizer let minimizer symmetry recalling equivalent value dominate weight circle blue red right
relaxed minimization nuclear sparsity typically incorporate norm trade alternate implement directional guarantee retrieval quasi polynomial sparse approach rely linearization constraint induce problem structure linearization sign original signal propose sparse constraint cone programming formulation extend noisy algorithmic benefit noiseless presence noise note system case term one phase clarity detail sect extend sect effect result discuss signal complex sect deal invariance measurement circular shift test matrix write bold letter bold letter matrix part transpose respectively j retrieval symmetry solution obtain interested solution condition method let ij j symmetric denote component phase rewrite jk nx objective reformulate nonlinear pose one recover relax estimate nonetheless positive group overlap aim surrogate diagonal cone j jj jj sign ji turn approach first sparse solution unique n solution least sparse contradict solution jj jj regard coherence coherence n recovery proof give theorem solution contradict imply solution multiplication unit thus z infer property invariance prove real linearization map x equal either nx I map complex jj jj jj jj square act positive number arbitrarily fix equation definition lemma jk jk jj ny jx problem rewrite nonlinear optimization relax substitute yield modulus show relaxation proxy sparse contradict unless jj therefore relaxation j r unique hold define index due introduce eq unique exist contradict definition imply perturb equation ie goal become eq sect convex form must omit except conclude detailed variant perturb noise via relaxation n j jj real addition path theorem adapt complex minimizer statement rewrite must satisfy replace due j j bound bind introduce notation relation I r j inequality eq let together constraint rewrite since positivity stability ensure consider dedicated sect particular allow order account appendix real typical subsection technique deal invariant circular shift problematic version circular shift linearize combine different pattern invariant version vector effective estimation retain define singleton make stand circular maximal shift linearize constraint ensure valid solving assume formulation convex shift shift additionally define x maximal apply name sum first first small shift magnitude finally give invariance circular shift invariance proposition x I statement invariant combination shift directly useful suppose check inequality involve square magnitude advantage shift reflect shift reflect feasible solution feasible kk show notation obtain I apply require map magnitude large compute satisfy implement shift unique correspond magnitude transform square issue shift determine jj jj therefore know dedicated measurement fouri restrict measurement shift fix zero j remove corresponding program lead final formulation modification approach greedy optimization implementation two method slight complex particular enhance support add normalize detect small estimate complex percentage carlo experiment distribution unit random nonzero
mix blockmodel allow membership flexibility formation analyst compare attribute post manner introduce blockmodel actor social actor form link whereby occur throughout aspect political method examine summary statistic occur frequently chance treat underlie structure example triangle could reasonably occur evidence whereby link actor link actor two represent blockmodel sbm membership two actor connectivity map actor onto form actor cluster extend actor determine sbm latent network difference model actor actor network conversely sbm whereby connect weakly mixed overlap actor membership actor interact sbm attribute explain occur school student likely gender play important formation belief reflect appear literature share interest status collective activity clustering additional hoc manner extend sbm incorporate link actor specific level covariate include gender age relate actor physical actor specification beyond link explicitly incorporate actor mix expert blockmodel terminology framework model covariate information adapt terminology incorporate membership model model thought actor characteristic network formation converse tie actor rest structure briefly review detail provide relational actor link present pair actor interaction actor represent link think share symmetric undirected otherwise say direct interaction refer school student interaction consider entry sbm assume membership group actor model interaction actor indicator follow multinomial prior ensure frequentist framework sbm variational collapse fully sbm substantial multiple group interact framework actor assign individual membership indicator actor interaction actor interaction model manner parameter distinguish interaction quite group exclude hyperparameter ensure mix beta treat nuisance parameter far covariate terminology literature refer covariate paper restrict actor incorporate individual mix hyper treat membership set beta multinomial decompose py z n np np np section sbm htbp b fashion employ bayes approximation previously useful membership setting intensity approximate concavity kullback true approximate distribution restrict factorize kullback multinomial beta introduce update extension ij gp ip py ij form make newton hessian follow np nr np nr np np experimental newton vary approach dirichlet wish estimate probability rather covariate prior probability intuitively think covariate parameter serve newton another encounter use separability model whereby pattern method suggest regression model prove force office entirely five actor location assumption hoc approximation making criterion criterion difficulty determining occur perform fold roughly drop fold straightforward value miss simply hold likelihood high uncertainty assess check total detail notable lack strong couple behave interesting environment formation link strong focus actor available previously incorporate conduct former office year impact position include covariate membership find evidence location affect effect network figure actor still form partly gender clear age facilitate covariate attribute rank low indicate associate gender school university office exclude fit value validate group somewhat satisfactory b b figure actor half actor display membership represented label accordingly overall weighted correspond indicate interaction include large font font half expect enyi whereby actor fit interaction occur check profile membership actor actor actor lowest seven actor involve respective exhibit membership actor belong actor exhibit three actor exception actor indicate full participant group membership actor appear highly social figure plot chart represent mixed chart mixed actor statistic popularity actor prominent green represent membership community red dark blue occur actor actor split actor purpose impact covariate facilitate obtain optimal obtain parameter inverse approximate fact firstly whereby twice create difficulty behaviour bootstrapping generate fit use record reliable outline worth reflect degree selecting appear impact bootstrap replication agreement estimate mainly significance covariate whereas status actor appear base quantile box parameter covariate perhaps obvious behaviour worth indicate group poorly explain four zero covariate term partly explain membership influential status year law school correlate parameter reflect inherently evenly nature covariate strongly positively skewed tendency retain term group consist establish actor actor actor strong group standardize mean deviation occur year age despite less age positively skew actor assign prior high explain actor probability within actor significant significant note upper parameter close particularly actor uncertainty would long exception school membership comparison seem quite actor appear impact range estimate quantile highlighted intercept status htbp dash occur examine fit predict link fit outline two observe link predict hold operate roc curve show appear well auc almost htb b checking
item well statistical pm option e difference exist carry post hoc pair result pm significantly improve half pm improvement wide item compare always statistical analysis right method seed select difference clique often sub clique seed high effectiveness attribute select seed criterion strict item smc semi superior heuristic list difference former item expand latter explain superiority come inference former chance weight heuristic treat equally c c run amazon expand experiment expand stopped ignore expand hence conduct intel ghz core g ram virtue ideal parallelism item large machine efficiently overlap call item present item devote efficiently quality seed range network seed method resort semi result advantage statistical demonstrate item run unweighted direct prediction also make item would like anonymous comment support grant china grant ac advanced chinese sciences china university chinese china expand scheme network seed non principled expand lead diverse propose new transform corpus treat corpus effective seed expand significantly improve performance complexity scalable large system elementary part link many network tend distinct subgraph community module occur computer reality active group friend algorithm discover belong community scheme step seed seed form seed select expand quality seed important detection lee clique seed select expand process method detect community improve give method em kind often seed select lead unstable due seed scalability algorithm may network node due pruning remove seed removal rare sequentially rank improper community seed select seed community easy end aggregate select seed rank network high rank seed minor diversity seed guarantee rank root rank drawback three kind mention globally expand decide community link community decide belong expand fast heuristic lack principle use lee fitness function appropriate datum drawback independently without highly community share post merge merge community merge difficult expand replace global optimization edge naturally virtue wide applicability propose theory em discover corpus edge community assign network belong extract effective drawback traditional expand treat seed set supervised edge item improve edge organization section skeleton item expand section experiment suggestion provide reproduce result publish notation g double subscript use prefer subscript edge clearly skeleton help reader rapidly ht cm terminology definition comprise network figure number single subscript double subscript nod v iv e e ice ice ice ice exploit mining community network propose key origin matrix motivate commonly similarity index gmm please table simply display follow list feature discard discard resemble preprocesse mining remove discard operation easy node similar document c p p p display item classify researcher semi supervise exploit na item seed use expand edge community expectation unlabele label community unlabele refine nb classifier item expand stop predefine match score middle color seed seed thin edge add two seed avoid drawback specificity local give information information compare filter select seed index filter second seed select fed representative seed candidate view lm figure bit select completely ignore specificity make measure link suitable seed specificity strong cause common edge suitable seed formal definition specificity common concept seed strength specificity get technology evaluate technology convert bit denote detail due information please evaluate number intensity specificity similarity neighbor specificity specificity mainly contribute locality comparing make edge appear seed split apart adjacent slow convergence process em filter select detail seed efficient selection method effectively representative term information lose merge virtual document document virtual apart try release seed create edge candidate e hence q apply make select candidate check include clearly scale th e network community narrow bold color final seed select get seed supervise add edge thin color edge color expand expand community lm select seed take clique classify community exploit topological potential include potential evaluate lastly community community neither adjacent include status generality include potential k ode j I figure triangle triangle give zero impose rigorous requirement order community treat I ensure ensure classifier belonging maximization divide kullback leibler expand resemble distance follow edge maximization edge evaluate weight occur term occur drawback color bias high specificity value respectively consist synthetic subsection subsection brief compare commonly algorithm clear benchmark control size govern distribution little community respectively network vary mix boundary edge community membership overlap fraction value generate figure average measure ground truth community normalize mutual use value facebook include year truth facebook lee detect value relate community accuracy indicate find well ht devote overlap prior 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nuisance construct value identity inference quantity alone let introduce predictive random valid predict valid center predict suppose iid unknown moment standardize joint minimal like latter satisfie nuisance eliminate ignore hand distribution namely central student centrality parameter student I q simple think box develop promise j generalize inferential share asymptotic base method study I differ truly property really school carry well base scientific effort focus develop fundamental building additional far read current still computation application believe development author thank associate suggestion research support national foundation dms dms statistical inference perhaps extension meet requirement free inferential I show promising generate valid probabilistic brief introduction principle principle I discuss principle I condition illustrate bivariate bayes belief inferential principle experience observe knowledge population essential part observable parameter mathematically distribution convert assertion concern plausibility plausibility meaningful probabilistic interpretation work make effort argument use change nothing well create confidence central frequentist great develop valuable live ignore yet fisher effort generalize inference reference argue mention free paradigm read reasoning seek free mainly principle namely principle upon principle basic marginal inference realistic knowledge belief plausibility stand belief represent support assertion true support assertion assertion plausibility function support also probability discussion total formally evidence available either assertion discriminate immediately satisfactory represent informative relative specify nevertheless bayesian probability proper improper constructing method close informative limitation latter discuss model take look precisely simple observe continue regard distribution distribution regard reasoning change status represent conventional goal view agree replace degenerate conditional must beyond inference formulation I framework know reach statement hypothesis interest summary observe support constraint general motivate principle next question measure bayesian approach meaningful scientific arise first encode informative prior sample view free probabilistic question difficulty fact basically bayes apply theory opinion correspond scale belief opinion essential success scientific belief desirable properly belief value need expand auxiliary design meet definition goal efficiency consideration though I explain prediction discuss principle quantity observable precise carry operation well valid random nested subset include serve satisfie simple example valid realization I make quantity eq valid predictive combine prediction belief plausibility need empty I set I valid stochastically assertion true meaningful scale plausibility belief value consequence test frequentist development efficiency principle validity efficient information marginal example difference I framework application future pose one marginalization handle unit size function auxiliary variable observe unobserved association build chi auxiliary freedom association argument attractive identity conditioning amount valid variable association rewrite term relate auxiliary normal partial equation give complement I technical produce exact multidimensional
mx mx mx dx mx mx dx v provide eq n next density eq xu convolution via function mx x xx z z aa z aa analogously turn right k k assume e turn p set p borel r complete evy triplet q eq q lebesgue obvious shown write hence positive constant q thm thm thm example thm remark cm evy embed estimator turn convergence consider problem evy support dynamic process laboratory method rf evy transform laplace change call embed solved formulate embed give stop integrable martingale se draw see item fact solution se currently statistical pose study call statistical consistently estimate closely multiplicative construct norm rate asymptotic normality brownian l evy turn use combination laplace construct rate basically coincide statistical already poisson let motion deconvolution variable near obtain example probability mainly independent probability random variable due problem inversion empirical fail operator support sequence tend convergence henceforth take simple principle decay fast throughout notation constant multiple arrive effort prove strong turn example density asymptotic let look therefore polynomially fast follow class assume choose eq closely relate context deconvolution n nn l rate factor infimum take sample q n h evy process follow independent evy situation since evy long evy triplet domain h follow let moreover bound eq real x hand straightforwardly derive replace empirical case need inverse transform define find evy two show logarithmic change brownian proposition moreover recover logarithmic fulfil exactly logarithmic theorem sense basically coincide logarithmic rate stress class know family beta rate mixture mean model mixture theory dependent heavy skewed coincide variable example density density statistical literature parametric treat nuisance zhang estimating mixing well knowledge first inference density generality minimax rate theorem variance mean evy change evy change evy suppose evy arrive follow exponent evy want time l necessarily property generalize evy frequency consider chen let evy high many therein al mixture normal distribution univariate member inverse function density density simulate bandwidth density estimate p tx procedure turn decomposition e km cauchy quite reduce compute integral nh realization xx run depict observe performance similar rate
precisely system throughout process generate instant subspace algebra take space mechanism unnecessary technical absolutely lebesgue theorem density surely constitute assume nevertheless state validity partially process define family partially markov markov state member family canonical constitute gaussian tx functional describe evolution state markov get equation z constitute process first hmm equation reduce arguably observable system encounter signal important application specific boundedness sequence reference involve sufficiently bound well continue observation lipschitz exist z lipschitz xx continue almost say modify replace lipschitz everything former everything consideration proceeding estimation later natural algebra variable mean square error ideally stochastic estimator interest one measurable nonlinear filtering frequently process concept stochastically hide system formulate extensively discrete filtering discover various markov purpose intuitive include direct far consider base responsible coupling measurable constitute tool measuring contain base serve channel result arise first ask assign another transformation second latter way behave constitute stochastic possible derive question key assertion induce process absolutely question conditional bayes density consider integrable base imply existence density resp support contain stochastic process ty actually coincide restriction denote collection rigorously absolutely exist characterize demand demand show derivative everywhere apply process comprise partially observe respect consider hide usual surely constitute alternative constitute constitute gaussian white noise identity top page additionally later first measure existence define measurable absolutely tx replace event restriction alternative belong fine since interested measure augment arbitrarily valid density equivalently statement demand density case adapt alternatively take iv use notion conditionally probability thus notion convergence consistently suit metric another limit borel converge equivalently random induced definition present rather present appropriately specialized later triplet space constitute nx constitute whose equivalently concept instance reader refer article let arbitrary limit possibly define base induce measure hold everywhere generic nonlinear asymptotically would like would yield practically present establish filter strong rhs constitute filter constitute mmse process side recursive realization focus rhs constitute process observation simplicity replace true employ employ thing argument classic filter way attention choice approximation coupling state filter stochastically make observation restrict filter change provide discover realization see treatment rhs resolution observe follow law course question operator member sense operator converge resolution appropriate formulate constitute definition ii constitute atomic equivalently pick either follow cx tx tt cn cn note begin filter operator fully devote facilitate subsection part useful scalar coincide absolute version consider norm present sake norm either constitute definition complete matrix multiplying yield temporal iterate proceed bind result preliminary towards state ii trivial member frobenius norm c norm eq trivially member must use product resp depend certainly correspond right yield may constitute significant throughout analysis simply constant affect constitute throughout effort reasonable limit relate member functional family euclidean proof square positive regard write next numerator express c tx constitutes remove equivalently therefore measure paper consider choose constitute conditionally admit pp existence supremum explicitly assumption rhs rhs however naive equivalently hold constitute mean matrix element make define complete leverage convergence connect weak stochastic tt member cx tx tt tt hypothese continuity set true x tt dominate would desire course member define bound lemma conditional must interest definition dominate prove result rather since constitute sufficient resemble situation equal unity strong condition next contrary convergence nice intuition respective sufficiently state intuitive closeness example base filter feed resolution parameter uniquely approximation additionally proof elementary omit natural circumstance ready establish subsection tx exist measurable subset measure eq concentrate determinant rhs rhs expression attention rhs know yield numerator bound member line rhs arrive equivalently eq rhs eq rhs expectation adapt measure statistically last uniform rhs ensure bound immediately exist subset either upper bounding comprise obviously define get either give since member also imply turn existence limit true adapt recall denominator q adapt least base trivially lemma put since nonzero tend complete condition measure filter compactly time occur nearly essentially framework nonlinear recursive grid perform appendix notational conditional existence
cone need separability programming et nmf instead scale algorithm e grant knowledge view preference go nmf algorithm reformulate detect exist constraint lp large data modification require extreme proximal algorithm lp addition entirely regime factorization extreme cause issue dd find elementary extreme cone dd computational advance help address issue dd organization brief nmf perspective explain propose algorithm reformulate lp paper conclude capital letter letter matlab transform argument non nmf nonnegative approximate solve optimization factorization negative column algebraic factorization refer geometrically generate depicted mind follow definition extreme convex combination exact generate unity entire vector index unknown equivalently constitute therefore efficiently context nmf lp belong incremental descent prominent exist large drastically constraint lp optimization multiplier dual I separability exist feasible selection dual imply program find dual use factorization identify factorization low zero let cost belong determine lowest readily obtain proximal algorithm pre column sure normalization rewrite lp set column stop update project constraint positive element switch experiment gb ram matlab version generate instance extreme create element column generate combination select carry nmf analyze topic allocate begin set effectiveness achieve list regime namely basically gray randomly image
sentence et achieve encourage likewise address memory sentence phrase base simply source whose model feedforward translation need close work show vocabulary almost simple lstm extent word sentence conclude great dependency much unable train rnn translation sentence although ability lstm translate initially long memory researcher performance yet dataset little difficulty work translation accuracy well xu google team google le deep excellent performance well sequence end make long short deep lstm english produce lstm achieve entire lstm additionally lstm difficulty use lstm aforementioned score task also sensible phrase word passive source sentence lstm dependency source optimization easier excellent powerful parallel surprising power ability sort neural conventional learn supervise backpropagation set result human solve rapidly backpropagation apply target dimensionality significant limitation whose length recognition likewise see word sequence therefore clear useful pose challenge input straightforward application long lstm sequence read fix vector extract output lstm essentially language lstm lag output neural map produce phrase system novel attention mechanism network elegant translation assume monotonic alignment ensemble far direct large lstm vocabulary score penalize translation cover relatively vocabulary architecture room outperform publicly baseline improves publish surprisingly recent experience well order sentence many dependency make sgd sentence source sentence lstm sentence variable tend sentence encourage lstm representation mean different qualitative aware word fairly active passive rnn feedforward neural standard output iterate h sequence input ahead length strategy target al rnn dependency short learn long dependency goal output sequence lstm obtain last lstm lstm lm whose initial softmax word vocabulary sentence end symbol outline compute representation actual use different output negligible lstm language second outperform choose lstm implement size lstm well lstm score lstm lstm solving discover learn much well sentence lstm drop score believe cause introduction short normally target sentence result minimal time lag target language unchanged source close target language minimal lag backpropagation communication overall early target confident prediction sentence sentence train see input lstm deep embedding input vocabulary thus deep significantly outperform reduce nearly state naive word result lstm pure recurrent lstm initialize lstm momentum learning learn half epoch batch vanish scaling sentence length sentence minibatch sentence short long sentence minibatch sentence minibatch speedup implementation lstm slow purpose different gpu activation gpu soon remain softmax gpu responsible multiplying english minibatch training take ten score quality score get report result initializations lstm pure phrase mt vocabulary list system lstm lstm ensemble ensemble et baseline lstm baseline well discover lstm show present du du il une des ann pour les collect es les du les du es est une des les une les dans air les un cr la les es
utilize cyclic add nucleotide template nucleotide dna signal know nucleotide nucleotide complementary template synthesis reaction subsection nucleotide nucleotide read dna template nucleotide depend nucleotide reaction condition determine incorporation study fix actually length cycle previous length natural previous previous yield investigate subsection old new model incomplete nucleotide incorporation eqs incomplete nucleotide incorporation eqs special complete nucleotide incorporation condition mention technology add four pre determine unnecessary specification name four kind permutation previously cycles cycle successive cycle nucleotide nucleotide template dna template dna nucleotide nucleotide incorporate certain kind detect incorporate detect nucleotide incorporate nucleotide ideally error nucleotide complementary template nucleotide add nucleotide complementary template possibility kind deal follow variation sequence sequencing incorporation sequencing sequence identical template sequencing individual template lose gradually decay sequence reaction synthesis incorporation nucleotide base incorporate basis nucleotide cycle template dna nucleotide cycle incorporate template nucleotide context flow order uniquely signal nucleotide incorporation strength signal incorporate nucleotide cycle use indicate cycle flow cycle example number sequence cycle first cycle utilize dna sequencing technology sequencing sequence instead template avoid advantage sequence reaction control adjust nucleotide incorporation sequence reaction incorporation basis nucleotide cycle region sequence technology try cycle incorporate incorporation rate utilize nucleotide nucleotide incorporate nucleotide incorporation delay next cycle nucleotide incorporate reaction flow incomplete nucleotide incorporation template signal combinatorial question complicate complete nucleotide incorporation tool solve solution complete incorporation incomplete incorporation united use template dna basis position sequence nucleotide flow nucleotide nucleotide incorporation flow fix length cycle incomplete nucleotide incorporation determine nucleotide incorporation generating length flow cycle flow cycle assumption nucleotide incorporation first value table arrange length cycle show nucleotide incorporation readily evident dp dl l cccc cycle base interesting cycle look table fix transform normalization factor find small negligible become big somewhat flow cycle sequence flow cycle create part cycle cycle long cycle flow cycle nucleotide sample example ba get get hence second length sequence nucleotide incorporation cycle finite subsection analytical enumeration previously check however analytical c program mention avoid unnecessary specification name four kind target sequence number avoid ambiguity add instead development nucleotide incorporation type delay number nucleotide cycle complementary template nucleotide current cycle incorporation special q situation extract coefficient power place detailed deferred flow permutation nucleotide nucleotide incorporation nucleotide cycle cycle nucleotide nucleotide incorporation conditional factor incomplete nucleotide incorporation flow cycle last nucleotide nucleotide cycle ix ix ip ix nucleotide incorporation recursive analytically transform exact cycle sequence incorporate nucleotide flow cycle nucleotide incorporation start nucleotide incorporation later nonzero evident probability flow cycle irrespective nucleotide part flow cycle must cycle nucleotide flow cycle sequence happen part reason first list table note factor table complete nucleotide incorporation factor list sum equation bivariate elementary flow similar generating function sequence fix distribution flow cycle nucleotide composition mean flow cycle close eq denominator factor small module series expansion formula close equal base difference exact nucleotide value flow approximate exact cc c exact exact calculate eqs nucleotide incorporation together normal variance nucleotide composition probability exact variance eqs normal slightly thick approximate number flow central cycle st depends last incorporate sum incomplete incorporation may incorporate give nucleotide flow incomplete incorporation term incomplete nucleotide incorporation nucleotide incorporation incorporation flow incorporation still nucleotide flow cycle fp nucleotide flow cycle nucleotide incorporate flow cycle irrespective nucleotide nucleotide cycle next nucleotide cycle incorporate cycle next incorporated cycle reason incorporation reduce factor eq
overlap bic tend group proper experiment report cccc group run improve hyperparameter set default sensitive hyperparameter show agreement actual point figure well find centre high dataset final respect probably separate distant galaxy lie understanding nevertheless affect configuration pruning also extend split latter classify observation position care leave extension interest know true acknowledgement author would thank comment early complete school sciences sc insight centre foundation grant research foundation grant ip proposition corollary statistical sciences centre school sciences college abstract criterion base clustering automatically effectively thereby include allocation selection practical use prior exact avoid one cluster observation algorithmic mixture model greedy search cluster crucial number reversible mixture alternative author efficient carry label similar approach context model rely throughout observed allocate categorical parameter approximate evidence maximize variant introduce base integrate complete differ complete use bic complete cluster framework assess bic latter include entropy group well prefer configuration good distributional solution appear indeed usually refer homogeneous point penalization discriminate overlap group become gaussian exact common thank conjugate distribution allow apart allocation block framework finite exact formula heuristic extend capable f return algorithm little answer straightforwardly allocation univariate situation explore section form framework routine drawback hyperparameter paper end remark ease framework univariate namely multivariate although applicability iid b focus context allocation call log define involve mixture dirichlet allocation iid variable every set hyperparameter model differ symmetric label wishart scale dimension modelling outline general exception assume shape crucial return complete already set collapse distribution group obtain factorization evidence take parameter formula final term hand centre determinant bic estimate hence every depend partition regard naturally take modelling among group fulfil exact yield advantage allocation indeed convenience depend suppose arise hyperparameter infer optimization simply restrict hypothesis value course overcome specify asymmetric similarly distributional possibility brevity leave task work result select cluster solution parametric include indeed paper serve mainly direct scale main global optimum second concern hyperparameter issue routine complete combinatorial greedy conditional mode mention rely configuration greedy employ block work number infer idea routine configuration informative clustering start random update complete loop yield stop configuration index change let number dramatically search usually reach convergence merge I completely end loop make soon group collapse get poor sensitivity rather assume split able mainly increase time obvious optima able leave must tackle issue author propose routine start configuration final merge avoid local optima merge current frequent finite mixture regard quantification wise state need less iteration merging cost interesting greedy routine objective algorithm calculate computational introduce greedy rather drawback local optima observation whereas change yield greedy gets indeed easily happen enough allow exploration therefore propose large leave well wide combine update multiple instead allocation belong nearest evaluation objective actually realization beta binomial trial group allocate need although objective trick reduce algorithm pool pick observation allocate increasingly number point combine thus dissimilarity storage proportional distance usually fine job proper transformation couple near hyperparameter version require simplify objective affect hyperparameter choose possess one interested specify informative standard extend mind limit interpret hyperparameter information symmetric make remove equal concern proportion realization value jeffreys whereas well chapter essentially small rise default hyperparameter choose observe datum constrain suppose narrow elliptical high default choice propose account many accord range choose diagonal diagonal describe cluster position yield combination shape identity omit univariate default wide case note concern parameter default hundred observation careful default however concentrated overlap may distribution routine update propose describe several clarity solution obtain greedy algorithm maximization model choose denote configuration expectation comparison create approach differently model procedure impose covariance simple maximization clear describe expectation bic suppose return contain thus compare maximization completely quantity proportional maximize obtain consequence update maximize one mean must suitable point exact version specify informative solution three mainly hyperparameter solution intend observation bivariate figure solution htbp right configuration correspond bic stress really agree configuration simulate represent value correspond show clear cccc
two treat extent relation continuous follow reduce write singular value decomposition closely principal approximate marker fix eq alternating step svd pre regression interpolation individual em summary project onto usually point marker vector scale find predict project q solve marker divide marker square length value label calculated average result like character individual marker generalize bilinear expect possible centre straight line mean prediction e project marker marker see obtain marker marker label reference practical view interesting point representation predict point axis explain observed trait separate maximize logistic column response column separate part row individual score package representation contain nominal category individual last use probability category present trait trait make identifiable log odd relative category b odd odd would interpretable probability category logistic relate relate component trait predict long straight make surface category logistic describe binary contain measure ordinal order column indicator categorical indicator expect individual value ij cumulative probability vector trait item variable define category intercept set response restriction probability obtain high dimension variable formally boundary scatter diagram establish search homogeneous characteristic multidimensional help search responsible difference individual htb logit binary categories eq jk contain odd ordinal cumulative share geometry geometry calculation category subtract cumulative htb equivalent fit proportional odd regressor response surface surface long particular category lie straight category item straight line project direction would segment except many segment separate contiguous equal represent point contiguous divide span region predict particular category boundary intersection contiguous must q cumulative probability hold calculate several category never rest category separate point calculate htb pt existence contiguous may example simple deduce combination equation intersection category solve solve equation j root negative intersect calculate transformation would calculate axis parallel prediction category order hide order category category high intersection back step start could equation calculate predict sequence precision step obtain previous search one regression ordinal regression individual estimate response paper regression interpolation change procedure quadrature integral procedure individual choose category individual part maximize maximize perform ordinal regression column procedure well quasi span explanatory see classify search responsible probable existence logistic see procedure remove problem choose simple maximize change affected way variable likelihood part could penalization posteriori distributional marginal gauss quadrature quadrature represent quadrature marginal posteriori score q quadrature individual dimension span among reference year organization department office european survey characteristic european members usa national institute focus effort carry availability resource science technology carry european union office european technology production resource technology survey try level group carry main international group quantify focus degree public private frame operation provide institute university thesis university database comprise belong international education award devote advanced study design region equal probability selection equal random select region assign rest measure module website http www process answer study attention module find aspect job code total algorithm two indicator loading table classification high challenge separation problem logistic interpretation high job security work almost condition job job status factor htb security job challenge degree social status benefit challenge independence status observe figure htb variable job security hide partially appear organization public job majority htb ordinal point representation angle benefit job security present first away aspect behavior slope although happen represent information challenge job security behavior category challenge aspect htb finally mind detail outside category possible challenge
choose computationally efficient sample query strong set strong possibly adaptively unbounded oracle simple differential theorem well differentially private threshold minimal notion privacy line connect technology bound private interactive analysis task appear hardness privacy digital introduce use code certain setting nearly establish introduction extensive code interactive give influential work interactive code name hardness false discovery give construction suboptimal code give code interactive guarantee intuitive first algorithm answer arbitrary adaptively query answer privacy black differentially answer many adaptively rich computationally inefficient accurate exponentially construct interactive code main ingredient code hardness non interactive code section read motivate definition interactive helpful review motivation code code movie piece company may copy company copy remove code ensure combine user create say trace construct key drawback single user prevent identify interactive company content copy distribute episode tv internet episode show combine stream stream company stream soon company identify continue copy another code consistency constraint say remove code robust robustness ready code game may user output vector empty user let c want consistent succeed recover convenience round execution notation notation user formally notation require user interactive say interactive robust error probability depend constraint call interactive code interactive adversary inconsistent may seem recover notice mean interactive establish interactive code every interactive failure bit traditional regime code code match code failure large logarithmic construction robustness give interactive interactive setting weak robustness completeness require high rather version set application false interactive code variant modify distribution support function pc nc pi ht issue receive j ii code figure addition precise set parameter help convenience intuitively call user correlation answer measure choice ever exceed mean answer thing never answer unknown answer must closely interactive random fix randomness unknown analogy one would round prevent like tail completeness gives specifically imply show interactive see answer score set ensure answer fraction round say force inconsistent eventually reduce answer prove equation simplify analysis issue fix force tailor concentration take equation choice follow suffice ensure order fourier always arbitrary fourier also density proportional handle round adversary round happen inconsistent answer normal round inconsistent answer round since round round mean small normal imply round conversely round concentration concentration user essentially form prevent instead proof verify desire bounding q ia add take ij answer draw likewise interestingly interactive still fail identify indicator event take adversary result code identify lower bind tail consistent establish good score adversary must constraint bias fourier relate round adversary adversary firstly lemma pp product fourier analysis bias expansion give expression effectively integrate calculus n nj jx accuracy n interested say oracle query circuit evaluate circuit attack triple take security randomize key message output ct message ct roughly security even access message security polynomial start simplify definition security need adversary pair ix interactive code robust fraction let let ct user attack comment structure attack oracle must true eq oracle oracle effectively adversary computationally oracle mean answer query arise oracle respect interactive th query false therefore enough require interactive theorem way adaptively sample start establish interactive however security enough user small security user whereas query user entry oracle formalize compare zero adversary attack without break security scheme let n interactive I otherwise sufficiently straightforwardly security depend adversary adversary efficient attack ideal high must hold event polynomial every definition defer claim easily eq answer consistent answer choose query every attack input query assumption query every eq second ct ic claim accurate error complete argue attack answer q deferred computationally adaptively choose every easily security answer polynomial put claim together main sake oracle theorem attack claim contradiction simplicity hardness prove theoretic datum level fact rely need security security message slightly discussion simply unbounded adaptively query adaptively private define notion appropriate respect change mind privacy game ht jx j qx n qx shorthand adversary q computationally accurate answer choose attack code user scheme let I ct query let start establish user small security interactive ideal attack parameter I column ct ct ii n ji every straightforwardly security query entry adversary access view adversary also fact argue attack attack probability must hold game security defer combine every claim answer recall number answer inconsistent adaptively polynomial claim terminate unless assumption sample argue attack answer ideal attack let computationally computationally efficient every definition security defer claim obtain adaptively every claim security thus oracle put together adaptively query contradiction reach claim terminate early contradiction theoretic hardness privacy discussion early interactive code attention claim claim security section claim claim modular claim section relate fashion omit brevity begin security via take key whereas security scheme choose adversary whether interact q claim claim let polynomial construct adversary attempt break security construct break security simulator new sx I ix ct ic occur claim efficient construction efficiency notice occur oracle return holds show query either unknown moreover message choose choose complete interactive interactive interactive answer construct interactive code parameter improve robustness interactive interactive interactive length user error specify answer c c continue terminate interactive code user error failure suppose non interactive interactive adversary adversary c receive c j sa interactive adversary round consider theorem lemma theorem claim corollary edu edu bind adaptively computationally efficient answer accurately unknown expectation statistical correct study al answer bound hardness assumption efficient give valid answer adaptively implication answer query choose call optimize hardness fouri analytic code simpler flexible construction query finite summary outcome unlikely occur chance discovery occur analyst incorrectly observation decade discovery highly influential control false scientific research typically discovery attribute possible inherently query interaction recently paper formalize give universe suitably provide answer generalize answer achieve analyst adaptive previous query answer adaptively choose arbitrary adaptive analyst answer answer query probability however situation turn query ask adaptively al computationally answer show oracle answer whether achieve oracle privacy assume answer importance discovery algorithm answer unfortunately assume function computationally query conceptually interactive bad oracle answer adaptively statistical private al query query sort restrict hardness whenever dimensionality requirement query
set estimating assess ij introduce estimator graphical neighborhood glasso precision entry correspond define family minimize ham stress standard calibration close performance besides lasso selection tuning estimate node correspond via similarly hamming highlight p c ij ji ji subsample fix invoke q ef pl km km estimate ij ji topology example confirm follow insight six graph topology vary range graph edge add edge e edge uniformly e edge uniformly number entire construct edge add selecting proportional current random consist set node set precision entry precision entry eigenvalue dimensional avoid new assess accurately strength neighborhood unknown outperform spectrum become pattern biology interaction expression protein sequence graphical represent underlie simultaneous gaussian know entry become particularly even precision global art solver graphical framework gaussian model parametric advanced calibration scheme approach practical introduce neighborhood particularly novel assess strength method neighborhood across wide scenario since unknown therefore promise
stimulus passive formula active environment active et implicitly consider explicit relevance handle active learning emphasize applicable handling miss case many convex machine literature class however approximation publish machine learn stochastic researcher expert common al objective important objective strictly example logistic semidefinite minimizer hold assumption satisfied important practitioner machine checking might require considerable assumption require literature conclusion obvious impossible typical mathematical statistic example state critical point desirable white theorem theorem risk al et type approximation review stochastic et variable metric stochastic converge white theorem risk wang variable bfgs random mass generative modern include mixture variable novel contribution situation passive assumption unique strictly stochastic descent proof theorem design interpretable theorem practitioner finally applicable five machine literature new minimize al design fundamentally development field ensure correctly apply specific also general handle involve minimizer exist necessarily sufficient condition continuous probability strong stochastic approximation partition let let number dimensional bound td bound satisfied subset value condition appropriate hessian encounter likelihood uniformly bound hessian layer perceptron evolve closed bound ensure solution practice empirically rather verify discuss analyze stochastic learning machine begin initial machine update guess refine iterate update assume mini batch identically mini value upon passive statistical function function attempt guess observation equivalently stimulus parameter give magnitude govern strictly search direction determine appropriate typically compute direction condition commonly ensure deterministic descent appropriately choose search relation mini increase tend law hold increase direction appropriate sufficient stochastic search type increase eventually decrease et specifie period stepsize period stepsize search direction size one call critical variable direction adaptive e et et al quasi likelihood cross realization estimation find v immediately use descent term relatively evaluate correlate whose computationally method evaluating expect correspond term multidimensional mini highly observation kk mini independent density mini initially increase integer environment update trial hidden consider variable characteristic architecture partition dependent realization visible likelihood rewrite derivative eq obtain substitution h imputation realization give expectation maximization define stochastic multiply integral mini trial sampling parameter estimate learn stochastic method analyze behavior positive integer case expectation algorithm use current probabilistic iterative machine environment characteristic deep representation learning machine implementation environment suppose episode episode independent identically addition episode episode episode machine passive specifie learner learning select action state machine mass environment characterize specify initial episode conditional state episode state episode episode j incur machine episode dependent allow possible adaptive machine formula derivative operator carlo approximation derivative gradient see episode since open system however episode influence next action methodology involve episode episode identically distribute interact environment episode sample overlap code dependent learn environment passive proof combination appendix et et review variation see appendix et al expand function theorem substitute identify condition bound bound exist iii piecewise continuous bound asymptotic function conditional respect
nature minimize complexity reason depth consider computational computation conference cut hyperplane line bivariate variety attempt take contour exploit circular suggest subset depth contour well depth cloud depth contour single complexity depth complexity line depth calculate depth successively update low coincide algorithm latter run sphere accord define region sequentially exploit depth updating continuously handle contour quantile univariate projection envelope region connection multivariate directional quantile correspond hyperplane depth contain direction hyperplane set algorithm hyperplane intersection form depth first search algorithm direction suggest depth special fast algorithm elaborate try save weak I small projection univariate depth explore generate line connect line normal hyperplane pairwise distinct hyperplane claim direction prove depth deviation real work try achieve precision exploit author one depth suggest framework computing affine tuple project orthogonal complement depth orthogonal tuple capable deal general tie lead theorem present issue non general notation hull complement w contain short depth eq depth compute integer remove add integer write subset position hull contain index denote contain lie hand contradict hyperplane whereas linearly linearly I every choose proposition therefore complete subset follow immediately precede projection orthogonal conclude simplification point map depth compute finally therefore independent fall orthogonal former category second computation precede space step arise section rise reduce dimension specialized algorithm bivariate choose result hyperplane hyperplane hyperplane general position never enter recursion hyperplane overall new min project point depth since subset complexity combinatorial combinatorial independently I new min yield outer loop hyperplane exception point map reduce algorithm recursive variate w variate variate stop case basis hyperplane j new min min section external library source code request easily implement orthogonal implement routine precede point calculated datum origin number store scale depth problem improve loop drop zero however experiment due independent operation algorithm base possess high parallelization exact execution depend step etc report table later remain performance differ intel core processor normal origin present execution cell middle line try try extremely vary increase exceed hour far outperform outperform former superior framework violate also call execution additionally heavily position unstable instead report effect dimensional design quick handling get large tie
orthonormal r dr jx u proposition function function perturbation assumption necessary perturbation identical alternative form functional taylor construction vanish segment positive third g separation eq need hellinger construction hellinger apply proposition hellinger divergence proposition far allow hellinger bound constant note n use contradiction guide framework desire conditional good grey sift project method hence hellinger construct similarity metric image denote sift image divergence clustering depict image class affinity exhibit hellinger patterns image nn achieve apply pixel intensity accuracy find perform divergence result imagine treat similarity metric classifier htbp p x f I iy nz I conditional divergence version rgb rgb pt false department computer science address address estimator distribution assumption statistic leave favorable theoretical apply derive popular quantity exist theoretic play statistic mathematical sciences addition analytical tool hypothesis functional algorithmic task develop influence range intrinsic several statistical recently gain object model mutual conditional divergence mutual information building graphical allow estimator functional post use von idea statistic study split setting function expand propose ds framework contribution however estimator perform analyse achieve parametric rate estimator normal sufficient approach estimator entropy functional list functional available despite generality image focus quantity relevance technique brief post hoc correction influence detailed approach functional knowledge paper post hoc follow paper integral extend functional density consider splitting build functional superior fundamental inspire split estimator functional analysis rely statistic look compact equipped lebesgue measure measure absolutely focus functional twice permit density ease exposition presentation definition distribution absolutely derivative belong development von distributional impose notion satisfie uniqueness domain consequently define term dirac delta sufficient assign term measure density control density functional satisfy form consequently write taylor expansion lemma appendix q expansion basis functional distribution argument influence satisfy estimating suggest construct influence right side expectation half expectation preliminary averaging confirm smooth datum trick commonly several work analysis cauchy schwarz inequality theory good stand decrease efficient except theoretically extend functional version cycle point however latter modification estimator entropy divergence mutual several hope good reference practitioner unconditional list expression software implement functional estimator property method several functional require integration estimation define r ds smoothness density kde bandwidth nh smoothing kde kde estimator form kde truncated kde use satisfied smooth functional table achieve square mse review note self contain directly bias follow schwarz attractive property agnostic correct rate estimator bound summation lead bad rate version bound correct couple use kde indicate limit challenge summation asymptotically rate estimator empirical asymptotic one theorem satisfy normal normality allow confidence theorem give valid interval use asymptotic differentiable technical zero order vanish rate behavior unclear degeneracy occur arise important two let h alone sufficient guarantee functional divergence assumption difficulty bound minimax functional define positive functional minimax optimality functional however gap ask improve rate regime functional taylor functional statistical gain believe estimator conceptually favorable functional assumption attention decade body work focus estimate shannon nice include enyi entropy paper method estimate kde near neighbor conceptually several drawback secondly kde obtaining plug require aware principled hyperparameter use optimal validation secondly method kde integration avoid functional see divergence estimate analyse rkhs straightforward clear convex problematic use weight divergence establish normality parametric strong work divergence method applicability include divergence compare software estimator estimator polynomial plug bandwidth perform plug make perform estimator inferior require software case poorly hyperparameter asymptotic dimensional hellinger divergence estimation repeat experiment compare asymptotic plot suggest hellinger whereas hellinger superiority focus density reduce expansion equal eq compact finally make stein analysis stein inequality intend estimator estimator originally von asymptotically begin proof lemma prove conditional bias preliminary condition consider derivative taylor normality addition begin around q step add chebyshev note step q ready kde achieve bias far bound bound schwarz n proof error bound kde integrate squared stein set sample stein shall note inside summation substitution lipschitz constants stein expectation first remove twice inside apply stein get complete analyse begin ip p bias bound bias condition first follow boundedness taylor hold asymptotic normality
conclude despite minimize good problem light consistency eigenvalue robustness certainly help behavior covariance build component soon choice quite within framework minimizer pr sr result lemma apply aim small critical threshold happen sense exploit everything polynomial package eigenvalue covariance course contrast converge estimator vary approximation consider autoregressive difficult construct estimator risk risk computation perform figure correspond red green stein good seem quite scenario ar bad frobenius little thing loss estimation frobenius restrict minimize risk invariant loss noise normal build robust good aspect assume work construction proof quite heavily outline quite depend behave wishart assumption restrictive high extension addition describe invertible absence risk mathematic method outline could consider might surprising present behavior estimator covariance zero since show risk equal contrast unfortunately wishart extensive behavior limit analogue frobenius allow unbiased adapt study tackle eigenvalue optimal recent look frobenius appealing account top eigenvector behavior perhaps know automatically condition covariance quite parameter interest appear reasonable although currently stein compute risk estimator identity implicitly unfortunately covariance satisfy compute pt conclude inside associate decompose pt eq get eigenvalue b notice generality would similarly iii loss possibility q iii joint define j pn pt polynomial happen l z dl proceed change pt respective dx consider first one l z l q integral converge long case b pt integral treat constant calculation old pt eq lemma collect define algebra part proceed weak regularity first smooth condition weak satisfie find variation stand calculus obtain pl notice estimator notice weak minimum simplify define p parameter carry argument notation p theorem weak rewrite eq twice satisfie therefore equal root positive yield obtain similar spirit therefore n mt weak conclude eq ii denote limit writing except cl almost surely divide eigenvalue therefore quantitie white f distribution q statement follow poisson substitution obtain variance contour transform form therefore go back bound lemma therefore use done obtain simplify decompose hellinger affinity density cauchy schwarz normals hellinger affinity supremum define note q contradict term conclude estimator definition nn l n k problem frobenius pca investigate use show use theory strongly consistent essentially asymptotically past year context study sparsity zero coordinate covariance distinct eigenvalue eigenvalue good high see also theory pca traditional retain rigorous require good estimation although covariance analogous context asymptotic regime strictly covariance frobenius essence really estimation level consider estimator finite rank finite serve principle propose restrict perform correction class unbiased invariant loss h refer noise calculus variation spirit idea directly risk truth turn depend dominant prove behave strongly estimator essentially minimax rate estimation construct worse worst remarkably show contrast estimation zero perfectly think structure generally accept unless robust encouraging construction simulation proof claim simplex pa norm notation stand wishart stack let distribute wishart construction convergent restriction eigenvalue estimator satisfy form expectation similarly weak regularity finite previous associate estimator similar dimension satisfy regularity regularity pt strong weak pt careful reveal spirit emphasize merely weak correction like performance common convenient move mention early loss thought frobenius stay unbiased risk rich shape part depend eq functional asymptotically sense explicit construct estimator
throughout especially towards illumination change sequence consist illumination frame sign illumination appear severe illumination change slight motion fail light accommodate robustness illumination whole book severe appearance change book book appear track tracking video drastically illumination pose purpose adaptive object appearance change argue model appearance apply template lead robustness record template novel manifold finally track inference propagate comparative video art illumination appearance unlike method motion formulate updating measure add bag resolve issue enhance introduce particle filter extension object track join acknowledgement department communication centre program figure track school technology appearance able pose variation approach base subspace several image accommodate use subspace represent object may propose approach euclidean inference filter quantitative evaluation challenge video approach obtain considerably track fundamental surveillance behaviour retrieval problem appearance design appearance intrinsic pose camera illumination rather rely object discrimination illumination variation achieve modelling subspace believe via adequate subspace origin translation meaning point shift small attractive purpose generally maintain location account generalise subspace affine subspace subspace track find affine subspace frame affine subspace novel distance affine via combination mahalanobi conceptual propose tb point subspace subspace measurement group consecutive object frame use frame third right region frame b object image dash wrong wrong location represent model addition frame distance result select candidate appearance affine object tracking approach tracking subspace update recent drift location object measure subspace subspace track update precede frames subspace distance measurement method subspace manifold wang online scheme contrast point subspace distance comprise block propose component take history previous create candidate object frame consecutive frame filter monte find probable candidate module encode appearance subspace achieve decision module bag object model encode affine module module history bag primarily drive attain variation desire appearance whole body encode object moreover person visible tracking appearance upon termination keeping model body whole tracking cope subspace scale frame blind inefficient since plausible monte space probable key idea become briefly algorithm track virtue probable candidate estimate I object recent bag frame object bag recent template affine track bag track challenge scenario rate use extraction slide template frame address particle appearance bag subspace module bag although euclidean distance minimum distance angular affine angular origin affine simplify address limitation manifold j mahalanobis orthonormal basis length geodesic geodesic q subject angle compute mahalanobis affine subspace distance distribution kl identify distance manifold define choice measure length mahalanobis result template tracking framework likelihood normalise subspace bag likelihood template opt sum frame update object state object frame state affine region calculate likelihood bag final likelihood eqn object framework generation geodesic operation compare affine bag operation tb box sake clarity demonstrate overall pose variation b appearance face various involve pose change analyse propose eight publicly project consist tracking task face tracking face face book frame video bit image sake efficiency affine candidate model template trade appearance precision affine performance model significantly idea affine performance affine center video subspace face assess method instance boost track collaborative publicly code propose box frames book
column work rank problem wide group analysis lrr solver focus follow compatible reformulate unconstrained low accelerate alternate sdp sized require continuous widely convergence divergence linearize suffer issue propose accelerate trick lrr drawback singular partial fast full problem reweighte relaxed smooth weight analysis converge indicate lead sparse rank iteratively reweighte work norm affine non robust completion proof norm thus application actually logarithm introduce relaxed iteratively reweighte least future regularize lrr theoretically general solve propose art avoid svd iteratively reweighte square lrr lrr low minimization lrr vision lrr reformulate smooth two I smooth call smoothed smoothed lrr lrr bring lrr smoothed smooth make easy convex globally solution easily convexity third order eigenvalue say furthermore solution say converge update solve separately break solve reformulate denote column svd qx z equation may matlab solve fix motivate algorithm separately treat matrix lrr problem lrr lrr compute art accelerate fast choice tune good accelerate worth though lrr structured group non overlap structured completion quite objective smooth popular norm variable main much guarantee logarithm induce norm derivative nuclear iw jj z g iteratively reweighte sum term square smooth proof show concave concave proof norm nonzero differentiable concave definite concave differentiable let get sequence limit globally convenience description implementation shall smoothed lrr lrr check worth nontrivial affine work unconstraine rank unconstraine minimization weight variable update weight usually easy update difficult updating also proof affine constraint square see handle square simultaneously also concavity synthetic real solve lrr behaviour converge appropriately decrease initialize norm experiment fix synthetic basis random corrupt gaussian show value lead inaccurate similar phenomenon lead accurate solution fast well denote type method svd matlab third party package default converge code lin set pc intel gb windows version time minimum time method datum emphasize lrr usually large solution rank plot computing see except linearize svd hence unstable sized high rank completely database face conduct subject face image pca normalize cut affinity lrr iteration linearize within iteration database draw sequence first onto subspace pca lrr project lrr fast c c subject acc acc inductive principal solve sum cause continuous smoothed solve iteratively tx face recognition learn training remove corruption two consist face acquire pose pose image subject svm recognition see accuracy different solver run much large figure plot recover obtain successfully remove face c iteratively reweighte solve minimization joint relaxed globally problem lasso overlap group lasso concave xx yy use solve dot together non eq minimum q imply sum eq globally converge j denote rewrite therefore mathematics university master technology china currently student
paper copula bivariate marginal square traditionally measure strength dependence index course order pose must impose require every reflect degree tail necessarily certainly interval property relate copula leave admissible admissible admissible motivated determine co risk admissible probability equivalently q independence representation diagonal serve tail path may maximize illustrate follow view index neither measure consider check admissible correspond right hand equality hold copula symmetric motivate definition give call maximal eq simple arise admissible path refer maximal employ variant classical index namely assume limit exist assume function illustrative concentrate conclude section copula example claim rely behaviour copula non rely behaviour copula explore model dependent inter amount et dependence w et zhang wu development topic excess economic pricing work research goal aim maximal role diagonal dependence copula investigate copula aspect reference let random pareto quantitie let copula et et al dependence newly suggest index define z z weight therein calculation noting dependence observe classical copula dependence formula next function unique tail maximal maximal copula symmetric copula whose diagonal copula give path k right panel none coincide low tail ordinary subsection symmetry imply path copula copula another example copula among illustrative example motivate present sometimes whether positively dependent specifically copula consideration maximal solve eq maximal path reach admissible fail cf end fr rewrite copula copula low tail maximally interpretation every admissible path generalize index copula obviously weakly maximally dependent compare copula get moment asymptotic formula c copula index interpret whenever course expression maximal demonstrate importantly close generalize eq closed path copula generator hold copula copula increase tail behaviour copula diagonal tail conservative dependence notion herein main assess copula paradigm modern management york much research environment research science engineering discount aggregate pareto distribution asymptotic capital expectation large claim york copulas tail dependence copula management ed pp mean copulas universit finance tail copula near independence extreme copulas density york nj characterization proof loss decrease achieve split decrease point maximal formula two index maximal conclude equation equation finding denominator xx solution derive form furthermore also lack close maximal expression arrive rx side tail proof tail li li order risk york mm section corollary theorem property mm path maximal department mathematics york sciences mm numerically measure extent extreme dependent risk paradigm management phenomenon hold asymmetric copula
et al identify situation simulation study set proximity gamma normal long central chi use circumstance brief overview issue involve population distribution like fisher justification approximate provide implement binomial test unknown estimation state satisfy demonstrating issue study several distribution simply asymptotic particular derive asymptotic variance large finite moment population density sample statistic nan distribution mean population mle remain rao justification asymptotic variance make accommodate total gamma variance measurable function translate know expectation easily obtained consider statistic nx expectation follow ne chi degrees prove theorem log calculate value simulate usage ratio respectively aforementione point incorrect exponential shape mean b location mean mean scale scale shape parameter way check fourth population population nan condition condition mean nd rd central ratio freedom like variance suppose differentiable nan apply moment population third fourth central population asymptotic normality smooth moment map neither equal otherwise write delta distribute asymptotically example view corollary poisson delta since nm delta n note theorem datum implement goodness kolmogorov suitable good fit implement context total incomplete gamma modeling purpose compute west implement basis low week total east implement series gamma nan associate china variance accept case adequate necessary result simulation study implementation nan however set statistic present accord freedom line nan expect demonstrate rejection nan actually equal weight mixture component mix mode variance specify rejection case nan lie demonstrate gamma illustrate false rejection acceptance nan obtain dataset http www observe goodness fit value gamma shape test statistic statistic true turn acceptance gamma fit figure rejection formal gain analytical strongly nan test pearson chi goodness test show conduct follow pearson chi goodness context followed however gamma largely shape test central chi square follow normal modelling chi distribution freedom aid article way check necessary applicability check whether ratio applicable well chi test class theoretically use fitting non kolmogorov chi goodness reference evaluation hazard journal pp square poisson binomial mm mm c daily series expectation method chi nj pp generate forest fisher statistical research worker company pp medical ed pp incomplete
risk difficult might label describe formulation seed one detector detector risk one frame negative seed generic detector obtained run binary aforementioned sample overlap ann car mean car incur classify regularizer calibrate enjoy optimization express equation minimization w x w impose task comprise model lose seed closely design streaming classifier past similar sense prevent fitting appearance individual object object appearance importantly justification category detector approach interpret category model average learning could difference datum benefit detector quickly seed thus category handle seed could alternative regularizer replace cost demand complex compatible regularizers regularization eq potential multi call average rate rule previous maintain update detector include detector one advantage suited rely practice mini potentially multiple pass datum share window also maintain depict simplicity issue unsupervised line hyper amongst mini appearance stream surveillance learn update scoring improve detection detector sequence annotate static correspond scene duration k allow equal self video detector domain evaluate video stream frame right frame ols detector detector generic detector pre online order detection detector detector experiment fisher reason art efficient category level highlight category appearance model learn able use application yield competitive result slide window ols ols seed algorithm wang detector without penalization detector self camera multi quantitative area detection report detector applicable set I alone video wang regularization compare unsupervised adaptation fix label video improve detector scenario confirm detector video tune object detector second video detector thousand substantial ol scenario illustrate multi tracking help well detector seed appearance trait useful handle intra small improvement discard generic enough video stream seed whereas must generic neighbor stand alone detector improve additional cost gray center amount label camera mobile drive paper detector efficiently stream sources ii contrary without manual intervention tuning objective recall detector exploit line instance level object detector approach world publicly mm learn identically distribute sample unobserve one explain collection particularly video typically collection focus unsupervise object continuously adapt rely related availability black classifier along differ adaptation easy inspire see discussion present mm approach along unseen video stream generate box score seed allow detect seed rest self adapt suffer transfer fit positive lack target address leverage spatio structure learn track automatically positive negative hard negative task yield gap instance learn propose applicable recent stochastic optimization novel parameter challenging world adapt detector domain approach annotate often pass annotate adapt impractical continuous streaming leverage unseen classified confidence classification minimize unlabele readily approach suited exploit spatio video collect sample learn detector rely set video move detector adapt video self learn fine target source use track inspire instance rely learn detector apply video stream track detection tracking detection zhang specific aim adapt also multi across category straightforwardly applicable category object detector parametrize classifier compute region category
algorithm generate posterior correspond full true particular weak scheme subsample interesting future development design likelihood hope researcher contribute area contribution fix collect complement decompose l term draw noise evaluation zero prior available likelihood contribution data point set covariance ard attractive choice error v mode hyperparameter optimize change compute multiplication evaluation optimally costly approximation spline spline surface radial approximate contribution thin spline coverage boundary predict likelihood contribution eq expand shrinkage analogously mcmc vb thin spline gp preferred refinement pair log contribution change drastically transform link logit categorical regression category I many category spline predictor dimension link datum connect tp k proceed since mcmc work computational dominate proof use om om ii straightforward taylor suppose prove r cauchy complete ii computed e part define follow q iid part conclude lemma assumption lm take I note l expectation follow similarly important ready prove stress notation make follow bound py pd iii py py part p py mp py py py mp part expression clearly iv therefore take theorem axiom conjecture mail liu se grant ce like discussion express author view markov many observation costly propose likelihood substantially scheme inclusion observation likelihood broad class approximation present subsample small adaptively choose apply bivariate half million observation survival data bayesian monte big proportional monte distribution early desirable full approximation furthermore selection etc costly evaluation pose however demand computing dataset become increasingly common computation abc vb stochastic great approach currently produce acceptable computationally demanding via augmentation hasting efficient show design crucial chain simple si order magnitude si mcmc draw time budget regular proposal introduce metropolis likelihood prove note even biased sample perturb exploit scheme subsample likelihood accurate application error focus effort advantage first adopt estimate sampling optimal us control improve mcmc budget organize subset section propose sample inclusion probability methodology application discuss implementation main vector fix augment marginal informally draw obtain move propose accept note arise regard choice imply intractable simulate h likelihood depend crucially however covariate observation log contribution log likelihood survey sampling problem total introduction likelihood unique piece example longitudinal problem subject individual series subsample indicator replacement easy estimation error selection determine motivate mcmc unknown true simplify hold verify kp iii motivate iv remainder assumption iii remainder check order taylor v hold subject bound highly posterior approximation chain observation noisy reduce per efficiency markov vice computing factor autocorrelation th lag limit factor mcmc obtain sampler produce vs efficient proportional perfect around fairly normal independent value high conservative obtaining assume approximately unbiased tune independently see tune variance likelihood iterate guess new size bring estimator particularly attractive sampling estimator guarantee crucially variance solve optimum depend computed tuning grid log contribution refer obtain surface mcmc approximation predict contribution approximation exact perform great care computational surface fit gaussian thin spline appendix surface fitting kernel likelihood multiplication appendix surface fit construct period year analyze set variable use bivariate hold control severe choose excess period bivariate probit multivariate probit augmentation illustration bivariate integral probit total draw discard first prior different sampler metropolis rwm rwm approximate log contribution probit link four response outcome separately put example vs proposal rwm rwm efficient draw rwm around plus thin spline expect around overhead thin fraction decrease negligible obtaining approximate rwm confirm decrease htbp also rwm maximum allow mean fraction rwm rwm proposal reached report efficient draw three rwm allow scale rwm show acceptance posterior mcmc nearly indistinguishable explore fractional theorem create subsample fractional error subsample choose average discrete survival illustrate period logarithm total sale logarithm age hazard probability tx ij ij obtain estimation parameter tp use rwm sample equation mode consecutive vast use effect give however illustrate dash panel rwm explore log panel logarithmic show draw vs rwm particularly size decrease
observe consistently classifier attain combine averaging provide classification achieve domain e acoustic representation combine approximately cross point db far g regime attain improvement classifier db snr conclusion apply front end speech operate acoustic address acoustic aggregation demonstrate outperform db perform gain level combination primarily focus term robustness implication extension speech straight forward approach combination classifier error code speech train alternatively propose phone hmm token passing speech former baseline hmm system first pass possible svm baseline though classifier construct solely due solely decision svms determine recognize algorithm subject confusion classifier snr confusion class confusion towards figure classifier attain confusion figure suffer high closure among group cumulative multiclass reduce use level follow instance condition figure meta improve gain outperform high attain classifier db complete hand level subset moreover classifier small adaptation ac uk automatic severe degradation presence additive human central behind art combine front end base compression modelling bring human effectiveness context model elementary gap human system human recognize isolated speech level chance snr db snr levels speech recognition exceed rate human error although conventional reach human attribute fundamental limitation front end extraction frequency cope environmental show take place front end severe degradation environment machine recognition speech front year variance normalization taylor front explicitly reduce effect front end distortion feature depend speech feature previous acoustic robustness front end end derive speech well draw motivation experiment linguistic message decision narrow reasoning accurate signal put consider speech beneficial exploitation inherently secondly sufficiently narrow approximated spectral additive aggregation selective de emphasis result improvement recognition band counterpart end resolution front extract acoustic retain speech potentially discrimination representation investigation propose front assess robustness filtering room cause spectra performance attribute primarily window conventional front end short room impulse response cause filtering speech conventional noise environment several speech literature scope channel filter robustness front compare improvement achieve speech framework hide hmms base architecture token speech classification term robustness linear classifier ratio classify classifier db improvement classifier section additive draw suggest future direction towards front continuous speech task machine increase speech use conjunction front fix front end speech hmms variable address fisher lie scope paper hence feature front length speech study possible extension front section future decision surface maximize two training training multipli bias predict simple k produce boundary linearly feature potentially therefore effectively classification polynomial sophisticated kernel acoustic describe follow svms combine code class complexity capture predict loss loss hamming hinge perform discrimination svms predefine code train code element predict loss classifier feasibility regard loss various ham hinge frequency maximally perfect bank decomposition decomposition cosine comparable somewhat inferior summary obtain different decomposition filter prototype filter bank representation coordinate bank primarily sampling avoid unnecessary limit burden believe redundant expansion speech signal filter could advantageous invariance speech construct capture identity even kernel sign speech polynomial kernel act vector baseline acoustic feature typically speech explicitly take feature evolution individual first divide energy form form dynamic correspond acoustic form component acoustic classifier obtain decision multiplier corresponding assign simple scheme score predict conventional majority voting map label alternative voting condition argument error probability voting contribution overall cardinality ideal decrease speech error majority voting may considerable improvement furthermore aggregation drawback specific use stack weight specific pair aggregation practical stack generalization hierarchical layer svm output svms aggregated meta level meta artificial real speech interpretation isotropic class domain find music fan nature etc sentence impulse impulse conference room spectrum impulse room substantial difference spectra mean view approximation reduce test proxy record location room convert dimensional vector second frame duration frame frames sec representation vector extensively art scheme estimate speech speech noisy relate speech clean feature gmm mixture spectra clean additionally fix variation training compute training consider classifier front end svm clean via perform via noisy use filter error unlikely case filter convolution particular svm exact knowledge offer expect scenario impractical nevertheless furthermore note acoustic ms centre decompose bank examine effect number bank accuracy speech effectively capture frequency relatively speech sufficiently narrow performance demonstrate increase bank reduce extract ms centre standardize within always perform meta weight scenario classifier meta classifier svm score clean meta vector contain clean db snr meta level svm clean meta classifier setup error stack yield voting achieve voting decomposition decomposition cosine bank achieve large composite therefore far list scheme classification white curve correspond combination stack scenario multi style stack development development consist clean noise corrupt presence linear filtering discuss figure frequency acoustic dash curve ensemble method vote stack meta level classifier section meta classifier stack train stack attain vote poorly composite acoustic stack quickly meta classifier assign robustness train base clean corrupt white explain show weight standard deviation assign stack binary classifier train style component style hold speech provide reliable amount speech observe stack classifier improvement match noise stack match performance acoustic classifier white result stack classifier match stack exhibit snr whereas composite db db snr stack range quite remarkably stack train test match db db snr train number weight small fraction style match db performance degradation noise stack figure attribute acoustic colored noise approximate narrow band white comparison report obtain variable condition stack classifier train match match match classifier db db suggest high
uncertain interpretable vice versa denote diameter fix statistic follow statement uncertain diameter dy interval interpretable interval axis indicate key proposition interpretable interval describe distance reach uncertain alternative may test determine definition percentile prove uncertain u u continuous member similarly continuity know imply contain uncertain infimum supremum answer since claim establish increase exist critical word equitability formal equitability interpretability term proof fix statistic fix worst interpretable proxy every base least equitability interpretability fundamentally able signal weak essence equitability interpretability power independence definition equitability extent iff exhibit interpretable confidence continuous base distinguish independence equitability interpretability power requirement interpretable detect independence ignore interesting relationship hope power address concern agree thing work greatly enhance art power think equitability base solely way dataset trivial dependence hundred thousand become manually examine define strength statistic rather power nan hypothesis independence alone paradigm sense reasonable ignore equitability statistic specifically broad equitability discuss detail concept relate implie property sample statistic limit definition estimator focus introduce prove discussion immediate section devote analyze state begin define sequel grid possibly row analogously let represent information population characteristic q refer mutual interpretable section characteristic name characteristic type relationship value let information rather abuse notation sample point order sample characteristic eq set define maximum edge analogously present variable statistic fact consistent consequence theorem statement equip projection eq continuous supremum realize maxima segment matrix consistent analogous corollary refer priori uniformly sample trivial therefore consistent consist normalize consistent true characteristic consistency abstract consideration necessarily suffice grow notice sample characteristic heart use grid quickly lemma build dependency grid consider master cell master sub grid grid seek bound grid variable consistency seek require entropy contain idea grid grid allow capture dependence distribute grid cell induce cell marginal argument analogously since sub grids master give grid mass two refer horizontal vertical line line add line line grid account eq triangle bound define concrete bias grid integer fix column away provide proceed show small allow inequality bind rest write I pair desire cn concave must sum observe horizontal involve observe give final grow fast lemma yield specify fix q sufficiently entry suffice difference n come hold least ensure desire ready infinite equip supremum norm pointwise sample show write n fm fr nm fm vanish pointwise adaptation continuous second map mean latter lemma tell probability consistent define essential actual quantity something intuitive rigorously previously choose investigation asymptotically different tradeoff easy relationship alternate alternate definition distribute normalization subject grid grid grid complex grid well show penalization also think necessary result prove relationship achieve statistical prove restriction grid ensure hold lastly simply consistent exist theory necessary equitability independence prove provide alternate jointly distribute supremum population characteristic interesting reason light characteristic normalization turn achieve mutual pdf version mutual variable abuse notation denote support observe consist pdf increase integrating set finite uniform distance pdfs function deterministic increase obtain uniform strategy characteristic resolution need issue possible entire characteristic continuous answer normalization move small cause however uniformity mass factor normalize matrix normalize information grid mass suffice decrease distribution distribution suppose move arrive write relate prove side lemma bound entropy complicate let leave cell notation obtain x x lemma appendix sum entropy combine line give resolution show continuity map complete form uniformly since consist give restrict ball around ball tell become map establish continuity see family finally within supremum norm give continuity corollary exist characteristic introduce continuity pdf normalization contain characteristic normalization normalization I distribute uniformly distribute loss grid place uniformly consider row mutual infinite supremum respectively view information supremum characteristic supremum population goal us foundation new introduce observation population let empty know k define boundary characteristic important boundary one partition dimensional grid former exactly proposition characteristic equal piece every partition column imply partition column characteristic rather matrix fix notice either case hand corollary supremum exceed follow introduce alternate characterization previous estimator approximate therefore efficiently compute wherein partition dynamic mutual maximize however rigorously justify give statistic consistent fact realize characteristic matrix easy compute consistently notation dimension column variable grid analogously define otherwise matrix supremum monotonically increase hold fine axis thus proposition non analogous order pair consider subset grid consider consistent estimator statistic consistent efficiently heuristic adaptation variance equitability stem expression maximization one grid type propose reason rigorously limit work estimate give datum formally follow theorem additive numerically distribution prove column dynamic programming row maximize row step bind show exist mutual achieve close use appendix row line probability lie cell contain cell line though grid contain row remove line fortunately though pair mutual merge leave obtain integration introduce choice numerical integration arbitrarily tradeoff corollary give compute additive q continuity give give rise density estimator function equip consistent sample variable continuous formalize develop equitability coefficient equitability specify interest reflect equitability statistic extent know denote show state power property independence equitability notion power equitability turn original quantity prove continuous equivalently open lead define new efficiently addition describe precision probability density analyze statistic density leave question valuable estimator variance difficult understand second statistic form canonical gaussian contribute understand capture notion theoretical equitability infinite limit highly desirable alternative characterization first open art size bias equitability go detail address idea author acknowledge constructive claim scale respectively analogously proceed proceed argument proceed term mean together fact inequality bind use e line give complete bind grid adjacent variable induce merge merge column since merge column identical merge expression ia complete non number equal equal probability function bind equal observe use lemma bind grid sub grid horizontal remove line mass cell union line column suppose lie leave mass right successive lemma binary apply next treat general case column l b lemma probability column contain vertical horizontal line entirely grid column analogously distribute upper hx hx hx hz hx z hx hx hx hx upper hz hx thus variable add mass lemma give variable q magnitude total mass mass go lemma b second please david relationships useful give similar equally equitability analyzing set formalize behind equitability equitability generalization independence us compute generalize mutual enable reason continuous canonical prove alternate estimator pair random variable hope provide rich theoretical foundation equitability extensive empirical discuss aspect equitability statistic suppose thousand association pair hundred million manually examine pairwise scatter plot context commonly take compute approach choose meaning score exactly question systematically relationship crowd potential list concern comprehensive trivial association care relationship detect sufficient power could reject excellent method allow exploration identify association sift relationship reject strength address introduce equitability dependence assign relationship type notion notably relationship cover similar note reasonable equitability statistic reflect coefficient determination regression possible characterize seem reasonable give perfect score relationship maximal behave desire original equitability much publish concern perhaps concern rich theoretical equitability main issue allow unified language use equitability limitation equitability formal equitability equitability equitability hypothesis whereas typical association yield relationship strength trivial relate concern power benefit consistency together easy trivially generalize estimator separate finite rigorously mutual information correspond orientation ask whether equitability power soon goal remainder probability relationship theory give begin analogous define call view supremum practically easy original heuristic previously proceed give rise sample density approach expect extensive analysis method thing equitability runtime introduce compare original statistic discuss exploration question compare size defer issue rich equitability equitability informally ability statistic similar equally noisy relationship notion rigorously equitability interested equitability variation adapt incorporate variable formally define equitability give formal overview brief benefit ask like tell setup set distinguished g quantify ask evaluate back finite want dependence criterion define equitability add particular strictly equitability equitability one equitability equivalent hypothesis though primarily various functional relationship change previously noiseless population one interested support manifold add perhaps simply mutual information goal make impossible dependence good equitability reason equitability functional motivate generic concept equitability unified explain equitability require matter motivation state perfectly make say equitability criterion equitability term perfect equitability informally equitability notion equitability equitability interpretability sufficiently equitability recent follow ed impossible nontrivial severe understand amount trivial proxy random arbitrarily crucially fact depend arbitrarily value may pointed issue allow relationship indeed quite translate thus apply address perfect primarily discuss section equitability give equitability equitability incorrect perfectly may impossible model equitability desirable remain provably empirically analogy science fact np want solution solution provable dependence comment publish discuss property interpretable define amenable efficient interpretability define separate incur performance cause choice equitability reason distinction need evaluate interpretability give definition analogy reliable interval interval whose value analogy depict figure table analogy interval would however detect good estimator subject care bias since rank p value value sample wide sample infinite relationship interpretable reliability reliability simply relax equivalent require concentrated area statistic x see look noisy correspond requirement exist functional relationship
crowd annotation sample annotation error problem mean cost structure collection formalize follow e image audio costly crowd worker sample nature procedure unbiased want difficult draw estimate unnecessary auxiliary arise statistical generally correction label shall se information hand machine insight simple correction independent construction annotate hybrid annotation auxiliary hybrid significantly utilize possible define introduce traditional collect annotate primary transfer design mining survey second moment denote per annotation annotation expensive characteristic primary precisely state design define number annotate sample population allow omit second primary unbiased third uncorrelated theorem prove sampling design unbiased unknown enough ensure p standard upper size annotate auxiliary annotated si expression derive si large annotate auxiliary first sampling auxiliary design auxiliary annotation traditional formalize p theorem collection certain cost general determine sampling begin eq trade figure less annotation achieve auxiliary annotation reduce pdf ad hybrid design indicate operate relative annotation operating point along curve relative follow hybrid eq annotation correspond annotation compare traditional sampling effective primary error variance hybrid primary annotation design annotation design mark black design cost additional design denote confusion value space derivation contain si annotate obtain confusion characterize misclassification binary matrix specificity unbiased recall annotation unbiased inverting abundance correct correction true value combine yield introduce abundance expression iy size give annotate primary transfer traditional transfer cost primary strong always rely hybrid commonly texture location annotate machine identify annotation estimate validation human estimate annotation pdf monitor community capture annotate automate annotation camera thousand image task daily abundance image annotate machine variance cross validation annotation annotation annotation task abundance day upper validate design carry survey collect annotate sample size determine hour draw replacement estimate investigate value plot cover image validation estimate survey estimate hour remarkably use sampling transfer sampling hybrid design manual effort equivalent upper calculation readily day manual annotation effort survey survey increasingly new automate approach occur coincide rapid automate annotation determine need error traditional still machine difficult subtle underlying density machine though many annotate traditional unbiased support hybrid mean confirm low sampling design b expect uncorrelated correlation verify experimentally correlation hybrid must take account simulation indicate validity assumption well option validation camera site visit satisfied would time affect appearance water vary due etc color third camera simulation estimate valid cover pp believe difficulty application differ estimate sample biased severe shift confusion valid estimation single g survey g sampling extend situation utilize separate machine matrix confusion unstable full rank modeling incorporate hybrid minimize procedure prefer hybrid hybrid unbiased maintain level base base survey audio survey population analysis survey corpus surveys medical solely design knowledge perspective future notably procedure like science california author acknowledge nsf national foundation division grant collection annotation survey wide digital form audio addition crowd utilize annotation collect population sample new novel hybrid utilize cost annotation key amount annotation need efficacy hybrid demonstrate application utilize hybrid reduce expert transfer auxiliary annotation
rich dynamic computationally tractable brief summary glm train latter section spatio temporal field number relate neuron stimulus glm relate precede stimulus former latter intensity glm spatio temporal field bold letter matrix point neuron poisson observe spike q expect spike train define spike brief resolution time spike train vary intensity entire log observe spike spike select calculate manuscript likelihood train solely intensity constant practice neuron fine effectively simplify form likelihood ignore optimization determine train stimulus spike glm intensity neuron field relate invertible infer stimulus quite provide neural response neuron glm predict current spike history stimulus vector count intensity stimulus stimulus neuron post spike history etc firing intensity decrease fire stimulus term select term simplify calculation interpretation individual precede time match increase gain likewise spike prior convolution occurrence spatial sample integration slow potentially exploit equation let simplification spatial component field algebra everywhere maxima exist particular lack concavity arise dependency model notable reduction replace derivative neuron neuron influence observation fire chance correlate stimulus reflect mechanism e electrical coupling etc naive implementation glm intensity fail distinction stimulus hold stimulus attribute stimulus correlate add post filter neuron fire rate conditional intensity subscript throughout sum activity internal neuron population subscript drop spike train clarity although likelihood implicit subscript spike coupling must generalize care spike conditional intensity preserve term pair appropriate intensity hessian neuron linearity intensity cross term neuron fit simplify neuron fit manuscript provide brief analyzing
wide composition library well phase presence phase signal library center diagram phase match ccc ex pt cpu integrate unsupervised challenging variety scientific discovery motivating contract sf suppose augment still subject admits augment constraint optimizing follow hard belong centroid non non assigning centroid close correspond initialized property proposition edu department science university van department university identify large noisy key material aim discover cell incorporation constraint addition solve outperform material precise enforce see growth many science combinatorial material discovery material property obtain hundred sample composition tool automatically analyze determine material composition important material activity cell evolution likely break material science role accelerate material accelerate discovery new material develop create library intuitively generate mix small promising library ray composition specifically x ray signal sample material material discovery call phase gradually composition term intensity phase underlie separation source basic ray source therefore factorization nmf formation map basis lattice recent enforce peak decomposition programming constraint peak respect nature lose become noise filter approach new integrate additional introduce decomposition allow specification negativity labeling general rich dependency example encode specify law propose technique call integer program constrain outperform scale world recover value compactly column ray pattern interested low input namely basic phase basic pattern point belong approximation datum mining denoise compression symbol singular decomposition produce approximation obtaining point interpret instance represent intensity ray intensity motivate compute svd undesirable example image superposition simple patch ray researcher nmf explicitly negativity coefficient negativity domain correspond upper boolean negativity hold science underlie structure chemical compositional variation lattice compound follow variation basis pattern define constraint negativity individually connectivity complex constraint motivate decomposition subject constraint minimize entry wise frobenius additional possibly require variable formalize value variable denote stack vector entry programming general encode combinatorial encode negativity supervise example suppose want explicitly formulate coefficient want entry column model application topic encode rewrite compactly appropriate semi include labeling belong q analysis basic grow framework supervise information incorporate typical enforcing point framework link constraint interactive scheme account feedback refinement merge supervise label alternatively desire temporal shift work literature single dimensional basic pattern incorporate logical negativity expressive class constraint specify among unfortunately nonconvex programming see progress counterpart either well even constraint rarely heuristic widely project apply call exploit advanced solve report procedure enhance sophisticated key quadratic operation literature leverage program integer solve improve warm search problem loop program presence inspire seminal coordinate descent upon number project quasi newton novel one take feasibility every iteration even fw fw w hold optimization problem feasible monotonically function monotonically non consistent depend scheme nevertheless evaluation play typically converge regardless initial provide experimental result capture label link complex logical describe high level knowledge motivate map nmf cluster range gene expression determine reflect basis obtain normalize entry belong assign consider information subset assume pair belong link uci repository truth generate various select point constrain first enforce non capture constraint approximately update supervision link constraint accuracy ac ic labeling match ground truth efficiently average knowledge intuitively properly take combinatorial sound take closure must link link implication deep reasoning computational overhead runtime couple second dataset g whether number class problem enforce negativity capture biological fact enforce kind prior knowledge totally technique ccccc avg std avg std accuracy approach average run show limited standard running second capture key law map identification temperature pressure phase occur chemical library involve pattern composition indeed compound lattice constant composition position ray pattern isotropic lattice peak shift signal vector free encode
beneficial close beyond limit besides decrease grow condition bind moment change turn resample unlike soon strong already note experiment common gain gain term term resample impact keep step impact marginally xlabel ylabel label anchor style axis cs south legend entry step legend style legend pos south west font opt table opt mark xlabel ylabel axis anchor south legend legend pos south west font opt mark table opt mark xlabel axis anchor north description cs south font step legend south width mark opt opt convergence xlabel ylabel description anchor north axis legend entry step legend pos south west font width mark small opt mark weight use optimize overall notice link descent update step early iteration behave like switch step crucial highlight try evidence unbalanced yahoo yahoo compose million triple ad click yahoo front page click rate row row click click compare incidence weight divide ten figure step perform well tend weight figure observe bias negligible obtain figure frequent graph mostly resample regime well divide yahoo divide potential expression consist infinite stream decomposition variance figure bias curve reach regime slope expect size difference iteration symmetry however dominate get become xlabel ylabel cs anchor style axis cs anchor bias legend pos south west style font width pos pos x variance south slope convergence synthetic xlabel iteration ylabel legend style legend south west description cs axis cs anchor south font table coordinate pos pos xlabel ylabel width south axis description description anchor south gamma opt sep mark table var regime opt txt triangle regime opt txt provide regard deduce sampling depend regime asymptotically gain however limit besides dataset difficult quickly happen slow dependency sampling allow take large extend simplicity focus least decomposition explicit interesting see logistic resample acknowledgement european project discussion hereafter otherwise thorough operator live proceed detail allowed derive need preliminary schwarz inequality one definite orthogonal eigenvalue eigenvector orthogonal definite form orthogonal form decompose g condition denote conclusion definite positive implie first long I also orthogonal restriction bilinear definite finally want direct let assume already tell fine denote large eigenvalue eigenvalue result compare true eigen divergence theorem matrix us eq matrix depend term look term always appear express operator recover soon never soon notation iterate exponentially bit bound exponential assertion proof variance indeed remove instance rest mostly bias term sup france sup paris consider average strongly case expansion exponentially decay new algorithm tight bind error may variance decay decay allow dominate dominate density lead gain term choose size significant improvement optimization together optimal scaling however convergence rate reach term come square split characterize fast show play strong practice behavior special sampling traditional bound potential gain lack asymptotic exponentially decay give tight variance decaying bias decay uniform choice dominate section dominate density gain dominate step size improvement denote optimization problem identically sample cover situation consider unseen select minimize often dedicate describe average gradient user step particular study center estimate recursion study cm gradient heavily mention convergence depend general convex logistic least sgd sgd general function partially without analysis decaying size dominant already view literature provide sampling optimize compute problem second large strict definite conjecture bind initial condition covariance bias give detailed know derive dependency term equal behavior decrease square also possible simple go assume usual rao also exact obtain decrease finally unlike bias cm expansion term tr tr behavior real often difficulty change several presence problem cost previous try optimize increase since wish objective
mean covariate eventually model model part covariate represent respectively zero base decision correspond coefficient zero first scenario important second variable rather qualitative covariate illustration interaction treatment correlation right panel choice coefficient interaction weak correlation opposite observational randomized assign patient treatment assign implement replication three aspect report discovery identify among tp identify treatment treatment estimate estimate outcome replicate selection table result estimation summarize follow first include lasso model ii small correlation average three get reasonable provide score small variance correlation variable much table treatment implement treatment especially approximately select important estimate treatment regime decision partly big treatment regime value among rule treatment lasso method iii method may treatment regime lead eight table magnitude nonzero treatment regime compare path trajectory variable increase path solution path allow ability setting number important large method include iii mis model method still lasso implement learn mis seem method method treatment select important moderate identify variable treatment competitive advantageous star conduct effectiveness treatment participant age initially participant level participant without satisfactory option acceptable level participant seven strategy switch augmentation participant cognitive ct available cognitive without satisfactory ct call level participant augmentation either li participant satisfactory randomized participant follow year base estimate treatment significantly bootstrap sample bootstrap value bootstrap confidence interval value article treatment important make select final outcome clinical trial important variable regime small value propose combine rank select decision rule combine comprehensive stopping tune maximize outcome treatment extended censor modeling expect treatment decision disease desirable stage h x different group treatment treatment correlation panel fit line triangle treatment dot treatment select nonzero score stand discovery stand standard setting size ii observational model ii treatment mean mean regime average replicate stand error treatment proportion individual value cs score model ii iii observational model iii x score identify variable tp positive correctly important value cs size tp randomize iii model model ii iii iii x opt treatment mean treatment regime outcome treatment correspond cs score iii iii observational three variable dash blue dot eight combination three choice black solid red dot dash three three choice solid blue dot h choice eight important combination baseline covariate red line dot dash lasso loss energy ability concentrate death patient friend patient complete medical private important able thing impact family status current status currently rate rating quick history diagnostic screening axis baseline axis ii hx hx hx abuse family abuse family intermediate medical patient rate percent sr frequency rate intensity score risk patient study daily care clinical optimal treatment clinical get attention focus ignore treatment qualitative interaction indicate characterize treatment individually article advantage method select qualitatively marginally versa optimal propose stop handle small perform practical setting clinical trial strategy heterogeneity characteristic clinical genetic paradigm duration type adjust tailor effectiveness traditional benefit interest regime clinical trial observational study optimal regime sequence stage latter treatment regime sequence tailor expect dynamic regime clinical trial study marginal treatment two popular backward induction deriving regime modeling value regime extend enjoy method outcome weight treatment regime experimental dynamic treatment regime star intervention disease study collect clinical clinical collect medical history effect collect clinical covariate decision interpretation treatment regime select could facilitate work alternative star star study clinical trial treatment strategy provide baseline patient medical intermediate medical decision next select useful decision expert regime area statistical selection technique focus technique medical make distinguished predictive qualitative effect variable vary method implement regime carry qualitative interaction design categorical covariate test conservative test least regime estimate variable propose penalize least correspond shrinkage penalty interaction select directly select interaction characterize qualitative patient treatment many covariate examine spurious advantage goal qualitative regime sequential advantage additional improvement treatment new advantage variable sequentially size another procedure include marginally jointly avoid redundant information propose stop criterion sequential decide introduce identify regime treatment decision demonstrate illustrate star clinical clinical trial observational exposure give subject interest option subject denote possible code accordance find treatment regime covariate subject response response summarize treatment mapping covariate find treatment regime treatment regime introduce potential outcome concept opt regime mapping assumption essential make identify regime treatment outcome influence straightforward show find expected response estimate need optimal current pay attention select information evaluate quantity characterize degree interaction score show possibility covariate eq tx eq note always covariate treatment covariate optimal covariate capture magnitude interaction subject treatment example treatment interaction treatment stand change knowledge reflect degree qualitative characterize help find regime limitation tend interaction nonzero covariate variable evaluate individually score variable crucial account base score sequentially covariate convenience arbitrary
gives project adapt increase achieve lm scheme slight overall improvement show effectiveness couple minute confirm effectiveness adaptation different pair translation score see hand baseline english language pair may complicated system day day lm adapt baseline n adapt adapt post addition investigate adaptation consecutive day day correction system correction new third quickly target list epoch day day create corpus day respectively remain randomly datum task incremental method yield english experiment five consecutive coming decide precede day day fourth proportion decrease combine various baseline day day day day importance adaptation domain available relative day vary project effectiveness impact adaptation various translation experiment provide score adapt sake clarity score system day adapt quality improvement day also score use three human translation provide european gain style adaptation beneficial improve task thorough technique adapt language want integrate correction statistical text concrete need propose speed work already perform document explore strategy lm weight network train combination sample original avoid lead fast language gpu experimental effectiveness translation english observe school science university le france fr fr fr neural outperform back like recognition day efficient adapt language instead mixture adaptation result cat environment fit lm important role natural back gram base embedding jointly language model popularity confirm many systematically gram significant last recurrent lstm translation system entirely network large corpus adapt new ability change property operational occur translate daily news article environment typical integration cat want correction finally want lack various involve neural lm representative overall perform correspond concrete need human human propose translation hypothesis post day already translation sentence propose translation perform translation day sentence adapt next task usually around next continuous space popular adapt lm corpus merge minimize domain development integrate specific linear system extract available lm turn lm cat environment investigate datum investigate neural community incremental presenting could perform adaptation namely recognition mention convolutional adaptation adaptation al explore output study corpora sentence three model gram rnn lm system variant adapt recurrent lm area speech early rnn speed adapt lm history automatically example want translate language align popular phrase base model translate short together translation lm gram log translation optimize idea speech build explore variant technique interesting nn work closely cat post provide human tool update phrase language european text resource summarize c en en l lm approach day day adapt system build procedure selection parallel extract representative development language translation method back lm base feature optimize score development source log optimize final translate human translate process day post hope rest translate procedure rather human translate approximately day percentage generic example per epoch none sec sec sec lm window experiment dimension layer neuron short short account token corpus converge epoch hour gpu adapt system analyze project document two adapt translation name limited could quite easily informative development ideally text loss generic adapt keep achieve always randomly
modify use context first introduce give exponent clearly away density exponent determine calculation consider boundary version describe bold width situation equal need ignore logarithmic rate rate typical rate assumption condition need impose exponent level smooth exponent usual begin data separation satisfy data set like yield provide n compare cluster exponent corollary extra corollary controlling sequence case estimation easy estimating illustrate estimation detail well estimate sufficiently small fast choose decay sequence corollary choose slowly decay decay motivate thus consequently exponent one hausdorff assess essentially however scope interesting whether true hausdorff achieve report last derive convergence case good rate sequence parameter little issue propose dependent recover without know recover present strategy user specify run family corresponding ensure apply end ensure elementary assertion sufficiently show n n yield establish observe right assertion theorem less b second smooth smooth exponent combine check end nn n sufficiently yield c sufficiently elementary calculation moreover always suitably guarantee apply n replace consider give proof use goal theorem hence sufficiently analogously g monotonically q estimate theorem proceed intermediate goal fix show use assume generality assertion write nn sufficiently restrict conclude n monotonicity q sufficiently c combine estimate obtain assertion case exist assumption analogously far sufficiently nn assertion finally prove recall already apply right side eq appear g n assertion proof suffice assertion sufficiently inequality consequently sufficiently third fix elementary yield inequality combine assertion theorem theorem issue propose generic small feed analysis show consistently small present strategy involve density class definition cluster propose I cluster set study reference unfortunately number cluster generally rule use couple create clustering dependent estimator useful tree avoid level focus identification structure level detail show linkage recover similar nn propose pruning remove recovery component level estimate set estimation article quality symmetric hausdorff two respect hausdorff metric clearly structure eventually fix contrast respect suitable set topological recent base use water modal flow suitable smoothness modal lebesgue single cluster one dimensional see none infimum usual avoid detail define make difficulty rule neighborhood mass topological connect zero address issue avoid underlie kernel compact distribution lebesgue recall functional remove whether cluster infimum component structure persistent persistence either implicitly g seems deal uncertainty namely seem opposite exactly connect compare consider look similar cluster modification discuss scope generic estimating level enjoy vertical horizontal estimation guarantee conduct level first derive consistency connected move main establishe well know describe mass boundary restrict boundary rate choose suitable therefore drive characteristic strongly build contribution consistency establish consider density know density new contribution add impose last least paper generalize feed description drive proof section proof auxiliary example paper refine notion definition cluster develop notation assumption throughout always borel fashion course interested lebesgue measure sphere possible interior closure boundary denote indicator set paper deal distribution density generality make density consequently notion somewhat canonical serve choice density long densitie distinct component become inconsistent neither two alternative readily available suitable relevant set horizontal thin line cut h issue fix bx define consequence call way whenever e level small closed satisfy modification say every continuous satisfied density density think range level absolutely help notion relate connectivity subsection make motivate definition generalize idea note partition informally speak break fine partition emphasize iff distinct iff relation case say empty set closed component well discrete connect partition call eq minimal figure illustration large horizontal order sense whenever aa dot indicate contour distance sup norm since component situation concept subsection cluster cluster normal three either cm condition see component two persistent gaussians together level coincide connect consider component open dimensional contour line line indicate thin line level level time uncertain vertical uncertainty cause subsection complement deal horizontal cause quantify horizontal denote operation closely relate operation base use estimate vertical sense ideally cl cm cl unfortunately absence cm cm cm cm cm cut connect connect component rest condition thin p rich lead indicate thin shape thin thick turn part summarize thick level follow statement hold add remove surprisingly meaning express even specify together effect significantly thick solid thin solid component persistent dotted line indicate within cm connect thin show dotted indicate within left thus behavior exclude help level summarize borel absolutely addition denote present analyze generic generic vertical horizontal relate component satisfied follow disjoint suitable detect identify identify c relate component step figure small stop soon ex thick plug solid line bold horizontal component vanish theorem bound component extend connect component satisfy dl feed estimate ensure partition geometrically well behave recall partition moreover example family partition partition lebesgue surface compact haar sufficiently manifold equip slight abuse dirac provide let c ph feed parameter satisfy conclusion estimator approach establish uniform unfortunately level become include approach open question differently address issue result construct density approximate grid bind cluster lead result modification c
relate university xu reviews american ann mi mathematics ann usa department computer usa global ny usa dedicate fuzzy combine diffusion map theory algorithm paper reduction achieve diffusion system representation description use fuzzy dimension reduction discuss describe nonlinear reference enough integer nn connect edge gaussian nonzero real moment property interpret scale result symmetric w interpret probability sum define increase local geometric datum integrate make possible broad path connect high path diffusion use aim find preserve optimally mapping distance preserve eigenvalue j choose element eigenvector point reduce dimensionality dominant correspondingly rate reason spectral range method correlation influence affinity metric interaction dominate affinity wide kernel kernel application sparsity handle isometry section describe second fuzzy real fa architecture consist node neuron field neuron activate pattern already correspondingly neuron encode layer weight pattern neuron neuron calculate define break prototype fuzzy subset input winner winning becomes activate determine match eq weight rate hand criterion meet back win competition project meet neuron represent art advantage stability plausibility advantage scalability speed parallelization art ability network complexity robustness map site origin tumor particularly important cancer diagnosis profile blue cell tumor publish diagnostic child widely extract information set hyperspectral band pixel hyperspectral contain band amount hyperspectral bring processing hyperspectral sample address hyperspectral identifying
highlight application easy programming library architecture share memory core cluster hardware graphic gate array integrated circuit distribute programming library various architecture inference underlie hardware automatically task system light upon problem memory medium low gpu low medium medium medium medium library machine core visible shared pass cpu core storing low meanwhile hardware reason write program towards drawback include capacity library support prevent resource atomic operation increment besides alternative framework multi multi queue library specific parallel pattern synchronization barrier green choosing device computer single gpu provide small compare processor easily easy maintain code dedicate device low power consumption pattern core ml limit cpu framework core core user specify gpu gpu move framework intel counterpart accelerate gpu parallelization population well smc develop hamiltonian demonstrate example accelerate collapse gpu framework particular gpu cpu use hardware investigate distribute user data synchronization process decide include receive barrier process besides framework synchronization handle message globally process invoke procedure process pass execution provide primitive library parallel review distribute framework execute task reader book large online computing read disk transformation disk machine key system pass key user generate hash intermediate pair machine parallel pass key user often often latent likelihood gradient aggregate ml ml open source collaborative dimensionality topic read disk overhead iterative well interactive distribute distribute disk automatically user store computation parallel parallel serial look iterative ml gb communication interact read key variational bayes require come compute computational show graph engine receive send last update vertex along gibbs easily conditional gps feature e compute asynchronous flexible scheduling pick queue pass vertex data vertex adjacent vertex finally adjacent vertex queue note long serial ml task matrix gibbs gibbs graph disk base version framework restrict communication worker communication allow pattern worker share count collapse lda asymptotic probabilistic derive simple scalable mixture go let progress dp mean extension dp mixture svms progress advance scale method derive back model moment always produce estimator algorithm matrix algebraic name moment perform extremely show computational markov wide domain include recent advance big subsampling distribute helpful exhaustive field learn database parallel system language considerable model widely become day science project google effort provide cloud service microsoft similar provide interface help language specific sophisticated automate process automatically interpret easy automatically report effort currently data acknowledgement support project cb cb nsf china project research program finish version put later section put concrete example independence base divide style asynchronous variational lda model get variational approximation posterior purpose collapse gibbs collapse conjugacy analytically k c z collapse collapse integrate ik quite optimize loop component count sparse reader reflect circle update change communication parameter however immediately train chain partly identifiability dependent style mh divide choice seem serial corpus computation worker count global fit server mention implementation detail ad lda counting aggregated update processor update incorporate parameter interesting look inference em step fix sample integrate well explain without account assignment processor divide infer lda aggregate local model optimize divide general share ad lda implement aggregate note ad lda lda drawback synchronization end sampling network yahoo lda issue replica machine processor avoid update put count finish document document vocabulary likely matrix yahoo lda overlap count local global yahoo lda synchronization policy yahoo background word w server server another direct instead want employ coordinate update eq document independence integrate ci natural reduce style mr phrase independently phrase simply aggregate triangle update algorithm integrate ci document wise variable ci integrate become style require algorithm core machine ad yahoo yes mr chen cn growth availability interest learn system bayesian scalable survey bayesian term include infer regularize flexibility algorithm subsample deal regularize inference live engineering massive stream become increasingly besides volume increase highly uncertain primary machine become field challenge big cover big problem big need deal learning dimension development spam explicit dimensional many scientific problem challenge regularization salient classifying image thousand million imagenet consist million concept average thousand wikipedia document category often category structure direct acyclic explore massive million become extract multi grain datum application speech model neural auto probabilistic generative model practitioner often slow factor conjugacy intractable integral several deal physical randomness incomplete principled combine evidence intuitively flexible offer characterize big elegant deal collect bayesian rule suitable deal big stream grow exponentially grow slow amount shannon increasingly leverage powerful computer deep capacity fast therefore serious effective become increasingly relevant big capacity address approximate learn term model learning must information change goal evolve must feed dynamic datum flexibility must flexible handling side structure digital sensor scalability scaling modify advantage grow article provide literature survey advance big hide variable specify core give datum likelihood marginal involve intractable integration th bayes sequentially useful formulation distribution make objective optimum identical fact add minimize equal interpretation significant aspect variational method make flexible incorporate soon bayesian posterior conventional distinguish bayesian post posterior bayes project density general bayes minimum bayesian ml range single variate regression semi scenario essence p p assumption assume ratio automatically model costly abc bayesian biased integral involve typically analytically category see method omit examine single integral variational history physics economic machine theory graphical reader seminal book nice overview cast feasible variational formulation posterior show objective maximize bind intractable target parametric e parameter solve descent often optimum integral approximate replace parameter variable assumption q em bayesian mc diverse repeat explore systematically method set common give density pointwise normalizing replace carlo converge number suffer severe limitation dimensional space book article detail monte powerful scale importantly advance later pointwise weight heavily proposal construct ergodic markov converge e hasting construct unnormalize prohibitive massive mcmc iteratively draw standard gibbs sampler sample draw convergence effort spend gradient mix rate langevin anneal sometimes handle mode gibbs sampler include sampler ordinary convergence replace ordinary conditional distribution marginal distribution question use method theoretical state property informative infinitely bag improper jeffreys prior use admit good frequentist contrast may since practical method hierarchical bayesian assume prior weak thus convenient put hyper empirical hierarchical dirac section progress make characterize empirical bayes empirical practice tradeoff computational conjugate inference belong dirichlet multinomial conjugate pair likelihood prior dirichlet z normalization factor posterior explore conjugacy allocation fig document topic vocabulary topic iw z lda popular application impose except provide impose correlation dimension topic model model infer correlation flexibility scalable prior integral practitioner including review much emphasis bayesian scalable review nonparametric bayesian infer improve flexibility leave section order challenge property learn requirement evolve feed time scenario objective handle rich input visual sensor scalability massive advance flexible scalable interpretable parametric pre specify matter may limitation especially priori may optimal datum change ideal figure latent factor structure factor abstraction level nonparametric bayesian elegant adaptation capacity define rich space dp ibp briefly review ibp reader article nice comprehensive dp first specifically concentration dp dp discrete surely atom independently base assign atom constructive stick fig unit break infinite segment stick remain segment beta break remainder dp insight develop variational algorithm chinese restaurant crp define partition integer crp derive restaurant customer restaurant customer subsequent customer table customer define property integer crp parameter dp crp step crp enjoy nice sampling dp mixture slice infinite sum condition assume single reduction analysis assumption weighted combination factor loading influence term usually ibp variant factor grow binary indicate latent ibp define process unbounded ibp derive crp infinite arrange customer choose first customer customer customer choose customer ibp role crp play unbounded latent dp de distribution crp mix distribution ibp admit stick length break dp stick length stick length development part machine extensive practical probabilistic function gps supervise model condition training matrix target value involve inner product feature bayesian simple example gaussian define gp characterize stochastic I examine requirement task conjugate review research process meet big develop sophisticated group spatial group different multiple work present ibp topological layer latent layer number hide unit recent extension concern dependency special hdp training framework review dependency nearby important dependent dirichlet dependent crp dependent ibp dependent ibp biological relational adopt ibp feature nonparametric membership discovery latent community advance extend scope equivalent variational build define nonnegative ordinary fig bayes question enforce constraint incorporate sparsity example max margin paradigm supervise lda additional design likelihood imbalance problem margin simplicity discriminant I ij average expect surrogate classifier classifier randomly make regularization adopt strategy relate upper bound lead imbalance issue relationship part choose likelihood second restrict regularization much flexible prior regularization bayes exist achievable bayes rule affect difficulty solve formulation duality theorem deal dd stochastic iteration fast convergence c approximate mh big scalable advance sampling multi inference optimally efficient unit computation model variational develop explore redundancy subsampling example reasoning create overview state variational descent sgd randomly update estimate gradient estimate appropriately infer update two global global lda assignment collapse illustration model inference consist draw mini parameter use search manifold probability tuning learn rate average stochastic propose use gradient trade variational model expectation method additional tight variational bound another monte variance auto bayes learn neural continuous ig gradient naive depend variational underlie effective technique need known parameterization representation parameterization cp exist minibatch estimate l maximize possess strength cp conversely hmc differentiable posterior dependency extend deep similar sophisticated applicable exist group three category sample mini gradient mix rate systematically various langevin example langevin dynamics mcmc produce noise isotropic p proposal prevent correction successfully monte gradient dynamic hamiltonian dynamic replace uniformly sample mh acceptance zero rate rigorously justify improve scoring fisher stochastic randomness subsample similarly riemannian manifold stochastic riemannian langevin simplex mh another category approximate subset eq linearly prohibitive set mh hypothesis testing allow reject fraction require mh theoretically compute easy frame replacement decide mean prescribed mh derive new stop bound bernstein adaptive mh target mh apply datum augmentation present method posterior novel augmentation indicate pz alternate update condition conventional likelihood random subset method stream mini know streaming go measurement update time tt role variational naturally stream rule perfectly suitable stream challenge evaluate posterior conjugate hide updating analytically kalman contrast complex bayesian model linearity close posterior intractable various develop bayesian p one streaming analytical form stream generalization passive result svms latent structure latent representation pa complex discover structure allow complexity latent hdp model resolve topics monte approximate smc resample large smc store expensive particle degeneracy processing bottleneck simple model derive kalman broad model density filtering develop extend basically approximate conjugate filter df df draw surrogate sufficient along df produce recent progress made distribute parametric family family solve tool
symbolic combine extraction symbolic computation symbolic context proof try often external select thousand theorem corpora ii internal reasoning evolve strategy corpora specialized work start complement first corpora main idea level library library attack library number approach number corpus contain million motivate lemma ai deal rigorous experimental evaluation corpora mining core scenario discuss conclude task automate large knowledge proof reasonably library construct isolate axiom previously prove theorem range thousand library parametrize proof number promise parallelization theorem measure loading library design fed find write theorem find already nothing current conjecture part far like include reasonable year experience library following indicate human name name considerably good library large weak various statement library library experiment ai library ai short turn hard name corollary many lemma omit prove depend focus variant complementary also necessary extent alternative ii far atomic inference library million hundred million efficient small orthogonal corpus name theorem corpora way quality reasoning system proof summarize tool initially art implement loop hundred lemma run index redundancy control age similarity lemma inference tool lemma produce successful run problem extract generalize thesis estimate dag acyclic inferences tool lemma lemma inference subgraph node axiom well ii stop minimal select lemma characteristic lemma characteristic include complexity implement idea experiment automate use lemma exist proof similar successfully mainly algebraic add large million early complete library library index limit style contribute importance graph try relative large popular appearance web network centrality implementation easily nod corpora core corpus corpus version core name algorithm intermediate name name initially theorem lemma put pass argument may common like alpha de represent keep operation table htb name name constant mb mb corpora intermediate style inference whole formula big neither disk trace obtain inference additionally intermediate obtain trace take hour cpu gb ram ghz consumption version intermediate call graph lemma trace table graph lr lr inferences inference trace edge lemma additionally symbol trace free type external together originally trace present trace theorem theorem explain trace alpha check alpha would obvious kernel keep trace lemma normalize variable lemma external produce version program replace alpha still keep dependency hardware core leave normalization process graph clear lost information lemma keep graph produce differently atomic construction intermediate drawback make mining big arise decision proof thousand core inference notably inference inference proof encounter step produce justification produce intermediate lemma justification level execute visible execute recursively step perform detail typical natural proof give trace small typical order magnitude order formal development decide look building gb million normalizing well distinct alpha normalization leave intermediate dependency whole post dependency edge trace format come trace theorems create goal proof user version proof interesting point view removal measurement version lemma clause operation like interact level record define perform efficiently define notion explore combine various dataset follow direct implementation modification way cut advantage tool necessary directly available define trace dependency iii use symbol general lemma name lemma axiom formally axiom stop axiom name lemma recursion define stop dependencie eq apart behind heuristic necessary hard need useful lemma conjunction lemma quality record protocol lemma express count main load hour take gb ram unfortunately experiment always integer quickly integer wrong extent chain inference q apart modification minor change need inference clause clause additionally create clause create artificial clause centrality graph dag neighboring node income node advantage minute ram disadvantage comparison previous take account already modification initial score advance perhaps could still keep overall reasonably another disadvantage weight quite counter base need important turn important reverse normalize combine sum try lemma come mathematic choose final dependency choosing theoretic cut cut library many cut cut cut htb graph library name mark gray give node name marked gray dependency newly dependency provable exactly assumption easy ai start name node dependency dependency choosing cut edge include edge edge make slow finding take previous subsection try limit lemma parametrize lemma already predefine naturally name name name name choice empty set name depend whether want lemma complement name theorem expensive second graph change mean lemma take cpu hour experiment limited core mine several scenario rigorous formal corpus quickly rank produce method look plausible scenario corpus compute corpus set newly theorem originally name theorem element way human ai parent ai direct produce train dependency precede new lot success first start metric precede equally use precede lemma consist lemma close lemma early newly parent whose take proof exist prove evaluation whole core remove resource hour take cpu hour evaluation core scenario lemma guess lemma still must provable lemma allow measure good new kernel trace strategy lemma trace hardware use server ghz ram mb cache lemma trace statement lemma store run part need feature lemma independently trace trace due intermediate implication take hour extract lemma lemma without hour take usual evaluation detect preserved version recursive theorem information library service theorem evaluate preserve name change preserved perform far choose evaluate ai method core union optimistic limited metric go theorem theorem scenario precede mining learn stack easy including solve together solve lemma show table old experiment cccc strategy theorem name combine evaluate trace theorem rate new alpha alpha normalize version come theorem seem table add big add million look whole next strategy support computation consider seem suggest focus either bad seem divide change real arithmetic big intermediate lemma success success create trace try translation hand semantic formula involve try name much structure theorem preserve preserve initialize translation formula suggest rather reverse opposite big come htb cccc reverse success formula resource intensive small core confirm method mean mining evaluation solve original inference solve divide evaluation middle good mining solve cccc unique theorem almost table various theorem comparison old
preserve view operation compute projection satisfied projection compare projection converge plain forest increase behaviour rademacher variant optimum improve random forest slow notable random forest theoretical sparse projection provide well tradeoff one algorithm popular dimension principal repeat projection generate decrease eigenvalue accord previously implementation package computing time mac perform mac os load memory execute condition output sub project projection respect time project output tree thus forest explore projection label study variant approach theoretically outperform term allow drastically size output time remarkably adjust jointly output forest improve adjust reach time tree prediction output similar multi classification sparse office sum variance pair eqn divide subsection adapt bias supervise algorithm carry assess effect projection obtain perturbation scheme e bootstrapping selection denote decompose l l rx bx l respectively residual variance decompose law first term randomization forest randomization decompose variance forest randomization eq ls ls fx ls ls ls ls ls e form I random different choose algorithm compute denote form projection like tree take l ls ls thus argument term decomposition ie l ls f l e l ensemble thus second put one ls rx b l rx xt l rx dataset biology scene domain music descriptor video domain classification study multi treat hierarchy dataset image infer feature entire drug drug interaction infer protein output characteristic htb dataset medical cv go go cv combine projection see brief description random lead forest ie superior dataset increase randomization projection improve forest bold last deviation bold highlight rf one deviation dataset grow decrease randomization output like drug robust however drug interaction really baseline forest tuning randomization may different dataset adapt projection output enhance complexity prediction different bias broad lead reduce burden supervise multi train label typical application determination topic address object category image many hundred hundred output pose address approach classification call relevance train independently classifier ie tree split sum score leaf label relevance build single account label dependency requirement storage compare addition output intrinsic irrelevant make attractive address multi problem complexity similar feature limit deal approach reduce label compress output original space decode compress explore compressed case linear learner times stage predict add decode error projection explore random forest label subspace forest score exploit ensemble decode leaf label empirically ensemble space reduce accuracy computational inherent tree ensemble different output theoretically idea good problem significant input randomization la output optimally large scale paper multi projection present property whereas discuss number sample nn supervise minimize output subscript vector input follow pre pruning split among feature selection subsample average leaf obtain aggregate output reach leaf output variance vector material q furthermore wise notice output multi statistic make thresholding tree build unseen predict aggregate learning generate among optimize split selection irrelevant simple price high high mind recall maps notation lemma matrix probability draw sparse rademacher obviously grow sparsity sense exploit random projection computational burden multi tree dimensional space idea subsection analysis point output single tree constitute bottleneck variance projection space modify denote project output generation projection project aspect empirically carry derivation supplementary material multi correspond random capture build bootstrapping error square bias decompose sum l fx ls ls ls algorithm supplementary material l l fx ls variable respectively parameter appendix l l rx xt rx result bad generate different problematic always prefer tree randomization randomization could nevertheless output term learn large variance dimension project subspace tradeoff randomization randomization low affect test ranking express label learn discard thus express htb curve represent value split ht label rank average display fold cross validation mean random forest standard deviation mm mm mm scene medical go cv drug behaviour feature cart converge around expense compression behave forest notice tree outperform output accordance inferior assess collect different dataset material reference range ten fold see compare learn learn subspace three value
illustrate consequence coverage confidence greatly bias thus coverage probability smoothed year nearly deviation fourth ps generally bias expect mr survival correctly extreme proper survival general give reasonable lastly censor scenario decision time point treatment assignment applicable bernoulli distribution probability baseline generate survival first function censor censor patient uniformly survival censor censor consider scenario easy year survival treatment time complicated function induction regime g x clear regime optimal ii treatment regime maximize year survival design finally base empirical survival search method st year treatment regime normalization scenario know simulation result summarize smoothed estimation nearly unbiased estimator treatment table survival regime se iii estimate regime nominal level iv smoothed survival largely coverage group randomize clinical trial four group plus large count treatment curve well survival give survival day treatment patient baseline clinical covariate historical found cd age may covariate goal survival notation cd come hazard counterpart study associate survival number intercept age treatment year estimate treatment early may treatment assign another patient treatment assign j nh nh identically zero process regime derive score specifically specify w establish consistency asymptotic respect n x e mean process delta ii theorem maximizer u iii theorem finally argument eq cumulative cumulative distribution function thus fr integration density process second taylor f fu combine p prove regime augment estimator survival model model numerator converge correctly specify denominator equal second term numerator equation term equal cs asymptotic survival u mean zero due expansion algebra note equal survival correctly I u weakly mean gaussian process iv establish regularity condition accordingly incorporate stage proof omit ps se cp mr f correctly specify ps ps cp mr f f ps correctly specify ps mean ps c c censoring rate indicate ht ci denote treatment approximation ps mr mr ps mr rate rate mr mr rate f ht ht low upper low ratio upper upper upper ratio ratio ht vs ht vs vs ht upper vs vs vs vs regime patient grow finding regime clinical outcome patient disease primary patient survival article estimator treatment regime treatment treatment regime index survival regime suitable conduct propose various clinical trial probability treatment survival disease cancer treatment favor heterogeneity study primary death plus plus treatment divide age plot treatment specific figure plus treatment treatment age plus probability plus old ht raise question assign clinical interest year survival probability regime decision patient complex disease addition disease may time treatment rule observe fast estimating treatment dynamic regime parametric call addition enjoy robustness estimating equation model value specify intend regime estimation learning method treatment weight machine regime treatment outcome clinical observational treatment regime maximize survival focus compare observational treatment assignment survival give treatment give doubly specific survival base observational patient predict level develop pre time recommend different accordingly optimal treatment maximize develop censor treatment finite bound policy learn proper time incorporate treatment covariate interaction effect method maximal year survival specifically survival regime regime regime maximize associate year treatment regime suffer numerical instability sample introduce value numerical survival investigate generalize estimate dynamic estimating single dataset clinical trial discussion proof option patient baseline covariate patient survival survival censor distribute risk treatment regime map simplicity index contrary potential counting risk survival time g maximize year survival year find treatment make uninformative censor survival censor estimator censor observe proper make causal unit treatment I assumption zhang et cast miss patient actually receive patient miss modify incorporate pa clinical need observational maximum specify derive censor clinical study censor e censor censor censor restrictive relaxed censor assignment treatment specific censor survival censor base censor censor inverse score estimator time numerator denominator censor certain treatment regime year survival rely specification improve proportional ph conditional cumulative hazard g I respectively two augment doubly property unbiased survival base fit censor base treatment regime year study smooth plot intercept curve maximization study conduct section survival treatment regime bias specifically g cumulative go bandwidth tc bandwidth ensure smoothed bandwidth red see curve regime estimation dynamic treatment regime incorporate point simplicity presentation decision patient covariate receive baseline beyond covariate ii treatment e coincide assignment consistent regime patient initial coincide censor survival patient treatment g commonly inference study dynamic outcome potential outcome actually receive consistent assign randomization treatment receive potential year survival inverse weighted survival regime patient censor take weight ia I ia observational say logistic smoothing h survival conceptually accommodate treatment decision may become reliable patient treatment regime asymptotic propose theorem regime weakly I
optimize log call encoder decoder kl bind ascent gradient straightforward gradient introduce trick variable univariate kl divergence integrate analytically refer appendix encoder set recurrent state vector sample encoding decode rnn hereafter update file binary know video game sample hz inspection become song optimizer make learn representation especially important optimizer inspire momentum bias create divide overlap instability learning decrease gradually final position song certain space modelling underlie also train time start learn generate music epoch show space order visualize ht latent decode train generate train overlap sequence yield representation dimension point music second encoding point randomly create call use possible rnn effective improvement dividing song possible point reverse step strongly time current approach lstm direct denoise unsupervised improve music addition complement supervise rnns com rnns variational auto generate facilitate rnns rnns exhibit suitable capture temporal music model development consist
study epoch act parameter universal sure risk sharp sample iterate towards understand multiple learning rely procedure massive deriving procedure generalization allow rather amount focus observation iterative data termination early regularization recently machine learn learn bound square updating process iteration large cost practical develop sequential aim develop keeping emphasize role complexity help avoid achieve achieve suitable original minimization restrict suitable alternative possibly way procedure process pick mention adaptive assumption strategy analyze number pass parameter trick example property heuristic online solid term towards excess iterate sharp matching possibly develop condition cover entropy dimension theoretical early stop incremental towards epoch rest defer material composition denote hilbert schmidt essentially paper hilbert rkhs consider develop functional reduce finite norm study distribution let define priori sized suitable increase fast failure pass averaging epoch recover recover gradient choose sure rely finite nonempty stopping help error excess since proof statement conceptually section great equation arise epoch optimize priori act suggest multiple pass beneficial rule cross adaptively low capacity sharp iterate sharp improve far therein non incremental incremental behave prove incremental capacity several square include different proof term build two quantity bound contribution prove due fact pass statistical iterate known expect iterate triangle q summarize error paper xt state main step equivalent step lemma derive follow recursive step initialize inequality assumption plug error obtain
specifically invert convolution convolution evaluate illustration uniquely show form time output multiplication matrix multiplication optimize multiplication less accordingly convolution multiplication mention early parameter mean approach care early however disadvantage forming involve implementation piece iteratively mini batch parallelism implementation lead multiplication effectively utilize gpu computational intensity write read memory traffic direct accordingly opt implementation explain another fast engineering neural use must input especially costly small compare often happen convolutional additionally early reduce subset nature follow step drawback although agree useful approach directly efficient specialized implementation handle many corner implicit implementation often optimize part convnet poorly batch optimize library maintain architecture something easy architecture routine fraction throughput size successively compute submatrix memory matrix multiplication take arithmetic although solution memory rather matrix routine require matrix routine mapping boundary convolution accordingly map load correct dynamically proceed convolution modularity modularity deep compositional schema backward engineering derivative flow device accord framework unify memory interface allocation raise framework make self contain descriptor function modularity framework preserve isolate layer definition purely additive development comprise implement layer protocol buffer schema include computation layer protocol layer scheme library descriptor setup backward call make respective layer implementation drop interface store device hold descriptor solely descriptor exploit reduce consumption group convolution filter backward pass gradient second speedup backward propagation testing illustrate gain integration schema unchanged layer implementation engine fall outside scope execution identical deep project integrate internal set convolution use domain besides processing speech language non square experience consumption multiplication mini batch integrate thank expand firstly convolution bring attain matrix multiplication hope gap secondly support useful speech video would like library multiple accelerate training library reliable provide require evaluate parallel architecture continue library provide library com berkeley berkeley present deep consume evolve make maintain issue long address library basic algebra analogous deep library implement processor implementation computational must parallel address optimize contain integrate exist framework convolutional reduce solve many processor computation arise network efficient implementation provide implementation explore significantly lead speech among neural implementation convolutional cnns deep neural network kernel differ traditional dense algebra deep framework implement operation activation execution deep community successful kernel architecture evolve must significant optimize kernel understand careful scheduling movement acceptable performance believe library computation several benefit deep kernel hardware secondly parallel evolve diverse diverse hardware separation concern allow library deep understanding architecture make framework take library flexible framework immediate rigorously maintain reliable processor architecture library minimum auxiliary case mini batch primary goal neural framework even provide abstraction low computational simplify integration primitive operation store keep low level support variant single arithmetic convolution pooling activation library allow indexing section image auxiliary tensor easy
vc e decision tree support smoothly parameterize relatively assumption dimension support datum vc principle estimate simple use rademacher bind allow remain piece absolute rl mapping complexity estimator equation use simplicity rl similarly I add additional policy provide note e indexing specify see fortunately example margin ordering structure rl consider consist combination may impose limit magnitude therefore may policy policy fix requirement form eq policy two maintain problem rl leave investigation automatically future maximization rl compute use therefore batch naturally grow follow demonstrate return mr learner variety domain world perform return evaluation comparison mr mr knowledge rl chosen approach discussion policy provide impose expressive allow mr maximize toy monitoring domain policy amount mr comprise radial function piece artificial trajectory episode toy understand reader world attempt presence dynamic evenly spaced radial impose performance solid mr figure mr fits illustrate red large mr select class fit place evenly impose figure toy fit early policy class amount datum grow policy monitoring world camera sensitive location camera observe locate sensitive additive camera dynamic take max radial basis limit red domain mr grow see relate reinforcement knowledge develop mapping represent rl structural rl additional theory bind performance extend policy aim prevent fitting amount frequentist lack true dynamic function deal grow see work require setting treat either class value indirect policy reinforcement appropriately sized policy provable extremely weak assumption rl allow theoretical previously bound allow structure maximize demonstrate mit reinforcement attempt choose policy return amount class size available principle structural statistical rademacher complexity identify maximize return policy class give unlike batch require system reinforcement decision reward agent straightforward batch rl data dynamic minimize error e rl prediction overcome limitation explicitly maximize datum explicitly estimate return poorly return estimate overcome principle instead return return control allow policy principled main contribution return weak standard batch rl result rl study single transfer bound return family policy review move rl return tie section provide bind policy discusse exist rl demonstrate build intuition reader discuss work paper completeness clear reader input class decision formally commonly solve give risk attempt principle analogy policy return rl hoeffding holds bound thought rademacher literature additional g rademacher complexity study dynamic constant mdp unable overcome interaction equation lie collect type policy episode use empirical n n n n hold episode policy call monte evaluation policy evaluate policy build section episode free carlo attempt artificial episode policy batch artificial episode episode start ns episode bind eq maximum see regard expectation begin move use equation least
remove data column project break equivalently projection estimating massive difference critical value large test deviation explore simplicity assume identity contain iid ignore follow pp km hold detecting differ coordinate relatively difference keep unchanged complete competitive next follow great consistency detecting search sufficient consistency compare asymptotic test first major compete numerator formula r sd z superiority diagonal sd test covariance p natural alternative coordinate equal non rescale alternative rescale two distribute choose dimension choose dimension describe section projection choice matrix choice theorem figure appear indicate invariance cutoff matrix power significance level nominal indicate level case monte first empirical test bs power two marginally large bs choice alternative b summary test choice bs matrix choice choice helps verify ccccc ccccc cp zero zero ccccc ccccc expression datum contain high minimal intensity gene derive pair filtering transform apply propose cutoff significance turn bootstrap p bs bs randomly choose repeat exercise median bs sd exercise repeat median respectively size compete paper mean population value indicate analysis illustrate practice compare compete asymptotic situation freedom centrality regularize incomplete beta beta project aa observe u nf n ks k ks nc property evaluate conditional identically variance central limit kn rr sr depend show kn sr inner parameter positive integer use depend turn imply hold integer v depend convergent abuse notation subsequence converge claim p n u pt dms usa university usa school engineering ny usa classical tend matrix overcome project multiplication est exact equality normal dimension often tumor normal multivariate generally derive either become poorly equality occur example limitation alternative work testing dimension work test equality independently respective testing sample pool use pool covariance become researcher extend bs establish normality statistic modify refer show asymptotic approach propose normality appropriate alternative test b early normality location transformation test base technique statistic transformation propose bootstrap likelihood moment degree p b test derive base high microarray hundred power upon absence structure clear exact preferred asymptotic preference reference projection space well know value upon nan distribution projection ignore enough tend infinity power past covariance incorporate test previously bs study biased situation power utilize projection previously organize test random section value test projection critical present apply conclude remark test solution project make arbitrarily matrix row project multiplication euclidean moment independent moment assumption pool covariance definite project randomized extension let numerator denominator freedom alternative converge b assumption randomize exact far randomize empirical randomly gaussian adopt standard conclusion projection value generate test recall matrix test projection projection satisfy project dimension
free illustrate singular svd allow unconstrained risk physical polynomial svd np ill nmf np also nmf ill pose illustrated fact decomposition therein uniqueness smoothness gain lee publish algorithm nmf multiple impose reduce freedom variation tv orthogonality development nmf provide issue machine one map nonlinear datum idea trick allow inner transform datum without map e reproduce hilbert infinite prominent machine support machine entropy analysis worth attractive underlie g classical employ recently attempt kernel nmf nonlinear nmf latter write map nonlinear transformation unfortunately first unknown space curse drawback reveal feature ill pose yield difficult deal tn assume propose nmf curse pre image oppose derive snapshot end explore investigate lie turn thank derive two additive descent kernel conventional propose tv approach hyperspectral paper introduce nmf nmf feature nmf several extension nmf incorporate illustrate hyperspectral conclude width old pre nonnegative notation f norm advantage iterative technique keep algorithms rule multiplicative column n scalar model n investigate derive illustrate nmf hyperspectral mean abundance decompose incorporate impose regularity overcome curse pre problem spectra mapping nonlinear transform product latter define nmf propose unfortunately machine feature lie drawback show side evaluate difficulty rearrange problem form simplify nmf problem element constraint drop tackle nmf relax semi nmf drawback give determined need call pre ill problem consist nonlinear ill pose base nmf pre image challenge attempt conduct homogeneous argue author map solve optimization problem moreover subsequently factorization base relevant curse kernel nmf model entry semi variant mean space oppose element expand expression take obtain nmf algorithm iterative additive solve gradient descent scheme accord similar update stepsize matrix obtain entry rule generally slow multiplicative derive multiplicative stepsize expression multiplicative stepsize multiplicative nature hand element trick decompose call gradient obvious equivalent input moreover smoothing n neighboring nmf similar express respect equal gradient easily multiplicative corresponding omit limitation penalty rule derive multiplicative expression term denominator physical often impose spatial influence study detail estimation namely norm spectrum tradeoff respect update rule multiplicative mt image technique variation tv penalty tv penalty framework worth derivation spatial regularization application extend direction image pixel th abundance abundance use four ni represent abundance pixel neighbor impose spatial pixel four spatial leave denote abundance spatial ratio leave particular abundance get cost respect expression update mt I nmf extension hyperspectral image band band dominate three material water water band remove yield band evaluate mean reconstruction state join extraction abundance estimate simultaneously spirit algorithms extraction abundance spectra since comparable abundance estimation abundance sum two additive nonlinearity yield recently generalize bilinear factorization require complete identify jointly dispersion dispersion nmf minimize impose convex basic nonnegative interpretation nmf kernel nmf least alternate constrained square curse explicitly oppose base nmf provide comparable nmf nonnegative embed provide comparable unconstraine nmf multiplicative denote since depend stepsize note experiment relate generalize bandwidth kernel stepsize wise involve lin lin linear lin gauss gauss abundance estimate algorithm image aforementione despite feature reflect inherent abundance whereas capable nmf counterpart analysis abundance map new kernel input explore nature curse pre multiplicative several incorporate regularity reduction bs china degree mathematic apply mathematics economics degree security france toward university technology interest hyperspectral receive degree engineering sc receive ph university france associate laboratory technology france interest analysis nonlinear wireless network process hyperspectral nonlinear co author award machine signal past publish review paper receive engineering spirit master degree control university master technology france security system technology france engineering research university diagnosis france email fr fr france email fr conference explanation e additional detail derivation update state hyperspectral image nonnegative factorization widely include blind separation hyperspectral sense deal nonlinear formulation framework suggest
htbp c c c applicability consider use researcher pl iii comparison criterion aic parameter statistic n x dataset wang appendix set appendix bic statistic system strength stress stress ii intuitive big equation stress strength invariance invariance property estimating sample mn eq interval generate minimum illustrate compare model e department statistic central ac central university generate offer limit investigate maximum applicability show stress reliability maximum attempt engineering literature clear distribution form additional attract researcher modelling originally exp detail al say advance point modelling generalize poisson distribution introduce geometric et convolution work reliability family survival give new variable survival family distribution cdf refer probability pdf eq follow cdf correspond random survival denote paper illustrate organize generating like show procedure performance assess simulation give method show shape pdf decrease rx f rx rx hazard proof straight hazard parameter quantile detail reader q branch assume take side get function real immediate check therefore property negative branch w substitute moment generate distribution give constant display mean central tendency also skewness c median median mode c c c c mode c c c mean c mode c derive distribution minimum shape parameter h rule follow al corollary constant physics entropy concept popular r expansion j reduce http com enyi moreover special derive q know value
generalization asymmetric know impulse vector play successful heuristic nuclear convex minimization nuclear norm reader compare nuclear version formulate reduction perfect comment weight appropriate weight constraint reformulate also outline regularization path call solve eventually far decide solve problem region optimal solution respectively keep fix evaluated decide point increase stop instead upper bound reach certain upper duality upper subdifferential solve particular inner product singular relax since side computed projection theorem bind become eq follow duality gap unknown integer solve solution solve record diagonal subsequent path solution solution I implement two choose relatively order order enough truncate impulse response negligible choose figure green line dense say black vertical line axis start gap exactly extent division far certainly truncation round interest approximated system exclude negligible user decide model order h promise show computationally approximate path approximate efficient selection g perform iteratively outline path explore cost possibly another input output turn subspace identification se dynamical matrix determine methodology solution path calculate base duality whole tolerance illustrate approach regularization minimization principle simple preferred engineering science translate intend preferred low advantage order include control implementation discrete dynamical impulse problem take balanced norm eq reduction alternative chapter problem problem find identification np relaxation explore minimization use nuclear define singular correspond nuclear attention aspect understand aspect heuristic work aspect concern often upper impact regularization issue
rsc condition largely inspire assumption rsc important rsc consider rsc separately rsc combine product could significantly ingredient apply bind satisfy regularization parameter bound dimension gaussian sample covariance sample choose combine verify conclude graphical use range financial recommender play central computationally unfortunately graphical gaussian latent marginally low regularize mild exist learn open possibility statistical distributional often ill problem particularly observation ambient arise recommender microarray financial inference dimensional structure distribute alternatively precision concentration non zero regime force impose statistically achieve complexity true approximated paradigm due enforce sparse dense suboptimal new extend model motivate many portion stock movie rating conditioning sparse marginalization observe regime graphical inference correlation conditionally marginal regularized previously utilize derive bound strong convexity incoherence precision precision rate zero effective latent general significantly offer structure section review relevant prior present letter frobenius nuclear norm learn covariance often glasso author selection consistency certain sparse latent tree inference tree conditionally consistency insight estimation error practice provide insight performance also derive fundamentally model model low salient video detect decompose low dense focus formulation importance formulate graphical regularize element edge connect include property respect ji assume generality property sparsity property statistical propagation portfolio financial precision unfortunately world capture global specifically construct precision knowledge variable covariance precision example structure variable remain matrix l marginal matrix write standard covariance conditional observe assume variable restriction dependency dense potentially property matrix recommender return motivate example effective consider tight frobenius improve upon effective assumption design similar regularize ml constant define optimization solver ml adopt decomposable prior regularizer encounter two derivation convexity incoherence low component fisher subsection necessary prior main subsection regularizer detail p decomposable function subspace u rank decomposable pt pe let perturbation nuclear decomposable respect eq norm small structural true sparse respectively pair shorthand later restrict direction interaction subspace loss also denote fisher evaluate define restrict fisher precision sparse structural error exist constant restrict sparsity assume eigenvalue information away identity property trivial denote depend tight sparse plus incoherence set ensure estimation incoherence interaction interaction inner motivate generalize fisher constant relate period true parameter constant discussion estimation fisher onto matrix subspace pair low detailed behavior project control page consistency contrast make explain bounding quantity frobenius estimation establish consistency algebraic precision marginal program estimate proven estimating superposition structure regularizers critical estimation condition log convexity rsc structural incoherence si rsc specify function hand si certain interaction element rsc si loss problem log previously establish si behavior taylor series approximate sum residual condition rsc cf lead hold detailed remark bind appropriately additive capture capture many additive derivation apply estimation sparse derivation largely estimation sparse pair approximate relaxed incoherence follow derivation incoherence assumption term vanish disadvantage overcome incoherence nature apply program regularization bind sample particular sample specify parameter derive hold high obtain inequality lead assumption constant regularization satisfy term bound estimation however requirement next disadvantage largely remove covariance matrix advance asymptotic obtain bind state assumption theorem give regularization sketch choice corollary end sharp spectral deviation covariance constant high significantly also requirement low simulation derive bound well effective hierarchical capture represent sparse whose global concentrated contribute eigenvalue magnitude characterization monte assumption latent dense submatrix sparse observe vary magnitude variable submatrix rank dominate ratio effective covariance local effect become observe effective effective simulate effective observe theorem hold precision matrix ii range regularize ml covariance predict pn b ac validate configuration rescale align rescale predict variable whose low likelihood extend grant nf author anonymous valuable along lee discussion use world example stock return motivate manually decomposition see whether
idea final component concentrate estimate state span grid require obtain span converge eigenvector span necessary irrespective wish true demonstrate information yield mixture additional average q separation necessary phenomenon happen variance span necessary phenomenon gaussian span though assume away nonempty mi coarse first single linkage group sample apart norm eigenvector project onto hierarchical contain close contain component ok perform exhaustive describe state start merge large eigenvector project linkage cluster g c w run result simplify run repeat chernoff precise gaussian tail give single linkage routine cluster linkage scheme point close specify single precisely concentrate respective separation hence linkage correctly identify within form apart divide perform accurate component separate cluster cluster single small weighted radius eigenvector give c within grid possibility obtain dense error k note calculate implementation linkage provide use decade estimate simple one mixture use span whereas estimation gaussian underlie therefore probability occur hence none none occur interval translate select candidate one close nx w triangle grid dp union bind nn bind mixture k state distribution identifying mean paper technique strong relate product bp parameter see hence relationship chi bernoulli coordinate normal bind gaussians approximate spherical distance provide gaussian extend spherical gaussian spherical gaussian around origin shift distribution shift separate last component p ji code distribution differ separate distribution mixture overlap least component distance distribution bind kl divergence convexity construction eq concentration distribute eq lemma equation minimum q similarly equation show equation equation proof component quantity hence equation relate cluster distance empirical sample average sample component use fact weight union bind triangle inequality hold component component eq hence inequality get inequality immediately follow sample component gaussian mixture mixture rewrite hence component error component mixture lemma term nj I I exist j j cn make subset prove equation error discard affect calculation cluster I loop exist c last inequality component probability eq inequality component apart non cluster cluster cluster concentration cluster differently irrespective sample total union show conclusion hold hold union satisfy ci projection eigenvector show probability lemma probability bound total exist choice w inequality immediately discard discard affect ignore lemma first hence would triangle enough eigenvector matrix prove g j eigenvector inequality fact single theorem theorem claim conjecture exercise theorem remark etc gray pt algorithm frequently much costly computation provide mixture mixture spherical use sample sample complexity previously know complexity near optimal derive simple contribution include meaningful band influence parameter document topic genomic consider source correspond document mixture distribution therefore central method mixture initially method maximization decade spherical gaussian consider mixture mixture component maximum sample polynomially dimension show use requirement complexity slightly relaxed notion sometimes approximate component instead derive give distance error letter seeks also often accept sampling distribute count topic follow mixture person gender presence various independent bernoulli coordinate mean observation topic hence population person gender gene independent population bernoulli product special coordinate variance extensively study separation provably reduce great document every human dna costly broadly recognize quite accurately factor approach modification modal concave monotone hazard rate unimodal compare mixture previous increase bridge gap near pac c bernoulli axis align mean spherical gaussians align algorithm divergence similar complexity kl divergence symmetry boundedness triangle inequality distance main pac dimensional near paper pac gaussian consider mixture product pac learnable factor eliminate probability show discrete pac learnable gaussian gaussians normalize deviation pac learnable divergence would similar also modification complexity spherical main contribution gaussian pac learnable theorem spherical learn sample ok contrast component mixture require sample addition one gaussian time provide estimator mixture gaussian basic one dimensional learn f estimator take independent construct underlie consider gaussian component coordinate show vector
read report management nlp extraction challenge focus much processing datum focus feed challenge relation two check global consistency present programming ip multiple temporal across algebra provide news illustration potential survey event graph annotate feed document diversity annotation feed diversity generally classifier improve rich reasoning expression composition set relation form event acyclic event graph must define closure define relation contain path begin head recent adopt temporal capture temporal arc event order event base mutually exclusive interval use variety machine inspire maximum entropy naturally one want performance score generic ensemble relation relation enforce algebra ip linear grid polynomial unless np turn practice implement ip solver significantly relaxation range semantic environment manually annotate participant classification dataset precision henceforth classifier notice ensemble whole set evaluation ensemble difficulty way procedure classifier c classifier score well individual table note quite fair procedure ensemble table detail recall notice score individual suggest ensemble often per ensemble throughout two procedure compose label use classifier enumeration notice u classifier albeit start outcome support assertion diversity recall c classifier c htb n c u u c u allow switch open source art solver intel processor ghz large large classifier htb cnn build demonstrate enforce consistency individual classifier improve precision overall practical direction research explore alternative mean classifier soft c f
result asymptotically aforementioned technique promising tool verification synthesis article far derive formal probabilistic tailor characterization dynamic multiplicative possibly discount cost work include priori additionally tight probabilistic avoid problem sample benchmark valid propose approach controller synthesis critical introduce theoretical characterization reach put forward approach discuss implementation error characterization reach avoid priori probabilistic posteriori bound general discrete markov comprise continuous space number measurable action denote py kernel admit xy dy policy horizon action process policy execution characterize evolve product endow trajectory condition state st instant obtain realization control borel measurable initialize cf consider horizon reach avoid safe horizon trajectory reach avoid property reach property avoid system fix markov policy reach formalize sample policy logical formula contain write indicator expectation trajectory else widely study theory verification simply obtain select dual define start reach within safe reach backward initialize probabilistic avoid property follow scheme prove function avoid analytical possibly discount focus synthesis seek markov maximize reach emphasize policy state policy characterize backward probabilistic avoid express eq avoid mapping programming reach write composition often reach time notice general solve point k exactly analytical expensive analytical reach want seek obtain take scheme curse markov exist k approach second learn particular algorithm consider suitable markov replace evaluation base evaluation adopt fit finite horizon achievable accuracy convergence bias conservative assign accuracy bound bound markov specific avoid priori notion possibly discount cost adapt reach avoid safe horizon base point base table let remark generation step backward mapping realization estimate eq fact base use independent distribute realization safe horizon set sample minimize condition initial computation rather single state adaptation sufficient synthesis synthesis policy estimate argument secondly classification use approximately say solution accuracy study error complete horizon bound model computable apply alternatively posteriori propose b well function weight eq optimal error space employ quantity inherent bellman cause point integral contribution transition dynamic follow subsection error introduce recursion inequality bound recall x ix individual state elaborate random quantity kx kx identically q error incur quantity extend express via empirical follow norm draw accord express probabilistic error uniformly reformulate empirical informally uniform standard related bound employ capacity concept rademacher number pseudo capacity complexity uniform parameterized generate appendix pseudo introduce inherent bellman iteration bound free define state space borel q approximation p assume solution quantity express global derive mapping single successive value precisely approximate th recall piece together accumulation iteration horizon reach problem safe table probability kernel initial equation scale influence error increase depend set transition distribution markov display density lead confidence less per event error large long value confidence dimensionality directly grid numerical space diameter grid furthermore memory usage add comment firstly inherent bellman dimension former directly low capable function good accuracy bias due bellman large minimize align reformulate polynomial lemma whole probability compute converge reach sample reach obtain dependent also sample sample manner k base collect step propagation composition term write weighting estimate bias ia bias employ propagation bias consider reach problem reach initial quantity accuracy size set accord two estimation k ar closed loop trace hoeffding triangle k x bound inherent bellman error less conservative iteration dimensionality approach temperature reach horizon affect study attain reach safe fix implement ghz intel core I gb temperature markov temperature possible configuration relate random characterized stochastic ambient heat room heat room room constant heat room process l multivariate give distribution transition denote determinant matrix mean reach obtain k x kk temperature within inside safe set solve less radial function uniform width toolbox matlab layer unit radial artificial require obtain state space probability characterize reach avoid contour approximation radial basis function safe reach approximate contour plot green action solution reach initial optimal action via employ correspond action temperature accurate flat blue far heat room high turn I shape stay room room interested performance compute last little fitting estimate namely fall interval easily later height ylabel xlabel blue marks solid forget plot crcr color green marks mark mark forget sep crcr accuracy w h ylabel xlabel log marks mark mark option solid forget crcr color mark mark option forget sep crcr iteration cause programming fitting consider grows exponentially cause accuracy good probability property specification reach avoid focus maximization horizon avoid approximate fit make neighborhood inherent bellman sample probabilistic error programming control assessment concrete approximate propagation lead tight optimize closely employ ensure exponentially deviation interest know hold sum inequality suppose support realization lemma provide base express operator event action follow point via realization kx kx jj jt hoeffde long sufficient chebyshev chebyshev alternatively bernstein bound whereas also variance range derive exploit function space analytical backward recursion notion fit hold characterize endowed pseudo finitely parameterize pseudo dimension instant sake substitute draw realization l l rewrite expect value cover metric function draw include finitely concept number cardinality evaluation l minimal deviation proposition draw trivial isometry covering number l independently tf xx dd cardinality therefore also pseudo natural sufficient pseudo let invariance allow invariant composition conclude real pseudo parameterize especially class empirical class cover average use option pseudo number overall gain option explore alternative inequality bound random bernstein tight improve inequality conservative sample complexity adapt fit minimize norm draw instant realize x sequence inequality simultaneously also inequality since union occurrence first always bind function follow w w w bind true hand w p observe define observe event define fourth depend backward
everything systematic resample ia nu ms nc leave invariant walk strategy particle form site exponential family cycle site cavity divergence global efficiently property write site gaussian equivalently ij ij multiply vector cavity must inverse latter compute equivalently cholesky multiplication cavity normalize ij property exponential family parallel compute cavity moment v cm marginal ep smc stem mode approximate slightly run ghz kb cache model c auc dna thm corollary section university paris scoring pac pac bayesian asymptotic spike make amenable tool gold expectation propagation approximate extend method essentially score label bipartite elegant way estimate threshold accord fine negative false receiver operate characteristic criterion scoring auc curve roc auc appeal auc score equal resp draw resp class auc base skewed positive much classifier small auc way instead bayesian consist pseudo exponentially auc risk bayesian establish bound part amenable powerful tool expectation propagation iid counter class denote notation hyperparameter respect lebesgue follow assumption hold bound density j I discussion less satisfied surely satisfy soon see proposition prove regard survey excess take optimal choice accommodate sparsity spike number non depend explicitly suggest lead performance prior one recover assign however pseudo mix dirac thus expectation depend way I latter appendix side respect hyperparameter commonly dependence recommend validation discuss practical implementation beyond brief mention difficult fix arbitrary make possible perform overhead smc start sample successive step weight proportional skewed particle resample replacement move kernel smc make adaptive numerically impose degeneracy always walk calibrate matrix product precise algorithmic ep implementation highlight site interpret dimensional gaussian experiment even drop depend global implement cross little extra adapt spike add product site un normalise site update straightforward advantage bernoulli dirac mass methodology non functional associate trick pseudo except apply straightforwardly smc sampler ep implementation simple possible implement site update ep match computed ep deal non identifiable auc hyperparameter maximize evidence ep site gp version exponential balance datum criterion unbalanced ep auc auc refer ep gaussian roc comparison covariate ep auc logit dna perform give approximation figure pac dna spike coefficient spike sparsity variance one decrease dna blue circle denote bayesian theory propagation fast unbalanced work rank multi consider hold probability density spherical bernstein bernstein upper hoeffding inequality prove permutation jensen lead sum version bernstein chapter k
unbounded price reality first arbitrarily algorithm proceed utility could price utility introduce yield price bundle price optimal substantially actually perfect extensive game pick simply algorithm first exponentially algorithm price specifie receive efficient utility except price price price gradually preferred note value bundle solution unchanged coefficient impossible learn preferred gradually switch necessary finally together price interact price set price utility face price vector quantity interest many mistake incorrect bundle analogy polynomial price mistake iterative polynomial polytope represent hypothesis mistake expectation function fix mistake mistake inequality get mistake directly reveal differ slightly query price learner et specify price budget main goal broadly reveal survey effort focus seminal explain monotone construction proportional generalize utility formalize learner pac distribution seek perform observation hypothesis restrict utility algorithm linearly concave utility preference choose without utility separable unable class function control correspond query wish price price adaptively arrive mistake result inspire classic finite majority remain discard mistake maintain hypothesis vote volume bound mistake learn utility specify power good represent normalize price good price preferred bundle subject bundle unique utility maximize bundle value utility maximize fix know budget attention vector assume discretized increment fractional capacity weight production bundle price minimize obtain maximum something utility algorithm select measure optimally round possibly set bundle predict get actually mistake sequence price begin consider first price maximize give utility efficiently maximize price combine yield pricing section optimal price perfect nash exponentially straightforwardly inefficient nevertheless observation family contain exponentially precise derivation efficiently let price operation ratio specify might production cost therefore bundle xu b x k actually therefore price attain nearly specify three optimal efficiently uniquely nk k kp compute price list establish pricing correspond bundle I note threshold whenever whenever bundle utility per least good otherwise also maximize solution lp straightforwardly characterize disjoint optimize set lp lp ip claim per good sufficient v v decrease price additional discretized bounded price consider fraction good cost order decreasing learn irrelevant always give price price output ratio price discretize increment specify bundle price make price particular price bundle information prefer good algorithm price good next learn ratio price item occur guarantee minimize binary search ratio ratio attempt originally learn already know initially set low adjust price point switch learn price v whereby gradually eventually reach preferred identify high must none good learn must quantity minimize increment linear search require identify manner preference search critical arise value always occurs set optimally unnecessary price price price vector good low still good become must price budget power running make complete algorithm remain approach achieve round achieve optimal generate optimal price approximately pricing learn could maximized bundle invariant scaling bundle ratio actual bundle optimally price receive approximately less opt query might regret possible regret model bundle motivate scenario force accord choice parent company day observe say mistake ever algorithm price call upper informally describe give algorithm maintain consistent see initially round constrain particular immediately solution optimal decrease polytope idea uniformly mistake eliminate probability exactly equal volume eliminate consistent mistake final volume however need way volume efficiently hence exactly coefficient dimension fix maintain consistent index among yet together new begin never go epoch fix epoch challenge I z ic ic z convex polytope round integer mistake mistake algorithm epoch final mistake bind time mistake per epoch epoch remain coordinate track volume set epoch stage epoch hypercube incorrect eliminate make mistake round fact mistake epoch find epoch mistake epoch linearity expectation therefore apply chernoff plugging allow leave whether discretized wish devise approximately finally different stochastic lemma california edu california institute edu research author university period bundle price observe seek adapt price perhaps follow online price purpose management work strong utility maximization theoretic reveal preference problem utility price sensitive period observe possibly fractional utility optimize utility mild objective price competition set maximization unit associate minus bundle maximize every round maximize bundle price budget optimally instead faces give price quickly learn knowledge instead
see distribute present new topology grid system exhaustive optimize name dynamic algorithm improve performance propose system distribute grid power device g unit advanced communication decentralize grid literature exist link affect optimize bandwidth power strategy exploit performance minimize mse associate dynamic poor select improved topology adaptation neighbor choose mse sparsity topology reweighte topology usually employ steady employ hasting link introduce combination neighbor topology adaptation neighbor specify estimate automatically performance poor mean topology distribute estimation rule coefficient incorporate algorithm organize describe letter inverse system instant system quantity measurement standard control pt focus linearize dc j branch angle therefore aim distribute ls name modify report measurement vector vary exist literature grid system communication cause neighbor performance link experience chance need dynamically topology estimation aim algorithmic give neighbor mse performance note describe combination rule rule cardinality satisfy distribute divide step combination step strategy first describe include combinatorial strategy strategy complete eq equation combination need low propose simplicity adaptive vary report system small result steady zero link poor performance follow adaptation combination norm task strategy combination excellent log shrinkage magnitude intensity stand minimum neighbor simplify describe neighbors eq include vector generate adaptation devise inspire combination change algorithm perform dynamic adjustment mse value
rejection bivariate criterion via nonparametric procedure control false sim multiple control false illustrate simultaneous testing become familiar field economic finance genome testing association typically hundred thousand image measurement voxel determine area cognitive false discovery widely massive interesting case meaningful structure microarray study gene structural usually valuable suggest likely false spatially hypothesis likely multiple attempt prior instance appear comprehensive find reference therein arise remove generate uninformative stage testing procedure pass filter quantifying test statistic weight affect choose filter weight loss validity recommend testing adjustment weight test hypothesis independence filter weight setting proportion incorporate information multiple testing bivariate hypothesis hypothesis primary unlike impose independence statistic scope filter wish explore region multivariate hypothesis project take interval single estimator comparison mild method substantially long component value correlate extensive validity rest paper review testing present control multiple evaluate section section end proof test corresponding outcome significance nan proportion incorrectly hypothese e rt introduce rt pt true frequentist group cumulative nan true probability equal threshold weakly dependent level hereafter control detail test l parametric nonparametric projection direction r p start intuitive rectangular rejection bivariate derive rejection region false rejection value b proportion nan part direction control discovery rate recall region rejection bivariate intuitively discovery rate bivariate rectangular notational simplicity bivariate define joint probability f bayesian rectangular rejection event infinite rejection region choose rejection power rectangular rejection region preliminary primary fdr insight multiple testing statistic compare find control filtering choice rectangular rejection high seek form optimal region let region f special restrict rectangular group discovery denote rejection fdr f optimal equivalently rate equivalent large propose constant traditional testing lem powerful hypothesis rate homogeneous version discovery distribution test correlate true nan bivariate nan true nan extension bivariate bivariate normality true transform bivariate bivariate test bivariate see general serve motivation develop rejection rejection bivariate normality take eq rejection formulate intuitive viewpoint researcher prefer reduce component aspect transform search eigenvector common covariance find direction project hypothesis act parameter define threshold call index bt comparison region stage addition procedure stage hypothesis filtering project bivariate index weight testing generate combine specific proportional parameter valid choice region testing form p multiple investigate follow utilize estimate direction sequence marginal distribution define rt ft rt hypothesis reject project role discovery theoretical true nan follow uniform nan hypothesis category uniform true far simplify correlation shrink project structure area employ image derive specific relax normality respect flexibility distribution example statistic appendix estimating relaxed cause achieve robustness normality hold true use eq parametric select come nan hypothesis close interval drop newly z follow distribution pi nan dynamically estimate unified dynamically dynamically nan dynamically choose sequence nan boundary procedure range hand verify hand condition pt consistency efficient f algorithm paragraph fix vary projection utilizing whereas inference generalize recall rejection value correspond shape value suppose pt differentiable equal constant particularly depend bivariate normally identical covariance bivariate figure vary slightly study confirm bivariate selection restriction impose identifiable formula equivalent ft plug direction rt parametric proposition p parametric control naturally project q reject comprise denote propose estimate substitute counterpart method control denote consist incorporate obtain control procedure investigate preliminary suppose bivariate calculate bivariate perturbation observe contaminate nan incorrectly question sensitive method carry wrong suppose primary side nan respectively p side hypothesis symmetric leave hypothesis indicate test statistic measure restrict situation true indeed method tail conservative asymmetric distribution satisfy fr x rp carry wrong justify power ft f quantify much figure setting control valid pt consistently higher alone remarkably contaminate procedure testing stage bivariate far illustrate compare multiple stage testing preliminary bivariate preliminary primary independent stage test significant procedure bivariate positively structure bivariate flexibility example except generate distribution degree freedom identical rest come hypothesis provide conservative panel ii conservative increase unlike large since freedom normality p normal independent assess bivariate set contaminated method conventional conventional procedure nonetheless information control case contaminate close illustrate stability appear method indicate advantage bivariate practically mixture comparison bivariate similar prior multiple conventional structure multiple testing two stage conventional three bivariate locate neighborhood bivariate cluster evaluate sided hypothesis test independently serial consist cluster mean conventional mean filter serve ii conventional information ii conventional alone neighborhood pt c microarray experiment detect differentially suppose gene obtain value chi degree freedom independent utilize bivariate true comprehensive comparison particularly case parameter copy parallel gene level contribute disease discuss study gene gene study analysis source interaction therein cancer comparative genome pure capture genomic datum gene population low calculate sided gene copy preliminary show scatter copy correlation motivate apply significance dna location show projection preliminary plot bivariate geometric location use significance scatter bivariate value geometric gene valid gene dna alternative dna find differentially express dna dna apply favor unbalanced copy gene serve scatter passing symmetry fortunately gene line come weight pass symmetry small gene dna level increase threshold gene perspective function simplicity threshold present scatter testing estimate comprehensive comparison reject three term associate organization go bind activity map mutation list mutation support gene notably top gene particularly gene identify integrate gene complex experimentally gene list perform gene map recognize gene david functional content term gene table report infer activate cancer propose multiple overall microarray imaging availability nan project quantify novel procedure establish projection mild operator index normality generalization random bivariate test thorough scope paper future spirit power maintain rigorously theoretically proportion uninformative reduce stage view independence aim test bivariate change hence increase change powerful proportion nan change beyond make sequence strong primary statistic publish handle multiple structure much strong ft rt analogously process f ft prove main involved go converge first regularity derivation exist g rational denote quantile continuity c condition continuity ft ft f c prove proposition normal h ft uniform directly condition consistency partitioning l give pointwise pick pt j j l km ft similar argument ft ft result nonparametric uniformly fdr derivation yield f fdr exist hand take side f f satisfie formula solution continuously g f differential intersection solution equation unique equal normality appear leave side true derive nan function nan nan use curve f
similar black individual heterogeneity mark median laboratory evaluate blue total black allow individual heterogeneity demand implement allow however properly perform approach worth accordingly impractical somewhat reduce produce history computing evaluating find implement somewhat slow likely correlate link movement account semi place likelihood somewhat many possible low probability instead draw movement improve draw correspond basis population e g record name match record heterogeneity integrate simple heterogeneity unify miss constitute step toward synthesis multiple auxiliary facilitate inference use multinomial unobserved probit provide sampler bayesian avoid need tune probit alternative capture logit desirable interpretation odd work potentially gibbs use logit capture abundance sensible heterogeneity survival describe model extend population formulation accomplish substitute desire relationship formally describe method analyse source arise dna method heterogeneity heterogeneity broad maintain reasonable sampling e g challenging need extension evolve mark examine allow individual acknowledgment discussion study finding paper author necessarily represent view service imply us public center usa aid department resource united service increase allow capture simultaneously behavioral individual heterogeneity parameter probit present metropolis hasting algorithm monte abundance model visit capture population usa find temporal behavioral variation estimate laboratory effectiveness technique commonly reliably estimate evidence mark broader explore properly account introduce detection convenient bayesian capture population passive become common largely less individually technique evolutionary hypothesis passive sampling study entirely match individual genetic observer design differential home heterogeneity detection abundance approach analysis capture individual identification occur contribution focus abundance temporal accommodate level detection survival develop simultaneously variation behavioral g individual heterogeneity bayesian probit augmentation technique classic capture sample three record encounter encounter first second history probability observe nuisance parameter case unique encounter inference never precede scenario may type identify denote encountered yield present whether one uniquely make assume close abundance allow generalize much accommodate behavioral marginal history frequency indicate record derive distribution describe applicable review could error make estimation adopt bayesian perspective use mcmc abundance survival encounter encounter history individual represents identify pt latent frequency denote column record history denote example record history record detection history population abundance encounter time record arise history rise record p p I I I I p p history history history encounter pr individual history record history kk implement method construct latent history indicate individual history record record table example correspond column simply replace row example row latent history record treat individual evaluate joint h deterministic rather consider propose accomplish utilize space nan solve vector heterogeneity explicit however allow heterogeneity explicitly population therefore abundance proportional mt b alpha illustration abundance temporal behavioral heterogeneity extension effect account modify adopt utilize augmentation formulate probit detection data treat binomial parameter indicator real individual individual individual remain proportional continue proceed j r rx respective j j never capture individual individual accept order conditional inclusion bi sampling page return repeat calculate iteration concerned heterogeneity detection quality artificial mark genetic material vary individual modify accommodate temporal effect heterogeneity specify probit individual intercept term individual identification latent u u u component modify model full pt tn mt mt u notable detect individual never detect analytical applied incorporate abundance usa occur hence addition provide error collection genetic visit dna capture close abundance allow behavioral heterogeneity motivation heterogeneity incorporate model rely meet capture fit probability method indicate behavioral heterogeneity allowing occur p mcmc algorithm accordingly reduce update distribution data sample correctly assign investigate sensitivity conduct uninformative specify programming pre post interface integer tune mh sampler divide acceptance basis accept million analysis analysis ghz intel core processor long movement rate movement result possible chain assess visual diagnostic package analyse diagnostic uninformative parameter prior credible interval find behavioral response suggest population report black dna low c individual heterogeneity cause abundance explain heterogeneity interval probability sample collect auxiliary informative analogous uninformative yielded record another way nevertheless informative little contrary take specify informative could sample degradation environmental could abundance prior conduct laboratory
despite simulation fair would reader expect perfectly fair introduce literature good reader situation important major aim implicit important software compete unify implementation give advantage incorporate initialization present soon package problem firstly fit plain namely treat improper density secondly decide exploratory likelihood way scope lemma di di universit di mail college uk mail c ac abstract introduction optimally comprehensive mixture density look treat idea benefit comparable fulfil apart evaluate standardized misclassification rate usual one one keyword cluster em improper likelihood robustness h optimally robust improper approximated distribution simulation currently comprehensive involve careful discussion issue assumption improper pseudo component define small capture outlier inspire show improper comparative study may cause problem use cluster certain maximizing give general multivariate shape want rely really generate method call reflect example use mixture fit way illustration use city discuss fitting plain outli collect robust dominate produce method recent overview level improper tuning introduce comparative involve introduce unified noise outlier prototype cluster set mention along song region issue additional dataset simulation study computation supplement property cite exist ix pp x ml mixture distribution mixture density parameter assign mixture estimate interpret proportion point covariance th cluster popular implement prove method assumption attempt deal ml mixture suggest outlier add mixture noise implement result uniform fix hull datum package include note proper maximize estimate point belong component v cluster drawback affect point reduce formal method replace density outlier noise far away area pre specify small central definite freedom consider freedom ml assign component result ml quantile degree freedom method cluster recent cluster amount index cluster model cluster triplet spurious outli cluster every point author maximize hand side outlier introduce method regularity propose consistency methodology partition proposal approach base find adapt estimation behaviour approach robust improper idea improper improper vector include improper used probability assign fixing define model region density distinction uniform convex hull cause problem require suitably discover easily extend well way prevent eigenvalue spherical relative scatter among study gaussian algorithm multivariate alternative see disadvantage affine would component transformation activate prevent still improper propose constraint quantity interpret point implement familiar half plain consistency topology extend result mle lack matter dependent choice quantity rather device enable good approximate cluster region look produce gaussian minimizer measure cluster prototype cluster th approximately good squared distance component indicate cdf kolmogorov optimally tune denote discuss good development version beyond paper however important precise normally bring see effect maximum enforce solution optima far compute recommendation fix initialization possibility assign observation time result pick comprehensive initialization consuming use initialization actually big outperform partitioning recommend spurious initial pr min attempt valid identify ml initial gaussian noise point partition program within initialization end happen enforce lead particularly set interval often contain large candidate discard later normally simulation assume true problematic consider ml gaussian method implicit way classify outli comparable assignment method triple permutation compute scatter compute expectation matrix study explore central student marginal marginal reference although case close non assume affine package mass supplement level idea decision p level play original freedom assume across datum covariance constraint freedom incorporate motivate consideration design avoid spurious ii design base extremely variability student decision mixture suggest initialization see notice allow allow initialization additional op op compare use cluster task estimate average standard code monte pair gray scaling emphasize difference high get precise misclassification online contain misclassification pair clear robust important perform gaussian work slightly bad time involve outlier suffer dimensionality u ml contain estimate generate rate generally though automatic always proportion p point compare suffer situation overlap completely dominate see separation student marginal perform particularly show good overall misclassification method comparison encourage see basically produce get case h disagreement substantial mainly sometimes merge cluster structure integrate investigate behavior independent compute interval add main produce whereas figure small constant constraint become active clear minimum minimum lie nice dimension core look become happen unless constraint latter section pattern mostly seem quite stable different h process produce estimate standard smoothly estimate noise proportion scale impact strong sampling consider rather discovery cluster change section example percentage classify involve central core distribution assign percentage point assign whereas noise really involve gaussian clearly separate separate need decide desire toward distributional dotted computed plot section situation affect dataset axis first separate therefore separate large clustering obviously analyze regard interpretation depend well treat noise outlier merge two cluster tune produce noise component artificial alternative real ground truth one class usually guarantee clustering reality prefer finding example give city originally competition conference fitting logarithm death birth balance divide divide number variable median figure death move figure moderate
moreover spirit dimension rather translate fast recently prove complexity still room third could complex value establish instrumental analysis describe approximate expectation function isometry existence around use cumulative couple use recovery control equality introduce limitation limited coherence level isometry control frobenius one dictionary coherence atom possibly sufficient desire quality would around potential algorithmic spirit alternate minimization limitation imply surely expression spirit improvement involve improvement convex recovery main consider triangle inequality assumption yield denote r r u previous rewrite w except handle thank convention j inequality u lemma denote p b pp j k take support draw since f I denote restriction j similarly f term f piece kp observe continuity apply I simple lemma term u proof far conclude lemma definition assumption consist signal select paradigm lead image audio argument sparse rely procedure analyze yet paper probabilistic sparse signal admit reference complete key quantity coherence combination field statistic learn line development framework algorithmic tool make design good prominent deal effort dedicate efficient wavelet notably many compression decomposition simply powerful classification formulate problem code bring play non convex success analysis establish generalization quantify signal reconstruction sample uniform focus aspect dictionary minima identify dictionary important interpretation call arrival modelling related visual accurately learn also obvious carry code denoise distortion denoise dictionary intractable heuristic e behaved characterizing help exist measure importantly early identifiability combinatorial condition involve criterion form basis identifiability noiseless outlier arise consider bernoulli model without analysis extend dictionary compose signal exist handle none absence straightforwardly take minima outlier regularize least cost truth relate consider minimization algorithmic demonstrating provably complexity initialization alternate open source implementation online approach available extensively exploit application htbp c noise exact admissible coefficient characteristic svd nb frames rademacher sparse response decay overcomplete dictionary presence characterize ground whether guarantee whether output algorithm resp characterize minimum exactly noise finitely upper provide sample complexity level allow brief description support select ii nonzero decay coefficient random recovery atom penalize penalty probabilistic signal plus noise loose coefficient closely nonzero cumulative see minimum generating dictionary prove blind separation understand reference tend variance hope nature algorithm sample rademacher average level cumulative coherence involve may level precise control admissible demonstrate relative amount robust material integer denote transpose norm frobenius f place exploit extract conversely b n zero index index denote complement denote linearly gram inverse orthogonal span column ba ba bb ba hold nm column dictionary learn sparse vector n ik reconstruct code definition denote typically dictionary signal q basis processing minimization regularization parameter control tradeoff unit image depend unit simplex characterize penalty generative noise contamination outlier state show necessarily frobenius future importantly generate signal discuss blind source separation invariant permutation hope specific transformation describe equivalence local invariance issue soon sufficiently generate noisy spurious support replacement available shorthand eq jensen entry magnitude small conversely marginal coefficient dynamic way measure boundedness complete handle neither indexed fact stem concentration traditional illustrated decaying sparsity norm relate cover control early field sparse specific express outlier train distinct property relate manner dictionary function represent training n number require namely complete minimum rely recovery signal almost surely control support solution regularize reasonably impose condition dictionary quantity unit term correlation exceed cumulative conduct assumption coherence coherence consider previous dictionary bind rely weak assumption context role define separately coherence would fully relax rip sparse code problem admit local neighborhood control regularization main building high dictionary sparsity consider eq denote deduce minimum provide local minimum problem consider minimize signal yet compatible small scenario hand large hand admissible regularization enough factor small coherence dictionary measure p least coherent quantity f resolution limit outlier discuss constant explicit aside consider assumption assumption constant draw f r robust addition provide right impose outlier refined argument frame I decay model randomly sign vector noiseless resolution infinitely slightly bad resolution noiseless indicate ensure minimum around fine choosing recover known optimization soon boundedness soon infinite sample quickly outlier control admissible energy e precision ratio f rr correspond minimized variation orthonormal perhaps orthonormal achievable impose constraint check limit precision local minimum radius around q noiseless resolution reach provide exceed resolution threshold base incoherent dictionary plain coherence eq hold soon incoherent union basis fulfil soon exceed consider satisfy maximally incoherent large read amplitude hand relax j j j existence coefficient satisfy soon much restrictive spherical ensemble spherical ensemble obtain independent dictionary satisfy soon classical continuous compact constraint radius sphere q reduce h consist ensure ball existence asymptotic analysis vector cost eq depend via covering show generative nc desire sample interesting negative target arbitrarily satisfactory resolution independent refined gain collection sometimes contaminate irrelevant consider sense share dominant property training consider resp matrix extract keep column associate resp clean contribution together n induce norm tx robustness context interesting regime arbitrarily robust learn outlier seem f technical arise implicitly define minimization leverage denote minimization always denote make pattern remark arbitrary moreover guess p linearly assumption light switch lemma signal uniformly addition suffice match conduct assume restrict ball reason motivate strong assumption involve coherence exact dictionary reader hence prove average rademacher average set usual eq rademacher probability eq z equation absolute real r respect probability bind conditioning
assign rna moreover annotation rna may reconstruct annotation seq reconstruction abundance estimation achieve interested reader refer comprehensive relevant computational many develop rna expression accounting method design group per sequencing mcmc second assess expression posterior employ likelihood approach expression testing gene multiple use differential gene expression usage usage refer total expression construct log standard usage sharing start differential usage square root jensen shannon first perform expression comparison negative group mean rna category relation construct form address uncertainty inherent two commonly accomplish g additive ab bb situation datum compare allele allele cancer patient situation rna statistical test although population limit test case happen seq population happen group hoc implementation specifically assumes differentially express conservative limited confirm study develop name seq aforementione accomplish method treat rna gene cluster overlap whether rna covariate interest would rna large gene burden multiple multiple major estimation expression separately problematic perform rna identify one differentially material know previously input file rna seq non adjacent rna seq usage adjust covariate htbp indicate penalize employ binomial regression rna seq binomial variation seq biological replicate differential expression testing adopt distribution assumption negative binomial rna seq replicate therefore binomial value adaptive lasso broad penalization categorical size annotation demonstrate satisfactory seq analyze seq reference genome situation part belong cluster far impose unlikely unlikely belong cluster cluster subset overlap size portion bp pair end assign I ia abundance ip th intuitively position sequence vary gene set contrast effective length nonzero include effective rna seq length seq supplementary material effective estimation binomial count effective covariate reasonable across configuration equation challenge matrix first candidate binomial regression candidate seq supplementary material database skip negligible length across candidate informative high effective length candidate important zero covariate often correlation among employ log penalty require interpret material c sample estimate expression read depth sample depth measurement rna seq rna seq first variation normalize n problem write impose penalty supplementary covariate snp focus linear additive aa ab bb b expression reduce binomial non impose solve material n iv iv uv g multiple covariate effect impose penalize supplementary material study express examine count across expect component differential describe respect set covariate alternative helpful understand special solve binomial chi model regression second category categorical binary g chi degree freedom penalize lr statistic log penalize nan follow asymptotic distribution penalize nan count step number lr procedure regardless rna seq small value study sample size vs valid population study calculate interest unchanged alternative calculate ratio repeat obtain statistic differential discussion include rna estimation testing rna rna variation expression situation expression gene low switch rna switch use refer relative rna usage seq rna usage replace discussion million rna seq read single simulator annotation simulate equivalently gene differentially term usage supplementary rna read genome next rna seq set rna seq pair read confirm effective supplementary candidate vast annotation restrict strong supplementary penalize cluster annotation supplementary abundance conclusion include seq htbp status ok abundance use annotation abundance next power testing usage file differential expression usage site usage majority file status ok ok file file status trust reason case combine replicate lead conservative implicitly case comparison gene favor power base proportion significant high figure attribute supplementary issue compare use roc supplementary estimate replicate replicate biological replicate due resample rna seq read fair recommend control simulation work challenge situation usage respect continuous covariate quantitative trait seq real european select minor frequency follow cluster version simulation annotate snp body rna seq across respectively select drawing assess differential usage nearby snps body multiple nearby permutation usage figure b simulation correctly detect differential usage respect treatment drug valid td rna seq vs supplementary list treat associated neuron dna protein knowledge software availability package expression usage intensive minute gene processor need implement material seq rna seq treatment discussion name assess rna seq categorical resample component distribution penalty resample first choice model seq data biological completeness binomial seq count lead severe lasso inaccurate supplementary apparent candidate consistent previous finding positive penalty well abundance rna seq distribute dna sequence affect abundance rna seq read bias likelihood rate limited impact testing rna read type error systematically account future assess usage pair individual pair multiple meta usage side near future plan include large diverse genetic collaborative sequencing read bp sequencing seq rna seq assign differential usage software development rna seq rna seq rna seq data count seq anti separately partially grant gm rna seq rna seq end sequence end discussion pair read read read impose upper th effective fr j shortest effective word rna seq effective seq summation weight ht discussion notation skip subscript length length fr consecutive whereas cover h r derive follow part r use observe even may sequence improve robustness determined count define end examine end identify break point read depth adjacent specifically gene overlap apply chi assess whether different length break parameter break default value cutoff default th break select construct consecutive gene among cumulative trial effective claim construct set set default change situation negative binomial dispersion discussion binomial extend situation j log penalize glm employ non intercept impose maximize iteratively update regression I I adaptive estimate likelihood implementation include loop combination carry loop square quadratic current square coefficient need remove improve efficiency initialize update n ij coefficient estimate little change crucial penalization select strong scale choose log large grid tune minimize rna often small de hypothesis bic log chen chen extend eq simulation restrict number optimal framework rely rna seq p suboptimal influence capture resample tuning conduct guide care laboratory laboratory resource national care university bl treat drug treat acquire bar maintain hour hour dark schedule room maintain cm laboratory water release day age collect tail drug day exposure steady concentration ng ml achieve drug bl nm age day drug treatment remove home brain product ok extract rna rna life rna verify library contain rna per library read bl use alignment experiment merge specific genetic alignment quality minimize cause genetic reference read segment bp per approximation snps bl genome variant report sequencing effort include release read mm coordinate update position string mm annotate pair pair allow pair read merged position tune thousand bootstrap parallel computing processing however particularly maximization numerically separate make dimensional log parameter factorial selection attractive seek penalize mm rely strategy concavity indicate support therefore update perfectly preserve iterate effect cause vice versa desire update increase parameter g five mm univariate omit htbp control case control control usage per cluster usage quantify total htbp htbp htbp htbp htbp outcome htbp mle glm nb htbp genetic map pass map bl j bb bl cg cg anchor anchor protein exchange factor exchange contain ca activation factor core domain contain protein domain box contain translation gamma domain alpha similarity member gm protein couple protein b associate protein cell like contain
causal worth note represent physical direction yield although certainly actual particle fail stand value represent choose measurement however categorization kind five outcome observation probabilistic incoming incoming link along causal quantum kind system classical correlation coincide usual one arise kind hide structure able general imagine every take concrete experiment need write outcome concrete tuple tuple term difficult obvious subset past collection distribution consider give causal define classical show actually quantum derive candidate sufficient consider recursively set disjoint get disjoint follow condition upon otherwise thereby subset causal write minimal familiar consider scenario figure generalize arm party source outcome party party party scenario equation conventional state independent assume scenario formalism outcome approach automatically treat realistic usually correlation probability setting actual distribution relevant processing application device independent expansion strong limitation need impose another device independent key even way highly part establish put disjoint causal past hold suppose thank suffice check maximal past w interpret party analogous know formalism box across party individually derivation carry induction party equation place arbitrary party show induction third worth note need event causal past introduction every carry variable conduct basic constitute representation physical modify system additional necessary reveal believe hypothesis situation biological try infer correlation behind action potential measure certain certain cloud formation possibility make hidden live edge determine outcome certain node operate take incoming turn variable outcome think hide live variable suppose physical one arbitrary conditional v particular different kind incoming sort hide live develop equivalent previous information processing fit standard causal behind thick thick blue node node approach probability variation allow reader non may restrict countable explanation order concrete hand v integrate result getting speak obvious ignore terminology suggest correlation actually node determine sufficient hide variable prove base case formula precisely assume big place likewise upon guarantee induction conditional v integral evaluates complete case evaluate keep already suffice u exactly party variable measurable conditioning make classical putting eq free use classical let copy argument quantum desirable result correlation classical conditional general causal good display special outcome node variable applicable inequality problematic aspect scenario polytope soon unconditional well question maximally hide space outcome scenario polytope general hide correlation able approximate classical correlation space variable whether finite space generalize definition infinite informally every realize finitely induction number node therefore put start realize hide variable node take together apply need careful way need subset finite replace conditional take subset finitely many possibility hence index measurable equipped distribution virtue coarse arise come equip tuple modify associated source word outcome new outcome retain distribution form outcome define induction correlation edge nothing except together hypothesis arbitrarily make second sum know assume variable structure also plausible randomness randomness parent additional one randomness generation back eventually end except parent precisely represent deterministic meaning v cover variable finitely reformulate v involved intuitive randomness parent vertex hide realize assumption nothing acyclic find start incoming edge reach node consist randomness inherent induction plausible technical quite demanding hide turn regard assign probability regard list say independently respective denote variable new u w f w stand tuple consist together construction incoming nothing manner define classical hidden modify u wu u randomness back rise overall result u coincide original upon component list drop make randomness along hide regard node classical induction situation property information root put processing complete virtue arbitrarily http net question measure measurable might achieve proof framework definition correlation form physical quantum probabilistic definition reason general rigorous quantum correlation piece structure previous correlation thing carry turn index individual success get preserve operation physical piece category gets label gets label assume strict play role carry summing outcome result normalize normalization preserve cone add neutral precisely module consequence mixture rescale simply multiply bilinear module bilinear distinguish normalize operation nothing nothing operation act nothing precisely scalar go real unit terminal every normalize thick circuit causality list requirement possible example next classical space determine correlation idea diagram object graphical calculus another reason believe live opposed thing overall composite specific node whole operation outcome e index operation marginalization normalize processing gate turn classical think operation realize exactly outcome every try label diagram indeed category formalize diagram idea appropriately label diagram product product factor generally unbiased category acyclic graph diagram graphical calculus deal per se rather appropriate piece seem develop everywhere difference causal direct indirect link causal causal comprise converse simulate indirect causal link link indirect ultimately link indirect causal correlation causal differ add prove claim correlation object extend assigning consider correlation conversely correlation turn correspond except assume come information pass subgraph imply end formalism correlation measurable whose operation space object operation normalize category notion coincide classical integral composition besides definition quantum correlation arise category operation seem work correlation contrast box every box theory box world sometimes one everything imply correlation try symmetric addition scalar come equip satisfy law requirement correlation arise object linearity imply object result linearity govern theory correlation consequence classical describe first option sense existence non broad existence locality fine tuning explain formalism space e hilbert speak correlation formulation quantum however operator work finite definition correlation prefer hilbert quantum state prove definition whether quantum symmetric denote infinite hilbert space trace understand quantum positive trace mind space form map operation operation preserve channel straightforward check correlation proposition definition concrete thank general obtain quantum proposition classical quantum know requirement assign hilbert space interested reason skip assign hilbert crucially measurable measurable obvious object finite hilbert assign eq straightforward assignment quantum operation operation operation map quantum would quantum advantage stochastic operation gets map quantum operation identity identity indeed classical correlation realize correlation sketch full go outline proposition situation turn stochastic operation indeed trick replace hide finite every v v corresponding outcome along diagonal canonical basis adequate party scenario free equation probability admit hilbert operator depend completeness implicitly assume since begin party detail label space incoming edge collection index outcome statement apply likewise similarly correspond simply relate tensor product state measurement multiplication encounter begin free guarantee hilbert since completeness I I behave rewrite trace hilbert jointly become eq dividing multiply hilbert hilbert basis index define straightforward operator quantum proof proposition translate back scenario formalism vast ordinary definition study verify comprise usual concrete quantum correlation scenario correlation party classical quantum explore like quantum hide extensively quantum hide stochastic recall variable index underlie causal precisely informally correlation causal must like definition like thick draw fill right right right circle fill blue l l leave informally speak every empty subset disjoint causal realization hide depict impose requirement hide machine biology become correlate hidden easy handle mathematically useful classical actually admits define take coincide output gate take incoming sequence case markov commonly illustration way variable hide come equip pair think result update operation hide finite direct acyclic correlation description distance thick anchor south east anchor north west circle fill lb aa b la lb aa aa ab al la lb graph generate measurable space finite joint distribution constitute network idea hide whose dependency arise bayesian actually represent word redundant start space old variable space basic classical correlation gate v assign hide outcome tuple old old gate coincide gate eq sensible left hand become need indeed original correlation onto tuple edge result start representation form one carry space gate follow income actually inclusion support end process eq simply far integral equal show use definition quantum measurable proper mathematic hide variable classical model like recall deal measure notion classical concept quantum set subset measure size satisfy axiom always mean actually measurable necessity measurable measurable theory hilbert quantum function pairwise want make allow case less letter interpretation measurable take form integral correspond paper understand notation integral symbolic since measure set regard integral refer text crucially deal produce exactly suppose analogue reproduce definition normalization line case space satisfy property fix fix measurable obtain abuse element operation measure quantum map trivial compose mean integrate trivial proof operation operation mean satisfy consider measurable normalized stochastic operation coincide product operation form product contain form need measurable operation definition see detail parallel analogous unique property assign leave note particular measure prove crucial refer additional terminology write difference algebra set measurable main induction nothing apply obtain arbitrary sp b desire measurable equip finite every measurable exist measurable q nx ps ns approximate well coarse measurable algebra b require many finite boolean algebra atom construction element hence product set spectrum unique atom tuple unique union suitable whole rgb thm thm thm conjecture thm thm remark section participant development innovation author foundation scenario small quantum theory quantum graph conceptual measurement role scenario formalism understand contribution latent thing markov influential interest theory lead recently development processing protocol security rely crucially realize illustrate non quantum correlation generalization include party party scenario additional subsequently comprise party party recently scenario add purpose present thick anchor south east north blue blue minimum size general comprise scenario technical source measurement measurement three pair point one event think event specific outside classical outcome way link think represent event interpretation equally mathematic thick scale anchor south anchor west space fill la lb lb interpretation connect typically random causal describe distribution become central causal assume represent outcome physical point outline sample sampling point reveal like uncertain complete imagine biological rarely also apply physics mechanic incomplete introduce degree represent particle event relevant party extension proposition formalism equivalent point explore plus minus skip node anchor east node anchor north west circle la lb la lb lc summary idea outline definition proof sometimes de force quite proof demonstrate trivial argument mark reader simple completely proof describe causal course turn know influence one direct indirect link equivalently order discuss quantum causal scale anchor south east time north west circle fill blue mm b z drawing look four event arrange structure indeed induce example spatially event happen suitably time continue causal outcome independent soon disjoint causal assume likewise potentially physical show reduce introduce variable paragraph pass causal process compatible unobserved propagate causal link give sense feature domain allow live necessary outline show fact characterize correlation
analogy setting unbiased blue estimator minimize estimator minimize variance strong request unbiased linear couple continuous couple ol respectively ols ols estimator mean replication unbiased proportional square residual develop hand side vector whose th element derivative prove imply q dt model hand process generalize tf ols unbiased proof appendix functional ol blue present work extension bayesian might curve might add weight instance distinct space generalization different world without loss generality assume delta symbol schmidt apply corresponding let kt w kt base couple last definition tf representative easily project ol projection projection linear crucial orthonormal straightforward take obtain blue knowledge common work setup choose condition support choice estimate estimator analogy functional design minimize design definition maximize herein study performance design motion location experimental location angle form right location central overhead panel available website compare motion seem adequate collect location trial support coordinate provide continuous optimum exact factorial batch trial experiment three trial perform thus exact optimum whole domain factorial different factorial factorial grid spaced factorial find design design goodness efficiency efficiency list low design accord separable usually majority differently proper space guess provide derivative roughly speak incorporate gauss functional despite complexity obtain elegant belong space experimental present literature instance rigorous model matter optimality criterion context theory set subset scalar ol map unbiased operator write vector couple vector tn representative introduce operator q thesis immediately tt prove linear equality five f nan f l pl h pt equality due couple ft ft dt ij matrix orthonormal complete representative kt replication relation eq assumption distribute depend neither given imply choice linear equation q tf equality operator lt lt set equation thesis side eq proposition definition theorem observation realization science economic field process procedure derivative reconstruct separately obtain hence gauss markov linear unbiased observation realization continuous science economic reason area book book parametric approach cover classification discrimination cover situation response functional multivariate repeat design remain focus functional regressor derivative exploratory high derivative reconstruct recent derivative usual reconstruct justification curve function course take consideration curve derivative reconstructed observe directly function reconstruct smoothing knowledge adopt analysis description work present consideration fundamental practical focus experimental design estimation summary final remark theorems regression response random linearly need experimental batch form trial repetition
second inequality due rearrange feasible objective feasible key strong optimal exist hold c xx nh prove em ready complete follow relationship feasible optimal last quantity auxiliary eq q proposition k f expectation side inequality consider sum together fact mf eqs lemma hold convergence analysis theorem interesting whether let satisfie obviously sphere satisfy assumption solution imply constrain illustrate convergence optimization easy certain counterpart specific q optimal solution focus set constrain extend building establish convergence constant verify w imply theorem even effectiveness constrain logistic label I review three multi sparse class positive remain negative conduct sensitivity study distribution step vary fix onto ball thus dominant computational implementation vs plot non much remark step demonstrate size robustness ht tf plot average run uniform parameter average comparison sdca sag unconstrained sdca adopt solve optimization accelerate sgd gradient suggest stochastic hybrid sgd pass switch kf plot set sgd sgd comparison behavior report objective gap plot initial outperform quickly gradually proceed phenomenon commonly hybrid report reduce stochastic solve constrain problem establish strong reduce analysis wide lipschitz continuity eq il eqs define obvious assumption exist know empty assume f lead hx f cx use optimization slow sub linear inherent computation many objective strongly project stochastic problem contribution convergence convexity variance reduce stochastic role big optimization solve prefer due stochastic gradient optimization standard draw compute stochastic slow full proximal rate recognize slow stochastic include sag stochastic dual sdca epoch mix gd prox linear practical square extensively even rate full address prox establish convergence convexity without although solution still convexity address establish relationship current linear solution objective bind establish objective function however paper suitable address adopt establish linear strong general mild satisfy present constrain follow paper eq continuously compact accord notice convex onto must moreover compact example objective dd logistic tx tp k np k let objective value achieve respect sampling remark eq similar prox term propose large choose outer compute single count gradient l proportion uniformly outer evaluation complexity specifically remark prox need evaluation obtain contrast obtain high eq section several fundamental key idea establish distance feasible rate recursive strong convexity lemma constrain optimization adapt corollary establishes gap solution function linear bind
interval use measurement include ahead instant flow therefore name lag smoothing may lag smoothing lag measurement arrive numerical investigate pos run choose interval conjunction temperature pressure accuracy flow however variance optimize outperform one manually variance accuracy suggest model happen pf step particle match observation compute intermediate k ip k sir obtain index sample set index sample associate particle eq previous involve need adjust end compute proceed previously rgb international gate mail international author flow sense problem previously module model certain temperature pressure flow leave well jump design capture adopt measurement physical process two approach approach sensor advanced field typically equip sensor control rate require flow expensive result measurement risk collect flow exercise soft soft decision purpose essential production tackle soft sensing base instance extend soft sense lift adopt recently enkf flow pressure soft filter example employ apart approximate bayesian soft iterative current work module framework include flow conventional steady state behaviour describe time reason dynamical jump capture rapid water operational underlie flow measurement one estimation g mention paragraph aforementione sense variance manually situation aim fill propose variance criterion problem proposes previously illustrate reader short introduction equation temperature pressure instant phase normally contain distribution simplicity estimate jump process flow follow probability predefine normal remark firstly dynamical model state operator noise term distribution unknown variance discuss optimize optimality assimilation flow rate gaussian gmm assimilation gmm pdf instance view density kde aspect convenient development later lead problem section form adopt non assimilation auxiliary rate sir mainly sir pf weight particle ahead sample particle brevity detail q j r k j jj nj weight generate rate tw formula manually trial final may automatic optimize sense later measurement collect interval type approach smoother optimal carry respect certain fix time ahead correspond smooth fix lag smoothing approach idea notational convenience instant contain jump minimize certain joint pdf solve line k large optimization process become need assimilation scheme often fix long minute study fix interval offline condition past instant particle dirac delta elsewhere substitute eqs obtain approximate influence optimization optimize lag smoothing use estimate measurement kl compare interval fix lag estimation arrive rather time instant current lag minute lag however beyond em algorithm construct function interpret cost maximize value initial one construct dependent condition maximize obtain optimal start construct optimal stop relative estimate j pre threshold estimate loss generality instant line measurement involve construct dirac delta function jump log joint pdf joint f calculate conditional pdf determine jump rate condition include instant situation approach filter approximated flow rate write ignore purpose maximization one need solve simplify discard irrelevant obtain analytical approximate integral carlo approximations instant associate multipli kronecker whose elsewhere essential thus drop q note multivariate draw variance likely relatively away case substitute eqs solve q obtain iteration square light eqs weight update estimate positive definite iterative weight need needs iteratively significantly reduce iteration f l easy verify hessian iteration formula maximize estimate example due behave slightly first flow run resolution md pressure temperature assimilation later pos run fine resolution synthetic course datum assimilation model run system apart case possibility assimilation produce purpose examine correction assimilation different number present respect seed configuration show vertical horizontal curvature degree vertical md label label md sensor collect pressure pressure k pa detail reader refer flow rate z pos measure pressure pos depict spin flow simulation reduce happen accordance fig pressure record place contaminate meaning pressure experiment deviation measurement pressure configuration jump specifically take set take true rate multiply result smoothing first variance flow jump particle sample input well pressure record temperature pressure adjust relative particle match observe pressure also profile true red blue dotted rate deviation pressure temperature red simulate flow blue flow simulated temperature pressure record flow flow flow appear true constant variance situation rapid flow true average minimize toolbox flow rate flow pressure panel use flow smoothing approach assimilation fig appear time rmse become conjunction rate variance interval spread implement em flow stop either norm iteration stop estimate variance leave variance cross order magnitude time consequence correspond fix interval variance flow variance lag equip lag pressure fig fig approach term pressure lag estimate less smoothing approach change true fig lag rmse become fix interval pressure temperature fine resolution study case uncertainty error purpose variance way pos uncertainty flow synthetic well flow assimilation synthetic resolution well md assimilation flow instead comparison well flow fine compare fig temperature uncertainty generate fine pos assimilation
inequality iff inequality statement eq put inequality subgradient q proof state technical use smooth denote get verify corollary schwarz plug obtain eq eq proceeding optimize inequality let first derivative compact domain lie enough need smoothness derivative denote put allow q eq inequality elementary q reasoning denote tb jensen need lemma reasoning put together divide everything take use jensen denote scalability selection ignore fact theoretical make example solution practical remain propose gradient perform selection tune cross build regularization rate prove standard fractional scalable optimize objective way come source moreover theoretically yield well yet rate depend use size reality wrong strategy attempt prior knowledge characteristic optimal cross validation slow partially exception keep epoch procedure number epoch decide set exactly batch regularization take role parallel study turn solve correspond stochastic nature different stochastic infinite dimensional plain gradient procedure achieve idea change dependent know algorithm provable rest sec sec adversarial set sec detail relate defer sec associate kernel implement inner satisfie reproduce property measure w consider l loss differentiable subgradient receive pick minimize difference statistical sample integrable respect binary infimum misclassification measurable kl kx k compact fractional indicate l big role analysis training use propose fast average immediate show result step risk average close moreover amount regularization choose infinity predictor infimum attain guarantee infimum vast examine infimum attain hold optimal suboptimal unfortunately able self tune indeed like kind perceptron mistake hinge similar step unfortunately mistake specific measure different loss algorithm equality return algorithm main difference past calculation computational let dependency outline moreover regret absolute gradient tighter smooth loss loss grow bad bound loss regret grow norm appendix kernel optimal rate coordinate truly misclassification risk iff exist relate misclassification expect special one agnostic regard low square interpret space condition discussion low term lipschitz unlikely bad case cover rate translate average establish novel approach stochastically improve suboptimal locally loss origin optimal obtain rate range consider without obtain excess lipschitz loss solution convergence specific dimension guarantee good misclassification risk perceptron weaker present classifier simplicity assume show hence risk batch obtain square loss infinity also tune achieve cross optimal core whose performance close batch first note propose regularizer bind prove optimal weight regularizer paper capacity independent regularizer exponent belong argue make svms indeed give parameter implicitly thank permutation theory tool convergence require data work also potential world hence concept method preliminary fold cross experiment three precise replicate experiment track contrary intuition sample probably finite dimensional seem bad fold time fast gain question open empirical w risk prove strong probably finally improve would result loss take gradient design analyze copy follow one sequence loss lipschitz algorithm hold importance difference dependency essentially second important assume knowledge
sufficiently separate extract decomposition note however scenario subsection rigorously atomic get dense definition atomic norm link correspondingly nd formulation generalize prove limit scenario grid dense correspond atomic grid sparse line base advantageous lasso practice automatically estimate variance phase common music independent minimize reference therein criterion understanding provide subsection note optimization covariance respect exist discretization frequency domain eliminate alternate name grid criterion solve discretization toeplitz sense represent see give retrieve use determine among choose always interest toeplitz mt sdp accordingly set sdp manner case sample adopt eq clean complete q give sum apply presence explicit atomic explicitly sparsity see term fit neither limitation determination motivated atomic explore connection atomic suffer two limitation neither noise variance reason inaccurate common method frequency accord datum lie range rank otherwise superposition equal split present component possibility numerical reason highly sdp set component bring frequency report grid I nearby latter issue certain contrast splitting cause version split grid without confirm detail phenomenon report previous lead frequency detect frequency splitting cause mean adjacent result bring challenge detection overcome propose line consist estimation covariance scheme framework carry solution main contribution covariance provide solution covariance exploit toeplitz choose window potentially resolution paper choose music frequency estimation study good sample model carry correctly estimate beyond provide frequency frequency effectiveness selection conventional length criterion aic well challenging require date available globally solve missing carry convex need initialization nd atomic norm q denoise correspondingly common call sr connection begin lemma q lemma h easily estimation formulate hereafter produce scale frequency variance omit brevity hold q noise variance problem convex formulate follow identify simplify noiseless atomic ensuring interpret different entrie fit sr whole reflect base version equivalence exist formally show equivalent implementation limit scenario problem obtain remark carry case result overfitte indicate necessity selection frequency process formulation section inherently amplitude derivation claim simulation utilize aforementione modify constant give elegant solver empirically problem meanwhile follow theorem edu sg method line concerned develop sparse limit approach dense datum atomic incomplete noise far prove atomic systematic simulation validate exist line atomic norm spectral signal process spectral estimation communication noisy index jj mf ks ji f available call incomplete miss practice cause failure weather physical frequency estimation line estimation paper mainly focus estimation incorporate many frequency classical music limitation difficulty worth approach nonconvex optimization require know easy selection frequency music theoretic eigenvalue predict eigen threshold outperform later compress sparse frequency decade discretize grid assume true practically close observation accomplish support prominent usually since cs finite dictionary discretization early dense almost complete adjacent vector intuitively reasonable dense since grid grid frequency observation naturally exist infinitely grid proceed worth drawback finite discretization coarse miss feasible multiplier admm optimization version analytically numerical notation number norm transpose semidefinite notational numerical distinguished rest organize preliminary extend case present extend systematic equivalence method present feasible conclude recover noisy knowledge dictionary sparse norm denoise recover play fidelity call denoise nd choice refer lasso sr lasso lasso sr loose easy concept atomic norm generalize nuclear convex compact contain origin function atomic norm dual atomic h write
onto exploit projective paper borel signal respective dimension typically represent vector filtering polynomially filter kalman recently hmms description hmm space stationary transition emission filtering involve follow act measure computable follow structure embed hmm exist exist markov evolve system ordinary death hx compute local sufficient special dimensional emission exist bayesian think label whereby admits jump gamma dirichlet statistical parameter k infinitely many type slight law refer mixture whereby conditionally conjugacy projective u process subspace purely atomic measure hence describe evolve review attention mutation generator x jump whereby jump cf project fisher reversible stationary kronecker delta act property dynamic property support alternatively diffusion function finitely many denote closure product reversible paper model stationary model exposition focus evolve countable emission prediction extend multivariate let coordinate show death jump diffusion generator virtue establish together lemma appendix assume conjugacy provide let lying consider generator datum close update operator operation sum let generator independently q tie law arbitrary class result cell k mm projection suffice merge denote turn marginalization multivariate single evolve mixture reduce stationarity distinct include distinct notation q dirichlet dirichlet measure iterate propagation provide lemma computable partially evaluate proposition consider family generator prediction theorem gamma measure think counterpart dirichlet gamma parameter denote representation conjugacy property random poisson intensity conditionally disjoint mass independent gamma distribution conjugacy finally know independent gamma process version branching measure space review interested branching mean per population process replace subspace compact contrary whose substitution describe branching account independent individual heuristic evolve constant operator reversible partition branching reversible distribution generator q act vanish process speed introduce independent generator proposition identify ease divide write I kx ik generator apply hand collecting note previous duality dimensional death conditionally jump deterministic driven type differential next propagation independence equal equal solve follow death jump occur jump occur iterate conclude jump distribution poisson observation hence version size process fix mm term leave denote respect multivariate follow argument step z mm respectively imply distinct mixing measure successive iteration filter family finite mixture generator application operator update operation sum eq derive filter conjugacy follow signal distribution law wise information integrate knowledge mean operator yield interval observation signal gradually approach ergodic signal acquire gradually weight sequential govern death deterministic jump continuously propagation gradually ergodic state dual thus able acquire information filtering concern prove due high generality derive propagation nonparametric corresponding parametric projection exploit support lemma provide probability death death start generator proposition propagation step generator q follow mm mm mm theorem mm mm mm evolve random indirect diffusion dirichlet point state obtain filter computable projective take mixture two mixture measure prior bayesian gamma secondary unobserved observation filter optimally entail extend key
definition q violate remark research study due nuclear norm dimension input linear namely show set nuclear rank projection onto dimensionality excess available constitute conventional complexity hope within reasonable cost difficulty might data identifying want analysis matrix accelerate calculation approximately matrix provide elementary scheme allow weak notion intrinsic stable later text dependence indicate decrease place nuclear c rank embedding operate dataset preserve euclidean random way adequate preserve construction high geometry hand leave space improvement main contribution manuscript multiplication notion rank rich reduction utilize wide cover highlight provide depend classic e lie etc theoretical special unit point strong et require preserve improvement nice exposition see improved bound matrix multiplication bound approximation exist well perspective preserve query describe polynomial time relaxation know development refer scalar span homogeneity satisfy suffice argument without loss I leave singular c triangle appear technical negative integer highlight lr analysis satisfied use triangle inequality define event probability positive accord l c useful proof provide fix tb conditioning l upper bind last equality sr obvious sr violate satisfied give appear union l rescale question construct heavily theorem propose randomize low follow specify follow practice distance row additive htp height n substituting write restrict search space standard become remove get eq complete novel provably multiplication drive
thick color color k prove theorem characteristic write always normalize respect prove implicitly al vector span e k hold proving every combination combination k eigenvector error almost show theorem fact column eq orthogonal moreover cf k eigenvector norm next eigenvector whose coordinate reasonably able characteristic eq ji orthogonal construct column eigenvector contradiction imply th somewhat us q due contradiction eigenvector approximate step function enough entire paradigm method cluster approximation translate suffice box manner point follow quick mean structure analyze spectral embedding give algorithm seek minimize squared center cost minimize close choose define mean normalize laplacian point embed group widely analyse result describe spectral show factor nice embed embed clustering necessarily kp embed point big concentration however point suffice take transpose proportional far another volume embed far embed misclassification big could large eq eq p hold finish imply case explanation mean partitioning partition return mean map trick bind one cluster vertex value technical reason optimal follow assumption approximation suffice permutation suppose index lemma ready contradiction contradict optimal achieve leave edge proof lemma contradiction approximate follow function assume index case distinction let eq prove I complete thick color red thick color thick approximate embed vertex tu probability distance heat kernel distance equivalent invoke analogous computation entry vector create additive give run obtain tu time conceptually framework step choose good different center choose result center additional embedding allow simple sampling motivate approximately show ensure close vertex removal step vertex form proceed grouping thank assign near center much simpler far vertex correct center moderate almost routine near framework combine however expensive become large however suffice mean space directly heat approximate heat distance approximate consider gap cost framework compute involve due lemma guarantee return throughout assume vertex proportional sample vertice routine approximate approximate eigenvector computation else approximate tc distance q averaging argument vertex core fu ir cauchy schwarz inequality inequality assume argument next lemma vertex vertice core one every probability vertex core z contain vertex cluster lemma use bind core come never core bind event core close core succeed triangle last condition give lemma fact hence exactly obtain vertex analysis belong assign vertex separation ask vertex embed reduce follow neighbor near neighbor grouping use proposition grouping analyze ratio return computed difference correspondence eq q early discussion graph circle matrix corollary eigenvalue easily undirected polynomially formally edge vertex define kernel unweighted rgb rgb theorem thm thm lemma thm corollary thm thm protocol email inf de visit institute berkeley inf de suitable admit present almost partition graph partition cluster cluster well connect outside key wide key cut eigenvector rigorous guarantee approximation heat embedding neighbor heat piece fundamental partitioning side cut formally undirected unweighted vertex edge hard time subset e partition partition eq clusters network capture notion subset domain computer vision procedure region split turn graph multiple subset key unique game despite various expansion eigenvalue lee high laplacian matrix informally order large partition bind informally close connection vector show variant rigorously analyze exploit heat locality hashing algorithm achieve cluster first eigenvalue span close achieve matrix characteristic statement k et prove think version motivate definition contain proof improve inequality application set span spectral theorem open overlap practical comprehensive extensive answer open question whether spectral mean algorithm rigorously analyze circumstance guarantee spectral algorithm let graph partition statement os ik algorithm euclidean different versus moderately e super run moreover embed obtain fast technique approximate via allow avoid eigenvector assumption hoc linear almost graph edge assume nk
reproduce rkh endow closure function predictor class classifier attain f boundedness satisfied efficiently prediction training excess surrogate g hinge exponential risk similarly respect convex surrogate define line research risk find crucial binary active theory last decade relate binary excess risk excess follow risk find quantitative excess excess establish convex transform closure bound convex distribution q extension view sufficient transform calibrate surrogate function loss iii insufficient surrogate affect account efficiency issue practitioner deal big unclear loss time family convex smoothing translation convex risk binary risk state efficiently function problem term smoothed advantage hinge transform popular surrogate hinge loss immediately excess affected parameter smoothness relationship obvious convexity surrogate statistical consequence follow smoothness step function follow smoothed hinge parameter transform complicated theorem smoothness bind theorem demonstrate infinity hinge hinge approach smooth smoothing correspond theorem smoothed hinge smoothed excess smoothed f indicate theorem smoothing approximation excess smoothing good tradeoff smoothing parameter risk analysis comprise bound excess bounding minimize f optimization generalization minimizer surrogate unify arise problem numerically nice hinge smoothness proceed follow immediately bound gradient rule understand generalization give generalization error recent optimistic rate yield easy lipschitz loss perform smooth constant problem error generalization bind smooth solution kn constant r understand smoothing iteration give follow assume characterize z z respect characterize converge see sensible perfectly classify point q satisfy perfectly classify percentage datum use characterize excess state eq complete proof failure constant otherwise excess e converge fast generalization limit achievable small investigate use surrogate excess excess risk provably previously relate make towards loss guarantee desirable property generalization excess excess risk result favorable smoothness achieve excess risk proof zero therefore eq solving verify eq compute define expression similarly e inequality q complete empirical convex risk bernstein plug replace universal noting remark thm surrogate loss prefer bring surrogate smoothness beneficial computationally optimal improve optimistic smoothness viewpoint affect excess contrast favor may binary excess risk motivate unified optimization generalization excess excess risk examine favorable condition convex excess instance product endow learner sample identically unseen come z label minimize excess study minimize n usually risk understand erm bound excess take function condition minimization consistent achieve erm convex efficient indicator loss function logit loss svm f adaboost learn empirical loss bayes necessary sufficient consistent differentiable origin convex
exclusive school evidence suggest cast label careful label type graph let infer leverage propagation label labeling relax node node possess quadratic configuration link negative find fix graph handle minimize eq full simplicity intuitively propagation friend whereas suggest reason school college city etc represent set respectively visible publicly type eq constant share type reason underlie r long sigmoid greater dependent threshold sigmoid enable control explanation edge allow extension choice explain really well maximize high require turn word control need force sure share one type match empirical suggest matching type enough explain eq thought type use model uncertainty type exhaustive reflect belief suggests indeed gain consider whose label early label propagation friend city city inference completely correctly infer city friend marginal extra benefit current city significant benefit city property try friend p enable type pair affect say city friend reflect match necessary eqs optimization spirit label define measure convex finding correspond friend restrict label convex friend form iterative direction back simplex f probability possibly improper could close eq label store type set label type project simplex converge distribution value require information message architecture generalization applicability reason formation mutually exclusive strictly school friend city black represent friend infer high school small maximize friend introduce part fold fold use fold various fold place difference variance inference rank label user rank top type rank actual present lift propagation iterative graph processing friend communication overhead retain top entry optimally b retain friend friend age high school college significantly improve run time significantly show friend decrease demonstrate importance limit increase friend enable well beyond point trend baseline facebook appear many friend friend become label friend add friend recall propagation benefit school college lift propagation city improvement school college easy latter figure propagation pick instead less common explain able infer difficult circumstance even label inclusion improve wide already careful type recall city college within group use college turn college membership extensive one addition membership impact label lift membership friend membership benefit recall lift label lift significant compare propagation merely scalability also careful make membership redundant priori impact membership regard type membership actually redundant type broadly sufficient true friend college correctly label share show top prediction type somewhat easy hard friend share explain lift plotted figure lift qualitatively offer within match matching need thus jointly mode particular user high college identify phenomenon create create reason necessary two carefully model method equivalent run basic label propagation facebook benefit model network primarily consider interested infer label type accuracy tackle infer label label label primary focus city connect network fail property label explicitly enable distribute pass architecture facebook label propagation infer label node predict low node say edu belong student node infer often partially fill label dimensional correlate ad search friend motivate problem friend contain city actual city call explain city one inference try friend essence common likely connect optimize category relationship address formation reason snapshot city know unknown friend completely know independently infer friend city common city friend city city among friend friend city happen type viewpoint likely share value type two user college beyond label item propagate graph inference try edge point reason friend college primarily infer label node knowledge link prediction task recommend college friend high school solution propagate cluster profile try deal incomplete allow believe readily contribution formulate one label belong whose property incorporate infer architecture scalability facebook profile real dataset recall improvement scalability usefulness survey relate follow generalization prove prior semi supervise relational model base view label quadratic penalty possible modify random interpretation label propagation handle large number assignment count none interaction hence fail formation typically predict alone relational use collective neighbor relaxation labeling well outperform focus good
ce drive specify actually ce varied range ce effective scale treat need energy experiment ce factor smaller effective restriction factor combine ce ce calibration exercise calibration cope check understand setting synthetic exploratory analysis necessary ce gain informative team collect prediction shape disk heavily field circular deterministic large input much physical denote calibration label new outline involve limitation framework couple deterministic computer plus discrepancy systematic disagreement model observation reality process review detail value via prior model computer output consume fit train simulation run recommend gp typical recommend joint observation coherent thing jointly give even lead mix substantial effort context couple simulation still previous work bayesian joint uncertainty bias justification coupling amount enhance joint despite simplify gp reason shall normal define stationarity simplify zero perform gp conditional integrate degree freedom component define correlation hyperparameter matrix posterior prediction require obtain point scheme typically restrictive ba schmidt modeling especially fast run computer fast big gp recent search estimation decomposition avoid sequentially carry library gp methodology develop provide gp inferential modular path focus rather input greedy fashion pair efficient update approximation local ultimately subset computational design neighbor extend correlation thereby calculation dense design size hour implementation provide package calculation parallelization yield challenge calibration way posteriori prior calibration parameter scheme involve cascade maximizing perform calibration pair vector computer input column design location model independently mx value great expense fit u measure well fitting train give prior field prefer therefore suggest maximize parameterization shorthand fit rather inner method discuss fidelity field x j j b f b f b fu inner max detail automate routine package loop predictive subsequent prediction via execution extremely example follow neighborhood implement model fit whereas sensible degenerate prior identity reduce work numerical stability term un inner evaluate search illustrate numerically approximate one implementation mesh interface successive try sequence weak regularity direct call popular many optimization derivative approximation numerical search small decomposition solver recommend posterior distribution give calibrate computer obtain predictive abuse let stand corresponding equation student equivalent one augment diagonal depend locally step design implement step save moment location combine de distribution I ideally full step lead student comprise normal necessary sum sample convolution prediction observe field still option might good consider identification retain desirable attribute evident concern primary agree alternative estimate uncertainty full counterpart build section adapt datum unit cube follow keep generation replicate broken regime unbiased designed explore efficacy propose approach scenario experiment motivate regime mc repetition use variation replicate realization model design begin four trial per value simulation mc initialize obtain solver boundary search generate comparison replicate variation example modular cube directly predictor entirely alternative error decompose level calibration generating estimating ht cm leave deviation panel estimate dash line triangle axis set arrange six panel group three truncate improve visualization arrange number replicate six six label prediction fourth lead nearly first cost u span alternative third leave replicate similar former indicate job surrogate replicate replicate imply replicate low thing quite panel recommend choose statistic min inter range ten pairwise number clearly variation variance stage random calibration great uncertainty come three trend omit clutter value along straight line go point densely near ridge combination confirm true far weak weak prior move uniform discrete nature smoothly vary value change change ultimately posterior motivating take average repetition intel ghz machine optimization span bias model quick right alternative option closely bottom leave even recover nonetheless expense estimating cm cm explanation matter fit stationarity approximate gp thus accommodate explain biased full joint exploit lack identifiability discrepancy parsimonious large good meanwhile summary mechanism second average correlation return motivate explore problem bias extent simulator substantial distinction synthetic experiment concern local unit cube response vary magnitude biased preprocesse preprocesse input experiment isotropic discrepancy restrictive field observation virtue version scale obtain subset computer specifically sample replication estimate energy thick ratio length limit observe pressure length diameter cope common suggest fast roughly short energy divide cube input square root small vector comprise scale perform monte hundred repetition conservative costly search field datum energy pressure column drop diameter provide variation report exploratory analysis aspect bias calibration insight difference variation aspect input substantial impact isotropic preprocessing regime average figure panel response average input preprocesse specification give influential input energy energy less sensitive input calibrate prediction rely model un report leave methodology gain turn bar subtract leave obviously mean pair fail difference predictive amongst credible visually predictor seem small understand however reject nan may due turn calibration exercise plot evaluation algorithm combine search indicate open estimate discuss minute machine take ordering would however fast version surface variation surface consensus value mean ce bias unbiased largely agree set however isotropic amongst attribute pre add close separable illustration discussion model synthetic profile log although highly informative negligible noise evident surface flexible weak biased yield much reveal right axis correspondingly dot bias surface stop early second uncertainty bootstrap review experiment estimate open circle figure heat plot right cluster estimate pair important bootstrap general surface large summary figure suggest representative amongst open colored converge biased noise suggest highly value make input provide team configuration nominal setting table design input nominal fill pressure ratio measure field exercise energy three provide variation experiment conservative accounting specification ask propagate uncertainty calibrate manner exercise propagation quantification uncertain input uncertain far bootstrap produce spread show explore calibrate exercise ht show plot indicate focus panel predictive four input roughly remarkably method despite choose estimating lead red degree nominal setting dash red providing spread skewed suggest prediction square preference allow show output nominal great agreement calibration methodology generally provide motivating approach calibration increasingly work processor core whereas increase although something salient feature essential process carefully extra package calibration leverage aspect calibration paper motivate poorly identify providing calibration output scheme method believe computation option amount high price great effort calibration yield excellent estimate flexibility nonparametric gps design calibration exploit idea model coupling gps discover calibration method decade
proof evaluate frequentist statistic inner p statistic corollary define large family extend translation lebesgue haar measure associate translation translation sum prove concern transformation typical invariant action mean modulus insight haar theory assume corollary sample replace statistic trick one dimensionality concern haar belong trick replace associate whole investigate testing different approach frequentist estimation especially invariance frequentist haar approach explicit central quite invariance bayesian invariant procedure haar sense right haar prior domain equality need first condition call stein common stein theorem satisfy rx rx assumption use mainly hold consequence invariant transformation group assume equivalent identity stein imply varie accord observe question domain underlie vs ie section modification presented namely propose illustrate see carry property belief ratio testing establishing pearson hypothesis test rely broad composite vs hypothesis data parametric merging test family vs composite hypothesis domain improper extension make bayesian type pearson indicate lr central vs define posterior integrate posterior perfectly allow define composite symmetry composite test sequel quantity interpretation far improper prior smooth side posterior less statistic general frame pearson extend interpretation measure seem sense mathematic interpretation issue need subset set joint replace role remark measure posterior proper evident world remain consider single weather daily describe daily five statistical whether prior enable simple hypothesis vary impact display explain improper simulation perform simulated reasonably alternative characterized likelihood frequentist estimation couple mcmc slice sampling implement simply order lr combination threshold evidence favor practice read great likelihood alternatively greater correctly consider cumulative sided whereas bf credible vs bf translate credible also natural credible inference hold inference generalize composite bf pearson bayesian version frequentist pearson bf pearson maximize frequentist classical unknown composite hypothesis credible bf relate hypothesis namely gaussian invariant frame stein theorem credible evaluate inference new example invariance likelihood conclude equivalence connect equivalence credible hypothesis frequentist invariance hausdorff continuous support invariant haar q invariant haar replace haar multiplicative right modulus invariant haar haar note haar haar occur haar definition haar modulus modulus invariant haar measure measure right haar right invariant haar haar statistics concept transformation family property action group say invariant connection transformation lead could differently define invariance family would long presentation group haar induce action frame haar turns actually note subset aa group finally define marginal density always finite probability even mean measure specify note action element element haar transformation lr rl measure posterior integral c modulus practice datum see theorem depend instead follow shall haar modulus imply haar haar relatively modulus since invariant g px notice simplicity transformation marginal way frequentist order side x px px corollaries corollary likelihood haar absolutely lebesgue theorem domain measure integration integration haar lebesgue modulus combine get p simplify sx sx sx sx sx sx condition see induce random particular threshold notice equal statistic frequentist threshold define pd h pd pd reformulate use appendix equation lr equation call integral b check inequality true multiply leave positive term integrate ix conclude implication eq b b reciprocal generalization distribution testing hypothesis test many threshold equal extend nuisance extend frequentist finite analogous stein credible domain confidence frequentist vs hypothesis measure nuisance two first improper soon second role lr discrepancy variable invariance pearson composite give observe dataset q distribution characterize alternative test great type make decision paradigm type lie fix invert test notation side bayes factor threshold straight bf jeffreys bf strong evidence favor probability bf simple hypothesis improper proper partial account prior proper partial bf bf initially vs composite mean gaussian powerful property x size issue see recent study consider unlike bf suffer idea prevent occur bf argue frequentist several classical unlike bf frequentist frequentist statistic ie integration contrary predictive value discrepancy domain choice discrepancy variable bit classical approach introduce derive simple vs test tool one mathematically propose compute q relative evidence belief evidence resolution large contradiction simple hypothesis posterior ratio unobserve variable namely bf ratio random evaluate cumulative compare pearson paradigm read likelihood fixing reject sensitive making threshold decision broad range grow typically clear computation display later propose generalize generalize add reference systematically deal case hypothesis necessarily list different analysis claim unlike bf improper posterior proper subject invariant transformation consequence function last example consist bf compare bf compare bf ie describe general alone show bf show support low general addition result example lr bf credible way bf thresholded seem invariant transformation indicate cumulative practice straightforwardly carlo markov chain algorithm almost histogram lr cumulative lr detection extra image dedicate large image available extra present dataset extra dark less star star degree classical potential study investigate play test frequentist prior hypothesis notice highlighted equivalence equivalence credible domain frequentist hypothesis discuss example discusse obtain frequentist credible extend definition frame yet composite hypothesis condition fixed parameter see although condition remain transition composite immediate hypothesis still generalization extension simply improper soon proper improper prior proper posterior probability bayesian pearson associate discrepancy lead conclude section essentially mathematical often think evidence expect equal happen bf highlight hypothesis raise question frequentist hypothesis agree interpretation could frequentist test unified consist analyze likelihood seem frequentist marginal hypothesis particular broad reference include modify p make exactly
indicator q refer whose entry easily see b cover nonempty precede argument full clique vector concatenation vector copy match denote clique marginal graphical correct graphical n correct independence central note part consequence require justification material tractable approximate inference covariance marginal fortunately leverage compute effectively covariance inference degenerate fix transformation reduce vector covariance invertible work propose minimal graphical full covariance project maximal maximal note subset linearly variable redundancy way choose long ss full b indicate representation conditional distribution degenerate observation clique example noiseless di c decompose count node edge retain count marginalization root node away marginalization edge technique u v u v tu row distribution v u u u conditioning formula v different way sparsity inverse statistic graphical factor density translate sparsity precision reasoning derive conditional asymptotic noisy throughout remainder tree edge notational simplicity noise unobserve represent drop factored denote observation determine entry reconstruct poisson long apply propagation laplace omit simplicity mean covariance ep density uv uv uv uv compute step onto laplace need mode variance approximate distribution solve optimize value form term since normal density bfgs work observation mode variance project infer inverse laplace take outer suppose ep message pass time maximization laplace infer count consume inversion obtain complexity model locate bottom leave corner make cell employ four feature cell cell fall lie toward encourage cell denote vector logistic domain count cell number move count generation generate vector count consider infer observation node count issue count introduce observation instead estimate burn million collect million iteration relative mcmc experiment run generate edge count introduce factorial show map much produces obtain task via compute table coefficient varied population approximation although significant make approximation map exhibit much ccccc inference experiment magnitude logistic extreme evaluate accurate although cccc ccc explore effect random vary size accordingly small method grows map rapidly ht ccccc vary true generate curve consistent population map map overfitte create match performance em iteration measure experiment measure cpu count count edge count compute edge require node count also much reveal computation consume relative match small map edge estimate good transition near conversely extreme map surprising much introduce collective limit distribution population size matrix maintain method inverting covariance develop efficient simulation show exhibit variance material base national science foundation grant usual writing replace show nh showing order sufficient clear inspection remain tree hard enforce consistency count sample equal integer count variable global consistency interesting corollary argument base detail converge replace mean property linear entry show recover invertible di dd definition equation valid indicator configuration trivial linearly collective individual collective statistic e individual intractable previous explore monte carlo approximations study maintain follow poisson accuracy mcmc map exceed set wish collective count might education census reason census education region concern anonymous location arise construct set individual copy population individual permit clique count individual joint answer condition observation clique individual setting clique clique model also difficult configuration take
dataset spend subproblem solver give gradient train crf gradient iterate solve inexact far experimentally tree much slow leaf strategy still benefit logistic learn ccccc linear boost zero mlp mlp boost mlp bound define lx h line zero mlp mlp boost mlp true mlp mlp mlp boost c boost mlp mlp boost mlp zero I boost c mlp boost mlp boost boost mlp proof successful write iterate observe addition entropy message non logistic structured exist minimize choose output major challenge solve high output np standard lp relaxation inference parameter alternate message pass descent mostly focus adjusted useful ensemble predict independently unary either hold adjust allow general relaxation problem alternate pass update major minimization loss logistic need function optimize experimentally test linear flexible predict fix sum function subset consider structured problem directly handle generalize energy reduce linear find logistic regression select loss concern standard choice slack rescale loss measure experiment ham eq maximum range general solve approximately motivation relaxation bind relaxation fx polytope agree region eq restrict value take q objective constraint lp practice preferable message passing exploit inference approximation pass guarantee use loss appendix result previously difference without smoothing configuration evaluate specifically eq contain maximization saddle inspire work message write alternate message ascent update message trivially fix message section regression initialize set multi ensemble solve albeit function minimize evaluate solve message pass optimize alternating optimization fix pass parameter thus concerned optimize conjugate message bias substitute simplify marginal inner maximization close thus equivalent eq term fact give result learning summarize alg depend situation local constrain function constrain image mapping energy would select experiment ccc boost mlp ccccc boost mlp linear mlp ccccc ij zero linear boost mlp boost boost mlp c mlp ij boost ccccc mlp mlp experiment respect bfgs layer perceptron descent momentum mini batch univariate tree stochastic boost loss fit one control loss multiply add classifier message iteration synthetic denoise visualization create generate sample feature add classifier combination nonlinear result low rate plot
spirit lead logistic label resp obvious treat follow connection deep log define converge pointwise uniform respect strong f randomness assume implicitly mle logistic well mild check function together divergence iid kn jt kk jt correspond odd ratio represent jt semi parametric practice semi extend iid logistic begin iid efficient completely parametric ignore severe make detailed replicate toy chain transition probability blue solid dot strong autocorrelation course constant use quite form pair replace k per either true reference log odd semi logistic effect effect add offset completeness practical variant intercept constant mean replace intercept another way constant simulate fix increase value pick value repetition perform inference enough ml perform show asymptotic bias project ideal possible nonparametric valid practice generalise model logistic parametric good parametric poorly line right constant convergent estimator thorough location link straight predict likely case country past spatial predictor availability net context movement predict visual reliably draw stimulus exhibit dependency tend move currently bottom corner take step go something rather motivate represent purely spatial interaction spatial dependency centrality bias preference take distance location tendency current axis horizontal therefore decompose sum angular function smoothing therefore extend straightforwardly explain turn r package component uniform example function variability replicate central location dominate although subject display include preference suitable format around minute poisson transform turn otherwise non generalise additive reduce display effect centrality bias subject gray blue way become parametric turn semi likelihood monte logistic challenge covariate dependency dimensional design non constant great logistic able leverage nonlinear reduction development challenge transform difficulty proof derivative ns nf f shorthand ns ne ns ne e ns shorthand ss obtaining intensity density confidence inversion equal aa ba bb ab aa nn exponential concave poisson concavity natural derivative simplify ss ts matrix derivative simplify certain rewrite odd since law almost surely surely sum prove function attain attain take establish previous interval assumption iii almost randomness absolute difference three term converge bind replace n third law I generalise book bound assumption independent may rewrite theorem supremum application spatial model involve estimate magnitude matter variability band smoothing spline smoothing infer report band smoothing fit band repetition thm thm theorem contrary specify likelihood popular lack infer problem poisson specify need infer another non iid include model dynamical binary turn show extend spatial poisson non parametric core tool modern deep vision datum appear function technical whenever estimate prevent direct many technique recent year difficulty non include model density intensity expand estimate iid optimisation turn semi arise extend iid markov description movement idea appear form learn transform family bregman divergence generalise kullback leibler divergence learn study place spatial far show converge uniformly technique time ignore indicate convergent framework parametric convergent section turn likelihood information call non iid still chain transition constant highly poisson q along line sum value mean previous space corollary turn model estimate corollary exist possibly ie function require need classical solve parametric part belong include penalty parametric uniquely suppose contain exists maximum optimisation
r estimator illustrate estimator affect error inconsistent appear analytic monitoring reality g light subject essential use calibration measurement inference bias trend functional propose measurement regressor response affect error aware reduce completely eliminate knowledge error deal measurement error include probably literature variable model van reference therein deal mostly parametric nonparametric method reference instrumental variable quantile ability interest describe recent treating model discover nuisance regressor kind measurement recommend region insensitive regressor invariance rank estimate nuisance parameter consistent every show estimator slope bias precisely even unbiased situation far present depend generate measurement distribution unknown regressor affect measurement observe ni ni ni identically error moreover thus variable observable predicting become ni u ni ni interested slope estimator ni ni e ni ni ni ni b statistic invert invert extension location analog error subgradient exist generally literature absence error ni ni normal presence asymptotically furthermore locally bias fix mean sequel take unless convergence shall need assumption underlie entity square skew symmetric score order statistic error absolutely tu ni v generally absolutely continuous density finite ni n ni ni regressor assume definite function finite give f normally prove subsequent asymptotic local magnitude response bias valid class demand location distribution method regressor non respective definite theorem observe instead measurement sequel confusion step follow ni ni nn either ni b v ni statistic contiguous sequence linearity case asymptotic ni n measurement response ni ni admit rank ni nr ni ij w nr ir nr nr ir px jx nr nr two absolutely hellinger array measurable measure van prove contiguous note residual b contiguous e ni case vector ni n nx nx fx dx tu u tu u du dt apply tu dt n yield fx ensures complete present case observe w ni nk ni ni ni ni du de n u x n du us contiguous ni ni n bound countable observe w ni expectation k n w ni together sake brevity ni w ni ni b ni use corollary arrive na ni na ni linear partial speak b ni convex function b convexity supremum take argument alternative equal asymptotically normally v replace complete illustrate measurement estimate empirical r bias square deviation compare deterministic regressor statistical software r
eq ccc mathematics department business usa il usa despite variance largely accommodate minor regression sparse variance post step employ penalty variance mean theoretical finding estimation high extracting regime extensively among prominent procedure lasso work low extensively address dimension square procedure guarantee mean correctly usual gaussian covariate unknown variance log explanatory variable positivity also capable vary order magnitude study optimizer assume procedure lasso perform update square procedure estimate estimate predictive aside provide estimate model variance provide estimate predictive covariate may scientific economic finance economic autoregressive rating dispersion generalize fall environmental recognize add activity extreme primarily distribution variance relation specific region asymptotically penalize optimizer establish property mean require examine complement finding scalar vector response respectively indicate boundedness sequence counterpart throughout index contain index submatrix extension notational simplicity small jointly indeed vice versa method approach simplify perform justification coordinate early suppose result close think initial enough pseudo perform procedure fix set work enforce result comprise stage solve estimate result pseudo residual appropriately choose finally compute reweighte differ penalty adjust mention stage think statistic minimizer differ pseudo act effect make pseudo likelihood close likelihood known closely differ choice oppose penalty satisfy sparsity condition property commonly lasso provide call minimal hope concave admit unbiased smoothly deviation scad minimax concave mc penalty behind form neighborhood act effect penalty value shrinkage reduce component generally q aforementioned concave satisfy likewise parameter balance value tuning choose minimize estimated degree aic bic property scad define scad develop substitute scad initial solution penalize maximize expansion penalty oracle property restrict scad proximal objective minimized sub decay definition statement decay result guarantee minimizer enjoy style penalty result type suffer possibility select minimizer converge minimum poor program set theoretical likelihood optimizer unknown unlikely derive maximum sparsity set minimizer examine elliptical contour list eigenvalue tensor large state oracle program require minimizer enjoy oracle simultaneous estimation limit theorem intrinsic low similar ease condition accommodate dimension generally necessary exponential either refer oracle property regularity low behavior regularity parameter variance fitting residual construct remove estimate observation accord mean likelihood minima address concern mild condition attain oracle possible mild course would mean knowledge unknown large minimizer assume similarly eq local minimizer enjoy notice allow substantial correspondence variance sharing correlation problematic minimizer recover satisfy concern solution satisfied design loading consider eigenvalue restrict eigenvalue specifically exist loading combine obtain enjoy penalize program stage reweighte variance precise addition access oracle mle fisher reasonable recover mild assumption consider convex stage assume minimizer enjoy normality property stronger prove specifically mle oracle mle mention rate penalty theorem would variance regard section conduct small study toy model jointly marginally remain covariate independent line simulation illustrate situation procedure iterate figure show precision coefficient toy aic st nd st nd st nd st aic st nd st nd bic st nd report second normal know support compare result simulation consistently scenario complex although furthermore demonstrate benefit first stage estimate nearly mse asymptotic hence theory analyze estimate variance quite show oracle estimate similarly guarantee prove assume log function fashion non section nonetheless interest sort guarantee could lasso family acknowledgement nsf dms complete resource university dimension quadratic let compact define objective accommodate norm radius construct minimizer objective estimator demonstrate refer oracle likelihood attain minimizer maximum stage result likelihood demonstrate regard minimizer pseudo maximum demonstrate mle attain sparsity set minimize hessian tensor lemma let unit sphere minimal net among cover constant apply assumption uniformly expand around value infinity curvature optima fix ball verify expansion remove prove strong employ similarly previous except perform proof proof oracle property knowledge minimizer precisely zero let value
vc improper learner small class exponential hypothesis instead sample complexity improve improve et show large proper proper private learn private learning concept class draw simple proper guarantee privacy fail accuracy al improper private infinite boolean return exactly privacy complexity prove sample disagreement hypothesis unlabele privacy requirement start relevant boost big error produce show boost accuracy private algorithm private show boost private boost mechanism al representation class improve consider probabilistic concept collection represent list learn select representation complexity private new privacy furthermore hardness assumption avoid size size characterize define domain size bound domain exist deterministic small deterministic class apply take solution quality maximize exponential database reasonable notion representation size database interestingly list protocol search bit privacy inspire record assignment satisfying least clause clause probabilistic representation assignment database individual maximize preference meet privacy another database database original record use size give reasonable query dimension vc lemma private c candidate separation negative cardinality element accord individual call neighbor one preserve differential nearby outcome differential privacy differentially database output take immediate concept label either pac algorithm accord target error pac concept distribution draw satisfy coin satisfying improper pac pac hypothesis predictor privacy private pac sample differentially use scenario choose approximately choice mass assign exponentially input pt leave define probability sf probability exponential private pn chernoff sum chernoff concept sufficient show must cardinality big empirical error private concept ready probabilistic representation behind sample hypothesis h dx representation concept private complexity cardinality hypothesis set boolean r concept choose place see characterize learner show implie private probabilistic complexity probabilistic arbitrary class every ca mechanism cc label choose sf first event return ensure mechanism hypothesis obey show happen probabilistic class chernoff hypothesis mechanism least learner class complexity prove learner learner assumption step set claim initially construct description short efficiently impossible properly construction inefficient learner defer probabilistic learner learn class exist represent c dx dc hc da h h h bind learner concept see learner size exist pair order refine class learn connection representation hypothesis construction deterministic representation claim boolean h extremely sized union representation contain straight forward boost claim use somewhat h non say fail representation one least bind hypothesis learner first size later sample h eq entry sample bit union event h dc two happen high bounding learner learner p pair dc exist b pair b concept necessary apply private broad problem scenario optimization universe choose refer function f choose solution reasonable database database notation necessary correspond big size private probabilistic private approximation big hard achieve optimization universe b bm bs bs publicly solution could qx probabilistic database f fs interested differential privacy universe record differentially predicate take query q cx et al define release mechanism differentially another predicate database scenario view database input quality eq every neighboring database element every representation bound universe database element every follow parameter mechanism eq exponential differentially fix bad bad event occur exponential mechanism least problem exist database record pair database element ratio fix b denote universe clause set clause assignment objective protocol et al notation represent represent database deterministic algorithm clause use deterministic representation necessary pick assignment satisfied clause clause pick assignment none clause randomly pick assignment database database differential privacy differential privacy requirement change proof valid private almost minor see apply must security representation element differential privacy sample whenever representation notation learner proper learner representation consider work boost use et show every identical lemma learn algorithm oppose multiplicative boost back replace factor boost capability ability let probabilistic first indeed exponential private fix step dc sf good event happen hypothesis hoeffding exponential learn represent class om representation q show every proper require still concept vc show separation point existence probabilistic representation description value class appropriate need process distribution return contain condition construct good probability contain choice conclude probabilistic hypothesis construct random draw description hypothesis goal care polynomial degree description pp p induce adjust inefficient improper learner necessary randomly every interpret e section definition remark corollary hard conjecture partially support foundation grant cs ac il private informally apply privacy al sample learner private combinatorial analogous know complexity private vc dimension representation concept probabilistic class private class exist sample similar
digits number double version black triangle law upper panel skewness panel panel panel g use maximal black panel determine various black line square panel c go independent realization occur complex often necessity distribution usually uniform limitation consequence arise tail moment sample size provide range handle library analysis finding numerical open search science observe quantity proportional say law density pdf analysis pareto much note frequency natural language rank frequency speak pareto retrieve store drastically last continuously researcher biology science economics finance social science law self organize critical multiscale collective intrinsic organization natural computer understand system power analysis power extract power law whereas consequence context statistic far none adopt severe pdf exponent equation power fundamental generation preserve restrictive suppose want extract whereas many concentrate inversion come inversion arbitrary motivated popularity extract variable interval variate invert certainly l previous write use generate power uniformly distribute typical solid computationally severe random role play period however precision number create finite bit satisfactory translate sample drastically synthetic bit precision simplicity close integer normalize number region green histogram presence correspond histogram circle histogram region blue region admissible exponent point visible region red area arise want accuracy space large h derivative pdfs valid random require consecutive admissible rise presence define overlap jump occur among consecutive discretization get bad wide law pdfs mean valid distribution network extract define fundamental epidemic fig numerically obtain confirm dy dash vertical stand predict color bit round method sum law random material moment tensor pareto due model sum law variable pay company individual pay distribute play role evy walk length simulation context behavior law pdf variance generalize stable deviation behavior exponent typical skewness excess decrease enough symmetric basis cutoff slowly normality small one dependent illustrate precision produce bit point machine precision power limitation approximately reach unit machine I able bit eqs limit truly instead numerical inversion suitable synthetic distribution principle variate computer store number principle computer algorithm paper explicitly continuous extend discrete limitation pdfs rather law distribution instance even strong discretization pdfs moment presence dependent cutoff law tail practical generation discretization tail outcome significance cutoff certainly analyse account factor concrete risk limitation numerical computer counter message increase distribution preserve sample
year cifar regularization maxout tree prior improve neural maxout product neural network image regularization use digit deep neural neural application application google speech speech hide hmms speech determine hmm fit short frame acoustic layer train outperform benchmark sometimes compression create low internal representation fall traditional image indirect application compression diagnosis vast medical structured design group specific high classification annotation medical image challenge deep neural effectively advance field need reliable grow crowd create big like removed manually crowd develop use vast amount unlabele develop huge resource make need neural understanding ever neural without entirely high representation key knowledge network neural area indicate far paper briefly describe history describe recent advance recently provide pooling dataset recognition field intelligence recognize classify image system accurate neural class benchmark dataset neural design image knowledge algorithm expert system create network effort engineering detector drive self prior approximate determine network posterior establish rule vast extensive make impossible review improvement deep neural architecture lead record break object recognition organization introduction brief history sub list commonly benchmark classification net network simple electrical circuit neuron input depend give computer simulate neural big scale theory switch circuit computer stanford perceptron patterns read stream phone predict neural world use develop research neural cat primary identify pooling algorithm separately decade artificial neural could mainly requirement lack train architecture bank read build convolutional network read machine reading motivate microsoft convolutional system include chinese character neural face record google face also vision mobile participant vision obstacle lot development improvement performance recognition image win image year win google accuracy comprise neuron stack deep face issue layer modification overcome come connect region neural net divide region map neuron connection connection feature advantage architecture instead connect layer low upper drastically cut weight entire connected layer weight back aim new patch sample use auto encoder back encourage maintain activation bias sigmoid boltzmann boltzmann rbm undirected graphical sparse rbms train divergence sparsity penalty autoencoder mapping encoder thus algorithm differ primarily around window pooling pooling activation pool additional activation window account pool non maximal activation utilize pooling pool output pooling region map split pooling contrast generate learnable region rich depend neuron determined acting linear sigmoid running function fast equivalent activation interpret approximation learn activation give input maxout implement large easily simplest square set predict increase forced become way consider output presentation hide unit omit modified mask element co neuron helpful several neuron helpful correct answer context dropout output mask equation hold dropout mask weight neuron turn inference average give instead massive twice essentially mathematically write justified use activation bernoulli calculate pass averaging present increase reduce overfitte improve generalization consist simple rotation field work object transformation augmentation augmentation horizontal extract extract increase size set prediction patch average prediction softmax patch augmentation illumination difficulty availability set image create rapidly meet demand microsoft common object contain spatial precise pixel distinction per aid contextual million store label abstract english list lexical note reliable
calculate original tensor cnn drop cnn record cpu base size rank cnn maxout softmax classify digit plus character pre similar layer constitute channel output channel filter result layer tensor growth rank accurately show accurate properly approximate error firstly make fine tuning one finally layer approximate significant drop accuracy fine tune derive fast accuracy drop time big speedup incur accuracy convolutional train noticed rank achieve table clearly fine secondly fine observation hypothesis large poor minima indeed cp minima bad minima random layer rank entire effect initialization cp fine c ft ft ft ft ft cp considerable comparison character classification cnn however determined bi cp layer firstly greatly secondly spatial variation tensor low decomposition improve support foundation grant consistently linear next tensor adding highlight slice check numerically tensor successfully pt yshift em convolution tensor fine layer compute cp tensor convolutional kernel replacement training process cnns obtain cpu drop class character cnn obtain speedup minor imagenet overall cnns vision computational notably mobile processor cpu processors layer cnn operation dominate cnn convolutional often expense consequently strong efficient implementation convolution major package work cnn tensor detail recall typical cnn tensor array dimension dimension output convolution convolution constitute dimension map output map exploit work speed within cnns apply filter investigate tensor cnn tensor algebra least square cp full ease decomposition cp tensor linear exist compute ease convolution tensor four cnn therefore package need tuning layer replace four convolutional kernel straight forward tune network back importantly cp fine tune previous method compare practically architecture ram storage valuable feed forward spatially kernel locally connect layer side confirm cnns modern serve facilitate minima cp cnns character discussion cc convolution box correspond tensor cnn side mapping demonstrate scalar box correspond spatial approximate composition mapping mapping approximate array either pixel spatial decomposition low decomposition accelerate convolution codebook bank filter combination share bank separable decomposable decomposition share decomposition suggest decomposition effectively approximate composition experiment demonstrate compute minimize network fine tuning inefficient well even cp suggest base cp decomposition tensor connect layer reduce rank cp decomposition part compute outer add tensor cp directly full convolution tensor replace cp square discuss fine network approximated conceptually convolutional layer decompose cp fine tune entire backpropagation necessary layer review cp core two tensor low lead separate many canonical cp cp minimal need singular finite canonical tensor error enough rank cp cp package choose non least square gauss newton capable much may cnns feed unit within tensor consume modern cnn use linear mapping tensor dimension third spatial
span consequently conditional maximize need redundancy code maximum instance family relative entropy distribution divide possibly sign integral statistic value match alternatively divide w possibly total likelihood combination see odd role maximize correspondingly arrange representation simple illustration negative require three maximum imply weight weight trial sufficient take alternative application algebra solve distinct linearly sign bayes however effect weight alternative quantile respective panel point right prior guarantee exact note divergence deal prior mixture kullback bad n early study show mixture mild subset model consideration finding sign bernoulli grid mm mm mm panel panel mass implication fold theoretical view provide bayesian demonstrate counterpart early bayes offer extract sample arithmetic explore family multinomial relationship support prior produce bernoulli match odd odd count neither proposition maximize likelihood provide minimax compression play essential minimum description show normalize though weight address marginal code universal prediction minimax regret bayes mass denote estimator characterize arbitrary range datum compression free minimax property compare parameterize exactly maximize trivial require work pointwise regret course achieve take begin compression achieve minimax infinite nevertheless continue distinguish particular comes think likelihood conditional sum conditioning bayes study regret study nature play role minimax consideration likelihood mixture simplification determination family bayes turn first recall arise redundancy compression giving difference kullback divergence formulation regret decision loss specify divergence procedure mixture kullback function limit bayes minimax minimax characterize redundancy call favorable achieve pointwise provide upper redundancy max also mixture see traditional role bayes asymptotic alphabet family role prior close jeffreys fisher asymptotically regret mixture minimax pointwise problematic bayes mixture sequence arise match information jeffreys fail problem bayes mixture slight family difficulty motivate consideration sign bayes simplification possibly sign code particular tool appear normalize require sum yet remain intractable mixture simplify equal conditionally multiply one ready computation ratio marginal sum predictive g case mixture marginalization marginalization paper whether provide maximized likelihood measure string px perform marginalization maximize get computational sign finitely iii numerical bernoulli trial divergence finally calculation observable observable outcome sign sign role string sign marginalization proper bayes sign integral produce valid emphasis
update update several need compute unconstraine project onto nonnegative advantage easy implement interesting solution properly drastically current recommended issue switch alternate method subproblem solve many method dedicate practice implement fast gradient converge stationary subproblem framework algorithm iteration expensive difficult implement initial guess rather refinement nmf algorithm mu solve use subproblem write form interact see describe thesis solve vector unconstraine though negativity advantageous optimize smaller unstable difficult see mild assumption guarantee converge stationary particularly otherwise initially recommend initially update gauss functions taylor display evolution describe section classic document datum set matlab matlab intel core ghz ghz ram display algorithm slowly data quite poorly classic initially solution objective perform poorly book reference therein e impose robust stable conclude criterion initialization usual scheme evolution optimality iterate termination discussion issue assess criterion similar criterion sensitive scale subproblem multiply dividing handle e scale bad type mu monotonically mu guarantee monotonically potential update order magnitude interval sophisticated strategy obtain ii come guarantee e nmf np general direction list centroid spherical scaling indicator therein use svd introduction one frobenius theorem denote factor al either scale initialize point initialization keep subset column generate goal nmf allow reconstruct column fact compute separability document topic transpose document word anchor assumption thesis hyperspectral section pixel pure sense spatial successfully develop far clear behave enforce suggest improve weight enforce sparse entry diagonal condition symmetry distinguish robust distinguish near matrix see use k drawback expensive class separable insight vertice hull column geometric find literature pure historical comprehensive focus effective successive moreover behind heart geometric nmf look vertex hull column work onto column implement use formula ht rank column let let prove correctness noiseless case induction necessary identify convex point moreover strict unless ht projection onto complement hence reason separable nonnegative w w improve post interesting purpose various schmidt column variable fact name come particular variant volume discussion behind volume hull processing noise combine make extracted span refined processing norm good column polynomial logarithmic closely relate greedy solve self dictionary reference exist many g vertex component norm strongly successive nonnegative include provably several connection mine nonnegative precisely mathematic science graph bc complete bipartite subgraph need check easily complete subgraph bc low polytope polytope exponentially find formulation importance turn polytope slack ia survey reference approximate related hence mixture nonnegative compute nonnegative rank amount independent variable therein know variant receive start communication equal combination input logarithm communication matrix closely rank relate find number nest geometry nmf easily interpretable dimensionality reduction bring researcher future publication lee author thank book kernel machines comment paper corollary question nmf automatically property nmf mining hyperspectral image nmf review nmf refer separable polynomial presence briefly describe mathematic via technique tool use visualization feature selection space span dimensional linear subspace vector rank column column assess quality depend frobenius popularity implicitly situation introduction compute efficiently value see reference therein svd principal mean center datum datum origin aspect pca assumption assume lead component nonnegative factorization nmf aim nonnegative wise nonnegative introduce article lee explain nmf aim relevant contribution rather nmf reason popular interpretable nmf hyperspectral imaging application air emission biology blind source separation single source separation cluster analysis collaborative filtering let gray level th nmf interpret image intensity image latter localize hence simultaneously several image dense face e nmf approximate correspond word appear document column word document sophisticated construction tf document associate document matrix factorization rank nmf generate q decomposition interpret word nonnegative original much basis number document interpret set document document nmf relate exist topic semantic indexing column spectral signature scene spectral signature light reflect therefore hyperspectral usually observe broad spectrum hyperspectral see illustration hyperspectral blind two fold material example surface pixel mixed material popular combination signature correspond road surface mix signature signature road signature basis weight rank hyperspectral illustrate six column signature contain abundance pixel note decomposition section hyperspectral survey see previous useful compute problem assume introduction nmf arguably matrix use kullback leibl music improve vision consideration survey nmf particular unfortunately unconstraine hence practice guarantee stationary heuristic application recently et nmf address et algorithmic exact nmf improve polynomial solve real high cost run operation usually nmf generate matrix practice satisfy generate interpretation classification reader therein uniqueness prior proper regularization popular usually contain plain poor blind spectral signature coherence pixel likely contain material usually neighboring value preserve image e therein algorithmic refined various nmf nmf projective nmf model different good look expert blind expert scene reference focus first issue standard section nmf design
joint term factor respectively score rao outside markov hierarchical observation variational parameter variational compute unfortunately require iterate every need noisy estimator sample mean low compute substitute gradient need iterate observation shape parameterization variance gamma posterior model variational require effort quickly develop explore present box many model optimization monte develop gradient maintain derivation correspond reach method demonstrate inference explore quickly evaluate model latent modern latent model infer inference summarize conclusion compute latent interesting computing practitioner resort widely approximate variational try find member simple close convenient exist ascent algorithm family close form expectation generic derive rapidly explore modeling assumption impractical practitioner variational apply method practitioner quickly derivation time adjust frame optimize adjust proxy gradient objective variational optimize sampling evaluate form stochastic variational perspective method evaluate calculation variational evaluating estimate library share reduce gradient essential develop control rao second emphasize goal box variational adaptive generic subsample gradient close coordinate natural way compare hasting require effort predictive quickly ease variational several method approximate kl include adaptive subsampling approach inversion impractical alternative estimation family set box variational inference approximate variable family variable goal variational q maximize divergence intuitively mass configuration explain reward maximize many configuration practitioner maximize expectation close ascent available conjugate latent variational analytic computation variational outside set overhead develop optimization maximize optimization noisy unbiased gradient update noisy distribution variational method gradient application maximize realization finally th stochastic widely found call gradient sample variational algorithm function variational build package variety make log joint reduce effort variety tb data field variational initialize maximize variance carlo large useful gradient require reduce rao exploit preserve black box rao replace respect expectation variable rao without specific rao replace conditional variable expectation place estimate variational govern characterize member seek gradient supplement say main step schedule second subsampling observation rate intuitively rate large vice versa issue let matrix iteration gradient per eq learning since element capture vary scale algorithm rate inference massive idea subsample gradient similar model variable place conditional noisy iterate noisy gradient variational use medical demonstrate likelihood longitudinal patients york disease patient k visit measure consist measurement take value amount particular come visit indicator health evaluate predictive likelihood visit carlo initialize randomly factorize normal real doubly size meet gamma normal series draw factor positively affect let process raw normal vector visit draw parameterization supplement black family tend limit parameterization allow visit visit finally emphasize ascent gibbs conditional metropolis hasting fail gamma instead compare hasting complete conditional predictive method hold test box metropolis hasting inside budget hour hold initialization model variational likelihood get hasting study estimator patient series versus iteration compare variance carlo rao rao drastically improve black box variational without fail make progress estimator rao variate rao estimator box variational consider factor health name gamma ts normal visit positive variable note expect factor gamma patient draw save factor expectation parameterization hard natural parameterization gamma draw factor visit allow propagate patient ts simple model
cover technique dedicated fusion trend introduce old work consist estimate high take correspondence impose fuse band band describe fusion notational follow band observe hyperspectral band denote observe datum resolution definition place hyperspectral model spatial assume circular account image assume uniform subsampling grid noise measurement matrix hyperspectral couple form highly correlate normally live column small translate relatively accurate estimation work consist linear building correspond discard former however see detail problem try solve ill therefore adequate regularizer use denote element product horizontal vertical respectively boundary condition variation impose gradient mean except detail regularizer detail align among vector regularizer hyperspectral sense regularizer def formulate term regularizer convex quadratic term direct split augment lagrangian shrinkage multiplier simple auxiliary optimization problem take lagrangian respect cyclic efficiently solve respect variable solution involve advance minimization operation approach employ primal dual unlike arbitrary problem experience literature fusion publish fusion hyperspectral publish real image hyperspectral university acquire ground truth bands band subsection follow horizontal hyperspectral band snr band former latter add db snr paris acquire observe hyperspectral resolution resolution fuse hyperspectral see ground literature angle base quality implementation consider work real life reference estimate image respectively implementation dataset truncate ten singular vector ten preserve original share note due average index choose situation yield index perform hyperspectral band remove svd project solve via work yield use gram schmidt gs intensity fusion technique transform bt box filtering result table comparative band show fig paris seen find seem spectral overlap hyperspectral channel compare zhang manner similar estimate spatial address level transform restriction implementation work part pixel leave run equip intel ghz ram memory take second see bt gs zhang flexible hyperspectral spatial fusion present hyperspectral large overlap high spatial instead one augment lagrangian shrinkage direction multiplier admm convenient variable exploit live
sampler stable choice prior dataset consist km galaxy survey currently cluster give choose new sampler ccccc trend time potentially exercise note stable report similar py gamma prior mode around cluster good condition consist profile profile within kind useful small explain encode recover multivariate normal unknown covariance mean wishart base measure currently assign denote variate mean denote wishart definite freedom wishart matrix diagonal dimension make weakly average probability prior class predictive mcmc report curve per curve per reflect grant agreement file multidimensional create plot file illustration txt file data multidimensional section mat size set mcmc array eq corresponding size chain burn average predictive probability dataset run observation probability leave table fold follow replacement point batch predictive probability exchangeable introduce variable induce eq plug evy exchangeable induced dp parameter evy gamma chain initial transition kernel proposition formula poisson homogeneous completely size pick atom q index easily measurable negative function measurable positive draw measure equality denominator induction hypothesis numerator use specifically theorem proposition bayesian mcmc generalize modeling important concern specification model infinite dimensional active possibly infinite exchangeable parameterize define infinite identify component formulation set introduce model apparent replace break notable introduce also alternative preserve almost normalize allocation heavily distinct component flexible mechanism several monte dirichlet process exploit tractable marginalization dirichlet overview development atom measure stick belong model stable poisson dirichlet process limit property stable dimensionality graphical rely task multidimensional recall mcmc present stable model contain conclude start completely reader let separable endow borel take space represent nonnegative random location characterize measure mean evy measure evy atomic homogeneity identically normalization homogeneous evy random variable surely homogeneous law govern l evy normalize evy comprehensive homogeneous index induce show exchangeable exchangeable partition law account induce biased permutation atom block increase block appearance atom biased atom tend appear early break construction stick break first length piece break etc form initial transition kernel supplementary denote random generative part mass pick mass conditional subsequent atom multiply probability next exchangeable denote order increase biased homogeneous surely absolutely dirac delta whose respect positive representation evy measure distribution throughout paper distribution homogeneous distribution govern paper popular j jt poisson evy poisson view distribution poisson specify ns py positive correspond positive see stable lt also show obtain direct supplementary material next stable exchangeable set one derive comprehensive exchangeable partition section sampler effectively apply evaluation hierarchical mixture extend joint mass mass distribution mcmc package integration numerically generator polynomially exponentially rejection proposal auxiliary augmentation describe difficulty computation value numerically address problem propose explicit rest stable nonparametric distribution smooth derive conditional maintain representation auxiliary final representation value graphical partition read conditional next cluster proportional exchangeable cluster variable study chinese restaurant describe partition condition auxiliary extension component lead chain involve update cluster maintain potential
empirical improvement small imply grow linearly alphabet prove theorem conference paper weak upper completeness tight report alphabet respectively range frequently last first sublinear sample use shannon alphabet language genetic shannon implicit fairly order imply furthermore suggest r continuous order order entropy may approximate could reason r enyi show infinity alphabet note achieve sample universal estimator know r enyi accuracy equivalent multiplicative furthermore accurate estimator sum multiplicative accuracy additive entropy range give unbiased exponent frequency use consider approximation early consider roughly part use polynomial appropriately table c determine enyi entropy illustrate integer show enyi multiple every bias empirical enyi table lead term approximation l k enyi approximate correctness eq infinity go infinity figure show performance draw illustrate empirical bias work empirical exponent attain exponent possible compare performance enyi entropies parameter significantly entropy imply fairly enyi constant theorem organize follow present power moment poisson integral enyi entropy establish enyi entropy randomness see follow inequality monotonicity every monotonicity upon inequality monotonicity old final follow upon rearrange poisson draw poisson simpler facilitate poisson moment start expect power expectation establish eq fact either equality multiply side expectation side q nonnegative schwarz review approximate interval set less require polynomial approximation minor polynomial bound variance approximation absolute follow serve degree performance estimator proof bound general analyze estimator separately symbol x nh sample corresponding estimate error estimate simplicity randomize describe q reduction q remain right bias n chebyshev reduce estimate repeatedly specifically hence hoeffding claim choose note half p must remainder sample appropriate estimator estimator derive estimator bias bind sum q triangle inequality independence jensen inequality variance satisfy bias subset inequality first concavity complete lemma reduce integer shannon estimator traditionally shannon analyze reliable shannon estimation similarly reduce correct provide choose remove bias alternatively bias usual sampling similar albeit complete form polynomial motivated paper consider approach arise symbol empirical nearly large interest estimate sum suffice similar power fact accuracy estimation power insufficient estimate enyi show attain brief random use symbol estimating sample polynomial da use unbiased namely combine rely bias bound complexity polynomial exist estimator least satisfy throughout satisfy bias q hold triangle lemma use claim low estimator sequence say estimate suffice consider sufficiently contradiction profile omit contradiction consequently follow upon correspond therefore entropy requirement difference sum construct require make ensure match need entropy sum positive respectively integer eq constructive converse lower integer q vector discussion verify follow must inequality hold theorem since alphabet yield sufficiently arbitrary lemma small positive root root newton identity polynomial depend power differ furthermore taylor small zero acknowledgement thank helpful suggestion apply estimate need additive technique show obtain sum namely accuracy enyi sum enyi complement inequality follow upper estimator sum accuracy comparable ok constant range multiplicative sample empirical version sample bias appropriately choose q since get proceed variance complete upon show note summation equal title theorem lemma conjecture remark remark shannon symbol grow replace sample need surprisingly develop
explicit cluster column know furthermore misclassification step conduct column assign augment multipli algorithm augment lagrange multipli rank structure nice multiplier rewrite indicator convex set aim minimization dominate fact subproblem optimization large hundred thousand svd scalability surrogate eigenvalue minimizer mean rank leave large scale future research minimize closed entry great lagrange convention multiplier j augment multipli derive summarize numerical political defer clearly implement ordinary clearly term guarantee procedure community explicit condition capable presence portion outli let semidefinite gap density denote eq guarantee solution belong group solely impose helpful work even good community adjacency sophisticated becomes allow second assume cluster community sbm literature node guarantee let density example modify condition relax detection plant within usually group density barrier plant clique therein involve section helpful behind optimization intersection cone boundary point hyperplane vertex programming discriminate clear observation clear affect result cluster specific moreover norm coordinate nonnegative apply obvious km b distinct majority misclassification j give example misclassification ball box box ball ball color assume box distinct colored misclassification prove misclassification control less rigorously speak easy mean r k j misclassifie previous base distance less hand community great comparison detect accurately without cluster whether approximation synthetic employ effectiveness community mean lagrange multipli simulation ghz intel processor algorithm fix illustrate realization know detail maximum formalize change diagonal n divided cluster community second accurately misclassification rate adjacency cluster comparable propose misclassification rate profile sbm modularity method misclassification indicate maximize criterion initialization misclassification modularity independent repetition classical eigenvector distinguish conservative political ordinary datum penalize apply nearly paper robust presence outlier propose strong mild condition density order grow adversarial consistent art feasible sbm extension current detection depend detection hold big capable must modify replace dependent diagonal adapt high correct sbm cl simulation establish guarantee choice much well open redundant procedure cluster usually group connection interesting write use coordinate nonnegative norm represent norm matrix whose correspondingly numerical constant algebra hermitian eigenvalue arrange cauchy eigenvalue arrange inequality xx consider symmetric whose zero nc sequel bind bernstein theorem total q zero pair zero determine leave article precisely sufficiently eq symmetric constant apply relax let particular outli node ordinary sbm art literature computationally feasible detection rigorously formalize follow lemma permutation give analyze several inequality one feasibility sufficiently utilize need construct show establish guarantee sufficient use lemma previously consequently three exist uniquely nonnegative diagonal furthermore eq aim must indicate suppose pt form intuition rigorous give cluster get small reason get choice intend constraint sure first projection concentration suffice guarantee existence moreover large numerical jk jk ii jk style thm section dms dms detection group undirected block allow adversarial outli node follow accurately misclassification set grow admit good outlier fast kind outlier community spectral adjacency fail retrieve major cluster portion political showing method exist feasible well fast outli node wide range engineer recent interest characterize approach reference network aim cluster node community observe undirected detection challenge spin perhaps sbm independent assume node assign label label adjacency graph respectively graph self loop symmetric bernoulli refer namely assume detection minimum tuple sbm detection study greedy see criterion likelihood stochastic pseudo gibbs monte belief gm cl lr convex accuracy fully modularity profile method prove consistent computationally hard justified theory community fast spectral cluster dense spectral cluster sbm generalization mixture connectivity different group latent membership easy detect apply graph single generalization sbm sbm first fit form arbitrary outlier important community detection arbitrary main question wish portion arbitrary rigorous proof begin portion node model cover range sbm suitable assume undirecte among sbm connect arbitrary node event matrix connectivity arbitrary restriction connectivity outlier loop equivalently entry arbitrary permutation capture connectivity correspond usual sbm bernoulli submatrix parameter define parameterize tuple necessarily randomness depend word allow depend generalization sbm connectivity stochastically pair applicable cover common name sbm assume node portion belong portion outlier ordinary connection specific outlier employ cluster weak difficult essential popular modularity outlier classify belong group connection refer object neutral political portion connection strong nod neutral even modify sbm combination neutral setting complex model overfitte sbm property political discuss sbm prefer significant cause lie outside cluster take community sbm model robustness portion result cluster robust good follow directional lagrangian focused consistency optimization inference group specify analysis proof contain additional technical proof material community detection computationally greedy justified maximum hard stochastic variational prove block fix going naturally likelihood propagation rigorous theoretical unlike aforementioned easy various section ordinary spectral laplacian datum type ordinary perfectly plot eigenvector absolute laplacian adjacency eigenvector combine capable cluster outlier suffice explain random independent independent adjacency eigenvector corresponding eigenvalue absolute graph adjacency matrix discriminate homogeneous behavior thereby unable distinguish laplacian discriminate two major simulation apply spectral laplacian percent certain penalize weak graph influence standard clustering detect kind outlier
rnn rnn multiplicative time rnn mn multiplicative rnn rnn rnn feedforward ff plain rnn regime rnn rnn see stochastic dropping test dataset initialization deviation sampling weight noise c c rnn rnn rnn rnn rnn rnn ff get spectral fix epoch try weight big lack store time delay see regularizer radius grow albeit give regularizer show logarithmic test fig trend indicate error income recurrent trend indicate increase exhaustive update base show rnns loss eigenvector limit low suffer store analytic presentation also explain regularizers rnns past believe gap sophisticated issue vanish term dependency recurrent loss rnn define pre activation income upon vector consider constant expect value compute hand deviation put form multiplicative pre activation equivalence bring form place eq analyse eq usual backpropagation follow post term deviation actual value analysis eight hyper range limit regularizer l regularizer e optimizer momentum h l rnn mn rnn ff regularizer dropout optimizer momentum size hide rnn rnn rnn rnn rnn ff regularizer optimizer l layer l x rnn rnn mn ff dropout optimizer step hide rnn rnn rnn ns rnn rnn ff initialization optimizer momentum batch layer parallel lead deep past rnns additional feedforward provide treat sequential rnn conventional relate dropout noisy rnns rnn improve capability aim empirically rnns model train norm performance initialize rnn advanced dropout work rnn recurrent network variation information language analysis financial domain mlp go technique train rnn perform time backpropagation suffer problem gradient gradient grow vanish gradient large fast purpose instability conceptually vanishing rely particularly property simple series need store vanish gradient become unstable behaviour rnn extensively sophisticated introduce feedforward rnns advance solution lstm task date study one regularization recurrent growth vanishing evaluation rnn derivative ability dependency discuss trend rnn like integration unit also momentum descent sgd powerful boltzmann gradient trick evaluation paper music dataset addition corpus consist denoise autoencoder step rich audio signal ed model key rnn typically rnn evaluate transition layer problem rnns variation conventional rnn term memory unit randomly delay increase extend apply complex corpus value improve dropout recurrent noisy mlp noise curve noisy descent feedforward rnns rnns rather solve delay widely long identification rnn really validate delay relevant however applicable temporal universal input rnn exploit technique fully connect hide layer therefore rnn rnn activation hide hide layer linear explain dynamic backpropagation rnns rnn calculate affected multiplication successively may lead grow vanish fast regime update gradient happen opposite eigenvalue recurrent much might vanish linearity might demonstrate recurrent simplified explanation perspective find come step become unnecessary dimension reach initialization represent represent matrix curvature illustrate thing routine surface face depend learn ground simple objective local optimize term quadratic normally expansion difference make involve inversion matrix make possible regularization particularly vanish systematically prove generation prediction dependency plain rnn modify lstm memory cell link conventional unit hide take flow hide activation layer vector determine activation fed unit sequence lstm time hard train gradient dropout neural every neuron incoming connection probability safe tend approximately incoming connection order dropout match plain validity dropout rnn hide single apply connection rnn regularizer act regularization dropout initialization rnn evaluate hidden output matrix rnn improve network gradient effect regime feedforward network tolerance unseen search coarse region rnn adjust gradient weight behaviour regularizer appendix demonstrate stochastic recurrent layer much feedforward multiplicative noise model evaluate recurrent preserve recurrent follow analyze restrict cumulative model backpropagation rnn train set increase noise space decrease weight noise time weight additive vector sample additive vector distribution recurrent noise update noise noise multiplicative multiplicative evaluate variant model additive multiplicative model weight matrix preserve even calculation backpropagation recurrent
sequential set adaptive compressed various determine location single adaptively identify compressed sensing design exploit structure claim near specifically tailor sense strategy compressive sequential basis information imaging literature sparse sequential sense various place example compressive sensing design minimize introduce sequential adaptive adaptive compressed sense mixture objective function sequentially rank gmm empirical measurement difficult devise compressive pick measurement amount previous measurement enable establish theoretical guarantee sense often resort query noise analyze measurement also develop numerical organize formalism greedy sense signal gmm finally conclude denote th let denote positive determinant mutual elsewhere quantile chi degree typical compressed sensing let measurement depend linearly subject sense vector set compress measured low model use non subspace gaussian lie union video signal general manifold exploit structure fewer early compressed regardless signal measurement sense eq either vary fix repeat integer power resource allocate extract maximize sequential outcome ax compress design row recursively view dynamic usually situation one usual approach operate core measurement probe maximize much formalize initialize return information natural useful noise x j either reach relate mutual determinant ellipsoid reach iteration avoid greedy discuss algorithm measurement outcome resource greedy sense nonnegative case correspond certain measurement sparse factor noiseless noise another finally greedy consist uniformly modify description characteristic location amplitude minimum signal n ambient replace remove noiseless exactly measurement n greedy algorithm lemma obtain measurement greedy possible mutual roughly measurement entry correspond sensor reporting count relax allow setup generalize sum recover non amplitude measurement vary repeat overhead lemma measurement incorporate gmm gaussian covariance consider measurement compress allocate allocate snr measurement measurement reduce communication q snr actual measurement proxy amount resource allocate colored resource sensing derive similarly snr greedy gaussian iterate adjust clearly signal decrease order establish white add accuracy satisfy measurement simplify hold see unit eigenvalue use eigenvalue informally measure power reduce sense gaussian gaussian eigenvalue w recover every outer drop black width color add x give measurement give various accuracy correctness white u eigenvalue u eigenvalue u e colored noise color add prior eigenvalue recover use noise model outcome affect gaussian advantage inaccurate bring demonstrate sequential new matrix measurement induction measure hence several thereby split signal implement sparse large eigenvalue method correlate change change sparse sparse therefore matrix covariance greedy gmm class gmm signal outline derivation respect linear transform mmse moreover colored determined instance demonstrate order formula color white colored greedy sensing error obtain error fall tolerance theoretically batch eigenvector matrix greedy outperform due method performance sense rank another entry zero greedy sense precision sense method perform identical use sense sense batch cc gmm assume gradient descent fig demonstrate cumulative mutual average monte trial descent heuristic fig greedy perform fairly compare gmm signal versus iteration associate measure cc average trial c batch descent greedy single measurement generate eigenvalue show entry greedy gmm sense mnist handwritten true label training gaussian digit handwritten digits picture digit measurement instance measurement gradient heuristic normalize ccc false recovery consumption california consumption year single year demonstrate reasonably good fit test sense year coarse well compressed collect consumption region automatically wireless sensor platform embed efficient monitoring consumption large power year versus number measurement framework compress sense maximize condition previous help signal bring robustness assume benefit moreover sense sparse demonstrate potential establish obtain error sequentially signal prescribe formally family I operation I single pair pick enough ensure invoke e uniquely variable algorithm greedy information sensing learn measurement measurement well colored full note write b w ba u maximum sensing interpretation color snr large eigenvector u w x u measurement density mmse condition measurement gmm close derive turn gmm weight q hence base close enable gradient draw computing carlo summarize whenever drop information condition entropy gmm overlap k disjoint covering yet start line measure keep intersect great subset determine coincide x k meaning size whole k measurement difference block reduce error every measurement intersect exact might happen l consist coordinate terminate n measurement end estimator coincide outside support x great r proof family signal consist signal measurement ax entropy apply size domain uniformly residual consider measurement measurement mutual measurement identical size reduce induction maximize use upper n query take explain eigenvector unchanged measurement apply several reduction combine power minimum integer direction measurement suffice ensure algorithm gaussian return signal distribution similar eigenvalue leave sum eigenvalue large similar difference canonical make switch orthonormal write basis measurement identity basis add measurement every I ensure mean return sketch first
relatively compare towards problem effective increment cumulative adaptation arise positive model constraint handle resample choose study consistent assume step constant adaptation mechanism es constant optimizing handle extend result chain algorithm attention distribution copula recently trend copula evolutionary algorithm next basic formally feasible section gives aforementione verify notation integer vector sure probability denote borel es linear sample es composition objective strictly hold handle invariant composition w g g call I j distribution stand sample index feasible sample candidate feasible update parent distribution step internal parameter adapt require es optimize define handle resample l respectively yield admit absolutely possess yield vector function wants deal es handling resample g k distribution copula regard g ff consist marginal distribution copula use size theory remainder recurrence divide left vector order apply rhs previous equation chain precisely geometrically j state es optimize handle resample matter homogeneous chain density chain ergodicity chain imply generalize proposition handle resample absolutely function lebesgue measure irreducible small set markov fulfil recurrent furthermore geometrically substitution denote l absolutely show function l infimum reach measure since irreducible finally take lebesgue strongly define v want drift imply compact want g condition integrable dominate integrable respect count dominate condition suppose compact chain number divergence es optimize side positive chain number hand side finite invariant lemma covariance constraint angle handle resample k kk n g j h e sign achieved state es handling resample absolutely distribution isotropic use proof equivalent isotropic chain use proof normal es handling resample multivariate hold nh integrable dominate theorem markov geometrically ergodic positive isotropic fulfil every obtain sufficient strictly advantageous pay copula e continuous decrease denote inverse reason copula invariant permutation permutation matrix hold permutation matrix first continuous continuous positive density continuous strictly moreover generator replace strict positivity respectively monotone copula q copula combine turn q sign opposite completely monotone define indeed copula express particular eq describe paper present evolution general distributional isotropic problem ergodicity accept ergodicity stationary monte opinion condition insight different play design evolutionary solid multidimensional attempt bring contribution applicability copula evolutionary copulas realistic actually
construct accurate marginal note road extremely largely abundance logistic additive additive smooth decision model kind decision case small scenario adapt high penalize regression drawback additive searching transformation available fail knot every additive reduce bias comparison increase variance cost moreover admit interpretation building besides reference dimensional classification discriminant organize dedicated study present describe variant original transform one give pair g jx penalty plug predict repeat probability increase stability transformation unstable density take kernel good mcp take final make split majority vote split splitting prototype marginal prediction procedure assignment usage make bootstrap misclassification reflect reader similar balanced switching feature run splitting perform transformation implement penalize employ level coordinate constant whole particularly repetition compute core implementation application multiple henceforth whole penalize leverage computer nonparametric classification computation various repetition column report argue contribution logistic transform odd marginal enter help separate reasonably compare additive model regularize road discriminant lda nb fair simulation setting sample five validation conduct need replication long f summarize standard use pseudo setting vector boundary due comparable pay use bad complex unnecessary decision surprisingly perform poorly bad especially common independent nearly independent nb ignore classification multivariate gaussian class boundary boundary nonlinear distribution oracle decision method fail classification report extremely fast similarly simulation computation cost demonstrate spam satisfactory road nb svm l svm road study spam demonstrate power attribute word character email length case letter letter rest split repeat summarize competitive training svm well dominate different training fail yield proportion due spline define penalize vector estimate identifiability margin condition parametric introduce relationship set specify notation continuously real taylor old continuously interior level condition assumption regularity estimator impose absolute assumption guarantee penalize un penalize oracle measurement compact j density f impose incur marginal strictly positive put bandwidth absolute condition similarly bandwidth exist n hold compatibility take specific uniform margin involve formal omit normalization p sample theorem excess risk control use next excess let sub pm version oracle assumption order density regularity condition component big density estimate strength need link density due regularize covariate bandwidth appropriately estimator excess explicit excess worth accomplish bridge regularize work change model note addition transform adapt establish omit propose new classification leveraging nonparametric estimator achieve feature penalization linearly transform original perform dimension flexible curse array demonstrate procedure misspecification well compete perform slightly standard additive standard ex fair rule insight size recommend impossible look abundance rough extension far investigation future beyond specific establishe estimator might near searching combination difficult task application ratio approximated snps could beneficial notable front independence sis marginally subsequently independence screen sis additive model contain technical proof f e p r since bind give rise play simplicity plug h mn mn mn bandwidth lemma denote result b b inequality n view follow simplicity order expansion pm pm pm c pm pm pm side q combine oracle corollary classification nonparametric feature augmentation know powerful univariate transform subsequently penalize newly transform augment train equip simplicity avoid create decision result feature generalize naive writing joint relate generalize model numerical domain real email spam gene expression implement parallel nonlinear boundary augmentation feature parallel identify category spam detection recognition high gene fisher discriminant lda near neural perform much many dimensionality microarray frequently thousand computational setup limited access conventional high new setting regularize refer classification augmentation selection introduce motivation suppose code pair variable binary dependent density ir py decision setting fisher addition rule structure among essential help abundance sample bioinformatics assumption correlation regularize road road plug directly classification advantage un regularize pool gaussian naive bayes naive bayes however among feature motivate ask question advantage precisely decision necessary thresholding good ratio transform future effort transforms build spirit sure independence sis use coefficient wish take feature
amount proportional similarly multiplicative distribution variance show distribution pair power recover want outline run output strategy allow large total conjunction terminal option either graphic terminal graphic macro ltb lt lt lt lt ltb lt lt lt lt lt bp r r r ltb ltb terminal either load package graphic need graphic macro ltb lt lt lt lt ltb lt lt lt lt lt r r r ltb ltb package color conjunction terminal option explanation load package package graphic terminal graphic macro ltb lt lt lt lt lt ltb lt lt lt lt lt r ltb rank recover claim quickly practical core intel memory netflix experiment collect radial five plot illustrate value illustrate varied value slow initial value seem netflix recover dataset reveal rectangular run singular algorithm singular eigenvector plot plot illustrate runtime eigenvector sequential allow good open version non descent low problem variety globally initialization rather account novel optimistic apply helpful thank advanced research fa fa science foundation nsf award fellowship author acknowledge contract air force domain indexing fa contract nf mid technique library throughput grant simulation software national nsf stanford program author acknowledge support finding conclusion express reflect nsf example happen stochastic descent descent update reasonable independently condition infinity iterate decrease inductive q therefore proof exponentially quickly stochastic enter choose start account converge optimum entry let try decompose rank e update eigenvector global problem mean global illustrate optimization give factorize constraint complete q relax entry dependence entry particular minimum boundary imply value imply np suggest analyze constraint case literature list sample iteration rank applicable assume factor omit scheme svd alternate minimization flow rigorous follow definition time space event encode monotonic call except call independently zero unless therefore recall q rational expand product bx cx b bx bx bc ab ab b divide side first symmetry uniformly normal purpose initialization component eq must independent convex inequality matrix jensen call matrix simplify evaluation rank rewrite recall substitution result q leave bind cauchy inequality since rank upper eigenvalue statement minimum prof apply expression event exist follow expand follow less function eigenfunction fouri x du apply expression desire next lemma case lemma make incoherence condition symmetric incoherent parameter symmetric matrix incoherent must therefore show show incoherent parameter let eigenvalue definition incoherence desire part uniformly choose evaluating desire lemma logic choose desire schwarz rectangular choose entry pick moment trace must derive behave uniformly unit unit radial symmetry denote moment chi square substituting sphere component sample schwarz apply definition desire part want suffice pick lemma evaluating apply desire subspace uniformly lemma project subspace span basis incoherent space note part subspace incoherent bind part evaluate apply rough rate hope improve linear assume error direction rate expand choose take inverse across symmetry produce q desire low approximation matrix want decompose angular phase recover vector analysis constrain appendix solve problem perform quadratic substitution case rip restrict linear rip definition norm rip matrix rip prove transform objective optimization rip parameter small directional derivative second direction fy cauchy schwarz optimum follow previous desire result theorem dependent standard convex method rate imagine follow angular phase algorithm state main body rank iteration coordinate precision scheme reason sgd scheme achieve monotonically linear approach coordinate additional electrical science stanford university stanford factorization matrix relaxation exhibit least square prove runtime analyze solve draw form eigenvalue transform application include tracking sample store operate factorization substitute problem drop store size people sgd standard globally manifold guarantee motivate converge globally establish rate optimum prior analysis previous noise problem previously observe select observe entry martingale space technique optimistic factorization semidefinite analyze rate sgd exhibit local solution optimize low approach correct riemannian back onto manifold order sgd point rate optimum algorithm involve study recovery minimization provide retrieval operation initialization sgd cover slow respective eigenvalue familiar adapt stochastic stochastic wide understanding paper analysis local stochastic iteration globally convergent provide stochastic analyze matrix completion retrieval tracking factorize function orthonormal eigenvector sample factorization introduce manifold stochastic manifold group riemannian size choose sgd size iterate think give intuition particular update q property care compute simple operate whole operate benefit rescale angular recover radial notice unlike independently independently r analyze low rank decomposition introduce family point globally whenever non fix sgd choose lyapunov show decrease time lyapunov function matter initialize rapidly regardless hold attack respect way angular close recover indistinguishable expect handle span eigenvector matrix angular success angular require member occur value satisfy satisfie noise analyze rank represent size slow define standard variable assume satisfy angular phase radial vary angular iteration rate complexity unlike prevent also recovery option explanation load package explanation terminal need graphic macro ltb lt lt lt lt lt lt ltb lt lt lt r ltb ltb l ltb proof document since use outline occur iterate unstable determinant intuition occur failure occur success
note reduction review oriented want misclassification projection albeit last part check whether statistic carry choice statistic misclassification respect vector datum space error dimension abc represent local predict summary eq besides rate upper bind bayes admit loss replace classifier eq q classifier support proposition minimize integrate know selection classifier classifier last perfect train provide tend size dataset numerical local difficulty classifier naturally local probability relative return substitute restrict mention set agree train moreover algorithm knn minimize beyond good misclassification knn calibrate produce reliable estimating must kind issue swap otherwise section core proposal method expect call already simulate estimate error hope whole additionally limit must reference table bandwidth estimator know table table independent database constitute reference validation misclassification rate call consider reference always abc simulate database display surface second help krige conclude resort krige comparable evaluation support reduce valuable point abc dimension statistic additional coordinate regard abc sort dimension summary suffice accomplished misclassification know attempt cost potential curse dimensionality knn ideal remain perspective tune assess proximity adaptive classifier validation reference initial adaptive classifier surface collection summary table reference qualitative trait replicate predict initial classifier collection compose qualitative trait agree return correct axis qualitative trait know classifier rate algorithm table accuracy independently adapt machine community discriminate material numerically structure neighborhood system undirected generality focus representative level difficulty index call site color model digital site lie undirected definition adjacent include parametrize adjacent site auto arise normalize call define edge realization directly grid color statistical literature drop height width height cm vb four define eight close field permit modeling encounter latent field index conditional py parametrize scalar hide graph face double intractable issue neither likelihood latent color continuous well fit give compose neighborhood system represent undirected simulation compose lattice except boundary lattice another integral beyond mention sum triple intractable neighborhood structure field obtain abc field discriminate neighborhood structure color proposal interval share color free preprocessing perform via color group color color namely summary become q color indeed propose appropriate number ccc rate axis respect horizontal axis line error statistic include dimension onto six axis section summary give beyond component geometric begin abc normalize table respect summary axis scale draw thank favorable via markovian prior hour cpu optimize time abc wang motivate cut sampler simulation extend calibration neighbor algorithm show evolution evaluate reference six three impact solid numerical good calibration prior really reduce obstacle large regard confirm reference training curse dimensionality size fig latter prior misclassification replace last independently three carry prior substantially classifier train new summary base help discriminate highly reference exploitation ccc fig display plan range plan full call validation plot part krige geometric summary highly table dramatically error classifier explain interestingly informative reference design limitation connect latent noise rely color indicate capture information geometric summary reference simulation diagnosis plot rate abc cccc b prior error vertical horizontal train solid dash error abc summary framework include noise diagnosis table difference carry connected component adaptive simple misclassification albeit positive classification paradigm provide error section derive classifier curse locally around trade dimension proposal dimension besides inequality complement avoid practically machine viewpoint give abc latent method construct statistic induce construct statistic approach intuitive isotropic averaged width length summary explain continuous quantization observe site analysis numerical demonstrate indicate limitation approach believe road add edge color group consequently reference table since field negligible misclassification summary able acknowledgment grateful feedback present like thank anonymous comment suggestion lead thank model hide abc new summarize procedure intractable aim like highlight ability evaluate local wide performance relevant relevant hide field gain statistic distinguish bring little information happens describe extra information aim precisely size might extension geometric paper extend grey subject theorem dependency hide markov challenge due answer computation paradigm sufficient statistic fall summary statistic cluster plausible abc evaluate via statistic approximate choice misclassification nearest gibbs field spatially correlate mapping genetic spatial random grey color undirected grid perform popularity major difficulty view choice intractable remark exception small however time deal explore answer try reversible follow important adapt context graph observe address question infer color algorithm extend approximate approximate computation abc address paradigm observe numerous simulation monte carlo probability difficulty highlight sufficient know consistency abc check article automatic scheme construct rarely review concrete accomplish abc apart reconstruct competition pilot abc also consume section present near neighbor base general hide paper analytically choice challenge abc model fit observe approximate review wide choice assume embed space space density respect lebesgue evidence define probability good fit perform maximum predict respect counting measure since invariant whereas drawback mode abc numerous approximate posterior simulate eventually good frequent decision posterior directly face curse lie indeed impossible ambient dimension abc perform dataset euclidean moreover regard computer keep track commonly summary iid replicate literature j serve compose bayesian distance term name
imputation discard incomplete discard incomplete remain proper deal miss imputation complete simulate miss purely stochastic analysis simulate pool well strongly structure video reconstruction video media contain frame reconstruct video security reconstruction refer incomplete empirically nystr nystr om allow approximate equation subsample replace quadrature subsample manually effectiveness nystr om formula projection segmentation extension eigenfunction outside geometric subsample consist record method gram entry semidefinite homogeneous uniformly gaussian positive necessary put method hilbert space nystr must consider lead observe side generate family maximally concentrate generalize wave cf f restriction sense formula recover basis elsewhere value characteristic column extension scheme stochastically precisely step restrict row construct note characteristic characteristic prevent introduce bias degree description attempt keep notation become heavy see account roll q add height generate evenly spaced dimensionality roll embed noise ensure lie spread rate distance height choose point avoid influence dataset assess visually diffusion introduce introduction excellent diffusion provide van toolbox reduction find visually original quite easy see computer denote track iterate connect plot plot stochastically version begin step final point provide move point original image standard dataset parameter case approximately slightly plot stochastically initialize plot plot graph test face excellent dataset comprise people pixel comprise people web run example show face pixel approximately nonetheless reconstruct accuracy people miss plot pixel reason apparent output display result method show reconstruct reconstruction make neighboring value infer entirely image word pixel adjacent effect image pixel increment report toy car object sparsity successive rotation sample every sample time continue drop point weather locate international record contain pressure wind velocity cloud cover day initialization stochastically draw determine exactly imputation note scale axis actually method difficult analytically represent denote single comprise eq remark appear whose extension maximally energy minimize update replace minimal incorporate contribution every restriction energy geometric linearly find bottom relative versus range percent trend less consistent matlab code write test mac ghz
relate latent image cifar similarity fold recognition performance absence modelling seem beneficial suffer svm gain compare b try linear svm similarity cell result increase believe boundary mention believe heuristic complexity case rand basis refer kernel basis om spectrum report rand trial sophisticated similar little deterministic difference normalization nystr om normalization tend case psd kernels h nystr om bad h believe reason lack basis alignment violate dominant capture useful eigenvalue basis eigen decomposition show ratio negative two column reflect entity nystr om normalization pearson linear positive normalization entity eigenvector spectrum flip square om generally provided slightly comparison spectrum c fortunately svm construction demonstrate fold augment similarity order similarity measure complementary resolution resemble process level fine svm fold support use approximately support resolution fold furthermore support increase compare red curve improvement psd two outperform compare curve model measure support sparsity roughly different complexity utilize try psd sophisticated competitive less cost define mkl consist perform alpha fold cross kernel contribute resolution result approximately use combine basis variance normalization empirical discrimination prior utilize regularizer consequence center correlation dimension help balance irrespective overall affect combination measure option combine prohibitive search combine similarity measure section normalization feature vector normalization scoring center scaling dimension z svm center center svm report validate basis sub sample similarity validate observe normalization work scoring suitable similarity measure svm score overall work single normalizing feature accord normalize marginally affect conclusion svm kernel combination augment resolution observe normalization much combine similarity measure motivate benefit least dataset remove validation similarity center properly solver unnormalized robust rbf measure analyze scalable scenario similarity measure competitive scaling datum name expand analyze extensively cifar intra expect play crucial role imagenet future scale object detection strategy major limitation psd many psd implicitly explicitly paper investigate approach framework show constitute suitable despite complexity experimental result cifar equip svm support svm classification problem success achieve operate become complex linear classifier propose non linearity see feature augment non scale mixture component high space maintain support grow approximately linearly time complexity scale support psd expressive enough various measure similarity introduce solver result convex unless explicitly psd eigen alternatively dependency learn mainly two drawback cost without aim address expansion show complexity model without removal eigenvalue require eigen decomposition similarity propose analyze investigate visual x learn learn xy positive semi psd k reproduce hilbert space rkhs vice versa namely linear svm objective ik kx kx learn evident problem product involve psd kernel closeness case frobenius negative gram point guarantee psd psd classifier impractical scenario reader svm different cost cost research dedicated test approximate restrict w method synthetic datum rank gram approximation complexity prohibitive large contrary low nystr approximate matrix eigenfunction expansion embed nystr om space method exist explicitly implicitly exploit test cost support subset measure matrix nystr om psd need psd flip eigen latter achieved find close close negative eigenvalue low rank approximation psd measure psd associate section negative maximal regularize value suggest metric distance clear work psd well expensive select svms mostly binary one v simple argue formulation conclude competitive complexity svm unnecessary overhead well class svm class advantage training svm summarize complexity computation k svm require evaluate measure evaluate svm max svm via map derive need non unnormalized consider condition I similar svm bx basis margin bx bx unnormalized square compare svm straightforward contribution contribution basis margin bx nystr method nystr svm kernels nystr feature therefore appeal similarity say assume basis basis result violate large basis covariance nystr basis superiority nystr practice nystr rbf follow b weighted normalization randomly basis accuracy number average scenario different spatial good exact measure non zero support sample need evaluate irrespective see construction mainly
account another discuss neighbourhood size control neighbourhood pc reasonable allow restrict independence test pc also pc computational complexity respect instance test indicate convenient quickly efficiently structure real scientific structure network repository benchmark network actual examine package structural previous focus seven scale create random interval generate variance number false positive increase path pc six particularly hc exclude model cross perform poorly roc reconstruction accuracy exception previously comparable rate high improvement exception pc confirm observe previously skeleton low sensitivity increase overall sensitivity calibrate properly comparison mcp former visible network comparable sensitivity positive obtain consistent low triangular b q expression equal positive uniqueness cholesky contradiction suppose regularity likelihood issue density needs check fisher dag definite suffice ij j orientation extra subgraph order compatible simultaneously equivalence order instead directly prove statement case technical ensure behave respect uniform law likelihood concave dag edge theorem pn pn imply pn check mn ij pn equivalent follow figure mcp hc mcp hc pc mcp text test use graph solid runtime runtime runtime respective mcp bar mcp mcp pc tp fp skeleton fdr fp dag skeleton fdr mcp tp fdr supplementary comparison sparsity recent accelerate restrict search thousand concave regularization generalize exist notably existence way matter comprehensive learning package algorithm compete high sensitivity rate generate estimate dag hour network penalization acyclic graph receive past decade application range expert system artificial intelligence graphical phenomenon certainly nothing new calculus develop estimate slow formulated distribution direct dag observational alone interested estimate purely experimental datum unfortunately difficult nonconvex scale exponentially realistic thousand thousand great new network critical likelihood structure observational property competitive traditional neither work however high whose thousand key challenge bayesian mind score space observational work effectively capable handle several various literature cover requirement none aware score development application regularization understand attractive computational allow penalty result concave scad mcp performance advance learn highlight regularization generalize theory class penalty conceptual deep recent development regularization vs long application insight algorithm hybrid constraint organization remainder review contribution establish approach discuss penalize necessary justify describe establish necessary material development theory complete outlined section empirical offer discussion future direction sparse learn rely interpretation bayesian term coefficient nontrivial address challenge dimensional translate family fast traditional estimating idea continuous penalty via show competitive enforce intervention proposed use observational explicit advantage convexity thousand minute propose rely traditionally network score base search optimize scoring description hill various network posteriori base test existence node dag structure long assumption satisfy tend constitute main drawback conversely approach fast spirit constraint skeleton remove edge possible step fast score search advantage traditional previously approach structure scale ever hybrid time assume exploit pc gain hill scale advantage distribute thousand category contrast present efficiently graph thousand knowledge purely rely constraint approach prune hybrid purely rely cholesky distribution equation reader recall completely acyclic maintain translation assume generate variate decomposition regard direct acyclic represent contain cycle slight abuse dag weight adjacency denote node distinction node clear simply thorough introduction graphical concept unless shall matrix denote column rewrite encode avoid unknown nonconvex play important vast assume penalize point much tailor logit generalization remain assume may reasonable see dag parameter structural determine instead rewrite dag normal connection normal equation dag shall call class different usual furthermore dag speaking entail consideration theory strictly speak encode refer ambiguity may pair explicitly adjacency give uniquely define inverse view define recent connection detail connect statistically identifiable stand difficult underlying covariance dag prove difficult exist method factor make order amongst cholesky important similarity markov field far justification fits establish via simulation compare cite discussion focus network leave future dag variable ordering denote existence edge sort general sort equivalent dag easy dag equivalently adjacency strictly nod match sort strictly purpose use dag permutation topological parent sort permutation convenient interpretation weight also technical represent diagonal rewrite compatible compatible easy check fact dag permutation dag dag permutation arise dag want presence absence dag statistically indistinguishable observational edge estimate sense motivate obvious connection give permutation cholesky nonzero entry exactly permutation oracle across permutation weight sort sort permutation dag dag variable sort highlight I reader dag obvious different ii dag primary amongst dag wish focus causal view generation finding alternatively could wish commonly adopt public structural relationship causal relationship observational alone causality identifiability framework question necessary discuss paper develop far approach estimator sparse dag sparsity avoid overfitte penalize nonnegative possibly additional penalty seek score posterior penalty recover differ aforementione two choice traditional approach optimization constrain include dag fix topological sort sort parent permutation neighbourhood determine project node establish one nonconvex minimization minimize loss interpret nonzero entry acyclic acyclic parametrization define parametrization program instance plug back nonconvex alternate parametrization unfortunately dag nonconvex behind allow exploit analytical gain shall gain indeed formal wish emphasize shall interpretation evident loss simply parameter ingredient penalty effect select bayesian whose coefficient rescale choice notable occur special aic complex traditionally computationally dag nonconvex lose attractive alternative introduce fundamental theory concave estimation three principle guide guarantee sparsity note guarantee parameter totally development theory perform detail penalty scad scad smooth mcp translate scad key mcp flat bias mcp mcp concavity mcp sequel mcp difference potential concave satisfy condition estimate motivate consistency strong require employ penalty assumption relax substantially generalize consistent estimation structure learn estimation support equivalence typically number general evaluate however attention satisfying justification truly parent similar screening commonly tend typically order thousand thousand happen expect parent practice constraint use penalty superior establish furthermore advantage allow speed confident admit justification novel insight simply guarantee mcp attain consistency structure high refer iii nonconvex careful discuss define minimizer furthermore complicated alone dag usual theory estimation identifiability true identifiable establish existence local true turn finitely minimizer dag equivalence class finite local minimizer candidate minimizer produce quantity distinguish properly control distinguish dag small minimizer local penalize minimizer hence dag empirical indeed representation another dag many remainder stay likelihood technical distinction begin state high scenario depend consider alone develop identifiability error development general gaussian easier single uniquely inverse equation simply matrix element maximize represent dag spirit maximizer maximizer reason refer say long curvature penalty tend maximizer converge additional include q local maximizer must careful imply consistent show right condition maximizer conclude local select dag grow super increase achieve theorem dags common topological sort choice parametrization dag topological dag isolate positive definite proof appendix theorems dag maximize amongst covariance theorem local maximizer pn na flat penalty ij act equally lie penalty define hope sufficiently continuity intuition may unique essentially answer dag approximate unlikely statement assumption impose combine parametrization give state theorem distinguish previous also denote maximizer give tt penalty function immediate adaptive result datum order identifiability observational notion oppose rule edge role penalty satisfy condition maximizer constrain penalize carefully show slope concavity concavity overcome local likelihood continue simultaneously guarantee parameter estimation vanish highlight trivially local satisfied penalty particular mcp interesting concern dominate penalize parameter true essential control growth grow dag long grow dominate estimate theorem remainder quantify penalty need maximum rate alone ij requirement require sufficient sufficiently hence course simultaneously mcp satisfy theorem factor parameter model condition extra redundant factor formula mcp may penalty mcp theory particular give scenario formal direction already remark equivalent dag reduce high course know advance optimal whole estimator advance show compatible consistency estimate edge dag penalty computation nonconvex guarantee minimizer guarantee minimizer remark estimator special concave penalization relaxation discrete light work compare good believe estimate permutation require careful type preliminary however remain work hybrid estimate initial direct partially search dag subgraph search undirected dag estimation restrict search idea constraint nonconvex perform minimization employ naive gradient employ cyclic checking properly perform enforce technical overview approach simple attention minimization holding hereafter repeat coordinate detail coordinate know minimization minimize consequence substantial check add estimate respect alternatively induce cycle neither edge induce simultaneously outline sequel function fact rescale cause computing hold ignore express implicitly overview must outline remainder implement concave coordinate tolerance block respect block minimize respect jk jk jk jk equation follow appropriately unit detail mcp show solution give threshold choice mcp q convert minimize hence mcp similarly ingredient reason choose mcp comparison function apply parameter strictly minimizer choice penalty concavity consider sequel regularization maximum estimate decrease scale guess nan minimizer model condition exist correctly work preliminary empirical provide section penalty discussion difficulty particular validation suboptimal avoid present several exploit greatly idea excellent implement use warm start active block parameter speed naive incur prohibitive bottleneck operation sum inner product hence compute cost operation million reasoning computation highlight require total cycle leverage become calculation number parent specify appropriate computing proceed calculating estimate justify know true practice specify sequel section update every active sequence concavity tolerance datum inner product I mm active final note estimate adjust order assess accuracy efficiency pc max hill equivalent standard greedy hill base pre mean intend exhaustive performance experimental thus consist method hc hybrid method brevity frequently pc method form regularization implementation mcp mcp gives offer gold dags dimension dag purpose hill slow move onto assessment advantage constraint discussion property outperform method fact accomplish search somewhat remarkable statistical pc hc package optimize implementation recently date publicly available perform ghz intel core processor gb ram mac os dag randomly accord os add independently equal array individual test test randomly generate sample structural choice sample dag generate dataset random dag instead exception hc regularization pc significance choose use start choice recommendation concern smaller long furthermore edge excess hour mcp concavity choose parameter represent fair also minimal estimate simply large algorithm iteration traditionally produce skeleton phase relation discuss case method dag ignore miss situation final orient skeleton undirecte ignore skeleton wrong direction distinction course regardless predict positive false positive estimate dag estimate log convert absolute far hamming distance perform dimensional concern accuracy metric imply false fdr edge sometimes recall estimate hc one pc use produce total average complexity processor time require define runtime detailed result choice order properly compare keep thing theoretical consistent choose accurate select seem artificial potential sensitivity nonetheless consistent compare discuss issue low mcp hc combination expect performance rely consistent setting test also increase mcp hc fp dag skeleton mcp hc skeleton fdr mcp hc pc tp skeleton fdr hc bad term structure far discovery simulated dag edge hc exhibit great hc middle pc edge represent hc consideration section difficulty exist overcome hc attribute overfitte though produce alone influence accuracy graph estimate algorithm evaluate estimate equivalence correctly may explain hc job oppose sparse constraint thus approximately mcp bic fact hc still perform well far bic estimate dag however hc omit term test fix mcp result test test show value test dimension increase remain across discovery comparable true positive indicate mcp mcp maintain discovery
different make plant refer every integral solution failure figure explore recover expect fractional relaxation popular objective relaxation thereby round step light relaxation recover regime distinguish plant cluster high enough quantify contrast sdp center plant sufficiently precise separation research direction come disjoint ball interest investigate certain point etc particularly interesting overlap accord truth observe median lp remain isotropic lp practical ground recovery phenomenon phenomenon context alignment angular refer relax relaxation another analysis median exact recovery guarantee relaxation powerful tool many domain ask various partitioning thank anonymous correction greatly improve work rw mathematical physics separation median lp property cluster rhs small ball around separation property see receive rhs enforce easy hold degree main unit respect neighborhood translation integral least proof proof show draw p x satisfy first restrict attain proof done use symmetric continuous thesis would respect lebesgue exist property random converge satisfy maximum attain step claim show number large enough center ball great zero probability nx jj hypothesis restrict alpha inside boundary demonstrate dc h circle dash circle intersect inside big position ball intersection let nothing rest partial respect z j absolute constant union probability sdp satisfy fix continuous center ball separate sdp integral intend go construct bad ball radius center away ball ball consider unit create copy pick center group center initially exist center center true example also initialization analyze cluster method choose center choose choose first ball event consider center lie assign center first ball get center datum center take newly form lie assignment center choose center therefore fail cluster disjoint cluster far fails assign initially copy assign center distinguish initialize incorrectly copy succeed configuration version prune instance initialization sample center center proportional remove center let say arrange apart group far away center fail recover cluster idea illustrate choose center prune unit ball distance center ball ball center select center select select center rest case normalization select exactly center arbitrarily introduction median primal yet play role variable amount pay thought amount pay dual median select element remove share remove contribute increase increase uniformly di si ji di di lp suggest primal point let rhs attain equal median argue set dual bx ax bm bx theorem assumption conjecture question relaxation focus median focus relaxation optimality tool relatively parameter free tailor distributional cluster center need relaxation integral relaxation tight separation lp separation yet psd center heuristic recover set cluster k factor recovery open suggest relaxation solve problem theoretical relax relaxation serve round serve true overall convex constraint convex computer science relaxation integer fractional relaxation underlie ground truth give model particular mathematical albeit approximate relaxation round relaxation round directly solve relaxed occurrence exact recovery phenomenon question motivate integral solution typical relaxation good believe examine yield strength phenomenon recovery understand say maximum minimum flow generally integer constraint totally vertex solution vertex relaxation study lp decode programming code recently signal community seminal paper compressive np case relaxation phenomena partition problem example include study partitioning clustering consider relaxation show recover optimal solution instance partition problem relaxation initial metric solve point disjoint commonly cluster point assign close alternatively distance cluster necessarily minimize partitioning point cluster read x mean lp approximation optimize problem objective via round effective although provable sdp relaxation mean previously relaxation geometric rounding appear recently lp median point admit distance also lp introduce ball position minimum center draw ball recover two point thresholding also contribute showing separation distance median mean objective standard programming relaxation integer relaxation programming relaxation relate relaxation relaxation intra separation center sufficiently separation decrease begin overlap transition relaxation begin fractional statement detail ball high recover relaxation separation relaxation recover separation variable point solution problem consist disjoint star graph show broad sense adjacency inverse vertice component theorem tight relaxation mean recover separation assumption theorem heuristic high arbitrarily separation fail high recovery section derive call separation see prove believe refined conjecture addition primal median clustering center appendix consist use scalar version construct sufficient relaxation probability satisfied separation mean lp exact relaxation study condition disadvantage heuristic good solution even heuristic combinatorial look relaxation heuristic solution property appeal iterative body stability tailor convex relaxation cluster ask used heuristic mean median heuristic even procedure like fail within regime relaxation guarantee probability recover correctly sbm communities community behavior enyi recover detection also know plant recently sharp threshold correctly moreover relaxation show exact threshold share fundamental objective point cloud obtain establish establish deterministic tie maximum graph moreover sbm graph random create distance technical difficulty study point though might comparable profile integer relaxation linear program indicate indicate whether unique motivating let ensure optimality integral existence feasible intend degenerate since zero observe solution indeed motivated observation enforce variable within easily identify median lp cluster optimal intend cluster lp eq point restrict rhs intend easy constraint trivially sufficient turn powerful separation possible exploit primal rhs get contribution distance way feasible rhs attain dp condition variable exist ball ball see point see statement assume remainder ask cluster ball neighborhood follow condition eq say satisfy provide big essentially say attain small center ball interval center interval q require recovery show median separate broad set cluster contain positive respect neighborhood center n independently median least find ball large hold drop expectation function attain close enough jk mean contrast relaxation median natural mean integral center exceed particular median lp natural lp use cluster see satisfy distance every cluster cluster solution two integral cluster complementary tell combine tight sense separation lp draw symmetric support sufficiently plant high plant imply plant point show p plant remain parallel feasible feasible solution know hold equivalently maximize maximize square unless trivial coincide ball since come symmetric relaxation semidefinite sdp relaxation cluster whose separated distance conjecture construct deterministic sdp plant condition high random matrix explain proof cluster total cluster point define matrix ease dual unit indicator index j ie dual otherwise cluster coordinate intend dual tell complementary tell diagonal shall switch length notation easier complementary specify dual intend entry ultimately pairwise know semi imply negativity essentially distance cluster finally average reasonable hold remain cluster greatly simplify z tx tx separation separation xx tx
rnns sequence recursive convolutional share whole see convolutional weight single mechanism learn sentence neural unit initialize project space nonlinearity activation think node recursion choice activation leave activation child convolution adaptively hard code network leave perspective sentence source actual sampling search corpus instance convolutional source sentence convolutional rnn rnn last sentence work encoder decoder source sentence length denote decoder generate sentence rank system approach provide phrase concentrate direct configuration rnn trivial determine target length unit encoder convolutional understand inductive encoder decoder measure encoder decoder english corpus combination corpus word respectively news news test set sentence parallel corpus english sentence word english rare word map token rnn decoder newly propose recursive decoder minibatch transition matrix spectral radius element wise neuron embedding case update update ht phrase phrase result encoder unit either step translation exclude select high scoring candidate reduce reach zero approximately rnn lstm search good use usual length prevent shorter e htp cm union phone number clarity china le et dans le une r la est de phone un une face est la position en la en sa en un de ce union et en une est union phone est de et une la l investigation complete end finding bank recommendation action reference la de du de la bank send ann conclusion send la bank te de ann e les pr send es le la direction des te ann pr send la bank des n des source question balance adequate within school reference ce des en comment le un acc dans une un de une cr questions pr se des dans les une de de fa il des pour acc des il la le acc de du une le still reference il vote il il les il il des source work reference es cr es cr ne es cr es cr source lot right il ce un large consensus il de le la il un consensus la et il pour se un en tr une pour instant translation specifically translation sentence look score translation change sentence sentence rapidly unknown challenge increase system although present along baseline phrase still translation word source significantly per model phrase recently improve translation handle obvious explanatory length sentence encode sequence neural phrase translation fig high sentence fact source reference respectively even train evaluate machine translation system test set choose one word list long short short word difference job especially short source sentence degradation machine ht le pr des est le des est le pr des le pr des des est pr des est le des est pr des est pr des pr du des generate additionally recursive learn united parsing encoder united correlate despite compare property automatically believe investigation paper translation purely focus evaluating encoder propose task sentence sentence translation possible encoder choose two differ newly english sentence sentence existence rare suffer significantly well future research direction machine translation purely find computation much source especially language deal secondly research prevent neural translation
box expand omit costly box datum output boundary box minimal cluster low boundary equation equation un normalize meaningful easily create performance experimentally area frequently receiver characteristic curve roc count positive form roc normalize divide cart forest hellinger distance tree among mining cart potentially use distance skew insensitive baseline rule induction box shape search iteratively maxima simultaneously list corner publicly breast datum uci repository obtain extraction evolutionary repository imbalance corner corner breast fast box algorithm imbalance rbf kernel width language separate fold fold test decision tree build prune research prediction control box expansion performance statistically significantly well use perform small choose good fast box often always bring several question question address question fast box perform scatter plot quality box imbalance ratio represents test horizontal negative fast intuition box expansion answer pose effect box example number might allow effect provide box around dataset datum restrict produce draw classifier generally seem cm datum good box box box perform cluster use fast box box interpretability set window process window window include fast box follow building window must form practitioner threshold box allow generalization box dimension count number place say place large place possible drawing classifier box box risk draw way namely boundary upper boundary box k matter divide tight box proper subset another set box draw classifier suffice draw classifier box experimental competitive challenge way one neighborhood warm close least dimension pass alternatively one approach cluster scan single pass keep center new designing setting box formulate mixed program act standard interpretable moderately sized box characterize discriminate naturally cluster failure benefit limitation hope insight box gold interpretable box interesting art cm p p cm cart boost rf box corner square corner breast breast mm vast majority world classification machine handle classifier benefit interpretable programming negative accuracy use specifically box separate method consider feature interest derive interpretable classification classification unbalanced domain range text prediction diagnosis trivial classifier use human expert form parallel call box drawing create classify box create box drawing exact mixed dataset gold substantial approximation performance approximation justify make box approach characterize class alone bring advantage fraction datum high imbalance create locally number involve computation though analytical may computation solution boundary discriminate local analytical calculation much simple decision choose namely classifier become imbalance approximate produce interpretable mixed advantage approach box draw interpretable generalization box drawing discuss approach make imbalance imbalance goal interpretable conjunction like introduce use liu et note experimentally work sensitive version decision sensitive seem interpretable rule patient rule induction partition like tend recover compose splitting criterion iteratively part describe patient slow neither box box though fast box approximation useful characterization approach start mixed programming act gold box box classifier positive correctly box majority classify give notation subsection notation definition parallel number index index boundary box box decision otherwise otherwise box classify correctly index example regularizer encourage box majority space box axis f negative box way gold compute say away resp upper boundary box definition rise box classify q give positive inequality formulation derive derive analogously make obtain degenerate boundary upper boundary number base computing environment full programming formulation size produce gold standard box drawing box determine drawing permit evaluate quality box operate much box use example cluster negative might cluster discriminate negative discrimination box around adjust locally power fast box seem box stage follow boundary set tight positive dividing stage partition negative boundary boundary expansion expand analytical stage cluster technique axis rectangle small parallel axis rectangle take minimum subscript part subscript subscript datum boundary th dimension th figure illustrate domain dash h cluster computation parallel discuss determine discriminate positive negative create exponential box
corpus alternatively set extract skip language literature max contexts context context share context entail impossible maximize switch underlie result embedding hold hundred thousand make tractable replace softmax elaborate direction present approach way derive word negative come corpus correspondingly probability come corpus observation q lead trivial mechanism prevent way random incorrect name negative stem become get almost al difference et corpus present construct specifically et time large equivalent draw normalization context text model relate fix learn representation context representation reduce regression model jointly make list software speak sentence context come around word window parameter size word window sample discard appear consider frequent define importantly word sub improve benchmark frequent another effectiveness window content away word make similarity really distributional clearly one
crowd worker ask look image example direct inefficient choose triplet collect grid format probe analogous triplet show grid image worker ask image grid allow collect probe yield triplet user change triplet amount effort crowd worker question grid collect triplet technique investigate effectiveness triplet acknowledge drawback triplet rely modification researcher multiply change crowdsource fundamental change lead embedding grid format collect several triplet triplet tight create reasonable low crowdsource budget collect trade user burden effectiveness triplet uniformly triplet quality decrease ingredient annotation triplet upon publication embedding useful search cluster find wish collect embedding author use triplet collect embedding work collect triplet collect triplet human triplet probe similarity use rather triplet pt collect grid strategy yield triplet good embed triplet collect separation half lie within area collect grid crowd active collect triplet triplet behind triplet triplet redundant crowd triplet embed space triplet rank object place geometric lie histogram occurrence answer triplet bottom histogram triplet individually triplet histogram answer triplet wide sampling triplet effect recognize grid triplet study human probe ask mark image human triplet per allow allow crowd worker load especially triplet crowd parallelism level human involve human right measure respect triplet take crowd worker pay complete author work quantify formalize hard answer author use acknowledge triplet histogram object answer object occur suggest certain recovered well random triplet roughly keep mind influence course collect triplet either one thick line batch color number triplet top appear triplet human triplet individually triplet error aim answer question run experiment show probe grid object choose probe baseline triplet collecting triplet embed embed embed effort worker task perfect proxy let validate approach conjunction paradigm handwritten digit contain digit generate comparison vector music similarity dataset point collect triplet present face dataset identity set extract triplet contain happen randomly triplet likely near triplet correct triplet uniformity create show selection small spread across music low synthetic select oppose image select close precise location compare neighbor worker perfect answer expect human reflect wide triplet publication validate human behave similarly proxy hour metric via synthetic embedding inconsistent triplet run image image contain roughly avoid shown allocate quantify publication collect triplet probe grid object probe varied three repetition experiment return grid triplet generalization triplet construct embed embed vary embed human grid spend near answer click bar percentile experiment triplet triplet view embed find choose grid cost triplet cost triplet cost triplet collect crowd collect cost result embed triplet uniformly embed unlikely much collect possible unique triplet generalize redundant evaluation reference experiment collect grid triplet collect time would outside budget strategy triplet grid sample unique triplet actual object give triplet grid task bold hour allow job generalize unseen constraint embedding low triplet human comparison grid triplet triplet come ahead view large grid even grid triplet sampling time separation size grid yield triplet answer triplet choose triplet triplet pick triplet ask gain outperform comprise answer collect fast complete large second vary fast question average grid per per able hour median even hour fast grid could hour trade important worker acceptable receive worker exploit hour complete worker pass give visible drawback take batch save recommendation collect researcher task identical trade issue researcher create quality create triplet researcher chance list collect consider grid yield triplet appropriate select yield kn work human continue investigate triplet adaptively grid converge random especially thank discussion support nsf fellowship award nsf google award similarity vision unfortunately embed collect task triplet technique effectiveness drawback triplet display collection task user explore collect analyze speed display create cost collect triplet
unity unity magnitude unity contribution decay x evident expansion constant depend eigenvalue unity circle imply number machine stack put number decay unity always strictly decay product exponentially mode mode exponential unless seek case decay act extreme mode whole numerically indistinguishable analyze broad range know perspective assign mode observe length report length derive magnitude understand cf eigenvalue amplitude contribution exercise ml stacking stack simple stack internal require pattern analytical statistic iid iid process generate string stack structure conversely correspond stacked us constraint stack thus symbol iid straightforward stack physical stacking recognize stack stack stack fashion must h namely respectively mix expand machine machine single expand directly expansion eqs state tm tm length et process much information especially apparent become sophisticated inverse upon expansion special subsection expression tm piece calculate effort find cyclic eq quick hold range become yield stacking absolutely stack stack sequence effectively assign coin tm repeat coin iid previous result repeat apply inverse tm identity matrix find eq final eq eqs show tm complex plane varied notice start cube root unity degenerate coin structure even allow transition underlie transition allow next ml ml probability see nothing completely expression previously however recursion relationship iid iid compute asymptotic although iid recently simultaneous c parameter hmm thorough process reader comprehensive convention adapt fig arc take transition state arc transition course illustration consider emission symbol uniquely terminology prefer represent nonetheless develop applicable hmms density mix number exist orientation panel al show list failure organization process failure result primarily sufficiently observe state give method applicable straightforward versus fig produce al appear iid fig clearly expect material generation specify challenging process use resolution transmission kind notation alternate block ab energy appear nine isolated instance sf stack external merging process process inspire machine recognize large process although state prevent occurrence inspire ask produce suggest show candidate must thorough dp reveal appropriate causal primarily give h et deviation domain self transition effect domain likewise increase domain prevent al interest maintain reasonably possibility predict sequence et propose al begin hmm machine fig case internal tm six state machine explicitly expand straightforward evolve blue root unity unity persistent four approach unity eigenvalue throughout eigenvalue upon eigenvalue tm involve root plot complex eigenvalue root unity transformation degenerate towards cube root unity eigenvalue tm solution part eigenvalue plot fig responsible decay constant decay quickly quickly slow cf indicate kind process fig implementation increase nontrivial cube root unity loop toward behavior suggest structure eigen structure apparent hmms calculation analytically mathematical assume hmm pdfs bring understanding material thought object importance contain representation directly demonstrate invert specify underlie hmm highly nontrivial considerable effort transform structural stand presentation wide stack possible stack contact previous applicability stacking rule position amenable restrictive sample stack method perhaps new hmms third dp machine hmms become describe structure present formalism framework identify mechanic information spirit theoretic sequel efficient analytical author thank external member laboratory office contract evolve vector stationary eigenvector normalize probability word convention denote object scalar concrete stack accord accomplish examine symbol symbol symbol symbol say symbol alphabet give internal tm gm h gm stationary probability probability long gm hmm fig circle arcs symbol upon probability machine spectral express stacking h represent stack term expand machine vary calculation physical h representation stack structure state ambiguity ml degeneracy representation size three transition distinguish among triplet label indicate transition label scheme store ml transition machine label stack transition take ml completely analogous advance gm distinguish fig state label choose give state satisfactory labeling scheme machine x h machine add three transition machine us self transition abc still induce transition advance stack transition apply transition able write stack gm expand alphabet six state b give tm completeness stack gm b state example connectivity presence self expand expansion yield sufficient component transition transition done determine calculate ssc define machine h previously ssc ssc call one ssc expand machine gm case ambiguity occur subscript symbol give ssc machine gm expand machine machine least ssc machine far distinction two effect calculate dp hide index integer p jx p transition abc language transition split label machine onto distinct indexing consistency machine machine transition map submatrix label mapping visually set statement submatrix machine state expansion gm express cyclic x cyclic among perform identity operation directly
unit suggest non trend mae rmse song year use mae quantify mae rmse song audio impact music aim evaluate collection consider single sequential complexity compression estimate aim descriptor string descriptor track audio descriptor descriptor similarity prediction song track popular web rating agreement perform control rating combine bag descriptor obtain performance gain song year descriptor scale benefit music content category concern quantifying receive field music digital web base music database novel music find retrieval manual annotation process latter infeasible amenable distinguish music audio identification track identify track similar collection track music similarity specificity similarity rating song particular temporal audio audio song descriptor temporal representation discard temporal sequential sequential complexity audio quantify string scalar summary statistic retain motivate involve rating song year descriptor audio relevant determining rating human rating year give chart assume chart entry song song determine thus might music recommendation song prediction might incorporated task year descriptor low specificity experimental account rating respectively finally conclusion fu track descriptor second knn estimate target li wavelet histogram knn classification spectral al construct decision tree classification et use purpose classification approach determine descriptor track pairwise track distance combination track centroid assume centroid thus close distribution contrast cross approximation centroid track pair apply identification histogram previously refer discard temporal yet stand contrast et representation mid feature widely version identification specificity approach involve intermediate aggregate locally estimate predict global window computing variance purpose local aggregation alternative al w alternative original result describing frequency whereas lee spectral al apply modelling feature generate compression lee semantic count likelihood recent attempt temporal representation bag feature al classifier classifier successive tag multiscale benefit evaluate aggregation base vary window size pyramid technique smooth structural change multiple modelling architecture aggregation tag resemble apply summary statistic since metric indexing retrieval computing resemble differ propose pairwise specificity prediction report shannon date music investigation audio sequence descriptor quantify compression require represent specify invariant measure sequence track audio compression constant original feature sequence frame compute due region fig obtain feature may encode efficiently conversely admit efficient invariant ordering observe apply specificity task consider dimensionality feature informative use alone task similarity rating distance descriptor track descriptor similarity predict rating pairwise distance descriptor vector denote th track available descriptor audio component descriptor track whose seek total similarity determine ps song year chart entry date track collection predict chart date descriptor specify method song year motivate multinomial straightforward determine coefficient use american popularity chart track annotate chart date chart entry extract audio version exception feature use th p feature component use select peak attack duration attack slope attack centroid moment spectral spread spectrum skewness skewness magnitude spectrum excess spectrum percentile magnitude percentile energy wiener magnitude magnitude spectrum amplitude successive component exclude coefficient spectral half wave centroid centroid peak predict centroid l low score track name stop I come time guess rise fall seven day I frank I country sign lee cat nothing gene track frame level descriptor descriptor compute describe vector principal component analysis fashion preliminary seek correlation coarse frequently choose string compression compression uncorrelated symbol string compression compression symbol observation uncorrelate distinct length unlikely frequently track alone average level obtain facilitate interpretation chart score report rank rank additionally low move piece contrast stand music strong power van track support exception track expectation specificity subsequently validity similarity capture beyond scope paper track analysis evaluate similarity annotation chart music pairwise similarity subject ask pairwise successive track point ordinal corresponding assume internal scale rating omit rating similarity five point similarity use absolute track rating rating quantify agreement addition music inherently similarity dependent internal widely verify quantify present select song apply chart restrict chart historical change production affect rating median rating per rating track display count rating less relative scale content might form recommendation track recommendation alone form recommendation interest track track evaluation five previously merge rating score discard similarity rating score perform evaluation result h count rating additional control subject subject assess training subject impose per collect rating rating subject control condition rating similarity coverage web rating control quantify control rating report five four agreement correlation sample sample correlation subsequently rate agreement coefficient rating aggregate analogously apply base condition balance classification interval predict rating distance descriptor vector use section additional baseline use account temporal audio feature sequence follow apply state space stack consecutive vector determine distance normalise compute distance similarity annotation square prediction quantify rating ordinal annotate rating term term pair term pair tie tie denominator geometric adjusted yield value difference pair versus accounting product moment coefficient separately unique rank tie rank tie contrast tie view compare ordinal proportion explain assign rank ba note contrast ba order ba rating annotation annotation annotation testing apply descriptor distance divide variance training across audio feature audio track thus distance vector descriptor among distance track combine obtain weight regression likelihood parameter norm penalty statistic rating accuracy hyper parameter rating incorporate p descriptor measure frame sequence error th c std combine descriptor five rating individual audio depict cross prediction yield similarly yield apply correction cross descriptor versus specific amongst observe descriptor feature describe estimate use ba consider performance rating outperform use combination employ alone incorporate descriptor gain ba respective four rating scale gain respective rating scale confusion annotate rating bootstrap turn post hoc reject hypothesis h c c c predict fig display feature descriptor perform rating magnitude across classifier one diverse multiple within song year rating use chart linear
direction many shall generate ad distribute sphere independent normal efficient generating direction moreover mention qualitative firstly large need geometry secondly trade conduct datum stem elliptical rather world need datum guide choose compare number calculation depth hand invariant separation ad set direction though direction produce depth precise keep increase calculation computation avoid direction calculation amount set generation univariate order univariate substantially see classifier stability classification phase direction heavy approximate depth minimize randomly direction depth direction direction much frequently computational classified depth calculate assign mahalanobi employ depth origin classify outside depth mahalanobi employ scatter within class alternatively mahalanobis depth mahalanobis depth avoid phase class center consider neighbor rule place classify hard validated computation sequel several well linear discriminant mahalanobis neighbor call spatial mahalanobi spatial use lda knn affine mahalanobis affine invariant appropriate version depth affine approximate td td zero class direction bad treatment quality depth treatment give separate hyperplane lda datum follow gaussian unimodal elliptical differ shift real approach assume different classified classifying class neighbor small see produce mahalanobis pool neighbor handle supplement rule restrict classify hyperplane correctly classify remain step determine parameter box constraint name whole panel together separate solid panel datum right quadratic symmetric k still kernel two class linearly separable hilbert correspond every determine margin figure solid plot discrimination small simplest separate select circle come decision exhibit right panel solid line indicate optimal left panel appear rule need besides select straightforward task say achieve tuning parameter computationally intensive determine correctly procedure take second several reach minute four classification phase point classified classify otherwise procedure calculation choose evaluate practical variety set methodology obtain partition diabete include ed take package constitute subsample diabetes multiclass split slightly process dropping object class description consider refer short description find page outlier see tie break treat knn simplify tie svm account tie approach three lda knn sect lda knn moment robust svm spatial equal depth classifier leave rate space mahalanobis depth direction cumulative task patient substantially attribute bold dominate patient classifier mahalanobis depth show table dominate also dominate diverse classifier classical different five aggregate measure task calculate lda knn mention negative value classifier relative small task value mention five table measure bold classifier none triplet satisfactory lda value visualize indicate well figure easily depth follow use estimate perform good classifier half depth part depth transformation moment mahalanobis similarly projection depth random outperform explain approximation direction treatment knn lda depth depth breast cancer vs vs cloud diabetes l r r treatment knn knn moment svm depth patient patient ed segmentation cancer vs vs r knn lda knn light nature treatment practical evidence number need see classified depth solution construct fraction rarely share amount base vanish version heuristic close separation come optimal separability separate rule propose cross svm separate really table frequency leave cross histogram bootstrap case two depth type plane find hyperplane small separation space separate cancer vs cloud diabetes heart patient vs patient segmentation vs vs procedure essentially feasible attribute datum transform distinguished depth vanish depth vanishing induce spurious symmetry non though produce region inefficient good shape exact version depth employ direction direction depth classify classify lie classified percentage substantially apply depth either near knn mahalanobis depth moment newly separate possible choice knn cover performance practically cross validation quick sort real lda knn none classifier depth dominate maximum moment mahalanobi outperform mahalanobis depth explain aggregate comparison greatly depth good mahalanobis depth vary goodness visualization experience problem tell stop span point separate investigate involve vanish hull application calculate expensive simple burden acknowledgement grateful master student university maintain package valuable comment anonymous greatly cm cm universit nonparametric fast procedure broad procedure first cube separate projective class alternative notion mahalanobis depth depth regard rate depth class dimension available alpha depth depth depth statistical generally class theoretical consideration procedure properly assumption real parametric classifying procedure due establish mainly evidence usually arise practical give field translate exist simulation really fitness demonstrate classification real procedure cube projective transformation reflect degree centrality carry depth function depth binary procedure depth separate linear plane contain origin apply alternatively study mahalanobis depth spatial depth everywhere vanish outside hull use depth finite outside hull practical question treat point represent portion classify depth supplementary classifier consider classifier compare mahalanobis spatial recall depth simulation study important broad experience usefulness far investigate space robustness apply substantial amount comparison indicator procedure number internet field evaluate standardized www package name three procedure classifier case include exclude neural network offer many architecture computationally expect datum adapt perform approach hand tuning set regard network classifier exclude svm sect describe classifier extended depth direction phase arise several classical mahalanobis knn introduce simplified svm sect task include sect extended depth property conclude problem phase transformation datum depth subsequent separation projective depth map coordinate transform reflect centrality subsequent ordering order depth depth definition current depth training first three version take value beyond hull vector variate observation r location scatter covariance denote univariate univariate minimal lie hyperplane distribution lie hyperplane obviously vanish hull mahalanobis projection illustrate mahalanobi
focus united plausible result test model disease context another internet trace evidence pose trace operational disease surveillance activity argue wikipedia internet source meet four challenge open reliable robust operational wikipedia thousand location need understand disease test disease model context suggest wikipedia access global monitoring outline plausible reliable sound operational disease surveillance forward scientific make reality mac discussion improve part gm technology biological number cb cb program program security department energy de public release file inter language article mapping input correlation file lead public health economic stability structure effort monitor risk focus monitor slow data media query effort promise challenge area disease forecast examine source linear language proxy systematic yet disease day test suggest location preliminary close overcome challenge monitor effective globally comprehensive impact united states surveillance health laboratory surveillance method internet yet scientific review disease forecast capability argue available aggregated online wikipedia statistical technique suggest datum forecasting model establish wikipedia outline reliable sound operational surveillance system gap internet technique disease extremely costly majority condition infection great united reduce economic effective surveillance detect quantify incidence save traditionally form laboratory follow report costly introduce lag observation surveillance upon internet medium datum mining technique health trace stream model truth health forecasting effective operational google trend challenge disease surveillance reliably integrate review improvement third broad applicability source generally available must data wikipedia knowledge disease surveillance resource usually simply incidence publish model flexibility test many context insufficient incidence train health contexts great new incidence dictionary census disease forecasting make complex limited context insufficient insufficient understanding biological internet stream forecast horizon evaluation approach yield fine one address approach wikipedia proxy hope available feasibility upon stream map daily access total context forecast successful context test day overcome challenge resource wikipedia keep date datum share adapt new context incidence article demonstrate several simple disease version suggest inter language article mapping readily translate forecasting short tight short argument first source wikipedia surveillance forecasting previously operational estimating disease incidence internet wikipedia access turn thorough set well challenge laboratory disease surveillance internet base disease surveillance traditional surveillance upon direct contact biological test rely surveillance datum include clinical call room example well surveillance report identifiable similarly resource surveillance essence department base department laboratory health exposure disease clinical surveillance disease example laboratory consist surveillance disease agent human mild severe clinical environmental public health school counter sale call early report alternative detection surveillance system notably visit publish wikipedia million language top website visit search engine roughly request engine wikipedia two key read change publish review publication vast article surprising seem manner abuse wikipedia effective deal wikipedia measurement popular dynamic wikipedia application order measure flow world popularity political economic attempt forecast sale stock application include forecast information prominent research assess wikipedia health e cancer drug information four health study wikipedia measure traffic relate wikipedia evaluate article disease relation news health issue find drug sale wikipedia traffic health article none article fourth recent united access broad use wikipedia interest wikipedia health purpose quantitative surveillance stage recently surveillance social stream large property complementary basic insight leave trace activity relate capture medium face web search health fact volume internet traditional surveillance effort exist internet metric exclude public evaluate party crowd source health disease surveillance root metric exclude metric counter drug proxy activity trace media message web server trace extract phrase metric occurrence create value train period internet true I future past case availability lag typically total vocabulary estimate metric accurate model produce correlation surveillance cite incidence variety stroke west simultaneous effort united wikipedia access study lasso year variation replicate article statistical key improvement disease proxy note briefly work detail reliable location forecasting briefly location result internet disease surveillance might systematic article article traffic traffic article proxy software open depend study statistical use available goal applicability surveillance operational public purpose surveillance base query google near google trends level internet surveillance well prior access query chinese engine google yahoo engine mostly english website payment index google trend google view level research scale effective model situation somewhat surveillance effort twitter certain outside company substantial share researcher media site consistent able find either extremely make wikipedia cite et highlight google algorithm little wide review improvement trend publish summary highlight well failure google resource trend else resource contexts google surveillance nearly effort small expand key propose base discover medical mention lda co occur list keyword health article build medical method discover coherent disease cancer topic correlate united drawback text require expert interpretation location knowledge measurement solely desire algorithm offer translate one context inter language link translation effort kind forecast even forecasting disease class forecast internet shift signal forecast lag al signal hand lag shift week forecast horizon indicator significantly xu et lag forecast et month query forecast appear potentially google trend forecast include sometimes disease important simple lag previously week daily independently day separate challenge surveillance challenge offer plausible wikipedia article disease incidence approximately location contexts acquisition processing use web datum variety include file contain hour time compressed request article request omit request differ human view automate request people read article factor commonly proxy analyze day data file hour miss gap hour treat minimal effect analysis normalize request yield article request hour fraction hour request language period request request file request retrieve daily daily china chinese united states united english china chinese total contexts list proxy resolution disease incidence goal evaluate broad disease across applicability mode transmission type similarly location develop test first reliable incidence specific disease frequently location disease health well health organization count information present wikipedia proxy certain language country english country united language need article disease enough evaluate generate traffic reasonable disease list disease incidence form file count infect b present latter plot translate diverse form format incidence map count set wikipedia scalar location article need concept english wikipedia link article select along link article biological article article low wikipedia article language article translate percent encode replace ne become language omit article merely point create article wikipedia http manually cause request target reliably map wikipedia leave traffic follow article goal selection article forecast disease incidence incidence sum day week month incidence disease frequency ignore wikipedia time temporal offset hour relatively scale ignore disease incidence series country article multiple article disease series incorporate multi article qualitative failure individual article supplementary repeat day increment forecast incidence day day count day match disease incidence day day statistical likely effective forecasting current day incidence incidence incidence day anti forecast anti model give still mechanism internet location predictive shift article yield version article yield article wikipedia correlation language two estimate model extend yield note evaluate model location test meta pearson score language compute disease language value opposite ignore article language sense favorable location apply illustrate e model fail subtle fail ratio snr wikipedia subtle exploration discover insufficient goodness complementary qualitative discuss fail evaluation failed forecasting omit forecasting brevity estimate axis traffic five wikipedia year period wikipedia series individually remain context file successful context evidence feasibility access success unite somewhat english proxy united high united couple former fail absence highlight noise source many noise article carry could carefully article rather china marginally successful successful capture baseline disease peak suggest peak model offset day dot anti forecast four successful context forecast anti forecasting context figure case significant forecast effective comprise forecast disease simply correctly vary reader interest due indirect news coverage find news coverage google trend failure cause medium disease short period day soon ill observe remove hypothesis forecasting
experience bad method par method rule full support indicate likely patient care service sign need maximum care top characterization allow quantify restrict spam breast nf svm cart rf display even severe restriction par substantially due benefit restrict careful c spam breast public reason rule intend decision nf rule select poor contain fact small restrict dataset analyze dataset p sec spam breast display run anneal step map interpretable potentially major benefit domain state national institute interpretable actually use well require medical trust decision help trust nsf classify ii kind list patient patient first traditional decision patient handle patient second receive decision decision maker disease first check serious disease etc paradigm naturally logic logic machine method produce design leave predictive directly align aim resolve problem clinical practice order rule type success list rule directly whereby classified second might patient heart disease high pressure high set stroke neither low stroke example decision construct part second result tumor tumor risk risk next margin age patient fit calibrate else age else age else risk else risk list serve dual purpose patient sort expensive sort naturally tree highest much currently use decision algorithmic manually assessment possibly score name practical decision course interpretability drive manual round gain popularity purely yield tree decision list collection logical popular inductive return example classified exhibit predictive inconsistent dedicated greedy decision monotonicity study lose enforce rather severe monotonicity often interpretable knowledge ic learn matter measure medical practice benefit risk list one care look top patient obey clause list start statistical build model help computation interpretable building discover modeling choose determine property monotonicity online goal binary represent risk rule h l mm elaborate desire user result b rule proportional rule user preference clause though letting independently distribute truncate permit monotonicity constraint diversity encourage large risk would space top list risk concentrate rule monte decision adopt l shorthand unnormalized equation note closely posterior optimize anneal namely objective subproblem find discrete space temperature simulate anneal discrete state search optimization order draw define neighbor new list length uniformly operation ex rule select uniformly draw cl rule remove mm optimize perform augmentation hasting describe variable schedule finally individual preserve enable experimental rule list medical practitioner place extremely restriction per predictive interpretability dependent substantial interpretability aim quantify performance specifically baseline publicly else else else risk else risk else else risk else risk goal predict day release binary outcome patient prior detailed aspect like status may require collect assess patient experiment choose condition list constant
meanwhile hessian expectation approximation sense reduction per measure euclidean formally strong gradient way motivate view descent negative usual one adapt descent information probability induce rise fisher fundamentally connect short quadratic euclidean make observe series divergence natural gradient space locally divergence kl divergence general argument p df obviously depend parameterization kl predictive parameterization smoothly psd metric riemannian distribution kl kl sense divergence objective discuss nice objective locally smoothly natural geodesic path riemannian towards jacobian gradient evaluate show matrix fisher give hessian even choice analytically expression neural hide unit sigmoid compute complex network hide make replace essentially gauss instead straightforward efficient one product linearize pass sufficiently finally multiply use pass fisher deep gauss newton matrix exactly least square precisely gauss gauss fisher hessian function equivalence hold fx matrix equal may actually natural important depend eqn normal normal familiar square familiar pay computation softmax layer part fed entropy softmax instead slightly close less computational linearize taylor correspond j z nice side make fisher generalize negative taking sometimes substitute standard within basic schedule rate choose heuristic nature guarantee ideally apply path distribution usually practice could negligible fundamental natural original argue taylor remaining word derivation natural appear along direction unit explanation might case natural optimally trade st vs divergence meanwhile change st approximation st st predict break fortunately equivalence know discussion serve reasonable proxy small tend curvature meanwhile update g equal natural factor important approximate accuracy approximation potential poor simply subtle gradient sensible conservative fisher develop discussion practical natural gradient eqn distribution use yield simple sometimes psd essentially already might way easy diagonal motivated free interestingly estimation despite various advantage theory exact use perhaps reason turn reasonable general section give expect due similarity eqn formula eqn turn useful approximation whereas moreover perform various concrete evidence fisher choice curvature convex rate comparable eq curvature experience learn recently form possibly modification maintain schedule rate also sophisticated combine diagonal quantity gauss newton correct fisher ultimately given accelerate improper serious due cg invariant overall compute automatically diagonal thank accurately diagonal gauss newton sgd sometimes incorrectly attribute actually eqn error network develop last couple addition diagonal base newton estimate history appear various work prove surprisingly old approach ng naturally parameter important affect characteristic eqn subtle gradient issue avoid phenomenon unlikely regret modify research eqn think conditioning curvature boundedness prevent optimizer severe quadratic k chapter add order stay radius zero approximation explanation single appropriate throughout course optimization local objective change adaptive adjustment one exponent cg justify curvature close around eqn important proving prove one scale step parameterized say whether parameterization smooth invertible examine parameterization closely elementary apply general kind rise invariance algorithm large step behave invariance parameterization direction note equal jacobian update curvature gradient w b g g inverting side give choice invertible analogous parameterization thus type curvature eqn case equivalent fisher fisher empirical except narrow case curvature hessian curvature sufficient j fx rearrange give relation hand situation occur affine practical step step update j smooth invariance automatically strong invariance whenever st words thus invariant affine newton method newton fail path fail order sufficiently invariance coincide quadratic sf direction towards riemannian optimal change taylor interpret square change add st giving interpretation expression large square negative interpretation improvement nd model scenario also argue computing tend report discuss aspect natural picture version appear offer insight contribution identification fisher gauss newton equivalence free actually natural method technique design break quadratic analyze parameterization step parameterization invariance characterization gradient possess feed forward circuit unit receive unit previous activation input denote output unit last call formally bias activity compute give monotonic document arbitrary call closely possible consist function disagreement guess use familiar encode predictive distribution could multinomial discrete gauss gauss newton
presence data occurrence contaminate proportion quickly large leading overfitte high contaminate contaminate give component covariance identifiability contaminate gaussian factor contaminate identifiability family identifiability establish package specie variable day upper height induce purpose respect evaluating cluster spurious refer bad approach model comprise bad rather illustration respect member real respect counterpart bivariate contaminate percent show contaminate analysis detect bad application factor consistent regardless perturbation section work facilitate contamination explore realize flexible paradigm contaminate elliptical automatic spurious herein start propose method reduction contaminate gaussian factor contaminate datum control latent outline variant expectation implementation illustration contaminate factor variate commonly focus elliptical widely theoretical problem tail distribution distribution elliptical represent contaminate component represent simple occurrence outlier point refer bad herein contaminate firstly ml expectation maximization stem contaminate scale illustrate proportion gaussian adopt contaminate factor elliptical error allow automatic bad contaminate mixture distribution improvement automatic bad contaminate observation parametrize parameter variant via g due singular estimate relative factor mixture contaminate factor contaminate sufficiently large relative sample cause potential estimate organize contaminate contaminate introduce mixture identifiability outline give graphic conclude robustness sake way w tail include location product focus mass contaminate contaminate typically good represent special bad via eq random cn expectation likelihood extension algorithm replace expectation maximization em ml characterization contaminate complete accordingly eq step calculation q note eq first calculation directly update perform choice algorithm maximize latter justify purpose could require proportion pre natural robust half good cm perform use package consider choice start constitute select different position strategy suggest contaminate form operational monotonicity observe great equal test assess contaminate acceleration use estimate estimate reach convergence whether acceleration give eq likelihood l l l cf converge k analysis relate financial index span daily scatter multivariate symmetry package reject commonly focus distribution contaminate gaussian statistically likelihood lr denote gaussian distribution nan bivariate result rejection graphical contour line ht ml arise explain variability variate random model variate pp analysis sensitive bad factor gaussian consider error factor analysis classical apply bad problem recall introduce contaminate contaminate contaminate w n cn term long contaminate factor value sake infinity satisfied ml estimate contaminate alternate maximization extension allow step partition cm step cycle step th contaminate accord cycle missing factor w n factor trace operator cycle q last row w ie k k
reflect meaningful occur disease nearby disease disease per similar majority reflect meaningful different significant algorithm robust choice estimate parameter also result scalable first patient efficiently vary b furthermore maintain characteristic challenge location optima guarantee scalability application hierarchy discovery patient patient nsf author nsf award author microsoft fellowship nsf award award recall categorical define notation dimensional rank therefore distance b b b second rank grouping divide recursive grouping sub sub keep track internal node neighbor merge firstly internal merged introduce path secondly merge recursive grouping merge method observable discover neighbor observation completely structure see internal sub note equal internal pair topological least node tree share define center pair path locally tree recursive grouping manner automatically surrogate structure relationship additive metric parent parent child child place single parent finally connect two internal local latent sub tree complete merging argue correctness exact work idea notion surrogate surrogate note know relate underlie latent tree show surrogate grouping procedure view consistent argue sub discover hide form maintain surrogate figure path path argue merge preserve path node merge structure correctness consistency careful hide latent tree identifiable satisfy variable neighbor node observation surrogate neighboring node surrogate along path surrogate connect observable latent surrogate correctness equivalence pair exist easy occur overlap path prove correctness latent subset small merge algorithm serial manner latent node visit induction iteration hide immediate consider implement visit contract surrogate neighborhood step merge know immediate therefore two triplet non node group three reference node parent group parent obtain alignment complete permutation merging merge align tree nod multiplication moment node zero hence parallelism method improve per tree local neighborhood process parallel size homogeneity edge tree triplet node product need triplet care triplet triplet merging consist latent lead worker parallelism require precise refer probability recovery tensor decomposition moment constant return satisfying eigenvalue moment eigenvector zero column drop svd scalar range one non entry entry probability find use computing embedding improve memory definition integrate tree follow divide learn model group iteratively operation span construction recursive grouping parameter decomposition guarantee correctly unknown low discrete implement parallel scale variable linearly variable experiment confirm health generate intuitive meaningful tree model popular hide variable markovian latent carry belief model computational expect hierarchical relationship tree object human estimation paper hierarchy disease health co disease patient patient disease identify latent consist tree exist hide observable long complexity scalable typically heuristic guarantee suffer optima easily integrate approach model simultaneously automatically learn method structure computational complexity via divide present divide applicable class discrete distribution mixture method moment tensor guarantee parallelism asynchronous aforementioned technical discover concept occurrence particular patient care task manual automate discovery clinical modal method neighborhood valuable correlation denote parent denote depicted variable variable neighborhood conditionally rest categorical use q j natural enforce parsimonious interact ia ik latent active carry triplet joint node active merge sub tree path final recovery depict divide start computation pairwise extent minimum span parallel done jointly divide merge group within group span operation algebraic alignment graphical finding underlie observed involve distance fit additive multivariate pairwise matrix dimension variable node multivariate heterogeneous latent expectation two distance along multivariate v cv cv compute svd moment distance span parallel carry independently group internal refer group sub hide grouping introduce use proceed proceed follow common expect distance l infer construct hide node multivariate discuss computation whiten huge computational latent structure sub tree iv iv accord equation node correction far learn tree neighborhood challenge combine globally consistent achieve span group possess local need align globally nature precise transition triplet find permutation transition guarantee final tree recall pair neighbor neighbor path moreover resolve neighboring node short union break node pseudo procedure reference list denote triplet parameter estimate triplet carry estimation triplet acquire issue triplet refer alignment issue two triplet triplet say node triplet group contain design triplet thus alignment correction challenging alignment solve become align recovered transition state node merge structure align tree degree parallelism identifiable tree dimension sample operate local recursive neighborhood neighborhood satisfy dl many natural tree bound hidden markov lead parallelism complexity parallelism e server red processor c couple multi capability version incorporate projection analysis discover occur patient record
future bid engine dependent behavior converge apply preliminary approach promise baseline introduce maximization call mechanism historical mechanism mathematically ad click ad engine receive bid th web user engine ad product quality bid ad common score compound predict probability ad example yahoo early ad place user pay j index utility pay I engine obtain pricing rule mechanism confusion engine direction bad symmetric nash equilibria public usually assume engine reality information ad ad period access rational response maximize reality diverse behavior highly capable place aforementioned recent year researcher try avoid assumption author relevance historical another kind statistical model learn training future either historical change framework work level optimize engine combine game call figure mathematically characterize rf g facilitate space base historical figure kind web click record historical click behavior period dependent historical bid mechanism predict future bid user click mechanism outer mechanism like emphasize fundamental characterize example cover optimal characterize equilibrium reasonable predict prediction period infinity way optimization detail introduction deal difficulty process change also propose find bid price hope rank high position receive click number click satisfactory per high bid indicate bid change depend bid bid denote bid finite bid search bid price time user issue ad click pay amount engine pricing begin period ad report change bid eq element change bid price I consider markov bid f b clear stream historical learn probability parametric switch bid eq learn bid price latter accurate former parametric paper behavior search engine compute mechanism bid discuss noticed bid change bid price period subsection genetic method optimize formally period stream engine probability proof prove stochastically positive u consider formulate f clear bid u f stream stochastically sample markov transition omit due restriction lemma theorem satisfy previous algorithm learn algorithm ease table query click parametric achieve give follow time achieve leverage learn predict period please initial bid profile sample piece ad mechanism click ad historical user sample bid accord learn period complex price formula employ method artificial intelligence handle linear introduce name improve bid mechanism may infinite mechanism predefine mechanism distance mechanism avoid greatly improve efficiency report behavior mechanism stream predefine predict stream use fitness genetic mechanism experimental generality quality task reduce impractical online response mechanism simulation widely generally collect click time click previous mechanism use mechanism remove bid click art engine keyword datum bid keyword ad mechanism day test mechanism different discount aggregate click behavior assume ad rank ad bid simulate bid three assume exactly basis take bid assume bid price strategy utility rarely assume bid uniformly multinomial fraction rest behave sbm implement baseline mechanism quality nash equilibria optimal mechanism learn historical gradient genetic individual mutation model set set show performance learn baseline mechanism axis avoid sensitive figure test stable performance well relative indicate approach theoretic furthermore well pass bad decrease statistically significant demonstrate impact effect experimental due response simply adopt classical mechanism deal optimize engine predict result effectiveness proposal plan consider factor plan comprehensive acknowledgment support edu cn microsoft com search
iteratively increasingly operator theoretical coarse grain hide coupling coarse grain spin couple system integrate physical minimize difference free physical system coarse grain preserve result map spin grain rbms rbms neuron couple visible describe restrict binary coupling leibl relative variational distribution hide like rbms less mapping thought compression expansion individual rbms stack output rbm next deep indeed rbms suggest implement extract feature organize begin variational context ise rbms deep stack rbms variational unsupervised dnn illustrate idea neighbor ise model discuss implication mapping physics block spin physical system coarse introduce describe block group term block create notice iteration physics one consider position spin spin configuration boltzmann hamiltonian partition function paper without generality typically hamiltonian spin q grain spin system end binary grain characteristic describe lattice spin picture figure two dimensional lattice block visible lattice interaction induce interaction statistical coarse grain grain grain interaction physics depend constructing depend encode interaction coarse grained coupling auxiliary visible grain entirely hamiltonian free coarse grain system usual ignore variational intuitively long physical invariant minimize grain notice transformation general move variational interpretation energy call restrict boltzmann rbms restrict rbms draw distribution binary image spin encodes ensemble handwritten digits mnist dataset rbms introduce visible unit interaction hidden parameter observe configuration visible visible reference rbm hamiltonian unit rbm unit rbm unsupervised learn rbm leibler furthermore visible datum minimize usually method rbm make dnn rbms stack layer rbm serve visible configuration visible via rbm treat activity visible ise ise tangent b transformation realize ise successive layer deep marked dot eventually attract stable point visible rbms analogous role energy object encode one scheme hamiltonian originally hamiltonian coarse grain freedom describe desire entirely language theory operator variational conditional exactly e hamiltonian language variational exactly approximation distribution work level energy literature usually make minimize divergence distinct gain detail examine carry numerically neighbor rbm dimensional describe along lattice lattice coupling neighbor lattice coupling perform calculation relationship flow coupling weak weak naturally layer dnn layer spin bottom dnn identical every spin spin hide hamiltonian coupling neighboring argument architecture implement interpret require calculation half visible couple deep neural sample critical visualization pixel depict material visualization effective field middle field move network consistent expect successive representative reconstruction numerically coarse ise lattice describe hamiltonian couple neighbor unlike ise occur set phase scale near grain mapping variational temperature periodic use rbm layer respectively see fig l layer rbm train divergence see method penalty serve encourage rbm prevent overfitte ensure interact rbm use explicitly spatial locality result dnn implement coarse scheme spin fig spin intuitive coarse critical hide fig fig reconstruction coarse grain dnn qualitatively reproduce despite compression deep successful recognition image raise question deep neural variational construct dnn examine ise self implement coarse procedure suggest implement scheme important physics quantum central dominate fix exhibit develop identify salient long interesting deep idea learn contrast often apply suggest develop idea entropy create amount open deep mapping space physical real problem give si material stack rbms variant com phase perform rbms divergence epoch momentum batch ise strength decay regularization ensure dnn reconstruction supplementary
maximize absence projection metric introduce empty link intersection common nothing inner adjacency complete arbitrary metric say geodesic subgraph single analogue one graph classical eigenvalue great one define maximize within observe link contain link sense informative cardinality principal analogously give q principal simple subgraph space principal principal ergodic sequence converge spherical principal direction informative symmetry whose fig link model project two component dot depend edge height axis sample project coincide proportion graph also year variability b variance close peak reflect tend together population depth random graph surely normality stationary adjacency graph minimize adjacency iff condition maximal iff subgraph version completely analogous proposition depth let stand start edge population therefore determine measure row eq unique finally initially provide elementary triangular arbitrary norm metric adjacency hold sake state distance distinct distance determinant odd negative eigenvalue apply setup matrix invertible determine characterization component stand adjacency one link family link link principal within otherwise objective reduce link find maximize ai link optimum proposition ergodic surely entail large enough surely entail principal eventually next analogous ergodic surely probability strong element strongly mix introduce prove instance strictly absolutely addition weakly observe center consequence mapping conclude es couple decade meanwhile focus statistical graph technique unsupervise principal well network last year behavior line stationary dynamic static growing threshold analysis characterize module develop particular technique community spectral among fit introduce dot mostly sequence parametric interesting discuss analyze network unique graph dominate label real connection financial market internet reason graph manuscript base nice measure depth analysis problem multivariate exploit determine question define several exhibit formulae calculate believe present extend important present graph link edge consider family describe consider diagonal sequence link connect path consider link transform inversion operator entry adjacency adjacency nothing correspond adjacency endowed study dynamic graph evolve discrete stand function link calculate connect graph belong expect median subset notion subset notion expect setup median usual expect network definition empirical precisely define network exist unique uniqueness central graph characterization maximum link call characterization central si j si sa la li subgraph empirical endowed graph law set element ergodic word central central homogeneity notion dispersion problem contain corresponding empirical scale finish present example enyi center empty graph complete graph intuitive maximum center introduce population version expect support notion maximize sample indeed easy hoeffding inequality enyi parameter thus arise normalize explicit present unique central easy empirical center coincide come section consist daily amplitude six location daily temperature range year month fig upper rank sensitivity extreme graph statistically correct comparison p share show construct finance graph month evolution year obtain http www central month central connect distant fig exhibit variability month temperature country correlate one analyze france united obtain depth graph robust know half depth mahalanobis different problem year metric particular define graph median precisely depth correspond population main maximize contrary require maximize fast monotonically month fig distance year exist four year graph exhibit b implication cm year month important definition measure distance setup allow statistical space write technique problem relationship first also
pattern dot percentage dot panel disjoint sub image plane panel st stand motivated introduce promise started adapt imaging modality ray would several article art attempt think framework extend work body electrical g department r department perform university security grant computer panel feature dual hierarchical use keyword patch frequency pattern keyword use sub permit discriminate suggest unsupervised topic useful wavelet transform tree topic year wavelet element wavelet powerful capture wavelet scale analyze extraction machine separate digital see image classifier seek provide north art five panel attribute di lack side combine dual wavelet patch supervise probabilistic meaningful style organize review analysis provide conclusion computer graphic color code color color often represent double double express coordinate wavelet wavelets resolution clustering transform provide characterize dependency tree complex wavelet transform decompose coefficient insensitive shift orientation local six basic orientation persistence wavelet intrinsic wavelet model mixture high smooth component successive scale narrow set divide patch patch image patch convert normalize domain complex j patch extract independently estimate iterative maximization method patch transition signature feature primitive building depend signature proportion proportion previous learn usage bag element word create text topic word weight bag proportion recognition five panel overlap image patch extraction independent style definition pt basic sub characterize keyword explain keyword style collection corpus kt generate keyword proceed patch assign quantization represent structure share dominant digit binary expansion pattern keyword style pattern sample collection collection dimensional dirichlet describe pattern proportion sub sample patch sub process choose pattern th pattern represent patch patch hide pattern assignment one need infer lda
crp respectively link ibp represent network call binary possesse attribute crp sf range logistic kernel rbf undirected meet use modelling phrase markov monte carlo inference crp number latent almost surely always store hyperparameter iteration latent variable case crp gibbs point difficulty add repeatedly sampling likelihood variable slice possible covariance covariance well assume extremely sensitive small batch fast due optima upon way interact extremely interact add batch rescale count count initialization step node interaction multiply constant factorization adjacency element either contain information modeling identify undirected edge miss network person know person could interact paper fine grain people people send message adjacency model way report log hold report test assign perfect ranking use assign independence sensible hold scheme hold representation consist count go contain count treat obviously scheme hold mean high p high probability node contain list enable extraction interaction employ publish reduce paper together count publish collect pair collect sample ensure hold interaction component crp gibbs crp node belong unseen monte integration corresponding entry gaussian dimensionality scheme hold might reason hold pair hold crp crp crp crp pair crp see crp model oppose gaussian towards closely little slightly different hold hold trace plot comprised noun word extract word identify interaction collect correlation crp hold hold negative significance hold computation seem favor gaussian model hold pair crp figure plot figure likelihood dataset hold type sample crp ht hold infer encodes pmf either chinese crp evaluate noun paper make main importantly computational continuous latent refinement enable streaming sampling scheme langevin gaussian descent might worse worth remove would put dimension dirichlet develop approach component extend phrase linear matrix also thank anonymous helpful comment concern modeling link usage indicator closely nod network model statistic exist chinese restaurant dimensionality apply social word solve observe input model predict interaction unobserve probability interact assign infer explicit describe weighted mean language word count count appear corpus unobserved word mean essential frequently bayesian mean meaning contribution since interaction expressive adjacency I contain node network interact prior representation specifically chinese crp finally refine exist crp gaussian latent noun extract corpus marginal fix put crp crp concentration parameter create unseen code class case multivariate variable covariance finally z matrix reason avoid slice put individual
operator explore execution solution specific type processor align memory boost weight input bit range rather approximation spirit weight element contrast approximation potentially could conjunction expensive operation relatively convolution multiplication field convolution layer importantly low accurately indicate parametrize exploit finally exploit tensor evaluation cnns character recognition develop provide evidence apply architecture way significantly approximation propagate great compression convolutional result addition technique weight tensor fully matrix representation permit storage describe construct good section tensor section describe convolutional maintain approximation efficient elementary linear assume direction improve keep efficient mahalanobi first propose seek coordinate whose less system let softmax image give know forward pass dirac center propagate mistake mahalanobis report use invert expensive use consider diagonal covariance approximate mahalanobis distance run tensor denote multiplication standard f convolutional efficiently approach need iterate linearly value ks diagonal singular orthogonal approximated keep ki I along convert compress even svd refer decomposition denote first svd alternatively approximate tensor decomposition operation square refer ht layer color onto color channel feature high convolution cluster sized cluster approximated tensor convolutional layer cnn dimensional particular project dimension filter combine find f xy xy basis vector constrain discuss u f approximation illustrate figure redundancy tensor approximate cluster cluster produce cluster original w contain easy gpu constrain count implement modify euclidean find sub either svd tensor could approximation gain cluster lc approximation h h many technique present degradation provide gain approximate tune train imagenet network convolutional cpu gpu k image present show gain overhead pass use imagenet number forward propagation spend convolutional layer supplementary layer approximation easily several cpu achieve cpu implement library intel mkl intel comparable matlab cpu speedup convolution baseline comparison code run find difficult gain base arithmetic material gain cnn different make specific however regardless implementation detail reduce often convolutional output channel approximate layer approximation span figure illustrate filter component point colored belong color show filter approximation project version st filter color channel performance begin point gain result cpu gpu corresponding colors cpu implementation relative baseline gpu color layer channel channel filter explore input outer gpu configurations cpu approximation gpu outer product drop gpu approximation procedure follow first layer tune convolutional weight tuning continue apply convolutional layer color decomposition pass keep convolutional layer manner overall great layer provide comprehensive storage central concern memory file neural product require majority describe use hyperparameter conv conv outer h nk nk bottleneck operation layer factor negligible reduce memory layer layer vast majority layer mobile convolutional technique evaluation quantization work hence use aid post projection learnable suggest generalization potential rank approximation approximate layer appear well table forward cnn architecture explore spend convolutional majority spend per fraction conv conv conv fc fc fc conv conv conv conv fc fc fc softmax theoretically achievable target
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rough laplacian see example embed cloud construct curvature serve background recall coarse space operator space operator intend laplace construct point serve give du du define curvature via laplace operator agree order classical du involve derivative agree diagonal analogy introduce curvature depend embed around operator life without take du thought laplacian eq parameter family du appropriately integrable function simplify eq fashion iterate du differ assume atomic call tt course du bilinear iterate e du scale notion coarse curvature empirical coarse scale application learn embed function volume metric induce adopt operator take ambient geometry priori knowledge geodesic approximation geodesic distance scale ambient limit tend sized geodesic analysis intrinsic geometry embed riemannian induce embed metric ambient geometry hypothesis curvature manifold unit volume suppose locally locally scale surely curvature simplify presentation state simple case relax distribution smooth everywhere space smooth closed theorem idea point recover sample force law choice datum size surely sized replace mention uniformly recover curvature smooth computation adapt q bilinear form rest l result leave give coarse curvature general measure converge curvature riemannian manifold curvature thought extension curvature smooth obtain empirical scale curvature assume fit recently fit sample another converse problem development point manifold even surface implicit surface use cauchy formula author like wu author express thank tool empirical process uniform law operator du iterate du operator iterate convergence large hoeffding lemma borel algebra borel involve trivial interaction hoeffding law uniform separable replace paper separable deal totally cover let totally particular finite class obtain choose hoeffding imply ambient lipschitz respect ambient function mean ambient almost weak function fix denote subsection function reduce number define ambient particular lipschitz fix class corollary fashion eq say relate ambient distance ambient smooth sufficiently smooth unique orthogonal projection embed function net ambient particular eq du sample inequality eq convenient normalization form q enough follow variable q use lead class positive notation function du letting class lemma bind simplify fix eq eq demonstrate decay rate quantity sure borel proof introduce measure embed suppose I let fix plugging expression thus borel give almost sure sure reason notation sequel function plug dominant exponential clear eq recall tf converge choose lemma eq introduce require almost lipschitz formula corollary corollary l eq increase view corollary corollary smooth space ambient uniformly take respect net grow bad polynomially q
label ref expert section generating compare bottom reduce good label ii perform top iii far risk notice decide good vote galaxy galaxy summary far describe galaxy perform ii pick label majority vote fitting compare minimize yield use use emphasize hence get model iii vertical line indicate solid bottom horizontal error classifier vertical attained dash accord respectively line indicate minimum attain solid estimate vertical indicate bottom bold number stand ccccc ii iii iii iii dealing build performance traditional sparsity avoid prediction error tune introduction surrogate variable big improvement compare perform former reason happen expectation sensible majority accurate provide reasonably many estimate lead well derive majority deal majority label derive latent perform procedure always use true majority base recover propose advantage latent naturally allow expert classify instance easily new expert tuning different achieve expense introduce introduce new logit dependency linear introduce useful help would show might use would find unknown loss function cost ann lee comment would member provide annotation partially support noted maximization first newton maximization rewrite regularize relate maximization measure performance first class finally assumption minimizer minimizer minimizing result find relate empirical arithmetic mean q iv cauchy inequality follow conclusion mean close minimizing vc v wish w qr v vc putting theorem get expensive assign highly expert undesirable situation train spam patient although expensive amazon sample unit classify many reasonably expert people medical problem desirable detect expert information adequate train crowdsourcing method predict though predict deal expert majority vote scheme label suboptimal trying find task train expert incorrect essentially algorithm amount method consist probabilistic unobserved usually root emphasis way usual observe label available literature term crowdsource tool try suffer crowdsource parameter sample increase substantially introduce parsimonious potentially bayesian shrinkage therefore hyperparameter valuable example new solution specify selection induce error label th unit feature attribute expert label contain summary minimize one mistake calculate error empirical difficulty closely depend generate score sample use validate compose value different contain vote input training obtain risk risk new misclassifie shorthand risk I give pick flip coin additional perform traditionally define em algorithm introduce improve prediction account role find introduce noisy optimum solve lead maximum posteriori correspond em iterate respect calculate plug solve regression map accord guarantee converge reason due identifiability consequently expert discuss em select agree next model tuning explore two completely response response calculate datum uci repository appropriate complement vote expert probabilitie misclassification misclassification follow misclassification vote describe experiment present expert response vote correct expert vote would probability correspond subset expert majority expert experiment compare em denote sparsity available comparison
car nine three learn part failure datum sensor datum service shape method follow sensor part associated actual part fail car go service fail car soon service record car occur service car use find number failure exception skip step rather failure form present assume network predict number failure first name scale shape learn failure due insufficient failure part fig approach part case learn result improvement accuracy inspire life similarly heart failure network present application cost part failure predict paper information like diagnostic historical future failure approach fusion diverse enhance estimate learn source rate predict future failure test failure service predict failure failure good scenario conclude failure sensor part failure plan enhance incorporate currently predict failure part next suggest optimize real world dataset com com multi attempt hard fail market lead company financial cost item failure failure failure play significant study learn rate parameter however available fused failure estimate failure cost estimate failure company service available fuse past part thereby improve data multiple problem occurrence service part failure rate sale change product conditional multiple source result estimate optimum claim inspire life give immediate next formal paper bayesian bayesian start learn later cost summarize brief discussion section overview predict code every service regular service routine failure observe service service part observe equip central offline build datum available order improve contrast past capture network dependency failure accuracy traditionally estimation component however divide sub level wise failure enhance occur take service occur failure observe service service failure part occurrence much early node node distribution define network dependency approach consider duration product index service source product cycle hour drive associated part fail part may part associate diagnostic associate record index fail first fail cycle fail contain cycle part contain contain cycle failure diagnostic service record interval present fail occur index part fail time contain cycle occur observe learn bayesian direct describe g encode variable rule decompose parent mcmc etc method e bayesian indicator part every dependency occur roughly failure actual failure follow goal learn combine theory present bayesian service record learn dependency cycle fail cycle variable capital small letter take parameter r kp take dependency mcmc hasting expect failure detail failure rate network follow shape failure part number failure car interval fail time affect failure analysis failure incorporate service bayesian network explain failure occurrence time observe time approach expect failure failure datum service service record failure three associate present failure analysis network variable predict failure failure record explain explain step explain capture dependency part bayesian consider failure part fail
gaussian problem link believe investigation ac uk cox covariate markov supervise case probability multiclass generalization supervise method classical nonparametric cut give supervise supervise despite simplicity specification still active explain part vast cox originally cox receive intelligence scalable perhaps covariate energy strong connection familiar boltzmann model prescribe field possible specify model avoid overfitte e mark cox spatial latent intensity model process relation near interesting often nonparametric intrinsic approximate close fairly parameter training validation could develop markov field already work build excellent contribution relate cox process supervise include time implement test mention bias logistic classification follow sentence fields cox relation perspective random min cut shall requirement field prescribe work classification despite discovery supervised method discuss additional connection commonly hard elaborate general link section propose semi min cut exploit supervise lie point mark pair constitute mark use parameterized intensity integrable bound borel set intensity borel borel borel cox random intensity point cox process process log cox exposition restriction paper measure surely condition valid commonly kernel zero square length cox show ht intensity cox square exponential unit describe cox density intuitive interpretation density kx x exist differ quantify occurrence quantify occurrence possibility elsewhere density product factorial moment lebesgue subject meet shall convenience density singular original although model model log cox mean superposition cox cox term superposition cox superposition cox cox population spatial whose speaking apply may superposition consider fix borel observe I observation product cox superposition density follow would limit construction equation proportional product product divide superposition interpretation independent contain point precisely borel partly result obtain cox example marginalization fulfil process knowledge affect desirable characteristic sense interference fulfil treat limit process standard interference meaningful supervise exist material new behaviour predict measure covariate distribution component softmax predictive datum new behaviour normalize function point give total wish produce similarity softmax choose distribution cross validation instance negative kernel kernel regime frequentist equally regime guarantee field briefly exposition augment generalization denote delta type fully model energy covariate superposition constant model differ process view simplify assumption ise ise model repeatedly apply graph min cut guarantee local optimum thesis energy posteriori cox semi problem condition q simply relevant know exclusive logical substitute verify condition problem subset theorem pairwise submodular solver generally fast solver former extra specific multiplying mean change increase likely category stationarity assumption category equation prior perform software matlab toolbox min paper comparison support machine svm harmonic supervise original demonstrate illustrative dataset generate two circle clearly centre perform use mean exponential see give sensible experiment min cut describe recover make class perform circle min semi six commonly diabetes come come brain interface single category movement either leave restrict mnist handwritten digit highly dataset number multiclass dimensional covariate diabetes aid paper literature randomly classifier length first randomly mnist set validation take intel ghz gb ram mnist use dataset classification variational inference logistic function optima random initialization take default software wide variety aware surprising significance result technical provide multiclass diabetes lc nn ff net rbf conv exclude diabetes free lag publish classical perhaps surprising training dominate term sum lie manifold relevance supervise covariate complex feed achieve neural achieve low require deal scope bias relax large leverage exist perform svm semi et al compare dataset synthetic et classification possible visualize geometric advantage semi supervise supervise give reasonable take digit set class call mnist first covariate consist measurement pass homogeneous flow two varied label randomized test existence trivial bipartite near euclidean harmonic length cross free validation harmonic ie length fold take second experiment counterpart supervise substantial majority perform similarly exception small double harmonic rate near separate surface show exploit advantage semi
last four fit gram fit indicate include aic law aic none differ demonstrate difference power law follow gram also lie ccccc gram gram gram gram gram gram gram gram suggest cutoff law assess superiority test suggest fit pl significance none power seem belong distribution dataset language fit well even mirror cutoff behavior raise question answer linguistic simply effect language whether answer make language profile repeat size profile figure suggest genetic profile sample compute sample curve except seem monotonically gram gram family trend result gram family unit classification power law law gram profile law follow power law cutoff gram find gram corpus internet specie law seem also claim law law word density normalize power law cx take member call point nlp cl predict small law language form family list language know language family tree classification indicate family language observe plot language family seem log deviation strict straight line goodness classification propose law language linguistic feature value response commonly package parameter package apart power law model author estimate significance superiority table recent apply linguistic color template meta series test show cutoff describe law ht name probability law pl exponential cutoff min testing law hypothesis law rank likelihood preference power significance absolute computed goodness low half world language investigate world language database word language word list least include experiment dataset represent language list name language family word include know mark word consist precede symbol combine click click stress gram profile family show quite language merge recall word list consecutive extract total gram profile gram profile family right gram power power fit gram gram gram gram value language type leave gram macro european na improve kind familiar word
section subsequently ability exploit improve domain four step represent hilbert reflect last thereby dynamic order handle real distribution work comprehensive algebra rkhs rkhs image I lose operation use term clear context take product z n empirical converge consist embedding kernel would require infinite method require embed evolve generalize situation output provide summary search lf f reproduce operator contain span operator product g g inner operator completeness regression order distribution variable solve follow functional st derivation consistent converge infinity step necessary component available simply learn result approximate combination observe distribution eq computed set mean sn potentially particular lie outside hull observe estimate guarantee lie available value form suitably original last rkhs map symbol compute pre act proxy inner rkhs embed drop replacement hand sample empirical similarly know consequently estimate fulfil small sample appear purpose margin one prefer weight variant construct sequence iterative greedy interpretation show embed drop depend concrete whether practice require pre second execute n sn sn size tn z treat desirable put emphasis achieve square problem belief early reliable contain ordinary square concrete expression remain structural modification curve learn sample body classical probabilistic technique aim form transition difference approach become particle learn learn mean covariance conditional infer nature subsequent minimal scenario car decade correspondence depict exist aim predict people situation trajectorie video possible separate probability individual report real dynamic highlight methodology additionally limit rkh embed gaussian train set interpret surprising combination allow show sample indicate autoregressive justified predict step distribution support location bottom illustrate nine new input mode predict quality indicate situation clear whether model gaussian call three step nine observe blue despite concentrated remove quantitative table correspond one leibler divergence besides last much observe distance set true suggest exactly residual compare pt baseline rkh see measure kl show segment real video sequence semantic source represent temporal interest video one long create segment sample detect interest video show per segment vary segment video vary segment movie measure actual segment table result choice z category baseline last segment merge segment global video close true video baseline tie sign multi correction except sequence vary application look drift step time choice tackle unlabeled target prediction set spam filter stop collect setup let sequence e kx joint learn adapt classifier look correctly available one map induce numerically surrogate therefore efficiently sample would goal classify right replace lead use weight ti ti cb play rest hinge correspond predict package demonstrate usefulness training classifier set consist car year car come source year decade give goal learn perform task order source source improve though affect visual
hypothesis say exactly nan hypothesis briefly suggest li specific explicit suggest minor rigorous detailed case augmentation modify high require argument order cx cx xx n k k event calculation joint partial exponential x cx xu I provide throughout integrate index density final approximation statistic modification statistic must implicit modify obtain modify integral subtract integral precede proof simplify appear complex index subscript dominate overall unable unable recently cite term four statistic ii iii statistic iv number repetition conservative derivation involve significance threshold decrease statistic hypothesis level application section become call question advantage hc hc hc hc hc hc diverse aspect mention approximation extreme attribute adapt original involve brownian empirical exhibit denote brownian bridge play derivation slowly enough rhs probability maxima long limit convenient independent b xt relationship wiener relationship define obtain classical slow apply seem reasonable correct modified inversion give suggest reasonably perform inversion frequently apply seem successfully four equivalent sense threshold datum shift nan mixture reduce suppose function value transformation notice directly transform nk globally argument concave besides conditionally nk k recursion level modify exactly term indice smaller high power poorly easy manuscript alternative numerical fast suitable hc hc hc hc hc hc hc hc hc discuss mixture offer estimating small magnitude suffice rarely side sided distribution statistic list threshold mixture parameter two sided repetition simulate number except hc statistic statistic modify power small significance level use statistic comparison hc recommend even consider give confidence choose suggest closely observe false discovery similar behave perhaps statistic subsection mixture version describe description b nan hypothesis omit suggest simulation compare bound calculate precede consider repeat configuration provide table low column true hc compare suggest far zhang method contain interpret contain li detect signal make method overall significance level test somewhat single expect frequency intermediate threshold give statistic correct exceed l approximation higher sufficiently practice comparison power statistic statistic goodness mix statistic mixing difference percent advantage significance level sensitive level intermediate power statistic definition capacity statistic value statistic relate goodness focused design detect excess small elegant large theory well receive focus center focus focus small cc cx give simulation behave third summation level appropriate threshold power interesting see whether context band suggest similar heuristic proof seem impose lower different somewhat statistic statistic remark mention modify slightly original modify although two simplify write proof interest exclude hypothesis similar proportional rejection unnecessary directly rejection region mean difficulty primary interest impose condition statistic unclear rejection correspond curves cx convex continuously increase n f nx xx desire converge go rhs expression event n c kf kf nc k n j n nc independence claim decrease enough suffice decrease inequality kn q uniformly achieve mf dy nf n dy dy b ki version rhs recall last k notation independent variable p r n rr mf n c first k k n n cx ny n nc rhs decay slowly n mf argument ny ny max c cx proposition upper nonnegative exist increase denote dy dy fm fm dy dy dy bm dy bm desire f ny k k om nc nc mf n om mf dy n dy dy proposition inequality nc n k ny k n remainder rhs equation rhs tend proposition high statistic n k binomial replace n thank associate reading suggestion improve style claim lemma e compare statistic approximations numerical show broad sample size higher detect sparse suggest confidence false whether nan high number consistency hypothesis statistic uniform hc modify term include recommend value reject global excess small value denote global hypothesis one false although confident study goodness function result adapt often cite poor value often alternative
face study control object time stimulus duration rest interval repetition meaningful face ds public format include voxel brain template without process head movement sensor etc study six subject include fmri brain area thresholde brain near roughly term fmri point voxel fmri acquisition smoothing order enhance voxel different spatial mm resolution mm voxel perform average version non version variant bss via ica section ica fmri realization generate specification describe realization additional verification slightly significant difference component dimension present ica blue represent eight true curve represent recover source figure ica spatially proper slice error visible ideal recover ideal leave corner external recover eight simulate fmri spatially brain slice leave box activation area pre external box illustrate reconstruct fmri versus ica complete ica component use reconstruction correspond sort component small rmse subsequently set start one increase add reconstruction plot reconstruction perfect ica fmri recover exactly number reconstruction versus exactly eight practically zero real fmri ds single employ full fmri voxel evolve brain raw background flat blue slice ds toolbox matlab illustrate ica course plot map experimental protocol employ dataset successfully recover external shaped see toolbox red actual human experimental employ recover run dataset converge fail component course evident third top match component leave match fmri noise ds ica component run respect reconstruction component illustrate rmse ica represent rmse final maximum low rmse slope connect signal eight match source component shaped course extremely experimental protocol brain construct brain dataset ds brain voxel slice inherent clear ica work satisfactory clear recover pre e real fmri never word perfect possible retrieve upper limit ds ds dataset smooth hence fine estimation realistic fmri consistency figure use intrinsic value dataset realization less signal original use sigmoid log clear selection linear fmri table estimation table mean range clear smoothed variant intrinsic span voxel furthermore expect consistent voxel enhance content activity area detail activation smoothing option carefully select specification sensitivity comment non smoothed variant space span voxel becomes apply plot use smoothed sm method reliable box reliable fmri mention focus level brain complex task minimum actual cpu entire perform fmri fmri dataset ds simple recognition expect distinct activate brain volume brain rather response inherently cpu core process simultaneously digital ica reconstruction plot human brain concern strict brain fmri retrieve marginal impact model short task correspond g correctly drive especially bss ica major combination parametric dimensionality recovery develop brain brain usa evaluation activity functional level voxel million several thousand art million parallelism human project purely fmri brain activity bss fmri try core cognitive task fmri smoothed variant real fmri complex brain task response theory equivalent brain cognitive structure require scale hence state feature real human brain assertion parallelism project acknowledgment functional analyze brain activity aspect human cognitive fmri human purely drive processing specifically fmri bss real fmri combination component brain process run visual although level process brain advanced efficient signal machine correspond total body yet neuron cm analyze especially cognitive relation simulate structure neuron digital infeasible functional fmri powerful analyze activity commonly bold contrast translate detect flow activate brain achieve exploit property order execute specialized brain properly detect activation constitute brain voxel constitute elementary spatial act resolution fmri voxel component temporal spatial potential intensive approach actual brain signal eeg cognitive task brain project brain effort project like million grid extremely power focus neural high cognitive practice hardware necessary fully simulate artificial equivalent turn machine still application artificial visual capability focus aspect human brain specifically cognitive task try cpu active brain volume perform fmri include fmri brain signal processing fmri study fmri define understand source proper reduction fmri briefly describe estimation fmri space briefly describe approach blind signal fmri study include experiment method dataset early fmri activation scheme paradigm subject specific color subject paradigm series input stimulus need primary area relevant task etc follow external setup previously activation essentially source fmri signal acquire fmri spatial temporal result voxel voxel x second time voxel brain snapshot practice actual brain brain area remain typical protocol involve subject exclude subject fmri create complex demand processing activate rich time low pass temporal fmri activate neuron flow series signal brain fmri spatially vary slightly subject traditional regression like model glm approximation since construct transformation shift derivative universal difficult locally due physical property various head fmri include task fmri isolate appear super spatial localization task relate fmri multiplication matrix spatial relate brain put brain fmri factorize map collect contain along spatial glm permutation shift regressor glm specific relate external instead component ica context blind bss glm ica dominate blind respectively fmri analysis volume voxel signal demand memory resource identification universal multiple experiment different subject multiply run case identify fmri activate specific property clear fmri identification lack specification background hence define standard glm make bss ica dictionary dl compressive cs increase interest ica drive fmri notably dl fmri analysis variation fmri deal complexity bss task reduce voxel consideration adjacent neuron correlate network scale spatio bold consider scan redundant voxel task accuracy identify inherent data fmri voxel sense consider however processing step conduct choose purely since currently study regard quality spatio statistical dependency essentially fmri dimensionality voxel retain voxel conduct information voxel inherent property retain external like recover bss signal formulate task assertion signal decay variance external reconstruction become separable bss recover e correspond fully reconstruct fmri brain activity drive fmri year quantitative randomness metric statistical various eeg etc financial time signal characterize texture analyze modality provide distinction e space algebraic apply dimension use evaluation useful regard redundancy dataset order intrinsic specifically quantitative linearity dimension mean dimension discriminative separately e mean parametric task analysis successfully previous prove valuable compare dataset extract feature qualitative clinical commonly calculate calculate closure cluster group various sample size approximate exponent way accommodate thousand however essentially thousand instead calculate grid equal sample cell occurrence count correlation ideally algorithm calculate intrinsic would allow totally uncorrelated discriminative specific set study apply set qualitative characteristic expert constructed employ sample available box order calculate slope plot sigmoid fitting parametric sigmoid identify axis identify scale specifically affect sigmoid linear central curvature fitness sigmoid assume uniform percentage bind low plot slope calculated reason factor fitness calculation parametric function axis range window term range completely triangular window rectangular fitness uniformly entire range triangular calculate fitness calculation window factor slope compute blind source bss fmri year consist different source activity bss temporal spatial exact bss ica identify gaussian source separate essentially reconstruct original signal combination discuss spatial brain activity corresponding time fmri ica dimension voxel produce either spatial conduct bss fmri spatial spatial accurate useful clinical fmri common ica ica widely fmri recent identify advantage brain additionally bss identify dl since ica dl assume specific source minimal ica fmri constraint maximum error ica error approach dl dictionary expect bss fmri fmri dataset ica intrinsic signal pre define fmri ica approximate quantitative track reconstruction change factorization verification estimate dataset differ track valuable tool analysis fmri datasets investigation fmri fmri fmri verify intrinsic activity task recognition fmri generator toolbox create fmri main source statistical underlie source component include super sub
expectation front logarithm may follow make may expectation entropy setting compute backward density emission distribution mixture maximization hmm multiple observation extension sequence consider two long sequence write simply dependent reasoning employ sum nd term way become reduction statistical arise hope new insight perhaps help discover hmms lagrange multiplier universal introduction basic hide stationary probability emission observation desirable produce maximization datum em find parameter eq rewrite term term
imagenet help improve cnn visualization generate varied synthetic cnn year feedforward networks cnns cnns investigate reason potential optimum large study generative modeling cnn define image category cnn final give image reference study differ non pre stochastic seek log approximate keep implicit image generative fundamentally discriminative share computational architecture usual drive discriminative criterion gradient require explain imagenet training help improve cnns generative explicit parametric white noise draw result distribution accomplish hamiltonian top deconvolution directly draw cnn extra meaningful varied synthetic deep imagenet energy product boltzmann algorithms deep relationship generative extensively cnns visualization image present deconvolution employ image define hmc category reference category score unknown yx w yx qx normalizing function flexible want reasonably easy generative underlie intercept train notational shall intercept already uniqueness mention set label maximize maximize log estimate class category perceptron top layer throughout accord expectation approximated importance set attempt treat weight gradient gradient yet difference provide drive easy adjust predict reproduce parametric generative especially pre current importance skewed effective updating lead toward importance skew start indicate discriminative generative appear expensive batch approximation specifically seek via calculation discriminative gradient induce specifically layer calculation replace sample form yx generative variable rule calculation exactly bring use f layer make ix yx imagenet care gaussian noise correspond hamiltonian specifically write physics context position function hamiltonian dynamic momentum denote physical hmc random evolve hamiltonian include computation deconvolution max un derivative visualization sequence three study discriminative gradient generative discriminative experiment identical experiment benchmark mnist handwritten digit imagenet natural study generative pre dataset utilize accuracy utilize deep distortion baseline set perform base weight max stage stop epoch discriminative start base rate table generative rate cm cm imagenet utilize testing train set stochastic descent batch decay momentum stage stop discriminative start base rate center corner training cm fast discriminative achieve mnist imagenet show pre discriminative improve update network accord toward generative gradient discriminative convolutional connect layer number par visualize train mnist visualize first visualize final drop avoid unnecessary visualization initialize hmc far visualize intermediate layer final fc visualize intermediate layer conv generative visualization
minimization least bellman direct function storage device possible keep due time final policy evaluate path discretize information record percent percent sample randomness state transition realization policy method implement policy use average percent optimal deviation budget simulate sequentially choose illustrate significantly direct percent least problem direct policy robust direct basis increase addition choose domain policy significant increase good parameter depict storage device price price storage htb reduce small post decision value three approximation three resource resource alone decision well overall although state advantageous simplify continuous problem available c discretization c number load wind wind ba ba full full instrumental dimension visualize figure policy price particular sample tend device high section promise scalable unknown htb resource level line variant bellman least bellman minimization bellman instrumental bellman bellman error instrumental establish bellman instrumental bellman method result strategy fundamentally approximate evaluate numerically control storage evaluate bellman instrumental appear basic optimal produce percent perform percent ideally suited call produce pure give relatively quadratic bit surprising exploration research direct challenge derivative limitation may need instrumental technique deal explanatory consistent estimate explanatory respectively probably widely technique since error easy least equation equation positively square inconsistent instrumental discuss instrumental follow noise ij unlike iv method equation e instrumental ij z indicate column il l il assumption trivially assumption across method instrumental variable estimator uniquely rank consistency instrumental give limit limit true covariance corollary definition recognize discrete curse know flat representation approximate learn hope problem function use allow approximate version algorithmic policy somewhat find success computer science community effort call state action pair value rigorous table representation scale problem science result establish architecture comparison perhaps approximate attract avoid action enjoy strong convergence although perfectly satisfied benchmark balance power load time purely suit realistic minor provide rigorous benchmark assessment produce algorithmic software benchmark http www edu focus architecture value receive attention literature steady powerful building temporal introduce instrumental variable overcome reveal modify bellman project architecture refer bellman chapter equivalent instrumental benchmark insight strategy choice importance example benefit storage wind policy gain assume load load price investigate discuss incorporate compressed air energy storage wind generation market wind storage horizon market address formulate market price wind study potential storage device thorough grow reader reference therein paper address close backward application create variation estimate bellman instrumental bellman paper hybrid approximate combine instrumental bellman instrumental bellman error hybrid instrumental bellman able time policy typically calculate basic instrumental variable search instrumental yet algorithmic fall direct policy outperform policy since bellman error overview base bellman minimization search gradient management stochastic explain performance dynamic policy investigate compare problem conclude rely bellman discount factor expectation change state transition throughout use convention index computing expect version bellman thorough discussion refer state pre randomness explain post bellman decision state inner maximization problem low dimension application variable multidimensional bellman field research widely column f post fix policy fix exist architecture weight post record determine decision record observation update algorithm post post decision cs get observation fix expect cs matrix bellman error minimize bellman bellman include bellman instrumental bellman discussion linear variant td least square temporal approximate tend td instrumental monte simulation cs estimation true value lie increase technical fulfil architecture approximate action geometric interpretation bellman equation least iteration q minimize bellman bellman span function bellman fix bellman bellman operator nice discussion address condition policy evaluation subproblem bellman function linear project norm state visit trajectory follow policy bias rarely visit another disadvantage policy important keep summarize algorithm bellman instrumental case subsection address bellman instrumental bellman subsection use section bellman bellman error instrumental alternative bellman find consider policy parameterize post function contrast value search vector policy solve follow policy challenging grow classic algorithm use sequentially policy simulate dimension optimize nonconvex easily computable derivative introduce fairly experiment observation policy treat objective combine quantifie much maximum get noisy value formally sigma updated condition normally value description implementation q maximize produce within converge asymptotically involve demand storage flow vector amount stand wind demand assume except refer storage grid htb demand demand wind device flow send storage store future storage wind upon capacity indicate capacity constant device lead maximum storage device must equation replace similarly ensure device allow demand implement storage fraction device full stationary capacity constraint note dimension dimension minus goal find policy ergodic horizon planning maximize accumulate discount absence load device price step uncertainty wind demand price subsection wind air square coefficient wind speed velocity wind square wind evolve ar model min wind suggest wind average price figure begin heavy tailed price adopt constant parameter deterministic periodic hour week month price model price jump jump interval addition jump size normal nonzero jump count time time jump occur divide jump jump direction magnitude jump consider jump parameter price outline demand demand highly upon figure peak pm greatly day week month forecast end incorporate customer make expert adopt indicate hour week month component hour week calculate load hour week load evolve degree load energy load htb programming variable post decision fraction storage device wind energy electrical grid td tp maximize discount policy may modify policy
recommend user compare one similarity dimension concept location social dimension sim dimension similarity dimension use least choose q idea rather situation experience curve evaluate user show memory mind document distance time memory environment weight user forget related situation reduce document risk explore situation exploitation optimal document multiply allow exploration situation metric fix sim case add compose preference sim situation current situation use algorithm recommend regard click recommend ap one day mobile priori horizontal axis month fa fa ts fa ts ts display click regard different fig fa ts effectively well term average click user improve explain consideration outperform document consider table significantly mean exp impact fa ts well precision without recommend propose document regard demonstrate significantly increase performance moreover conclusion recommendation recommendation follow researcher recently start user aware exploitation propose user bandit introduce name aware sampling fa ts document intensive evaluation result exploitation behaviour mobile make collection recommender identify recommend relate situation friend document recommend document recommend document short recommendation need balance actually great exp perform ts drawback strength amount experience recommend long document name aware thompson balance adaptively trade situation ts strategy explore situation paper organize follow related evaluation illustrate section conclude refer dimension dedicate armed bandit rs contextual recommendation document analyse ts contextual balance document recommendation author arm arm need continuously explore armed content author consider music recommendation long risk criterion take account additional standard deviation rs recommendation strategy recommender study recommendation aware thompson propose exploit consider curve recommendation situation home user office focus introduce enable human vector nc represent accord situation concept location user activity attribute preference click document failure spend document recommend situation correspond user recommender system propose user recommend exploitation orient important like use home situation add cv c cv cv cv document game choose reward thompson
detailed generalization perceptron online using provide theoretical accumulate loss major ndcg map perceptron different ranking measure ndcg map numerous seminal survey ndcg induce ndcg make subsequent write emphasize rank easy eq reader note receive learn whether induce induced loss eq also convex algorithm require r relevance document since predict rank score sort perceptron algorithm initialize predict receive def else provide measure perceptron euclidean unless state let control norm subgradient feature sec subgradient set main propose though perceptron receive exist meaningful relevance ndcg induce perceptron perceptron conclusion let linear parameterize document correctly margin holds accumulate ndcg instance list document definition max I max ndcg relevance batch set learn solving analyze learn minimize main generalization ranking take close look actually dimensional parameterization sec rank sort via scoring map parameterize form parameterization scoring actually map generalization bind class parameterization linear rank surrogate notation relevance surrogate generalization bind restriction sample sample mean duality w w b immediately w point w constant lipschitz continuity w comparable exist literature surrogate lipschitz r norm generalization inherently technique force thereby avoid price pay surrogate generalization surrogate rademacher surrogate general surrogate show family condition ndcg map family ndcg set though parameterization scoring correct parameterization invariance dimension property mean independent list formally score rank permutation invariance permutation permutation invariant dimension full parameterization permutation translate ac pp preserve create permutation column except column repeat w position hence column match column match first matrix rank check satisfy surrogate rank represent surrogate relevant calculation margin surrogate design relevance conjunction calculation show surrogate lipschitz lipschitz actually bound margin perceptron gradient gradient calculate make surrogate realization prediction surrogate theoretically relevance distinct space relevance space remove simple minimize design relevance gradient hence provide like algorithm guarantee induce measure ndcg map loss suitable provide surrogate analysis role generalization surrogate introduce modify perceptron cumulative ndcg loss generalization lipschitz surrogate third batch imply good possibly kernel tackle preliminary case scope subsequent acknowledge nsf pointing theorem unless understand cr formulate crcr g sort relevance relevance relevant list incur irrelevant document place document sort irrelevant document score among map irrelevant document highest irrelevant relevant map case need upper bind v r relevant document irrelevant place map bound r r likewise repeat logic equality calculation thus however j j ks iff document place take ir ir j l l jj I I li nd last thus document great minimum definition r x max st nd x tt tw z max tw proposition fix immediately tu originally take carefully derive definition minimizer eq theorem lipschitz rw rw er plug back get theorem thus relation x thus r constant eq index query relevance w sum pair query eq index index index correct query choose om surrogate score come normalize guarantee first truncate ndcg gr mr ii important property depend permutation thus document relevance document rank sort r gr ndcg positive weighted mean less relevance keep mind property come ndcg hence property directly note thm proposition ranking supervision relevance score contribution rank batch surrogate generalization independent object query propose rank surrogate surrogate obtain large margin surrogate structure cumulative ndcg induce also novel surrogate satisfy generalization supervise frequently rank number query relevance rank learn hope document respective relevance level rank list performance ndcg map performance measure convex reason exist loss optimize broadly three predict relevance document document take binary entire document associate take surrogate minimize major usually rank use exist base publicly moreover conjunction surrogate question algorithm provable guarantee remain large surrogate surrogate use surrogate supervise exist popular margin surrogate rank literature standard supervision supervision relevance map good lead define surrogate since ranking arbitrarily investigate develop classification large online perceptron surrogate special allow loss perceptron develop different perceptron algorithm extend set loss measure ndcg follow modify rank rank surrogate vector give ndcg unlike yield varied purpose sample unknown iw mr prove appendix definition understand similar induce relevant fail less relevant modification correspond thus loss document rather weight rank near must loss penalize much perfect penalize weight vector document entry weight thus even loss intuitive truly emphasize surrogate structure framework surrogate margin extension framework
memory requirement form prox multiple phase accelerate discuss due section projection multiplication step cost cost discuss role feature transform solver several mapping suggest literature sparse kernel memory treat box highly modular kernel sparse dense input kernel map vector appear solely block treat ij know scheme map split like additional entire transform generating construct independently monte result row primary map solver family shift mapping appropriately cosine entry transform group collect inside operation algebra store costly avoid implicit generator part need map fourier feature laplace mainly focus fouri experimental recognition digit comprise test comprise instance derive intensity classification comprise comprise report distribute environment core per cloud distribute resource hinge fourier see store dense run processor size strong notion parallelization come tradeoff time admit execution little roughly accommodate memory requirement long input model explicitly tb multiplication dominate run test solution communication cost gb plan ccccc avg percentage communication transform step step solver solver solver first solver solver first widely solver compute parallel gram incomplete cholesky use accelerate primal interior version binary create version divide class comparison core solve large speed make computationally though fast solver rank compute locally cccc testing time cccc time feature classification reduce pass dataset core solver hybrid parallel capability lose default running include transform solver attain accuracy comparison solver method solver memory demand contrast never form entire resolve scalability challenge conjunction optimization lead involve implicit dataset handle splitting approach performance term scalability implementation various modular stochastic update theorem lemma claim problem remark definition algorithm section google research research high optimization randomization propose kernel randomization variant multiplier carefully memory parallelism modern performance support enable loss regularization dense sparse library keyword method input training process loss truth convex prevent tradeoff control enable unseen test large big impose structural smoothness practitioner strong constraint theoretically big tends quickly consequence practitioner turn million estimate often carefully design loss regularizer trend recent success intersection numerical indeed scalable implementation play rapid massive well truly big effective constitute mathematically elegant dimensional linear parametric span series testing model central kernel define domain define turn procedure directly poorly train cubic parallelization poor pose barrier acquire scalable algorithmic environment algorithmic distribute describe estimate example unify approach block admm much design highly need proximal regularizer admm kernel environment well indicate scalable capable return library vector machine highly favorable acknowledge technical admm paper influential framework entirely empirical problem carefully necessary block splitting partition consumption extremely large example stress modify admm become quickly experimental scalable solver available maintain rest article organize follow various discuss transform widely machine learning speech brief brief reproduce hilbert equip act hypothesis stem expansion optimization solution plug solve linear learn rise model suitable kernel rich imply capability still price scalability dense incur randomize key algorithmic device dramatically train linearization linearization method ridge operation require dependence furthermore show distribute algebra efficient improve modern view attractive randomize ty l linear return regularizer reduce solving choice solve prove extend reach recognition state application number speech challenge optimization transform though original review direction multiplier admm informally take heuristic building big environment partition build model admm similar presence model variety admm operator admm split function involve tie admm rule gauss update cyclic coordinate augment iteration admm penalty cast admm eq add augment lagrangian n j proximity prox projection operator constraint regularizers efficient dual section solve row random towards setup distribute computing comprise ram distribute across node assumption scale semantic collect cluster aggregate distribute cluster memory simultaneously store disk restriction read block transform block produce I e generate process discard construction operator variant admm suit partition towards operator graph k computationally preferable linear set derive matrix partition partition r evenly pass interface I parallel computation relate set option multiple imbalance interpret admm semantic agree ij ij I j separable block j ml see rewritten view constrain turn average eliminate turn similarly also eliminate imply derive step unfortunately
substitute eq almost ignore handle conclusion permutation contain apply permutation vector bin store empty denote extra feature one hash nan assign value empty towards offset show assign red empty along offset bin circular bin empty bin offset value bin finally proper without offset would multiplication empty bin value ensure simultaneous empty match new bin number rotation ensure empty consideration newly near non empty bin circular final equal irrespective prove fact lsh indexing sublinear search generating processing traditional hashing hash testing hashing require exist simultaneously bin eq event theorem eq event interested compute convenience expansion linearity take term expectation remove dependency box bin bin simultaneous picked simultaneously occur bin occurring randomness bin simultaneously actual simultaneously bin empty bin simultaneously empty bin locate space simultaneously empty simultaneously empty bin pick perfectly empty bin equally e pick randomness selection directly close argument change right randomness leave provably improve procedure simultaneously close circular go add randomness bit circular empty new empty figure I bernoulli associate check circular empty use circular improve bin use non empty bin circular circular bit offset empty bin move bin offset final empty circular leave circular bin empty continue bin offset empty bin bin bin go circular empty bin value remain bernoulli hash bin complexity hash storage bit hundred thousand practically difficult satisfie lsh q unbiased simultaneously empty square mse respect hash variance estimator unbiased plot summarize theoretical improve well variance scheme experiment two scheme lsh neighbor publicly train query parameterize lsh generate meta function different realization hash parameterize lsh hash table store hash report union choose base recommendation show result recall please lsh point near neighbor standard retrieve point since result run retrieve summarize clear around improved need point query improve provide balance retrieve achieve point well clearly superiority indexing improve hash indexing retrieve lsh directly hash number point moment reasonable estimator unbiased regard three rhs behave fourth necessary empty bin dominate happen practice hashing reveal sub optimality add randomness provably especially dataset come evaluation hope improve scheme ph partially li fa nsf configuration empty empty ball exactly non bin empty bin likely involve combinatorial argument configuration simultaneously way term empty note nm n random empty remain bin likely randomly bin bin empty empty bin empty bin I replicate correspond nearest bin circular come eq desire simultaneous two case close simultaneous bin towards circular else empty
foundation integral domain ready exist continuous tell term conclude continuous finally martingale martingale since since conclude ni f prove complete proof carlo sample exponential insight sampling exploit convexity monte independent identically iid reduce q reduce monte difficult use priori manually finding improve simultaneously importance generate form past adaptive determine choose fully instance distribution define density serve finally exponential per sampling variance sampling speak efficiently find applicable evaluating insight estimate eq furthermore per family perform adaptive importance sampling suffer become minima importance establish important convenient variance infinite importance function old log side pass strong convexity soon differentiable integral interior x dx take euclidean onto appropriately sequence intuition towards step reduce call x nf tx nf mention unbiased adaptive importance third alternatively view second estimate course operation easily evaluate family result mathematically proof compact finite conditional sequence n nf x however lead geometry eq q indicator otherwise see compute choose bivariate variable speak density inner parameter word update ns nm nx improvement distribution initial ever sample gradually match price arithmetic option asset asset asset discount compute choose contain normal shift run end asset price importance fact prove count adaptive importance stochastic setup make sample theoretical variance surprising beyond convergence previous subproblem update especially case solution exploit establish subproblem storage requirement represent subproblem grow size could separate point generalization omit sake density lebesgue exposition adaptively divergence estimator special enyi divergence method enyi
transform manifold autoencoder mnist digit pixel dataset leave bottom interpolation line stack autoencoder image back text look suggest manifold near volume representation close easier separate possible manifold flat manifold estimate low transform moving correspond hide signal change unlikely configuration picture basically amount distinguish probability manifold answer question ht concentrate representation transform datum factorize space capture space elsewhere parameter model put lot mass outside get parametric density see put probability mass hence encoder one element case direct q call decoder decoder capture conditional distribution put probability role training sure preserve role training sure criterion achieve unless enough condition unnormalize start multiply first condition term vanish consider conditional maximum use contain satisfied mean decoder prof capacity increase become factorize net capacity neural may desire make unimodal strongly keep least fit estimate recover normalized option associate could importance knowledge basically add kind encoder optimize learn maximum proxy example dataset probability simple counting maximize neural factorize regular input challenge deal encoder optimize want output distribution want keep reconstruct although gradient direction optimize encoder cost similar eq encoder linearity apply discretize interested pseudo gradient back propagate straight pseudo gradient idea bind factorize binomial value ht encode feedforward decoder loss factorize decoder direction decoder compute pseudo back propagate pseudo inside encoder encoder transformation autoencoder factorial experimentally without anneal greedy pre previously network stack rbms deep autoencoder stack autoencoder consider function term zero loss autoencoder considerably trade sometimes descent perfectly prior map point use tradeoff schedule rapid growth forget schedule thus reconstruct usefulness trick also value tradeoff parameter difficulty stage weight fitting stage unity loss must perfect information otherwise never recover decoder special autoencoder version reweighte encoding give training fact two log attempt maximize encoder I encouraging much close encouraging contract noisy equally factorize provide evidence feasibility technique use mnist handwritten digits mnist validation split compose consider code mnist factorize binomial minibatch minibatch cost increase momentum hide layer sigmoid output input sample bit select probability change randomly autoencoder encoder decoder bias treat estimate unnormalize perfectly practice find function allow unnormalized distribution sample partition estimate importance took expect centroid layer train give reconstruction deep deep factorize qualitatively mixture incoherent mnist decoder unit necessary encoder match input binomial digit due autoencoder reconstruction digit factorize entropy factorize necessary encode hard factorize thus dimension characterize align practically happen constant sign weight make table contain bit flip column table sufficient entropy low autoencoder prior factorize encoder dimension dimension measure output rd column table remove factorize avg output datum avoid perfectly likelihood world go small ball
encourage causal already capability distinguish capability looking consider first preliminary result start detect improve like cause markov search pair interaction descriptor distribution theoretic distinction within letter markov distinction make mixture first cause respectively cause second third moment mixture identical third instance associate population I cause moment descriptor replace descriptor obtain population I k encourage though able quantity informative statistic g insight explicit rely make consideration light dependency secondly expect second layer eventually descriptor configuration c link create relationship member link use markov evident cause expect rank second previous asymmetric descriptor quantile approach associate rank strongly rank term member position associate absence population descriptor introduce j j I k j ik create descriptor distribution population denote return observational descriptor paradigm induce quantile term effect note would informative term cause major c improve appropriate selector like difference leave estimation easily approach approach consume dependent suppose perform user perspective node existence link step markov information filter mutual may take consideration considerably approach package base l penalization greedy hill potential parent restrict constraint hc hill incremental mb min hill hybrid dag pc use configuration reason si discovery score structure experimental synthetic setting dag example pair direct descriptor return denote link result training forest preliminary perform compare art package grow incremental mb constraint learn si learning algorithms hc hill base structure hill version training compare figure medium sigmoid series consideration make experimental variate obtain several improvement art move accuracy improve increase competitive package c assessment simulate datum use causal portion portion section second include portion goal never encounter training implement unlike return rank gs grow pruning phase phase technique return rank area curve assess different comparison train synthetic gs c c gs number also take availability causal interest table filter least inaccurate algorithm result return algorithm outperform respect drive belong think causality leave stochastic retrieve causality observational challenge preliminary confirm existence causality link descriptor research degree causal relationship indistinguishable configuration improve address assess classification extend exploit relation extend dataset bioinformatics de email ac statistical causality lie heart recent show infer indistinguishable thank propose machine infer causal link rely relation supervise successfully extract descriptor variate lie statement dependency causality many formal approach causality justify detect infer causal observational influential rely independence detect causal ic accurate reconstruct pattern restrict configuration causal define independence triplet conditional unconditional sound slow development indistinguishable pattern opinion mean aspect notably conditional unconditional independence distinguish configuration result prevent indistinguishable configuration evident appearance tackle cause pair additive information geometry common feature causal reduce uncertainty direction recent organization effect pair pair idea success random indistinguishable rank competition common bivariate causal supervise feature describe dependency link pair variable encourage cause another need multivariate rapidly information existence causal variable return dependency remain one evident dependency dependency lead learn link rely relation member markov create relationship asymmetric descriptor classifier link assess competitive effect hundred causal engineering physics control artificial cause effect mixed produce outcome challenge took rank eight rely transform classification stand link direct input bivariate particular association residual nu nu auc confirm relate copula redundancy filter random forest regressor posteriori improve final four subproblem invert inverting accordingly present approach exist variate direct estimate multivariate partial arise causal configuration value difficult distribution notion parametric aspect take since want cause quantitative distinguish characteristic causal relationship expect asymmetric quantify set asymmetric continuous define pearson coefficient nan conditionally independent structural term mb belong operator independent theoretic say satisfy effective propose selection algorithm notion relevant causality science notion life remarkable dependent variable density conditional theoretic dependency dependent
position nucleotide x example g number give unique equal small rewrite reduce eq item unique example nucleotide position nucleotide force add significant reduction apply single variance sequence library library language sufficient answer library numerical many handle well program link library package numerical calculation furthermore symbolic calculation make web server user library nucleotide mixture separate handle exclude although include zero automatically library user ratio format artificial example nucleotide randomize nucleotide position library set library effect nucleotide library average standard library figure nucleotide ratio mixture impact library ratio library member librarie library ratio include library library mixture ratio mixture need library unique skewed ability one accurately complexity nucleotide mention nucleotide thing degeneracy standard deviation behave differently sharp ratio deviation broad multimodal peak shift library peak deviation distinct distinct peak peak right peak number increase sequence peak equal peak deviation support school author thank protein engineering statistic library guide library unique library handle equal mutation site formula calculate unique library mixture nucleotide computer utilize library statistic library nucleotide effect skewed large library expect unique library protein biological property protocol library gene incorporation degenerate synthetic dna sequence usually equal mixture create region growth protein library protein stand equal mix mixture mix library ask formula library variance within formula calculation library formula huge way calculation keep big library library usually huge library nucleotide basis sequence possible associate either respective give
g patch shift skewed pooling require stay invariance nature transformation identity change exploit target label subspace complex cell assumption possible approach way autoencoder natural autoencoder parameterize decode function input producing training minimize manner autoencoder autoencoder reconstruct corrupted denoise autoencoder implicitly corruption denoise provide probabilistic representation connection autoencoder feedforward option autoencoder deep architecture stack train greedy decode simultaneously intermediate depict encoding start decode layer autoencoder odd discard goal autoencoder higher retain abstract decode recover discard three autoencoder basic denoise autoencoder variant connection definition one multiple autoencoder identity meaningful denoise element hadamard learnable bias sigmoid ensure stay bound stay element decode motivated denoise source autoencoder learn good connect mapping inside sigmoid motivated encoding decode connection abstract connection drop redundant three connect input additional small compare million batch low denoise implicit probabilistic fair layer model size model scale datum find autoencoder notice weight weight begin denoise improve affect design tie tie denoise feature easy visualize exist computer vision refer limit million mini mini batch rate adapt million update analyze representation material division try stack fine tuning phase update equal million use global stack beneficial layer reconstruction time initialization sample standard low autoencoder dash two layer ratio since autoencoder information ratio far lower effective low beneficial model mod connection benefit well cifar benefit add connection significantly mod tb represent denoise color mod scale horizontal significance negative blue side mod practically neuron studying find typically several qualitatively layer neuron leave kind example neuron selective three selective orientation selective orientation orientation detail supplementary material tb c depict column follow layer neuron identify well view procedure feature depict figure show invariance autoencoder increase towards layer multiplicative connection encoder decoder discard level well able manner direction discrimination color orientation summary early autoencoder layer ability autoencoder therefore combine autoencoder operation way explicit much deep dataset std cifar mod add cifar mod na mod continuously image put aside generalization last allow million overfitte problem preprocesse apply cifar dimensionality reduce match dimensionality reduction retain input use adapt normal activation bias nonlinearity center mapping way turn proportion understand well go question form frequency invariant interesting form pooling neuron look connection connection figure order invariance leave color significance variance generate neuron initially try weight neuron connection remove take neuron neuron receive depend neuron small proportion variance output income output name significance significance depict coordinate turn neuron strong negative weight color connection significance invariant tend since nonlinearity unit negative tie learn add concave like generating detector form convex concave weight truth invariance rotation invariance sample image image invariance figure translation figure impact neuron stay even l strength significance neuron good relu function relu activation b c w b operation tb significance cifar red phase link identify pool layer belong visualize layer neuron link neuron mark group color phase perform encoder allow layer autoencoder regular autoencoder connection encoder decoder pressure abstract translate reconstruction allow strength connection structure connection use world verify representation whose invariance fast layer formation denoise autoencoder tr error autoencoder build corrupted decoder map back autoencoder need store recently become dominant large available difficulty autoencoder learn autoencoder try retain supervised image activation away detail perspective clear unsupervised must semi supervise variational autoencoder raise autoencoder connection level focus abstract efficiently result back store close select relevant investigate connection extend early comparison autoencoder network denoise way represent change balance bottom heavy invariance level regular invariant feature detail connection guide pool qualitatively selective aspect input size right layer ratio typically irrelevant recognition source orientation recognize
denominator theorem differentiable global give side condition believe thing denominator show replace q since function proximal optimality schwarz paper continuous close domain nr e iteration due eq strong incorporate recursively expectation proof fix compare prox mini eq notation need second value n h figure close figure close proposition question united pa university united pa mini scheme improve composite number nonsmooth computation gradient objective start stochastic step repeat last iterate becoming start gradient show predefined mini implementation acceleration parallelization interest average close continuous gradient subdifferential parameter equal activity solve problem many fista impractical process coordinate stochastic paper technique reduction particular mini batch variant gd motivate typical stochastic sgd limitation inherently sequential parallelism combine analyze ms gd proximal mini enjoy apart parallel hence speedup attain specify formalize predict batch employ gd upper bind stepsize equivalently proximal gd prox old reference past unbiased eq gd prox point reference outer ensure ultimately extremely gd max stochastic per epoch minibatch ix analyse case get accuracy guarantee decrease epoch define stepsize inner loop computational minimizing reach gradient decrease attain fix target evaluate give present parallelism http www tw dataset compare ms gd circle mini parallelism green dash parallelism divide stepsize star show formalize threshold straight ideal speedup ms gd lipschitz euclidean define modification lipschitz strongly standard norm define convex collection n nice suppose k strong cauchy schwarz define monotonicity prove obtain subgradient write change apply
iteration exist sis marginal statistic sis iterate iterative applie broadly screen fashion save decrease obtain recommend decay schedule algorithm design attractive update time keep drop way computational load large scale constraint accord analogue screening follow fine situation conservative screening stage role reduce fine prohibitive contain interpretability consistency dimension concentrate problem counterpart attempt insight principal component column wise write without solve q dimension input take cf sparse formulation coincide reduce pca greatly inner pca simply r get sparse pca employ loading multi sparse pca spirit procedure concern pca body share similarity subspace estimation criterion convergence terminate work ad easily self cause ambiguity free submatrix extract index production obtain vary code high level twice observation recommend predictor explanatory assessment raw keep ability identify split whole subset tune sf median show error yield factor however careful pair observation design response value data outlier varied factor perhaps say correction cc number predictor split design median model interestingly identify anomaly value low dataset summarize adjust variance variance p seem property load r soft show various loading r respectively cardinality mild extend rr r r e r due substitute remain p jj aa j aa aa aa fall c aa j restrictive satisfy oracle matrix exist e example random sub r jj j rs onto p rs rs cs cs j universal term side handle lemma instance r follow norm p j p rs p aa l obtain achieve j j avoid mn p form follow c jj mn cc nc b aa closure thresholding thresholding regularity rarely group kronecker p r rt apply get penalty give loop k triangle solution rotation step justification hence time beyond running algorithm handle non mapping composition map r f characterize accumulation proceed similar line theorem minor modification accumulation point boundedness close ft fix globally line omit proof h rp h imply rs r line tf tf surrogate f thresholding induce cf continuity need j b dd x r obtain hence sub increment entropy e dd column motivated space contain must standard volume universal manifold denote norm universal q cauchy schwarz integral freedom detail theorem perform feature extraction unsupervised propose multiple explanatory guarantee sharp reveal predictive develop algorithm penalty theoretical achieve efficacy simultaneous reduction modern analysis offer projection subspace variable pca n predictor regularization obtain x explain variable pca typically life irrelevant loading prefer fail principal moderate sparsity individual loading still employ fashion guarantee sparsity explanatory guarantee variable toward extraction desire row even construct drive perhaps theoretical simultaneous main sample yield tight provide unify able inequality convex penalty show past suboptimal implementation ease meet challenge iii setting come universal predictive framework extraction perform rigorous tight regression problem signal information criterion scale free develop penalty theoretical local unsupervised analysis conclusion begin illustrate motivation component vector r pca special section plain drawback conventional attract lot attention reference strongly fashion meet sequentially pca certain optimality orthogonality hoc ii conduct burden dimension unnecessary remove loading guarantee get may employ unfortunately optimization scheme construction synthesis perspective variable projection addition j previous discussion parameter bring extraction sparsity facilitate favor form refer vanish flexible ideal enforce elastic mcp also develop nonconvex penalty furthermore doubly datum point new factor column refer decompose matrix efficiently decomposition either case difficult reveal joint inequality estimator multiplicative constant clarity constant necessarily assume enough inequality hold type penaltie universal apply c c penaltie p addition form cost spectral applicability also extent provide appropriate tune obtain show lasso gain low suboptimal error order j large first incoherence remove penalty purpose practically however universal choice two parameter rank aic none job novel perspective among principle avoid ratio assumption share concern may notational denote model sufficiently penalization offer fact emphasize coefficient cover reference assumption give degree second term characterize risk response model q roughly finer interestingly multivariate df information familiar unknown could sparse pca cf supervise however could challenge scale simplicity iid entry suppose parsimonious define sufficiently enough prediction c c sf constant value experience know well sf recommend address issue rank constraint nonconvex penalty interest light recent strength penalty relax thresholding rather consider real iii moreover r
dataset laboratory model standard lda word assume offer solution enforce constraint word unfortunately hundred enforce alone sufficient induce achieve interpretability specialize control structured token comprise vocabulary disease concept dag keyword mesh mesh retrieval organize pathway interaction summarize human thought deal effort structure necessarily property window expert interpret understand structure modeling equip control propose exploit dag structure interpretable summarize annotated article mesh hierarchy word inform guide topic expert sparse patient spectrum diagnosis concept annotate structured subject mesh lda meaningful mesh term find lda latent ibp compound allocation provide along manifold summarize form patient instance draw align font height minimum width parent path south pt pt style style pt path parent document bag representation model datum consist model generative comprise represent statistic lda build upon ibp compound dirichlet addition unbounded introduce three process preference describe topic relationship dag nearby respect graph associate tree drug treatment anti treat paper investigate treatment sub tree intuitively summarize word many child modeling nearby thought core model explain replace lda word hadamard product ibp document topic represent vector ibp concept mask represent relationship word distribution concept form use length observe sparsity dark view allow variation document describe vocabulary primary care hierarchy patient expert think describe could cover introduce layer allow explain observe word sparse generalization time procedure additional metropolis help sampler move matrix specifically mh prefer proposal knowledge mcmc use move encourage novel rely intermediate assignment tensor count assign topic topic nk k multinomial count topic slice give topic multinomial assignment know never way document concept count derive assign entry nk nk nk concept entry k q objective concept induce prior procedure fast mixing count reach unlikely sampler concept fast mix document introduce mh topic word ratio mh sparsity induce prefer equation get term dominate allow toward sparsity lda recover course layer sparse topic information point incorporate control vocabulary art interpretable tb occur patient receive diagnosis organize structure cm hierarchy disease recurrent mention diagnosis one independent run divide mean graph lda predictive however summarize clinical topic lda corresponding discover correspond use hierarchy probability rather word topic severe publish clinical tb library maintain control structured medical mesh search sr look summarize evidence clinical question consume reduce involved mesh helpful annotation systematic review researcher decide relevant term manually assign article inherent variability specificity make leverage difficult identifying concept nearby mesh interpretable provide retrieval tb p blind double blind channel drug dataset document annotate mesh systematic channel produce concept lda mesh rapidly report trial investigate use without topic comprise hundred mesh concept sample topic discover know article report control trial concept instance systematic anonymous confirm evidence retrieval topic wide popularity flexible corpus assume consider scenario idea coherent interpretable identify word among automate interpretability rate human develop topic work focus indicate link human work predictive summary kind describe disagreement result non probability interpretability nest chinese restaurant learn topic specific nonparametric learn also use kind topic interpretation complicated require human concept sparse encourage part use graph guide formation concept interpretability concept structure also simple interpretability expert define context word sense tuple incorporate hierarchical supervision improve come relationship content website summary jointly word show hierarchical exist forest enforce topic hierarchy label specifically treat label assignment document probit assign parent capture hierarchical structure contrast focus prediction graph use rather manner sparse generation much consider allow concept imagine nearby word nearby entirely nearby difference model neighborhood hierarchical prediction classification enough structured knowledge basis often scientific domain resource exploit achieve state interpretable graph control vocabulary structure induce interpretable bayesian nonparametric topic leverage interpretable maintain ability
gap use matching map use sparsity choose perform slice posterior normalization run special hasting update acknowledgement thank device award markov general tool practically apply prohibitive iteration present auxiliary mcmc query likelihood potentially small proposal approximate asymptotic fast method feasible bayesian probabilistic appeal output uncertainty make provide selection often model form distribution use inference monte persistent challenge coherent evaluate evaluate target update similarly typical variational bayesian procedure approximation intensive online procedure subset make procedure build optimization achieve result chain monte carlo consider data hasting mh recently mh move stationary condition effort exploit mcmc leave posterior introduce collection effectively turn parameter improvement structure issue evaluate discuss limitation conditionally target notational convenience term sampler hasting unnormalize evaluate seek auxiliary following eq distribution auxiliary wang hamiltonian joint remarkable given evaluate form minibatch subsample much computational evaluate family compute statistic need computed make likelihood run generate alternate update conditional emphasize partition part remainder bottom bernoulli ignore chain tend iteration iteration markov likelihood dark likelihood evolution detail proceeding section picture illustrate version toy implement likelihood whole likelihood bottleneck mostly structure chain convergence regular average computational summarize important determine iteration posterior chain number important tight put easy family summarize set consider either vector negligible n stage scale describe parameterize tight bound cost tight example choose bit front well tight place perform quick approximate explore resample take drawback visit resample work practice bottleneck usually simple overhead replacement hard efficiently choose line descent optimization long access chain still satisfy seem unchanged distribution z n metropolis accept efficiently datum matter whether accept geometric tune evaluation iteration point course markov leave something ht nz valid markov minimize assumption likelihood dominate step linearly constant choose storing operation scale store value need return th track cache store array dark index keep track thus dark assignment maintain record array position useful mcmc certainly regular evaluate per expect mix slowly favor iterate much slow answer question set depend give mcmc likelihood iteration accounting autocorrelation offer compare regular experiment classify mnist use principal use evaluate hold metropolis yield metropolis choose tuned optimization summarize mcmc autocorrelation per na I perform bad mcmc per much burn give map tune poorly reverse true rl c speedup
classified weight learner accord learner weight learner mistake expert consider total mistake base learner therefore weight least side obtain conclude mistake learner learner predict label total expect mistake algorithm aggregate expected mistake randomized condition expert expert learner classified mistake randomize majority expert convenience learner complete proof compare bind rwm mistake rwm mistake expect mistake rwm addition need incoming less expert suppose mistake similarly mistake consider fact well obviously also consider instance instance expert mention hypothesis expert region generality fact rewrite eq simplification q rewrite true mistake rwm complete weighted bag boost uci repository evaluate aspect instance effectiveness framework classifier update bag boost naive bayes balance breast diabetes letter rwm experiment rwm depend experiment dataset near ht bagging boost rwm breast cancer diabetes letter another dataset great method rwm dataset support result although increase rwm power difference arrange ambiguity table pair test indicate show difference arrange red cell black cell well table cell confirm bag vs boost rwm draw breast diabetes letter view time difference rwm bag exploit label justify great overhead create factor balance scale breast cancer diabetes value well among base green fp good value cell true group base breast diabetes mistake bind rwm mistake mistake calculate formula mistake experimental confirm accuracy mistake small mistake rwm large mistake mention c mistake mistake result diabetes mistake well rwm increase addition reveal rwm input show new online among superiority different learner class specify class correspond dynamic powerful clearly base classifier utilize cause great base affect whenever important use base learner volume exploit algorithms ensemble classifier prediction well online ensemble randomize majority rwm expert define converge region well resolve rwm propose novel prediction expert rwm result also well sufficiently expert randomize weight classified identifying belong label algorithm performance paradigm prediction good study extensively lot practice spam detection object bag know weak satisfactory performance online bag handle stream mention consist learner select learner input classification recent machine recent area lead predict goal predict close expert majority rwm present mistake bind expert exponential fundamentally rwm instead zero rwm exploit definition well good expert expert error expert necessarily negative true reveal separately base improvement rate call cascade expert theoretically expert tight exposure sufficient practically contribution rwm know rwm consider rwm apply data number output w trial mistake make majority mistake expert far large rwm tends decide accord opinion expert mistake compare discover sure algorithm mistake say try good datum instance one opinion numerous fp false rate low fp rate fp rate either look expert low look expert low fp rate lead three classifier learner rwm exploit factor every classifier predict weight learner prediction responsible
separate protein achieve cb dataset challenge secondary structure prediction great challenge computational accept predict protein understand protein drug protein determine structural state thus use algorithm protein extensively study protein close since early neural core component many successful significant leverage information development capture use recurrent neural probabilistic graphical graphical neural crf secondary commonly classify combine secondary coarse grain structure prediction achieve grain state secondary reveal address address protein introduce protein crucial improve performance secondary secondary formation depend secondary far apart protein still limited capture spatial knowledge various structure speech successful lack necessary mid work tackle challenge secondary broad supervise markov output avoid marginalization deep layer advantage crf field mrf versus generative classifier dependency important supervise structure structure protein enable hierarchical hundred introduce multiple convolutional allow high feature suit make informed ht utilize without generative training train computational reconstruct generalized difference input intermediate boltzmann avoid marginalization explicitly graphical learn directly enjoy feasible back prove autoencoder irreducible ergodic chain converge introduce latent denoise iteratively px converge example sharing seem leverage supervise supervise generative network supervise analogous corruption let denoise auto ergodic estimate provide procedure minimize reconstruction reconstruction generative corollary corruption process reconstruction train regularize triplet assume train h xy p flexible noise avoid marginalization hide benefit capturing task distribution ht contain convolutional layer layer computation layer therefore location making sized gradually simple convolutional consist input channel convolutional layer feature convolutional convolution thus connect visible unit filter map bias visible noisy activation pre z calculate straight layer label focus secondary challenging problem structural simultaneously predict structural sharing position secondary previous protein inclusion program package content sigmoid original encode binary input feature encode protein improve performance protein training commonly retrieve remove protein chain state secondary label infer structure discretize absolute result secondary label cutoff coverage majority protein chain short protein short aa zero contain protein perform cb far filter cb performance measure run consecutive reconstruction randomly add post obtained reconstruct consider reconstruct multinomial secondary binomial number network motivate arbitrary away prediction trick start activation sigmoid layer gradient comparison epoch implement library train gpu segment layer sc sc noise sc sc state sensitivity loop turn bend bridge description one protein dark color indicate strong versus achieve layer structure conv conv pool conv pool type convolutional denote pool window layer architecture use channel convolutional consecutive feature run get close experimentally incorrect later organization secondary achieve major less frequent state unbalanced label specifically unbalanced effort identify extremely rare improvement public benchmark cb validation train sequence homology ss wang discover success architecture vary try start gaussian convolutional original good layer performance dramatically prediction start learn reconstruction layer seem necessary reconstruction validation reconstruction error go vs representation generative sized secondary structure protein structure network structure stochastic capture datum convolutional level structure sensitive inform high distant architecture structure bioinformatics scene parse segmentation architecture hard code organization
center reveal embed lose take singular contain add row important capture scalar basis feature span justification theorem proper complete orthogonal interpolation theorem say decision function generic kernel orthogonal intrinsic g kernel pca alg interpolation generative learn degree linear notice b approximate basis generator generators projection address degree increase computing power exploit power generator project approximation sequentially approximate rank onto general fix hilbert approximate strategy project feature various alg lie degree generator ideal generative learning degree generator read notation projection matrix threshold apply add kernel compute interpolation basis row maximum threshold generative discriminative entry thresholded return singular generative time discriminative compute interpolation space alg analogous consistent estimator storing parameter repeat complex form evaluate maximum evaluation one vs width thresholde singular purely linear polynomial gauss overall misclassification second subsample vary runtime detail conclude competitive handwritten close discriminative implicitly generative discriminative discriminative scenario decision generative separate hyperplane uniquely generative learn manifold discriminative way manifold moreover dual discuss principal idea root statistic symbolic practical reading symbolic major kernel trick reproduce duality algebra duality rkh consider principal learning feature explain detail pca learn embed also algebraic topic overview directly applicable understand structured hand algebraic structure scenario theory build polynomial outline symbolic back al polynomial svd vanish basis basis variation symbolic vanish learn coordinate duality duality symbolic algebraic outline theory ideal duality simultaneously component learning stand open method conceptual lt european european grant fellowship ex symbolic algebraic inherently dual structure algebraic generality kernel main kernel illustrate propose simultaneous accuracy propose synthesis symbolic algebraic inherent duality method kernel fundamental machine method kernel trick g major drawback learn kernel principal learn symbolic algebraic inherently structural representation major directly interpretable seminal allow transform easy major numerically unstable scenario address attractive major issue symbolic applicability issue symbolic tool simultaneously generally argue discriminative world discriminative generative allow combine avoid considerable introduce relate duality polynomial object involve treat paper field usual convention divide change show object link duality vector polynomial space homogeneous polynomial polynomial nx dd space decision let k independence claim pass limit usual map elementary map identify explicit dual namely f canonical identification canonical scalar product compatible need scalar property k must description product fix exponent extend hold outer product orthogonality let less express duality algebraic reproduce hilbert additional evaluation symbolic polynomial equation could obtain identification section purely algebra next beyond usual rkh alone concept algebra proper object feature multiplication infinite says admit multiplicative generator generator need class space polynomial analogue manifold relate algebra geometry relate decision generative reveal duality introduce matrix resp kx concept ideal orthogonal scalar intuitively basis vanish algebra statement interpolation let generic let hold ds n nk ik ik x x nk remove iii equivalently row enough contain kernel degree yield statement instead grow whose size even claim ii claim vanish therefore carry consist feature algorithm introduce subsequent enable variety vice versa kernel estimate feature manifold generator conversely estimate pca relate obtain discriminative permit inspire model vanish sample hausdorff ideal partition noise act dependent task classification irreducible name irreducible sample principle ideal take manifold label part could give th entry compute output entry sufficiently large noiseless alg discriminative vary
follow result call spirit suppose strictly ty ty scale model dt ty ty drift brownian correct density px easy researcher often marginal dynamic residual nonlinear reference example uniformity check transform process residual box al du serial independence although insight misspecification sequentially correct moreover mistake kolmogorov type functional test iid use residual require examine simultaneously uniformity independence uniformity kolmogorov control nan show analytically design base incorporate lag product standard distribution alternative take account parametric implementation smoothing piece hinge integration simulation nonlinear simulation form avoid tool evaluation misspecification simultaneous uniformity correlation independence inconsistent li uniformity imply distinguish depend procedure impractical close spirit goodness hinge original detect misspecification rest test bootstrap justification provide daily stock index briefly deferred generalized joint uniform motivate incorporate product uniformity independence implication theory useful goodness test univariate empirical inconsistent discussion issue bivariate result illustration stock return risk define quantile tr e exceed unconditional ensure literature risk therein unconditional rarely metric aggregate von kolmogorov statistic avoid process wise check write interestingly hold q univariate account unconditional lag likewise information lag aggregate test account unconditional misspecification combination sum scheme possibly drive address lag care lag power sample pairwise five equal already moderate box provide dynamic correlation propose box draw across account one want nonnegative estimate asymptotically practice know monte simulation parameter statistic recursively statistic empirical distribution percentile much bootstrap utilize block directly since nan misspecification nan necessarily transform case block bootstrap desirable costly many parametric block model application take simulation take second thus speed bootstrap small reality univariate substantially lag enter dimension discuss detail fixing along omit relevant practical asymptotic impose conditional form cdf dynamic increase first weak element covariance bridge brownian bridge parametric cdf composite nan hypothesis effect differ ba mn mn following truncation r f g empirical still assumption generate abuse unconditional last choose additional average misspecification aforementione extreme misspecification elliptical report carlo control dynamic uniformity alternative projection exist ergodicity p consistent might distinguish alternative might case say solely consistent whole complement dependence serial might together assumption imply necessarily bootstrap critical bootstrap prove analog test setup rv assumption restrictive linear expansion state denote critical value critical come bootstrap critical approximated repetition assumption carlo propose test repetition critical save detail misspecification model simulation request technique stock exchange index maximum therefore require assumption ml fan mixing applies specifically design stick evident misspecification easy introduce usually transform residual test univariate test goal one hypothesis order make martingale transform modification leverage experiment examine finding application test see eq take thick triangle marker circle marker lag dash line circle marker plot sample panel student triangle marker lag circle marker consider dash circle marker panel change size panel test show marker plot fill monotonically close decrease misspecification capturing case also dependence lag instance aggregate lag wide alternative effect box include repeat experiment power obtained report save summarize nominal similar normal student equal span apply transform reject significance reject example test table line subsection generalize n jj von test show panel von kolmogorov panel jj von kolmogorov ar von kolmogorov c lag generalize ks suggest capture test reveal ar generalized reject significance statistic equal ar reject mean ar ar type equal ar statistics literature multivariate vector td f past parameterize conditional dynamic copula joint specification test follow multivariate transform univariate df joint formula probability chen dynamic copula transform constitute series iid apply statistic model important multivariate series bivariate two reject nan ar misspecification undesirable financial fill literature helpful process paper useful develop theory include grateful question suggestion anonymous comment possibility foundation school economic acknowledge financial ref cm economic school graphical tool quantile duration etc properly control dynamic smoothing test new integral transform establish alternative justify effect often ignore monte carlo finite sample property test check popular stock exchange datum
es n pr name ensemble list illustrate experimentally base sub find combine give surprising similar user combine weight fill achieve model name inferior simple plan present rank recommendation name occur name name name scalability issue future parent name thank valuable feedback european community university student thousand name choose pick parent inform decision recommender ranking produce collaborative algorithm list experiment world search explore name discover challenge intuitive consideration parent parent relative pick name actor rule mean name family belief role choosing mark name stand crowd avoid source email address address thousand name parent pick right present parent take collaborative filtering give name pool name study context offline phase main easy simple perform letter user name item recommender use bold bold scalar item name occur name occur name item occurrence aggregate bag order search name finally represent user action occurrence decrease order query explore name collection comprise dataset contain activity name figure name observe distribution concentrate per name show type interaction enter website order click name website link search page user request name available name category name correspond challenge given recommend name search recommender evaluate respect name user restrict enter activity user activity name display set name enter activity list detailed assessment recommendation left name position order list recommend name precision position might happen handle name clear activity type concentrate name name name user name name user median name name provide challenge user name recommendation representative user remain transaction name ignore impose g recommender depth al network recommendation name name name scale predictor collaborative filter collaborative cf web site online input interest recommend item collaborative approach compute collaborative filter amazon com match name combine list determine match name construct co occur name collection behind many name co name transaction thus process memory name recommendation list name occur name popular correlate name name recommendation name occur number recommendation max max max neighborhood approach collaborative describe recommendation capture name recommendation ensemble quality value combine estimate within reciprocal individual filtering recommendation boost model describe create name occur name test enter name bag randomly select name give user specify proportional user positive towards name user name consist list name sequence co occurrence name sort use choose name bag occurrence assume respective bag recommendation choose select pick frequency would recommendation iteration include list finally recommendation correspond follow occur prediction activity occurrence name name interaction
network topic friend site facebook twitter ad topic describe classified ad relate engine optimization content describe rich article likewise link building site add link email google email post market black server month spam corpus relate particularly interpretable topic email strategy spam describe way ad google web corpora beyond corpus insight hierarchical corpus tb type program ad category topic profile type try sum topic proportion job table show result nearly third com abuse majority appear involve insight analyze example com project wide topic concentration reflect proportion email opt list topic concentration focus stream explore generalization introduce finite dimensional technical use dirichlet main scale corpora result corpora real world nonparametric become complexity stick break construction scheme arbitrary version potential batch framework recently explore also worth facilitate theorem derivative k gain term convex flip essentially concavity establish tb concave concave fx application jensen yield right side establish upper crucial variational need maintain overall equip give expect inequality analytically statistic beta distribution eq tb show conclusion c q tight important conjunction bind tb tune alternate appeal upper recall obtain specifically occur expectation set upper equality x last four recover factorize well around tight sufficiently factorize lemma detailed procedure log respect study corpora document know hierarchy newly available corpora security deep network computer decade topic framework bag word explain frequent occurrence corpus dirichlet great deal topic intractable remain research devise approximation paper topic corpora document hierarchy structure corpus subject matter news article business sometimes e international idea topic model simplicity topic estimate say validation node corpus hierarchy leaf represent reflect ensure topic proportion category parent variation hierarchy special hierarchical devise new child believe broad demonstrate subject corpora field security seven year job crowdsource view corpus site corpus derive internet create break ground model number depth corpus moreover corpus depth variational approach demonstrate describe detail em parallelization additional probabilistic corpora inference model describe procedure evaluate corpora result section conclude inference derivation algorithm tb illustrate seek corpora child root top level parent individual corpus essentially extension lda model proportion weight topic document leave level proportion associate non proportion root corpus topic sample corpus parent tree figure dependence scalar concentration topic proportion category variance formally denote likewise assume document proportion denote namely conditionally topic proportion category corpora document begin recursively lda refer inform tb proportion dirichlet proportion draw topic proportion draw multinomial bit topic model however occur corpus hierarchy level view case flat corpus dirichlet show perform demonstrate advantage symmetric draw dirichlet serve nonparametric lda topic generative special base allow possess application inference seminal develop gibbs track conference sampler collapse spirit drawback fast gibbs develop framework collapse level deep later framework truncate possible dynamically vary successful variational corpora actual corpus involve variational inference complexity auxiliary stick construction scheme parametric already sometimes identify go wrong direction average preserve lemma concave lemma prove sketch concavity lemma obtain term likewise obtain apply inequality concave equip intractable result value dirichlet shorthand appendix emphasize naive jensen hand direction need log thus rigorous low shall surrogate inference detail respect parameter perform coordinate ascent corpus variational different category bottom corpus corpus update topic reader recursive call hierarchy recursive perform subtree start child category inference node corpus em factorize approximation updating variational coordinate newton corpus refer variational double naturally tb recursion variable prior proportion train assignment compute evaluate corpus manner node corpora uci repository vocabulary token vocabulary news vocabulary experiment batch article million token publicly batch implementation former hdp gibbs latter hdp prior initialize hdp topic hdp per corpus report hdp hdp summarize experimental bar fold fold hold testing range fold hdp corpus explore hdp slow hdp corpora corpus appear order explore hdp much corpus certainly scalability demonstrate exploit corpora corpora corpora paper category million token job token vocabulary crowdsource collection token vocabulary seven year job crowdsource site internet corpus lda attempt job hierarchy interior job leaf
al agreement reasonably insensitive choice al assess carlo label pool population primary namely label many monte draw aggregate calculate monte p c c al meet meet meet se h c meet meet meet meet se replicate average table illustrate six good aggregate classifier averaging group six good lda meet meet meet rs rs se rs met meet meet se se c na I bayes meet meet rs se se lr covariate denote pe py py appendix diverse choose two class prior use source uci provide wide term covariate variety sensitivity presence absence problem dim class name dim generate sampling gaussian mixture prior problem multipli multipli multipli multipli multipli red decision boundary classifier implementation detail describe lda standard implementation apply covariate scale I implementation package predictor assume ideal svm package radial calibration mle fitting computing optimisation define al systematically train label use raise stochastic optimality produce reduction central whose selection define reasoning abstraction theoretical heuristic make construct experimental classification seek select example example diagnosis review performance label select central motivating specify loss function describe suggest selection classifier great expect construction great thereby sense work present theoretical quality eq estimate motivate loss reveal whose maximum abstraction abstraction generate insight fully al method difficult analytically performance since source variation comparison make shannon entropy motivate development issue far experimental evaluate al literature explore source binary eq al al al behaviour label label behaviour raise robustness optimal label examine structured illustrated follow conclude background brief review somewhat notation categorical model covariate response denote bayes thereby give classifier classifier class class theoretic allocation denote misclassification allocate j index indexing dataset training division show division training subset discriminant analysis regard length fitting notation slightly extend non object node near produce performance classifier assess quantify disagreement prediction define allocate log empirical generalise expect denote loss log hereafter classifier label example typically discuss classifier discriminant nearest na I nn na I independence give standard classifier label abundance good algorithm human expert label systematic example improvement pool example may covariate relatively label consider common examine repeating generate curve amount label grow repeat application dependence iterate al may reason turn rs example rs label rs benchmark experiment al benchmark performance assessment much rs hence address rs rank performance number uci base entropy second seek metric comparison desire goal label need level often context certain analytically heuristic example choose classifier decision idea uncertain tuning uncertainty classifier allocate shannon define j justification search hypothesis efficiently rs loose search label prediction disagreement vote predict kullback version control sense search valuable insight motivate pool cluster suffice optimal optimistic illustrate gain theoretical error example author examine experimental spirit current classifier label section examine construct account fitting datum example error rate define way replace approximate efficiency approximate rate pool calculate total problematic classifier uncertainty question error uncertainty hard reduction form dependence critical notation intend base already train much single choose examine define label give label reduction actual denote actual goal great turning pool unknown take expectation label loss capture difference loss exist loss improvement since define section extend al reveal bad expectation show selection calculation curve smoothed nn al behaviour examine label condition primary take marginal vanishe denote marginal optimal behaviour target maxima reveal behaviour illustrate raise behaviour address motivate introduction generalise central marginal illustrated create shift illustrate fully specify covariate infinite pool assume allow explore q cx knowledge allow method popular shannon binary univariate balanced later target define section rate assume split class size hold full calculation consider j decision boundary denote calculate examine decision rule c equivalently boundary lr class wrong around second straightforward give denote cdf result appendix al oracle new denote update f directly equation cx cx figure great close figure case classifier fourth great improvement toy example complicate j solid green case great move boundary close analytically examine two se rs pool assume al rs contrast se problem functions maxima j j black se rs show blue red estimate dotted green improvement three rs se thereby selection select whereas case classifier se never optimal se never great se suboptimal turning rs se stochastic nature rs far se explore method label se rs start variant average draw label comparison rs improve location quite b c covariate black se rs green case multipli multipli multipli multiplier label sensitivity fix rank b always sample equation conditioning rarely make explicit examine motivate address dependence raises label alternatively sensitivity second draw low pool similar visually statistically similarity pool ranking great sensitivity problem classifier discriminant single pool grid imply rank pool ranking test correlation correction c closely relate draw similarity toy dependence dataset case turn near different section target theoretical present different estimate label dataset method compare benchmark rs describe equation include three primary estimating estimating raise interesting statistical estimation choice I implication dataset parameter na I disjoint partition figure scheme directly take term I ignore estimation term label problem train estimate optimistic I motivates term thereby estimate subset partitioning subset estimate classifier parameter arbitrary random perform time result study fold pool computational explores section focus al varied know classifier diversity
minibatch divide result sgd sgd ss much sgd two adopt minibatch rate significantly small sampling improve observe significantly strategy unbiased mnist epoch minibatch size divide row summarize objective algorithm row test four variance dataset term gradient demonstrate propose sampling descent provide rate strategy conduct extensive traditional uniform sampling promise validate effectiveness technique thm zhang university nj optimization task neural network sgd minibatch often sample minibatch lead high variance whole technique significantly improve encourage experimental confirm extensively community every method example random minibatch uniformly minibatch unbiased gradient estimator relatively propose dataset reduce key idea unbiased end relationship strategy minimize sum standard rest review work study minibatch descent empirical conclude finite rate linear rate researcher return average last fraction previously similar polynomial sgd extensively study unbiased explicitly new call finite moment still employ uniform importance sampling consider variance minibatch idea complementary reduction importance function useful throughout paper please lipschitz function gradient strongly co standard multiclass predict classifier label solution regularization describe rule calculation large popular modification draw equal rely derivative descent disadvantage randomness initialize ts r ti propose similarly compute v accord maximize reduction minibatch solve dynamically gradient cluster variance derivative require clearly impractical cluster relax k r correspond optimization calculate iteration calculate relaxed simplify follow algorithm cluster provide present notation begin analysis suppose satisfie nb expectation obtain conclude lemma smooth suppose nb verify satisfy sum inequality ht ht inequality convergence smooth function gradient well technical inequality h simplify p use convexity property give conclude another propose smooth p final use inequality iteratively fact
independent purpose plain represent straight confidence rate provably pac blue symbol provably theorem green symbol rate appropriate stopping time symbol approximately slope visualize guarantee see gain arm add mention probability pac illustrate huge gain keep mind time really prohibitive nature second term negligible implicitly know zero multiply pac probability error need average twice large deterministic sample budget relate pac design uniformly good across consistency requirement fix draw often preferable bad observation sequential difficulty impossible predict efficiency experiment consideration low give relate different aspect distribution denote round draw let increase right hand large q correct one possible satisfy right proof find exploration normal let independent ts suffice tend context involve theoretic appear gaussian bernoulli match tight budget confidence numerical significance exploration deal scenario suggest implication testing like match reasonable strategy stop rule context mostly give lead regard proper provably pac exploration use sequential stop assumption exist bandit density introduce log key follow classical whose relate stop event also quantity well let lemma jensen lead introduce successively log rewrite apply tn aa conclude together region state technical partly omit eq accord subgaussian super martingale eq let dt give proof quantity implication eq conclude lemma follow apply suffice ss ss ss ss x conclude arm rely follow lemma generally family proposition apply respective two often refer chernoff xx derivative use one show sum de universit option alternative dependent bound performance testing improve currently confidence equivalent alternative provide stop terminate budget alternative though practice criterion identification popular website empirically preferable user value page respective standard determine high user become user b either number fix pair determine schedule determine source algorithm take past display sequel benefit adaptive ignore term present armed consist unknown motivate arm expectation choose receive arm resp law resp belong identify expectation agent define arm past word respect satisfy triple determine sequel correspond strategy bandit two setting consider choice draw draw almost surely recommendation rule strategy compare identification resp budget follow bandit use error budget determine fully alternative law variance pair h among sequential ii expect require test minimize order probability gain randomize conclusion bandit introduction set aim maximize horizon equivalently introduce understand parametric proper leibl analysis include ucb ucb ucb goal determine arm try observation arm identification interest problem go armed good arm parameter recommend sequel advance bandit work study armed bandit arm empirical arm strategy derive pair model budget strategy wrong strategy arm ucb strategy pure exploration involve divergence lower easily armed set confidence obviously always suggest sub expression specific bandit see obtain c equal case match indeed reference q shorthand early relevance quantity arm analogy conjugate bandit define prove reach recommendation differ rule reduction draw arm time arm simpler introduce uniform sampling optimal strategy collecting pair arm algorithm sequential propose stop empirical belong introduce sample pac lower apply case elimination pac match rule interesting exhibit pac explicit iterate logarithm propose elimination pac case sample govern ensure function elimination exploration pac match appendix elimination exploration elimination round elimination feasibility variance class bernoulli bandit define arm parametrize kullback leibler either express static theorem budget strategy dependent directly closely algorithm use set
right class arise notion probabilistic state mark finite initial finite alphabet specify symbol recursively extend arbitrary follow impose state string nan unity specify however strongly remove ergodicity next formalize generator initial induce function countable imply space initial probability generator let probabilistic recursively nan index imply yield mark immediately space ergodic whereas marked representation mark transformation entry symbol fix initial sense stationary state note exist stre beginning stationary equivalence unique canonical induce canonical ij satisfie property canonical initial mark canonical representation independent canonical representation copy exist q begin define stay within mark extension ergodicity state mark representation initial mark latter may distribution connectivity induce remove state arbitrarily state fundamental importance entropy synchronization fix determine current state analogous rgb rgb rgb rgb distance scale font fill minimum text draw yshift draw xshift yshift edge aa south anchor xshift east auto distance font fill text text right edge bend edge xshift yshift xshift yshift edge draw bend bb south estimating rate generator translate synchronization problem symbol determine finite history probabilistic graph arc probability trivially thus stre state contradiction generality x string ij tx j consider contribution arise nevertheless qx complete theorem induce string existence string may order string alphabet construction scale string string computation string symbolic note state state symbolic string specify alphabet set symbolic count string overlap string count occurrence imply symbolic symbolic derivative symbolic distribution empirical since I read guarantee complete describe string sample theorem arise geometric vector construct different string generate derivative geometry let sl hull recall lemma claim vertice state stre convex hull string string derivative hull e denote allow n kullback stationary complete easily bit hx I always conclude stationary time symbolic directly employ possibility entropy generative entropy probability finite entropy q perturbation cause change turn maximize perturb entropy cl establish let claim perturbation differential perturbation entry perturb small form perturbation note attain imply within set admissible perturb perturbation attain admissible perturbation monotonic globally admissible difference establish note complete deviation symbol stream string extension unknown conclude establish claim write x stationary time correspond thus hx hx hx h note hx complete next modify symbolic derivative generate pr limiting x chernoff e pr pr denote rt dropping first big hence lead complete continuity entropy hx x pr hx h x complete string alphabet string hx e hx hx hx hx stationary alphabet bit rgb rgb rgb rgb rgb rgb anchor south title title yshift align legend style pos north east fill gray top dash gray width thick color xlabel length symbol xshift ylabel entropy letter ylabel style yshift xlabel style yshift xshift table x figures south anchor yshift title style align legend pos north east draw fill style axis grid style dash gray width height top color gray symbol style ylabel letter ylabel yshift yshift axis axis cs shannon figures table east anchor west title yshift legend align style pos east fill gray style axis grid gray height color gray xlabel symbol style xshift ylabel letter ylabel yshift xlabel style yshift format format sep axis theoretical south anchor north yshift title title legend cell align pos north east white gray style false gray true height grid xlabel style xshift ylabel bits ylabel yshift xlabel yshift scale false false format sep axis cs figure c east west title yshift legend align legend pos north white style style axis gray xlabel symbol xshift ylabel bits letter ylabel xlabel yshift scale format fix format sep axis cs value table x figure south anchor north yshift title yshift align legend legend pos north east fill gray fill text style grid style thick grid axis xlabel ylabel letter ylabel yshift xlabel yshift scale false style format axis table south yshift north english text yshift anchor south text black yshift anchor south east xshift south east auto black minimum width scale edge draw bend xshift yshift xshift yshift bend leave south xshift south east scale font edge bend node yshift xshift yshift edge bend english experiment subject put bit letter achieve entropy generate e symbol stream generate probabilistic alphabet lead significantly approach finite string string satisfy describe hx occurrence corollary hx e complete binary rhs bound alphabet relationship alphabet figure band capture data length rhs confidence band function step propose stream importantly string effort concern level contribution constraint lead uncertainty rare ignore string occur application english shannon experimental english letter alphabet size verify corpora letter collect letter example allow comparison shannon author estimate mention assumption converge trivial get lyapunov finite ergodic model directly probabilistic e look walk somewhat compare application insight alphabet stream generator history tell show quite converge contrast base error symbol tell precisely indeed symbol rate symbol generate stationary process correctness exploit probabilistic finite string importantly free converge confidence bound bad confidence requirement compete approach sequential datum stream naturally quantify tool recognize perturbation source carry characterization make practice fundamentally demonstrate converge fast effective range english text additionally algorithm bound connection input uncertainty require pre symbolic dynamic kolmogorov complexity kolmogorov estimate insight drive dynamic tool detect dynamical ergodicity stationarity relation symbol stream dependency long decrease entropy pre error know algorithm compression rate string distinct phrase sure optimally one end string length source entropy parse report instead occurrence contexts string occurrence count interested former answer possibility exist rate limit analytical guarantee unable finite convergence trivial report algorithm issue theoretical entropy follow distribution free characteristic consequence uncertainty specify bound finite length font auto distance scale circle draw dash align fill xshift yshift west xshift fill anchor west anchor north yshift b south align anchor text south align estimate north align state hide anchor north east font yshift xshift south west align stationary entropy stationarity report converge slowly rate imply
work whose eigenvector discard two j follow j equality due orthogonality j begin relationship j derive j theorem prove theorem suggest discard low error actually shrink turn connection sense view say truncate distinct say tend explore connection associate distance truncate similar transformation matter j pair associate say shrinkage point bound error simple reconstruction accord equation discard eigenvector truncate pca lead perform aim addition check classification pca far result publicly machine repository discard distance pair accuracy state pca discard eigenvector eigenvector row discard eigenvector cause eigenvector discard pair norm wise mean square discard wise distance addition check effect accuracy pca publicly uci repository eigenvalue distance neither accuracy denote write may q fundamental zero invertible write converse invertible fundamental algebra orthogonal therefore invertible transformation system homogeneous augment matrix discard algebra state homogeneous system infinitely many
offset improve sequential scan require huge similarly scan due alternate mcmc intuition conditionally independent mcmc proceed gibbs mean scan useful determine pseudo fit value amount value deviation mle display along generate mle indicate room mcmc mle scan gibbs indistinguishable confirm asymptotic poor foundation inference massive restrict significant improvement close close mle full estimate explore novel augment new theoretical concern original provide cd moment cd visit advantage pseudo show cd like conducted validate cd promise intractable propose cd provide cd perform apply family cd inference turn quite mix perform present challenge traditional quickly alternative cd relatively stochastically result become context foundation devote family generating offset e intractable except actual evaluate every also parameterize inference equivalent kl p unfortunately impossible evaluate normalizing problem cd introduce gibbs term likelihood deviation chain start objective function form observe nearly minimize clearly necessarily becomes imply maximally suppose represent result mcmc regularity though may time equilibrium implementation result belong term rhs use third problematic dropping suggest small divergence theoretical restrict boltzmann gradient maximum gradient infinity upon main development whether ignore principled approximation index b ib b iy step unconditional satisfies distribution unlikely step choose appropriately drastically reduce augment subset marginal let equality term discrepancy distribution reduce lower perfectly far justify cause result close implie augment minimized log within combine generate since obtain aggregate information indicate poor find good number gradient zero yield mcmc reasonable restrict occur approximately condition exponential moment analog become iy moment composite objective become pseudo form composite cd maximum computational alternatively mcmc equilibrium write composite either direct sampling sampler cd kernel arrive identical pseudo long additional gibbs likelihood objective wish index choose random conditionally mr function weight among let gibbs sampling back expectation derivative express complex long chain case use cd moment family approximation geometrically computation perspective though mcmc make require chain limited approximation yield quick expectation approximate expectation regularity upon condition convergence unlikely meet unless mcmc run similar gradient gain approximately within composite hessian newton eq hessian expectation approximate take similar effort algorithm quadratic convergence family interaction typically fit mcmc study inferential mle typically represent people connection
induce automatic efficient bayesian inference linearly characterize tuning infer multilinear factor provide predictive distribution miss entry extensive simulation synthetic intrinsic capability recover truth miss synthesis demonstrate outperform approach tensor tensor rank determination inference synthesis array structural affect involve video represent pixel person pose factorization capture multilinear among therefore theory study apply social video brain process popular tucker cp exist arise world attract great research year tensor multilinear partially factorization missing formulate least cp conjugate riemannian optimization tensor incorrectly severe performance another technique exploit formulate technique nuclear norm yield split nuclear also scheme tensor define straightforwardly weighted norm mode addition completion factor simultaneous completion technique combine tucker auxiliary strongly also nuclear affected emphasize cp tensor np bind tensor ill rank tensor investigate fact determine miss attract interest tensor factorization bayesian monte carlo mcmc variational inference extension robust include bayesian rank computationally inaccurate issue either slowly address issue tensor factorization multilinear noisy incomplete tensor miss rank automatically specify induce individual hyperparameter latent variable place due resort characterize approach effectively extensive illustrate determination robustness application synthesis preliminary multilinear cp specification inference mixture datum follow conclusion dimension tensor g order denote capital tensor denote letter ni I tensor sum element product vector size hadamard without hadamard kronecker rao reverse define q order entry tensor observe entry tensor noise assume shorthand term interpret rank tensor factor row wise vector cp factorize parameter nr n index essential factorization multilinear model general latent selection computational costly elegant automatic infer tensor overfitte hyperparameter minimum hyperparameter rank determination relevance determination ard ard weight principle analysis consider parameter sparsity place factor govern precision share latent matrix far factorize dimension point effective dimensionality bayesian prior yield write parameter posteriori extent squared impose impose develop eq hyperparameter analytically vb framework cp seek low occur assume factorize assumption form th factor family parent form parameter graphical inference mode message co parent parent term factorize see n denote subset associate column accord q whose mode need introduce random n sec appendix attempt compute multilinear quadratic length efficiently fix rao imply interact take account simplify multilinear finally posterior approximation update moment intuitive give follow update information tradeoff fitness prior posterior firstly coefficient similarity scale fitness note via posterior crucial automatic incorporate posterior see sec th mode hence posterior intuitive sum square lead turn perform message parent include incorporate however posterior straightforwardly vector leave inner matrix expectation outer th expectation quadratic n see entry intuitive relate number residual fitting square entry essentially take follow eq intuitive related factor also rest get solution initialization point initialize strategy draw singular rank upper value tb incomplete indicator update zero component computation entire summarize automatically update new prior affect hence posterior becomes force prior information component unchanged iteration use posterior approximate yield n n sec appendix input size tensor generally complexity w automatic reduce rapidly complexity polynomially optimal highly tensor parameter avoid procedure require predefined completion factor miss entry exist point estimation deterministic develop powerful satisfy cp account local assumption rewrite q probability adjacent define coefficient sum firstly keep unchanged change conduct synthetic world fully cp factorization completion base scheme aspect reconstruction performance entry world image intel memory synthetic procedure factor n entry uniformly mark video material tensor factor component tensor monotonically indicate effectiveness capability denoise estimation evaluate tensor e extensive vary condition size matrix svd result tensor group tensor size evaluation initialization incomplete tensor snr db evaluation initialization see incomplete tensor evaluation vary miss tensor snr db evaluation perform ratio rank miss miss rank initialization initialization term tensor complete detect snr db decrease deviation noise miss high missing achieve db ratio note achieve even snr true fail ratio determination primarily true level occur may helpful determination tensor generate rank db miss statistically consistent evaluate repetition fig ratio perform achieve miss precisely rank outperform miss extremely completion sparse conduct additional snr experiment see h image benchmark show method image tensor conduct four missing pixel condition snr pixel noise free c experiment text text miss entry image pixel miss mask compare describe ratio completion tuning ground pixel lr lr nf nf c c mp runtime runtime runtime ratio superiority additive obtain noise obtain smooth color recover predictive obtain clean remove effect appear method factorization significantly overfitte result observe mainly intrinsic low natural pixel recover image cause
reconstruct clean noise white representative voxel corrupted choose comparison learn contaminate tensor reconstruct table summarize reconstruction set enable regard image measure value slice fourier result stack choice radial illustrate fig regularizer read compare minimize consist square total utilize adaptively synthesis slice noiseless mention table fast suffer eight time volume b separable operator multidimensional separable nature learn reconstruction reduce operator larger design multilinear demonstrate corollary tu project team de nd conference publish representation certain research representation recently gain increase multidimensional thereby drawback inherent structure achieve enforce separable learn deal multidimensional operator code great popularity resolution compressive sense last combination atom synthesis counterpart sparsity transform domain nm encode e operator prominent analytically specific signal representative follow give short separable operator algorithm framework separable learn alternate step current approximation noisy regard row operator signal orthogonal requirement cost design include ensure incoherence sense unit structure matrix signal ultimately restrict limitation memory computational tackle enforce additional synthesis examine combine author analysis svd separable introduce separability maintain inherent imaging capital letter letter letter index accordingly necessary tensor section able deal multilinear call product transformation separate mode j nn mode eq result n n I offer understanding constraint rewritten vector kronecker product interpret operator additional extended certain operator regularizer demand label row operator maximal row trivially linearly condition realize enforce operator enforce point separate mode unit norm rank kronecker trivially neither property kronecker operator fulfil n iii incoherence regularizer barrier barrier within propose control able gradient base manifold many range medical one represent mean hyper third encodes scene encode information homogeneous kind extract flat synthetic parameter operator act local signal appropriate entire constitute fidelity serve optimization approach multidimensional operator minute c method
extension set trivial intractable support foundation project research contract college fellowship supplementary material bring autoencoder learn fig h separate thin black draw dynamical e case system dynamic challenge measurement mapping practical identification technique additionally identification inherently high identification mean deep demonstrate enable predictive system stream eeg sensor network dynamical desire design important role translation surveillance identification mathematical dynamical measurement functional relationship different measurement study standard exist expectation maximization method linear inherently difficult active last decade state survey smc dynamical hard un identifiability local overfitte high possesse dimensionality dimensional purpose automate art parsimonious dimensional deep architecture stack auto convolutional audio product google amazon feature auto generative mapping encoder mapping reconstruction vast number study latent kernel two mapping tb model use deep encoder network low nonlinear system dynamic learn illustration encoder map low map subsequently predict access encoder decoder motivate encoder contribution paper dimensional representation dimensional datum experimental embed latent performance compare separate consider image low compactly represent since insufficient capture dynamic feature g velocity map feature dynamical turn instant neuron none height execute x try neuron miss neuron miss neuron every neuron try neuron z try center leave dim align align center input leave x neuron text height execute try every neuron try neuron neuron fill z every neuron try fill every neuron align dim center align control state input feature encoder feature map decoder identify dynamical prediction extensively identification decade measurement ahead achieve predictor relate measurement difficult special state past nonlinear corresponding predictor long somewhat work autoregressive predict predictor nonlinear function network parameter function normally estimate back additional access value compute approximate detail final illustrate none height cm execute begin miss neuron try try count miss neuron neuron miss neuron neuron try neuron neuron try optimize encoder model propagation compute auto strong encoder pca exploit encoder encoder pair encoder component layer good auto encoder machine pca dynamics link arm horizontal plane velocity serve dynamic solely pixel nine consecutive ground truth frames instant ahead prediction joint bottom plane pixel speed dimension velocity model layer neural evaluate prediction prediction precisely iteratively feature instance identification illustrate assume image row prediction auto encoder sequentially model obtain predictive step ahead prediction train optimize encoder perfect job frame ahead reconstruct ahead prediction see model prediction training separate auto perform model reason bad predictive believe auto tb joint separate training image display red validation display space separate present separate training enable dimensional dynamical feature even place enable extraction model behavior compact manner datum one dimensional would insufficient period far along output tb xlabel ylabel pos north east width height plot plot subspace separate training compare display blue great image learn parameter sequentially predict correspond poor joint give naive predictive slightly horizon compare subspace linear model restriction capture embed sub optimal frame pixel increment direction learn autoencoder use network dynamic long term illustrate validation frame accuracy datum display encode dimensional four corner corner within correspond case separate exhibit material display achieve fit error minimize fairly control interested long horizon
bandit policy round case run algorithm policie abstraction ability rich notion optimization oracle also greedy suboptimal class thompson confidence bind efficiently maintain compare algorithm barrier restrict access via use key similar distribution policy convex oracle rather algorithm program sparse general base warm start total call oracle round certain arguably simple variant contextual bandit technique let context policy joint context contextual bandit vector observe action receive observable record result round ta ta set interaction record round rx instantaneous reward maximize regret round tx differ interaction also reward estimate h tx assign possibly greater detailed agent probability condition pick history policy maximize score one policy enumeration impractical general work learner reward r r return choose randomly recommend drawing x maintain necessarily policy pick like schedule epoch schedule epoch solve constraint require rescale version regret mass exploitation require place action exploration side control estimate accurate good greedy style section constructive feasible solve require round eq epoch schedule allow initial action reward history history q algorithm analysis potential progress analysis substantially iteration give iteration express theorem output weight unchanged loop start call loop loop start historical context vector involve support compute long identify identify eq action check tx td recall round dramatically reduce epoch schedule call schedule epoch algorithm present call practically seem epoch precede computation nothing intuitively expect warm call warm start different schedule epoch schedule warm start call complexity also example cost scaling observe start represent context round operation function need obtain involve oracle cost update rescale step store constant enter scan rescale run attractive call policy specifically schedule weight epoch similarly oracle ever weight sampling action construct substantially low formally schedule q compute put entry epoch schedule deferred bind call give mode defer appendix policy control variance hand side x discrepancy compare informally ok constraint q summing apply martingale regret optimization weight write simply distribution potential epoch unnormalized vector nonnegative might context ignore combination measure far regret proportional thus encourage action regret aim algorithm appear definition intuition calculus partial roughly constraint negative decrease increase weight constraint fully minimize corresponding derivative large negative analyze argue sum weight step must weight remains challenge show significant reducing argue respect nonnegative prove substantial execute suppose weight copy potential maximize use taylor fact give cause decrease least lemma number execute call approach combine round round mean oracle less fact warm epoch epoch potential end epoch start epoch early round large least potential write epoch epoch change advantage term change relate specifically expect reward optimal high along use intuitive detail defer potential increase decrease update round round require lemma explore first cover bag dim minibatch algorithm several baseline overall still plus total section online oracle classification take class action set call answer question thereby complexity track good sample full maintain fix upon suitable square amenable implementation file due public document dataset tf treat action reward evaluation take report well explore exploit explore powerful baseline bagging predictor example replacement predictor evaluation impractical run dimension hour alternative hour somewhat surprisingly occur decay sampling impose adequate large cover baseline simple modification report use default contextual use doubly supervise multiclass simple rate report effectively loss achieve algorithmic statistically general call round remarkable work believe scalable contextual bandit directly acknowledgement thank discussion part microsoft inverse score transformation action action tv epoch particular proof sketch union probabilistic reader probability distribution nc schedule epoch high epoch epoch contain round policy epoch epoch follow apply union imply union choice allow epoch schedule q increase observe event statement probability epoch round union weight epoch constraint define need require follow outline large much hold epoch achieve satisfie inequality inequalitie reward together variance induction base triangle inequality km epoch q round optimality exist epoch hypothesis rearrange give moreover display simplify yield optimality epoch inductive imply eq apply display simplify yield complete step optimization low epoch epoch trivial lemma straightforwardly translate involve break round epoch epoch third simplify step evaluated recall q epoch schedule whenever probability r ta mt inequality union bind hold least double lemma whenever epoch constant th definition moreover epoch imply last elsewhere assume achieve km bind execution must already equality exactly recall let handle jensen concave compute change define direct taylor choice since break piece specifically algebra k throughout vector produce algorithm mean depend proof note thus lemma combine eqs statement rewrite cumulative weighted reward eq nonnegative expression epoch expression note nonnegative tx ta
convenience monotonically collection monotonically modular modular propose objective induce ground feasible set focus start exchange pose function induce detail trick build dissimilarity number image category synthetic gain fig display field demonstrate challenge categorization due foreground attack select rf suppose foreground object solve rf candidate measure building nontrivial call pyramid error pairwise distance sum pyramid simple classification foreground improve categorization image foreground object sift image region match alignment sophisticated mid level adaptively pyramid layer pool salient structure enhance classification handle gain foreground encoding segmentation essentially cast call find field pyramid mid pool predefined region however mid pattern category fail handle prominent image handle propose framework discover field mainly category merely say location find rf category accuracy note highlight mean receive computer capture foreground construct rf submodular greedy guarantee rf nontrivial mid layer pool usually dimension quantization code pyramid image low sift preserve meaningful foreground object design nonparametric match image benchmark database essential elaborate pyramid nonparametric concluding keyword framework field multiple vision field learn aid image detection discover salient region salient something meaningful require van use salient scale detect object propose spatial pooling location location foreground mid learn detector location et pool predefine pattern notable translation foreground improve performance introduce address pair require wide submodular finite fa fa fa fa aa state add help add please therein similarity recently mid svm mid generate pool pyramid demonstrated learn convolutional neural pairwise mid reliably quantization mid level sift descriptor motivates propose collaborative weak fed category rf capture foreground object training query preserve meaningful foreground rf candidate vision formalism selective suppose category generality template rf candidate candidate candidate overlap grid foreground reliably correctly part many select desirable cover worth difference rf pool grid desire rf method pool image explicitly consider scale translation extract size object detection region proposal rf force fed object appearance contrast method preserve valuable allow rf capture object salient image intra inter prior similarity pairwise besides multiple category least capture principle inter exploit similarity graph similarity candidate large measurement nontrivial elaborate put index sum meanwhile index minima follow indexing maxima sufficient h follow submodular property exist monotonically submodular benefit hereafter explicitly enable extract balance help overfitte call balance image positive understand please gx I demonstrate add achieve prefer another result balance follow monotonically bias location specifically image mild center mean search center still capture fidelity intuitively position descriptor point rf furthermore fig pyramid arrive rf index within pyramid third rf analyze descriptor grid define similarity accordingly two parameter control transformation actually similarity gaussian rf dense connect uncorrelated candidate threshold large nn similarity costly computation adopt dense extraction scheme descriptor extract complexity construct extraction descriptor consistently extremely consume either fast approximate construction incorporate detection descriptor extraction sift sift descriptor resolution calculate rf candidate similarity dense extraction produce unnecessary meaningful sift descriptor instance reflect principle bias intra field classifier incorporate rf put grid denote descriptor feed image sift rf category exploit tree image qualitatively validate effectiveness discover public benchmark evaluate object categorization generate first object meanwhile away intersection expect point intersection please center bias prior set control parameter gain bottom gain point intersection gain inter image demonstrate effectiveness approach correlate foreground object contain category intra variability location variability scale resolution aspect benchmark class randomly exactly extract split r lc several one mid level represent image network lc base locality constrain linear feature learn mid pyramid kernel svm lc deep classifier dictionary mid level codebook sift encode codebook quantization sparse code feature pyramid pool image neighbor closely relate use sift comparison illustration select outperform sift detection remove noisy descriptor position sift descriptor fail notable change translation reason object find behind small similarity construct find performance visualization suffer guess objective small contribute moreover image merely ensure center constraint foreground image help construct essentially normalize divide transform control plot database meaningful outcome curve demonstrate work image foreground object category exploit model select vertex suffice guarantee propose pyramid pairwise merely preserve foreground object final try result research similarity feature propose even construction sophisticated representation consideration efficiency learn deep
entropy play role non mechanic find field hierarchical structure e et al zhang document classification community shannon address nature specie interaction specie individual system thorough interpretation diversity effective specie index replace relative maximum likelihood bias community rare specie unobserve perspective diversity hypothesis specie first shannon symmetric first technique analogous prior neural response dirichlet impose narrow shannon priors diversity exhaustive intensive general prior specie possibly infinite preliminary contribution variance distribution et able result moment prior parameter refer extremely modern tractable dirichlet relate bayesian et posterior discovery large diversity notice briefly class infinite tuple atom exhaustive account random partition infinite satisfy relation usual specie observe th new unobserved specie kp partition belong symmetric poisson family mix discrete distribution characterize formula dirichlet prior exponentially normalize generalized belong derivation result paper stress allow nevertheless explicitly two moment shannon entropy induce et poisson infinite dimensional prior belong recursion index easily shannon generalize already function r derivation third omit size specific prior stick break kind j second moment shannon adopt poisson dirichlet prior inverse breaking recently size atom difficult close entire allow gibbs weight dirichlet xx arise respectively follow et stick break biased atom follow easy eq shannon dirichlet numerically prior suitably formula comparison purpose standardize shannon prior guess index diversity effect prior guess concentration around behaviour shannon al poisson dirichlet combination ht prior standardized index dirichlet stick break biased concentration symmetry produce fisher prior family law random discrete gibbs mix simplex shift species parametrization bayesian application formula index shannon calculate partial derivative respect sake brevity omit explicit formula prior shannon fisher characterize prior put population appear flat uninformative suggest shannon entropy parameter theorem et sect moment provide inequality study posterior moment binomial specie observe relative moment generalize entropy general substantial theorem moment close extremely calculate application uncertainty derive high density interval simulate observe relative dirichlet index shannon n variance moment mind agree moment posterior prior arise formula variance may mind generality evaluation frequentist bayesian already nevertheless provide procedure conduct bayesian count collect site refer hill count formula realization size observe give specie interval estimate shannon interval likelihood ml correct fisher table nonparametric thompson estimator adjust coverage adjust specie preferable frequentist fraction parameter nonparametric derive sample thompson correct account miss greatly choose robust conclusion population independently stress propose account presence unseen prior place theoretically relative finitely update unseen fisher prior low suggest posterior moment posterior summarize shannon index kind use notion rank atom discover th least eq still almost surely th integer pn v determine integer moment recall h recalling eq limit theorem specie prior relative specie moment follow one particular allocation integer q integer sum partition function analogously integer suitably countable cs intensive bias indicator species pr partition nonparametric prior cm marginal gibbs sample diversity specie cm neutral species dirichlet break representation prior discovery formulae gibbs ann relation species cm shannon species finitely exchangeable partition bayes entropy description study site size shannon wiener generalize cm hierarchical model inverse nonparametric discover specie
invariant geometrically transformation could expect world set sized texture mix obtain patch descriptor gray value class absolute sized descriptor use length histogram histogram bin size vector texture retrieval representation patch texture patch descriptor enable fig texture represent color visualization pca texture encourage cluster enable quantitative assessment texture patch compact descriptor accuracy accuracy lead provide comparable magnitude descriptor effectively obtain accuracy apply consist sampling patch multiple texture represent use descriptor color texture compare transformation descriptor assign texture inferior texture significantly texture connect mathematical transform class particular pool thereby develop histogram learn great discriminative illustrative world texture definition matrix canonical basis combine standard invert determined write pair recurrence deduce combine similarly zero da da da da tb prove result deduce meanwhile property trace make definition rotation achieve vertical coordinate horizontal coordinate index tuple implie combine translation coordinate give cyclic translation writing meanwhile imply combine give patch give cyclic rotation partition eq equivalence coefficient class claim mm mm mm mm mm thompson edu thompson ac uk transform transform hadamard multiscale dyadic show appear different phase change effectiveness demonstrate invariant invariance coefficient basis algorithm thus autocorrelation covariance sign group permutation view discrete unitary great approach autocorrelation describe multiscale texture operator harmonic continuous study ambiguity important role code new connection signal describe hadamard autocorrelation powerful tool detect impose shift invariance variety code removal include face texture pattern recognition autocorrelation cyclic dyadic suit represent multiscale texture hadamard suited counterpart wiener dyadic autocorrelation aforementioned autocorrelation invariance transform patch follow color patch sample texture though exhibit obvious translation able texture display sample patch break patch transform absolute patch display transform patch section transform invariant dyadic illustrate theory example transform pooling invariance significant distinguish class transform summary theory transform result invariance multiscale transformation inner section hadamard transform show material suit application give tuple multiscale group detail example matrix ht isometry whenever unchanged sign permutation property representation detect illustration figure simple texture identify come texture transform demonstrate ability classify patch capture describe moreover multiscale ensure transform patch texture exhibit sign matrix function transform equally view familiar tool mean hadamard inner sign detail think multiscale rotation four symmetry multiplication set label vector positive example permutation dyadic multiscale term give coarse give fine first display change product second row pattern hadamard third fig example multiplication associate inner frobenius b da b q signal matrix expand connect autocorrelation convenient expand orthonormal isometry next bridge autocorrelation equally hadamard transform index autocorrelation band information invariance binary subset hadamard index index tuple characterize combination hadamard autocorrelation define appendix fundamental transform way exploit transform equivalence coefficient exploit transform coefficient significant distinguish class ht absolute coefficient invariant say allow absolute value cyclic absolute pooling build invariance transformation often pool provide principled partition within follow absolute coefficient give equivalence average
huge assume infeasible homogeneous kind audio stream divide duration feature extract frame model compute segment acoustic pass slide window similarity measurement segment identify coarse present news language air air access language follow describe section consist acoustic call acoustic change common detection divide audio segment small speech bic adjacent high segment acoustic source adjacent segment duration different news audio air ground segmentation obtain toolbox news bic audio segmentation audio bic literature face automatic segmentation inconsistent result identify acoustic next news audio news length read propose measurement propose two technique news audio anchor identify change stream duration second use point detect pass depict pass feature vector feature combine together group second group model calculate criterion extensively segmentation metric efficiency window co vector segment dissimilarity show audio audio segment see image represent acoustic audio evident technique reliably audio stream detect change pass proportional segment point obtain second perform second pass detect news actual center stream audio extract audio literature automatic segmentation pass pass coarse acoustic point acoustic self identify acoustic propose audio segmentation audio news music actual news automatic audio news time align build corpora address read map text exist sub solution segment acoustic reliably acoustic pass audio process audio audio stream audio pre index news movie etc segmentation corresponding audio music news addition speech automatic extraction news segment correspond speech
avoid avoid explicit covariance enkf kalman filter forecast observation ensemble filter seek specifically posterior ensemble inverse minimum operates sequentially apply forecast eq scale perturbation ensemble covariance square background speed carry variable ensemble k solution read perturbation represent analysis operator linearize consequently linear operator jacobian observation give multimodal carlo mcmc algorithms metropolis distribution complex density invariant mcmc generating proposal accept reject generally powerful may hybrid present filter assimilation hybrid carlo hmc know hamiltonian monte physics attempt drawback reduce explore sample hamiltonian operate phase total describe hamiltonian dynamic differential evolution map computation flow replace reversible integration method five st stage stage take abuse solution draw probability make analogy hamiltonian auxiliary momentum hamiltonian logarithm target probability auxiliary momentum mass matrix hamiltonian canonical equal show momentum variable hmc algorithm build initial summarize state issue numerical represent mass impact final affect diagonal efficient draw numerical current increment energy hamiltonian stage stage integration discard draw variable accept proposal continue many distinct draw filter enkf representative member even assimilation forecast linearize member draw estimate filter principle remove logarithm alternative enkf describe assimilation sampling stage k forecast member next result forecast pdf providing give ensemble follow chain forecast enkf acceptable choice stationary calculate ensemble base forecast frequently emphasize build full definite vary member warm entire sample covariance increase state necessary typically fix ensemble member ensemble matrix flow lead water set diagonal variable describe x index circular fashion experiment component range simulation algorithm synthetic create reference background system different observation operator complexity level linearity six cubic obtain trajectory magnitude state square component operator differentiable absolute et highly nonlinear scaling control nonlinearity model step size sampling sampling first test guarantee satisfactory take cost tune trial observation chosen number calculation filter numerical realization potential acceptance different metric analysis observation reference assimilation trajectory span reach perform burn noticed converge number burn stationarity work burn member generate decrease generate ensemble retain number usually inter chain parameter control requirement upper need order step consequently stable ergodicity markov chain length begin hamiltonian step benefit ensure result analysis system instance filter median instance central variance vertical central box length outlier plot outlier exception hilbert rmse enkf closely indicate inter instance box rmse central represents extend height number reason size enkf unit chain rmse filter plot represent median instance blue height central box consider plot show representative sample satisfactory rmse red failure outlier hilbert space enkf suffer outlier indicate panel inter rmse sampling filter plot median variance vertical line height plot red scheme rmse make divergence high continue define hilbert fail rmse tune give satisfactory indicate panel time indicate inter rmse sampling filter box times length plot cubic enkf due sensitivity level nonlinearity operator filter sampling converge stage satisfactory h indicate inter filter rmse across central blue vertical extend outlier plot red fine convergence see reduce size figure lead use indicate step rmse instance filter box plot red median rmse represent vertical extend height red observation operator jacobian operator sign experiment mostly filter hilbert forecast enkf analysis almost identically quadratic operator h rmse filter show rmse central represent times box consider outlier result obtain high bad rmse increase panel time indicate step red line instance blue height central box observation perform sampling filter fail observation uncertainty level linear ensemble small reasonable panel inter red line rmse values box vertical times height box plot size four linearity outlier give satisfactory even analysis select h use panel indicate inter chain step box rmse central variance line times height box outlier red jacobian difference alternative factor differentiable perturbation nonlinear enkf performance performance factor indicate inter rmse box across instance central represent variance vertical time height box outlier plot tuning notable improvement stage behave change panel inter step plot represent across central box variance time central plot red sensitive uncertainty degree nonlinearity test observe test performance performance level observation level state frequency frequency different indicate background observation htbp frequency state background observation standard different indicate observation deviation htbp panel deviation htbp frequency background operator parameter burn tune step refer step test optimal suggest realization performance integration outlier infer principle enhance inter indicate careful tuning step number step lead filter deviation error panel box plot line rmse instance blue represents extend height plot red controlling setting number use validate ensemble hilbert suffer outlier assimilation several exclude combine ensemble care density alternative future test capability challenge factor observation perturbation change correspond measurement enkf increase length well short hamiltonian four prove observation satisfactory large step size indicate indicate panel inter instance box median rmse vertical times height box red hilbert rmse relatively acceptable reasonably closely plot chain figure obtain good result hilbert low stage outlier chain summarize enkf observation operator show size setting name short version respectively sampling filter include deviation step filter water sphere water equation provide simplified model describe mechanism angular longitudinal height homogeneous wind discretization discretization longitudinal vector combine wind wind height integration adaptive reference trajectory synthetic add three observation wind magnitude wind create reference noise model deviation background magnitude reference condition background wind wind model background account create ensemble kalman hour ensemble create add condition covariance ensemble base covariance covariance method totally perform well algorithm make future propose filter burn stage step step enkf sampling order outperform enkf filter study moreover forecast covariance h hamiltonian number step c observation min std quadratic min std observation threshold min max observation operator min std min std h c cm std std std std std propose assimilation posterior sampling avoid need develop adjoint solution operator nonlinear variance offer analysis implementation require matrix attractive assimilation operational experiment carry linearity enkf outperform enkf continue satisfactory case enkf assimilation machine challenge failure subset terminate member run ensemble member considerably herein replace implementation enkf ensemble member enkf member posterior probability add new direction computational integration successive ensemble tuning context operational resolution perform comparison acknowledgment fa computational present five numerical position sensitive choice position lead test design state space subtract infinitely total system hamiltonian system time experiment whenever target numerical nonlinear operators equation time stability achieve step one three advance hamiltonian time advance
small approximate mechanism generate matlab simulation numerical device hold device memory expect use large whenever measurement interestingly bad moment execute communication author binary nonzero normalize eq although conduct result confirm e perfect measurement use general recall collect measurement I ratio consider procedure variable compute derive formula tight eq turn expectation derive let complexity require write choose appendix bad analysis interestingly attain nonzero suffice measurement note hold know know bad would still change binary e convenience error binary essentially least simulation study least poisson use replace require measurement suffice hope demonstrate poisson illustrate reader lemma along poisson confirm accurate unless low case tool closer range bottom curve low figure small panel fix closely interestingly useful symmetry attain confirm suffice choose compress compressed sense stable interesting nonnegative highly entry nonzero entry stable maximally skewed theoretical extremely away preferable complete proof similarly conclude note furthermore thus increase complete computer university statistic university nj usa department nj usa recovery often nonnegative develop adopt compressed design maximally skewed average fraction become stable summary dense
original description kind scenario occur search efficient code case describe analytically happen sampler ask produce program originally dataset text generative text list lambda assume assume assume apply observe flip bernoulli program flip predict program text program result novel argument program abc lambda std std assume poisson lambda exp begin expression expansion program univariate fed illustrate sampler account endow argument entire family box parameterize refer conditional sampler abc begin assumption truly improper program argument generalize abc time summary individual summary take value correspond statistic hypothesis hypothesis equivalent coin turn penalty accumulate line aside probabilistic programming language program expressive level environment signature express random type constant integer crp discrete process prior base real mixture uniform common primitive compound compound sample representation discrete dirichlet base generate compound procedure count compound production rule environment incorporate input name name type current compound avoid program return possible manually took write translate common sampler example require single production corpus sampling corpus production smoothed prior couple inference programming perspective result approach abc penalty source final preliminary probabilistic production employ probabilistic six histogram sample program randomly sampler domain variance mode repeatedly blue histogram exact leave plot infer inference converge code abc lambda safe par par lambda stack safe par par bottom infer program text assign use sampler program continuous feature infer two kolmogorov test vs analogously histogram program program express program salient characteristic text bayesian costly posterior particularly repeat posterior predictive inference representation option particularly order language program aim probabilistic interest encourage take beta metropolis give successful trial probabilistic probabilistic repeatedly produce statistically distribution infer probabilistic analytical include program text generative tb observe flip flip observe flip program abc safe safe beta safe safe predict beta binomial interest top salient probabilistic induce probabilistic program bottom exactly analytical posterior posterior sampler indeed close exact novel synthesis raise answer key really synthesis synthesis inference goal program search single generative program possess characteristic length etc program text intractable future basically available employ latter whose match know actual temperature schedule ergodic require result certainly surprising good job report open well particularly goal genetic way cumulative incremental program convenient text normal already learn subroutine inductive gain structure continue internal experience ability text match human seem something powerful piece intelligence inclusion generalise human reasoning suggestion comment van david college conclusion author necessarily reflect air agreement fa u reproduce view conclusion herein policy express air laboratory false frank department science united institute mathematics program encourage empirical suggest technique probabilistic language sufficiently powerful enable also might future program complete high probabilistic language simultaneously procedure procedure text programming description merely particularly degenerate probabilistic programming language possibility text generative high paper account effort directly sampler similar observational datum potential collection distribution box sampler automate discovery might perform leave bernoulli impose hierarchical sampling procedure fit human random variate somewhat mh probabilistic forward generation generate ideally program result program aim program evaluate generate generalize posterior expressive family suffer distribution valid marginal high programming probabilistic program effort step probabilistic program relate former treat text find exactly match latter generalize either introduction modern program specify equation use traditional force enumeration find inductive logic genetic constraint language support choice make logic lambda theoretical program inference unlike learn program sample observation input pair generalize main objective statistic field learn text say parametric program code structure sense manner observational
collapse implicitly dimensional come automatically quite approach number low way crucial computationally require runtime obtain train dr seek see dr nets model dr learning metric dr projection quite rely objective lda input space class scatter minimize scatter maximize latent among variation manifold space use dr fed disadvantage filter objective dr proxy show filter place corner closely dr equivalently project nearby latent apart achieve metric however metric solve semidefinite learn mahalanobis optimality long approach dr unify far generalized hinge svm extract closely dictionary learn always latent implicit new need contrast explicit mapping test operate linearly nest linearity svd would train mapping reduction regressor jointly nonlinear result iterate provable objective filter approach secondary dr mapping separability scatter little classification optimize place collapse maximally dr algorithm generalize specific dr combination extraction mapping net jointly optimize classifier acknowledgment part nsf award section lemma prop thm corollary definition conjecture conjecture electrical computer science california false dimensionality dr use preprocesse classification first learn obtain optimize classification jointly particularly dr method algorithm train rbf svm step svm usual close art runtime dr mapping latent class train jointly dr extreme tends manifold centroid linearly maximum dr preprocessing task low dimensional costly importantly learn particularly dataset avoid reason dr remove uncorrelated mostly away dr adequate freedom label dr inform call supervise dr input dl vector supervise learn train input supervise dr sense minimal intra scatter scatter supervise usually encourage separate input manifold make filter even pca proxy real objective minimize particularly dr filter considerably nonconvex particularly important question arise propose generic jointly optimize loss rbf armed dr classifier latent apply filter nonlinear svm achieve art short version appear conference paper describe svm pattern dy n want optimize usual separate hyperplane give term weight bias linear slack difficulty heavily simplified use introduce nested idea break functional follow prove seem trick optimize original target rbf next algorithm svm svm ordinary work exist scalable svm train independently regularize low generic net focus special include radial universal commonly unique memory mainly drive involve gram exactly reduce warm cholesky run include slack alternate scalar cost kkt lagrangian kkt pass reduce kkt express dual optimum achieve solution summarize step contour increase red set plot corner dot denote dot step several vs multiclass vs decision label determine svms objective function parameter slack svm consideration variable typically active solve matlab toolbox binary class high test involve step vs take vs svms jointly optimize classification becomes iterate subproblem rbf form update remarkably although optima initial hyperparameter margin reduce initial quadratic parameter increase time early iteration validation go improve fast iteration suffice hyperparameter usual mapping massive parallelization suitable latent versus class indeed processor center latent state art consider ideal infinitely l summary approximate ideal extent close maximally separable finding map trivial seek piecewise constant hard learn pca lda test split standard neighbor kernel svms inputs unsupervise pca kernel lda neighbor hyperparameter algorithm svms choose mac hardware remove word appearing reduce extract pca create split item validation hyperparameter try margin mean standard rate classification dimension bring cost fix dimension superior outperform consistently binary mnist include balanced validation set separable infer svm nonparametric nonlinear dr explore two pca validate pca lda use demonstrate incorporate regularization improve generalization neighbor bottom projection training algorithm perfectly use due power lda svm though remove noise reduce show pca training c error projection classify digit original evaluating require huge store solve center choose hyperparameter unnecessary hyperparameter careful kernel width parameter width width svms explore lda svm mnist experiment lda poorly perform pca initialization rate vector use kernel time basis obtain large speedup explore experiment lda poorly svm similar fig bottom different dimension error quickly much configuration different projection bottom visualize latent lie dimension overlap completely separate see view nearest pca gaussian gaussian processor speedup function early latent avoid subset digit visual comparison vs parallel toolbox scheme alternate scheme optimize rbf progress actual runtime mnist odd minimize svm quadratic dimensionality weight solve solver nest iteration spend alternate alternate iteration suffice find optimal small eliminate increase progress alternate slow nonlinear gain algorithm nonlinearity thank auxiliary coordinate
edge hypergraph incidence none algebraic constraint finitely individual become interested characterize incidence framework check independence variety compute gr adapt polynomial incidence hypergraph show subspace take system unique incidence incidence hypergraph learn give point datum conversely support tight hypergraph finitely quantify corollary pick word straightforward set pick random sphere denote index row minor column represent minor set deriving equation solve span incidence constraint k x sd ss underlie incidence equivalently independent set parameterize put space give subspace degree freedom leave potentially remove section incidence generally existence system exist finitely real complex solution system incidence become I framework characterize incidence check ideal generate variety computationally well linearize singular algebraic incidence independence maximal geometry property generic intuitively dense variety generic property framework hypergraph property framework hypergraph alone framework framework variety avoid framework relate restrictive ideally variety possible necessary explicitly easily give appropriate hypergraph area purely draw combinatorial I variety combinatorial go theorem follow combinatorial incidence hypergraph furthermore capture incidence existence jacobian algebraic matrix space trivial incidence take jacobian coordinate jacobian corresponding give lie let oriented form coordinate add v td notice volume coordinate ct kt q incidence j j three jacobian jacobian form incidence jacobian row row per framework realization way subspace chooses pick simplicity pattern show regular sketch regular state exist jacobian algebraic isolate could explicit component imply take jacobian less corresponding hypergraph theorem prove follow define matrix graph index index accord map graph one correspondence loss switch variable name determinant notice summation ready expand hypergraph union incidence determinant trivial trivial subgraph edge copy arrange pattern hypergraph group belong coordinate laplace rewrite determinant coordinate separately row observe zero expand hypergraph decompose map contain entry recall decomposition always correspond row zero copy particular pick row row non zero lemma minor non observe full lemma incidence expand hypergraph form inside decomposition expand hypergraph behave case non combinatorial characterization pure calculate open avoid pure span otherwise expand tight pure give notion dense generic expand give hence pure subspace position original jacobian solution converse since generic independence datum point sphere corollary dictionary quantify pure fail framework subset solution system pick sphere less construct dictionary pick major construct underlie hypergraph subspace ss stage expand construct minimal hypergraph small integer vertex constant least subgraph word structure verify vertex construct hypergraph conversely hypergraph put vertex remove vertex tail contain one move one slightly expand hypergraph add edge copy additionally count expand construct put add move one shift move inside take os entire hypergraph iterate therefore regard construction underlie hypergraph get incidence arbitrarily incidence arbitrarily coordinate full row maximally pure theorem algebraic jacobian fail pick find similarly entire take time complexity paper point incidence completely characterize hypergraph recover number corollary main pick algorithm additionally independent claim theorem wang sparse obtaining upon geometry span hypergraph characterize underlie specifically incidence isolate specify combinatorial systematic algorithm performance datum point know dictionary satisfy consideration interested relative dictionary arise context process machine vector recovery minimize represents lagrangian iterate start solve pursuit update convex overcomplete ill point difficult reduction exact problem dictionary np directly though np learning solve produce dictionary selection alternate mod iterative formalism mod posteriori recover iterative truncate singular take atom atom form orthonormal sparse stage relaxation minimization well theoretical true several algorithm strong constraint minimization find dictionary require basis al provable overcomplete via iterative svd dictionary provable however overlap dictionary frame dx xx x vector resp resp dd hypergraph fit point dictionary hypergraph machine complete characterization yield solution dictionary e isolated dictionary specify size related however highly pick dictionary sufficiently sufficiently provide systematic relate together approach require follow combinatorial character incidence system dictionary dictionary system another know characterize hypergraph give call pure dedicated specific framework instead subspace directly linearize jacobian uniform generalize non impose systematic increasingly constraint classify whole independently interesting learning find restriction dictionary dimensional subspace point dimensional union frame satisfy lie basis deal find subspace residual iterate point robust method pca algebraic union homogeneous fit determine obtain small span subspace set specify give intersection necessarily general closely relate intersection condition come dictionary union small span subspace outside pairwise intersection directly recursive small set dense problem follow step solve decomposition always apply iteratively high step problem follow span class subspace
nucleotide cycle positive identical number equivalently seq length original sequence express nucleotide probability constraint signal range infinity frequently expansion power series bivariate positive denote nucleotide flow cycle show nucleotide flow flow cccc specify complete nucleotide incorporation nucleotide flow recurrence seq cycle cycle seq nucleotide flow seq obviously recurrence seq nucleotide flow cycle nucleotide cycle reason solve close form solve initial condition extract appropriate symmetric nucleotide flow cycle table together nucleotide probability elementary focus symmetric value add need probability give second sum sum probability become seq seq row factor expansion nucleotide extra small practically ignore big contribution clarity value eq converge upon dominant eq quickly expansion q availability make include number cycle cycle variance positive cycles eqs calculated series eqs dominant q interesting show eq signal twice cycle probability linearly flow cycle linear growth size combinatorial system govern naturally distribution detailed variance plot nucleotide probability nucleotide eqs exact calculate eqs distribution respectively exact accurately normal distribution variance nucleotide probability reach positive nucleotide broad nucleotide probability fact reach eqs normal equal nucleotide cycle number use fix calculate point dominant keep term ignore depend nucleotide probability linearly perturbation theory nucleotide perturbation theory cycle nucleotide nucleotide eqs curve calculate nucleotide accurately equal calculate eqs curve normal calculate nucleotide respectively instead fix seq cycle process create infinite let flow cycle seq flow cycle seq nucleotide incorporation infinite nucleotide flow nucleotide nucleotide nucleotide within nucleotide cycle length seq flow seq first signal must happen signal flow base stop nucleotide base part ix nice property mean calculate coefficient differ long nucleotide nucleotide linear still limit gaussian linear check analytical develop program seed give close simulation website simulation derive cycle signal run variance average put relation example first eq put fact get e e acknowledgement clinical science rr national center resource health sequence distribution signal sequence cycle nucleotide derive cycle nucleotide software development next generation research next sequencing length sequence dna kind add nucleotide nucleotide complementary dna incorporate intensity complementary template dna template nucleotide flow axis nucleotide signal correspond reflect nucleotide incorporation activity nucleotide incorporation unnecessary name usual flow flow zero three third flow signal use nucleotide table q call consecutive object various field r sequence assumption actual follow similar signal study first proper function readily yield realistic instead seq fix
pattern accuracy classify datum label fp limitation test set model yield accuracy parameter reaction system reaction trajectory optimize model include synthesis slightly abuse terminology say imply maximize range maximize find greedy quantization expensive simulate quantitative checking particle fitness operate particular require fitness consider reaction diffusion system formula l design implement induced processor ghz simulating pattern ls pattern explain maximize induced pattern optimize parameter simulate set optimize optimize formula negative new newly ht l cm hx xt maximize produce formula system system optimize repeat process user terminate iteration similar one optimize simulate simulate terminate superposition whose interpret partition efficiently quantitative semantic combine develop supervise synthesis experiment version biology several direction moment version exploit semantic plan multiple branch third method locally interact expect experimental technique biology circuit section assumption propose dynamical central novel superposition logic semantic image logic perform integrate checking algorithm particle synthesis reaction form single everywhere nature formation origin biology self though diverse biology physics pattern recognition usually formulate characterize structural area pattern formal foundation formal semantic pattern locally interact synthesis rule interaction strategy draw checking follow locally interact dynamical system give pattern network state base superposition spatial superposition logic tree partition decision image descriptor infer example parameter desire optimization fitness function semantic logic formula propose logic encounter pattern principle locally interact reaction diffusion pattern produce automatically paper organize section semantic formula generation check optimization remark technique class goal main input represent several survey accurate descriptor choose depend related pattern descriptor concern intensity gradient appearance interest rotation feature descriptor contour verification pattern verification possibly behavioral pattern logic formula descriptor verification logical descriptor spatial characterize texture combine pattern descriptor use modal logical operator intuitive inspire author superposition logic existence representative representative logic captures pattern consider rather oppose logic fitness search produce quantitative semantic discount inspire notable difference metric main pattern recognition quantify produce desire pattern quantitative semantic pattern denote negative integer x spatially rectangular specie define n dynamic system diffusion specie j indices vector reaction diffusion jt interest specie observable q example protein infer analyze observation steady steady check x n tt system trajectory steady observation trajectory reaction diffusion reaction diffusion location depend concentration cell fitness inherent ability operate space fitness finally solve application step decide user reaction diffusion element small tuple represent concentration observable specie within sub select row indice tree vertex sub child vertex example represent tree direction sub child north west north south se htbp rv se ts ts vs lf l ts v b ts v vs vs fs lf ts ts space vertex equivalent observable species superposition region avoid observation inspire author aim would four child hold proof easily expand matrix equal tree htbp notion transition classical allow check ts ts sm se ls start tree labeling ts generate self give represent direction give label path example bs ex set eventually eventually globally operator logic resemble main operator resolution select spatial direction allow work operate qualitative semantic spatial formula pattern yes qualitative semantic write follow qualitative semantic check spatial violate satisfied may guide exploration generation reason quantitative measure spirit region discount transition take quantitative semantic define follow semantic q proof structural
researcher extensive community challenge free free problem employ quadratic problem develop experience rl constrain zhang robust scheme nonlinear dynamic programming base estimate derivative construction introduce control problem work rarely system system arrange present subsequently brief conclusion appropriate definite appropriate dimension function derivative compact x tx dx x eq x continuous closed loop asymptotically generalize horizon functional briefly convergence establish optimal start admissible admissible respect continuous noting notation side along optimal nonlinear equation observe system prevent approach control action name function rewrite note represent action control control expression pt initial solve equation control involve basic operator improvement current control policy implement learn optimal policy design system approach refer exploratory exploration insensitive behave collect method demonstrate control ix ix q yield improvement mathematical induction accord satisfie hold lyapunov derivative along admissible policy admissible u generate ix ix ix u define follow similar eq expression sequence consider monotone always part ix theorem substitution u prove demonstrate residual develop linearly function solution iterative q tx x estimate truncation yield residual ix l x ix force weighted integral residual lx lx name substitution notation integration competitive integration computing lx lx similarly substitution domain accordingly iterative set vector index else go rl involve framework policy iteration section model control let policy continue generate q ix u induction stable policy accord mean mean hold hold similarly theorem proof complete solve algorithm avoid repeat omit derivation update law lx lx jx law implementation procedure present compute positive stop else go back implementation convergent employ real admissible control necessity converge matrix block denote policy give solve free policy continue pt generate iteration iterative equation solve unknown rewrite collect system least scheme vector square similarly error expression function expression solve pt q unknown continue algorithm expression equation equivalently vector square implementation omit accuracy vector angle attack unit quadratic matlab care parameter update figure algorithm achieve show dot gain observe obtain algorithm model vector converge figure gain dot line gain simulation fast converse follow function policy iterative ix x k free system convergence vector converge iterative gain dot represent note good convergent show close loop employ figure
hence weak task obtain open problem context recovery natural suffice question negative guarantee exist property claim appendix ht ccc dot line fraction solid line vary entry fraction recovery spectral success phase transition r success much ht frobenius vs gap prediction plot norm vs comparison compare synthetic mention operator gap block basic cluster sample spectral gap spectral small cluster case depend fix vary give go spectral gap go demonstrate trend augment lagrangian alm nuclear success ratio figure plot spectral gap dot success line color gap trajectory gap increase color indicate sample positively correlate exhibit type success ratio conduct reduce end generate gaussian let frobenius gap noisy lead error output temperature day test matrix algorithm singular output spectral rank compare closure guarantee proof hence adversarial contrast work dual dual true matrix optimum optimum notation require proof note generalize span projection operator follow write orthogonal complement dual construction show operator stress proof dual later incoherent generate satisfie pt characterize incoherent incoherent eq satisfy incoherent e z lemma provide characterize dual let respectively unique satisfie ready construct recover scheme construction give c r lemma satisfy trivially inequality q r tw uv tc contraction inequality prove recovery spectral guarantee strong sample result matrix analyse strong incoherence index give spectral result recovery incoherent alone property information plan guarantee signal definition conjecture edu microsoft research microsoft com matrix lot new provide universal scheme come spectral exactly recover satisfy incoherence uniformly recover matrix require several recommendation system quantum recently provable solve incoherent significantly second relatively might desirable processing application rank reduce large bipartite similar result large require explicit vector later strictly weak coincide psd certain strong incoherence require nuclear gap graph suffice apply matrix incoherence spectral exact incoherence universal alone universal incoherence property really particular block show success irrespective capital letter letter th represent format frobenius represent unit context organization discuss define bipartite use require require storage individual sparse application result universal critical difference approach highlight algebraic analyze contrast minimization sampling generalization exact recover index bipartite bipartite hadamard recovery algorithm completion recovery regular bipartite graph g note regular eigenvalue adjacency definition term singular property graph property decrease bipartite study generate briefly couple
due inherent social graph individual create identify interest although label security exist technique anomaly limited graph unable reason identify easily internal ip external alarm common contact save unnecessary anomaly level give datum three probabilistic rely recent enable improve structure detect estimate see newly observe anomalous simulation streaming parameter detector new update define new detector detect anomaly important inaccurate use share copy distinguish evidence node subgraph accurate detector establish detect conference application naturally detector node team conference finally interactive visualization analysis enable easily focus critical change datum identify graph common problem neither anomalous part static since availability label problem transform g et disjoint union subgraph anomaly finding anomaly technique compression rely minimum detect subgraph anomaly search subgraph almost hypothesis broad work residual towards detect mat dynamic include connect detector design anomaly cause gaussian anomaly focus kronecker work change generative introduce anomaly change overall structure detector tool explore nature visualization multi tool explore inform tuned star pattern region integrate multi extend anomaly flow et al method observation new anomalous sufficiently value value new notice side anomalous streaming model light acceptable positive identify operational user utilize simultaneously anomaly community significant devoted develop model capture broad require accommodate stochastic introduce membership generate intra community os enyi er community membership flexible degree os determine world occur er intra follow degree size describe version original generalize level os r enyi degree partition subset community intra sample os enyi er formally ic specifically note edge define degree exceed cl calculate community recall set exactly let denote see different community edge original great internal er allow assume depend node expected occur probability expensive anomaly describe sequence label technique infer input assignment edge density model probabilistic anomaly detector vertex community scale apply graph construct occurrence weight suffice survey scalability grouping world edge density estimate model os enyi er seek within subgraph assume beta lastly estimate poisson yield ig mode posterior gamma define leverage anomaly graph subgraph define directly inherent multi subgraph node intuitive limitation multi second detector build define probability external subgraph member nod multi baseline detect fitting baseline anomalous discriminate anomaly subgraph section datum first detector anomaly given value detect anomaly graph subgraph subgraph hence allow anomaly particular anomaly upon poor mixed geodesic employ geodesic strong baseline natural application capability model anomalous graph anomaly perturb anomalous graph anomaly streaming anomaly anomaly graph value anomaly detector label anomalous fall similarly detector lastly detector include conduct use ten sized degree vary eight accord create anomaly experiment three community six anomalous node anomalous anomalous see hold constant change node decrease intra extra four community together anomalous anomalous node anomalous receiver roc curve area auc display contribute community level r pr roc precision see dominate category inferior winner superior expect cccc cccc team team acc acc st big pac pac east e big pac big big big east mac st pac st pac acc acc st pac acc st big acc sec positive entry false ccc ccc acc pac big pac east sec mac mac acc pac big east illustrate scale real world statistic represent graph division team game play detection fitting year detector newly observe update newly observe detector apply two ground conference schedule within conference change graph produce value expect scale discuss community detect markov weight appropriate markov clustering identically posteriori refer conference name discussion detector detector identify anomalous anomalous graph numerical attain statistic detector indicate sample probable graph addition identify anomalous conference graph anomaly rank anomalous detector conference membership detect maximally anomalous negative conference precision detector anomalous decrease membership notice anomalous short graph anomalous anomaly scale detector user focus attention community interactive visualization tool fine grain anomalous graph figure illustrate prototype visualization figure community visualization provide little insight anomalous section alternatively indicate community figure allow conference name display contextual domain conference display inter conference indicate interactive anomaly west conference graph immediately apparent conference figure outside conference change confirm conference detection readily interactive interest resolve hypothesis anomaly occur force discovery anomaly team conference post color address identify anomaly emphasis anomaly context identify anomaly hierarchical allow community hierarchical streaming anomaly base describe community detector produce accuracy truth additionally detector gaussian insight scale capability superior multi experiment accurately anomaly subgraph expectation visualization inform give sample enable discovery anomaly occur scalability address community agnostic mention community context secondly estimate require calculate space aid optimize gain
via twice complexity variant introduce regressor performance twice robust also proof hierarchical describe string node label emphasize string child refer child string say stre empty string string string l nod leaf node give regressor presentation start root entire space regressor small region observe child regressor incremental hierarchical regressor e hierarchical embed regressor linear output model expert efficient assign tree present output significantly reduce certain regularity piecewise incremental tree adaptive incremental beginning combination output expert suffer sequentially deterministic upper introduce twice universal e universal though fine appear increase parameter incremental start begin single root node instant find increment generate node divide region disjoint plane child regressor child accumulate regressor vector child child regressor evolution node dark light regressor depth correspond regressor divide find regressor incremental linear regressor accord regret region minimize issue section incremental structure regressor incremental represent linear observe piecewise whereas perform combine piecewise piecewise model incremental output set expert achieve model well differentiable piecewise increase exponentially sense optimization framework final w iw hence combination entire e force online practically considerably problem assign instead illustrate calculate entire new leaf assign performance represent regressor regressor inner definition construct datum require use universal maximize respect adaptive incremental correspond weight performance optimal piecewise next sequential achieving end demonstrate great conclude sequential structural update growth complete regressor compactly root node regressor letter reveal weight root calculate mean concave obtain regressor performance final estimate calculate eq performance incremental regret batch region regret follow arbitrary model incremental th piecewise regressor vx appropriate sized identity xx regressor calculate region nonlinear note organization piecewise algorithm batch piecewise regret prove bind conclude conclude letter find calculate weight estimation p w incremental leaf fig see fig partitioning method light therefore tree case length theoretically complexity life regressor stationary practical bound discuss order achieve require sufficient regressor evenly regressor remark algorithm piecewise partition algorithm regret indicate introduce asymptotically however intuitively justified regressor fall piecewise mention piecewise tree limitation accord divide disjoint force computation accumulate regressor since process remain asymptotically implementation provide evenly regressor neighborhood multiply total accumulate regressor node index fine node partition vector create node introduce partition regressor accumulate regressor advanced anomaly method straightforwardly incorporate framework begin regret prove higher define suboptimal affine taylor twice differentiable function affine apply lagrange remainder obtain conclude algorithm regressor regressor slide spline update square state knot lr basis represent window provide computational emphasize create tree overall due regressor regressor computational regressor straightforwardly nonlinear regressor use update regressor update computational original accord regressor illustrate performance synthetic match signal map circuit life various benchmark mean white circular represent hyperplane desire regressor fig normalize accumulate propose performance include figure experiment illustrate normalize accumulate algorithm performance batch observation fine region datum increase highly circular hyperplane algorithm highly nonlinear comparable sequence extremely fine early processing unlike algorithm introduce fine e hierarchical universal increase limit number expert hence observe fig produce discrete generate fig normalize accumulate emphasize nature observe uniform curve piecewise priori partition limited basis fig algorithm partitioning regressor partitioning underlie regressor see illustrate relationship rest inconsistent prediction generate circuit drop simplicity circuit accumulate propose nature algorithm omit achieve average gain algorithm accurately predict subsection namely involve realistic link arm target use involve realistic simulation arm angular acceleration arm link medium accumulate respectively regressor achieve accumulate reciprocal result first b hence achieve desirable structural assumption nonlinear regression signal incremental regressor partition independent regressor base sequentially increase nonlinear performance define incremental demonstrate superior algorithm series benchmark edu tr study nonlinear sequence result guarantee statistical assumption address regression hierarchical incremental present partition regressor drive gradually drive sequentially asymptotically optimal length provide description demonstrate significant incremental study sequential aim sequence x find exist assume possibly vary nonlinear life capture salient desire use either extremely filter spline different scenario hierarchical recursively regressor drive model structure piecewise prove achieve twice tune algorithmic upper modeling regression algorithm literature accurately represent differentiable perform particular sequentially space disjoint create amount partitioning regressor rely hoc strong extensively attractive tree hierarchical piecewise tree yield satisfactory regressor achieve accumulate loss region minimize regression depth tree computational compare particularly however model structure learn locally introduce modeling twice minimize finer necessary create piecewise model degradation partition keep fine aside nonlinearity nonlinearity modify author technique straightforwardly framework introduce doubly example regressor region correspond interval internal tree exist leaf node internal regressor leaf union child region regressor represent scenario partition depth tree construct depth decision modeling limitation increment incremental decision potentially length achieve power infinite piecewise twice model certain
respectively equation choice build multimodal update rule build markov target think metropolis hasting introduce count straightforward check slow density thus markov instead markov ergodic markov chain modify update jj ix ergodic easily ni n use family deterministic consist iterate accord kernel simulation implement metropolis hasting density need cover case view physics factor numerically investigate visit favor transition penalization formula assume converge expect ix update heuristic expect metropolis rigorous see wang algorithm update stepsize weight formula stepsize sequence goes vanish ni original complicated change related quasi analysis back set check explain see wang stepsize sequence choose adaptively build establish extend wang deterministic update density imply hasting wang update wang change rule linearize satisfy wang converge sequence ultimately address wang stepsize random dx x positive stepsize sequence suppose meta obtain practical say n function measurable function sure random stepsize increase wang assumption update sense discuss stepsize n wang ni check purpose one sequence decrease moreover theorem easily definition allow eq algorithm corollary stepsize sequence large wang stepsize theorem sequence convergence result see update eq recurrence relation sa point sa raise past control whole trajectory come randomness subset step prove sa recursion establish recurrence infinitely let crucial difficulty fundamental address dynamic follow proposition sign total variation present recurrence mean weight accord weight recurrence existence sequence converge recurrence prove give markov state performance wang similar wang lebesgue measure target read temperature normalization present potential locate construct isotropic move distribute accord hasting reversible lot leave right precisely locate around state temperature main leave leave enter conversely numerical quantification point typical leave thank adaptive wang see asymptotic metropolis differ sequence dynamic potential position saddle point let realization chain report wang visit much prove stepsize already almost bias limit corollary leave around perform value trajectory parameter leave well compute quadrature integral stepsize bottom exploration influence time concern multiply perform independent realization dynamic large wang independent realization start library week machine shortest check large confirm simple limit wang stepsize convergence visit allow state respect wang type view wang aim stepsize decrease mean decay still combine averaging view natural modify stepsize scale update rule positive deterministic call choose iterate compute draw wang algorithm stepsize particular stepsize relationship relate update notice notice rule addition consistent obtain behave wang accord care avoid fast stopping weight stop stop define evolve accord logarithm subtracting exist stepsize count index logarithmic normalize visit number computer behave metropolis probability accept propose move draw accept effective behave proposition modify increment proposal move hasting attain leave precisely fit law law wang simple system claim one consider agree well various parameter law line observe power ccc study convergence logarithmic empirical modify realization scale result prove plot fit confirm variance except around asymptotic regime attain long decrease decay decrease average version decrease plot decay behavior value iteration bias time fit measure fraction density q walk favor unbiased well follow reasoning section r compute ergodic respect eq follow reasoning section new enter modify update main discrete reaction well reaction paper make stochastic stepsize version visit fashion wang vanish stepsize less dynamic method penalization sample become speak parameter free version wang explain specific stepsize penalization time wang linear deterministic stepsize explain adapt wang sequence look show sequence converge prove possible couple geometric cumulative measurable second extend main fact necessarily lemma expression successively sufficient give convergence sa continuously v assumption verify assumption stepsize almost surely imply sequence compact let check hence n h introduce whose state assumption equation admit additive hold first check sequence nk km integrable conclusion square integrable martingale converge hx thank hx constant k detail omit combine easily unnormalized ni ns proposition deterministic remark use induction deterministic ensure let n decrease small possibility recall c decrease notational next ni bound go lemma sequence increase g g gx n ns sharp prove denote simplicity monotonicity convexity imply go proof show concavity logarithm monotonicity set one set c deterministic constant writing existence variable first right converge since choice ensure converge conclude universit paris est la self sample multimodal probability measure method variant wang adapt convergence wang modification exhibit similarity field molecular consist building dynamic ergodic langevin hastings average trajectory ergodic interest measure multimodal region probability ergodic dynamic high region average converge slowly ergodic difficulty modify target order enhance average respect recover biased devise importance sampling
residual small throughout hc ica brain subject fmri hc fmri signal subject matrix subject spatio term ica fmri explain probabilistic first hc ica source signal network assume subject observed fmri mix iv spatial across independent voxel stationarity prior ica pre whitening perform variability across voxel follow isotropic hc ica subject combination population covariate covariate group biological trait effect th ic voxel adjust assume level hc ica adjust primary treatment control important benefit neural many clinical gaussian population level desirable modeling fmri signal location brain activate area exhibit fluctuation well suit mixed pattern capture type signal tractable estimation fmri background negative positive fmri bold interpret background rest facilitate derivation involve latent variable state voxel follow z q hc likelihood v I likelihood voxel ml estimate parameter hc ica conditional log v web supplementary material marginal step probability finally distribution analytical conditional purpose main notation tractable need step update estimate formula update rule supplementary material summarize section material obtaining level signal variability base fmri thresholded ic maps activate voxel supplementary value ica software marginal three k v evaluate k major em exponentially exact evaluate sum space variational tc derivation depend heavily specification tractable require numerical cause convergence model whole latent small provide show r rp j z qp supplementary restrict fmri characteristic state specify background background fluctuation voxel activate sparsity fmri hc ica implication chance voxel activate overlapping activate support finding propose em subspace subspace z supplementary material subspace approximate measure lead simplification expectation q latent reduction step specifically update compare result use moment j j summarize algorithm start k p v expectation regard v counterpart base modification replace expensive hc ica huge secondly involve mix ica high challenge hc ica connection directly rewrite hc ica hierarchical level side iv iv mean major effect ic em exact ic exact fmri two manner computation subject signal noise signal hc ica regression simulate covariate use versus voxel specifically hc ica conduct effect post regression estimate ic type voxel level voxel result ica method hc ica inference tc h v type voxel power voxel cn cn l ica hc hc hc ica hc ica hc ica hc ica estimate difference demonstrate strong visual two area demonstrate strong connectivity subject examine temporal time experimental supplementary material correlation result suggest task significantly particularly strong coherent find signal become prominent demonstrate connectivity region suggest compare strong functional estimate whose calculated permutation visual identify little difference posterior fail reveal central node language hc ica powerful detecting effect brain compare dual adjust multiple across conduct fdr correction hc ica testing network voxel significant fdr thresholded fdr correct hc test effect potentially help understand clinical characteristic brain develop hc statistical covariate covariate exist ica hc ica help finding regard brain challenge model heavy efficient procedure hc fmri base em dramatically ica state theoretically method fmri result correspond support conduct simulation spatially source signal moderate method spatially covariate effect affect sparsity covariate obtain shrinkage dr voxel mix relate additional experiment supplementary material evaluate provide mainly focus across subject second trivial u canonical general g q nee u p apply z z moment analytical update update p moment ic prove lemma independent interpret activate versus ic lemma odd lemma tv q qp conditional vector approximate exact conditional evaluation moment estimate identify activate goal naturally probability indicate voxel v within specify subject hc show task brain compare another three scenario contaminate snr randomly initial change sign scenario necessary change correlation group average across result optima group map covariate add
noiseless sketch completeness noiseless walk noiseless implement call upper bound volume body q total call base ball walk noiseless lie misclassification incorrectly classify outside optimally intuitive statistical confidence adaptive full inside let denote standard variable make decision decision decision decision dictionary take stop decision either outside inside dictionary probability fx query ensure query fx decay constant big enough say arbitrary decay algorithm oracle combination walk illustrate noiseless establish inside algorithm oracle geometry level ball area inside body geometry convex need crucial draw position behave isotropic involve term constant body isotropic position current vertical inside call hand probability close give alternatively bad thus save call hence give band illustrate direction vertical impact vertical cube final epoch boundary analysis mass error area avoid probability part come band statistical testing way step big ensure statement sharp body cone error noiseless query total behave choose error query complexity body lemma conclude query example propose attain query less noisy noiseless obtain minimizer reveal model basic learn need quantify number call know information lipschitz yet comes obtain desire dimension seminal work optimization dependence author extend stochastic dependence leave polynomial noiseless extended author consequence upper averaging decay method progress classical category distinction attack yield second noisy yet leave low hope walk reason optimistic ideally randomness asset disadvantage informally start body form convex see walk spirit obtain continue fashion analyze ball restrict verify current inside outside resolve mention briefly work whereby noise may gradient another assumption literature additional constraint objective boundedness observation average mean affine transformation fy check convex let vertical lipschitz noisy call align epoch remain contain introduce modify property walk algorithm round start warm maintain provide cut region compute third step body affine transformation near body affine calculate near nearly body start since seed run mix n back guarantee take isotropic call near isotropic position
scenario crowdsource task whose reliability priori appear economic significant ix hold prediction arise error accuracy unsupervise ensemble important obtaining collect instance wish pick one second improve multiple simple majority voting perhaps define crowdsource expert system year reference yet address em maxima propose perfectly totally develop binary limitation actually specificity consistently accord balanced accuracy assume classifier balance ensemble suboptimal classifier significantly make follow focus simple sensitivity specificity classifier imbalance scalar imbalance joint tensor share eigenvalue extract sec devise imbalance restrict imbalance dimensional make consistent also unlabeled elegant expectation maximization multiclass devise probability confusion prove multiclass moment confusion classifier motivate example datum ensemble learner competitive even scenario classifier make crowdsource study case distribution observation building estimate confusion matrix center tensor closely notable center class need resolve matrix even decompose divide simple imbalance totally tensor covariance optimize second accuracy tensor finally consider let realization class imbalance let set th fully specificity future balanced totally predict classifier assume knowledge accuracy consider specificity readily tackle problem instance I marginal classifier conditionally pair well variant ambiguity fully class imbalance imbalance certain disease population predict presence individual genetic profile know noise em upon approach motivate unlabeled mean know contain imply balanced accuracy classifier show appear appendix sign ambiguity balanced accuracy imbalance b entry classifier balanced practice quantity eigenvector plug matrix classifier estimate r cast rank construct resolve inherent follow prove property iii assume imbalance computationally explicit multiclass discuss unsupervised make label predict label ix specificity convex former value assume classifier via taylor show plug eq spectral motivation consistently classifier plug directly improve linearization around inaccurate may guess maximize imbalance approach estimate estimate imbalance tensor second exploit computationally strong consistency derive conditionally triplet tensor follow tensor imbalance balanced accuracy correspond denote equal unlike ambiguity eqs depend class imbalance q invert determine imbalance practice though maxima imbalance classifier classifiers eq q converge maximizer b consequence consistency ml estimator note g concave find global maxima operation consequently grid computationally class imbalance instead specificity confusion confusion classifier probability k regard error employ consistently confusion confusion build upon develop section split empty disjoint subset classifier consider vs diagonal estimating confusion binary classifier pose estimate confusion high order dependencie three tensor beyond scope simple method multiclass imbalance uniformly iii balanced vector class imbalance generate standard deviation imbalance unlabeled imbalance improve instance mse scale show accordance accurate tensor total dataset repository due page detail dataset additional appear appendix distinguish background classifier randomly thus ii ml full label stability realization choose balance upon approximately vote vs realization improvement particular observe classifier result unsupervised ensemble learner denote situation independent exactly several work direction relax strict assumption classifier instance difficulty direction prediction realization follow vector half recall diagonal element correspond q relation give plug combine number error method give imbalance transformation output new classifier classifier rather classifier un eq latter equivalently plug collect note p replace incur hence delta square delta estimate delta quantity beyond study dependence classifier accuracy asymptotically eq v writing error deviation gain insight comparable accuracy balanced b mae base true curve nearly accordance eigenvector jointly covariance tensor beyond scope lagrange multiplier constant k possible expectation equal possible choice attain outline follow probability couple probability maximizer fortunately sufficient property continuously derivative corollary continuously differentiable eq differentiable hence also inside logarithm hence satisfy equal confusion shall classifier confusion matrix nonetheless lead subset end first confusion small confusion
discuss selective test time model event ignore irrelevant necessary selective respect dominate family fact draw family inference nuisance exponential nuisance parameter exponential family sufficient dimension nuisance conditional eq letting eliminate alternative test alternative alternative unbiased selective among satisfy unbiased confidence region invert selective test accurate unbiased definition thorough review family model simply law selective consider selective selective worth mind way choose example tailed test law equal tailed rejection region side way test imply selective level selection another selective splitting nearly always suppose wish abuse generality result convex completeness admissible apply exponential datum govern parameter response linear regression different coefficient splitting stage assume define cutoff large acceptance conditionally note depend neither test cutoff randomize technical unless copy independent independent copy could must occur event event l natural occur illustrate theorem bivariate selection splitting could interval available fisher matter selective increase expect interval tail interval long need together plot consistent event unlikely discard unnecessary stage splitting use information effect law leave concrete exponential section selective arise multivariate model consideration unknown generalize selective test coordinate ordinary ol convenient remainder adjusting denote onto statistic respectively distribute henceforth subscript ambiguity selective test sufficient transformation selective q conditionally independent distribution observe ty test law recommend serious construct selective case natural equivalent testing selective good linear accord functional point lead poor job select particular adopt avoid need several article selective selection work assume variance know obtain square select nuisance correspond rewrite eq event base base unfortunately line sphere equally hypothesis conditioning insufficient carry meaningful carry test nuisance write choose whether increase conditioning lead per selective case make mutually selective conditionally play role determine conditional great deal whereas condition contrast suppose matrix sparse highlight whereas conditioning set union realize especially important step model subtle deal conditioning plot value interval multi adjust likelihood ratio density choose inference conditional uniform thus window logistic linear glm glm represent difficulty control variable realization selective trivial like gender conditioning constrain promising approach may asymptotic though selective illustration selective matrix column snr magnitude choose splitting half yield instance split partition contain data procedure select select left inference inference lasso test lagrange sign test distinction procedure selection likely select superior model noise respect selection chance fdr aspect stage performance incorrectly power conditional screening well select quality goal outperform fdr intuition use information stage perform stage dominate improve drop seem successful stage slow surprisingly hold understand tradeoff work check draw student five rigorously nominal alternative appropriate scientific propose control q interval control closely selective countable define enjoy interval author address propose chance incorrectly fail construct confidence interval least square parameter regression ever consideration matter always singleton selective condition clear rate control converse relevant still question consider also scientific bag fdr proxy goal genome wide associate diabetes quantity vary interpretation gender control title job control gender question selective inference carry question ask frequency selective ask principle simple matter diverse still selective design price little improve challenge difficult take procedure reality procedure lead property ahead key challenge research balance choose realistic article represent repository file first website support foundation grant dms fellowship stanford genome fellowship taylor support foundation air office helpful select show expectation kolmogorov independent sequence z arise one wish region randomization order region implement rejection region test boundary randomization q correct sequence would monte algorithm property carry test define family approximate specifically cutoff cutoff acceptance possible n region nz pair right quick allow quickly upper confidence search nonempty dimension intersection unit selective sample hyperplane selective ball radius weight sphere outline selective conditioning draw fill text width em text center corner font taylor stanford model selection control selective recover property select analogous context closely intuitive justification exploit unbiased selective inference selective generalize test think consist analyst choose unknown analyst informally determine ask answer choice model use formally subsequent inference prior collect govern physical least partially example often exploratory decide predictor interaction include properly property file suppose significance level intuitively recognize still high among nominal conditional control presence simply value cutoff control selective valid ask simplicity example imagine scientific estimate ever demonstrate compound choose analyst decide roughly scientific question address result publish claim lead finding explanation effect extensively simultaneous several author adjust construct estimator genome wide pass fix gaussian large conditioning drug clinical trial adjust file effect meta selection adjustment interval construct false expect fraction non cover amount coverage interval see relate selective employ control propose region brain view classical exist analyst classical model usual rejection describe statistical check model leave open possibility argue selective type hypothesis practically base random scientific typically random classical control implicitly randomness eq viewpoint science split scientific aside selection depend nominal nominal selective meta selective nominal splitting practitioner identify separate popularity justification imagine though take ahead temporal actually solve control selective amount available selection furthermore always series rule part article directly selective splitting treat though reveal treat paragraph term denote informally everything know complete think one decide two discover knowledge stage everything reveal stage fair control conditional prevent appeal surprising reject unless sense conditioning carry hypothesis interest discard little carry formalize selective property selective control key conceptual question major selective us test even exponential selective briefly derive unbiased selective exponential model conditioning stage selective computing prescribed focus regression generalize recent proposal derive powerful selective require compare post selective selection initial second section compare selective conclude conditioning arguably observe function parsimonious identifiable researcher predictor probabilistic coefficient selective select conditioning many range aic minimization selection cf selective lasso solve term encourage eq notice correspond selective event lasso figure partition different screening imagine stage datum package remain fall test careful specify consistent interpretation coefficient adjust effect effect adjust education condition test follow otherwise concrete mind framework selective broadly allow random carefully inferential goal discuss selective development assume measurable analyst carry base tackle pair nan hypothesis mean guarantee necessarily beyond design loss test alternative countable question question abuse refer selective inference possible analyst select analyst shown select completely explicit correctly specify contain importantly candidate analyst poorly formal guarantee perform misspecification rule exception whether adaptively adaptively analyst probably wrong collected experiment issue take case discrete randomization necessary level adjust event ask question ask question ever select conditioning selective interested test selective selective countable selective design valid concentrate mutually selective selection denominator countable dependence test devise selective test concrete selective whole long run control nan wide operate scientific share countable question research apply iy I research probability grow long control frequentist independence generalization multiple discovery rate fdr rate even group fdr aggregate across convenient think contain set selective establishe selective inverting selective analogy duality selective event suppose selective selective confidence selective coverage cycle cycle cycle circle conditional variable even selective procedure view conditional used selection whose informally condition fine say control selective error rate give take baseline selective type selective confidence fine suggest refine fine monotonicity selective type w fine control type choice control variable extreme flip proposition fine computational reason additionally nonzero event convex another reason refine inferential guarantee meaningful coverage rate control splitting correspond split informally information mean quantify amount remain decompose expectation eq average conditioning quite consider selective gaussian conditioning highly contrast law practically lost conditioning confidence invert interval
energy residual iteration step candidate time much measurement sparsity iteration estimate identify k pt iteration stagewise omp orthogonal super etc identify candidate correlation measurement residual pick magnitude index candidate predefine find index correlation greedy adopting include match subspace sp thresholding htp propose recover rip satisfy rip vector isometry exactly obey isometry conventional ol ol slight ol fail algorithm converge iteration computational ol compare ii notation lemma section give study performance conclude remark vi notation useful rip order refer isometry constant consequence see begin interesting identification th sort element kl one correspond implementation computationally expensive require construct desirable effective substantially simplify iteration index equivalent q noting decompose k relate q simplification offer mention geometric interpretation project orthogonal study convenience state success select one clearly make index observe first correct contradict select previous iteration eq correct build condition ensure select convenience notation large k contain select appendix note monotonicity slight isometry g fail justify eigenvalue rip ols incorrect index eq first study differ find weak case definition identify index th holds respectively combine obtain adopt testing recovery reconstruction signal construct draw variance choose signal amplitude mention reconstruct particularly omp ol comparative approach simulation omp programming er gauss eps eps plot reconstruction sparsity sparsity call critical sparsity signal exact reconstruction algorithm fig critical even method exhibit high bp bp h eps gauss ols run measure matlab program core processor ram window reconstruction much accordingly much less ol call extend ols allow candidate list method fact ols identification reliable utilize energy analysis recovery iteration kk coincide omp ols addition empirical conventional improve empirical promise recover full value definition minimum diagonal replace reciprocal singular lie together singular partition inverse triangle respectively lk finally combine set k k k together lemma due lk hand eq check equivalently q definition corollary remark deal year recover signal call orthogonal extend least square ol choose index support much improve ols perform sparse restrict isometry rip isometry demonstrate compare art algorithm cs pursuit orthogonal isometry recent attract attention processing main system signal recover minimization q intractable combinatorial involve impractical realistic devote efficient algorithm recover rely search principle combinatorial computationally pursuit reveal bp
association data auxiliary follow could observe lead next thing accurately predict intuitively clear predict advantageous auxiliary much distinct reduce auxiliary example I straightforward estimator residual minimal point variable variable typical regression coefficient possible far reduce marginalization association rewrite ignore auxiliary scale dimensional replace inference convenient rewrite diagonal diagonal consider furthermore one I statistic value admissible predictive step uncertainty probability assertion agree assertion adjustment formula need plausibility partially agree I success I validity desirable property I assertion I assertion plausibility essentially condition set call hold nest either satisfy excellent reference function validity natural one optimality consideration first optimal simultaneous multiple application selection association observable space sampling association relation assertion exist make belief stochastically formal give association least predictive respect predictive name result set simple define impose assertion assertion section proposition show optimal random relative satisfy support satisfying example assertion simple see write event random give versus insufficient assertion generally might even assertion I consideration think simultaneously understand handle fundamental section extend development normal interested assertion disjoint simple predictive inefficient likewise predictive possible general assertion write reasonable strategy element intersection random two complex assertion predictive intersection appendix simplify complex assertion resolve choice set intersection sided interval optimal interval symmetric asymmetric optimal predictive corresponding intuition intersection individually justify way measure multiple assertion respect intersection efficiency disjoint predictive intersection previous section simplify set resolve assertion variable resolve ambiguity element make use transformation fundamentally inference impact support write problem assumption association mapping implicitly act auxiliary relaxed group act act directly notational act variable fit usual transformation assertion write concern assertion unchanged transform help assertion change sign affect take mapping move property immediately property transform solve transform random focus admissible admissible probability useful ax concern unchanged reasonable require display belief element aforementione invariance hold balance reasonable interesting practically beneficial balance check calculation however transformation model balance make belief stochastically assertion maximize element definition main notion element particular balanced association center I subject collection event predictive sub collection hyper theorem predictive assertion intersection half simultaneously intersection shape box towards transformation variable invariant coordinate irrelevant addition label multiplication product marginalization specify box optimal support shape box invariant box balanced sense definition besides property hyper cube axis balanced cube random plausibility henceforth drop plausibility away cube shaped contour plausibility plausibility plausibility region frequentist coverage plausibility length characterize quantile norm consequence validity across plausibility region new hyper cube plausibility nominal frequentist conclusion consideration context argue turn identical naive plausibility region sub message probabilistic automatically problem demonstrate I drive start correspond truly coefficient kind assertion still cover selection use cube stochastically lead calibration stochastically plausible fix claim I drive procedure control error equivalently certain validity validity tb apply implement plausibility plausibility look sort permutation rank magnitude p clear large rather plausibility plausibility formula include assign table get q left cancer analyze examine association clinical among receive include transform seminal response compute plausibility see accord lasso select I cccc plausibility simulation study autoregressive correlation e six scenario vary result I variable section hypercube display figure percentage parsimonious parsimonious procedure aic cross validation tuning match configuration range plot parsimonious include variable give result validity I curve except lasso panel bic adaptive true parsimonious panel variable consistency I fix select attribute consideration I come multiple theory apply valid uncertainty important drive I valid post naive plausibility region base develop addition I base notion pick plausible connection I calibrate control family rate simulation demonstrate I emphasis meaningful probabilistic summary development application I already problem involve multinomial genome wide multiple testing expect consideration problem improvement extend development paper interest principle develop complex initial step case step complete acknowledgement national science foundation dms dms suggest comprehensive treatment goal amount technical focus random support algebra subset close topology mapping measurable define forward separable stochastic distribution include default predictive member next function rich consequently I nest nest usually simplicity move terminology nest support demonstrate random simple distributional restriction set give subset equality form assertion auxiliary equip measurable subset contain closed key relevant ax relatively condition proposition ax x ax since belief attain collect intersection define new predictive support random satisfie condition state admissible intersection random direction easy clear either case ax ax st disjoint simple optimal predictive index intersection closure vx vx j candidate predictive assertion admissible theorem remain theorem show handle split st ax ax theorem ax measure predictive element define core balanced inequality element attain balanced balance resolve ambiguity shape make predictive get assertion imply random support sign magnitude symmetric one depend invariant sign understand differently decompose assertion random balanced beyond balance optimality normal balanced unbalanced unbalanced balanced sign take side small unbalanced predictive random belief large say balanced sense q uniformly admissible go assertion transformation pair maximize suggest connection optimality symmetry I book multiple assertion application want assertion balance disjoint sense respectively complex assertion make connection need symmetry property index predictive optimal complex maximize assertion elsewhere towards assertion disjoint decompose ax unique connection simplify connect condition generalize assertion continuous sense sense get helpful equivalent admissible random nest index write include class care make clear rigorous measure ax measure stochastic address fix investigate index symmetry condition respect definition write generic optimality admissible prove state maximize contradiction balance condition set increase union get example correction might need put contain r intersection inclusion write easy immediately former less latter claim define similarly leave involve non evaluate respectively generality q h key motivate construction leave right hand equal fourth hand q contradict predictive balance maximize complete plausibility plausibility fix size understand read plausible plausible addition plausible really prefer one want select overall variable pick complicated assertion procedure plausibility reject coefficient predictor plausibility remain seven enter zero model predictor still therefore continue remain plausibility additional predictor stop also table eliminate prove dimensional present though symmetry proof take non generality distribution centrality optimality consider assertion association equivalent assertion say side assertion predictive random plausibility u u sa plausibility function optimality want show predictive random u u admissible plausibility nest large complete parameter zero mean multiple really assertion expect plausibility believe marginalization ignore leave association propose plausibility behind expression like variate student see singleton follow plausibility plausibility I suppose sa l claim compute predictive obvious demand instead assertion specific hypercube predictive case complete nan apply incomplete stochastically large quantile efficiency note whereas empty assertion assertion prefer proceed explanation discussion tool argue
bound really easily lasso risk predictor opinion work property computationally risk good linear three different stage study property think useful type study weak discuss develop call employ variant half agnostic lasso net half distribution inference question predictive contribute risk predictor use interpret detail let interpretable produce interval thus exclude question infer linear deviation validity confidence essentially purely avoid asymptotic could define obtain square procedure summarize figure pt n split select subset forward lasso confidence similarity inference valid inferential statement dataset thank provide select ph percent interval percent figure percent parameter property explanation scope current datum predict gues augment residual permutation test easy interval validity depend desire device neither method residual prediction change variable remove assumption free interval free interested task infer attempt assumption regression indeed tendency three idea correct careful interpret article describe change hold change seem like pick change refer change put assign otherwise claim causal world alone eq change prediction change bring I exercise user likely interpret advance understanding inspire model easy low much hope author low assumption world I important acknowledgment thank helpful taylor force together array test automatically adaptive remarkable author make strong quite make advance understanding procedure paper error design weak form incoherence eigenvalue certainly place indeed think also highly exception exception design matrix random design matrix
set computational overhead herein approach determination finally iv end dissimilarity representation cost dissimilarity measure term generation respective euclidean second quantifie give cycle iteration derive induce cycle grouping compute putting depend derive good cluster determine membership derive overall bad main determination modularity system operate cost significant objective capability blind derive suffer simple model induce community one form identify especially maximum modularity force derivation additional read account measure performance combination final necessarily focus perform well recognition choice characterize modularity also code herein train scheme exploit derive remove edge likely induce form edge higher set fuzzy construct get increase remove derive consider group fuzzy substantial change scheme modularity additionally test see cost affect effective running start sec sec provide explanatory take uci repository lastly sec dataset contain pattern synthetic multi target non target specify train non target pattern system variant denote genetic mutation dissimilarity implement check fitness change vertex membership intra distance membership parameter intra cluster distance define perform base setting determine preliminary fine software implement library equip gb ram class area roc compute average target pattern rank evaluate confusion analyze precision measure measure define report average run execute seed significance class separate spherical test green actually belong blue target solve membership plot modularity demonstrate euclidean apply select target target nonetheless reliability achieve define taking target non indicate describe dissimilarity euclidean herein reference ref show auc uci performance variant seven seven achieve auc normalize usually affect exception worth note still sp dataset degradation three bad nonetheless worth point pdf observe severe initially degradation grow ab bc contrary demonstrate dissimilarity base adopt deviation breast diabetes pp ar normal breast cancer diagnostic bc bc sp gauss nn som gauss auto som dataset na nn nn auto encoder som na auto encoder letter letter show adapt target process graph mean distance solution solve greedy characterized parameter control importance substitution obtain knowledge result consider applicability herein result confirm hard p notably dataset auc classifier consider gap accuracy number solve e letter l letter p novel classification design make dissimilarity employ e classifier decision form concept modularity decision equip suitable boolean pattern validate two type benchmark base uci label comparison uci demonstrate several art result dataset prove effectiveness less pattern term dissimilarity representation allow accord suitable hand direction usual herein change graph map dissimilarity representation volume therefore future span optimization technique interest g membership devote usually pattern recognition pure viewpoint generalization system produce easily human expert insight since future goal viewpoint rgb name outli anomaly target pattern term recognize classification base primarily approach input euclidean derive effective region vertex dissimilarity optimize scheme consider design boolean decision test allow description contain either pattern effectiveness technique involve pattern orient character class deal involve real target instance determine device work properly correct device trivial instance model method take rooted decision adopt real video medical end develop dissimilarity although allow cover context adopt etc embed dissimilarity ds input edge normalize ds us ii define additionally represent ds relate embed span minimum far analyze induce concept modularity membership soft decision classification attribute experiment offer comparative benchmark follow overview provide clear introduce technical background material use successively detail evaluation conclusion provide direction iv reconstruction finally information theoretic generative parametric include describe operate suitable distance measure input technique category group neighbor approach drive base around vector model region surface optimize finally theoretic entropy mutual one important support training svm like particularly domain employ hyperplane like forest refer comprehensive survey state aforementione categorization herein intersection distance information theoretic notably exhibit sense substantially theoretic fuzzy graph partitioning concept moreover dissimilarity aspect however programming prototype cluster popular abstraction sound mathematical entropy model literature fuzzy establish one classifier subsection modularity discussion dissimilarity entropy finally modularity dissimilarity characterize dissimilarity rs determination span prototype strategy embed row dissimilarity fast way dissimilarity dissimilarity detail datum I realization nd gx ne ij enyi span literature connect enyi length e vertex edge edge relation normally value determine weighted degree ij weighted intend partition establish measure quantify cluster vertex compact intra cluster great inter modularity formally follow partition rewrite edge intra cluster modularity heuristic propose modularity assume normalize input domain requirement notably implement embed sec dissimilarity accordingly rs pattern distance suitable representation high determine rs prototype small informative euclidean see entropy descriptor also guide synthesis construct framework develop synthesis module I consider concept modularity sec vertex induce partition whose edge denote modularity contribution modularity therefore need take account boundary efficiently boundarie fuzzy decision synthesis optimize model objective optimization combination calculate two spread separation ds modularity group solution validation effectively instance contain considerably stage fig intuitive
time faces student linear subspace dimensionality speed access project reduce storage requirement quantify reduction projection ssc dimensionality order significant degradation engine behind quantify subspace change reduce challenge datum find low dimensional lie union extract formalize assume l subspace refer literature hybrid application inter unsupervised disease may dimensionality reduce speed acquisition even directly often desirable lead reduce reduction appear general cluster privacy widely dimensionality reduction linear euclidean distance property map popularity purpose characterize namely sparse ssc thresholding subspace subspace intersect quantify impact dimensionality cluster letter letter matrix denote ij identity refer stand sphere random subspace first point segmentation ssc segmentation approach perform high dimensional incur operating quantify ssc characterize specifically ssc apply reduce set quite provide contain impact explicit ssc even reveal affinity dimensionality reduction engine state project subspace space increase quantify impact briefly ssc adjacency construct cluster ssc lasso construct point spherical set subspace perform z z element spectral subspace discuss step j j z segmentation connect adjacency ce misclassifie ce inherently quantify work albeit sensible specifically absence impose reduce chance splitting increase select drive ssc connection virtue automatically I statistical connect ssc subspace estimate insight eigenvalue start throughout randomly point element j j direct say property inter c c subgaussian store high dependent propose matrix hadamard hadamard result moreover subgaussian isometry property rip sense conversely establish randomization column satisfy rip subspace k ks ks ks ts start main result ssc adjacency ssc adjacency apply cn state affinity ssc high small hence ssc impact quantify affinity reflect subspace reduce dimensionality orthonormal basis basis subspace suppose x j probability proof give ip reduce project orthonormal generalization basis basis dimensional randomly project satisfying formalize u result theorem imply I obtain case estimate obtain standard adjacency set n j reflect upper violate I start I high dimensional e j probability cause projection theorem accomplish perturbation concentration result violate false connection impact dimensionality ssc problem cluster image pixel acquire
evolutionary represent model higher imply long matrix mutation evolutionary therefore alignment type evolutionary abundance protein result usually rescale express specifically form choice find entry matrix score align sequence element integer similar sequence give could information match unknown large distance treat knowledge commonly think principle knowledge evolutionary naturally allow uncertainty evolutionary protein alignment evolutionary distance describe similar h influence volume long accept previously suggest evolutionary model match match mean respectively increase previously value h alignment account configuration must preserve new model widely penalty match also protein structure alignment model quite new small additionally view laplace marginal distribution uncertainty uncertainty numerical comparison body impose flexibility handle transformation challenging find sensible easily incorporate evolutionary protein inference distance cm sequence alignment ex school sciences mathematics university sciences technology school mathematic abstract know protein protein also influence bayesian align protein protein gap incorporate insight protein bioinformatics gap bioinformatics alignment structure protein aim determine sense primary protein position may sequence two complement alignment evolution protein position exist structure remain essentially unchanged well closely protein protein available bank reliable becoming developed ce design uncertainty uncertainty alignment allow quantify mathematically protein point point alpha configuration protein rotation configuration main interest shape protein alignment viewpoint essentially two integrate alternative consider model manner uncertainty correctly underlie flexible body demonstrate similarity transformation match application demonstrate range applicability alignment protein matching setting match every consistent match consider protein think therefore impose constraint prior incorporate fully section describe sequence illustrate describe bayesian constraint challenge previously literature measure evolutionary conclude pair element letter string protein alignment align type complement informally score type align give score entry express score score score overall providing nan observe g e score necessary gap sequence show alignment gap pair evolution instance mutation occur one position type largely region sequence alignment highly align region pair gap achieved interpret sequence figure alignment gap allow alignment matching penalty number gap alignment imply gap impose preserve configuration label body transform form configuration observe observe regard location spherical model therefore mapping point particular impose constraint configuration configuration row one match poisson point integrate support match normal align meaningful ordering preserve alignment well necessary gap sequence alignment match alignment extension illustrate gap consider sequence alignment match align indicate gap gap create figure first sequence length three gap count number gap would gap alignment configuration consist penalty start sequence formulation way decompose sum contribution pair indice form alignment extension e multiply q haar special unlike prior discuss sensitivity generating metropolis hasting alignment acceptance probability match suppose gap comprise form current alignment random n select propose match however currently match say switch proposal currently match configuration match match propose match acceptance say propose also match add accept currently match interval match switch current retain otherwise match reduction negative add match term q note perturbation alignment remove match switch number propose iteration mcmc posterior distribution programming analogous sequence generating computationally intensive converge method move deal joint integrate rather treat treat place q hasting proposal lead summation programming appendix extension similar alternative integrate adopt alignment recall total p ga h ga use mid quadrature grid stability rather costly evaluate scheme integrate new methodology analyze study align structural analyse pair analyse identification give sensible investigate abstract represent representative also tune great initially expect prior use suggest gap extension gap example case parameter parameter state mcmc burn translation take centroid set keep inference match typical match match probability probable least already appear match match align region uncertainty evident gap create diagnostic bioinformatic deviation median corresponding posterior report correspond modal use three different value specify match match describe principle obtain define cost match specify incur match point regarded note match incur relatively miss true assignment use give match distance guide obtain plausible uncertainty alignment converge chain run value value trace mixing value log manner good benchmark determine subsequent perform start angle draw nine run mode therefore discard remain top probable match st probable additionally run evidence converge plausible confident include match converge confident global dominant time prior expect value previously respectively posterior interval previous appear compatible distribution suggest sensible value match similar top posterior agree previous consider range evenly spaced range careful inspection investigate wide trace chain chain alignment pair match th probable assignment match give match reference match appear give report situation type number give context convergence value apart median log run diagnostic trace together convergence reach good range converge give probable find initial evidence reach plausible quantity sample posterior interval estimate match
eq find negativity triangle inequality evaluate dimension feature directly learn database z match want match pairwise must link modality modality want generate similarity form item logistic sigmoid function often hinge function indicate pair specifically stand deal unbalanced experiment divide norm formulation treat scale domain nuclear make desirable machine discover modality gradient find order regularize non constraint eqn accordingly eqn point entry product feature sign positive contain modal definition simplify remove term soft thresholding technique thresholding eigenvalue assume rank accordingly descent update way step objective omit algorithm iteratively search combination solution alg eqn low bilinear similarity popular modality image medium retrieval field pls microsoft generalize cca cca space different modal match pls another classical method cca semantic gap correlation cca pls cca important label learn discriminative great recognition cross orthogonal modality aim point unlike cca pls limit paired feature network cnn cnn code namely research layer output dimension pca method cca pls query cca pls wikipedia wikipedia article article build select associated word total document derive cnn feature pls dimensional thus experimental pca pca preserve cnn image feature dataset wikipedia wikipedia back number blue bold precision scope propose algorithm good database pca pca perform comparable outperform pca reason wikipedia may less category globally note fail work feature need reduction well without without lie reduction remove redundant pca may discard time pls algorithm pca table also supervise clearly outperform one pls partly reduce semantic semantic work find semantic fig precision curve curve text retrieval performance bit display pls database wikipedia high especially wikipedia similar retrieve experimental learn modality algorithm accelerated explore modality database document show objective consider suppose subgradient function subgradient nuclear svd part great denote turn bound institute chinese sciences ia cn medium receive year internet modal feature thus get heterogeneity match different hand metric explore learn heterogeneous heterogeneous modal penalization accelerate proximal gradient text medium database performance compare internet decade display audio requirement modal researcher media retrieval image several recently new author feature key heterogeneity modal common latent modal match classical solve aim modality mutually maximize similar cca pls semantic correlation semantic suggest combine correlation work helpful reduce image level beyond method analysis weakly analysis learn study near neighbor relevance component margin et aim gaussians traditional suffer difficulty modalitie metric similarity medium modality algorithm solution
cca literature involve objective really due replace problematic control concave monotone solution compress problem penalty eigenvalue basis pursuit lead iteratively reweighte selection problem surrogate function function surrogate fig three surrogate approximate original still continuous original section concentrate algorithm note exposition matrix pair matrix hermitian value approach complex ig px modulus applicable construct complex method minimization expectation em approach principle behind transform reader therein sequence objective start follow easy scheme decrease monotonically every I first inequality follow apply surrogate maximize produce iterate refer problem appropriate solve e iteration quadratic due differentiable concave surrogate ng px c compactly k follow equation htbp quadratic fact idea penalty quadratic propose solve iteratively weight problem reweighte sense quadratic function example coefficient tackle imply iteration minimizer fact potential function surrogate tackle propose incorporate systematic differentiable function follow aim function lead surrogate smooth huber penalty huber smoothed fast htbp smooth become smoothed smoothed surrogate quadratic irrelevant function view incorporate issue k I k I p x k p pp px x pe p apply smoothed smoothed answer define everywhere concave monotone smooth define low quite smoothed smoothed f ng ng solution smoothed original high smoothed say general global local advantage gradually large begin probably undesirable maxima success numerical ready differentiable smooth smoothed nc objective smoothed quadratic follow ignore eigenvector k tw iterative since base iteratively reweighte repeat eigenvector iteration generalize eigenvector since iteration drawback make attractive become ill condition suffer extremely slow convergence difficulty ascent free ill conditioning let maximize l l search scalar achieve go infinity follow lt l lt lt lt lt lt let xx lt rr direct accord easy thus ascent worth multiplication product efficient though slow become accelerate convergence large linear widely ascent introduce multiply ascent leading summarize practice particular positive direction ascent direction reader refer book ascent usually converge decrease minimize similarly scheme preserve ascent need objective ascent initialize generalized eigenvector ascent property guarantee choose repeat l l ascent assume special notice special sparse min admit q equation otherwise return smoothed use iterative algorithm generalize eigenvector iteration another exploiting algorithm close term suggest solve accord problem nonconvex let diag repeat proposition mm fact apply diag diag rewrite diag diag quadratic plane quadratic term form need back summarize require positive general diag easy require diag diag repeat absolute diag problem thus accord sequence generate algorithm compact thus guarantee algorithm sequence maximization smooth neither convex concave shall surrogate maximize prove prove function present useful later differentiable surrogate table show continuously everywhere except let satisfied continuously see continuously differentiable continuously f show smoothed maximize compact ng sense solution ng continuously differentiable lipschitz proposition ng p ng vice speak nonconvex hull attained supremum attain relax set optimal let referred sequence limit equivalent solution objective q n converge subsequence j easy necessary construction imply generalize exactly recall guarantee lead generalize compute stationary long guarantee exist experimental pc ghz ram subsection propose generalize complexity extract dc c maximization knowledge case problem solve dc solve iteration experiment set subsection dc algorithms ascent compute eigenvector matrix identically normalize size average trial see fast dc note implement attribute iteration evolution objective one figure much converge notice run versus average trial eigenvector diag sparse generalize generalize eigenvector successfully recover list result respectively initial regard three plot chance recovery parameter recovery versus versus chance achieve dc become exp stay decrease lp lot surrogate function two surrogate much make easily exp seem choice choose gradually probably smoothing fix inspire apply decrease smoothing solve less specifically step apply parameter decrease step choose random scheme decrease setting surrogate show decrease exact recovery htbp chance special eigenvalue pca receive attention covariance vast literature essentially generalize benchmark code website surrogate function call refer need eigenvalue cholesky decomposition matrix subsection mention entry smooth four different penalize propose fast four norm penalize may specialized algorithm deal penalty average subsection generate sparse achieve covariance diag orthonormal eigenvector randomly four eigenvector successfully recover chance successful wide range plot high chance achieve chance versus regularization dna allow possibility answer complex experiment usually make component potentially subsection breast set five scheme explain pca eigenvector scheme keep entry value explain increase cardinality explain high htbp trade cardinality allow generalized pair approximate nonconvex generalized turn regular problem point numerical propose outperform include special closed solution derive scheme show similar easy q maximizer know q p conclude complete notice define different case optimality left notice guarantee satisfy still satisfie satisfy arbitrary rewrite eq form integer get necessary integer satisfie need
variable originally propose extension single elimination originally omp ol aim yield homotopy algorithm regularization reconstruct homotopy procedure pd algorithm different minimize maintain maintain list recently approach optimization address together list handle gradually continuous hyperparameter opposite context involve hyperparameter solution warm value homotopy exploit affine track consecutive piece spirit interpret homotopy procedure solve pseudo convexity metric consider resolution warm reconstruction rarely gradually increasingly measure choice grid grid modify definition value rather adaptively similar homotopy pd propose nonconvex penalty difficult inverse additionally automatic cardinality rule usually approximation observation assume independent submatrix index rank hereafter notation forward selection introduce notation stand frequently resort slight abuse terminology line path may regularization take constraint penalty generic path statement greedy refer convert stand support indeed read minimizer minimizer minimizer penalize curve concave envelope affine contiguous case support ks set minimizer provide property set appendix path envelope may coincide state reverse inclusion penalize search notation deal output compose support pseudo index geometrically support segment fig lb lb lb lb lb lb sort order support associated extension start dedicate equivalently extension decrease adaptive dedicated three sn removal replacement eq terminate replacement function subset removal coincide ol generally replacement trial compute fast stable cholesky ss stand submatrix active unnecessary standard version implement ns vertical top bottom iteration select update support new dictionary support equal iteration none font lb lb lb lb lb lb lb lb lb lb lb lb replacement top refer selection four support de inspire homotopy minimizer continuously homotopy denote optimality first condition main homotopy solution minimizer terminate illustrate line black point separate eq atom support font lb lb lb lb lb lb lb lb lb lb decrease represent line cc lb lb lb lb lb lb lb lb lb lb lb lb ss axis without output meet I limit replace list support illustrated line lead first repeat call whole appropriate deal overcomplete dictionary early stop consider rule rule minimum step lead process plain repeat terminate compute concave might jump slope increase structural gradually candidate impose curve fig concave curve envelope domain font pt lb lb lb lb lb c font lb lb lb lb lb font lb lb lb lb lb lb font lb lb lb lb lb lb lb lb lb lb lb lb lb cm font lb lb lb lb font pt lb lb lb lb font lb lb lb lb cm font lb lb lb lb lb lb lb lb lb lb lb font pt lb lb lb lb initial configuration b line concave interval support edge update pd subset include concave decrease illustrated fig c envelope line remove new subset compute line concave empty support subset cardinality assign exploration removal keep similarly iteration attempt possible illustrate call explore support include pd concave reduce horizontal correspond extension iteration explore compute either support remove low cardinality explore list call sort decrease lead cm terminate support decrease replacement jj least cardinality candidate correspond alternative adopt call state empty detail omit brevity reason concave j firstly notice j prove sketch stress identify pd iteration concave next atom pd upper reason value improve early within computation hereafter height fig mm fig plain deconvolution spike db impulse dictionary sparse deconvolution problem jump db jump db kind involve ill condition pd analyze simple example detailed mm width sparse j line deconvolution impulse response gaussian convolution impulse thresholding toeplitz dimensions gaussian b respectively define jump atom code jump match height jump sparse generic either dictionary overcomplete dictionary deconvolution neighbor highly correlate condition dictionary may recover difficulty deconvolution width impulse overlap detection atom support pd result sparse first seven detect jump may model category ms cross derive expression cm cc pd data signal spike pd fig solution support cardinality pd rp rp small pd white curve almost coincide pd curve deconvolution cross spike contrary yield accurate short record moderately q number spike spike find spike db spike typical pd curve circle replacement return white pd coincide continuous low pd grey grey bar reach provide insight pd deconvolution horizontal axis single initial empty successive pd support include candidate effective increase cardinality fig improve solution decrease fig pd early subsection setting ratio cardinality width impulse deconvolution sparsity restrict f generate overcomplete gaussian impulse cm jump jump focus penalty estimate penalty algorithm simple ol reweighte ir cyclic ls cd smooth sl resort penalize least algorithm ls cd simple thresholding compressive sensing behave ill l cd efficient thresholding cyclic descent become popular although hereafter allow rough initial fast measure suggest choose less sl increasingly penaltie low relative nonzero sl implementation dedicate noisy work efficient inverse replace strategy solver crowd homotopy limit homotopy crowd mainly matlab implementation crowd author large problem compete grid grid vector randomly simulate specifically location nonzero I trial pd value support cccc time second width width fact snr mm snr width snr width mm k width mm mm snr ccc width fact snr snr snr snr first cpu pd viewpoint viewpoint propose solve either minimization apply output norm strongly ta average another represent separately pd cpu evaluate support negative st positive st average se tp order false positive negative fp tp analysis se tp algorithm likely perform use provide additional pd subsection reconstruction sl strict output se score run cyclic cd nonzero regard perform norm post processing interpret iteration cyclic l squared cd towards minimizer cm cccc cpu second width n snr width mm snr fact snr mm snr mm snr ccc fact snr width fact width snr mm snr width mm snr fact snr pd clearly group cd sl hand ols pd discriminate accuracy behave contrary outperform obvious advantage ir homotopy adaptively output relate whose ir solver tune stop pay burden fig two line pd horizontal trial computation start termination pd expensive however want reason draw pd viewpoint pd pt l pd ls cd ir se tp order true se tp ls ir se tp pd ls cd ir se tp true se time depend many implementation storage stop rule follow comparison two depend medium relaxed avoid huge stop pd medium pd l amplitude sl step last nonconvex dimension remain reasonable arbitrary rule comparison trade favor pd ir numerical ccc mm width snr snr mm fact width mm h snr width snr mm snr fact snr width mm snr h marker appear l sl performance noisy often specifically least always exceed discriminate positive localization non account wrong estimate cardinality subset order quite jump true support partially detect pd well correspond free I tp ls cd ir se se tp pd ir tp pd cd se tp tp provide propose algorithm competitive overcomplete detailed experiment space reason deconvolution overcomplete although qualitatively good pd se tp hard discriminate often consider spline generalizing jump detection piecewise jump think piecewise inspire regression jump shift version side overcomplete soon competitive carry pd noisy matlab choice desire ready suited induce highly correlate minimizer algorithms usefulness extension low range gradually greedy improve large enable classical selection criterion estimate include backward potentially pd refer removal dictionary atom remain testing simultaneously become carry replacement test omp ols spirit consideration propose path
new criterion explicitly account dispersion study performance traditional numerical regressor propose less costly joint proposal perform precision find computational cost greatly present dispersion fast two monte simulation model evidence conclude section random beta write eq dispersion beta independent differentiable function link function logit log complement cauchy link parametrization also carry inference find dispersion statistic final last fisher likelihood usual regularity thus upper nominal importance typically iii regression include adequate also link assess misspecification outline selection propose regression widely coefficient determination regressor add selection aic model introduce criterion good identify asymptotically model follow sense alternative measure relative location focus beta correlation denote maximum likelihood covariate pseudo take pseudo pseudo estimate log model measure goodness quantity eq measure goodness propose modified additionally recommend selection far goodness criterion define criterion minimize estimator dimension candidate say minimize aic instance asymptotically leibl accurate introduce unbiased leibl distance linear regression regression autoregressive aic regression bic include correction namely consistent autoregressive incorporate correction account dispersion inclusion extra covariate shall two inclusion goodness way account second selection dispersion penalize regressor approximate propose model dispersion eq numerical quite inaccurate moderate computationally dispersion beta mean regressor dispersion introduce sequentially outline dispersion select regressor adequate selection regressor selection regressor dispersion two entail estimation different figure time intensive approach ten covariate dispersion regression entail model computational efficiency suppose scheme run day minute combination criterion shall carlo dispersion regression parameter use implementation step file computer datum use different monte replication draw uniform throughout ht identifiable easily identifiable dispersion weakly identifiable identifiability approach influence mean intensity weakly identifiable dispersion easily identifiable weakly emphasize usual relate monte give respectively logit link generating model model case since regressor likewise take evaluation criterion use criterion able replication criterion select regressor correctly covariate dispersion dispersion variable assume regressor first four strategy frequency additionally discuss implement pilot simulation great weight one inclusion regressor heavily model model aic model aic joint accurate weakly identifiable correct small sample dispersion identifiable reliable selection weak identifiability small accurate good criterion nearly scenario well balanced display selection achieve criterion top dispersion identifiable recommend regressor notice finite performance selection heavily dependent identifiability table present correctly dispersion dispersion identifiable dispersion identifiable generating process good good winner figure correctly specify weakly identifiable perform reliable monte obtain dispersion constant focus select mean interesting correctly especially correct correct dispersion nearly dispersion far indicate regressor dispersion well come dispersion explain criterion regressor quite naturally selection scheme combination implementation propose ps monte present table nearly scenario model propose among implementation dispersion identifiable accurate emphasize specification dispersion vary dispersion dispersion dispersion identifiable identifiability perform equally even numerical practitioner recommend use misspecification school response read index student capital eight year transform original code dispersion regression inferential inaccurate dispersion dispersion vary regression logit consider covariate candidate hypothesis test mean include covariate logit reject nan constant nominal notice sample close evidence perform scheme arrive covariate dispersion namely diagnostic correctly aic arrive regression covariate dispersion covariate statistically nominal std constant considerably happen dispersion specification assume correctly
still ambiguity make identifiable estimate positively compare gradient bfgs newton implementation adopt bfgs describe bfgs glm software simply ignore r r compute subject extract r md normalize positively glm voxel take core intel ghz total four hour discuss section pure package validate bold presentation natural bold fmri descriptor publicly twice image per second rapid acquire comprise image within image time align run alignment manually additionally datum preprocesse detail perform window extract training original result beta map proceed encode handle image spatially smoothed pyramid modulus scale generalize learn bold original full necessary overfitting assess method otherwise coefficient activation yield activity present prediction bold highlight unseen prediction leave fold dataset task subject ask reject gamble chance gain independently varied level potential gain loss label challenge predict brain publicly available mixed gamble task slice correction segmentation normalization interface subject consist tr stimulus perform run across create correspondingly run predict gain correlation true metric well suited regression order label sensible occur always lie interval perfect ranking perfect disagreement fmri previously encode activation beta method standard glm glm design glm separate form dispersion size second form reference across region decode decode voxel true glm estimation task variant glm see display count identify chance identification algorithm map intermediate however expect correct directly translate estimation separate outperform classical range r glm worth score whether statistically test success recover probability recover method hypothesis probability equal alternate probability tail np p distribute performance glm identification subject count correctly total chance metric less voxel benefit range score basis subject bold average voxel outperform use identification subject glm subject study glm voxel voxel wise encoding score time voxel rank separate design element axis dispersion derivative axis trend suggest improvement basis peak reference around canonical score basis design give sign test leave one average voxel superiority design estimation separate generate gain basis axis time dispersion derivative abundance superiority color peak observe local model design matrix voxel score basis axis give sign leave confirm superiority design voxel acquisition slice estimation pearson correlation column voxel thresholde contour value show green area produce highlight visible r produce glm method bottom see perform voxel follow gray matter voxel wise acquisition slice plot voxel thresholded testing contour line top voxel matter relate shape canonical decode compute decode univariate selection parameter voxel cross consider assess superiority encoding difference score fold great performing sign report value together encode high basis basis size tb average consider well second dataset estimation outperform fix reference glm basis follow software sign order examine generalization task linear separate voxel constrain basis omit efficiency reason possible bayesian spatially adaptively learn subject work latter case level analysis consists reveal possible boost appropriately metric assess estimate metric identify encode predictive activation use novel activation compute benefit range method r glm voxel full observe increase homogeneous already glm good design basis provide find variability subject constrained decode classifier long basis ten derivative difference region involve observe correlation sensitivity incorrect used generate bold test signal bold natural correct estimation high impact decode evaluation accuracy stimulus type decode sensitive procedure encode activation glm spatially condition voxel previous cast efficient newton glm design quantify encoding decode glm outperform compete grant le de france france despite common canonical region subject datum lead power fmri constraint yet differ across voxel exploit model glm glm improve decode activity compare decode compete decode functional fmri machine response machine technique predict cognitive functional record study decade cox bold task stimulus although possible bold signal common consist extract beta bold analysis voxel base model activation coefficient bold addition third know quantify quantify activation mean linear glm wide suffer limitation glm commonly response activation know substantially suggest improve overcome aforementioned limitation finite impulse propose within glm estimating modeling e generalize general model characterization study primarily focus detect activation chapter freedom propose possibility combination three consist time derivative nonlinear desire space long overfitting approach require level share even choose inherently costly case regularize explore reference focus basis hyperparameter goal increase brain advantage fmri estimation voxel development voxel development voxel naturally translate robust voxel method simultaneous voxel previous smooth newton briefly conference experimental presentation glm separate design ten brain two encode drive decoding provide comprehensive glm glm improve decode computationally tractable open letter denote size kronecker concatenation notation k concatenation slice array first describe extract coefficient bold stimulus trial presentation stimulus signal voxel acquire glm response linear underlie matrix convolution stimulus software amplitude one voxel consist glm design temporal basis form respect work refer stimulus correspond taylor impulse ambient shape canonical stick duration basis generally give form stack element successively stack regressor element size kk instead long activation possibility trial single assume peak bold possibility coefficient peak amplitude glm reliably large perform poorly limited conditioning increase sufficiently similar robust spatially estimation unique obtain column event estimation glm method stem glm across translate rank amount enforce glm glm coefficient estimate bilinear nuisance regressor ambiguity positively sign cost feasible jointly convex practical formulation advantage contrast glm rank estimate coefficient factor parameter subsequent analysis rank equivalent latter beta method normalization average project matrix readily prediction non unseen link unseen occur encode trial part response formulate analogous discuss classical glm estimation correlate voxel regressor regressor glm estimate rapid design boost decode task extend predefine function predefine set basis construct concatenation
firstly inner integral calculate analytically principle break curse integrate analytically secondly two k page see useful factored reinforcement freedom function correspond therefore restriction inner unnecessary freedom unique constraint k ol due nonlinearity induce greedy algorithm basis kx k fit prior immediately enforce especially lack derivative uniform discuss limit infinite eventually example uniform correspond reflect frequently yield shall high formally generality cast project regularization diagonal element call derivative equation matrix page input factor everywhere uniform smoothness everywhere kf kf page basis initialize optimization improve cost equation calculate change add factored basis function factor x dx consume gradient precise equation proof similar basis state optimization solution statement empirically however improvement stay minimum optimize factored solve derivation inner influence randomly cache loop dimension fall inner loop change linear problem propose newly basis linearly test dimensional toy sparse traditional test artificial toy label variance plot factored basis randomize sample region prediction difference corner algorithm converge runtime influence strongly quality bad approximation converge efficiently data gp benchmark repository mm compressive dimension describe various concrete cycle describe value variable output dimension describe white red sample quality scale dimension value factor toy draw construct factor factor fitting taylor virtual rmse comparable set factor considerably compact seem face challenge function convex local control performance slow runtime shot cut preliminary experiment number adjust noise factor ideally principle dimensionality enforce paper poorly result function predict mistake difference sensible update algorithm area break experience regression sparse calculate product expense multiplication allow inference summary factor basis promise perform gaussian less space potential extension improve upon runtime benefit greatly algebraic author thank helpful science input dd k factored factored function integrable factor though factored product trick integral heart factor k j well choose kx k kx j discrete kronecker universal combination denote k marginalization basis analytically solve product wise elegant basis low pass filtering optimize randomly ex lf derivative algorithm fourier function product parameter ex ex lf k set unconstrained solution normalize g z z x z z kf twice g f ex kf kf derive unique diagonal absolutely continuous training fourier l covariance kx cache sample b improvement kf k lf h k eq function optimize basis outer new df tu face curse integral class regression factor structural property allow point product application reinforcement break curse speed computation derive greedy factored basis regression perform benchmark factor compact introduce competitive process yield factor like analytical wise product marginalization kernel suffer curse computing network application like belief kernel like support vector de mainly due sparse classifier call method gp average wang choose svm task restrict everywhere function require equally state exponentially lead bellman curse effectiveness function due though small take thousand construct propose basis directly support pose select former function factor factored solve analytically
news autoencoder tree yield autoencoder capture method map hide decoder reconstruct autoencoder perceptron neural implement idea stack autoencoder representation work autoencoder use decode soft tree soft split output leave section soft tree use simultaneously hidden decode pass encode layer external autoencoder autoencoder hierarchical decision decision leave internal decision give child traditionally implement soft probability decision consider soft internal child name left outcome univariate single split geometrically speak though split orthogonal split orientation make applicable region right child two classification implement logistic leave child xx equivalent supervise predict scalar sigmoid nonlinearity convert nonlinearity output want soft decision give response splitting hyperplane tree supervise square order backpropagation efficiently gradient parent decision structure soft train supervised well layer follow chain autoencoder back autoencoder tree encode hide representation decoder want initial e update additionally derivative decoder representation decoder layer encoder level slow layer autoencoder decoder tree epoch leaf continue depth increment update introduce allow split additive random mnist handwritten digit database set mnist handwritten image pixel output output denote matter category sort two one map nonlinearity map decoder stack perceptron autoencoder gain gain result extra level depth see especially dimension digit result representation autoencoder perceptron tendency nonlinearity sigmoid corner autoencoder observe multiple leave hide rather tree assign representation size fashion closeness important behavior autoencoder hierarchical soft small gain increase dimensionality layer distribute high encoder depth histogram since soft every count leaf leaf blue include digit learn locality region locality learn mnist decoder digit certain child phenomenon digit autoencoder reconstruction see tree bag representation word path omit clutter capture fine fine model map leave leaf distribution extension tree response modify rule modify response input autoencoder local linear projection degree locally partition assigning distribute digit dimension indicate effectively autoencoder tree representation move away intuition incorporate locality get reconstruct extension representation digit see class capture autoencoder tree provide smooth representation space move like addition locality small reconstruct figure soft encoder reduction comparable autoencoder autoencoder decode autoencoder dimensionality within opposed apply reconstruct centroid process hierarchical autoencoder
stand source thank parameterization equation accord operator hadamard vector parameter estimate updating turn update mix look term fidelity differentiable fortunately proximal interested calculus quadratic differentiable entail forward backward splitting disadvantage dramatically increase therefore subsequently rather resort simple derive subdifferential subdifferential follow subdifferential admit explicit would source independently expression proximal w important equivalent amount thresholde quadratic fidelity burden orthogonal consequence updating provide rough might prevent later assuming perform admit algorithm estimation problem nonconvex minima initialization diversity equivalently threshold value greatly improve minima algorithm towards point computed guess source improve update source choice discuss thresholding thresholding hard main drawback soft substantial improvement consequence update replace stand operator name stem well discriminant verify algorithm detail k therefore relaxation precisely nonconvex cost sparsity constrain blind source separation alternate converge critical firstly decrease threshold strategy minima minimization guarantee step threshold help prevent secondly update iteration prevent lastly spirit reweighte update motivated numerical tends display distortion ratio sde evolve hundred iteration subsequently transform frame assume orthogonal transform allows transform appropriately extra freedom help produce bss redundant transform yield improvement make redundant wavelet type wavelet translation frame practice dominant tight imaging amount allows transform tight case choice role concept diversity true amplitude discriminant sample source hard select threshold initialize maximum amplitude initial guess source decrease final threshold noise practice threshold stand th gaussian guarantee noise coarse fine property source estimate amplitude ii spirit simulate anneal minima entry choice rely weight low sparse source trade lead penalization provide sparse desirable separate source begin access value mis mis start test turn lead good carried bring evaluate bss rna newton blind separation general mixing source negative ii quasi disjoint support separation complex etc negativity performance bss p make monte various call spc introduce entry activation process entry source draw mean model gaussian law laplacian width half mixture resp right redundant translation invariant wavelet pick level actual spike combination follow interpretation ground decomposition need account part due noise stand neither global name distortion performance denoise effect method ability follow mix matrix criterion introduce inverse correct permutation performance monte carlo simulation unless tb visualize bss display source observation source turn active spike bss seem prominent spike exhibit certain poor visually source retrieve rna paragraph level correlation coherence varies distribute opposite source accord source correlate source performance sparse experiment channel db display leave bss rna quickly sensitive behave show much entry correlate higher behave consistently magnitude bss display bss behave quite source db algorithm source spike source tb tb performance bss precisely bss sources diversity significant entry source claim early high share dynamic correlate method number proportion entry leave resp amplitude standard bss behave db dramatically db bss tb dr recover generally grow limited source entry source limit next fix db panel source performance source recover less high performance decrease rapidly keep mind sample say amongst independently source discriminant increase simultaneously grow might source explain decay mix method seem improve behavior notice mix average scalar product matrix matrix involve source measure source precisely distortion source actual bss method scale impact weight algorithm estimate source source via precisely across turn rather perturb large amplitude weighting source weight proportional column favor entry affect presence prescribed level comparison demonstrate contamination whenever discriminant turn noise discriminant consequence impact proportion entry fix evolution reveal noise increase tend method surprisingly algorithm presence bss behavior matrix naturally step order reject eventually amplitude detect algorithms tb long recently modern analysis method separation play recent light crucial play accurate background multi observation component foreground decompose source contribution introduction entail component blind separation focus datum remove component prominent emission impact blind source separation study carry simulation observation ghz model simulation major center simplicity prominent area source locate region correlate figure translation wavelet signal bss default separation bss noise column source resp nominal noise db feature mild value snr exception emission spurious source residual belong emission seem correctly define estimate figure display error keep reveal difference error free tend emission evolution bss normalize varie actually bss seem db already view algorithms matrix suited separation figure confirm criterion regime value noise source detect likely origin partial correlation tb input scale tb scale ff residual scale residual tb rna ff residual scale rna rna rna residual tb ff scale residual tb ff code introduce article available blind separation partially correlate bss tackle article retrieve emphasize discriminant propose adaptively weight adaptive component bss experimental correlation source slightly entry source retrieve finally apply separation suit nature blind bss technique context retrieve purpose bss discrimination distinguish reveal rarely valid practice partially bss novel retrieve partially correlate source precisely correlation source technique field source source rapid development extract development dedicate survey blind suited analyze linear observation source quantify source term stand additive well row source blind separation infinite problem require information source purpose classical rely ica far context differ focus separation source harmonic analysis apply attract lot compressive reconstruction dictionary sparsity source nonzero coefficient source generally entry coefficient vector negligible representation example signal representation wavelet adaptively learn signal sparsity exploit blind study rna source orthonormal adapt retrieve source generalize source support active one source partially disjoint bss series signal complex especially exactly sample source vanish alone introduction diversity md concept source amplitude verify source salient source share coefficient building concept diversity show bss developments bss emphasize negative matrix focus set matrix necessarily non life actual neither statistical source may dependent source particular research development dedicate already correlate discuss tend source paper good result alternate algorithm iteratively admit close project square th known thresholding operator classical least partially correlate sparse blind clarity transform essence low limit problem find jointly possible precisely norm source bss tend mix matrix source radius notice sparse independently source
finitely f fix sentence exist finite finitely computable validity unary predicate binary e validity hard rational f language duality imply number unary predicate finite validity countable infinite unary predicate countable hard rational iff countable look favor possibility countable proof carry replace complete work alternatively problem countable requirement predicate predicate logic requirement logic seem play theorem unary predicate logic logic aspect property fully sentence partial join tree ask meet operation deep information regard fact reduction fundamentally require rational inter eliminate unary restriction crucially force equal reduction validity solve arithmetic operation rational language carry case fortunately example illustrate magnitude hold motivation logic see theory classical theory concept mention two benefit technique convert version probability expression bit canonical ordering counting etc analogue complexity abundance like indeed plausible one could description class logic game another develop complexity I cs equip final thank throughout away start work discover I logic make smooth finally importantly mathematical detailed mathematical world world able subsection thm corollary thm example thm theorem thm title logic paper first order logic equip interpret previously rational hard general language logic complete countable logic countable remain largely individual language unary predicate requirement language language finitely unary predicate countable artificial neural ann age big increasingly ever technique inductive logic attempt pac pac logic context investigate meaning hold measure logic logic inspire logic measure interpretation keep asymmetric logic learnable learn roughly error bind universe logic property seem carry turn reason much hard order logic fact arithmetic analytic decide language whether sentence table knowledge validity definition logic tuple language unary predicate binary predicate infinite number type empty tuple suffice denote valid coincide sentence logic language e validity also complete open logic unary predicate language logic valid iff counterpart validity countable model answer answering question vice versa assess calculus e c validity complete complete tuple notation denote language sentence coincide regard validity ordinary general mechanism finite rational relational language unary predicate function symbol equality e logic transform sized sized countable knowledge countable countable validity complete notation logic logic whose interpret analogue straight motivate research logic assumption edge artificial prove countable sentence development tool idea semantic e allow form sentence rigorously inter different rational equip powerful tackle completeness finite hardness utilize perturbation simplify value distribution last validity coincide countable model countable valid sentence valid validity countable finally mention possible letter underlie write countable signature sequence free variable formula shorthand shorthand sentence formula convention formula addition use place subset recursive hard every computable many hard resp resp complement resp resp complete please concept denote algebra define boolean additive algebra boolean say set least implicitly discuss e measure letter start etc context please concept theory triple variable linear satisfie equality two equation find nonempty iff concern feasibility program identify division feasibility arithmetic rational path program consist inequality concept regard formalize definition logic e logic dual logic abundance let countable signature universe algebra x logical example distinct iff treat interpret q universal variable general split x implication symbol boolean combination symbol reduce implication eq implicit make sure sense impose refer model ordinary language countable signature possibly language formula relation measurable include equality denote iff formula iff call model similarly concept analogue countable thing likewise make satisfie analogy normally record often countable possibly contain write interpret strictly formally every atomic q logical distinct variable treat eq iff interpret split atomic implication implication similarly logic q definition duality iff states table validity complete iff iff show model theoretic property exist f logic model counterpart theory automatically countable let formula complexity involve measure middle derive fact agree finitely additive countable iff finitely countable notice atom restriction reasoning must atom nan atom extend measure inconsistent measure countable countable contradiction everywhere treat make element measure main object e finitely call validity resp x countable validity countable likewise sentence finitely satisfy countable distinguish finitely formula finitely first would validity logic logic goal concept develop section possible motivate point mention later reader check apply respective exhibit example highlight logic let true equality iff singleton also logic nonempty true also countable let logic parameter q b common follow axiom sentence resolution analogue logic logic iff logic record ia x x I ib logic conjunction sentence rational number sentence finitely though expression graph vertice graph artificial example logic ease read logic context logic note sentence z boolean symbol define shorthand composition symbol rough picture implication follow identify indicator oracle randomly return accord pac accord concept return take note see language express assumption order universe everywhere define concept word concept assumption relational hold eq represent iff class collection assumption conjunction iff conjunction concept list triple decision procedure proceed returning represent proceed iff quantity mention illustrate dimension express formula hand logic express strictly straightforward generalization free part expression complicate boolean combination irrelevant label depend note along would concept apply establish kind question unary predicate also express size parameter sentence weight vertex language logic pair vertex loop moreover vertex vertex edge thing population express undirected complete unary express unique positive vertex vertex graph graph express existence carry vertex logic make statement element element undirecte complete apply subgraph valid iff claim countable model size universe measure interpret ik v kk reasoning fact modifying imply let universe subset finite place countable measure meaning realize arbitrary valid q universal formula equivalent valid certainly satisfy contain f universe number time obviously finite restrict appear appear desire tuple language valid yield language set finitely valid formula finitely coincide sentence unary predicate call countable f well essence predicate number indistinguishable part elementary unary predicate universe subset partition immediate interpretation measurable elementary equivalence strong sequence iff construction iff eq x imply converse apply start sentence node subset child child child call denote level write measurable additional simply uniform z x distinct valid e label q level measure level f reader choice node across apparent statement ready tackle relational language e countable particular suffice universe everywhere sentence free iff height finite devise suffice universe iff interpretations structure level program lp replace rational rational finite effectively logic proceed involve strict lemma vector exist iff rational feasibility elimination exactly whether affine plane index yield rational feasibility equivalent feasibility solve ready characterize relational logic roughly relational universe everywhere property check universe restrict everywhere interpretation condition verify universe probability everywhere iff existence strict duality language rational countable finite validity like express order string form eq variable form sentence weak sentence likewise sentence order sentence clearly every sentence term q differently tree generalization tree tree generalization analogue well pair level leaf node expression eq singleton tree x x order tree sentence verify definition concatenation eq thus property hold immediately simultaneously iff subsection rational would like countable symbol equality language unary predicate iff reduction work computable sentence sentence exist apply full preserve q finitely rational finitely repeat simultaneously lemma therefore logic case former half completeness countable unary predicate predicate rational reduce specifically show proof proof hardness reduction finite e rational initial respectively interested input assume represent symbol break section finite section describe first language section construct finitely indeed vocabulary vocabulary constant unary predicate state still intuition vocabulary formalize linearly order element roughly logic specify measure use around element avoid force equal become encoding encodes reduction iff finitely formula recursively free x nx consist conjunction interpret deal model positive sake clarity axiom atomic element measure respectively eq initially function shorthand conjunction iff great say shorthand implication shorthand expression shorthand p p transition construct machine head point cell head explicitly head character belong conjunction condition symbol conjunction exactly head cell symbol hold q x least conjunction write clarity set take element formula rhs assume shorthand probability say satisfy finally sentence parse therefore sentence strict segment purpose use conclude reduce satisfy universe natural iff iff position define iff symbol iff state since measure clause thus fix q conclude probability q e therefore assume sentence encode chain element axiom restriction relation element rx thus clause equation hold n mm satisfy ny
eigenvector score associate division operator take b work researcher log aware blockmodel spectral necessarily highly likely misspecification regular secondly previously cluster several cl concern robustness blockmodel bic blockmodel community choose community cl bic bic composite cl paradigm cl relaxation complexity relational complicated implement working estimator misspecification likelihood density statistical inference capture cl bic relational datum go would like misspecification joint consider stochastic blockmodel ng parametric impose specify full access univariate blockmodel family composite marginal first log unbiased usual regularity associate composite minimize blockmodel datum replicate common form individually regularity arise include argue much correlation among composite score retain good correlation consistency asymptotic normality scenario take context composite likelihood consideration k k result community eq different estimator distribute replicate asymptotically normally establish model universal though misspecification blockmodel replicate bic true community whenever correlation severe consistent asymptotically consistency form blockmodel composite bound community conjecture node increase community number leave work treat composite work assumption variable denote derivative q matrix u complexity specify cl bic reduce bic estimate cl bic naive vanish composite likelihood let adjacency ab ab ab la l n see multiply ab parallel partial ab cl bic number community simulation dataset blockmodel set independent label contaminate bring degree binary independent variable regular stochastic blockmodel thresholding gaussian correlation il w l correlate correct blockmodel carry matrix record criterion agree apart cl integrated likelihood bayes vb estimate community select bayes value true real datum set additionally incorrectly community median indicate correlate decay respectively correlation bernoulli cl bic cl cl bic prop proportion deviation deviation bernoulli w cl bic bic cl bic correlation common whether belong collect table il collect table c cl vb cl bic vb w cl bic vb cl vb cl bic cl bic bic bic allow topology connect form community size result network blockmodel identifiability replace unit q slight abuse expect table contaminate impose world cl bic correlation large instance community cl successful community generate purely blockmodel correlation cl bic vb select impose vb cl bic simulated vb yield median translate selection noisy estimate nevertheless consistently correct bic setting addition present stochastic simulation correlation simulation tend even grow community bic true vb measure quantify assignment community represent pair estimate agree assign second ratio within community imply detection cl bic grow blockmodel record score median ratio c c bic est md mr prop md md deviation goodness fit median ratio est scenario community label community community grow blockmodel cl bic across result aside fact decay scenario potentially indeed label ahead exhibit obtain evaluate cl whether increase number cl bic line international trade originally contain trade cl bic bic pair blockmodel focus year form weight country country finally show bic community correspond blue european medium south select south american split traditional yield divide community bic community little obtain longitudinal health http edu student school cl correct blockmodel figure show component result select community actual misclassifie score cl bic community still cl black cl criterion extremely even correctly specify fail black community student close among student goodness cl bic assignment slightly mr cl bic indicate superiority pair example bic penalize cl bic cl bic robustness issue due misspecification underlie capture amount recover structure especially literature study property blockmodel misspecification blockmodel blockmodel likelihood select simplicity robustness work spectral interesting explore cl dense real another whether correct blockmodel supplementary material file replicate simulation upon request read among community relational blockmodel inherently mixture raise selection community bic stochastic conditional assumption community different edge usually violate propose composite bic select blockmodel approach simulate material contain relevant code online community correct blockmodel blockmodel network analyze interest researcher study underlying structure world
stochastic gradient hessian product form replacement classification establish algorithm descent hessian vertical label objective dotted black mark cd outperform sgd objective h epoch epoch effect setting epoch memory helpful stage article manually economic market convert represent appearance word extremely sparse nonzero fourth market accordingly function numerical quasi aim curvature sgd method choose report iteration objective increase iteration fast initial eventually variant size figure mark axis epoch illustrate vary batch fix improve computational effectively hessian choice e g b parameter value blue spike occur term bfgs formula lead monitoring say indicator update skip small memory size consistently beyond improvement comparison reason deterministic bfgs observe poor unstable varied eq entire indicate therefore ratio computation display error norm batch exhibit non sense less norm gradient batch great show gradient decrease batch tendency inaccurate hessian stay accuracy need fw calculate form differ bfgs perform curvature calculate use gradient evaluation indicate sample compute algorithm numerical effort limited poor quality implement follow setting unnecessary rescale fw iii hessian updating average last recommend compare realistic observe batch h speech set stochastic entirely bfgs implementation latter use bfgs enforce uniformity resample datum evaluate present sgd diagonal rescaling quasi newton similar update rescale hessian noise stage online stage stage minimize use step take newton employ product compute derivative newton approach iteration minimize employ idea geometric presentation method seek asymptotically method concern address fisher context network interpret direction maximize outline implementation empirical additionally maintain argue contain hessian needed cope hessian improve al employ hessian curvature note nature regime paper operate stochastic convex quasi curvature interval gradient incorporate useful say make quasi product may essential goal require uniformity prevent task establish convex indicate quasi solution analyze applicable problem provide mechanism ensure satisfied thm conjecture l incorporate stochastic application quasi update lead curvature effect robustness propose quasi efficient scalable employ formula beneficial collect interval sub hessian product arising learn method machine massive impose batch feasible however scale suited computer sensor operate approximation limited bfgs produce regular hessian vector product uniformity avoid potentially gradient consideration convex arise simulation instance refer machine input typically take form collection refer objective empirical amount mini batch gradient yield operate stochastic substantially entire employ hessian hessian stable efficient manner scalable operation limit quasi optimization method solution hessian gradient gradient affect ill conditioning completely remove appropriate choice next present quasi employ limited bfgs correction recently define correction corrupt effect gradient numerical arise organize discuss experiment illustrate contribution paper term stochastic sa use programming standard machine sgd discussion quasi newton method fact curvature good enough bfgs minimize correction always strictly scale application scalable enjoy linear rate scale direction evaluation extend quasi newton stochastic update one quasi inherently average reflect hessian entire something stochastic achieve hessian calculation flexibility add new curvature emphasize curvature update schedule gradient curvature iteration eq define potential approximate taylor sample define training example emphasize code regardless give length gradient fw define average essential iterate bfg update correction mathematically describe update new matrix bfgs newton form employ formula loop step correction strength bfgs classical pair interval extra hessian iteration method sgd even early stage newton representative namely test give gradient hessian evaluation require multiplication evaluating batch approximately product assume total operation sgd appear report use choose logistic nevertheless newton method loop recursion bfgs employ limited bfgs updating outer product effective around compact main iteration since suffice curvature iterate lag product several select minibatch experimental environment well experiment cost namely cost range choice bfgs similar good value range continuously function strong slow employ iterate remain take convexity h fw entire nonnegative regularization assumption satisfied show hessian generate assumption satisfy analyze allow literature newton formula iii definite update zero denote boundedness set determinant show since eigenvalue use away therefore satisfied next establish beyond contain objective assume bound case follow assumption well hold iterate q fw
arbitrarily determine feasible separable present error induce separable roughly sample indeed contribution work compute contrast deal aspect contribution induce bayes consistent denote main formally state follow bayes consistent give unit diameter lipschitz refine enable generalization optimal property dependent expansion computable finite chapter rule terminology extract induced assume compression optimal generalize compression discover nn various heuristic leave reach nn label extension classify decay oppose compression space normalize diameter lipschitz small lipschitz denote metric cover ball radius finite agnostic whereby label example nh bayes classifier define infimum surely slight abuse distinction space partition positively subset margin minimum opposite label naturally induce h sx margin agree previous definition binary double style double double distance pt scale shape mm fill ps ms ms leave none dash p fill none ss minor terminology connection make member extension actually explicitly minimization paradigm minimize penalize explicitly term motivate analysis sequel nest inner n slight propose analyze sketch ff ss minimum cover routine matching computable ff bipartite graph compute opposite label minimizer total small u formally n penalize empirical risk ff f margin optimal r risk n fx pt technical sample break basic decompose excess decay surely proof proof appendix order connect rf concentration bind unknown would overcome introduce surrogate follow illustrate empirical common double deviation nf term decay almost surely index risk rf nf r rf rf nf n nf since nf nf n nf l make form penalty yield nf l nf inequality follow approximate margin particular hold r n margin surrogate loss every hence large r estimate analogously bind run various convergence compare actual take put x illustrate compare four classifier classifier rbf validation describe cover matching searching runtime nn satisfy contraction lemma essentially contain idea l writing bound term lemma write term since l take r recall define n diameter imply totally cover finitely ball diameter imply continuous set generality normalize pointwise lebesgue dominate imply hold choose sufficiently small prof claim rescale ng r decay lebesgue
plot atomic bandwidth figure repeatedly increase separate observe distance choose range number dx figure curve cluster sigmoid bundle bandwidth resolution shift identify atomic cluster dash line could potential outlier similar signature signature simplicity signature acceleration tangent signature imagine signature produce unimodal distribution acceleration acceleration signature signature good replicate original signature acceleration multimodal suppose different e signature signature output surrogate assign acceleration close allow signature cluster formal analogy simultaneous test local mode identify mean therefore effectiveness curve strength reduction choose priori cluster furthermore functional tune distance implicitly scalar functional shift properly select selection shift algorithm bandwidth drive propose optimally estimation bootstrap selector arguably bandwidth shift incomplete proposal strategy introduce selector propose maximization mode significance maximize hard come selection regard investigation datum bandwidth interesting research gate obtain explicit highlight correspond ascent analog publication r estimating mode functional bootstrap estimate methodology shift algorithm sort e shift original signature shift draw figure depict function distinct unimodal density intuitively shift axis direction ascent converge either leave leave local cluster idea generalize figure unknown empirical corresponding empirical density paper idea modal euclidean natural dominating find local surrogate estimate due surrogate require existence dominate open space approximation call fast local mode density repeatedly close use bandwidth update density local mode eq density activity apply signature mode unknown early several asymptotic normality mode derive risk obtain minimax dimension mode attain propose root seminal shift algorithm science segmentation mechanic shift arbitrary repeatedly bandwidth generate ascent close unknown precisely line repeat solve unknown also call discussion ascent ascent corresponding find justification density critical ascent see initial assumption distinct intersect gradient naturally form collection equivalence inferential viewpoint sound foundation definition mode correspond associate estimate connect component px estimate definition depend severe represent drawback furthermore pose often different persistent although euclidean challenge functional branch deal datum dimensional surface last decade nonparametric fully euclidean realization theory therein attention principled vast density attention difficulty define probability address define algorithm euclidean towards let x x shift gaussian euclidean local mean profile shift consist update simultaneously datum tend converge number algorithm perform cluster converge repeat update generalize shift clustering consider isotropic allow presence mean act opposed version apply simultaneously entire generate multi algorithm shift keep operation call equation depend position rewrite unitary q estimate density position gradient ascent large correspond position direction size visit repeat update candidate view ascent allow set trajectory equivalence class unknown define shift number imply ascent unknown recent consist suitable variable potentially dimensional measurable space induce generate pair difficult natural dominate replace radius induce function behave consider process decomposition value cumulative px bx fx fx op lebesgue dominate measure propose population surrogate profile distance two assumption henceforth shift exist worth surrogate space may define basis component small help mean numerator turn interpret population informative flexibility user pca force towards shift incorporate prior implicitly many save example uninformative semi avoid alignment fact systematic among curve show shift hilbert ascent gate profile notion ascent characterize element maximize gate derivative proceeding recall definition gate gate banach let gate differentiable gate differential gate differentiable gate linear gate gate estimate gate gate differential bandwidth possibly entire profile satisfy truncate instance gate derivative still surely fix long verify among purpose ascent say matter coincide gate functional rewrite unnormalized estimate surrogate operator analog equation eq ascent conceptually conceptually fix point update see mean ascent functional problem generic hilbert starting lipschitz gradient map therefore surrogate gradient flow estimate converge enough flow guarantee flow population surrogate regularity address address sequence shift sequence element approximate think associated flow fix adaptive bandwidth incorporate unnormalized analogy evident eq root bx hx bx closure open local maxima satisfying consider local bx maxima functional shift shift local develop hypothesis functional next impose profile profile ascent scheme profile differential ensure right table satisfy another useful shift automatically separate respect atomic p bandwidth note functional obtain iterate q simultaneously functional principal intrinsic coefficient th functional essentially projection mean produce clustering cluster know elliptical shape speak cluster cluster principal score elliptical shape yield similar however intrinsic dimensionality functional great evident space span principal oppose functional coefficient component top dash elliptical combine situation shift depict lie fail circular pattern seed suitably depict obtain cluster shift meaningful effectively modal previous show shift surrogate satisfied statistically functional mode whether point second gate derivative definite assume unknown surrogate twice gate gate analog hessian point surrogate negative pointwise constitute instead test return point similar adapt functional natural test statistic p give gate differential surrogate gate surrogate gate order optimality profile involve whenever element adaptive divide size shift subsample step subsample next subsample test statistic second subsample statistic unknown use construct take purpose
ph co analyse optimisation un mod un de est co en un pour des code des es le dans pour la la code la de la dans une base un les en pour la de pour est simulation tool study computer time consume run study sensitivity reliability molecular convenient code since suppose take less actual say design already widely study deal code numerical code code consideration call conditionally function code set type literature stochastic shape distribution normally author output approach variance author model generalized quantity process propose estimate one moment hypothesis output estimation moment carry computer consideration input return computer single support access probability input density I represent real choose non density bandwidth estimate output stochastic relevant express coefficient link apply analytical carry application consist choose classical regression problem consideration kernel estimator hellinger suggest I four dark light along black hellinger vector major drawback kernel curse quality dimension sample average poorly variable bandwidth kernel technique vary parameter several adaptive numerous use capture curve quality form function regression goal form former sample function datum lie provide build basis coefficient f I therefore integral way adaptation orthonormal experimental subsection negativity account adapt function property function experimental impose orthogonality ease prediction experimental coefficient among know build orthonormal onto maximized maximization problem spline propose apply discretized ensure sample decomposition ensure propose however approximation et al propose put negativity let free first basis force non function interpret section analysis build basis interpolation denote define function uniquely pick parameter g f specific hereafter mp interpolation point basis compute heat dependent channel algorithm therefore mp approximation interpolation lose basis greedy associate f coefficient solution quadratic
rate estimator sparsity favorable aspect simultaneously convex discussion section beyond bandwidth precisely aim beyond regard penalize group restriction generalization lasso nest structure group parameter zero hierarchical convex employ hierarchical penalty tailor semi extension penalty connection also write datum formula efficiently relate establish treat population recover minimal make employ show frobenius multiplicative logarithmic factor appropriately define population thorough demonstrate example use covariance discriminant describe estimator define desire sparsity pattern triangle form index notational useful express group example extreme may seem outside indexing triangle element panel depict index frobenius second express toeplitz w already bandwidth corollary matrix positive high hold moreover version guarantee course behavior give w estimator group lasso notice traditional act size penalty zero recover bandwidth estimation norm minimax logarithmic factor pattern hierarchical term employ triangle recall note principle weight consistent selection minimal weight refine norm particular fact include penalty penalize hierarchy within exhibit hierarchical factor appropriately covariance estimator diagonal pattern coordinate convex guarantee separable separability block correspond update involve ellipsoid explain ellipsoid remarkable proved weight start covariance large triangle pair thresholded triangle ever step bandwidth next show estimator dependent contrast estimator bandwidth toeplitz dependent begin adaptively estimator section property begin state assume marginally sub exponentially true nonzero enough demonstrate estimator recover need adaptive hold result high see proof assumption couple inspection need directly see avoid probability regard ij distributional marginal x continue hold possibly bandwidth prove mild next intuitive bandwidth able detect require measure root size noise high require sufficiently might weak showing exceed threshold previous may excess requirement however without bandwidth bandwidth small theorem apply scheme requirement positivity unable either minimax logarithmic population study begin state general optimal minimax immediate corollary bandwidth positive covariance deterministic theorem result weight achieve variance trade probability price constant multiplicative motivated general class henceforth trivially notice magnitude entry class vector assumption covariance would shorter tailor penalty adaptively minimax summarize suppose either logarithmic similarity base single realization discussion distinction penalty optimality weight suffice one must resort previously consider relate interestingly exactly force neighboring decay necessarily contrast type matrix neighbor decay immediately far enough specifically appropriately constant factor minimax respect frobenius adaptively approximately semi convergence frobenius require prior rate frobenius block statement class minimax covariance minimax sparse cite inspection state follow clarity estimator condition even illustrate theorem recall imply op estimator focus behavior matrix norm instance approximately immediate adaptively slightly suboptimal strength optimal rate class adaptively sure criterion bandwidth choice adaptively block minimax study positive quite circumstance prove assume assumption either provide reliably definite definite suggest choose conclude note light corollary property ten spaced value simulate move covariance frobenius operator quantity vary linearly frobenius side align closely pe simple weighting weighting scheme vary along equally space simulating suggest corollary f pn pn phenomenon phenomenon suggest section performance band band band nest lasso cholesky series estimator vary procedure element evaluate way frequently lead semidefinite require section definite problematic consideration suffer three description approximately take increase particular lasso outperform simulation scenario matrix parameter scenario likely nonzero find block still covariance nest lasso slow rest frobenius norm find band perform operator bandwidth essentially note poorly contrast poor two op increase frobenius require convex observe label quadratic discriminant assume assign estimate binary consist goal classifier automatic predictor intensity within class inspection size five parameter estimate rule regularize typically kx lda regime introduce kind covariance practical estimator problem admit result adaptive rate operator matrix multiplicative logarithmic matrix establish allow bandwidth grow estimator rate optimality truly contrast propose guarantee exactly version guarantee finite estimator fast linear indeed procedure require semi appropriate package name implement code method equivalent minimization primal duality dual function eq coordinate clearly lagrange multiplier equality make replace make rhs decrease simplify r h turn w numerical tuning range wish eigenvector op kn constraint may drop meaning update involve project onto semidefinite cone similar explained initialize positive part subroutine r return proved inspection directly place avoid ij distributional assumption marginal c q next lemma exist constant let n ij j jj constant find find lemma eq easy q proof equivalently hold k theorem wish recall separately theorem show w exceed follow requirement establish complete theorem q suppose recall amount net amount weight state prove instrumental first matrix newly contribution treat differently arbitrary second set denote vector penalty term write new sub b gradient monotone dual cosine simplicity focus constraint hold eq follow next equality eq finally let
high confidence propose reach prescribe hoeffding matlab part guarantee integration library variable quantile integral optima case ensure almost interval know confidence conservative bernoulli possible include failure process complex able iid simple analytically tolerance tolerance confidence publicly next automatic integration present review constructing rely theorem mention know get variance need desire width suggest construct confidence interval proportion add pseudo failure adjust since well interval carry suggest tail calculate exact uncertainty confidence interval however small follow inequality need algorithm prescribe cost computational confidence interval end discussion sample random variable review chebyshev mild chebyshev inequality random variable interval know letting mean width interval eq costly quantile gaussian variance bernoulli satisfy slow term chebyshev interval guarantee need provide conservative width confidence random use inequality inequality variable suitable chernoff special lie hoeffding inequality eq construct width uncertainty iid mean computational hoeffding algorithm release guarantee integration library library automatic pp replication confidence answer return show tolerance replication replication result guarantee replication ask exceed encourage answer case conservative least loose try construct guarantee uncertainty ratio reasonable price pay htbp recent practical carlo monte carlo numerical require input practitioner justify motivation construct bernoulli soon matlab toolbox
answer differ problem advantage towards challenge dataset latter learn mapping linguistic question answer task end enforce task learner benchmark base answer task exhaustive symbolic annotation answer question large answer task world real scope beyond tuple recognition dataset rich suitable retrieval challenge reflect represent content contain different question answer stanford consider question segmentation method v discriminate object category color stand refer thing color annotation heavily concept frame include language work number predicate word word error reliably common cutting cut front restrict sum common treat logical database assign similarity aforementioned requirement metric quantify architecture motivate accuracy via membership answer produce respectively bag author membership suffer aforementione whole question recent work direction improve score collect answer first run answer answer answer many answer answer agree answer average answer call extension potentially base coverage issue problematic concern abundance improvement consider similarity success accelerate believe consist learner limited hand build learner source additional vision resource contribution architecture identify exhibit answer challenge architecture task scenarios max institute inf language visual rapidly observe increase modality allow towards open hope achieve ai machine towards quantify increasingly ask open summarize challenge answer answer task base unique truth annotation carefully drive force benchmark provide output challenge progress machine understanding task progress inspire researcher architecture language sentence alignment question answer argue open answer attempt tool benchmark quantify carefully well generate grow benchmark evaluate increasingly ideally yet want metric evaluation assign domain challenge base limited coverage third aim issue bind reference frame answer task answer inconsistent obviously human inherent compete true truth answer true look consensus take multiple answer interpretation metric idea entirely build open aim demonstrate challenge exposition helpful building challenge open work modality either world machine aspect open answer deal challenge prominent order guide scalability natural reasoning serve human spatio architecture reproduce diversity thousand ambiguity category grow boundary become inherently instance sometimes difference reasonable architecture create human prototype concept limited category color learn noun g white white ambiguity reliably human depend context may observer frame moreover unclear predicate aforementione adapt symbolic reliability
contamination asymptotic standardized contamination replace collect require form influence robust base idea robustness corresponding estimating technique bivariate random marginal z ne bn k contamination point induce proceeding estimator approximate simplify influence imply corresponding hill clearly unbounded nature hill robustness correspond er assumption suitably follow derive influence similar non term detail index case distribution may transformation use approximate contamination sample also contamination simplicity come let regression tail dependence coefficient tune contaminate contamination bivariate could contamination assume contamination say contamination influence distribution influence estimator correspond approximation distribution boundedness influence supremum decrease great robustness extent contamination hence increase quite intuitive robust contamination contamination influence derive contamination point influence boundedness present illustration first robustness contamination consider bivariate coefficient stochastically independent bivariate distribution variance coefficient compare tail dependence standardize bivariate tail generate namely mse base performance brevity similar report far also quite similar p th er er er f er er er mse increase bias decrease near near give exist estimator hill extent increase mse slightly long propose estimator depend robustness simulation say observation er er p shift contamination robust tail copula close er er er er er er er th er er contaminate empirical bias contamination heavy contamination interestingly contamination case consideration contaminate model case mse increase contamination optimum become choice increase contamination bias slightly mse mse increase estimator quite contamination contamination structure performance contamination extend mse increase pattern even well exist finding clear perform value mse large remain near respect estimator hill small bias influence robustness near however dependence propose univariate generalize hill structure density achieve illustrate extensive estimator exist estimator contamination member outperform bias note performance parameter choice real estimator bivariate future consider view statistic moment estimator tail sensitive presence propose robust robustness classical estimator illustrate extensive bivariate extreme economic finance become day big loss predict case may help analyze grow trend value model model generally tail rare event heavy univariate characterize tail measure tail multivariate marginal characterize distribution effort dependence linear market return mostly tail function specify coefficient dependence parameter extreme see work al exist care estimator coefficient significant portion external ignore factor recommend bivariate proper outlier tail drastically objective big produce estimator tail univariate see lee receive attention bivariate dependence see well result illustrate bias mse consider kind structure presence outlier start brief dependence coefficient influence section examine though section finally end short unit fr pareto tail assumption obtain limit threshold depend bivariate extreme asymptotic assume probability vary slowly vary function txt transform pareto transform technique univariate property tail dependence index researcher hill estimator classical hill basically logarithmic likelihood regression alternatively univariate recent attempt exploit weight density robust lee hill achieve proposal recently assume exponential regression apply study bivariate tailed consider bivariate tailed unit hill lee fr transform marginal view tail tail index excess exponential distribution exist positive number identically distribution power divergence outli lee routine show estimating estimator estimator
predictive work dataset inter one machine strength predictive six find reliable predictive model efficient var k provide much wide band inefficient var ignore reliability conv conv var sometimes band interval cv conventional solution confidence interval attempt compare effective nine benchmark distinct superiority var precise conv conv value small envelope conv interval wide general trend regression square interval conv conv fix var hand quantile general trend estimator efficient quantile reason superiority occur variance global function square base size confidence inter response global introduce interval model proportion specify regression literature extend find properly side introduce statistical interval rate efficiency envelope prediction perform method rank provide figure compare art interval locally idea behind exploit prediction interval contain desire could bandwidth technique parametric interval bandwidth bandwidth differ conditional response variable find confidence interval inter quantile distribution conditional quantile response obtain estimate quantile propose predictive contain least desire proportion variable quantile suffer quantile occur conditional predictor conditional conditional estimator converge important note suffer base reliable member take local conventional asymptotic conditional remark limitation list reliable prediction square regression bias error address desire
procedure w nice error show initial f projection procedure nice get rotation simultaneously diagonal since norm bound know rotation invariant imply triangle project convex column initial within column thing prove multipli never initialization slice expansion enough eq tensor let run satisfy incoherence least norm probability least last lemma proof show consider incoherence correlation condition draw component argue bind combine calculation incorporate valid draw dominant good initialization propose lemma tensor tensor assumption denote capture correlation weight suppose relative define probability expand prove accord span first column row orthogonal operator similarly subspace svd ii therefore top leave right singular leave singular spectral norm second inequality equality b equality lemma therefore apply I I I inequality auxiliary entry last inequality taking next h maximize finally noise high tensor
fx hx agnostic learn hold restrict version agnostic hypothesis boolean p agnostic draw randomness pac agnostic say differentially neighbor satisfie definition change hold neighboring refer justification say fx differentially pac refer deterministic denote value characterization agnostic characterization learn let coin way produce communication bit randomize protocol input public coin one way algorithm algorithm output coin private protocol coin protocol public coin protocol compute require notion complexity protocol unchanged protocol
low near high gradually reduce noise conceptually overall step step anneal schedule target length sample improve anneal length anneal anneal fraction spend anneal run handwritten digits mnist train validation architecture hyperparameter agnostic deep layer train decay schedule epoch fig subset collect sampling baseline assess anneal configuration sampling empirically confirm burn run chain
error system examine whether able detect cascade large environment epidemic source truly close almost principle might seed rarely epidemic graph entire truly report truly report node epidemic source internet internet good ball infect report reporting finally truly report degree closely law network test algorithm report classifier uniform report classifier frequent turn increase achieve rate uniformly report truly reporting scale free positive carlo infect node whereas node report state often implement term environment finite optimal optimize preprocesse theoretical adequate section thm thm epidemic extremely human reliable phone infect people home readily secondary stay home stay home precise identify epidemic local require knowledge
observe generalize secondly observe consider estimation initially individual sample size deal problem incomplete maximization recent successfully maximum article branch organize describe subsequent study devote study obtain develop section conclude remark provide readily read devote theoretical control branching mathematically growth define follow variable identically dimensional common respectively denote control particle generation law control generation process individual remove presence etc type add population etc population probability surely verify development law control seem formally belong power I subset shall henceforth index regularity condition exponential family include many poisson binomial negative binomial belong distribution depend term
sense occurrence assign preference switch switching incorporate go already active base problem finally analyze problem classify sense comprise instant basic equipped solution include switching field divide discretization utilize gradient time programming select priori simplified determine end algorithm network switch initial cite initial select solution optimum trajectory start drawback especially base fact lead great potential horizon cost horizon bellman approximate state potential motivate author study solution interesting feature development condition
deviation c c c statistic carlo simulation sizes square corresponding deviation error theoretical previous confirm monte small size favorable regardless gap scenario preferable geometric transition relatively square well estimator seem impact establish reveal favorable case consecutive observation binomial rare remain consecutive somewhat model statistical queue queue time every hour transition consecutive observation assume iid maximum queue
input sampling step compactly store find rank provide run input sparsity time weak frobenius e replace spectral large low directly find first weighted alternate different sampling alternate algorithm need length though alternate run spectral application matrix small processor good burden distribute guarantee tight capital typically denote th denote unless denote denote operator denote frobenius norm denote principal angle distance span denote constant
expect cumulative regret gap dependent tight constant gap upper finally practical important minimum cost span optimally algorithm solve optimization formulate find modular work variant modular unknown episode episode agent observe weight zero receive product basis payoff maximize cumulative return equivalently expect cumulative formulate delay network initially unknown basis span delay span delay contribution bandit bandit successfully optimization extend combinatorial problem solved propose explore uncertainty episode gap bind also gap upper structural bound semi bandit problem synthetic diverse movie range problem write write
classify hierarchical classify class associate example train case misclassification design classification multi temperature conjunction object benefit advance area occur acoustic scene update category signal likely either car strategy employ acoustic environment environment multi acoustic database database environmental emphasis computational design perform introduce propose modular present challenge fail significantly outperform significantly comparable benchmark acoustic misclassifie correctly scope reach human environment produce anonymous read early manuscript substantially also technical audio frame spaced bin map band human meaning capable well frequency result logarithmic result process constitute capture spectral envelope periodic exhibit spectral peak frequency cosine encode include frame govern discrimination feature belong relative belong rather perform aim avoid variation accomplish subtract global extract global deviation vector normalise standard infer interpret generative distribution notation identify extract model generate operator class equation accomplish parameter rule indeed must sufficient component generate large tend spurious variation datum
recursion depend alpha add unless three coincide full method largely stem subproblem describe alpha comment key depth proof alpha subsequently alpha special simplified benefit reader form write implementation avoid alpha general minimization comment alpha establish alpha assumption first describe identity product space equip norm definite hadamard abuse section alpha solve domain vector block fact terminology relate broad proper uniform serial choose start generate block sampling enter nonzero proper move change
upon close map image result count perform capture assume window grid flexibility like actual observation counting explicitly often count histogram consequence live conceptual matlab grid page parametrization histogram representation task art extract generative improve reason apparent need concern outperform bag word computer consider latent community recent scene image originally capture count text image constrain way text building drop give way find possible constrain property location window capture count model grid count need model window feature count count demonstrate representation count combination accurately traditional come camera model variation represent bag due computational achieve ignore spatial patch bag arise variety extract low image cluster discrete assign descriptor codebook ideally categorie sufficiently discriminative count become exist histogram multinomial validate bag feature extract natural relate image consequence classification
rate convergence rate approximately less smooth result alternatively process rt dominate matching possible term large rl match third convergence rate excess risk b ab b b b ba bc bc bc bc derivations supplementary material x excess derive excess bind z obtain hold argument bound x mu bound counterpart e upper combine bound union bind simplify bound illustrative supplementary comparison justify experiment alternative probability measure endow sound ridge regression embedding rkh special prove regression kernel distribution old
supplement data indirect supervision pair collect website tag distinct label cause noisy hence consider token appear size end symbol either vocabulary concern question scoring approach label image replace question triple intuitively consist project treat gram hand triple share scoring eq absence map kb triple sparse absence relationship might seem however uninformative form relationship another entity besides entity bag word lead lexical counter answer add embed modeling entity appear hand triple sum triple embedding encode entity easily question triple triple kind rank hence consist question triple
discuss scalability graphic upon permutation marker computational burden become prohibitive randomly calculate statistic percentage desire extremely many simultaneously lead resolution permutation make moderately sized association partition marker represent e effect marker trait employ marker use novel simulation marker perform costly permutation half million marker recommend quadratic snps popular notable tree decision appeal assume little phenotype complex generality box little typically study black box little relative variable measure occur without permutation association study issue previously task significant bioinformatic direct deterministic posterior inference parameter walk hasting rw mh explore slowly predictor
remain hold try fit full misspecification benefit generative robust misspecification goal dropout dropout difficulty analyze dropout tell generalization interested term measure conditionally thus analogue section classifier poisson model dropout eq binomial generating whenever large average average node thick pi connect z connect connect box theorem provide appendix heuristic intuition fix test central roughly j distribution
neural current analogous connected architecture layer deep network note convolutional attribute architecture standard code code produce explicitly assumption process easy analyze activation composition exhibit connect layer kernel fix finally model tractable dropout give desirable suggest architecture acknowledgement van helpful appropriate architecture problem construct infinitely architecture capacity capture degree degree limit suffer covariance infinitely perform dropout architecture neural without training relate examine
case expect necessity error neither require q likelihood truth px logarithm ignore maximize x prove achieve completeness minimize regime node decay consideration observation question whether precisely recovery graph sufficiently least vertex admit recovery whereas intuition path noise extremely arbitrarily constant contain edge imagine adversary truth pick give edge consistent rest inconsistent guess mean distribution error respect choice thus interesting grid central machine poor pose interesting technical computationally matching maximize cut plus edge get node among possibility choose hamming observation compute label agreement observation agree majority precisely otherwise predict incorrectly note edge v graph reduction weight hard full
fa state action represent feature linear slightly abuse action fa bootstrapping difference resolve issue keep per slowly behavior gradient reformulate ensure td td maintain along regardless follow stream experience bottleneck reliably order run scalable et formalize parallel learn orient w reward single experience successful take angle exist literature power reward additional originally
insensitive presence however typical reference reference elliptical tail occurrence point contaminate mixture one represent typical observation prior mean represent consideration contaminate gaussian contaminate gaussian interestingly density commonly chapter distinguished failure vertical leverage regression distinction good bad bad leverage outlier fit improve indicate yes yes bad leverage datum probability classify category proposal regression distribute introduce mixture define distribute approach allow actually could stem square mahalanobis observation group leverage point belong class covariate
distance otherwise least update cluster member least cluster history whether every stream distance distance distance exceed historical deviation call begin slide advanced stream treat cluster examine old long slide removed refer old empty remove list aggregation match assign move close assign standard deviation historical historical keep begin present evaluate dataset experiment assess cluster purpose comprise monitoring appear twitter platform minute trend work extract contain public tweet day ground sharing cluster volume tweet per hour dataset tweet volume series twitter bin one hour algorithmic cluster assignment assume availability ground cluster operate produce cluster define whose correspond confusion
repeat objective nonconvex cd iteration different ordering test penalize one j warm calculate computation instead active criterion full ji generalized model fix however existence work estimate estimate penalize ji criterion purpose dag select induced maximizer difference ratio dags tuning parameter index thresholding accept substantial group assume rearrange individual component p hereafter pp k edge point matrix likelihood h likelihood group influence likelihood distinguish follow
w ds get therefore q get upper size boundary control claim term claim recall ty inequality partition length sdp easy triangle balanced cost exactly conceptually important main structural every graph uniformly every feasible solution main exist plant cross cut kolmogorov bit uniquely number bit store permutation encode string kolmogorov unlikely identify set whose I store size graph e w right constant subset belong belong edge hx hx binomial distribution therefore bx bx v v bx gx x gx xx x exceed space transform pick embed encode satisfy permutation restriction bit encode bit use xx bit eq bit encode extra finish low lemma nd nd encode bit encode kp r g kp kp kolmogorov complexity use structural immediately
maximizer step henceforth call lie form em log unique estimate estimate care estimate justify find un subspace original em find maximizer
learn researcher practitioner fewer publicly successful application machine speech recognition surprising kernel theoretically model highly connection note nonetheless impossible deep address sample barrier benefit tailor aim automatic innovation advance much propose fast hundred million hundred recognize thousand scalable multiple way representation multiplicative well additive validate extensive benchmark effectiveness counterpart accept provide kernel important light readily hyperparameter architecture valuable test comparative view method original improve either suggest two yet believe line theory scale organize relate account approach report extensive automatic speech future direction training example summarize implementation computation keep portion inside early cope
standard demonstrate variance standard obtained need em observe satisfied truncation criterion change less verification converge local checking maximization routine converge maximum likelihood choice estimation gp algorithm advantage program frame teacher include handle package complicated program build matrix contain although tailor code mix matrix iterative nonlinear mix nonlinear integral also autoregressive toeplitz serve mention update computational lack readily available nr solve initial far stability routine modify hessian hybrid reliable observation student year gp fail gp run much cp fail fail fail supplementary material mathematic standardize test score student large pre process remove student link observation teacher link student
satisfy cumulative risk start optimal deterministic convert second recursive union version fast rate analysis order rf reasoning moreover convexity rf q combine new proposition convexity end argument convexity list rate sum n tx converge thank loss nx fx partial sum negligible fast price study aggregation tune large similar list contrary price constant grows extend precede iid setting condition cumulative risk adaptive n c n order provide iid eq
memory split subproblem requirement reduce generic solver solve optimisation bregman proximal therein admm useful solving optimisation handle formulate reconstruction paper interpret reweighte lasso solve problem base admm optimisation apply reconstruction dynamical input equation n gaussian white noise positive covariance description cover nonlinear form satisfy assumption system
grain sum network coarse grain evolutionary dynamic matrix simulation discard detail would lose coarse grain presence multiple pathway summary artificial trait acquire order trait trait acquire simultaneously trait acquire trait inferential point determine sensitivity accurately determine trait acquire I I inference comprise phenotype summary component sample summary infer matrix encounter compatible intermediate trait miss biological record allow place unobserved trait compatible possibility infer trait biological intermediate prediction predict inequality deviation strict explore trait compatible single transition compatible several account possibility trait acquire acquisition trait trait acquire first tracking network acquisition give higher acquire vice versa rna www com use life rna rna spike measure abundance normalise describe dna bind green sigma usa species www parameter minute second follow
assumption embed identically error result achieve order mu selector discuss introduce well adapt beneficial difference fix particularly order take mu rate assumption sparse selector assumption follow bind lemma small yield generalize selector design modify selector admit follow
interaction core multiplication min distance cubic short efficient algorithm level already message short total supplementary material complexity definition em pattern alphabet index integer symbol arbitrary word empty always clear context empty word e position pattern equivalent condition let empty place cost least derivation pt end pattern belong construction direct check forest tree treat cost efficiently edge root leave go index supplementary minimize dynamic
projection onto algorithm f k stop appendix proof theorem cg convex curvature subproblem respect norm h triangle department electrical computer superior email lx support ci e grant recently good statistical property regularizer generalize call algorithm regression contain contribution derivation atomic derivation onto conditional cg also wolfe problem accelerate projected evidence project considerably atomic norm dual
interpolation bilinear coefficient scale paper transformation scale dominate increase please refer supplementary convolution modification propagation implement pooling transformation bilinear use please material derivation refer si convnet baseline carry architecture except convolution replace open code si dataset come variety scale invariant scale variation unfortunately category handwritten digit foreground pixel architecture convolutional map kernel fully regression architecture model art pre parameter note augmentation convolution convolution convnet dropout convnet convolution weight parameter mnist
input datum criterion reconstruction dissimilarity quantization pairwise misclassification method effective low dimensional code generate binary code huge projection overcome method high bilinear projection shape respectively bilinear variety work impose fourier extensively process propose enable embedding slow table dependent time frequency alternatively optimize domain extensive show fast
pac bayes data experimental indicate analyse algorithm select vector machine svms well bound early rely consider pac learner involve pac probably correct bayes reinforcement data rely signal encode knowledge increasingly enable signal big datum different
ridge shrinkage approach collection mixture prior collection model behavior predictor mixture general version partitioning predictor block submatrix subscript prior distinct differential distinct amount govern block reduce ordinary prior design x block orthogonal block allow concept analysis motivate measure construct indicator essential block affect block prior recommendation choice hyperparameter default specific view prior block criterion hyper limit predictor orthogonality asymptotic simple summary essential rather orthogonality encountered design successively condition prior use variant result elsewhere subject mean component posterior g block orthogonal block similar satisfy define result behavior avoid hyper prior satisfy condition theorem prove say coefficient display relatively block drive yy lower avoid main hyper block satisfy away block hyper
property projection detail order logistic observe log l ny solve subject write square subject old order apply logistic approximate algorithm subproblem extension model version application dynamic outcome generalize high dimensional notion monotonicity implement
description scope paper focus mainly aspect click count j dt n j j quantify r decay influence choice I example simple view connectivity parametrize social recently lot see emphasis theoretical lasso process application structure intensity allow direct intensity baseline self influence want produce achieve dt empirical assume ground intensity model easily lead
state product gaussian view prior depend approximate sequential monte carlo posterior particle represent particle approximate weight observation parent propagate particle approximate setup learn hyper represent particle filter state filtering particle one particle receive resolve regularize introduce work marginally markovian therefore markovian four main part auxiliary importance consider would represent second chain line represent forward
behind aic rewrite account maximum hessian continuous ni definite proof modification start x vanish equivalently generate q rearrange term represent conceptually new rest derivation plug normality first grow prove adapt choose call term write chi degree free multiply take side know affect second equality entry identity increase surely design surely definite equality requirement surely em corollary theorem theorem method combine prediction implicitly make input hard prediction substantially differ aic useful shift suggest substantially aic bic averaging function
feedforward complex map region lead compositional layer output give replicate computation input expand basic definition intuitive composition layer unit define weight l activation precede activation activation lf drop subscript activation maxout network refer unit arrange specify number width classify function structure choice region piecewise connect input linear full depend hyperplane come linearity n distinguished hyperplane hyperplane distinguish hyperplane hyperplane several complement point hyperplane towards characteristic well hyperplane region attain hyperplane identification neighborhood formally two neighborhood input say identify carry layer feedforward region
univariate linear predictor expression response non randomly snp treat response analysis build three control adjust shrinkage layer layer control specific loading third control selection loading column wise avoid shrinkage dense loading dense require genomic capture effect batch effect factor couple pair simplify ard priors column resemble induce wise loading recover substantial dense loading recover match genomic expression observation capture co gene sparse gene module annotation observation component exposure specific gene expression include data loading association perform association low projection high trait result interpretable affected modeling gene jointly covariate datum supervise guide space maximally wide status may include co vary uniquely scientific maximally application identification subset exclusive direction carefully identifiability recover recovered scale genome wide association gene expression level averaging maintain interpretability enable finally extension allow heterogeneous currently heterogeneous believe
cell location know step time worth spend connected component plot policy video method synthesis temporal logic develop control temporal logic focus motion planning possibly ad hoc method extension game also game challenge player game product synthesis synthesis synthesis memory besides logic minimize method underlie maintained gain past synthesis objective retain interest requirement base share state obtain transition notational step iv iv ix I q term
laplace unknown furthermore integrated nest laplace approximation nine demonstrate acceptable self similarity natural flexibility modelling variance auto gaussian mean homogeneous specified variance nest perform carlo need precise second minute marginal parsimonious largely handle call use report sequel code kind multiscale transform standard transform direction wavelet model discuss consider denoise concern test pixel represent pixel model wavelet
participant survey assign likelihood relie assumption normality violate risk implement statistical software package likelihood function function maximize estimate variable upon number complete determinant discrepancy calculate assume conditionally
scientific separability lead insight approach algorithm typically optimize keep seminal lee alternating base least architecture entire optimization disk fit main contrast separability big nmf goal apply reduction execute matrix new motivated section leverage algorithm near nmf deal generate cone extreme deal separable since column find extreme successive alternative base find residual index subsection devote find column cone extreme
disagreement failure probability rx without classifier draw replacement validate rx derive unconditional integrating combine union complete query part h precision classifier validate px unconditional section pac combine get state way extension match match score match extension rate matching field different network explain node verify sample actual section validate single application produce match match score indicate part place identify match simultaneous
around constraint arise construct outer set instance separation relatively optimize property get upper cover space constraint ellipsoid possible ellipsoid dimension convex constraint outer relaxation depend rademacher duality multiple constraint upper rademacher rademacher q rademacher tight geometric point infer sharp imply illustrate single region cover set circle make large intersection region value version recover offset attempt relate compressed involve various assume former context deal whereas intersection multiple ball aid compute subject survey fundamentally semi supervise supervised exploit distributional unlabele distributional unlabeled distributional manifold restrict empirical focus zhang rademacher ball researcher arise unlabeled knowledge base simple modification incorporate kind knowledge focus algorithmic constraint modify
repeatedly problem either gain one always distinction unique straightforward utilize step denote initially look complicated problem exact outcome dx possible substantially simply criterion lasso augment rank various implementation address arbitrary somewhat naive sequence problem denote require qr initial change solve take maintain qr improve naive strategy essentially order magnitude detail implementation qr section implementation filter lasso trend laplacian fuse lasso computation generic system least offer considerable boost various complexity proportional iteration across always point complexity nan unbiased degree freedom lasso fit mark formal solver sparse cholesky tight empirically linearly discuss general operation qr outline aside overhead implementation filter fuse straightforward dx long trend filter fuse problem fortunately specialized implementation filter fuse general note ever early termination complexity various note implementation concentrate track dual enter coordinate leave enter boundary set step greatly exceed exception signal never boundary sign throughout
multiply initially take place basic alphabet space vector letter inverse dimensional cosine angle sum information pair store utilize summation preserve unique easily sequence fix permutation store say c see cosine uncorrelated show represent appear vector show
define bandit measure mean linear notion regret motivate framework usual distribution one risk risk root variant mean variance present logarithmic continuous function face principle sublinear another use kl use usual good
divergence initial target quantity horizon choose lead run example iterate multiply second carry experiment conduct pc follow intel core tm ghz ram use parallelization example density fact strongly lipschitz continuous constant explore section sampling bernoulli gaussian z z c experiment table vector histogram v qualitative add histogram equal equal density direct dimension overall generating explain fact consideration logistic feature label conditional estimate logistic parameter rely covariance logistic last perspective seem geometric justified rule especially ensure introduction
setting publish paper configuration basis basis configuration patch show dense original sample class tree codebook representation scene lot large high computational image level auc label c iteration scene scene yet solution even image capable comparable rank base treat feedback report close converge result
learn order upper train gradient particle stop perturb take local optima reach use perturbation multiply perturbed noise go back temperature
challenging vision widely adaboost result adaboost htbp cccc toy eps width pos eps eps width toy width network procedure understand reasoning logic advanced field boost variation boost large weak classifier overfitte problem train perform classifier combination classifier may answer yes widely weak speed small discrimination comprehensive boost tree network correspond thresholded adaboost remainder display failure adopt boost author
iv cf part part second part verify additional eigenvalue eigenvector assumption clearly absolutely positive open neighborhood origin nan inspection element impossible establishe claim argue nb positive large eigenvalue dimensional span unique eigenvector large eigenvalue impossible nonnegative w f symmetric nonnegative assumption absolutely r open origin nan ii limit large eigenvalue limit large eigenvalue kb b x assume part proposition assumption b belong x eigenvalue coincide lemma know continuity square always find subsequence subsequence assumption observe random converge distribution precede subsequence limit claim immediate one consequence claim w z z nothing lebesgue surface cf observe b b establish claim every borel far gaussian show borel unit various place particular fact origin obtain bb b n nan ng square random freedom gaussian joint mapping non borel measurable establish part prove measurable part lemma see hx g x gx must hold assumption obtain together establish denote measurable replacing argument part almost root freedom obvious concerned distributional probability rich allow independent simultaneously almost everywhere choose theorem theorem claim ac behavior autocorrelation spatial study literature test circumstance build finding unfortunately portion serious build framework specialize indistinguishable keyword autocorrelation correlation hypothesis important test autocorrelation time regression ii autocorrelation spatial overview autocorrelation regression low alternative note early seem show limit autocorrelation one become follow autoregressive see either depend certain observable intercept span regressor intercept typically integrated context test general power covariance responsible
e skewed sentence regardless alignment hence eqn simple solution effectively uniform learn fine grain embedding frequent illustrate figure subsampling subsample english train extension source implementation online asynchronous gradient time simply individual per improved pre wikipedia training perform alignment train directly raw text file obtain standard preprocessing gaussian eqn naive advantage multinomial setup next sentence make update parameter compute due log
technique efficiency publication result big work quantum amongst thing circuit heart many come note boolean circuit study result decide concentrate supervise straightforward studying compare neural net deep net despite establish technique add hill focus finally detailed analysis color course part benchmark benchmark refine optimistic fact vary engineering believe preliminary justification paradigm section describe framework binary boolean circuit short vector classifier circuit could else encode classifier obtain
weight net hide detector useful neural molecular descriptor encourage net train baseline least list use multiple family aid inactive c group expression cell protein alpha identify specific identify specific channel interaction cell cell rna generate molecular descriptor descriptor exclude molecular descriptor descriptor score neural generate binary select threshold ensemble limited inactive formulate screening ultimately rank optimize performance relevant virtual hold leave learn baseline forest rf ensemble lr fold model report result validation datum well particular extent baseline tuning g performance network include optimization long stop train net neural
overview optimisation brief survey process gps optimisation offer belief behaviour process prior exponential opt mat ern automatic determination square scale hyper parameter completely characterize mat ern make exponential thus optimisation corrupted arbitrary marginally gaussian specified behaviour acquisition function acquisition exploration
employ low estimator compute propose square mmse start carry follow bar optimization fix ki k ki ki ki ki ki ki arrive ki ki ki ki ki ki reduce
enkf consider possibility formulae formulae divide framework introduce coupling system derive joint mathematically possibility divide counterpart whenever convenient joint conceptually still aspect may appear attractive term far organize enkf estimation framework multi divided framework condition extension divide finally discuss potential development enkf example illustration extension divide differ thus focus hereafter ease drop involve quantity couple sub covariance overcome technical describe unknown observation operator e affect convenience vector respectively assume different sub separable corresponding sub say depend certain system augment become separable noise observation uncorrelated scalar formulae assume I still derivation let I member background
illustration link shape piece connect fit entire appear dictionary although symbol write boolean algebra non diagram comparable graphic recent os ds ss ds cat additional show indicate connect noun part image take pt formalism dependency idea tie formalism take translate relatively difference linguistic choice algorithm compact converted phrase believe linguistic phenomenon natural entropy mutual word pair dependency work viterbi discussion formalism control thought piece valid syntactic order single piece sign indicate connect indicate discussion marked head become head dependent word lexical word lexical entry group lexical entry conversely allow lexical noun different lexical past lexical lexical rather fundamental single link fine grain object object indirect grain reasonably take serve rough rapid extract syntactic sentence assume capable dynamic criterion semantic guide parse parse static mechanism outside external realistic viterbi operational general viterbi plausible apply analysis limit oppose roughly human dependency limited include semantic class role partial relationship syntactic criterion parse inter relationship inherent parse assign source inherent strength relationship inherently long well possible formalism semantic semantic lexical absence entity different syntactic expression particularly special regard relationship predicate predicate atomic predicate semantic sentence predicate subgraph hypergraph clear implicit require entity action formalism structure topic linguistic appeal algorithmic computer text base transformation transformation thus short appendix formalism ultimately nature extend describe syntactic relationship rather point one relationship internal constraint specific relation rx graphical summarize software linguistic exercise summarize viewpoint linguistic capture piece reproduce concrete write denote current might write
imputation implement causal nonlinearity special package imputation package matching fail solution imputation predictive fit check visually lead estimated spline smoothness estimation repeat ten analysis benchmark estimate fit observational distribution estimate multiple case systematically demonstrate variability issue choose nonparametric successful effect show causal calculus multiple
penalty penalty j could penalty elastic net class penalty r generalize relate p differentiable lasso derivative elastic net mc penalty fail unbounded create path previous theorem say
difficulty give table core believe incorporation sufficient test text learner separable varied experiment paradigm quantitative ordering see difficulty see cause discuss order correlation actually human order slight prediction neither consider difficulty observe differ complexity boolean complexity moderately boolean find fairly combination low case two triple one identical match htbp cc cc prefer complexity bold depict coefficient determination boolean depict boolean give case note discuss prediction experiment predict set child categorization difficulty identification please see explanation subject implicitly explain categorization good category
feasible quickly demonstrate linear classification classifier specific p later exist demonstrate algebraic tie geometrically way interpret dual setting relate ball classical like insight perceptron von concept get unit length represent surface ball obviously equation angle boundary allow give interpretation since instance condition feasibility feasibility ever extremely popular literature summarize sec margin turn
inf mh monitoring convergence compare dominate iteration runtime high acceptance proposal approximate derive chain case take chain much chain indicate sampler modal assess report expectation generate add expectation would agree individually value differ choose high sampler material follow simple graphic scenario modal room light center room viewpoint camera room camera describe position orientation roll angle estimate multi room symmetric camera result position camera roll camera resolution single core ghz take gaussian deviation infer location angle inform histogram orient descriptor image feature cell mh compare overcome improve technique inf quickly experimental setup mode analyze sampler visit mode ever pairwise mode change mode way correspond random
measurable spatio space space concern closed ball satisfy many construct point reasoning give underlie temporal information mark explicit summary measure regard measure lebesgue usual process spatio temporal point space section irrespective mark measure borel discuss turn functional induce probability e consideration copy may choose specifically stochastic discussion close wiener measure e induce type definition count e distinguish whereby thing probability simplicity measure irrespective recall element g l consist point take tie g call enumeration vector geometrically spatial functional aspect note support later process however already stochastic path tm possibility mark turn explicit deal spatio simple classification thing begin q location occurrence mark functional mark connection apart non spatio enumeration assign occurrence support may write l far require constitute process simple part ground constitute say additionally stationarity intensity notation let completely uniquely bounded irrespective usual marked process invariance case stationarity say rotation rotation temporal stationarity stationary refer stochastic auxiliary mark support I furthermore g x I r
base method note level propagation reveal worse mae find visible emphasize long propagation accuracy strategy filter prediction almost good suggest believe propagation aggregation method information incorporation trust factorization improvement improvement achieve type affect incorporate memory base exclude rating rank relation devise aggregation method mae rmse rating include rmse mae rmse mae mae mae experiment start user new recommender huge many system handle challenge randomly start include training table clear affect negligible exploit trust reveal lack incorporate social trust propose datum trust sample trust perform rating l trust relations mae mae mae rmse mae rmse rmse mae rmse rmse mae mae rmse mae potential trust relation relation lack trust compare utilize setup subset trust gradually effect relation trust table number uniformly rating remain feed reveal trust enhance utilize trust trust relation exclude summary enhance rich source mention go triplet could due triplet overcome efficiency turn try subset triplet gradient stochastic derive learn test evaluate mae terminate mae start reach dimension gd mini batch sgd sizes gd min batch mini sgd gd triplet sgd computation updating rule first gd simple name use size simply exclude figure gd although gradient suffer slowly comparison time individual gd sgd need gradient iteration gd take less iteration iteration accuracy attractive gd sgd least gradient compute finally progress make beneficial propose matrix factorization incorporate trust relationship potential overcome traditional summary incorporating indicate
subspace optimize single shift standard share follow literature use define subspace empirically meaning projecting maximize minimize shift leibler minimize result convex need derivative iterative subgradient terminate iteration binary multiclass benefit classification score multiclass stage target basis differently exist representation learn directly classification generalize target batch converge calculate derivative I v validate approach adaptation follow experimental report detail
deal proceed either consider optimize new objective ascent equation clear hold constant concavity hold constant expression concave due constraint describe handle ascent concave eigenvalue minimize distribution dpp compactly simply lemma formula dpp e dpp update eigenvalue need derivative sized impractical lemma dpp dpp marginalization marginals derivation self normalize explicit unnecessary exactly practice keep slightly turn eigenvector respect sum simplify gradient simplification contain identity
proof example part section continuity assumption entail without continuity entail eq equivalence modulus continuity fix expansion r bound put prove crucially use uniform continuity show strict take pick sketch case know discuss importance part know convex game payoff maker require knowledge subtle program depend indeed singleton crucially actually adapt case yet prohibitive beginning constructive well prove definition lack continuity theorem regime ensure randomize pick finitely element actually calibrate simplex norm norm space dimension auxiliary calibrate q use definition inequality norm substitute well example toy perform simultaneously incur player opponent combination opponent index
leave reader constant completeness inequality consequence bound since assume first triangular concavity use triangular combine immediately r argument f f apply second independent conditional vanish contraction finally eq n imply claim proposition bernstein give
template fitting example patch employ regression forest vote haar like refine guess location guess initialization critical contrast deep take pixel importantly template build fit fitting detection pose differ task task apart pose attribute gender useful robust detector difference feature rather face use formulate face alignment coarse fine cnn cnn pre partition face part cnns output layer successive auto encoder coarse alignment pre network lead still achieve computational scenario embed addition task reduce overfitte model local place method extraction whole region simultaneously aim mutual old machine task since allow objective task prove difficultie rate across work pose regressor body part detector optimize
random case give f ds ds exceed finally ef e e frequently property large tr spherical lie hilbert eq embed feature map product clearly tr see large time matrix r tr gaussian decrease course decrease nevertheless seem suggest conventional replace scale covariance
nonparametric fit additional complexity however note fit range quantile addition readily interpretable calculate margin form discuss mean correspond study choice amongst al incorporate variance stable development year effect dominate different distributional feature take utilize year incorporate quadratic gb purely perspective sampler rejection hasting pp easy satisfy variance gb variance give al dynamic function variance follow gb good complexity gb similar fit besides clear trend development covariate believe largely suitable tailed run heavy tailed variance clearly choice pp analyse go gb h quantile gamma pp al standardize fitting looking display gb provide good standardized display predict gb compare prediction fit percentile percentile arrange losse al close gb h model gb report fit predict loss level present quantile level tail range increase fast across quantile figure figure percentile quantile plot percentile quantile line gamma moderate gb reasonably observe percentile quantile gradually quantile al
leave note leave represent leave article represent heterogeneity image representation lot investigate heterogeneity incorporation use covariate information develop longitudinal patient task remain direction event say homogeneous intensity variable coincide condition critical reduce tree condition condition setup mean attain unit attain correspond eq suppose unconditional map coordinate clearly continuous density permutation unchanged ease tree leave question question whether projective basis probability conditional ignore kernel density first need establish determine law moment scale normalize positive project correspond implication theorem statistical analyze goodness test random arise model process basis distinguish population generative easily interpretable simulate tree statistic thorough heterogeneity brain novel representation wherein develop test heterogeneity brownian tree goodness test wherein underlying euclidean increasingly encounter several datum tree structure hierarchical include database involve detection record structure rna human brain task protein treat tree observe observable atom tree acyclic distinguished represent ease topological tree
marginal latent realization less z ij ij observe response sample weight group trace latent trace plot trace mcmc loading parameter ordinal wish college numerous suggestion university grant grant grant health surveillance asset economic status population south survey ordinal nominal absence explore homogeneous group asset status variable ordinal item nominal survey item factor nature probit use underlie structure exploit combine hybrid provide mixed nominal mixture md survey cluster homogeneous group within paradigm monte algorithm economic within region surveillance continuously south early south since goal contribute status asset index way population study accounting survey landscape explore recent survey result datum contain binary nominal item concept literature cluster analysis explore
feature space common square euclidean leibler divergence objective desire structure activation whose control euclidean problem could minimize fix minimize invert overcomplete recovery greedy separation exist example activation spectral speech solve set classic use together code discriminative aware discriminative sparse propagate solution reconstruction ground depend dictionary typically would need
interval affect population close enough use reduction conditioning differ expect uv uv independent variance range accuracy conditioning actual rejection skewness central moment skewness x z z moment enough moment skewness large skewness hypothesis test z desire rejection probability skewness simulation close interval reasonably sided population theorem act correct correct procedure bootstrap order bootstrap percentile interval error skewness statistic test obtain equation interval skewness estimate bit non respectively small comparison percentile third discuss application except constraint size procedure produce sample combine conditioning come subtle way context rule sampling draw except suppose store basic bootstrapping bootstrap residual give bootstrappe row bootstrap include bootstrappe square left panel bootstrappe residual predict residual original prediction residual sample randomly bootstrapping correspond bootstrappe principle bootstrap formula standard se practice bootstrappe huge bootstrapping observation say resample level software high factor interaction combination sample combination bootstrappe residual bootstrapping rule estimate bootstrappe fit calculate helpful bootstrapping residual behave lack refer lack systematic random bootstrappe affected bootstrapping observation variability result slope large relatively small slope height value residual prediction bootstrappe linear resample residual condition two bootstrappe help line help student slope intercept either variable help interval narrow variability individual constant much effect confidence interval parametric scale model parameter bootstrappe introduce bias bootstrappe bootstrap reflect smoothed nonparametric population may empirical datum positive transform smooth smoothing common rarely bootstrap mean bootstrap practically continuous except one situation data procedure one draw bootstrap systematically remain add right amount exercise mathematical original bootstrap effectively correction factor bootstrap error create population copy bootstrap selecting
remain time seven
kx kx result kx leads trajectory vi compare side sum similar except trajectory composition finally state remain remain origin theoretical admissible guess challenging approximation straight boundedness evolve improve mathematical field engineering south school rapid city sd edu remark control vi theoretically aspect include stability limit effect error involve boundedness vi system evolve estimation initial within region remain reinforcement rl dynamic programming powerful obtaining solution mathematically tool attract researcher application need analysis convergence learn besides hdp vi investigate pi despite vi remain pi seem adapt pi initial drawback vi learn
complexity dissimilarity implementation straightforward penalty constraint vanish feasible solution write row rewrite lagrangian term iteration consist fix variable update multipli matrix implementation ds define q update multiplier respect shrinkage onto ball notice minimization optimization parallel resource simplex constraint done randomize notice program thus parallel resource reduce expect time similar column propose resource provide solver cubic generate dataset vary server cpu gb fact admm framework efficiently study ds propose program regularization change representative possible base dissimilarity ds find partition set important case implication put obtain select less increase put emphasis compare large certain dissimilarity follow proof theoretical material representative different dissimilarity dissimilarity
attribute description benefit visual category visual research recent discriminative typical assume class scene setting meet rather closed detector like effort essential cope tailed object dynamically define datum shoot train learner mid semantic human teacher semantic predict category amount define attribute signature etc interestingly support cognitive literature researcher explore human object natural category conceptual evolve cognitive effort category human associate predicate biological attribute would offer elegant novel perfect attribute accurately often abstract linguistic property diverse visual road attribute shoot proof alternate transfer propose account exist value
necessary regime regime ptc regime uniformity least uniform sufficient prove tt nc tc bound rely slightly count sample property kk size linearity expectation minimum number exceed uniformity run draw chebyshev far focus dominant fall directly implicitly uniformity matter choose output achievable repeatedly run failure majority vote uniformity vote draw failure probability failure uniformity bind prove pick sample case small low follow bind give give metric behind proof pick term fail characterize sample uniformity regime tight remain small upper require norm whenever prove follow uniformity logarithmic possible guarantee output uniform opposite bind uniformity necessary theorem version prove mention guarantee correct guarantee lower prove splitting case probably condition indistinguishable uniform adapt tight distance sample relatively conjecture regime small seem
underlie black grey process true branching dash solid line correspond median dot poisson significant estimation case observe negative exponential strong instance ratio median bias exponential efficiency exponential density step complete dataset standard e point estimation around bandwidth mass one mass interval solve outside interval another select nice unbiased error branching simulate density branching simulation intensity model realization false true homogeneous random follow allow percent convergence criterion cumulative absolute difference summary estimate across fig branching grey summarize consistent branching branching branch approximately inter process apparent estimate become short consequence dispersion cluster long attribute
pool region show mit coefficient cnn score part correspond illustrate clear clean filter come part even training become selective face localize c c test image cm cm cm cm illustrate score part image though consistently multiple sharing appear capture image belong conversely multiple capture concept part capture seem specifically respectively object object object composition part appear game reveal high weight identify low suggest part negative class rather surprising people examine image class image visible face cm cm part part detection slide window fashion purpose part filter weight drive hoc discriminative diverse part concept base perform previously cnn accuracy improve selection level
sensitive admm slightly slight freedom experience rarely always correct grid check axiom theorem condition criterion author large fdr automatically find test region elsewhere manner discovery power separation signal optimization augment lagrangian demonstrate fdr exhibit simulate fmri work plausible fdr control smoothing multiple concern nan simultaneously simple problem summary statistic nan ensure standard testing control successfully apply across notably analysis dna sources genomic exhibit microarray include fmri statistics brain allele population correspond physical fraction spike location lattice environmental network spatially localize method learn exploit fdr find spatially statistic discovery rate pre increase signal raw score distinct research incorporate popular advance composite review multiple group wide composite regularizer recommend
bx q assumption uniquely proof many g van lemma definition separate separation result section rely mle although separation property strong condition common support compact compact strongly separates consider continuous separable since compact fx fx n value banach number banach similarly complete assume two strong guarantee analogue however separation estimator strong solution property inconsistent counter chen wu let px px px unknown lie whereas always produce value likelihood subsection conduct great density q estimate I normal degree remove select repeat estimator well mean median inference unknown parameter parametric base observation use assumption describe mixed probability huber draw population eq q contamination simplify randomness take observation asymptotically equivalent
exact grow approximated issue arise continuous economic university science department multivariate time series circular rely project skewed cluster relax independence circular burden involve justify carry data scheme focus recover finally bivariate time hide frequently natural multivariate series longitudinal generation unobserve hide modelling tool education modelling multivariate series component type mixed mixture univariate notable conditionally application properly accommodate complex distribution moreover unnecessary number often reasonable price computational burden result circular
process observe converse true learn learn stay dictionary learn serve realistic algorithmic process filtering take consideration examine explore interaction example investigate upon heuristic examine relationship encode environment generate element dictionary activation norm end wish posteriori calculate subproblem q mod alternate locally
basic requirement learner stable cart base subsection bayesian aggregation simple canonical scenario component remain vary let minimax well setting shrinkage robustness empirical regression lack justification sample sample calculate root rmse iy mcmc iteration burn mh acceptance ridge ridge cross coefficient covariate moderate lasso comparable predictor nuisance la la lasso ridge dramatically appear second htp htp dot dot display burn although fluctuation negligible like fig whose magnitude typical la predictor however predictor suggest coefficient coefficient truth covariate affect impact predictor ns predictor response ns la ridge ns lasso comparable moderate non excellent comparable la ns
every selection combine corollary distributional spread conservative quantification precise term parameter distribution j cf mode laplace possesse attain automatically make standard object correspond speed therefore quantification know minimax euclidean loss nearly minimax sparsity regularity parameter choose next show choice lasso put ball substantially big intuitively explain prior good lasso due induce mode mean vector identifiability posterior base solve design inspire full pac paper modelling prior model combination choice prior coordinate heavy tailed general kullback measure shift leibler might kullback divergence heavy signal quite q constant pac technique constant pseudo take posterior address question achieve slight large show corollary slight prediction dependency natural corresponding subspace sx collection distinct subspace define
dash performance reconstruction together baseline essential directly quite expensive predict neighbourhood fig fig rotation cell array impose priori fs nearly equivalently neighbourhood cell sparsity preferable result feature paragraph relate measure practical learn experiment qualitative use technique indeed nearly identical evaluate discriminate cat face template evenly rotation classifier nearly classifier rapidly fail rotation conclude transformation encode visual effectively rotation oriented dash line representation cnn use composition group comprise relu three layer relu convolutional conv conv linear layer relu hard negativity method learn mapping fs neighbourhood sparsity sect orient report sect imagenet report validation
average speed first benefit em rely insight noise sometimes lead well apply detail apply sufficient benefit quadratic define speed em combine noise iteration fast reach stationary figure ht noise lead general tool extract algorithm iterative show benefit chapter backpropagation feedforward neural network backpropagation theorem proper noise feedforward neural backpropagation log detail backpropagation backpropagation backpropagation ball illustrate ht noise sphere change benefit boltzmann rbms depth pattern speech deep deep consequence pt black circle inner neuron fill red neuron text width cm dot layer network cd give noise benefit rbms deep rbms backpropagation bayesian effect function model statement likelihood domain pdf also approximate freedom quality approximation model fuzzy system tool uniform linguistic build approximate fuzzy discuss fuzzy address hierarchical iterative context contain minor result maximization mm mm algorithm mm extension mixture alternate algorithm end benefit medical incomplete automate speech image genome denoise disease track prominent even analysis researcher em ten step current use likelihood distribution chapter em mle motivate generalization formulate chapter end notable estimation method preference preference quantify well summarize likelihood fisher fisher formalize use evaluate estimate rv pdf low observe likelihood contain assertion often implicit statistic question observe simplify class parametric pdfs provably convenient rigorously give parametric describe likelihood estimate e joint pdfs optimize log transformation preserve point log ml incomplete fisher attention reformulate outside bayesian year state mmse method select criterion finite minimize mean square estimate suppose sampling representation identify moment mmse invariance fisher argue invariance hold change alternate maximize pdfs formalize measure probability foundation likelihood integrate marginalization em marginalization unobserve maximum estimate probability weak mle eq mle normal likelihood analytically numerical method root derivative newton nr series derivative score nr use experimental corruption group additive extension mle complicate ml observe datum basic treat augmentation complete fit address sequentially well guess likelihood equivalent imputation nonlinear good guess compatible statistical deal complexity coherent idea decade ad problem use truncation censor information family little standard behind hoc extend miss synthesis field datum formulation array include censor group truncate mixture censor algorithm schema family handle incomplete simplicity schema least old iterative ice backpropagation schema cause explore surface log fast em boost em enhance fast subsequent enhanced like bt log likelihood likelihood maximize apply incomplete instead corruption loss complete lose corruption likelihood optimize address derive surrogate replacement ascent lead perform ascent result step iteratively suitable output remain ascent surface stop successive give tolerance converge data likelihood complete likelihood specify random likelihood random crucially model careful analytic convergence algorithm identify corruption pdf pdf mixture sub population population illustrative exposition observation explicitly another transformation complete right censor gamma give censor analysis random subject medical procedure right experiment keep track unobserved exceed time main change generalization whose setup assume select admissible reduce delta function eq allow transformation transformation admissible space add flexibility admissible speed ascent em first proof statement condition observe q q leave unchanged maximize force term kullback leibler divergence negative ascent relative e ascent produce limit limit mean point saddle iterate point example converge convergent maximizer guarantee log present apply map point set
sufficient element span partition span dash span assign assign dash assign left side remain polytope thm pt minus pt pt plus pt minus pt er analyze reweighte compact ground propagation lead marginal
see perspective sparse previously efficient gradient update remarkably without ever factorize invertible matrix initialize update update change computationally manner maintain version representation propagation principle target square see direct naive way prohibitive rewrite
considerably higher obtain accuracy comparable statistically another advantage second use rest text run entropy na recent benchmark author identification task training text english author word long data section problem write latter distance document give language report correct solution well rank english set gram second
sensor grid diameter star fig cycle eq spectrum laplacian star graph eigenvalue communication star cycle hence eigenvalue therefore take complete use derive require iteration near star cycle rate diameter network signal present transmission via receiver channel end cause channel resolve channel I copy digit respectively receiver recognize digit accurate digit receiver digit observe state none solely digit formally simply however identifiability information generate connect matrix exist discover true word agent equivalent
fix every every average radius let point large infinite result preserve preserve original let say preserve error distance least preserve distance hold isometry preserve pairwise multiplicative interesting property application addition product additive say literature say map multiplicative error subgaussian space isotropic subgaussian briefly linearity isotropic subgaussian equip euclidean isotropic subgaussian set subgaussian map mean subgaussian particular due occur less storage
expect overfitte feasible htbp ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc range ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc size ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc remark minus em possibly great obstacle human simply machine intuition could validate decision primarily focus design complex spam vision turn create interpretable system artificial intelligence decade even interpretable likely accept numerous capable produce insight score national medical diagnosis scientific input predict response create understanding inherently highly affinity datum address process practitioner adjust cm cognitive entity refer constitute standard sparsity drawback interpretability think complicated limited estimating association medical scoring assessment tool enhance coefficient adopt human tend aware illustrate idea believe sign expert correlation predictive produce constraint relationship match view domain build interpretable primarily accuracy surrogate interpretability proxy heuristic fully accuracy interpretability practitioner training practitioner perform tune simple accuracy interpretability introduce interpretable predictive framework integer ip classification primarily help accurate flexibility practitioner optimize produce accurate completely achieve theoretic balance via meaningful regularization without sparsity monotonicity difficult produce exist design scalability mean ip solver polynomial time linear argument flexibility building
jx mx ip efficiently take validity easy imply know completeness decomposable node completeness sum completeness necessary ensure validity completeness density probabilistic reason appeal brevity completeness univariate identity say none child validity value equivalent decomposable modify product remove redundant require introduction polynomially many additional light consistency interesting know node univariate function normalize associate interpret normalize weight interpret top child correspond factorize distribution accomplish top like acyclic formalize paper old justify definition cite generalize take function choose integration generalize observe always decomposable separate additional useful come section tuple formal arithmetic term polynomial term polynomial whose associated collect say function sometimes zero domain distinct negative negativity circuit monotone arithmetic variable factor arithmetic scope scope polynomial relate multilinear polynomial multilinear polynomial multilinear include determinant matrix multilinear arithmetic circuit arithmetic circuit compute multilinear say child node arithmetic circuit multilinear circuit open whether arithmetic circuit multilinear without case formula multilinear multilinear syntactic identity scope become syntactic multilinear arithmetic circuit view decomposable somewhat less obvious useful later decomposable view multilinear circuit similar affine univariate monotone multilinear circuit node notation partitioning polynomial collection circuit consist scope e multilinear polynomial multilinear multilinear determinant non multilinear polynomial row multilinear circuit circuit multilinear circuit node child concept circuit relate hard dependency scope become concept multilinear circuit usefulness remainder connection motivate scope dependency member depend denote scope novel relationship completeness circuit review give quick behave differently sense expand tuple fan moreover decomposable completeness free one way affect expressive validity definition
different network success adaptation overlap da method meta observe source classifier correspondence adaptation adaptation successful low classifier pick difference layer layer label adaptation incorporate train blue source one correspond adaptation method experimental case domain considerable shift domain summarize mnist mnist experiment deal mnist digit patch color pixel channel patch invert pixel pixel digit become hard dataset digit still cnn train domain distinct long poorly lead successful adaptation consider quite difficulty adaptation task synthetic address common synthetic house synthetic
remark subset usually advantage compute line result tree interpretable last process forest full create tradeoff interpretability score predefine number decision three subsampling column leverage feature replacement test policy development geometric subsampling norm column
product paper mathematically equivalent distance discuss mmd experiment version independence latter indeed mmd sample look except pairwise statistic statistic match use informally definite formal iff suffice kernel property base proceed calculate independence keep dependence randomly observation like time accurate calculate reject prove fix alternative sense type go calculate fix draw independence two repeat number conduct independent section test dimension
set I support lemma put pack vertical case pack next near long random q must exist make vertical horizontal packing vertical finite act horizontal take packing pair sphere row uniformly intersection control first intersection bound hoeffding replacement reduce live random span precisely
correspondence parameter mis see correctly also large plot none perform positive change log involve un
recently machine learning regard conditional describe conditional condition kernel survey private author characterization dpp question arise naturally mind model
vast store explicitly store value library store cross heterogeneous hessian heterogeneou two level figure dot ram nine million unique add element linearly hessian figure hessian cost log graph group color one affect derivative variable still able recover hessian number conditionally independent need add figure four heterogeneous across group many reduce sparse describe appropriate group hessian substitution grouping classic software difference pattern estimate package interface large finite gradient explicitly user gradient store hessian densely attractive assume log twice unimodal use algorithm approximate curvature log posterior available need quasi newton format trust region exploit memory cost predict convergence log require extensive long optimum find fast specific application optimization difficulty find mode hessian mode generate cholesky solve
improvement potentially edu theorem gamma involving mcmc stick constructive correctness stick break use process random measure explicitly poisson truncation variational dirichlet variational bayesian structure likelihood finally nonnegative factorization task corpora beta prior pure increasingly popular bayesian exchangeable ranking prior infinite latent indicator matrix
local clustering maximally cluster arc helpful exist assign noisy problem convex hull generate reduce figure contain non realistic reduction region discard convex devise middle distance measure artificial variation hull w w si extreme far must avoid middle ready set distance via set realistic variation control cluster threshold convex basic step cluster complexity evenly compare pair comparison run ie apply adaptive clustering cluster
significantly train example training help ensemble svm ensemble scale aggregate implementation implementation framework primarily nonlinear novel ensemble software svms equation default
fc learn machine memory experiment include encode ten category near machine svm category retrieve base encode mesh multi voxel category aim predict type cognitive process pattern activation brain acquire fmri learn major fmri cognitive behavior human couple scale record brain interaction spatially distant brain connectivity describe process outcome element brain connectivity physical connectivity dependence brain connectivity effect activation path decode selection define neighbourhood neural analysis correlation partial causality functional partial elastic net combine correlation imbalance connectivity correlation cognitive construct increase subtract connectivity matrix pearson graph modeling process structural brain voxel intensity connectivity utilize within connectivity form neighbourhood local
improve interpretability technique increase applicability technique high figure omit htb artificial htb htb dataset sensitivity obtain sensitivity map space sensitive analogously input artificial visualize ability decompose knowledge acquire train classifier powerful methodology classifier literature ability reproduce base human supervision
xx symmetry obtains directly behave permutation obtain output quasi precisely posteriori function addition similar xx dissimilarity inverse minimum infinity chain analysis asymmetric case q hence show complete quasi diagonal operation matrix equality inverse limit dissimilarity u census year construct asymmetric dissimilarity function similarity dissimilarity scale proposition form quasi particular choice decrease dendrogram quasi analyze highlighted proximity preference blue cluster six cluster east green west plus us dendrogram pdf influence proximity cluster record interaction input economic economic interact focus use production north american production interpret direct closeness decrease input come combination input rather economic dissimilarity input production use production similarity dissimilarity though minimum height cm anchor thick method quasi algorithmic facilitate quasi ultrametric node quasi partition focus economic four code extraction pc rl mp service support service dendrogram capture influence singleton dependence except service leave mp mp resolution formation singleton sc explain diversity service economic engineering service production engineering pc financial
subsequent cutting plane post dual optimization calculate minimum obtain search method far upper repeat mode refer iterate mode upper flip portion local solution bind consider greedy depend highly initialization repeat several method cut refer refer relaxation approximately simplified difference impose list consider follow produce relaxation introduce triplet long cycle cp author website add triplet plus cycle add cp cp relaxation art interior binary submodular energy approximate energy trust tr principle evaluate b refer bb upper branch tr ed bb default initial iteration implement branch b mini limit quickly paper limit set program evaluate experiment perform fair implementation compare website toolbox test ghz memory gb evaluate cpu ghz procedure processing reduction processing write mainly improvement speed implement state branching cut cp sdp produce interior cutting give bound superiority unary strength connectivity dense sdp sdp lp class impose scalability cp lp
principal component additionally grid performance implication functional rgb quantum mechanic know prediction cause basic quantum mechanic instead greatly reduce dft exact theory prove chemical solid matter middle body onto interact reproduce exact density reason ks dft successful interact approximation body interaction much reliability dft scheme build sensitive approximation functional past four decade extensive improve empirical functional require well trial great success dft bad dft continue exist accurate approximation ks ground enable dft calculation dft fraction benchmark free calculation capable treat unnecessary dft ks standard approximations ks dft utilize however comparable total euler equation energy functional
sis elastic complexity cv majority vote insensitive predictor b outperform conclude rank predictor lead robust cv bt method selection tune challenge validate speed accuracy selection b bootstrappe proposition claim bootstrappe moreover argue set biological insight finally yield contribution challenge lasso valuable wide guarantee toolbox make thank
material analytical landscape often sec modeling tend topic sized topic sec detail thing asymmetric discuss hierarchical topic science sec english wikipedia sec technical usage computationally landscape affect performance generate accordingly measure consider examine generative section show topic well prior assume commonly asymmetric prior language likelihood generative model limit document even increase relatively estimate topic topic sake word topic actually deal language document word across topic generate language let document later stress log model language english language say merged topic word english divide two english former general word might fit overfitting improve english portion enough word vocabulary log document generative big regardless per vocabulary document precisely english document difference english topic pay language merge per compute log likelihood know language english fraction document pick symmetric asymmetric point treat next section pick make uniquely therefore limit equal
cell expression low separate add variation gene cluster gene practice normalize deviation spc result spc gene cluster shown address proper cluster regard include side cluster among cluster assign column cluster merge form gene remain gene gradually converge cluster c identify cluster path ht b compare spc good value produce inferior add category calculation section near neighbor result sensitive identify parameter search value resample result gene classify require run spc c cluster spc tight see table none misclassifie spc add cluster gene spc table random gene spc spc spc attribute arguably separate indeed big gene start confirm specific biological role small three chance specific dna spc expression possibility discover far examine even strong fail small big cluster leave ht p dna specific dna binding
consider subproblem hx rx x tt k k concavity encourage reflect force solution x expect equivalent concave general dc consist hx program new dc problem function dc dc q k solve consistency approximate problem bottleneck start apply algorithm section machine class generally respectively seek discriminate set separate use adopt notation introduce take nonnegative slack value lie side hyperplane hyperplane first objective remove select observe special q sparsity induce problem n strongly simply solve briefly let compute nx dc dc dc generate converge critical finitely nh sect local solve program case dc apply critical nh differentiable local iteration consist k nx solve convex solve compute nx program n dc generate finitely nh x tolerance sect approximation beyond threshold proposition general enough sect exactly
letter use n answer follow datum highlight concept regime adopt coherence property parameter identify second identify collect behavior could confirm exploration mixture structure go second keep coherence parameter cluster please thing reveal remarkable useful coherence behavior affect analogously coherence general coherence large answer highlight beginning illustrate large accordingly might assertion experiment e dropping additionally reflect
detection sensitivity parallel resource neural networks cnn achieve great advance imagenet study cnn imaging classifying cnns cost implement computer graphic hardware investigate feasibility effective reduction facilitate object classification preliminary detect candidate ct volume automatically compute voxel heart
user proposition type development adapt machine set statement set disadvantage theorem dependency automate avoid first globally constant dependency set allow etc dependency minus name proof heuristic statement theorem work development manually check behave obtain lc
image transform rt rt rt orientation texture essentially need rt discrete grid ft rotation
expression represent replacement accommodate window follow operator kernel kernel outside certain region experience typical follow operator introduction currently rigorously ability randomly equal training testing datum half square divide small box version perform outlier mkl despite introduce parametric language would shape inference fully inference location broad trend explain parametric form raw supplementary acc interpretability spectral sp bayesian mkl exponential
message pass terminal assign variable show node correlation distribution ia terminal interaction message argument message pass one run dense norm close message pass terminal message follow residual respectively abuse notation important mention field approximation pass whose remove correlation measurement estimate system size evolution se output equation threshold pseudo surely satisfy noise conditioning single terminal characterization add get square noiseless mmse mmse
classifier svm classifier make sample pt propose mkl rt nn dr perform rt mkl rt dr sample class mkl dr overfitte rt mkl c class nn rt nn c rt mkl rt svm c rt svm svm rt svm mkl rt mkl dr svm c c mkl rt show mkl rt split consider contribution see mkl rt clearly approach successfully dimensionality dr select kernel great lack
implementation resort multiplication require hereafter memory bandwidth fair way non selection regularization run setting manner cost value exceed training optimization therefore times range website source solver cd iterate vary zero less cover complete picture wide relevant summarize news benchmark experiment cc cc operation cd list factor zeros cd slow cd obtain solution indicate mark significant extensive svm baseline permutation addition shrink heuristic remove word solver frequency tie adaptation list range medium r exactly differ coincide significant compute comparable code report compactly include fold interior completeness list mm algorithm red circle cd parameter curve curve validation percent green configuration well choose cd logarithmic dimensionality highly imply optimal represent subset adaptation coordinate overhead coordinate adaptation trust cd method baseline set runtime font mark finish l solver cover type training runtime second small font finish
corpora language successfully directly representation autoencoder english train autoencoder dataset compose million sentence translate relevant language use english section corpus provide news english category create top hierarchy document raw form setup preprocesse classifier document embedding heavily
sparse precision past recent directly involve distance incorporate discuss denote f partially empirical observation remain clear lead reason matrix pair remain sparse effect three point conditionally autocorrelation cut ar complete matrix stationary empirically location magnitude diagonal decay sort element compare functional approximated covariance matrix c ij sort plot equal fix increase increase diagonal precision fix illustrate numerically precision generate domain scale diagonal certain j comparison asymptotic become close much estimate different density point fix nonzero truncate
psd uniformly without order eigenvector norm since directly discussion skip divide objective difference convex
experiment star assign experiment uci census example node kernel fix training objective also drug discovery challenge irrelevant value method decrease slow due irrelevant section admm popular compete method admm discuss data atom draw true zero norm try relaxation figure versus require tune admm edge properly seem perform communication cost practical conduct experiment compare admm well communication leave synthetic finding overhead irrelevant generally expensive solve gradient idea spend ghz
notice case recover know proximity operator exploit regularizer positive satisfie v leverage note show step obtain furthermore term permutation denote writing assume decrease
clear passive learner learner draw draw draw label learner set label property square logarithmic splitting several perform ol result useful appendix universal constant follow sample return explicit wise label gain execution instance guarantee might ol
hide layer well hide propagate consist user compute sigmoid predict click layer record sequential span make rnn applicable sequential represent correlate represent behavior historical testing carry crucial achieve prediction group consist ad display text time whether head diverse large setting fairly capability predict whether user information temporal dependency quantity big rnn sequential behavior ad display order averaged likelihood sample label click ad feedforward backpropagation basically
consist combine begin recall principle sketch functional excess even dimensional return dimension later follow yx x intuitively value unless
risk satisfy say alternatively scale shall theorem establish asymptotic elliptical hold remark fulfil e derive asymptotic shall restriction differentiable function von formulate assumption ultimately monotone
vector e addition feed conventional machine stage softmax already column hold logistic advantage unlabeled paragraph bag word powerful paris paragraph gram gram lot paragraph gram gram representation tends generalize consider concatenation next window input force paragraph reality iteration text window text paragraph bag paragraph version dm addition conceptually softmax oppose similar skip gram paragraph one learn paragraph dm one distribute dm alone usually try strongly recommend benchmark paragraph
try small learning momentum setting get level paper feed restriction improve efficiency around integrate restriction like belief propagation model inference mean approximate marginal update getting convert feed forward iteration tie layer enable extension paper tool discriminative preliminary mean
make learn mapping approximately preserve theoretically recover local validate employ image nonlinear encoder preserve auto encoder make generality generate learn layer deeply couple transform handle complex auto encoder couple cross local consistency discrimination stacking gradually share experiment task superior ac cn comparison heterogeneous sample extensively image simple couple autoencoder seeks couple every stack auto margin intra inter penalty makes simultaneously preserve consistency enhance capability
control kernel weight define learn less generator prevent overfitte nominal attribute sect preprocesse attribute nominal sect return normalization later instance kernel line store learn center weight activate gaussian illustration extract consideration attribute normalize normalization kernel weight store center activation estimate width std discriminate kernel learning instance activate one kernel overlap compete narrow near instance take dimension width present spread dimension return consist task nominal pseudo code preprocesse I preprocesse imputation encode attribute x j j x binary accept miss line several advanced classification accuracy importance base category attribute nominal nominal line integer problematic converted attribute category convert mean close problem parameter line nominal attribute parameter category attribute encode nominal equal category
match various age status euclidean distance result principle define entry adjacency course manifold see illustration cone manifold database wish age group site necessary group different average understand broad almost surely consistent half provide general state develop analysis network state population appropriately behave manner possess distribution normal center define expectation equal pre specify correspond reference connectivity large assume true moreover target laplacian know drop subscript nan whether sample asymptotic every sample subsample asymptotic chi evidence state population special evaluate whether eq pool knowledge traditional unstable definite estimator area research year regularization strategy frobenius solve generally choice term assume covariance also substantial closely inverse network interest roughly magnitude find covariance entry procedure assumption possess briefly independent identically thresholde thresholding follow denote ij nx il
change step numerically learn close type initialize paper differentiable simple perform empirical search proximal fig illustrate basic idea algorithm proximal another direction ideally direction gradient preserve geometrically line try towards optimal line search proximal solution
work fairly experiment projection asymmetric transform perhaps indeed perform slight advantage transformation universal tuning reason prefer simple option previous exploit hash asymmetric lsh normalization lsh beyond theoretical user interested give item hash symmetric consider asymmetric unfortunately free asymmetric extension call inner summarize previous hash lsh lsh definition similarity apply contradiction hx hope valid
capacity partition depict figure key propose interaction potential constrained key function sequential carlo also design target smc direct chain generally see smc undirecte smc estimate limited channel introduction sampler known theoretical result
acyclic dag vertex correspond vertex literature notation random probability parent arc vertex simply denote joint indicate indicate parent probability n parent goal maximize posterior probability equally likely simplify account prior certain modularity dirichlet hyperparameter draw e yield condition encode bit define pseudo boolean encodes anneal analogue classical simulated drive fluctuation interpolation ground state encode solution desire guarantee state formalism useful hamiltonian hamiltonian final monotonic real monotonic function vary slowly
previous practical third distinction endow case initialization report superiority unsupervise scheme initialization small aid potential determining channel layer focus community theoretical capture typical acknowledgment thank dedicated carry partly intel grant center reference need follow induce orthonormal hc ng psd I n vector follow stand transpose psd non hermitian readily circle eigenvalue definition get j small inequality define v eqn main mapping hilbert z u u u eqn obtain I cauchy vector real w dependent multiply apply vector equation fix z take logarithm outer z equality reduce di eqn fix stand contradiction x u u u z u arbitrary contradiction accordingly I I create vector indeed enough exist j z orthonormal constant inner lemma impossible contradiction show state theorem locality pooling reduce vector rather enforce locality
total consider parametric easy linearity expect alternatively tight expectation formal method testing happen statistical viewpoint go hypothesis occur background model word procedure instead detect eliminate need event eliminate vast clearly poor value critical inherently robust show false positive relax allow preference validation preference considerably eliminate keep elimination eliminate preference feed speed claim tight preference readily bi number keep cluster minimal consensus discard relax value bi discard meaningful fix bi belong bi next present experimental multiple build non object specify several example propose bi job recover linkage see j linkage tendency line circle compare b linkage assignment assignment residual find linkage obtain decide discard stable decay overlap limitation linkage object figure algorithm recover scene build reconstruction correctly detect detecting discard base element line segment detect distance equation pixel notice adapt would far refined cell leave ccc segment assignment thick green green thick model assignment
jj c consequently resample mechanism permutation carry child similarly proposal proposal access child c state I c tree structured step resample node child particle weight ratio recursion leave sub root child weight situation execute run via computing requirement stack child note usual internal procedure describe step merge via resample care well study two result justify appendix first normalize constant c unbiased exchangeable consequence second particle list q strategy building structure graphical appear although tree come come undirected graph salient present discussion give situation collect hierarchical structure collect school school integer assume correspond variable specific hierarchy leaf internal parent encode tp dimensional ise vision encourage nearby field see grid bivariate connect actually describe collection index example integer encode hierarchical integer grid formalize encode configuration least back unary add unary factor tree start illustration self exclude unary subgraph distinct consider factor graph subgraph fact copy subgraph formally family respect variable lattice least without pick tree decomposition recursively construct index tu indice sir
chernoff kullback leibler two namely observe get subset law q adjacency namely vertex line lemma similarly lemma thresholded exact bernoulli diagonal center invoke lemma covariance page lemma rip universal rip numerous instance rademacher satisfy follow let bernstein hold proposition article concern large
svd approximate spectral ice edge deterministic scale matrix product tolerance approximation expression term discard eigenvalue small low approximate note gain filter consequence choose prior spectrum retain eigenvector prior hessian inform low hessian several mesh dominant eigenvector filter independent since surface refine mesh invariance mesh refinement also dominant content dominant inform portion truncate compute efficiently free slide parameter field satisfy forward incremental stress incremental adjoint adjoint stress incremental incremental adjoint linearize version forward adjoint counterpart linearize operator amount adjoint linearize equation since hessian linearize solve typically order magnitude linearize solve inverse characterize construction rank observable construct linearize forward compute linearize similarly adjoint usually forward adjoint apply dominate linearize svd require product component scalability therefore product discretization situation typical pose problem neutral content ice figure uncertainty plus carry algebraic scalable covariance inner quantification ice freedom discretize incremental incremental adjoint uncertain slide core laplacian take parameter characterize velocity magnitude observational iteration residual outer problem solve inner solve decrease newton
child implementation q lr learning extract child signal refine another signal time
jointly model document modeling allow model individual approximate eq demand one need store evaluate issue sample retrieval illustrate achieve eq weight ignore effectively observe even without impose favorable simplifie considerably weight concept explore research retrieval usually query utilize concept posterior experiment rank ensure preserve rank translate satisfy binary constraint top top rather experiment therefore focus preserve pairwise preserve rank experiment reduce
level rely framework tune aforementioned gold comprise relate match last baseline rule extraction seed report entity extraction table tweet label entity treat wrong h distant distant crf boost entity distant supervision entity separately incorporate tweet entity entity extract question tweet word skip gram draw context embedding similar extract entity manually sensible tv movie book identify match human label like attribute extract network extract twitter follow learning reasoning logic weight logic expression team world predicate convert predicate node framework optimize denote logic rule iff predicate propose effective another sort logic truth logical conjunction way formula say distance far rule term distant weight inference calculate predicate compare framework efficiently distinguish feature use soft one extraction list attribute preference
dimensional version solve regularization address six fista investigate problem letter one briefly let everywhere proximity define recover matrix q seek data fidelity consist encourage
know matrix element sparse become evolution instrumental analyze uninformative completion informative noise lead informative point stable coincide bind noiseless completion matrix uninformative initialization time evolution equation give uninformative verify fig ht phase transition count completion treat low concern mse phase perfect recovery phase mmse position mmse function transition mmse fraction signal variance suggest case much set count threshold case g present pca evolution condition small small low completion uninformative evolve uninformative correspond observe ht situation rank case second phase mmse beyond count mark short presence fraction phase count smooth decay mmse respect case difference matrix intuitively easy transition coincide mmse largely explore mmse however seem performance g dictionary factor variance index index modification state expectation maximization scheme conjunction obtain line mean element matrix denote analytically solve specify state mmse dependence mmse analyze stability uninformative number uninformative initialization evolve expression coincide transition mmse consistently analysis uninformative unstable solid line transition point initialization correspond expression stable factor peak qualitatively phase analyze obtain measurement computational concern scenario independent bayes us dictionary big amp discuss present form section amp algorithm bethe bethe evaluate equation approximate log large bethe mmse bethe free section alternatively direct expression investigation rigorously approximate compressed sense derive factorization statistical mechanic particular cavity section replica derivation rigorous conjecture asymptotically include compressed main evolution simple equation mmse reach amp obviously concern mmse amp interesting factorization asymptotic diagram dictionary blind matrix calibration
optimal representation compute average decrease shown conclude outcome useful include survival outcome increase retrieve investigate separately show beneficial repetition try overfitte dataset dimensional train individual time subsequently predict mse mse increase kernel increase generation also corrupt alone also examine noise mse increase level value dimensional h generate datum infer investigate
noisy experiment gaussian digit experiment normalize randomly sample cross purpose repeat noisy trial learn mc fig error exception mc well know feature space denoise particular train infer performance comparable outperform small relatively digits produce miss fig entry miss lot extension term subspace mc ambient complete method high mc complete superiority mc ij ij ij possible sum v q processing edu modern rely axiom structure drive geometric put describe relate suit term mc two term mc generalize regard follow motivate mc mc second present mc outcome numerical superiority literature drive mean analysis subspace subspace subspace decade modern statistic
generate space phrase similarly visualize phrase word visualization clear rnn encoder decoder capture structure phrase phrase duration cluster plot phrase right neural able length sequence possibly different encoder decoder pair term sequence novel include gate gate adaptively translation rnn score pair linguistic rnn able propose phrase encoder improve translation find encoder decoder decoder net language capture linguistic suggest language application encoder decoder phrase let decoder phrase note language future application acknowledgment bm cg thank compute cifar partially
approximate function central component paper find sign proximity basis method chebyshev distribution somewhat flexible similar theory tool construct polynomial lipschitz polynomial additional component simple small marginal consequence know close learn close instance box localization algorithm start minimize yx label moreover
compare specification genetic framework genetic reject weighted package assess goodness manner function fit routine closure generate fit specification accordingly goodness fit model along reference reject mark fit family version much well model indicate goodness structural accept reject visual help regard come case absolute mean comparable goodness fit propose star construct serve addition process network tie fit h simulate node star tendency star tendency fit among red similar nan tie node major aic see
update svd except local site keep derivation atom dictionary respective computing singular dominant vector update respective reduce set computable site dominant eigenvector collaborative power classical assume eigenvalue eigenvalue span eigenvector paper interested variant eigenvector tn site end site initialize site site attempt site site dominant eigenvector precede iteration popular consensus average doubly design rely topology consensus average initialized consensus consensus carry consensus iteration communication neighbor iw system z nz ti denote one imply consensus obtain nz j within power consensus standard consensus iw consensus iteration estimate finally carry site successive eigenvector fall prescribe local doubly site solve x k z iw ref r k full collaborative term initialization cloud differ svd site reference
user transform become interest new real significant lastly quantify dynamic lead descriptor management database application term network online site share site friend user content post choose connect information type connection underlie social second dynamic flow network user well new examine study consumption body quantify user however less flow along particular user create content flow user drop piece content get find might decide connect original access content consider interaction sharing event change detect predict challenge establish require explicit trace traditionally share hard quantify fine grain effect diffusion rich mechanism content connection information user examine change old complete subgraph english speak twitter tweet million new million one twitter highly connection change month month overall slowly background get particular information
belong class maximize sparsity decomposition datum propose search form svd singular nonzero result form call heterogeneous approximate plus identify identify fail identify simulation exception exist differ variance svd identify presence arbitrary method identify contain primary misclassifie false false positive misclassifie significant vice versa iteration later propose r algorithm genome edu recommend available http www http impact impact pre easy accuracy care feature default setting force use ghz intel processor base variety simulate describe method identify three observation misclassification sequential simulation simulation previously correct sequence prediction specifically simulation identify percentage instead record entry simulation study
recursion generator movement stream anonymous file feature use prototype contain seed generator create seed start stream advanced initial generator sequel simple seed working environment represent share parallel processing frame omp h include omp omp long seed omp omp std std std header file processor respectively seed seed desire array object default design seed seed state default call seed array share memory random subsequent benchmark actual generation number parallel correspond unique execute simple master worker receive seed advance involve package
value posterior measure measure monte estimate speedup target study speedup full drop speedup dimensionality grow become computationally expensive thus gain limited algorithm advantageous value keep computational reduce low bias accurate small result depend full sample drive inverse construct adaptively construction simultaneous exploration full accelerate approximate together sample posterior attempt square carlo adaptively algorithm preserve ergodicity sample posterior accelerate expensive distribution approximate comparable build offline vector order magnitude reduce build solve drive preferable especially though concept build orient surrogate polynomial li helpful comment discussion united department mathematics er mathematics integrate capability inverse department institute usa de er sc inverse govern equation repeatedly pde monte drive technique tailor use distribution evaluation couple together
average record reach nmf algorithm utilize graph contribute exhaustive comparison report algorithm randomize achieve run number base uniformly news mnist k provide algorithm obtain f table sense magnitude take add benchmark graph benchmark community edge node community node share become increasingly hard datum author equal degree node algorithm vary construction mix c c l
justification divergence follow bregman convex n l require discuss pl obtain eq slight improvement pi useful potential hull k q follow know proof observe corollary dimensionality notice standard simplex difference get standard loss k radius one union plug bound corollary detail interpolation norm choice depend choice give q show plug achieve form norm interpolation norm additionally strongly function private extend show descent algorithm matrix loss draw privacy guarantee matrix refer hull rank matrix bound immediately get excess excess empirical guarantee purely mirror consideration noisy mean
recover example def sigmoid poisson def gamma gamma support inner product control gamma gamma need distribute put mass soft spike spike unsupervise discovery gamma spike def weight constrain positive probability mass move poisson gamma draw close soft spike figure visually demonstrate plot gamma gamma concept corpus present portion hierarchy sigmoid use bernoulli combination pass bernoulli derivative def identity sigmoid weight factorize share model interval family deep
simple meaningful form go share dimensionality reduction resemble net autoencoder scalable involve discover difficult justify hand scale theoretic nevertheless hope preserve challenge diverse source biology method perform unless encode complexity make difficult free successfully coarse grain representation fully dimensional seem diverse without label system able robust face redundant start several stand preliminary follow neural scale g side generalize non representation acknowledgment thank helpful w nf discover explanation
extra additional pass constitute singular randomized range indicate intuition relevant canonical space canonical question extent randomize range effectively
logistic gain albeit high gain click statistically logistic regardless click agree lr lr fig support click question improve world performance increment measure report seem small third ad put ad support often also identify change level improvement remain consistent hold auc testing technology production consistent line live review combine output probability share numerous strength approach consecutive day transaction manner reflect world show predictive use performance set particular involve
follow avoid correct give validate frequency module module delay window require implementation module perform fourier establish propose detailed module comprise act behave cascade depict complex novel nonlinear recurrent neural show note remain neuron layer increase computational scheme improve predict recurrent cascade scheme detail cascade provide process validation
triangle bound
exact help gain insight recent participant conjecture nonnegative nonnegative kronecker nonnegative rank result usual multi ms counter one fact rank slack nest outer slack decrease square fit hence large existence nmf nonnegative rank kronecker two relate rank question tool address conjecture explain introduction slack matrix regular polytope give extension circle allow approximate cone program program nn hybrid observe slack matrix match covering improve slack exact conjecture never able rank run display nmf initialization rank conjecture interesting increase slack illustrate rank slack due important nonnegative slack proportional increase
tail near satisfy theorem note gamma satisfie requirement assume value either fix link logit probit basis h old true lipschitz continuous n nf w p relation hold j taylor hellinger easy covariate respect spline identity expression simplify use ia ib poisson away zero infinity link monotonic lipschitz root poisson parameter absolute choice hold fact constant grow polynomially kullback leibler divergence fix lie compact n contraction contraction
turn state seem write opinion far pixel hence individual template look white favor black encode opinion symmetric part look object part half turn rule observe argue noisy part composition rule motivate drawing probability compose responsible generating responsible turn noisy template part unless opinion extreme tend focus aspect likelihood improve
discuss ise mrf ising inspection relax formulation mrf precisely cg review cg mixed mrf categorical cg model besides mixed mrfs potential application return motivate particularly useful binary mutation point binary snps form edge mrf graphical contrast permit value real domain consider linear say need crf mrf first crf eq conditional gaussian long previous mrf distribution x eq normalization crf mrf count value gaussian crf discuss trivial dependency value mixed mrf allow interaction term count value continuous log word poisson exist product continuous implication dependency possible form homogeneous ising specify conditional poisson poisson ise mrfs ise value distribution random x xy value mrf distribution permit node exponential interestingly class model value conditional positive real conditional specify exponential xy homogeneous pairwise build within could expand formulation poisson homogeneous pairwise heterogeneous pairwise may count node bernoulli conditional gaussian conditional
include maximally effect classifier histogram analytical calculation potentially improve risk proof assertion optimization risk formula need possibility estimation risk estimation statistical modeling practice empirical theoretically problem recommend rule pattern supervise space value algebra subset probabilistic
reliable recover procedure oppose theorem pair leave form triple arbitrary leave specie gene exp observe proceed conclude reliably recover bottom agglomerative dissimilarity tree less long gene dissimilarity agglomerative triple leave let triple look second condition leave gene ab ab equation upper hoeffding eq follow ab substituting error pick define ab prove tell dissimilarity leave show ac definition gene low
observation follow instance notice default case choose episode vanishe default value specify study cumulative regret roughly worth result bayes theorem problem bound gaussian distribution speak cumulative scale dimension indicate linearly hence suggest tight figure vary vary show robust choice perform wide range identify people accept subject representative census feasible person offer people offer age whether person education year construct dataset offer generalization offer report
grow affect map jump coincide p accord set minimize direction four winner minimize treat reference contain attempt detect right check come move direction interpolation cost new original
position equally discovery background classification remove anomalous discover systematically learn task job behind capable increase amount type phenomenon outline deep anomaly light term worth note anomaly big datum many outli unsupervise technique information comparison light curve anomaly subspace massive number object describe method apply outli area variability class unfortunately massive set explore outlier create deal point discovery challenge contrary example approach advantage anomalous certain unsupervised anomaly prior whole outli many find would technique supervise outli obtain meaningful anomaly illustrate green two space grey isolate outli method red middle outlier region point separable outli product probability adequate outli joint occur probability build advantage precisely vote training classifier confusion assign possibility feed classifier attempt classify object anomaly previously mechanism outlier
cnn seq region size neuron pool seq table exceed supervise effectiveness seq cnn predictive seq seq seq layer neuron entire bag gram multiplying nb performance lm third layer vector variable size vector gram learn seq cnn table except region seq layer improve seq learn effectiveness indicate sequence gram size might focus baseline sentiment reduce vocabulary gram lm practice improve gram categorization gram lm effectiveness dependent nn nb lm seq seq cnn seq error comparison bag sentiment classification document categorization k training gram indicate
rate count area typical application intensity poisson process bayesian form intensity mode spread posterior explore early gaussian smooth spline cover paper also potentially inaccurate finite approximation transform process gp multiplicative endow hierarchical satisfactory aim
else numerical far know despite attempt proposal high alternate method multiplier optimization trend problem particular parametrization leverage choice big admm parametrization computing trend filtering linear operator specialize implementation strength admm implementation reliably wide tune small size value situation admm however specialized produces visually perfectly cover setting achieve converge specialized admm display speak admm consider behave order implementation routine specialize quite considerably extend trend trend filter trend filter worth extension univariate readily block generalized additive reader well iterative much motivating illustrate inferior trend short heavily difference order specialize primal dual conditioning subtle ever
line form work focus set mean perform inference distribution moment play stein straightforwardly moment stein infinite work measure sample draw set gaussian contribution shrinkage theoretically improve however require relax require well estimator continuous reduce norm bound implication propose theoretically practice wherein shrinkage different cross validation shrinkage specific present consistent shrinkage refer section leave cross parameter open dependent difficulty answer complex construct empirically shrinkage scenario include window discriminative shrinkage estimator already appear extend provide justification contain estimator rest section present various include aa continuous hausdorff say vanish borel banach lebesgue f l df endow semi definite uniquely rkh
plot crcr nan nan green marks mark option forget nan nan color marks mark mark solid forget sep mark mark mark sep crcr unbounded scale xlabel ylabel title red mark mark mark option forget crcr nan nan nan mark forget plot crcr mark mark forget sep crcr nan color blue mark mark mark option forget plot sep crcr color mark mark crcr matlab height scale xlabel ylabel data red marks mark mark forget plot row sep crcr color green marks mark solid forget crcr mark mark solid forget sep crcr mark mark solid forget crcr mark solid plot
complete theorem normality divergence estimator root divergence asymptotic dimensional variance divergence slight modification lemma belong family asymptotic turn independence derive regularity interestingly minimum estimator expect additional along variate variance although estimator bandwidth converge value corollary simplify case correspond clearly relate condition explore detailed consider simulate minimum mse divergence estimator kernel normal reference bandwidth smoothed density density choose smoothed pure without contamination density report clearly mse slightly estimator increase quite application reason divergence option estimator influence function suggest study several
relate electrical load learn device user aid interaction test system subject robot trial human basis feedback significant compare feedback device contribute initial benefit expect acceptance active learn affect artificial device birth course adapt device job independence major current insufficient properly lack area insufficient need control channel device channel alternate location body acceptance clinical lose device lack especially prominent versus type despite potential challenge device learn interact motion play
observation carefully grow baseline make characterize value regularization classification expect advantage smoothness improve classification smooth respective infer smoothness balance modification may idea develop restrict assumption set margin behavior mass around hand complicate large closeness sake convenience satisfy assumption turn hypothesis mass assumption guarantee possess balanced sense reliable lipschitz refinement assumption pointwise discussion infinite dimension worth point continuity setting since check eq soon assumption obtain dimensional compact support possess regularity compact support set follow example satisfy difficult check covariance satisfy belong dt taylor still symmetric laplace small laplace belong standard cauchy distribution possess close setting compare analytic ensure assumption set although minimal base restrict neighbor minimax provide margin smoothness away suitable choice classifier compactly context near neighbor neighbor
count match comparison handle particular instance broader ranking preference suppose pairwise c define match sign sign count comparison reference item compare cc similarity easy player player player show comparison consistent identity assume item rank replace property pairwise permutation r rank tie definition diagonal hence glm outcome pair comparison item increase define observation j since mean item order decrease infinity glm enough get low remove absolute k n q similarly participant play several time spectral originally introduce vector symmetric nonnegative irreducible irreducible pre value respectively permutation decrease matrix next technical lemma first irreducible matrix monotonic use irreducible eigenvalue since subtract positivity apply index value
fourth gradient state satisfy smoothness limit perturbation form theorem hold recursion derivation show specific square order examine recursion dependence matrix I r expression shall provide hessian hessian value notation next explain steady approximation sufficiently network hence hold agent type remark manner theorems establish define variance depend whether agent group refer therefore one addition entry influence agent belong level subscript denote hessian weighting scalar hand belong agent combination sub group argument consider dramatically agent play observe belong belong hyper connected topology careful deal agent scalar effect evident fig able determine
come solution original problem subproblem fix subproblem fix second subproblem subproblem decomposition imply take subproblem th derive matrix one converge stop give matlab g g direct propose version year admm algorithm apply estimate cite
assume problem component appendix theorem exploit recall attain selector suboptimal simplicity show optimal logarithmic sphere motivated presence deterministic mean nuisance parameter risk one possible deal result measure suffice example detail gaussian summary assumption triplet jointly minimax lower exist constant denote infimum estimator denote selector benchmark selector know selector ignore error study use generating element
rate boundedness depend occur e analogous fashion derivative evaluated rearrange jj eigenvalue whereas mle true similarly expansion contract exactly proceed identically change one part ii state dispersion proposition ii ease drop conditioning log mode vanish point ensure model regular hence infinitely ignore set direct hessian hessian unless ensure converge regard term prior plus converge element h pn p pn pn pn pn whenever conclude recall modal hence approximate add across mode hence I pn pn pn drop although part condition make explicit orthogonality assumption proposition posterior mode condition guarantee regularity remain add proposition x probability mode element hessian obtain proposition mode value maximize expansion mode n nh
use datum consist audio model gibbs truncation sampler fairly binary mask multiple mask correspond heuristic component corresponding track signal standard evaluate sir ratio sampler yield separation performance
method respect estimate generalized criterion apply grow observation method comparable cv expect ol drastically condition rise tune comparable modeling hand outperform local
unit per dramatically time description examine toy showing section anneal case describe cross categorization million subsample generalize commonly consider separate operation subsample schedule operation use mixture motivate section wide latent combinatorial nonparametric assignment cluster detailed list set represent disjoint subset chinese restaurant crp base call component recursively follow assign cluster draw probability clustering crp index order exchangeability suggest simple gibbs sample posterior clustering
use detect community though measure modularity focus graph overlap nash equilibrium experimental result relate material article draw distinction approximate nash equilibrium apply detect partition direct community prove bipartite experiment author community reach modularity represent adjacency weight kronecker elsewhere herein interpret community modularity link numerator margin cell many satisfactory example tends merge bipartite formulation consensus nonetheless regardless graph remain type graph transform bipartite action formal bipartite belong belong bipartite margin margin transpose margin margin block square symmetric node distinguish community diagonal bipartite detect detect determine validity partitioning liu introduce block bipartite notice author take consequence modularity graph partition graph
balanced sbm attract tend block small interpret sbm nonparametric general way piecewise approximate reasonably popular estimator propose histogram appeal control robustness noise case inherently measure preserve map restrict ambiguity preserve map histogram sdp relaxation like organize introduce sbm relaxation compare pca consistency result brief discussion implement sdp devote network histogram conclude discussion ease cm vector inner cone natural matrix confusion act square act produce norm kernel nan space th sub indicator elsewhere vector index kp formally sbm simple adjacency edge belong exactly community belong community sbm form independently bernoulli variable write think operation pointwise write sbm symmetric derive two sbm plant pp determine identity pp plant
square least propose briefly algebra advanced read several conjugate modulus say root unit unit angle zero geometrically rotation vector pure unit propertie eq calculus general derivation recent elegant calculus comprise derivative derivative
exact approximately stay try formalize fuse tend follow go ever interpolation deterministic must stay boundary outside change go lead change issue fuse consecutive spurious section
policy reader actor mention programming set architecture policy technique instead discrete govern incur translate map action identifiable paper policy go discount sum receive start initial discount rl cost go however challenge instantaneous next govern hessian cost optimization minima go project iterate projection iterate hence scheme cost go adapt monte horizon discount step transition artificial could obtain carlo simulation discount cost trajectory use estimate build loop direction expression bias
international student anonymous helpful determinant simplify specify width demonstrate error recover main normal identical assume get find collect identity infinitely broad depend text condition fulfil q result correction inference department college laboratory road centre mathematics university department
fluctuation player requirement rarely wireless general presence payoff perturbation payoff specie fitness variant aggregate weather effect fluctuation jump incur dominate strategy eliminate strategy likewise strict nash equilibria stochastically mild aggregate presence surprisingly variant deterministic counterpart dominate strict nash equilibria stochastically stable irrespective perturbation broader stochastically reinforcement learn derive player become explicit strategy focus long stability stochastically lyapunov trajectory nash equilibria stochastically irrespective principle distribution play equilibrium matter vector dual real span denote slight abuse delta simplex shorthand tuple dependence write consist per player player denote profile player space otherwise x account mixed regard variable payoff stochastic dynamic process employ cumulative
mle never determine interior require polytope polytope claim empty degeneracy condition definition add vertex give add vertex necessary integer difficult simplex conv ne n r sequence omit polytope polytope hyperplane integer correspond lie boundary interior observation large mle exist check sufficient statistic sometimes graph positive indicate performance graph convenience produce
optimize optimal generator optimal conclude interpret maximize whether minimax game eq reformulate global virtual see subtract recognize previous expression shannon show global process enough algorithm allow consider convex supremum attain descent update converge conclude practice adversarial represent excellent guarantee adversarial
reach curve panel ratio number negative quantity curve show close unity estimation curve right panel addition red solid maximum roc curve terminal roc roc keep increase low parameter illustrate see value training well end compare result regularization coefficient indeed change
source give train distance simple powerful online iy n consist class riemannian essence trial trial classifier defines alternatively difference classification involve class eeg class output class trial riemannian mean discriminant indeed matrix structure signal covariance temporal knowledge signal pattern processing stem covariance matrix embed average response index trial belong target trial super covariance decompose covariance trial covariance however cross absence cross eq
law subsequently exploit several previously ergodicity overall indeed online state reveal rely forecast optimal control law provide sharp sensitivity law misspecification beginning require perturbation result advanced program contrast without inspire less type control law preliminary version appear conference publication limitation mdps leibler online offline setting addition perturbation state cost omit treatment demonstrate role strategy type compare paper report thorough evaluation tracking graph particular carlo simulation bar compare strategy policy choose pool randomly policy without cost strategy passive r good grow simulation strategy well randomly sample policy frequently formulation mdp theorem contain control include policy analyze contribution direction future work result index stochastic cone row variation leibl divergence sup f
pca direction researcher center information extract gram favor motivation application many thus provide sort non pca central signal dictionary towards keep component analysis motivation measure hyperspectral computer signal processing machine learn gene bioinformatic computer human recognition online handwritten character numerous nonnegative consequence lose center mild negativity unique large eigenvector positive component center issue center versus pearson versus coefficient pca matrix counterpart center product examine counterpart devise bound connect matrix large gram examine outer matrix relevant eigenvector non matrix way center base center beyond extend machine eigen gram multidimensional centering extension conventional centering discrimination
z ji kx sdca exact coordinate sdca brevity sdca originally output expand prox stay incremental exact sdca perform current option prox operate dual zhang conjugate conjugate operator primal proximal basic sdca conjugate completely eliminate sdca ensure trick interpret intersection algorithm primal sdca convex rather strongly sdca sdca variants sdca expand perform coordinate operation interpretation nf
design thank guarantee mechanism optimize know therefore exactly optimal guarantee mechanism bad mechanism equilibrium particular bad equilibrium sensitive algorithmic intractable running mechanism algorithmic preserve section introduce task standard definition example function could clear context omit subscript sometimes shorthand use shorthand except omit whenever notation estimator property readily available expert return point expert produce unless expert worker effort worker characterize effort estimate minimize effort unless otherwise worker payment amount randomness definition problem estimate access worker suppose access know mapping worker estimation test worker payment produce worker produce minimize mean square
series indicate state emission multivariate covariance element interpretable minimize extent fusion eq weights control contribution fusion weight eq learn absence motivate freedom informative state apply reversible irreducible mathematically eigenvector evolution complete hyperparameter subsequent independent recover independence modification enforce adaptive identical affect th update solve th identity approximation close approximation stability eq emission irreducible must satisfy detailed balance ik
variational fast netflix however research property suggest depend distribution feasible vb prior dataset introduce conjugate posteriori discuss simulation study strength prior dataset test netflix dataset denote column respectively matrix entry define decompose together observation suppose summarize rather
label task gradient propagation technique due vanish gradient addition limit rnns range dependency step relevant elegant lstm design special unit memory connection store temporal multiplicative unit memory controls activation memory control output flow lstm continuous stream memory forget gate internal cell adaptively cell internal precise conventional rnns sequence well rnn sensitive language
edu weakly label demand object cope supervision discriminative visual formulate benefit discover together weakly challenging image cope minimal amount supervision prominent availability image annotation consequently handle need annotation effectively ever grow annotate available addition robust noisy ill motivate explore rely supervision early idea successfully albeit learn data set intra appearance variation background clutter cope difficulty multiple retain one frequently positively
translate piece clutter clutter ram reach outperform ram ram convolutional attention deal clutter look learn policy include supplementary material intermediate reliably explore avoid clutter translate mnist piece clutter table similar improvement ram convolutional capacity amount computation change image convolutional appeal recurrent attention interactive input dynamic bandwidth play binary pixel agent aim ball pixel bottom gets begin mean capture know precise position location attention game action nothing softmax action
alternate pair parallelism work give compound topic count document denote dot subscript integrate show dataset lda mini batch strategy mini processing read process block mini x subset mini period global across mini batch update determine accord explicitly denote pass stream examine validate gibbs virtue parallelism throughput thank hardware accelerate must accelerate bottleneck early online
I resample filter theorem filter exist condition constant unnormalize x qx x hold necessarily wise cope combine tx tx qx identically distribute
initialize code initialize near indicator update fix rank fixing code n tf nd time propose explore popular representation important connection code neighborhood unify construct ranking regularization
gaussian n follow assumption omit deterministic apply quadratic objective convex suffice ensure hold system geometry concrete acknowledgement office grant national foundation grant dms support microsoft fellowship second calculation vector j embed complement thereby rademacher width claim consequence expectation empirical randomly remain contraction inclusion put piece begin rademacher j guarantee accordingly inspection definition claim next vector shorthand jensen inequality ij j g claim convex belong closure eq z observe put piece previously hull define xx frobenius nuclear unitary take generality let index write section
matter fulfil let arc nice matrix ix x candidate rank decomposition hold function point indeed analytic sense map derivative u nontrivial kernel satisfy derivative already point obvious notational modification exist rank obviously neighborhood neighborhood u decomposition hold consideration imply contain requirement apply corollary fx choose find rank analytic sequence possess due instance corollary part possible tangent assume g angle share nice abstract convergence slightly fx g iterate possess rate abstract theorem act validity case leave
fusion base signature compute feed determined briefly image verify person test architecture signature representation take dot exceed identity train depict feature video face considerably locality sensitive hashing less unsupervised image choice temporal window confirm main resolution clutter class test assess transformation table compare perform dataset align version really state appear model fusion create randomly scale temporal image
maximum framework yet optimization method capability deal follow introduce propose cauchy simulate conclude method non approach weight influence item replace sensitive large iteratively method eigenvector weight corrupt large noise alternatively remove ability integrate sophisticated probabilistic popular laplace formulation alternate solver component error magnitude noise subspace small recover claim choose pursuit also poor incorporate
record need interactive explain concept world web far content internet medium video student allow frame medium video change face education providing fit hundred specific varie similar format content question advance online resource assignment interaction attract online circuit student first least student pass pass score complete first complete perspective student take course year mit education year online people complete considerable illustrate scenario student drop make course course finish analyze hand understand student usage response online thereby feasibility analyze rate student student student may lack motivation leave reason rate exercise believe understand student student reason prediction increase provide motivation prevent certain student able predict accurately base week imply manner design accurate statistical accuracy student student wrong accurate predict student would due reason paper three tackle predict student persistence focus aforementioned course circuit student indicator present comprehensive produce consider yet tune ask accurately persistence first week course week course history many accurate week ahead organize describe model build machine belief decision summary play role
log sample number follow solution numerically fisher method asymptotic derive asymptotic confidence denote l n fisher definite mle arrive theorem straightforward value q covariance result invariance asymptotically normal transformation mean variance
site fire solve multiple learn square root rank highly scalable one data rank future also scale berkeley edu berkeley edu leave share root sketch data low retain sometimes even real efficiency arise typical many problem
satisfy odd dependent model response monotonic transformation marginal odd ratio category produce remain collapse
stationary point window write cardinality arbitrary window simplify unit canonical gibbs model ph core direct intensity convention h u estimation standard configuration set location intensity datum configuration keep indicator well logistic offset computationally hard
power replace simple convolutional output apply input maxout weight reproduce hinge index filter dramatically unit suffice thing output multiple architecture replace experiment perform use hyperparameter regularize scaled optimizer choose large value balance lead instability add
need dataset training datum expert wish scalability curve time second ylabel legend pos south east restrict title header false sep comma txt table header col sep txt core linear plot well early restrict advantage namely solver optimizer solution hand partial optimizer saddle show derive optimizer solver wide applicability experimental competitive art sgd optimization asynchronous version algorithm equivalent serial
tree serve discard tree dependency word dictionary word dependency matrix fix unsupervise et al bias nonlinearity validate sentence wish describe work description attribute context scene naturally motivate context particular convolutional neural imagenet detect location activation parameter cnn choose discard simplicity cnn architecture describe contain million resemble parameterized sentence similarity sentence turn compute intuitively match rise criterion design objective image sentence
operation incorporation extend maximum perform mutual discriminate candidate evidence discriminative power train allow spectra rigorously paradigm l begin protein mass mass round roughly represent spectrum identification sequence database responsible constrain denote mass spectrum identification construct comprise let shift relate offset mass atom offset water plus atom unity mass unlikely search doubly unique convenience benchmark algorithm heavily ms spectrum calculate p mass number score spectra benchmark variant implement ms begin spectrum vector equal length construct denote shift foreground e fit
develop aggregate loss loss shall case characterize mention density consistency several reconstruction reconstruction observe keep mind procedure whose respectively measure mae distance reconstruction histogram histogram notice density apply throughout cc true density mae mae rmse mae c cc norm l method probability losse interest functional total distribution reconstruction laplace real besides quite reasonable admit analyze analytical solution hard variety determine probability loss record pass quite error datum operational loss frequent situation amount account certainly besides potential method lie explore operational risk problem tail economic
feedback present gradient descent representing initial interesting deep method author organize formulate follow extension section follow conclude remark f state time integer dimension definite definite first argument assign scheduling may latter weight usage resource set continuous feedback policy asymptotically system minimize order study e simply assumed copy policy delay motivated operation bellman equation sometimes denote incur throughout remain consider recursion x bellman state simply give feedback eq eqs desire utilize parametric e approximate backward fashion use find detailed select interest use repeat process offline conduct correspond nn provide continuity continuity switching versus early simplify
finite mdp special terminal episode starts end reach terminal transition loop mdps th layer move consecutive episode reward end learner reach belong equivalent transition mdp denote policy follow
discuss issue square problem though generic solver fast obtain design dedicated algorithm leverage active benefit subset leverage subset strategy detail bt j q j open solver slow solver toolbox ii iteration quantity implicitly work update active computational algorithm linear active theory much practice set shrinkage experiment
complexity summarize note present cause slow coefficient obtain emphasize effect allocate size outperform complexity em energy market http com use repeatedly n compare complexity section design w w w outperform initial aspect respective counterpart exhibit performance size kernel allow representation approach outperform approach would effect accord experimental study behavior size subset realize section efficacy selective strategy complexity em propose orthogonal projection multiple include
mit mit expressive scalability gp leverage low full input result approximation guarantee close kullback leibler criterion subject considerably refined gp utilize low conditional trade order markov represent vary markov approximation gp two amenable parallelization core thereby scalability evaluation real dataset significantly scalable rank achieve rich provide measure uncertainty gp poor scalability limit scalability gp suit slowly vary correlation require high capture e correlation fidelity localize gps compactly support particularly rapidly however utilize
concave oracle rate target absolute constant agnostic number dc c disagreement bind address agnostic directly priori achieve roughly bind disagreement one provide disagreement lot progress decade amount annotation search hypothesis hypothesis achieve excess minimizing query study noise advantage inconsistent agnostic consistent agnostic disagreement analyze notion coefficient disagreement passive practical give noise label disagreement classifier sphere limitation apparent apply rate relatively development include recent learning address rate version space measure agreement extend zero error generally fact plug recover formal disagreement confidence
north e north south cl cl south north cl north cl north south south message schedule use consensus send consensus send contextual message desirable point message reduce consensus iteration lead accurate remainder unchanged manual highlight latter hierarchy make send variable immediately variable simple layer much less layer layer final regressor capacity regressor make perfect enough heuristic way complexity capable regressor datum important predictor layer wish model latent message income consensus message task parameterize regressor context consensus pair discuss data message run collection message pass consensus message message
execution execution sound costly copy program copy memory program rewrite use branch execution explore possible path path process space handle inter communication must store handle mutual object particularly useful counter barrier prevent proceeding barrier execution trace statement observe run program call constant appear choice exclude manner code sequential trace
cluster appearance gibbs become evident value group demonstrate topic corpus group th group document vocabulary unique dirichlet topic express construction topic j topic model also poisson count factorize n full conditional hand hand lead directly bayesian collapse gibbs form collapse sampler topic dirichlet topic nonparametric remove tune may hdp assignment sampler process globally share dirichlet distinct hdp lda additive combine ji jk
transform laplace zero every fourier remove logarithm measurable minimizer l confidence e f z f l give every xt inequality give density function bound prove help logarithm minimize denote function end error equivalence minimizer estimate depend c leave zero uniformly change variable de de lebesgue theorem find position know proposition
chi freedom rv support mutually consequently j b function stationary u u u follow constant inequality hold p establish generic stationary satisfy triangle chebyshev inequality constant u claim application two additional condition process recall independent constant large constant
measurement prediction possess gmm exact algorithm depend application simulate cross discriminative class unlike cluster due uncertainty instead dependent reject everything sure massive classified even rest naturally object hence never undesirable svm classifier hard uncertainty fuzzy bias uncertainty another discard surface figure boundary fuzzy together scatter random draw randomness formation account effect nature even uncertainty study two physical mass properly use temperature limited drawing
steady mixture outcome unobserve markov concern make idea appear think idea represent sense realization identify decomposition fair coin equip correspond coin notion appear satisfie reference terminology asymptotically identifiable implication identifiability every update agent learn state belief equip weak element necessarily agent outcome consider introduction dirac product suppose agent belief parameter agree belief accordance agent gain insight learn outcome make run observer ergodic typical infinite decision past observation
relative error confidence tight take per asynchronous mdp states action wherein action pair choose relative action asynchronous slow relative clear significantly asynchronous fast ball frame theory k contraction k b define deterministic affine contraction martingale difference matrix matrix mb decay third represent noise converge absence slowly middle desire counterpart tune well surprising iteration initially sure need scheme increase practical using enough new offline online discount
concerned quantile curve associate asymptotic scale deviation empirical process concentration inequality general factor main estimator proof defer brief limit constant hold k difficulty impose methodology necessary context set compact subset term expansion much stochastic center side give standardize approximated dominating term type process field symmetric q hold hand therefore statement xt xt consistent estimate xt xt convergence sup converge sup pn additive error detail definition lemma quantity quantile differ find approximation regression constant estimator section limit hold scale u defer explicit cc theorem xt xx
purpose ease attribute file format essence comma file attribute comment meta datum explore effect mostly set give meta predict class hyperparameter hyperparameter hyperparameter entry seed level meta datum instance provide bp predict meta act bp bp general fold set meta meta fold cross hyperparameter set set meta study use hyperparameter accuracy hyperparameter setting learning denote refer hyperparameter set bp set aid high meta
method nuclear minimization rank soft thresholding singular analogous iterative compressive case vector counterpart robust solve formulation avoid mc mc pg approximate soft mc warm accelerate pg accelerate alm augment mc descent factorization pmf mc mc alternate mc mc mc online match pursuit hard em vb bayes develop decade comprehensive solver low noise provide detailed comparison reader experiment present publicly reader modify code evaluate relative low stopping merely see closely estimate ground algorithm run show convergence average problem low binomial location outlier sample test alm solving vb test noiseless noisy addition test vb author matlab alm inexact adopted provide practically svd singular accelerate svd solver implement coordinate algorithm update update experience simple implementation package author default paper outli entry input simplify help tune guess rank os
point letting svd eq equal eq complete solve svd singular newly rate algorithm compute application tag gene website separate detailed htbp instance cardinality biology complexity parameter validation ml trace solve multi problem norm
effective relaxation numerical comparison subproblem acknowledgement give helpful comment closeness image cone suggestion subproblem discussion linear anonymous carefully read valuable corollary lemma exact liu abstract problem penalty induce propose decomposition minimization handle weighted subproblem develop partial proximal point subproblem limit bfgs bfgs newton finally penalty subproblem l bfgs newton cg type penalty decomposition propose term compute minimization vector product minimization feasible nonempty nonempty set globally wide application sparse noisy difficult addition feasible solution optimal bring solve favorable minimization regularize
resort trajectory completely underlie random measure illustrate mixture estimation key measure definition quite since evy simulate jump suitable sample conditional aspect address paper organize section provide explicit random survival framework gamma process dedicate moment section methodology survival use survival section two sake exposition simplicity full presence censor within multiplicative arise end recall random disjoint mutually random importantly reference existence say henceforth correspond identifie call evy distribute hazard censor
addition decomposition author report factorization function matrix factorization implement author pca implement default value require metric absence matter variable infer scale go decomposition scale limitation absence permutation matrix row correspond row quantifie implement first zero likely vertex unobserve require diagonal corresponding pearson find bipartite unobserved equal number unobserve allow correspond
discard hasting tune trial run conjugacy normal innovation sample truncation bottleneck execute ten mh draw worth point collapse straightforwardly original filter innovation let innovation ratio likelihood simulate system initialize innovation histogram true fall well credible region identify month disease activity subsequent year prediction sample running particle particle prediction disease activity month line kernel simulate trajectory enable particularly useful complex markovian two main sampler correct second movement drastically show empirically mix exploit pass compare conceptually require backward pass markovian effect error weight arise also worth point markovian model storage quantity need discussion storage future ergodicity ergodicity encourage particle find informative interesting work also different amount dimension ease
application defer chapter main low case identical distinguish draw randomness internal describe uniformity testing adaptive support uniformity outline turn principle deterministic carefully pick proof argue deterministic behave regard around sequence family due query us triangle inequality variation adaptive sample apply alone enable uniformly uniform pick uniformly argue distinguish case indistinguishable would indeed differ distance uniformity apart generality let loss write preliminary expectation subset query group intersection choose eq result roughly tail proceed contribution query recall partition say intersect instance sn j j
hide unit output give output eqs lipschitz q rs n ne I define similarly rs rs ne ne lb b combine completes consider hidden layer three dropout complexity three eqs lipschitz dropout lead complexity within drop improve
query categorization train correspond text linear detection task object class top convolution operation slide image window product model match semantic rely match propose product approximation composition code use compact compositional code database efficiently estimate query compositional approximation compositional dictionary compositional show dictionary motivates generalize yield learn source large compositional compact code length experimental sift searching interest similarity search near study research computational geometry computer vision paper neighbor inner study analyze product product
norm use slow applicable motivated propose scalable convex researcher value rank involve integer nu nu size tensor unfold nr j small size ni trace unfold norm formulate svd burden tucker rank usually sensitive liu al norm splitting technique auxiliary g nr parallel admm problem formulate follow sufficient matrix orthogonal solve analogous b problem orthonormal proof material order propose
tune vb accurate mean agree quite fact vb seem reconstruction suit reconstruct sharp density estimate example small map work produce smooth solution noise level smooth indicate tv prior produce preserve large reconstruction become smooth converge reasonably iteration need need large iterative gradient also notice amount like parameter converge infinity noisy converge test mostly good encountered converge either image illustrate might improper computation case gibbs vb prior parameter convergence ensure value avoid study prior make choose crucial
additional benefit alignment naturally source phrase length counter word nan propose long sentence perfectly accurately part sentence sentence admit patient medical centre un est un patient centre correctly translate source medical sentence status health worker sentence de translation preserve meaning sentence un patient un un centre kind series build translation type exp pour la de des les mean sentence word phrase quality translation mistake lack mark translate exp des efforts de cr des conjunction quantitative confirm reliable translation google reference input
follow use sample exponent thing concentrate obtain improve improve bernstein replacement oppose detailed eq follow assume rescale excess first page rescale excess ef fx ef ref ef f gx q consequently notice either proof theorem ef define notice convex jensen assumption eq rescale excess way achieve definition great use q otherwise certain combine probability recall put conclude proof variable repeat obtain sub root let f r rescale
week order accord appearance texture individual time texture name f f f f f stable f f concentrate f f f concentrate fit model effect occur trait htbp dim treat treat observation group consist mainly product mainly consist mainly consist product four mainly product missing find similar accurately consist mainly thin texture overall mainly colour among group average overall consist thin texture product texture observation fall therein response attribute
dataset conjugacy effectively show quickly accord becomes collapsed vb report find superior find occur illustrate level nature pass algorithm ability good merge good solution use run algorithm convergence also result merge trial often procedure hyper vb ten find periodic group capture periodic advance light dark effect model gene rbf hierarchical allow introduce far inspire model methodology incorporate share series series currently explore motion modification variational speed implementation merge collapse collapse wide implementation website http uk publication bayesian innovation enable structure wish intra group variability dp fast collapse
library online attractive reason single process distribute environment online consider small extreme avoid second require iteration produce classifier consume training classifier observe instance learn learn classifier notable pa linear passive correctly classify weight weight current training margin passive consistently outperform numerous algorithm task binary passive propose pass online typically pass training convergent set train size restrictive algorithm instance train
phrase maintain active list contiguous pattern frequent algorithm active index addition assess document considered phrase certain document guarantee pruning phrase early termination search phrase space ni p h algorithm p di c increase slide phrase obtain aggregate iteration fix phrase index counter algorithm candidate index refer implementation closure line indice prune search natural termination require occurrence minimum grow linearly minimum support frequent transaction pattern mining search exponential candidate phrase candidate phrase entire minimum document space prohibitive ensure separate segment phrase invariant allow effectively phrase closure mechanism serve far reduce runtime traditional phrase extraction reflect key keep rank phrase external basis nlp filter phrase phrase phrase implicitly document segmentation return construct phrase phrase intended contain number phrase mining enforce bottom merging decision agglomerative construction significance agglomerative phrase quality phrase upon document employ agglomerative phrase merging construct phrase phrase phrase induce valid implicitly pass chance aggregate obtain frequent phrase
manually annotate interpretation point class ground water bridge road construction concrete table ground rare class design experiment sample sample metric type random replacement straight second sampling try
connect convolutional mnist digits informative one image gain severe alternate next measure belong category computationally forward pass moreover sequential fraction benchmark encourage I work also suggestion gpu project accomplish google brain team google vision systems appearance computational pixel limit resolution image appearance constitute major recognition make address challenge object series system rather moreover potentially change handwritten dataset nature vision optimize
instance descent lipschitz function convergence bad smooth function sdca sag lipschitz dominate substantially smooth loss w respectively support many find good moreover question work improve well dependence condition size establish modulus work online importance average individual bad lipschitz explore reduce number contrast work technique technique use
b combine two claim combine say eq pa ba ba px b mu cx mu cb px px c px px c mu acyclic order eq node rewrite additional argument follow argument one possibly
underlie dynamic drive thank helpful acknowledge fa air force office scientific research project people represent provide snapshot analyzing evolve identify scale pattern interaction fundamentally change occur formalize network framework reliably combine hierarchical point occur resolution evolve social align know frequently framework use large highly interact functional approach non system interested understanding identify social change periodic case result response change network could stress introduce infer
start let eq rank rank notice target form prove proposition arrive line triangle algorithm never fair let incoherent approximated subsampling probability appropriately rescale n phase approximate distribution multiplicative f effect uniformly rather average norm weak coherent bound lemma dominate return substitute consider low completion sampling overcome uniformity focus former contain direction know absence uniformity art uniformity conceptually simple direction discuss adaptive low approximation understanding broadly adaptive unsupervised practically acknowledgement nsf grant award fellowship completeness apart concentration proof similar decomposition provide invertible follow control inequality result notation application bernstein coordinate absolute inequality plugging ensure bernstein proposition denote orthonormal probability q translate
effect discuss health big track rt walk fellowship university student health aid collect thank valuable comment ram tucker e medium track population determining develop novel detection individual develop accurate diagnosis twitter sample disease twitter develop combine text anomaly
vary section weak bounding annotation box define box subset annotation define bound z else union box label category include bound third column bounding box category map coefficient assign corresponding column depend box specific profile bound z z z make coefficient work really try linearly affect bound box label outside bounding box decomposable w r clique bound box generate intersect treat way modify label unclear penalize intersect equal bounding annotation infer category label
ty accelerate provide deal multi categorization problem option class th ti ty py iy conjugate categorical induce th entry set similar proposition class ic partition I ic reward opt challenge calculate efficient dirichlet reliability multi contextual crowd popular crowdsourcing require worker label crowd label worker reliability first incorporate worker reliability reliability true introduce worker reliability set model worker reliability multi utilize two static basic building several devoted allocation crowdsource particular assign instance worker accord although minimax rate new labeling repeat worker incorporate online dual assignment investigate guarantee error gold worker reliability mdp address decision crowd maker instance whether worker fix amount budget budget amount could knowledge characterize allocation mdp crowd labeling level budget crowd fundamentally noisy active difficulty instance crowd labeling model worker reliability require crowd label label learn budget allocation crowd many instance essentially horizon bandit horizon bayesian mab mab cost bayesian ucb policy different mab problem reward note stop optimal stop horizon must conduct study interesting opt soft lead adopt unless h h thus omit visualization first investigate worker
pc admit equivalence many fail generate causal structural modeling form ica optimum joint external enable identification causal identifiability often search contrast recent derive uniquely identifiability e variable bivariate regression advantage principle additive identifiability condition variable e identifiability identifiability unique identifiable order another propose regression causal additive noise introduce hilbert schmidt unique solution convexity issue discover assess cyclic modulus bivariate find mutually independent limited
uncertainty either seek optimize mean two objective solution select single offer genetic optimizer pareto minimize objective typically pareto front pareto another major depend pareto front multi sensitivity direction uncertainty difficulty multi objective strategy receive formula entire system engineering structure example cite use fidelity physics take minimize make reliability various source reliability formulate reliability practical structure main tail lie moment metric fail service behavior express multivariate limit
factor q eq estimator modify sure denote suited remark bandwidth operator experimentally bandwidth excellent estimator operator bandwidth thresholding tune sure modify covariance either compute mean square estimator
classifier regressor least square lexical convenience text technique kernel time support force linear reduce suggest significantly rbf task approximated et provide conservative empirical speed significantly number approximation applicable popular package derive svms applicable process kernel identify vector belong increase rbf kernel prune explicitly low therein
learn comprise filtering comprise step prediction filtering suppose px qx filter qx z pz px pz px dx pz respectively I nx mb operation datum I z algorithm summarize result case filter kalman kalman filter kalman particle restrict observation observation permit belief prior ni z experiment mb filter sect describe ground truth mb difference present filter synthetic nonlinear exist bayes filter incorporation application world vision validate mb concept sect mp ax gx ia analytical rkhs mb top misspecification scale bottom misspecification vs right combination mb I r mb mb mb probabilistic describe well learn mb show norm degree
key determination combine sec posterior update explain magnitude gaussian noise interpretation residual explain minimum explain information expensive represent increase incorporate show also factorize independent parameter zero versa small magnitude force element large enhance essentially observe datum incorporate message distribution sec appendix update compute straightforward need assume sec posterior expectation evaluate explicitly sec detail posterior residual measure square frobenius lower bind expectation entropy sec detail small leading automatically hyperparameter type achieve w guarantee get important initialization result initialize n initialize scheme n diagonal simply sparse tensor e efficiency manually procedure summarize bottom fig message start indicator evaluate
coverage size cp coverage cp unbalanced present compositional proportion team attack serve error team regression transform transform attack serve mean team effect game error vector variable distribution estimate proportion e relation proportion transformation inference covariate probability parameter n use
th sum substitute error derive follow symmetry summation set depend get note replace establishe encode message differ submatrix n x exchangeability px px px n x py result achieve derive exponent argument taylor series decompose use eq note lagrange taylor value derivative evaluate ix preliminary necessary due sparsity ix rewrite note incorporate consequently analysis herein channel code code candidate true set item candidate hold partition equation derivative px px n py py yx simplify second equal add subtract mutual independence necessity suppose element salient index error variable
image instance tend segment follow distribution paper propose law cut whose law fix achieve goal treat partition objective objective solve locally objective performing receive considerable attention computer vision science graph cut cut ratio one utilize clustering cutting accord theoretic objective cut problem cut circuit show approach despite spectral algorithm suffer
art building noisy house still recover manual outperform tolerance another drive effect large possess strong power collection conclusion encourage news match object match relaxation provably error free achieve remarkable efficiency greedy round strategy ability even turn perfect long partial severe occur situation ability nearly broad finding combinatorial integer programming perfectly semidefinite relaxation appendix algorithm simplicity matrix operator matrix lagrangian program operator convex negative whereas represent penalty optimize primal dual close number iterative update procedure q operator resp project reasonable return even dominant portion behave small component randomize procedure generate ni absolute constant since rearrange presentation diagonal perturbation include block convenient decompose comprise check minor come row column index eq quantify eigenvalue eigenvalue column universal constant positive eigenvalue
reduce computable effective class dimension vc denote subset computable vc converse obvious index effective note computable far concept computable statement condition dimension comment definition completeness finish vc complete index effective concept vc suffice property sequence function take set serve
backpropagation order visual word could path image top bottom image randomly implication effectively overfitte well visual outperform piece wise gradient input compute parameter efficiently parameter gradient descent ip target I I part image car appear lot exploit successfully seminal histogram substantial follow whereby visual appear specifically represent decompose r implication binary leaf visual computation grow deal modality annotation annotation note annotation annotation image annotation mixed bag annotation framework treat annotation word annotation word joint annotation relationship spatially annotation annotation give possible annotation path leaf annotation decrease order predict achieve model previous work capability base binary derive propose autoregressive
domain basically find ks fail reject significance nan solution verify random reject cdf b random line empirical cdf grey around discuss clarity empirical latter address next employ inspection introduce inspection unit comprise day inspection carry give inspection day inter arrival inspection reject associate plan classify infeasible plan
ise strength ise time achievable strongly model exhibit use strictly challenging e warm hard least assign vector goal give access h es every sufficient edge set actually note ml estimate add function independent set thereby likelihood follow hard many correctly complexity least observe
hash table store location hash asymmetric use preprocesse create phase query report hash table hashing sensitive ideally operate diverse rank neighbor ratio query hash effect good run hash report per recall recall independently fraction gold neighbor number retrieve query scan recall dataset scheme hash varie level see much compare hash scheme asymmetric outperform hash irrespective superiority indexing product sign perform traditional news worse plain mnist look mnist fact mnist plain close course negligible effect penalization ep news variation hence poorly variation always product difference plain asymmetric proposal require implementation adopt hash widely popular indexing practice originally
covered process assign asymmetric cost tune relationship attention discuss eqn fisher eqn like two conduct view mirror set negative difference criterion aspect epoch score hamming distance fisher criterion epoch cost view reflect matter contrary test drop gradually converge epoch gap obviously big binomial epoch distribution generate hamming rank performance figure ideal negative wide coincide fisher nearly different suitable affect solve leave configuration fuse connection function repeat rate list c c distinguish modify training speed improvement table outperform include
pz z set variable pz z recall pz conditionally
number topic next done learn moment complete full bernoulli gaussian entry span combination sparsity problem relax programming row explain output l sophisticated find vector subspace deterministic version require quasi problem setting assumption weight learn function ax score bernoulli deep activation layer assumption ax
h hz quantum exception infinite temperature success initial temperature often field state typically employ reduce fidelity coherent gibbs state fidelity least protocol use rotation apply success failure branch successfully fidelity state trick e h gradient state order estimate via optimize utilize quantum amplitude search repetition need distribution field efficiently true give state uniform ideally result final field state performance quantum sample boltzmann machine connect edge number operation visible use gradient see draw boltzmann expectation likelihood state quantum success average algorithm reduce repetition success call require compute configuration rotation visible rotation error energy thus claim contrast optimization scale boltzmann machine constant follow asymptotic advantage practice two objective train model optimize finite advantage quantum mean perform assume require arithmetic computer development quantum synthesis could use remove store consider access training require require bias regularization learning array contain bias bias em em k train train j estimate one access quantum oracle either efficient boltzmann memory quantum unitary procedure ability superposition seem powerful resource sophisticated need wish quantum utilize provide circumstance compute visible hide boltzmann scale scale q
general profile location scale multi overall point approach belong method explore information availability multi tend reveal accurate information global single matrix capture geometric eigenvector infimum eigenvalue scale neighborhood al support svm rule distinguish two domain surface image later classification scale necessary section feature homology review persistent homology good reference homology group make usually topological equip integer summarize appearance homology take closed persistence diagram appearance merging interval turn persistence diagram make properly
report superior max threshold multi base stop min classification change consecutive learn min satisfied criterion report stop operate stop min far applicable stop learn annotation label begin require counter al example prediction performance versa annotation sp set prediction occur stop type example encounter factor check quickly make stop perform stop stop select informative preliminary made performance
opposite opinion choose outlier whole foreground attention lead attention notice background lead dominant explore dominant agree removal point another less way select subset clean validation result highly competitive alternative rank exploit reliable sample national year match match regular pair comparison comparison head advantage home capture intercept team home support inspection regularization path reveal outlier outlier select lasso outlier team rank conference rank conference regard return see could lasso importantly large unbiased score big c team team c c name net intercept match least match winner tie eight outlier return
root localize anomalous could anomalous due rare attribute typical situation anomalous corrupted provide pattern consideration corruption anomalous contrary corruption child however corruption attribute localize right child pattern b realize corruption accept reject continue attribute child fig recursively alarm reject e anomaly search tree stop parent branch corruption corruption action correspond node continue correspond finally anomalous opt accept leaf favor well cost alarm illustration progress fig corrupt attribute locate attribute region partitioning improve complicated nonparametric situation accordance regard localization plausible density mmse averaging generate solution computer denoise instance smoothed mmse imputation cause fail satisfactory detection reason novel map imputation technique generate feasible likely approximate map size estimate original attribute training replace attribute statistically treat corrupted attribute attribute note sufficiently model accordance estimate condition localization tree attribute certainly detect cf introduce novel map corrupt attribute base rank implementation output corruption localization phase therefore computationally imputation phase require computation attribute initialize
value state ill many modern domain utilize extent often available covariance computable setting albeit practical pairwise environmental science field often correlation distant spatial modify bivariate covariance readily demonstrate formulation organize follow brief inverse methodology follow consider extend generalization numerical life section inverse portfolio portfolio provide appendix primal proximal belong subsection briefly coordinate block coordinate box qp al notice qp solve block coordinate descent correspond primal dual theoretical proximal newton author use nesterov algorithm converge provide enough optimal proximal global moreover operate outperform glasso alternate composite variable z z iteration
current modify recent work label read introduce classifier support correlation problem seek graph dependency limit markovian relate theoretically sp algorithm capable problem mod example sp mod assume correct whereas algorithm output state mod relation approximation learn relational relation mod correct traditional definition take account label correct dependency output part training contain
mean semantic semantic website associate query website associate query also compare sensible associate website e com dnn semantic website shot improve margin like hypothesis zero shoot compare svms shot supervise task identify semantic label label datum achieve label well low costly class h svm bag word dnn dnn embedding embedding semantic
possible state optimize contribution account map average significantly svms sample framework smoother finally unify special insight practitioner follow introduce unified algorithm model naturally extend numerous general perform hold extension structure svm application area human action recognition sentiment rely joint perform well framework discriminative graphical hide incorporate framework entropy minimize
suggestion start relate text removal provide classical backpropagation actor back respect e verification support human assessment name classification control make human decrease backpropagation implement feedforward train gradient neuron continue train discard learn classical overfitte show develop reinforcement line represent fed send mean modification represented modify adjust epoch represent delay module delay architecture test several collect past text extract
feature selection method call mr method select relevance mr base simple applicable use relevance input focus manner large nn lars lar criterion lar similarity output measure couple lar large feature get select regularization analysis implement practitioner benchmark propose compare exist lar dataset select feature focus select redundant manner scenario lar nystr om theoretically justify feature review feature n
k iterative understand yield mean high gaussian provide synthetic datum capacity know mean penalty short k unsupervised technique include cluster cluster spectral rapid automatic acquisition encounter problem huge attribute weight grouping portion responsible example responsible activity synthesis activate relevant make inefficient feature eliminate automatically importance feature approach dimension reduction principle nonnegative principal cluster detect log fitting dimension framework optimize particularly statistic zero weight feature work still keep word many final example seminal cluster relevant cluster relaxation improve put penalty cluster intractable overcome
successively tight bound high dimensional pseudo code accord dataset iteration update marginal calculate mini size stop iterate typically sec build construct label empirically quantify small total among stop explain criterion explanation similarly contribution factor quantify iy iy add factor beyond certain write ratio use distribution ratio construct implement accord domain factor binary specify special create recover correlate outperform task imagine independent bernoulli half define
negative kl distance kl represent concept encode imply soft gradient yx dot product draw two gaussian gaussian percent get choose principled bound chebyshev landscape regularization differently grow hard constraint keep reasonably sized hard lie constant diagonal involve keep hypercube energy score small dominate rest worth qualitative quantitative task asymmetric linguistic word diagonal note learn corpora token match aside leave publicly token appear less drop
stationary point discussion shall clearly consequently theorem applicable generate immediate nonempty closed algebraic algebraic algebraic practice design guarantee sense describe similar discussion initialize exceeds iterate huge finitely many boundedness clustering get stationary point guarantee solve feasibility next consider close algebraic certain exhibit lemma homogeneous proceed suppose letting eq second shows contradict conclusion constraint nonempty algebraic suppose generate addition limit cone assumption also proceed without z z conclusion z z consider precede follow
cause binary constrain baseline constraint extremely compare run objective net dropout layer layer momentum momentum choose size hamiltonian carlo mcmc take gradient rapid hmc careful tuning basic hmc include introduce parameter experiment optimize hmc spend measure correspond minimize sample absolute across worst large chain across variable package threshold base hmc integration also additional constraint datum repository normalize unit initialize deviation input compute
solver outperform year integer create program cut iteratively new optimum cutting solve simultaneously score build formulation unconstraine maximizer polynomial require cut technique clean call optimizer understanding approach prefer relaxation estimate stop computation solution cope even obtain solution minimum cubic large probably little situation observe report require large amount much deal domain devise approximate computable sample appeal whose take relatively empirically double tree variable effective optimal set conclude note background learn direct acyclic graph dag node categorical value
obtain manner site square gaussian burn previous convergence result shannon diversity illustrate independent horizontal axis represent shannon represent model highlight covariate red credible interval seem restrictive goodness sort remark help regard impose mentioned distribution stick important smoothing operate dependent clearly visible dependent smooth fit regardless make dependence diversity estimate proposition ht posterior diversity dot shannon dependent weight name construct thank stick breaking bring flexibility define define kernel whose allow dependence flexible counterpart advantage ready marginal scheme conduct mean compare sample arbitrary
ensemble neighborhood moreover optimize intrinsic induce reconstruction since geometry tangent principal obtain singular scale look reasonably expect correct scale approximate differential implement manifold sample manifold gaussian result pair heat approximately equal test order magnitude logarithmic point evaluate pointwise svd size cccc standard vs noise svd indicate interval could optimize distortion dimension metric matching interesting thing typically synthetic explanation near tangent select
score direct translation penalty phrase length section single translation reverse short phrase importance translation translation model incorporate short denominator score penalty model help see observe sentence length word similarly propose existence mean sentence avoid effectively deal htp early turn american year old increase health segmentation early modern turn american old department reference
query suitable query retrieval distinct multiple image pareto front manifold individually database dissimilarity query sift vision image dissimilarity computationally intensive dissimilarity sample query manifold need efficiently underlie traditional manifold image retrieval query next rank produce create pareto point dissimilarity sample pareto point compute pareto one pareto depth front dominate remain sample return pareto stanford scene dataset key front importance retrieval illustrative image pareto front forest accord within tail locate front query image necessarily image front contain desirable retrieve pareto pareto characterize convexity pareto database establish pareto theory connection front depth admit maximal
search ad user age gender search display query query user title appear title keyword keyword ad unique assign assign click number ad number click label six indicator age year old age third imply unknown three component unknown gender gender make next component display ad make use component display position likewise word appear element encode create bag bag word bag bag th bag indicate component bag component may nonzero much bag encode keyword encode title ad component encode keyword search title product keyword distinct component encode observe important cost regressor nonzero estimate regression contain describe section classifier want take ad ad observe write read click contain give mle model mle find minimizer likelihood loss see follow datum determine query user feature ad ad display say offline former use stochastic computing infeasible recall performance element online update whenever vector adapt preference process metric numerical select sample select consecutive
extreme statistical law identically distribute frequently extreme extreme write mean minimum extreme coincide useful distribution reliability survival log independent systematic component eq q unknown continuously possibly nonlinear finally monotonic respectively matrix respect analogously value substitute maximum extreme result adapt minimum index extreme represent scalar nuisance parameter similarly form drop column additionally symbol indicate sign likelihood
table use network good behave realistic imagenet example validation imagenet reference experiment validation convolution mlp relu use prevent overfitte parameter parameter fully connect take reduce imagenet ad ad mlp convolutional weight transformation mnist extract last pooling layer common convolutional imagenet layer connect investigate train mlp reference activation setup incorporate conclude imagenet activation nonetheless able comparable mlp adaptation
high array outer array th comprise vector outer product arguably tensor direct discussion tensor decomposition consider analysis social time straightforward survey decomposition reader refer tt tm compact decomposition term slice recall apparent represent constitute cf likewise write diag n diag n slice first dimension stands build matrix feasibility fundamentally assume couple entry well effective freedom rough idea obtain ensure imputation completion tm otherwise incomplete compactly see also generalize completion tucker argue frobenius outline rank completion provably encourage rank tensor decomposition note tune appropriately stream real setting incomplete slice acquire sequentially depict right leverage subspace devise tensor factor minimizer cf normalization coincides adopt upon obtain accordingly minimizer cost computationally nonconvex bilinear could instead instant namely update ii procedure necessarily recursively impossible aforementioned challenge big requirement sgd alternative
begin constraint z setting optimization eq subgradient subgradient compute r method three iteration optimization introduce refine initialize tc z constraint grow link constraint hence fall unsupervised membership de via coordinate mahalanobis many traditional fast near neighbor hashing require explicit address near advantage greatly reduce need near forest leaf seek begin point identify parent yield full hierarchy
considerable inherent biological camera dependence sort view assess assess different size place randomness variability region mm voxel mm averaged error strictly simulation deconvolution sp curve worse show completeness technique detailed deconvolution order negativity known perform show reduce estimate spectral spectral analysis pre process methodology consider interest study method structure assume useful something assess size parameter coincide compare mse indicate spectral five even though perform well rest surprising favor spatial reconstruct parametric competitive assume however involve level ccccc sp perform brain gamma pdf incorporate identical pdf region make real
super know present term ccc std model train half recover model figure see good due orthogonal ica point fail quality orthogonal one always two far mix order large mix anchor likely activate activate hide unit reach
allow median condition space follow step proof strong theorem exist aggregated subset imply follow lemma determine subset choose satisfie particular recall assume reasonable possess assume ols select square eq q notice select letting correct probability choose c k essentially result subset update
consequence eq minimizer existence determinant establish suffice expand eigenvector objective unbounde unbounded boundedness unbounded contradict unbounded unbounded frobenius eigenvalue great accord partitioning permutation principal submatrix irreducible simply permutation irreducible correspond conclude unbounded reasoning unbounded scalar e diagonal kk pattern entry finish remain note occur eigenvalue satisfy hold bregman q virtue bregman divergence duality opposite note condition hence parameter ar satisfied decrease order optimality immediate consider inside recall monotonically induction hypothesis verify check kkt optimality cf define feasibility remainder diagonal consequently concern converse stationarity remain follow note small eigenvalue matrix hard concern block particular block minimizer limit sequence existence contain establish reasoning proof general maximizer sample certain
able recognize place task instance recognition categorization place recognition component detection scale qualitative type place recognition work discriminate place configuration place analogy model analogy integration method formalism gram extensively aim structured modelling conclude base approach
generated implement software generate exhibit scale define result model time sample normalize diagonal test sequence weight sequence random test result roc reconstruction present superior dataset encourage particularly weight scheme use give inferior reconstruction force reconstruction original graph roc gene association leave submodular middle reweighted covariance gene experimental gene activation
partition file reduce introduce node indexing column trajectory name name really recover correspond name recover object efficient entity system indexing static create name string key key indexing accordance singleton pattern e name node name array index argument select name specify argument name two indexing index synchronization depict simplified node interface interface define abstract class implement related implement requirement quantitative intensity discrete class implement implement continuous node continuous e example create string string string add add state add add node state parent new double define interface classifier implement class implement time specialized process rely example indexing string nan state string state new state add add add false new definition double double double model classifier separate format format start bayesian node follow separated node define bn start list parent format format contrary yet exception specify class compose sign file distribution bn parent sign bn file format currently nevertheless format support
expand show kernel svm away two message quadratic enable discrimination object prior noise preserve able suggest image order principle accurate control learn predict performance attribute edge encode figure emphasis object boundary around svms visual preserve heart finding visual assumption combine perform well figure
equal factor reliably bind se find wide close possible improvement moreover c spirit problem latter agent repeatedly select arm observe reward ask high reward elimination find single gap well fix particular elimination algorithm section arm algorithm arm
number score calculate return quantity composite may resample outline far variable band light scatter correlation carlo resample perturbation
annotation pixel semantic label presence insight convnet pixel learn learn pixel model image cast bag pixel need annotation rarely segmentation
ask fix reach max eq factorize problem eq eq reach restrict bp write believe bp bethe multipliers eq q lagrangian enforce lagrange multipliers lagrangian read lagrangian derivative impose use way q bp try actually reach
k tensor nice f element assume j k j j kx assume predictor index belong order replace distribution follow basis active predictor spline stand prior suppose distribution harmonic rate isotropic logarithmic variate harmonic prominent obtain strictly obtain naive application smoothness condition co co interestingly automatically level ambient affect exponentially integer smoothness integer smoothness restrict case empirical n define hellinger distance space rate predictor density satisfy respect say predictor
speed apply tuning compress dense connect quantization convolutional layer facebook ai com cnn recognition repeatedly demonstrate result image detection year cnn many layer million storage extremely deep resource hardware embed device tackle investigate cnn particular term demand layer clear lead good balance recognition accuracy category task imagenet compression loss classification progress standard object almost cnn
x bx complete ii eq bx g bx lie follow clear equation g x p iv equation I p p therefore root therefore hence respect bx bx proof complete distribution eq I unique n h unique since h
say formalize entry constrain say constrain picture iff q explicitly conversely fewer free arbitrarily infinitely q nevertheless infinitely subspace free last dimensional essentially explain lead direction directly behind every contain equation contain else behind know determine clear entry disjoint set row edge look like bold correspondence title eq q label j vertex line title title title jj title title jj label label vertex vertex label j width title jj title jj j title title font line edge determine conclude entry respectively vertex width pt vertex title font pt edge edge come direct must despite highlight generalization start loose really lie span subspace course could ignore way generality trivial discard easier determine determine subspace satisfy subspace subspace fit obvious essential single lie use generalize behave dimensional subspace
h argument popular intend highlight admissible place l produce cluster embed consist point perturb simplex lose however generalize symbol work spectral connect q size connect isolated vertex let partition mi jx symmetric instance proposition yield v v orthonormal hence give point ray mutually orthogonal directional ti isolate accordance lemma main function demonstrate continuous accord local maxima g imply maxima outside identify simplex strictly give contain maxima suffice weight apply notice piece u ij bring equation maxima note mapping without system let sphere suffice maxima open strict convexity
sampling e region posterior sometimes indeed draw method via exhaustive interested detailed discussion bayesian hereafter prior pd information simple chain monte step start step increment new calculate accept repeat termination control begin burn discard prevent h
within belong eight global group choose least ols ol commonly model comparative base stable solid versus curve axis positive false positive show estimate base five repository alarm child adjacency supplement additive stable assign coefficient dataset sg weight perform correspond skew however relate infer since avoid sign estimate q convenience alarm comparative infer direct figure true negative percentage learn blue performance ols method result line difference varied away true ol decrease remain reliable infer positive similar figure show comparative
combinatorial signature log correspondence root mathematical dynamically generate move contingency statistic r f two cell thus move connect table add move contingency call walk start table contingency exist existence generators algebraic linear equip walk result irreducible move metropolis use adjust return exactly dynamically log linear restrict appear parametrization mention specifically useful decomposable divide apply statistic complex log table encode hypergraph vertex one normalize convenience log denote parameter set edge hypergraph instead usual list represent understand parameter vertex hypergraph collect map contingency hypergraph familiar two discrete see hypergraph hypergraph complete quasi independence probability hypergraph complete vertex partition remove hypergraph hypergraph complex
shift sa sa sa sa cycle sa sa sa sa sa sa shift sa sa sa sa sa cycle fill align quantization xx shift scale xshift inner sep size shift fill drop minimum size shift cm shift fill drop size pt thick draw east fill draw west sep font text width leave self check fill drop minimum thick fill green sep east green west inner font draw circle drop fill draw xshift east draw west sep align self check circuit shift auto node cm font b bend left node loop yshift draw loop draw bend node yshift align group inversion align stream solid cycle cm edge bend leave node node edge bend none none none align sequence align draw drop sep pt height grid gray axis anti stream xshift xshift yshift thick background layer yshift xshift sa sa sa sa sa sa shift auto font bend yshift yshift yshift bend leave yshift pt distance scale bend edge loop edge bend node distance scale edge loop none axis axis align center sep minimum rectangle fill north align thick scale scale style dash axis center draw height grid major dash thick axis align sequence thick align flat center flat white black xshift yshift background yshift xshift sa sa sa sa sa cycle symbolic causal assume existence probabilistic algorithms b sum purely circuit allow table stream conclude length circuit carry rgb rgb rgb rgb rgb rgb rgb rgb thick green drop operation algorithmic stream generate independent pt stream symbol move position go operation stream thick fill drop size sample path copy read symbol position pt thick generating source read symbol write move go fill rectangle drop deviation symbolic stream symbolic section hence evaluate choose correctness see symbolic derivative rigorous actual implementation compute deviation since stream anti quantify dissimilarity similarity stream classify efficiently stream near stream ultimately similaritie structure quantify mutual dependence compute generative mutual datum identify stream stochastic stream similarity clearly stream find reveal without knowledge reveal detail stream confidence minimal length reliable scalability learning task depend notion identify instant predict step series strictly superior certainly well claim outperform par yet set applicability system assumption notion dissimilarity individual measurement set possibility metric universal least nature process sequential observation discrete symbol quantization range symbol represent slice range slice quantization alphabet consist represent fine large stream thus symbolic alphabet symbol alphabet quantization fine expense increase scheme
incorrect incorrect nearly appear many purpose suit world application letter solve efficiency series stochastically embed operation fluctuation algorithm adjust often dynamic volume physics analog variety
manually initialize weight bias add weight mlp predict good decoding reach development finally include pre maxout dropout decode error development point error analyze importance decode narrow width reach decode width slightly replace put used decoder acoustic language important really learn align sequence rescale predefined rescaling failed converge low stage training address stage procedure affect rescaling
cardinality example cp generic mean structure provide output compatibility iii retrieve violate qp fundamental please output boolean word definition concept incur formulae f indicator boolean evaluate compatibility incur world contribute additive formulae carry contribution formulae total maximize opposite
simple vector make lsh lsh satisfying feature call neighbor search arbitrary extend lsh projection appropriate item database publication computer vision community relate build suffer drawback approximate construct become projection even use vector conceptually simple section new performance runtime accuracy crucially reveal boost particular improvement examine nystr om discuss subtle nystr om aforementioned demonstrate advantage nystr lsh example lsh lsh projection improvement draw two largely lsh schwarz fail
posterior first investigate update covariance matrix optimality lead likelihood class leibl distance propose prove optimality approximation offline manner costly approximation particularly require demonstrate theoretical ray heat example exploit indistinguishable inverse inference approximation approximation risk optimality approach treat endowed distribution encode model interest incorporate forward distribution bring structure encode kind among inversion parameter action operator feature may identify approximation bayes lead substantial approximation inverse prior define approximation characteristic storage requirement fast form negative covariance matrix class structure also arise kalman challenge update within suitable optimality symmetric metric broad relative update class along lead generalize eigenvector hessian assume posterior extend kullback leibler divergence hellinger low yield optimal optimality especially linear inverse regularize estimate matrix appropriately action hessian precision regularization effort concern exactly easily square precision million context bayes risk square exploit optimal become minimize bayes squared analytical square weighted direction low approximate posterior fall posterior
error construct give sparse factor first construction define statement prove reverse induction pi di refinement degree root I back affect multiplicative define claim linear factor find factor pn contain entry factor identity square multivariate normalize multiply side clear range
h p p p edge word select probable meaning word average close compositional deep compositional sentence phrase furthermore strategy word second measure sentence vector compositional reflect sentence towards purpose
version cox modify derivation new estimator adopt measure bias equivalently application conclude scenario tool system operate several returning capture characterize illumination require analyze image complex wishart successful equip parameter covariance look former hermitian mean statistically relate severe various exist approach method choice mainly estimator usually bias indeed bias size
high binary fast automatic discovery perhaps datum meet cumulative present handle variate variate boltzmann inference procedure people across competitive art collaborative filter public rbms extend diverse pattern recognition university ordinal feedback preference datum boltzmann machines rbms variate variate ordinal able opinion profile competitive state technique public extend recommendation review boltzmann attract due deep bipartite undirected enable sampling
detector still fourth plan use scope improvement category begin target domain assess discover multi belong classifier kn ny classification use basic writing svm projective variant flow require target domain include margin learn discriminative use target domain feature analogous c c c svm source domain combine overlap computer video terminology adaptation internet google come domain amazon image resolution resolution digital camera one pose illumination experimental follow source number per category vary category category source example role select per target domain testing without split source
ideally tailor simplify come information identical reverse causal need future causal predict future minimize expected distortion equivalent causal distortion minimize distortion forward minimize expected state measure prop square error distortion measure emphasize reverse causal another even infer maximally improve nearly distortion though treat reverse leverage retain future causal reverse statistic use statement appendix directly thm seem clearly good knowledge prop prop intractable tractable causal reverse practical information function finite countable simple allow shape though yield illustration substantial display elsewhere distortion entropy causal state single state provide distortion limit monitoring calculate feature ref self one solve maximize iterate iterate eqs explicit find maxima thus choose initial force analyze suboptimal solution sophisticated ref carefully state use equivalence sec distribution analytically produce entirely deterministic expand codebook necessary start codebook decrease allow usually maxima zero start increase key difference compressed condition calculate maximally describe phase curve describe function feature time describe symbolic dynamic retain length calculate sec process
formula low identity factorization compute combine recursive strategy interpolation radial basis dynamic mechanic hierarchical covariance solver process multivariate tensor efficient dense factorization large important several field others monte carlo dimensional variable obtain covariance normal symmetric symmetric definite factor computational deal conventional symmetric base cholesky rank represent symmetric factorization diagonal e decompose major versus cholesky long block
formulation disagreement loss extent discuss metric define full laplacian matrix well frame rotation nontrivial robust visual surveillance summary end disagreement loss view version correlation cca certain kind transformation rotation translation scale different visual desirable
gradient descent equation compute w implicitly proceed update ann accord old weight without parameter pair set pair u ij increment ij l u b via increment b update ann via ij l ij repeat step method extract inter variance fit graph embedding class within school extract projection
h c u u also convergence tn arrival consistent update may estimate time parameter vice term broad applicability algorithm conjugate compress linear response accord proceed ba center normalize diag draw associate yy f xy xx proceed successively conditional jt yy xy high autocorrelation face mix joint parameter space partition proceed observe yy tm inversion b close form expression update jt jt c jt ts j modify surrogate c l jt k draw df dramatically reduce requirement see shrinkage bl infeasible mcmc iteration suffer severe particle dimensional statistic particle store propagate impractical approximation n ga density j kl df numerous arrive sequentially predictor jk
task access device capable algorithmic task capacity recurrent neural additional keep time give constant challenge result evaluate error method different analysis time refer normalize square square normalized root variant cover attempt normalize distance output deviation produce percentage value throughout plot surface create accord equation point blue solid line depend optimize offline validation
correction second order third order write wiener weight wiener pass vanishe choice posterior minimize theorem integrate argument uniquely kernel arise part match major community basic method define wiener th choose meet conceptual third solver return main aspect wiener integration simultaneously add improper wiener wiener linear exist infinitely often coverage ode posterior context wiener prior
associate statistic record divergence arise substantial abc mean allow substantial speedup p p preliminary kl kl approximate yes abc mf mf comparison posterior approach preliminary trajectory trial residual proportion mf protocol inference statistic inference kl divergence kullback divergence use approximate abc protocol demonstrate infer experimentally study number copy quantification well mean copy infer ref behaviour induction per produce geometrically paper avoid cycle rna cell
demand take site compute life necessity reliable mobile device explore speed cnn approach speed cnns redundancy cnns consist convolutional bring performance boost exploit redundancy within keep accuracy approach redundancy conventional filter convolution channel vertical accuracy know cnns express visual interpretation similar rather redundancy unnecessary feedforward backpropagation cnns sparsity accelerate computation align neuron iterate successfully filter locate position due filter
converge functional instead prove dense easier deduce every exist converge deduce say functional hold increase natural bound previous necessary restrictive notion useful property minimizer approximate minimizer minimizer guarantee energy energy statement convergence say convergence functional identically compact minimizer functional index namely functional say every functional convergence functional interested functional probability functional f n dependence understand section consider remark namely riemannian structure metric remark precise space borel allow measure product space contain graph pd remark introduce use identification space note contain product opposite couple g equation hand great conclude space lebesgue imply sequence convergent converge true impossible convergent completion pd pd dirac delta dense pd dirac
analyse size approximate complexity affect rich space accurate hand move core issue appropriate count simultaneous low bound bind process ratio difficulty require boundedness generating function indexing satisfy truncate ratio prove meet requirement study devoted topic main rate tail bind multinomial make metric kolmogorov deal upper approximate metric finite approximate lemma devote hellinger distance partition partition j translation calculate metric unit nu nu dependent f accord construction euclidean nu nu order
learn variate setup require sample accurately extend obtain derivative use purpose fisher derivative tensor representation refer thus decompose svd case term present analyze hand unlike matrix tensor large regime less overcomplete lead rich obtain option perform discriminative prediction
limited therefore fluctuation state propose mining discover machine anomaly density anomaly many develop mining deviation detection density extend previously distinguish trace average state quantum include circuit mine key tool broad quantum broad necessary diagonal
log population two side n u generalize nuisance iii hypothesis
match perfect significant neuron understand relationship connectivity processing either graph chance conclusion consider iii inference information figure match extract statistically neuron connectivity classification neuron il neuron neuron vertex number four category employ four category proper match measure match second seed category seed amongst category match match vertex correct category indeed seed greatly match plan heuristic optimize seed summarize mc replicate graph contribution least
electrical engineering university respectively associate electrical computer currently electrical engineering company foundation frank distinguished fellowship award research interest high adaptive communication imaging edu acknowledge support nsf large low step adaptive sense theoretical show achievable summary original rank plus logarithmic experimentally robust collaborative task vision automate surveillance investigate sense pca paper address suppose admit decomposition nonzero ultimately interested identify location nonzero column particular focus may identify investigation identify array arise anomalous pattern component task increasingly leverage dimensionality idea line utilize reference therein compressed burden inference operate task surveillance application ideally region numerous give map aim map perform image linear wherein interpret overlap patch image equivalently matrix version patch previous effort natural subspace approach estimation salient region model common subspace efficacy visual recently rapid security surveillance imaging task entire successful salient
virtue reveal analyze apply orient laplacian even subgraph induce remove index remove connected component let eq lemma describe smoothed piecewise assign multiplication interpretation term electrical perspective graph freedom number estimate k xx number support reporting degree freedom graph filtering reveal mathematical estimate aa interpret electrical perspective graph describe go induced accumulation current network q say form number node solve current repeat current new induce current iterate even say node assign current potentially overall structure estimate informative tell rd order induce piecewise vector nonzero induce current piecewise extension propose filtering trend perform weighted incidence forward recursion could loss order datum say explore imputation investigate potentially penalty extension wherein graph convex principle moderately sized reliably variety order handle problem describe procedure take
start first moment uniqueness equation eq gaussians excess recover imply coefficient nonzero equation solution lemma gaussian add construct mixture moment gaussian differ contradiction repeatedly add take explanation bin setup alpha beta beta alpha gamma alpha alpha beta gamma alpha x alpha beta gamma beta alpha alpha alpha beta alpha x x x alpha factor alpha alpha goal alpha alpha alpha beta factor gamma eq b z alpha q gamma gamma c b divide unbounded exist per lead lead sufficiently polynomial normalize compact root lemma homogeneity polynomial coefficient magnitude regardless conclusion bound lemma I perturbation loss normalize combine root desire since max therefore max max c cr max suppose good approximation therefore conditional roughly would max x follow cc branch whenever take whenever remainder take condition branch setting clause
correlation carry hierarchical recommend linkage within correlation cut dendrogram cluster linkage minimum correlation cluster highly assign
interact economic connect indicator extremely mean number indicator indicator indicator theory prove purpose monitoring forecast proportional identifying vary order average averaged indicator equal pearson product indicator comprehensive dot country position node centrality economic edge colour code pointing regressor colour quantify relationship centrality connection high capability might strong connection span magnitude call financial tracking inherently couple fit possible account track average maximally error along aggregated regressor forecast exclude operator column besides forecasting indicator turn exclude forecast capability especially international trade flow lag
rating without b particular technique greedy achieve netflix dataset superiority previously work shot variant first variant active sequentially rating second variant one news recommendation arrive short require identify user tradeoff select complete present multi explore user review item offline setting work item start essentially mainly whereas tree solution start currently implement however optimal user potentially lead empirical theorem thm remark conjecture
weight finding problem algorithm stage initialize item add weight graph cycle edge edge may network span delay link initially unknown variant maximize modular address problem formalize th entry weight item stochastic item negative associate arm assume unknown generality stress equation episode episode goal basis minimize episode optimistic maximum weight statistic et et
robot trial reach number trial reach run map million evaluation map contain behavior extend behavior arrange shape heart shape curve place robot black draw position degree freedom angle theoretically achievable performance performance adaptation trial goal cm test save reach often specifically iteration second explain median except trial optimization robot continue reach scenario level never classic scenario challenge trial case substantially case median accuracy pre post behavior post able cope able behavioral descriptor choose simplicity sophisticated likely cope experiment show behavioral descriptor able position traditional extended fig value art policy search three map find behavior physical robot prediction note perform search prior alternative variant variant search evaluate search search randomly select test well one keep variant evaluate prediction performance map performance trial bayesian process randomly let choose evaluate bayesian classic obvious policy behavioral search directly dimensional dimensional dimensional variant map ahead time search directly algorithm automatic experimental variant gaussian initialize constant select experimental dimension automatic variant break modified simulator experiment section simulator involve remove create variant map simulation case replicate eight map six replicate variant time lead replicate per experiment roughly real robot simulate perturb multiplicative noise analyze fast speed variant number case trial trial trial experimental trial robot outperform demonstrate perform variant variant variant work policy search map contain result search evaluation nearly variant randomly policy design trial prediction introduce trial process learn trial low six
give sometimes give supervised dataset drug target network one network know disadvantage predict give recall ls ls ls ls laboratory molecular biology uk network largely partially framework train separate node train node systematically theoretically formalize problem pair bipartite discuss global approach extend later unseen drawing carry biological network highlight relationship biological entity gene protein micro disease network experiment practice entity researcher take interaction experimental prediction formulate inference consist pair adapt consider input feature second train predict direct neighbor exploit
algorithm inspire cluster cut different accord gap statistic candidate comparison follow refer introduce present simulation compare discuss complexity microarray conclusion general resemble size candidate spc rank consideration candidate determine combination gap statistic conduct high across global remainder detail cluster inspire cluster propose split hierarchical recursively stop gap certain produce label input dendrogram result hierarchical bootstrap gap default reference clusters remark examine dendrogram dendrogram assign root gap current
compact also return range learn affect generalization tool continuous hadamard matrix word could expect per concentration need continuity refer continuous constant normal invariant around mean showing lipschitz function cosine construct combine choice play condition inequality lipschitz union order magnitude exhibit significantly memory simplicity focus penalize able compute benchmark cifar achieve art expansion even begin investigate purpose uniformly see tb fidelity imply unless assess carry experiment uci e least instance investigate approximated whenever completeness method exact rbf kx albeit practically desirable due matrix recent achievable theoretically advantage function exponentially retain
take advantage consumption day day separately big markov hour day begin day remainder consist column vector time capacity take level take amount unit grid update dynamically interval poisson number average time estimate monitoring
reliability contribute bayesian integral predictive numerically expense ignore probable value expectation yet also several principal parsimonious suggest information high live subspace dimensionality less find practice combine involve possible integral analytically without simulated curse conclude model base discriminant implement discriminant applicable class
arguably inferior avoid fail put configuration sampling limit degenerate zero avoid respect mixing specifically marginal two quite little iteration method approximate randomly generate rapid mixing approximate compare chain calculate marginal exact tree project divergence avoid algorithm minimize lead exact useful belief propagation
drastically varied way ground example assumption deal metric dataset reconstruct query break neighbor one digit figure able recover density axis agreement axis globally query top behavior unweighted know short path near neighbor region ht life production fall place finally metric
cross parameter compute predict classification heart covariate set covariate observation hold ten see result h logit breast heart subspace something table state ccc post class dim ccc dim post present infer subspace sphere utility sphere angle avoid represent different dimension embed allow sampling well dimension great efficiency model acknowledgement would useful acknowledge grant biology gm grant dms mathematics institute nsf grant institute environmental health national health proposition corollary
towards goal issue address understand generalization appear functional room thus h notice come size rhs eliminate qr proof aspect question answer recommender estimating automatically training often reduce issue design effective rank functional correlate metric evaluation technique loss wise loss weight derivation piecewise loss theory markov field answer web retrieval answering recommender typically keyword return rank relevant potential recent survey approach automatically train relevance context profile object estimate query return object reflect query mathematically set goal sort list
semi nmf ill pose optimal nonnegative semi nonnegative rank initialization nonnegative factorization q frobenius nonnegative use focus nmf fact column input matrix decomposition column interpret cluster centroid reconstruct approximately column indicator discussion nmf motion super hyperspectral nonnegative also nmf compute semi rank usual nmf rank unconstraine r unconstraine necessarily singular abuse language rank sm procedure nmf decomposition svd exact nmf sm vector column sm nmf irreducible generalize belong class well often semi section semi already light nmf much computing nmf nmf ill pose exist
compatible ignore site copula raise influence typically site ignoring author impact alternative establish asymptotically joint instance comparison censor historical copula case study context g highly maxima development peak still active spatial modeling would allow joint advance give cause expect future development peak estimation semi parametric generalize pareto margin augmentation enable inclusion censor datum apply france historical objective historical quantile site model version ignore historical local strongly version illustrate extend historical implement estimation ignore historical quantile result likely complete
pathway conditional type model differently type protein dna bind co attribute data domain main obtain aim measure physical protein bind reduce function pathway count edge false positive aim measure factor dna bind peak seq bind event bind event mixture pathway bind whenever distribute nan hypothesis relate beta change gene variable existence direct gene gene mixture normal standard deviation expect
market index conclude research appendix paper analytical separate univariate structure correspondence multivariate copula construct flexible f normal possible term copula consist part multivariate dependence principle margin margin split distribution flexible asymmetric normal special split degree freedom skewness margin link covariate covariate margin margin split model link mixture split therefore split also margin multivariate involve quantify correlation variable usually methodology dependence extreme express bivariate principle dependency attain copula specify whole copula strong relatively fr hoeffding bound help leave
similarity fraction overlap chance feature calculate index health patient total model heart failure top list table discrimination respect auc hyperparameter auc hyperparameter control select
step variable lp double denote derive necessary sample w ij w j ratio competitive permutation ratio cr table average ratio report take matlab linear matlab toolbox ghz cpu gb ram problem bid match achieve ratio datum table suggest order achieve despite competitive r cr cr
identifiability model uniform mutual undirected graphical modern probability underlie capture aspect task partition computation graphical case markov variable ise long history start spin domain finance computer processing sampling significant relatively easy hence largely liu paper give markov node weight tree thus find liu mutual graph loop much reason neighbor indirect correlation distant discuss next subsection structure ise node search possible neighborhood whether imply conditional hold show greedy
study merge property propose dissimilarity agglomerative modeling aggregate agglomerative heuristic first one one new cluster denote merge cluster model less explanation accord factor infinity weight leibl bivariate brevity mainly rewrite sum kullback good choose merge dissimilarity infinity dissimilarity leibl difference property leibler impact unbalanced merging different code plus principle agglomerative cluster merge successively cluster dendrogram dissimilarity measure dendrogram euclidean criterion dissimilarity measure dendrogram trade merge similarly merge agglomerative dissimilarity partition sense less distinguished
kernel change frequently case large sign away mapping magnitude depend consume prediction produce rbf network network compact reduce still quantization approach idea constrain growth quantization quantization codebook size codebook vector close input quantization adopt codebook size allocate similar code add center size update convergence nonlinear regression give convergence successive converge noise zero sequence independent ai assumption two steady
rectangle rectangle right green rectangle dot red color width join central concept nash nash play equilibrium precise equilibrium perfect agent strategy ne discount solve formulation cause problem significant global via conference detail formal proof presentation sub ensure bellman particular agent game equilibrium henceforth refer sg sp game sp direction carefully choose descent point sg sp propose aforementioned ensure ne follow centralized base scheme assume known localize one running require action observe policy strategy irreducible rl good converge self play discount aforementioned desirable property follow economic learn play convergence nash meet unlike discount possesse guarantee style node cm policy actor bend east bend leave bend west employ actor conceptually nest operate fix value update minimum mention simultaneously albeit vary loop formal considerable ode previous paper usually show track ode reach solve hand adopt show
risk road road reason shorter narrow road scenario policy go expert make previous narrow road eventually fall share provide policy convenient deterministic fact coordinate frank collect explore distinction policy execution action interestingly go expert approach theoretical justification previously conceptually make policy expert go collect current detail alternate available addition general policy spirit dynamic programming proceed provide learn time step potentially time justify state ideally policy expert distribution policy determine exploration proceed follow current
reaction prescribe law proportional product reaction eq reaction reaction rate reaction bind quasi constant simulate use numerical integration step formal analog perceptron input similar early perceptron capable feed network model produce external internal analog yes analog control weight consist vs four implement weight rest species weight target input
generalize limitation accord theoretical spectral frobenius yield bound thank david nsf grant u advanced project social communication program agreement number view conclusion interpret represent great follow term great term choose third probability term less great formula third less probability eq permutation
information ground segmentation use combine mt trend accuracy stage stage division essential dataset image method mean roughly bound box compare method baseline obvious combine consuming performance well without trick feature study ignore svm vs training image explore vs one besides class one class image compare little unable supervision class imbalance improve increase capable stanford specie use
prune eight wise column wise eight wise call calculate full version operation ccc ccc ccc c shift add shift add shift dct definition dct l cc nz nz chen dct image employ set
negative approximated therefore performance necessary base population goal notion distance clustering well know notion population clustering recall method produce space deal clustering survey alternative apart proposal partition interest lie develop notion population clustering clustering moment introduce distance measure hausdorff set compact try proximity set set refer content identify although return hausdorff approach primarily practical seem cluster desirable assign portion correspond close clustering sensible distance cluster counterpart component minimization linear problem comprehensive assignment resemble adapt possibility add describe equally possibility
feature contain four attribute wavelet transform skewness wavelet transform apply method point cluster examine confusion purely apply uci machine repository white detailed ph score score blind range bad dataset rule selection pick sc plot figure notice fourth appear sc visualize involve rating scoring mode confusion cluster interpret like large center cluster good inside cluster score seed uci repository seed author mode mean type soft ray technique image image first normalize bandwidth pick confusion parameter seed
cluster ellipsoid hyperplane solve format particle search well programming particle constraint maximize publish simplex solve use simplex hard could ellipsoid algorithm simplex prove iteratively ellipsoid barrier interior one area way label store near learn optimal formulate technique solve svms bag neighbor model network approach nature minimize
effort level know maximize agent contract observable agent contract long minimum work ensure effort principal effort observable principal pt theorem complete microsoft nsf finding conclusion recommendation alone crowdsource market platform matching complete particular set quality dynamically adjust whereby effort observable pricing armed bandit represent contract cope propose new algorithm contract space treat single arm discretization adaptively region contract eventually discretize sublinear improve discretization compete several crowdsource market dynamic pricing subject descriptor analysis computation abstract behavioral crowdsource agent pricing armed bandit regret crowdsource intelligence complete computer alone crowdsource market microsoft universal relevance pay worker course human worker human english require less effort produce dedicated make sure complete properly encourage payment worker valuable view worker contract specify quality examine dynamically quality task evolve payment specifie payment output worker market whether task effort level observable observe worker worker contract contract worker formally expect make worker choose effort effort complete incur bring quality worker observe unknown worker decision treat dynamic contract mab problem arm represent contract structure algorithm region choose region treat promising area action promise general discretization appear discretization observable structure analyze propose width horizon round bandit illustrate via corollary special low theoretical worker choose accept task special dynamic pricing prior practically crowdsource market novel derive corollary specific
person voxel world resolution camera compose calibrate video moving employ voxel frame rate second voxel figure frame voxel voxel set quite sequence motion link nevertheless qualitative evidence able sequence truth could compare seed pass ensure perform time k time analyze eigenvector desire embed neighborhood resolution voxel finally topological assess assess segmentation produce analyze ground truth voxel link model voxel g figure work firstly body compose cut link belong unsupervised contain secondly body counting segment unsupervise label retain voxel obtain score take cluster segment score finally obtain segmentation store link voxel link histogram proportion cluster segment body p run consistently six comprise turn type boundary unsupervise lie correspondence score priori score sequence link rather point embed bend correspond consistent clustering dramatically bad segmentation segmentation appear relatively favor arm
summary say theoretic reduce sufficiently theoretic noise reduction formulate regularize solve computation illustrate propose future extension support aid scientific aid scientific research
low sake find aim work finite observe receive reward variable good suboptimal arm sample recommendation generic identify arm eliminate batch elimination depict integer elimination maintain remain arm within use functional sample define later arm functional proceed generalise sequential arm eliminate round k sequential eliminate remain round round arm
reject retain see result vary vary lr statistic confirm obtain parsimonious gain note report likelihood good model information concern note practitioner randomly close lr assessment adjust eigen orientation ratio version remain without reject motivated consideration extend test multivariate family eight gaussian extreme configuration estimate requirement beyond seven test likelihood ratio overlap
fig unseen semantic neighbor semantic class unseen connect class exploit unseen bipartite semantic bipartite weight store parameter shoot connect bipartite direct class unseen short graph image probability unseen fig classify unseen end unseen class start terminate class unseen class include bridge connect unseen unseen computed shot find label high linear neighbor unseen make unseen
c c c task visual system sequentially promise biological sequential step far propose internal image terminate return presence measure compatibility evidence region step set image region action three decide terminate four maximum termination learn sigmoid search select target image offer bottom effect evaluate region define evidence use normalize bounding box corner policy express execute repeat action detector bag maximal strong supervision find model maximize confidence
width width pt rgb line round line join pt color pt pattern color pattern node shift point note think intersect study translate relate distance whenever close ball center ap angle pair face along ap ap ap bound later point bind angle face statement hold replace corollary angle relate certain convenience though implicit characterization subspace idea connect angle eigenvalue orthonormal number
diversity gain pair decrease pair proportional rare analogously label cost label transition count coverage
single point different concave concave identical induce set tuple sort magnitude within break know boundary active subject either convex subgradient least know construction differ I
also large matrix center sparsity store plus low format ideal rank package present demand empirically consider estimate rearrange proportional deviation get eq say likewise iterate four residual zero fast slightly matrix practice subset omit center skip iterative guarantee center center likewise row center unbalanced way gauss certain degenerate guarantee formula simultaneously learn problem solution alternative solve row costly solve operate strong advantage time early make imputation alternate iteration hence acknowledgement thank author discussion lead center dms national science foundation begin concern regression eq follow observe bind clearly
knn strong often multi modal result use framework adjacency graph many contrast edge local neighborhood superior robust graph sl ne neighbors consistent requirement knn learn constraint four knn sl ne sl comparison list discriminate grain subtle grain category vs vs effort optimize sl embed high accuracy knn metric organize propose similarity sl ne discriminative similarity sl ne learn use relevant covariance matrix tie mixture localize fisher affinity note neighborhood dynamically update metric sl neighborhood
augmentation nearly experiment library reproduce acknowledgment upon national nsf research acknowledge code thank helpful discussion suggestion institute advanced technology il edu recognition number recent advance augmentation dropout improve unsupervised advance incorporate recent unsupervised analyze unsupervised augmentation cifar unsupervised supervised finding discover pre unsupervised surprisingly ratio unsupervised color training deep linear unit augmentation convolutional cnns
x x algorithm read strict lagrangian augment generate respect separately long augment lagrangian alternate lead direction back r ax ax r ax prox r ax r overview parameter consider admm e convergence averaged find program depend variety course program addition first admm modification inexact first handle parameter parameter vary quadratic augment lagrangian replace receive r ax ax r respectively dependency flexible monotone admm also consist find consider triple understand subdifferential maximal monotone vi problem solve new parameter admm vi ax ax equation sub problem study reference overview find want see rule subdifferential calculus find minimizer nice minimizer prox point rise prox r prox recent minimization prox prox g algorithm dual relation algorithm first hand equivalent rule add apply step prox iterate give definition general solution linear equation avoid modify step taylor method alternate relation straightforward equation x using rewrite prox several modification basic linearize admm multipli primal hybrid primal
approximation diagram region provide characteristic affect correlation standardized fluctuation feature divide two feature irrelevant feature note coefficient relevant ratio standardize always strongly regime order give general feature selection statement satisfied gap relevant correlated note indicate regime accurate sign much strong estimate absolute approximation square rmse probability function vertical variable red increase pearson coefficient health measuring special pool http edu percentage body regression body index upper arm purpose blue red pearson percentage body mass index red figure demonstrate
slow dl present supervise net direct early supervision constraint formulation performance exist justification stochastic gradient function loose pointing promise particularly worth wise minimize classification reduce error individual backpropagation unsupervise semi carry deep classifier instead standard softmax function framework softmax classifiers supervision softmax cnn softmax mnist cifar cifar technique maxout careful main style introduce classifier model layer early combine dl
object track use tracking collect structure represent location associate descriptor frame typically template calculate distance parameterize correspondence tracking frame template adjacent frame construct task adjacent frame learn collected frames w column concatenation part process template learn verification structure compatibility scoring method compatibility specific eq transform template spatial norm
write q censor augmentation manner censor ensure form nature conditional standard see make predictive want r rt us hierarchical generalise becomes hierarchical sample censor r location want use sample use subsection briefly describe ahead temporal ahead forecast two observe want temporal predictive calculate forecasting already forecast predictive daily fit discard mcmc chain converge quickly brevity mcmc table autoregressive autocorrelation successive autocorrelation reflect fact skewed simplicity spatial decay ci vary km spatial dependency
iid distribute n v generate loading noise performance z repetition loading fit range fit fit success fit ordinary correlation correspond effectiveness regression ridge regression surprising ridge look though bad focus essentially give naturally suit canonical cca center cca maximally correlate mathematically cca constrain close corresponding role cca tool find common
duration give effective mass barrier produce effective axis auto dynamic purpose delta intend dynamic sensitive convenient follow stationarity energy analytically solve constant write ref dynamic transform ref place motion reversible explicit force problem hamiltonian hmc table effective size force duration mc mc effective center table indicate propose hamiltonian method sensitive effective hamiltonian effective sample force evaluation c employ hz vx I
completion possible agree produce completion span clear nonzero discard residual zero validate em drop incoherence validate disjoint column state correctness completion whether agree ideal incoherence
generality classifier reduce rather volume reduce per classifier cross computational problem enable execute brief present brief overview classifier motivate test discuss classify reduce
hide observe use variational illustrated collapse parameter maximum support annotation lie bag positive bag negative necessarily level absolutely bag false practice instance bag svm constrain hyperplane maximize among bag use principle max sim softmax softmax sim yield max score super partial annotation different transform bag bag label logistic stick link instance maximization generate consider independently relationship solution pl instance address aforementione framework instance membership single instance deal miss amount generative outperform develop discriminative annotation finally like maintain bag bag dataset bag bag instance include exclude annotation bag bi bb b c class simplify notation annotation map graphical illustrate notation bag instance bag bp next label th feature vector weight th practice indicator take argument discriminative
attribute dataset attribute input root square error convergence part cross validation rmse ten example improve confirm theoretical exponent significantly quadratic growth rkh lead compare proceed grid vary rmse select ten fold grid likewise validation logarithmic spaced rmse std important validation take capable achieve good performance decrease practice range ccc rmse std
partial remainder roc analysis roc false various threshold curve unit variable two diagnostic population convention patient great therefore roc give give survival function auc summary index define conjunction carlo valid
distinguish input iy label generation term normally variance inference proceed impose specify typical non squared rule simplify process integrate term free function function variance cdf likelihood couple function adopt equivalence probit accept latent role slope cdf sample describe next section paradigm unseen instance exploit additional need
kernel fact less various depend assumption non partition hence method jump boundary effect cast multivariate represent expect case express information still nevertheless investigation generalization investigate statistical inference respect estimator consider profile profile square fan investigate whether limit hypothesis alternative x fan profile regularity nan q test chi square fan demonstrate testing nonparametric constant price introduce nuisance parameter statistic bootstrap j j j bb root component nonparametric component categorical efficient series example
focused task alone without example relationship norm bayesian inference via nuclear popular nonconvex minima nuclear norm application domain hull rank far generality finite dimensional seek optimize ex ex nuclear index unconstraine feature take approach illustrative mn mn kl mean constrain direct storage inversion become infeasible prior computationally feasible approach optimization cost independent gradient gradient compute use collect term convex unique identity function replace norm regularizer equivalent ex ex ex valid variate posterior representation theorem parametric function ex ex strongly rely kolmogorov extend use extend consistent covariance sample requirement kolmogorov extension training set covariance trivially representation optimization addition index
couple beta cause peak mass near yield tail express gamma gamma hierarchical layer tb member represent mention know half cauchy consider normal layer beta near zero cauchy compare distribution tail inference calculation intractable inference utilize expectation unknown subsection
heavy kl member associated member regret whether entropy simplex motivate look closely log sum conjugate property similar sum series go tool able express bregman notation analysis semi continuous suitably relative make bregman divergence subtract
propose consist straight tuple systematic position architecture involve tuple also investigate tuple move circle place piece piece k player perfect piece white black form player put piece color consist place piece diagonal piece piece piece pass game pass piece position reason become popular domain intelligence goal player evaluation player simple current player apply position desirable determine move play tie position piece counter
outperform classic tn tn step square svm paper acceleration proposition sequential direction abstract overview write style research introduce unconstrained subspace optimization usefulness signal compressive imaging machine explore combination separable method obtain art plain trust region easily invertible truncate nested improved merge accelerate lagrangian constrain science dimension use interior applicable problem exceed several become possess bad fista
utilize density functional probability induce shape maximizer excess probability induce geometric information shape gr class comment reconstruct probability avoid literature set assume priori disadvantage mode ms value analyze top vertical clear unimodal ms ms
label q ray stand usual significance range give complement interval proper stand observation cf first centre see divide residual well e presence high iid probability iid finite statement sequence iid finite infinitely many strong number condition remain combine fact bound contrary infinitely e imply
amount compare rnn actor actor task difficulty actor actor question object already statement question oppose rnn table conclusion similar outperform rnn support time c difficulty actor actor actor time conduct answer sentence answer rnn module rnn fed rnn effectively feed output module see difficulty object module feed module generation perform choose correct correct question answer incorrect answer incorrect believe correct indicate strongly rnns variant rnns c rnn lstm com reason component long jointly write context answer long effectively act dynamic toy task world latter reasoning support sentence question understand machine way read write large long memory combine modern
effective number test correction define eq controlling hence effective particular significance level label although iteration overcome drawback consider subgraph since ignore subgraph control permutation eliminate subgraph throughout recommended occurrence investigate gene study database subgraph large distribution study membership drawback threshold difficult graph knowledge subgraph use feature correspond informative subgraph subsequent view false positive detect positive domain correction prominent correction correction exact test correction expensive
name collection average sequence converge linear th convexity turn
seek output implicitly primarily interested approximate employ evidence dimensionality present common wide nonlinear vb density model parameter iterative identification information theoretic determine dimension demonstrate problem mechanic relevance call assess motivate mechanic describe equation calibration readily forward utilize engine response quantity measurement include material behavior coordinate configuration static case gradient lagrangian tensor primary consist body force second stress tensor material value point stress find aforementione condition encounter material far solve analytically vast majority resort numerical field prominent employ well interest reader book available term solution algebraic equation r r mesh shape discretization value frequently assume finite number discretization discretization aim infer variability discretize fine might
queue child advance resample particle child child simulate particle leave queue fully I queue particle create child release queue seek queue full instead particle particle single virtual keep track propagate particle multipli average account preserve reach complete particle reporting initialize particle initial constraint operating optimization demonstrate overall validity particle hmm associated emission second linear gaussian
brain graph hence graph degree normalize improve section colored brain region additional node plot brain spectral rsc spatial ba tuning value ba yield spatially cluster densely spatial spectral densely coherent regularize spectral two covariate cluster less node coherence increase uniformity size demonstrate spectral cluster brain regularize govern region brain brain brain graph treat brain covariate discovery highly brain break visible importantly approach align region connectivity ba rsc ba adjust ari quantify partition cluster brain partition sc covariate spatial spectral spatially brain different configuration ba ari brain similarity spectral partition alignment brain conduct
bilinear extract five store recurrent allow several appear record break neural demonstrate effectiveness store layer major two secondly store image rnn model word utilize capacity achieve recurrent strategy lead good evaluation tb start sign model layer word recurrent multimodal softmax deep cnn frame view recurrent type layer layer activation recurrent denote current representation representation vocabulary calculate vector element learn adaptive connect accordingly backpropagation propagate recurrent time multimodal recurrent rnn two recurrent multimodal softmax word embed one encode syntactic relevant word euclidean
phrase exist perhaps label review name movie sort infer sentiment word sort sentiment embed compare belong five subsequently choose highest predict output negative class ignore prediction test high negative use strategy achieve instance achieve precision neutral approach boundary sentence fall sentence multi learning obtain transfer learning require supervision obtain approach
propose probabilistic valid correctly therefore first low purpose examine bind obtain posterior vb leave one solution sampler leave manner inference normalize constant namely serve pseudo pseudo likelihood figure evident quantity converge iteration naive low increase proceed decrease unfortunately update one cost extra computational would technique inference monitor solution low technique emphasize technique applicable nd equally burn gradually decrease completion burn variational th variational burn ratio fall predefine final anneal concern evident take iteration variational eq automatically difference posterior bayesian clear whether true stationary average stationary want stress remain unknown true solution point convergence knowledge naive solution user enable speed one drawback vb dynamically cardinality computational sampler maintain heavy computational cost intrinsic simple ignore inference cluster accordance stick cluster avoid evaluate truly set intrinsic complexity proportional experiment implemented eliminate cost vb full variable eqs need expectation apply object remarkable relational almost beneficial
type tuning tuning selection challenge also domain interpretability assume involve arise biology super network may role fundamentally degree connectivity scale network fact exist free accommodate dense intend entirely almost order encourage effect enyi formulation accommodate idea estimate model graphical work graphical convergence degree whether graphical lasso network perspective publicly author acknowledge nsf dms fellowship nsf mf mf sl appendix recall scale augment lagrangian follow derive update separable update update depend address respect lagrange multipli bit algebra c permutation row
one achieve learn classifier favor body denoise autoencoder know stack denoise autoencoder able across allow domain explicitly learn identify origin neural include hide towards connection predict membership objective domain adversarial confirm experiment toy performance neural network performance representation improves rely robust label provide marginal classifier tackle adaptation approach method intuitively risk focus
description stable spline system property highlight alternative close factorization highlight computational associated computation stable corollary university united ac uk impulse process spline impulse surely stable entropy property stable spline independent
inverse statement true universal barrier describe time distribution define via technique simple computable barrier universal barrier make optimal briefly whose lebesgue family inverse mapping onto strictly elementary recover exponential family differential entropy barrier moment x easy exercise partly function satisfy function
word lda early vector use category clarity index subscript style comparative classifier grind end respectively consider date time different distance distance high distance subject matter similarity showed figure figure lie dimensional dissimilarity denote geodesic along short distance short ht similar ht mention similarity define need encode work define influence influence strong similarity mathematically define asymmetric distance link influence tendency measure alternatively think hausdorff measure link hausdorff unlike point point view context close capture take favor vary spectrum asymmetric hausdorff evaluate
overcomplete tensor optima characterize true efficient guarantee establish moment suffer spurious optima long overcomplete spurious optima especially grow compare dimensionality recently overcomplete random incoherent tensor component orthogonality correlation true model overcomplete hide time dimension improve analysis iteration initialization require mild overcomplete spherical naive bayes multiple view conditionally hide assume component convergence mixture mixture model spherical dimension vector sample correlation overcomplete model direction almost noise establish level gaussian low use spectral clustering impose tensor orthogonality separation additional work model satisfy noise provide cluster spherical mixture mean vector tensor turn tensor
solution example bound strictly semi online cost stream expect viewpoint reasonable rough dependence semi order operate set number guarantee learn powerful par neural example intuitively assign surprising therefore powerful importance recognize retrieval investigate argue require online yahoo user
degree freedom dynamically tolerance tolerance low tolerance negligible change step curve iteration current find value matlab find solve degree freedom I use contrast discrepancy mdp lc average regularization parameter final one ms indicate c mdp lc result mdp lc indicate entry lc invert contaminate regularization mdp respectively right either consistently poor except either respect appear noise
analyze rewrite proceed algebraic relate j
adjacency regular regular map perfectly regular never always distribution world eigenvector informative alternate time update link include hyper induce ignore number obtain tell permutation affect generate associate row vector look mathematical describe particular find motivate section encodes graph informative contain small perfectly eigenvector always fluctuation connection example equal compute fact eigenvector property polynomial expression recover polynomial equation form theorem value although may hard characterize many satisfy reasonably think sharp concentration adjacency random power eigenvalue obey capture different polynomial expect different link spectrum far know graph triangle
one number item default worker maintain simulation task worker b behave differently increase worker worker accurate boost know worker reliability htb variant specifically task bar iteration step figure accuracy fast take step run similar confirm change omit htb strictly worker simplicity compare worker violate toy suppose worker true error achieve rate misspecification misspecification extent publicly sample label independently vary see error clarity figure result usually among class worker web search first identify worker label potential e expert htb ccc visualization done conduct label independently vary proportion vote em label generally dominate run label maintain compare em bar impose label achieve algorithm misspecification worker set complementary worker example misspecification ccc percentage assignment increase task error impose bar language collect worker task e question worker binary second sentence label pair similar different assignment
study linear bundle formally utility generalization decrease separable utility call decrease concave piece denote segment specify rate agent derive good define applicable utility proportion utility next economic capture range substitution utility assume let h far behave utility derive zero regardless complementary utility set precision define rational size formal utility reveal preference introduce label measure multi say distribution informally amount low error statistical vector input bp output pair learner consider learn say reveal preference price utility bp reveal preference correspond demand sense actually notion query learner determine utility instance access query price vector bundle reveal
hilbert sensor sensor infer sensor formulate generalize design trace expect value character seek solve measure pde solve independent parameter facilitate construct posteriori randomize trace include characterize describe posterior vector configuration penalty newton adjoint pde function elaborate determine sensor configuration coefficient pde log medium flow pde require gradient essentially candidate location quasi newton exhibit invariance design trace q bayesian nonlinear inverse govern sensor experimental collect field minimize sense subproblem experimental challenge particular discretization map require underlie problem turn forward problem algorithm maximally exploit tractable large scale state datum include development concern inverse problem increase govern numerical nonlinear pose large frequentist objective finite inference amount condition address mathematical incorporate operator infer objective entail optimality e second order I scalable cost forward pde solve effort area include author sequential quadratic different criterion govern nonlinear system equation govern paper
assumption discount establish govern almost surely project bellman equation bellman projection detail transition component diagonal matrix diagonal policy eq aim asymptotic bound quantify convergence td well td asymptotic challenging reason fix bellman underlie mdp influence come mdp difficulty number mixing occur td start stationary rate amount underlie approximation order
find persistent diagram one persistent mobile sense environment distinguish assume motion agent subsection weak localization probabilistic movement characteristic group behavior sake behavior individual boundary specifically boundarie orientation characterize line change average velocity line segment length angular isotropic switch velocity follow leave central partition model second tangent boundary rw movement agent period time movement stop model memory
also normality expansion specifically formula taylor residual
immediately measure frobenius logarithmic rate bit minimize max matrix completion consider likelihood family instance logit belong bind achieve frobenius extent determine minimax integer infimum let observation bind logit probit theorem lower square low proportional multiplied sample
exploit word machine baseline correlation embedding correlate cross bag gram like possibility language amount language allow build level corpus explore use autoencoder language word representation align language rely simply bag align language quality since autoencoder word certain correlation maximize regularizer improvement empirically classification english approach state art percentage good english annotate processing language annotate sentiment already english language situation language dominate digital content elsewhere however
term fw sum lipschitz derivative explicit bad case pass oracle may access copy subset I
representative inherent challenging goal control capability recognize significantly hard difficulty complex linear noisy wrong expensive controller reinforcement orientation keep close episode location stationary absence wind episode end pre specify whose cyclic cyclic collective collective adjust pilot around axis design use action detail discretization process apply adjusted action discretization dimension equally sized interval spread process decrease agent sample transition batch transition representative representative state interval transition episode length decrease since step execute interval decrease episode clear significantly cut episode approximately argue nothing slow part consequence performance episode computational peak model begin considerably come warm start policy update section compatible computational task version address literature algorithm guide datum evaluating problem definition start discuss theoretical guarantee assume uniformly restrictive collection direct interaction environment strictly indeed relaxed assumption reward require policy collect choose section good discount case assumption regard kernel unfortunately usually empirically speak problem perturbation perturbation balance task problem like task representative result computational crucially ideally state would row hull contain state representative state problem state representative experiment cluster state center representative show section possible simple perhaps seem reasonably together average resort quantization approach update learn fit state prior know reasoning task policy vary space regardless representative add representative use representative resource strong task varied course good problem theory practice decrease number transition admissible proposition
ty ty conclude alternate prove rate convergence hessian descent order ie ie w w subproblem step lasso problem solve package coordinate p w ie ie k theorem proposition corollary stanford university department department stanford graphical penalize though scalable advance optimization address pseudo base propose minimization
vb robustness model drop expect overfitte increase cause free em variational overfitte integrate prediction performance vb tensor miss vb stay approach link attract date restrict probabilistic order deal variational bayesian matrix method derive variational form relate motivate variable conjugate et naturally tensor tensor well learn maintain distribution put
rather forward g field vector modality early situation also detail one want try image box annotation occur recognition challenge incorporate strong annotation form goal per image object categorization similar supervision distinguish available first tracking use time might capture auxiliary additional information image available interest different kind appear literature modality learn compute share alternatively image fill improve object categorization show training image author term gaussian noise approach specific applicable instead applicable framework individual framework formalize classification multiclass versus rest training vector vector encode computer vision image histogram
source representative subsequently process subspace filter process slice sample dynamical observation transform spherical across proxy uncertainty information available datum resolution segment associate use divergence bregman stress uncertainty
mean index represent label label vision derive mrf labeling energy correspond mrf decade technique computer vision cut belief propagation convexity method performance minimize extensively study study conclude visual labeling statistic solution benchmark pair suggest complicate sophisticated high high clique potential handle difficult visual labeling function exist optimize mrfs arbitrary pairwise potential boolean approximate extend submodular type enable automatic transform provide graph cut property scope
reduce multiclass newly category rank assignment rank category person express like music create tie assignment tie complete tie occur possible extremely order important partition choose possible model tie intractable resort pairwise comparison separately denote replace comparison ic il I translate original preference drawback relaxation guarantee hope enough preserve specify adapt comparison il occurrence lead estimate visible dimensionality extraction
segmentation reach image g crf crf crf traditionally field smooth segmentation typically contain couple node spatially qualitatively clean spurious classifier hand feature classifier produce score semantic prediction qualitatively illustrate homogeneous result detailed conjunction range thin require discrete connect crf employ label assignment unary label pairwise potential otherwise pairwise matter fully feature pixel adopt kernel pixel color intensity hyper scale crucially pass
numerical original cover originally miss complete observation htb uci uci heart uci yes yes yes uci uci uci yes uci uci burn yes uci uci yes uci misclassification scatter plot none besides see majority accuracy dominate perform fall time acceptable accuracy dataset good accuracy hard scatter glm penalty reflect score p p bias applicable variable omit comparison give may dataset table lp se se se se glm model se se se se glm table bar dominate find apply plot se bar expect although model model majority rule simple ht omit bias correction correction visual se provide prediction categorical bias improve se se work general se se overall correction algorithm necessarily section negligible bias htb se se se se se se se se se section verify great interpretation ability illustration employ analyze section census uci repository extract census database original current survey characteristic census split training machine learn leave
handle depend label label third assume depend previous labeling process crf toolbox brief description label noun noun head noun noun phrase task assign sentence test parse noun phrase phrase etc sentence phrase etc choose segmentation meaningful text sentence etc word segmentation chinese assign denote begin inside word test choose sentence consider word sentence input component alphabet component template crf package property summarize
map patch patch input image rgb characterize feedforward number subsample ultimately cnn convolutional kn patch shape map filter parameter k p l define activation z operation convolution non pixel pooling pooling recursively hold b sufficient purpose approximate approximation provide use replace finally grouping multiplication weight pixel principle anti limit result give product z replace substitution z z approximation z k regard
understand although stress redundant merely mean change correlation even growth static distribution constrain type type scenario v virtual likelihood approximation systematic analysis virtual less static forest datum constrain fix calibrate growth considerably substantially constrain calibration derive datum expert calibration supplement table virtual supplement mostly correlation correlation particularly supplement understand hierarchical divide growth diameter high grouping structure strong interpretation compete difference parameterization evident value high growth specie point difference mid think systematic estimation reliable manual fit fixing
recover consist visually entity dc air dc air chemical family balanced test class confusion importance partitioning discover control top multiply improve base ground significant
calculation expect another propose signature variance n var signature small variance similar index denote base distribution signature boundary use sort absolute signature signature follow
outlier effect equally include inducing show boundary potential induce boundary boundary cause true induce boundary misclassifie precisely differently effect backpropagation mlp g great effect classification large particularly stage mlp calculate propose method mlp simple gradually increase deep localize assume contain instance induce instance datum set instance determine instance exhibit examine misclassifie misclassifie b misclassifie weighting instance motivate misclassifie
usefulness use error differentially express patient spectrum available selection want modern throughput often detect control number approach unstable automate bootstrap penalize elastic selection widely use network genome association study stability combination stability component specify descent introduction introduce section common evaluation boost section examine patient boost conjunction aim phenotype measurement try pathway generalize outcome response predictor covariate fit aim fit example square linear obtain analogously
vc exist l rhs entropy logarithm number metric relate probability least thus equation eq entropy upper bind rademacher originally n lx lx thus upper bind solve suppose approximation give hold exist finite n whose na bernstein arithmetic mean least focus event auxiliary n n q xx maximizer action x second third x x likewise write get optimizer apply finally n n n n event least lx satisfied space add lemma l r c minimize property bernstein n e arithmetic l separate term l inequality upper imply convenience bernstein number random pn range functional fr ec minimizer however error prove l least estimating problem exponentially supremum policy evaluation extend similarity proof considerably constant x bernstein appendix show arithmetic focus inequality auxiliary
eigenfunction langevin case dramatically eigenfunction operator langevin indicate weighted outperform langevin index eigenfunction less align cs observe th produce rw slowly sampler plot chain provide proposal shorter langevin rw langevin rw sampler direction smooth eigenfunction langevin langevin snr significantly outperform langevin high eigenfunction langevin rw langevin langevin proposals comparable parameter surprising move weak weighted proposal well langevin lag autocorrelation b star symbol langevin symbol langevin lag langevin stay constant prior eigenfunction lag langevin eigenfunction low average langevin lag eigenfunction operator proposal brevity h useful quantify impact contrast weight proposal posterior
document represent high lda first network capture document however relation representation representation well visualize similarity measure topic represent capture similarity topic word kullback comparison triangle define base complement version kl divergence employ hellinger hellinger q topic network node represent discover topic similarity topic edge even collection place matrix format currently employ format store nevertheless parallelization implementation computation break parallelization core structure case cell column row word appear key key comprise tuple map key input key value topic appear operation complete expression hellinger every represent simply sum pair topic hellinger multiply subtract network
open bias fix train compare wikipedia token build occur corpus word technique
outside control kkt center combination multiplier boundary solely support allow hence ball triangle centroid radius f density cluster account method ball ball leave ball cover since cluster address inspire always mode shift centroid well formulate lagrange multiplier kkt lr recalling reformulate j programming towards name difference right model worth play role would enter actually reduce cluster discussion modify triangle encoding encoding kind encode pilot experiment conduct particularly atom randomly face encode learn encode encode atom corresponding feature last largely understand phenomenon plot treat patch atom one local appearance
normally direction model discuss calculation normalize existence section idea regard likelihood goal probability network degree consider node minimal parametrization parametrize q statistic arbitrarily remove sufficient information non normalize constant calculate directly normalizing correspond nan graph equation imply pg k n
definite one easy without identifiable matrix induce focus model simplify pair minimax known let minimax bound ml three setting pair bound combination technique main technical constructing packing semi induce pack cc main packing constant ensure pack detail impose constraint bound gradient negative log f cauchy recall induce put arrive effort convexity parameter et main focus walk case theoretic study analysis however discrepancy minimax us estimation error ignore specific see eigenvalue standardize
assignment identity matrix specify censor observation e survival person age interval specify great point tracking become standard rbms ordinal ordinal observation order observe threshold read offer alternative ordinal rbms categorical observation category associate category I large utility fix treat logit suppose categorical observation observation ask team person pick without z z translate rank category stagewise well category illustration particular tie impose rewrite z z mcmc inference alternate eqs inequality limit assignment remain unchanged due conditional independence
policy require induce inner actually hilbert allow function ms exclude however probability notational restriction practical future policy accord learn future value examine possible distribution mdp mdp episode time let make reinforcement sequence episode incur regret th episode q note regret expectation review thompson reinforcement later regret begin mdps episode
multivariate covariance allow size sum simplify kind kind index force pair pair nine index kind six pair remain last expectation vanish unless euclidean rank invariant orthogonal distinct way quadratic analyze sampler invertible identical choose loss put see carlo powers lemma term leave assumption expansion power calculate expansion enter combine substitute collect equation yield
set partition call subgraph consist undirected graph node chain line order fig b separation simplify induce prove provide type walk connect walk section choose suppose walk connect outside walk suppose keep non
eqs small ignore expression practically average increase flow cycle equal sequence variance cycle read length achieve probability great sequence nucleotide length base cycle nucleotide flow nucleotide probability right nucleotide eqs cycle previous distribution cycle two nucleotide perfect normal exact distribution long tail slightly shorter
trading literature short base temporal variation price stock without mainly limitation market individual portfolio optimization company level finance comprise mid portfolio financial decision company entire york stock new public manually handle company technique portfolio company stock
pass concave region expand bind since quadratic second ensure follow compute finally derivative prior range follow b c ccc ccc c ccc ccc c microsoft microsoft com draw discrete convert work sampling convert describe mathematical algorithm search maximum correctness evaluation closely rejection draw sampling predict work generic exact interest probabilistic build case desirable exist specialized rejection chain discrete return configuration perturb work sampler energy force resort observation perturbation infinitely many perturbation perturbation determine one irrelevant
correspond slow signal spike occur transform time series backward q small threshold yield indicator process clean spike objective step procedure network spike part fire observation make low additionally well activity context neuron fire activity signal output q spike case global activity processing simplify apply show figure neuron
subsample among versus across combine benefit production subsample census notable production fairly analysis suggest conclusion adjust month home researcher record across two benefit tend pool error benefit rely error population exhibit variable sd compare sd production age compare sd production number contribute substantial census examine communication demonstrate serve reliable imputation outperform modify incorporate dataset good experience rely potential default substantial sample appear misspecification default joint potential systematic future optimistic computationally time roughly likelihood large require much categorical clearly simulation dimension continuous likelihood mixture multivariate normal specialize exist leverage adapt extend context fully design fitting
dimensional except sign recovery observe sign sign prohibitive generalize computationally even check uniqueness application rotation specifically difficulty basis randomly select position rank dimensional decomposition principal problem name inverse efficient sharp inverse problem well study see elegant characterization theory modulus theory rely convexity functional estimation space convex present highly technique readily inverse early exhaustive lead statistical prohibitive problem solve prove case appear mathematics statistic generalize geometry notion low show width size study phase transition paper suggest geometry local sample ensure successful noiseless setting decomposable norm set focus detailed minimization erm loss excess risk subgaussian class proper localization radius convexity erm bound need
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dd trivially integral reduce consideration via zero multiple implicitly device determine theorem detail involve convergence k r obtain combination quadratic remain complex j employ deduce establishe state derivation conjunction r j n denote similarly estimator eq theorem quantify property estimator process demonstrate alternative frequency domain maximum likelihood converge representation provide theoretical estimator mis model demonstrate pseudo long distinction frequency domain domain accuracy replicate match overall double likelihood mis specification design domain long true primary secondary paper property domain sum square employ mis true generate long history series back dependent well
square goodness fit apparent goodness purely weight replica calculate inverse apply simultaneously figure equivalent give base accurate square systematically shift towards value potential review odd application discuss four area detection characterization scientific molecular characterization biology power deep understanding system scale range write dedicate j meet influence student clearly train responsible member scientific award university associate physics usa journal company experience entropy physical science search characterize receive start research biology biology max institute interest receive ph university usa laboratory california institute development temperature product interest include phenomenon development sense
various help evaluate vote necessarily adaptation optimization mean map training subset associate prediction function fusion majority vote measure function j j evaluating map rank mm positive prefer pairwise preference instance know performance map notion loss multiclass follow idea relaxation fusion pairwise rank pair want q force
character symbol language extend previous assume length message know advance code always know many last limited guess average string independently alphabet length string note string perhaps suppose random independently x simplex count vector count model form total count accordingly target mn poisson equivalent provide scheme string also assign string produce length use logarithm regret regret redundancy string q redundancy pointwise provide x become
k mn upper kernel operator l moment eigenvalue oppose th moment th moment x integral involve fourth dimensional learn grid agree dimensional diversity use weakly informative q extract color descriptor feature assign coordinate color sort axis produce histogram color process set sift descriptor descriptor commonly recognition image descriptor give sift sift feature histogram near cluster descriptor process feature scene dimension normalize norm combine dpp partial dpp dpp choose
consider input noiseless exactly evaluate expensive ir obtain initialization except nb portfolio ei poorly decision space ei es nb ei nb well treatment parameter novel theoretic optimization predictive maximize step since function approximate original entropy predictive evaluation produce entropy acquisition easily approximation es synthetic world es observe produce ei popular greedy decision tends result ei simple often get approximate path gp maxima derive approximation formally definite
screen respectively good dct table approximation structure assess coefficient transfer function relate could square expression select transform dct approximation thus comparison fig strong similarity spectral dct present actual
figure htb h htb construct time thresholded graph note screening via perform spectra series relation specify note threshold critical performing correctly discover series frequency diagonal covariance discover screening resolve synthetic pass gaussian band see complex screening correlation time chapter present time series focus identify thus statistical screening variable discover threshold positive negative significance quantify consider small number experimental validate screening partially fa circle corollary rgb rgb screening series chapter discuss series domain goal time time show time asymptotically statistically permit challenge series correlation screening accommodate fouri component degree regime specify threshold significance usefulness
nn sec sec v nn sec sec classification collect reference divided macro nn nn perform develop dissimilarity perform ds nn equipped euclidean differ third reference nn rule equip configuration label consistent additionally e genetic omit deviation sake classification gives expect inferior regardless system general variant affect significantly performance accuracy inferior r valuable absence operate search ds deduce l h dataset r consider configuration comparable reason yet require future deviation always small regardless demonstrate complexity cubic quadratic move effective computing calculate serial cpu show whole improvement exception especially speed properly ds reduce computing evaluation report specifically calculate demonstrate superiority cpu viewpoint variant operate synthesis employ process stream focus perform big operating especially I h involve
normalise give track candidate rotation determine track use retrieval pairwise report track track song accuracy precision ranking track interpret average precision track track query track compare accuracy distance across query adjust error comparison query precision determine song th measure transform distance distance apply outlier ensure rapidly track similar monotonicity rank combine average pool vary basis rank probability cover identity probability form utility versus straightforward yield baseline include correlation random examine distance measure relative codebook exception codebook size range improvement codebook size relative compression average result qualitative whereas loss relative appear advantageous compression whereas advantageous compression rely markov
e e p ce reduction design least favorable consider case j choose later b g ec j rip leibl combine low treatment omit p j lemma involve favor certain logical hierarchical focus hierarchy mean existence attract lot slow meet challenge big importantly study hierarchical difficulty sparsity type structural simultaneously reveal purpose strict iterate efficiency efficacy propose notice additive include effect adequate helpful sometimes behavioral science full n n hold hadamard pose plain variable
difference lstm exposure amount content gate hand control location input gate gate lstm unit new memory content control amount lstm forget gate information add tie gate difference alone type unit would well although preliminary translation apply motivate thorough lstm
profile concern depend profile candidate little benefit profile page short profile length candidate length correspond page something act short text acceptable accuracy candidate three profile word accuracy correspond text word case profile short text opinion piece couple page play claim accuracy profile correct rate achieve profile contain rate well statement evidence nr c text thousand rand c l text thousand rand l l texts rand profile length two length consider increment resolution text permit estimate short text attribute accurately experiment perform table last accuracy report end correspond page long middle act play column table correspond scene page novel article text text accuracy increase medium decrease mean decrease short text profile available achieve accuracy short text acceptable two four length
classifier kernel histogram overlap considerably reject scatter plot contour fit gaussian algorithm knn figure plot univariate algorithm post hoc univariate knn sort method low difference rf rf pairwise clique lda knn significant also post hoc separately lda order knn rf clique give information cc b b plot contour respect another type tp tn learn
incorporate common learn number one effort learn space number per metric weight information expensive prevent generalization near method learn attempt propagation unlike discriminative class code author website matlab code generate locally region select neighbor segment letter dataset uci reduce use normalize except letter split reduce overfitte training tune use basis dataset global global letter misclassification perform dimensionality high fast psd compare learn near neighbor c avg bold misclassification along
sampling carefully proposal low model operator standard probability rbms deep agree closely rbms compute agree full rbms one optimistic one encouraging suggest conjunction probability mrfs mrf boltzmann rbm mrf bipartite unit purpose exposition assume binary case distribution write q visible bias weight bias rbm v tr tr tr likelihood intractable exactly persistent rbm average v unnormalized intractable rbms
quantile measurable quantile function loss l use empirical interval figure follow accord along contain probability optimize conditional achieving contain variable intermediate interval intermediate prediction intermediate good approach aim prediction good training prediction class hinge use support set take thing empirical compare come mass around scalar trivially ideally know still conservative formalize equation capture let derive model residual good residual two member show know singleton contain construct define quantity depend capture section probabilistic solution depend able closely capture necessarily loose definition quantile outline quantile scalar eq robust equation conservative possible probabilistic guarantee sufficiently small empty conditional quantile function
improvement prediction u denote pdf evaluation point extract process function evaluation improvement consider permit underlie common define may variance process practical
dominate purpose follow respectively recall martingale generalization analogue uniform bernstein sequence uniformly probability therefore control martingale condition order iterate bound counterpart hoeffding inequality uniform constant p theorems identical appendix unchanged wiener far proof detail therefore defer super resp give low identical bound immediate martingale concentration without explicit basic
know margin penalization lagrangian problem primal reliable class assign eq associate positive class majority class solve tucker point rbf smoothness suit demonstrate much long rbf classifier difficult set reinforcement tuning method tune consume e rbf drastically classifier employ nest design supervise search determine close parameter
gaussian gp would derivation acquisition show six acquisition target surprisingly portfolio acquisition evaluation thompson mat ern form purpose sample minimizer need hyperparameter amplitude prior ei pi meanwhile keep thompson thompson namely report propose split equally among gp draw sample experiment evaluation optimum performance time performance repetition one standard commonly optimization three respectively
purpose text file consumption computational fairly long combine write library excellent convenience format file handle call currently file normally treat rest request call likelihood file discard alternate allele alternate retain alternate call miss variant alternate possible code support allow arbitrary name handle also improve expand mini manual search mini manual associate keyword keyword comment issue comment analysis trait respect quantitative transmission likelihood advantage speed increase meaningful procedure al robust certain permutation applicability law combine vast speedup decide rate table collect
improper obtained finally use walk rw acceptance likelihood mh gamma proposal moment instead prefer walk rw conduct specifically log recall ratio approximate mh metropolis gibbs fashion primary appendix update memory far distribution amenable gibbs particular use dd may inefficient less regardless refined method truncate little coefficient approximate use one variable sensible necessary value big context sampler backward observe length current reverse since sampler acceptance section primarily auto acceptance e mathematically convenient determined although assume short motivated specify precisely infer parametric bayesian modelling alternative primarily relate alternative model whereas approach allow computationally spend towards primary interest develop efficient properly reversible jump mcmc chain within marginal integration therein previously apply
svd attempt solve local optima lack relaxation problem notation nearby error notation linear constraint heterogeneity quantification know exactly e entry toeplitz element measure high metric mean error huber difficulty case structure exactly tractable formulation convex relaxation upon nuclear matrix singular nuclear vs norm choose large small nuclear robust problem dense theoretical next plain effective come weight describe relate
outlier amount event scale detect specifically separate separate tweet cache due lack group tweet take square square pm indeed semantic link event tweet quite text mostly term separate term square cache cluster mainly similarity graph cluster mainly construct two similarity similarity computational graph total filtering term popular frequent substantially affect reduce algorithm largely also filter scale event term share pair similarity cell compare keep computation due daily stream area york city experiment take second finish construction similarity minute code mid core due social medium online decade rise series research event user event use popular platform twitter attract significant due early approach detect specific rely stream keyword indicate wavelet developed well meaningful reduce noise medium analyze event tag associate propose temporal spatial tag author tweet measure detect keyword temporal hierarchical procedure twitter temporal similarity tweet co occurrence keyword
optimization guarantee find optima consistently estimator convex year advance toward zhang zhang show optima leave find optima fan et optimum wang establish guarantee lie substantially simplify work establishe agree well regularizer advance nonconvex problem agree establish recovery understanding point nonconvex objective objective technique variable tucker optimality primal state certain class nonsmooth theoretic smooth possibly nonconvex regularizer allow nonconvex regularizer suitable mild condition early additional minimum signal strong remarkably regularizer include mcp regularizers usual incoherence condition guarantee recovery provide nonconvex establish several nonconvex estimation weak nonconvex mcp develop theory absence incoherence regularize possess optima nonconvex however paper wang homotopy obtain homotopy oracle paper purely concern theoretical consistency stationary finally zhang show eigenvalue weak restrict certain nonconvex regularizers estimate provide approximate early however stop recover vector organize follow material estimator primal method concern corollary graphical case regularizer regularizer implication support contain contain illustrative simulation confirm theoretical universal write simultaneously write frobenius mh subgradient write radius also technique proof
use slice argue previously space fit memory gpu additionally divide step correspond implementation gpu cpu gpu pair master pair care computation master cpu communication node configuration must merge special configuration model update via node also dimensional index pattern novel implement flow node implicitly communication distribute implementation node configuration htb initial guess divide direction probability copy cpu execute gpu execute gpu direction execute
yield notice convergent subsequence boundedness take side hence hold kx q limit ii proof fact suitable termination accuracy parameter establish define proof q ready second minimization em pt define arbitrarily solution subproblem go go inner termination satisfy inner sequence accumulation stationary accumulation stationary moreover nonzero inductive argument imply p proof statement fact outer subproblem value arrange accumulation subsequence
background dominate residual capture proof appendix residual graph bernoulli graph subgraph include unit positive residual assume implication subset power vertex involve subgraph vertex concentrate foreground intuition subgraph vertex mean detect always relatively subgraph embed activity relatively remainder subgraph easier detect put language much easier detect less working communication enyi generate subgraph embed horizontal expression hold order maximum within subgraph closely scenario random subgraph tight subgraph provide good detection desirable detect small subgraph may stand eigenvector technique detect anomaly outline symmetry enable detection subgraph stand graph project principal rather entry enyi demonstrate two anomaly subgraph stand apart background compute detect presence anomaly chi contingency number point chi calculate favor radial symmetry anomalous spectral reliable anomalous behavior small subgraph identification complicated
direction class distribute sub develop allow uncertainty scenario traditional approach improve optimization method uncertain sub gradient know despite explicitly address aspect optimization uncertainty demonstrate time impact machine incur cost online regret sub multi study introduce decentralized sub interact path neighborhood online prove regret extension convergence regret aforementione corresponding algorithm used centralized effect favorable due rate failure inter sensor link uncertainty distribute fixed switching graph use
appendix offer insight student equation submatrix n marginalization sufficient kn k gives require k elliptical collection analytically theorem analytically solve n able analytically derivative hyperparameter carlo derivative eq wishart wishart generalization positive definite multivariate function q recursive generative q marginal distribution equivalent thm thm remark thm student nonparametric student integrate away wishart student derive inverse process overall student retain attractive gaussian
loss produce almost surely converge condition thus corollary proposition show uniformly bound uniformly conclude converge accord imply eq converge surely accord surely tends subgradient continuously continuously differentiable continuously j accord continuous hessian shall eq sample norm follow bound accord eq lipschitz finally taylor equal since vanishe prove proposition converge surrogate since vanish lipschitz derivative bound first taylor tend infinity fact since multiplying follow tend hold minimize implie tend investigate recovery expect online pca corruption justification whose pca draw norm change
vice versa citation separately identify original citation top corner exact match identify microarray laboratory generate laboratory responsible much laboratory affect retrieve laboratory correct drop effect laboratory six five paper act date six link cluster cell line cell
challenge spam semi supervise filter show supervised differ classified challenge collection email task filter cycle combine challenge challenge remarkable classifier train test perform filter semi filter support dynamic compression logistic spam spam train publicly test attempt whether semi report replace challenge challenge delay message six batch message message keep reproduce train task experimental outcome show version respective delay hand supervise perform
actually upper argue encoder provide close reconstruction reconstruction error generative elementary generative actually closely small check cost draw negative represent w r contribute evaluate generative quantity easy introduction bring inverse auto structure minimize depend feature reduce upper two f term minimize use compact representation reconstruction reconstruct auto another tight feature space involve integral dirac infinite encode value underlie network activation layer bernoulli distribution covariance matrix intuitively encode accuracy reconstruction
alpha plus collect high possible index high index index mean vector represent count number k u care look follow plus log logarithmic time summation running become hessian run mle precision initial run equation newton step newton put order bottleneck read run
program robust available rest notation diameter ix describe similarly primal rely guarantee target parameter infeasible ti mp u infeasible robust infeasible terminate call begin rule update apply directly instead require observed dual variable note primal variable together obtain condition variable z u ix lemma summing inequality prove return infeasible exist counterpart return infeasible recall
constraint completely analogously remark remain see input approximate note formal learning perhaps surprising provide justification tackle task inexact without manuscript aware hardness approximating case two unfortunately contain write case general f k problems dictionary relate potential dictionary admit approximate within factor problem preserve slightly weak
category property improper make health system benefit collect analyze increasingly find property correlate time may prefer member population receive benefit portfolio risk student college include code fitness predictive power category use machine naturally dependent everything place exclude physical fitness vocabulary success character trait power contaminate predictor underlie mechanism access problematic
sample n fp p ip evaluation turn predict input convert output dimensional project crucial speaking onto function nonlinear operation risk embed develop triple novel scale first kind massive furthermore risk assumption lastly improvement magnitude estimator well world datum set nonparametric perform hilbert noisy henceforth work give empirical
show unique extended driving discussion multidimensional rbm dedicate follow brownian motion deterministic simulate without definition rbm increase behave like brownian motion appear drift generic transformation difficulty acceptance multidimensional rbm dominate directly multidimensional rbm absolutely note rbm challenging arise one reflect core lie observation rbm us density contain accept key simply decide direct n x
use relevant spatio temporal classify spatio temporal behavior spatio temporal covariance avoid impose priori instant frame temporal denote grid perform regime exceed covariance order know undesirable poorly poor inverse
nh nn map eq h q claim x problem important integrate knowledge structural contribute incorporate algebraic algebraic independence trick demonstrate usefulness ica specific constrain underlie hand hand cause respective invariance transformation corpora transformation invariance need ambient noise speech robustness translation transformation handwritten digit recognition g popularity knowledge bioinformatics amount fundamental formalism trick feature ask kernel invariant sign mirror common complex phase factor rotation algebraic call semi suitable invariant trick twice namely invariance
filter element shorthand much degradation supervision incorporate idea discriminant compose ease enhanced supervision classify index patch image spirit intra pixel wise covariance likewise inter class lda eq q trace know pseudo full handle lda mat deeply build repeat propose variation verification digit texture discrimination face person pca b face illumination select subject expression pose pose manually corner image pixel corner gray together subject subject four remain neutral illumination classify variation test illumination pose illumination illumination pose illumination expression pose cross pose pose impact filter cross illumination one network set accuracy similar however observe randomness impact block robustness illumination set artificial observed percent translation direction plane rotation suggest various block may histogram aggregation
ba z n r
successfully apply topological count number ideal partition solution discard lie apply strategy number ideal vary odd number partition da di partition critical ml ml degree property hold ml degree ml generic cubic notice cubic surface equal suffice choose equation degree partition minor solution ml instead trying determine solution sign topological euler possible distinguished hyperplane degree distinguished degree intersection lying solution conclude root intersection correspond root f dx thin w fill black circle fill circle inner sep
discretized proposition loss consider tuple nonnegative integer since partial define since constant let assume side center z every pair discretization particular long precision threshold define combination shift center negative perturbation support exist vanish f always discretization put bound introduce know degree gaussian function bound proposition close look high tangent span top span output output simultaneous utilize weak specify convergence extended dimension size add adequate universe neighbor denote let dl sd ls orthonormal let kk dl sd
estimator obvious equal distribute strictly preserve protocol protocol feature v x differently distribute invertible information protocol map exist extend yet support intersect everywhere similarly imply differ b let integer protocol xx value identical definite thus invertible mp finite disjoint linearly independent v invertible position show periodic differ v argument yield determine analyst extend subset analyst select user repeatedly interact analyst feature p b estimator r v x able generate rating negligible determine preference restaurant rating movie readily item may private rating give modify protocol reporting rate reveal rx ensure subject respectively rate item profile information extract comprise privacy pair rate jx x jj gaussian item protocol summarize bias ratio item include reveal construct reveal rating subtract mp analyst feedback mp behind immediately privacy preserve rating since formal reveal rating establish among attain minimal protocol
handwritten incorporate target multi special handwritten example fall category group whole membership yield weight put membership weight non yield superior estimate treat old shrinkage available target shrinkage multiple constant finance one structure constitute choice base expert superior slow cross computational validate extend multiple section introduce quadratic program intensity optimum sample fulfil asymptotic observation limit structure theorem capabilitie letter denote symbol eq unbiased general index index matrix datum denote p follow omit obtain less setting est I give consider assume assumption increase dispersion eigenvalue increase dispersion assume behaviour imply
affine connect differentiable work directly pose challenge address manifold tangent rkh embed space considerably simplify preserve manifold burden extend euclidean rkh present random projection first hyperplane projecting present various vision task superior discriminative typical outline space discriminative completeness random hyperplane recognition person texture svm riemannian locality preserve relational bring power
estimate add preferred accepted compose hence add likely accept cluster wireless estimate factor solve determination cluster wireless security propose attack device communication wireless serve wireless framework develop deep nonparametric structure correspondingly factor estimation error
data foreground video entry sample performance fw fw prominent grow frame visually appeal fw video medium frame grow background illumination rotation weather fw accurate iteration significantly overall still fista illustrate plot increase fw c video cpu cpu square visually recover background capture foreground per iteration fw linearly advantageous take illumination superposition sparse matrix form stacking term capture smooth term represent cast full assume experiment summarize fw fista clearly fw scale r visually rank smoother condition scalable call fw wolfe fw norm combine frank
representation document wide variety modelling include classification model create sentence simultaneously explore representation exploit direction future rgb university united ac uk capture compositional word central challenge language retrieval represent embed low preserve crucial capture convolution document lexical concept model achieve compact advance vision present novel technique network text symbolic decade network model translation name entity research compositional space algebraic approach simplicity arguably network use great
pool magnitude velocity fashion distance maintain trajectory energy coordinate velocity equation move along along trajectory hamiltonian reverse together use move transition hmc understand operator act operator hmc cc b b involve hamiltonian monte carlo hmc base represent momentum momentum dynamic indicate dotted randomization momentum horizontal movement occur momentum vertical movement hamiltonian flip
dataset face individual capture condition expression size select half per person project zero row dl common sub include subset contain illumination sub round class satisfy great variation illumination viewpoint different sense adopt feature descriptor performance dataset leaf dataset use carefully flat clean background setting make easy application advantage focus shape specie art testing sub size dl descriptor relatively visual difficulty benchmark point fast image chain
heterogeneity also alternative pc assume necessarily principal discussion within covariance across major smooth common pc percentage care take dimensional slow lee et al insufficient complex covariance characterize partially approach surface although currently whether lead estimate eigenfunction regularize two step whereby curve compute pc compute eigenfunction pca smoothed cubic spline pc sparse involve wavelet pca unified pc advance allow pc grid model source variability lee surface quadrature score grid across develop sparse curve eigenfunction pc lee level di et present level multi functional method densely observe di al random al extremely et b al al pc score functional introduce encounter functional functional coefficient response many work introduce linear orthogonal discuss truncation penalty several reduce interpretability coefficient make across choice depend function basis purpose measurement involve non reduce measurement bias al accommodate predictor vary predictor general strategy regularization relate model flexible beyond present response inclusion fix extension nonlinear introduction highlight roughly develop basis sample common grid regression ridge result multiple orthogonal principal regularization necessarily first pc regularize pc et discuss smoothed regularization truncation basis remove error pc principal across curve estimate pc ml di subject predictor subject pc basis truncation truncation estimate eigenvalue still bayesian classify functional predictor transform time frequency construct logistic predictor function regularization structure incorporate estimation weight point inherent datum functional spline assume common spline basis regularization introduce use design use spline subsequently way additive scalar fix model multidimensional natural cubic spline basis represent regularization account
hardware gpu decomposition accommodate add combination nearly core alternative leverage library together compactly correlation small reduce pseudo input truncate expansion like parallel option extend magnitude illustrate big capability sum tree allow interface million size hundred core reach method whether surrogate show magnitude fast krige focus quickly kriging involve subset base give typical rapidly decay correlation simple fill response sensible fast accurate full high work choice scope sub spread optimal design criterion would search design building criterion lead prediction nn scheme design iteratively calculation regular exhaustive green roughly split comprise near one even relative early location exclusive iteration search choose variance much make design attribute aspect novel exploit trade
describe utilize uncertain information describe paragraph carry look credible improper combine b parameter reduce posterior base vector independent real improper attractive feature tail credible know b density easy marginal posterior credible union interval illustration tail short credible interval posterior unimodal credible program matlab scale offset contradiction follow scale offset
team play difficult college arrange highly rank contain mostly low rank percentage centrality specific short path node metric conference team closeness centrality also particularly graph team connection instead centrality contribution connection centrality centrality therefore rank great influence graph metric centrality assign win address limitation centrality win many high graph node adjacency centrality node eigenvector centrality neighbor notation identical equation place eigenvector represent calculate final numerous intuitive calculate highlight proportional
sensitive accelerate robust specifically attempt employ representation dominate robustness enhance speedup solver via alm derivation theoretically extensive ten verify outperform fast increasingly wide highly sample visualize huge text video greatly additionally outlier side outlier reduce subsequent significantly promise accuracy sample although efficient effective
face inherent extensive diverse accuracy label face face verification face verification face topic computer decade surveillance retrieval mobile device visual verification face dataset face large complex variation pose gender prove difficult automatic face verification work accuracy improve establish study close gap human verification human reason verification drop however scenario cross appearance collect training highly verification domain face verification modern face verification category extract low building classification exist face flexible deal level even projection center need specify similarly deep layer etc
form attention case practical lower entirely lie bound incorrect framework practice issue assign miss term although density remark miss miss beneficial assign miss discuss suitable improve chain model movement density miss miss prior density assign density offer aspect fact target infeasible introduction thus vector augment augment distribution follow distribution multinomial probability full conditional distribution I often mix denote
resemble linguistic simulation even able truly low unlikely give identify number demonstrate amenable combined tree furthermore retain proxy distributional look explicit moment high variability include language least mode language explanatory power linguistic apply positivity four positivity constraint explanatory identify false suggest identify tree amenable truly component amenable develop insight language separable hadamard product language dimension matrix indicate overall co projection particular standardized illustration second language remain language many contribute overall whereas distinguish component hz hz hz hz show range interpret hz range likely relate hz relate portion spectrum hz round frequency around hz likely human speech data frequency affect show interesting range difference particularly effective separate language effective numerically course require feature exploratory place especially distinguish variability acoustic evolutionary interest identify prominent feature effective
satisfy convert theoretic assignment seven relation table relational logic learn reproduce behavior show boolean atomic symbol logic output accurately bit demand present atomic symbol value interpretation formulae l randomly generate formulae contain logical operator compute relation discard seven relation formula partition operator formulae bin implement similar statement formulae six relation almost balance without basic task modeling pair statement six variable unseen structure logical yield short k example across bin
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vary rise marginal market market determine adjust five minute accommodate real constrain typically formulate lp determine incremental min index minimize achieve demand balance via flow cf bound bid solving determine lagrange multiplier associate lagrange multiplier define express eq price practice transmission loss calculate correction consist price flow line complementary imply loss ignore would either readily latter effect isolate entry argue subtract way collect
computational amp group test fig projection constraint amp additionally closely match transition performance amp algorithm match amp requirement practical projection case superior convergence amp burden r evolution thorough analytical future part union th grant triangle
among produce processor compute avoid processor processor function fortunately processor hypercube score processor keep take compute note processor processor compute collective execute compute processor processor evaluate processor high computing constant message hypercube processor number processor hypercube hypercube hypercube processor previous definition denote binary string denote string include partition hypercube lattice hypercube processor function hypercube parallel processor hypercube cluster string adjacent bit processor responsible transform subset bit string processor hypercube encode bit string subset processor processor processor dt kt subset encode processor hypercube encode string processor processor processor processor
relevance purpose unfortunately metric estimate nature interested click click may different art user click highly evaluation model always control randomly split statistically control baseline search engine group modify engine baseline component metric click system reach statistically prove successful allow engineering business decision manner nontrivial engineering resource consume effort need optimize click often guess like ndcg engine later control proxy determine modify indirect inefficient offline evaluate log
still spline lda cccc ccccc performance lda spectrum categorization speech consist frame dark frame dark water frame widely website contain measure nm interval nm low high spectra use face recognition task contain gray image individual normal dataset categorization contain category contain object view align pixel database probe randomly spectra remain spectra remain set contain recognition spectrum lda function radius recognition method baseline lda available deal image svm different linear radius basis
lead lsh desirable query dominate evaluation section examine cdf q figure apparent quickly keep intuitively undesirable code performance similarity sublinear time search especially basically minimize region figure plot region value normally pre
arbitrary compact solve apply classification case consider setup scalar array array non kernel compactly measure reproduce rkh closure one point trick index embed index space trick continuous compact hilbert define closed kernel define maximum margin u u solution project fundamental separation empty separate support integral characterize x write difference probability measure integral v k support hilbert project onto ray favor second label index set back kernel classifier coefficient convex coefficient find solve tucker infinite technique infinite control necessary kkt maximal principle subset equip hausdorff ct c ensure svm good classification ol krige unified ols gauss imply geometrically svm classifier geometrically margin suggest ols leave open article present approach parametric formalism deal infinite applicability theory measure vector ordinary ol conditioning uncorrelated covariance ols krige array arbitrary index support classifier hope deep connection extend result equip ol joint ol long version long version space banach
receiver look receiver post receive processing square mmse channel adaptation abstraction quantization feedback indicator feedback calculation map bit feedback map value feedback map index value index compute report delay henceforth delay delay lead problem alone feedback mechanism prediction scheme change reason interference gradually effect change active change different band example couple stop inactive band case net interference macro dynamically dynamic frame employ macro become dominant due fully resource load lead improve link perform channel employ treat filtering square treat effect partial loading algorithm transmission different user compute technique invertible furthermore unknown hence select rate wherein temporal build exploit technique come feedback ms sequence predict
ssc overcomplete sample theorem matrix union position point sufficient use sr use ssc algorithm cluster representation task assumption like face recognition segmentation see make use dataset dimensionality preserve dataset aforementione without precede allow fast compute result datum expensive cubic projection sample nature dimensionality tool projection become essential technique efficient evidence cosine projection
fidelity investigation different negativity row induce drive relie regularization positivity otherwise impose form lagrange penalty flexibility fact split subproblem lagrangian quadratic independent reduce minimization split q admit proximity proximity indicator projection positively extension
variability fit design principal pointwise pc pc solution pca towards pc target knowledge nan subspace toward pc checking opinion interested sampling component calculate design section specify also useful may interpretable calculate bootstrap explain span projected bootstrap procedure pc rotation towards eeg true scenario sample basis vector denote measurement simulate ik kk true set eeg dataset draw empirical univariate score pc random noise score imply proportion explain score variable coverage simulate simulated bootstrap sample comparison increase measurement eeg principal score variability fit population score basis variance score eigenvalue eigenvalue consider eigenvalue measurement conduct total hour simulation management job simultaneously job gb gb virtual depend scenario simulate pointwise coverage right coverage pc pointwise coverage close third consistently give coverage percentile give poor percentile interval skewness interval
metric rbf propose similarly justify small volume bounding mahalanobis idea preprocesse characteristic optimization researcher investigate fusion svms approach vector count centroid k completely reduce remove sample additional another related classification exploit independently split analyze classifier propose gain one rbf building htb allow dependence point transform generality calculate
learn adopt cnns imagenet much alone comparison newly train annotation classification moderately improve sentiment analysis far sentiment useful include business sentiment sentiment image much relevant propose sentiment noun vs svms consider base leverage concept mostly activity trying solve fine grain recognition organize try non concept deep study computer
degree correction line community separately event almost non partially overlap corresponding benchmark bipartite usa datum collect class interaction perfectly match literature partition al dash fig largely group modularity type community consensus slightly bad modularity list appendix find consensus minimal benchmark review ref majority identical empirical human system via var create constrain genetic giving structure type connect network somewhat document word cover var gene broad make corrected force fig correct recover community adjacency gene nearly gene overlap community degree community correct analyze find gene analogous finding bipartite maximum correspond classification broad heterogeneous degree movie actor directly edge exist actor actor movie show database movie study
simplify cost evaluate totally second difference easy rely flexible position capture position example expect movie encode prefer e therefore energy function un step potential resort distortion current permutation move pick keep order rest unchanged place new permutation move operation relative preference order randomly pick two item swap swap
recent result indicate bayesian coin outcome coin coin weighted tail coin use specify beta new coin give conjugate binomial nice posteriori equation give equal probability q correct bayes pm similar inform model inform prior ignore uncertainty uncertainty greatly affect characterize choice ideal draw use simulate eight analyze dataset prior directly simulation population divergence similar model bias time difficult dataset model method summary contain minimal correlate four default statistic contribute information time uniform summary prior limit five unit little divergence time assertion entail statistic coincide analyze draw vast yield cluster around text plot select avoid analysis much argue dispersion time plot essentially
preference try find human preference alternate robot determine action observe human together find human type cluster expectation maximization rank distance criterion integrate another cluster chain partition partition transition correctly sequence hybrid framework learn robot act robot vice versa uniform partition rather use bic ideal use cluster x human robot turn transition act robot action vice versa matrix must sequence parameterize denote otherwise repeat step assignment ix j converge algorithm call transition em nz calculate current high return maximum transition matrix randomly
explanation visual cause term operation macro target image assumption exclude target behavior black white relation unobserve discrete short possibly contribute omit simplicity noise incorporate behavior stand relation generative model image group observational observational clear observational observational associate label know observational image allow image take excellent predictor weather weather cause weather particular read whether visual cause ability visual pixel pt generate standard causal distinction detail target image space target behavior causal cell partition consider equivalent partition visual visual whose stand relation visual image know allow visual long relate observational observational observational among induce observational almost
epidemic likelihood paper observe set share lead leveraging solution one spirit leverage seem advance brief example continuous infer bernoulli bayesian prior posterior deterministic numerical integration detail example abc range traditional recent reader library quadratic discriminant matlab employ regularize polynomial library matlab interface regularization value classification implement nine chebyshev multidimensional project principal prior rescale one amount whiten multiply fold max try several give accuracy move average exclude pool computation iteration propose simulating base comparison user discrepancy implement way perform comparison use tie abc monte know carlo abc start sample weight implementation threshold quantile accept schedule pre schedule quantile quantile accepted slow take choose schedule epidemic transmission inside center uniform work assess expert rooted proportion infect
subject censor rmse forecasting table low apply motivate percentage force volume cf patient rate differ infection patient model establish association subsequent patient acquire patient index patient recent choose subject focus share variation hazard negative worth association logit link keep ease clinical hazard show smoother estimate penalize spline hazard flexibility hazard capture need assess hazard function end may forecasting
metric root rmse higher well quality rmse approximation include another pursuit fr extend match pursuit method give input initialize matrix set u k construct fr mp optimal weight similar svd matrix fr mp find know algorithm necessary complete netflix netflix dataset movie netflix customer dataset applicable characteristic propose increase iteration netflix limitation mp logarithmic number iteration verify convergence speed propose theorem theorem singular though bad f empirically different n mp er axis axis plot residual
vector anomalous instance assume generate different inferring latent dirichlet prior multi anomaly stochastic effectiveness demonstrate term anomaly view anomaly great interest information source wide variety naturally view page represent word occur page audio visual anomaly multi task horizontal anomaly view anomaly anomaly multi instance view detection anomaly anomaly figure multi anomaly view
nevertheless process meet challenge people conduct reduction advance robust high properly direction burden reduce valid robust appear dimensionality serious ideally mean direction presence back transform estimate coincide loading reduction contain much row corrupt tp true subspace orthogonal curse surprising finding connect outlier much severe subspace roc roc pca fashion significant loading vector inversion update contain drop batch batch size satisfy procedure roc significant intermediate kp pc direction column increase fast formula recommend assume unless speedup scheme tolerance computation batch share similarity problem svd accordingly cost affect roc pca subspace generate vary simulate denote tr denote use measure detection probability fraction label ideal estimation serious serious distortion matter simulation intuition choose combination small true
algorithm area input threshold classification specification algorithm likely recall kullback divergence random bind success probability minimization hypothesis sample indicator start last determine algorithm threshold binary misclassification training involve measure hypothesis family accord fix know convert error generalization part kl u sample make average learn algorithm h td difference triangle since low substitute low bind generalization complexity look reveal relative effect
asymptotic expansion iii assumption likelihood simplify consider expansion method error expansion likelihood unified replace expectation expectation asymptotic express high difference see include maximum connection lemma bayes method denote relation ii iii table type iii corollary summarize magnitude base leading term magnitude type
hmc hmc size tune acceptance step time time effective efficiency ess simulation min ess ess hyperparameter tb run ess ess min ess hmc hyperparameter ess ess min ess hmc lag hyperparameter low ess run mass matrix px semi step level hyperparameter burn iteration ess
k nj regime expert also hybrid vector direction multiplier follow interior phase adapt incorporation propose address challenge automatic high effectiveness compare synthetic real include natural language despite present subset particular observe good interpretability popular support svm devote order first memory optimization problem major disadvantage identify second memory requirement demand since store newton
claim exercise arguably american option exercise option round execute trade decision natural constraint turn exist american option continuous exercise dynamic american consider round adversary exercise option adversary also movement specify movement lie extend upper american ordinary gs reach gs gs gs gs gs gs gs gs unique round american pricing set option paragraph need iteration never exercise need elaborate little move round american option payoff controlling option need write round american option remark applicable recursive option price pricing option game binomial american type minimax payoff converge price uncertainty step st underlie asset decide nature lipschitz payoff bind price define motion volatility control sequence intuitively continuous option upper gaussian convex volatility surprising time counterpart start denote markov chain take lipschitz locally x hx xt adapt call condition control value namely value continuous process speak measurable martingale martingale put delta continuity compact discussion discretized variation continuity condition exist subsequence uniform match control consist dimensional wiener process control value p three first lie
flip occurs condition independent draw independent plug finally event distribute twice chernoff union substitute back pick early deterministic ensure randomize first exist utilize lemma constraint rewrite similar absolute simplified consider occur occur back third strongly attain strongly use fact maximum lower straightforward expression minimize show hold finally constraint low eq desired define sized block suppose target induce diagonal compose equal use block symmetric well notation write coefficient k exercise
mathematics nj semidefinite relaxation mle tight recovery problem noisy mle recover sdp regime
cnn availability multiscale convolutional cnn object extraction proceed multiscale version plane plane descriptor descriptor dimension convolutional conv conv maps image size result feature conv conv conv conv conv eight pixel incorporate local contour multiscale train test original two result contour fine tune pixel contour detection exclude two fully imagenet pre five layer top
turn biased estimate mmd mmd spirit take estimate square quasi sequence sampling sample q amplitude linear il basis efficient calculation mmd similar unbiased mmd eqn provide appendix aforementioned mmd describe equivalent appendix htbp shift denote mmd approximation set k I ia k ia l speedup bring gain mmd original aforementioned approximate time calculate complexity entire speed basis mmd mmd utilize mmd approximation consideration sample accurate thing calculation compute stream usefulness prove mmd
strongly proximal ascent prox le propose sag complexity sag variant et store gradient prohibitive reduce favorable machine zhang call method employ stage scheme gradient complexity avoid storage past analysis sag analysis extend prox solve prox incorporate weighted proportional lipschitz complexity upon one substantially prox uniform slow large much work explore
global rd ct important tune validation pre define consuming model adaptively resample candidate discard illustrate resample understand resample bootstrappe machine neural parallel use past inferential create process focus calculated datum effectiveness measure refer model fitness square rmse determination categorical outcome predict rate might create machine tuning data near neighbor model grow maximum fitting complexity pruning alternative pruning factor cf parameter determine depth partial pls
e simulating convnet clean good class likely label model ground imagenet clear behind superior training noisy information table error adversarial overall level table learn superior cm none none error true simulate outlier cover class chance cifar class training image outli know clean cifar amount outli significantly reduce without nevertheless outli reduce particularly hyper eqn right explore ability train noisy softmax
first pick pick gold standard side get consider gold standard result question gold decompose question include second first gold standard second gold satisfie induction hypothesis remainder include gold evaluate eq q question complete compatible mechanism free payment gold incorrect proceed standard gold first question incorrectly question evaluation proceed worker question free payment hypothesis gold induction since payment must non induction hypothesis furthermore permutation payment answer incorrect gold payment form algorithm payment gold gold incorrect answer let remain repeatedly apply answer wrong payment argument question uniqueness add payment payment mention proof desire sake brevity must property first proceed prove separately involve l induction rearrange get q rewrite expression simple desire l consider subtracting get q subtracting rearrange opposite sign consider know lemma recursively give payment answer evaluate ensure satisfied compatibility sn si sn level answer confidence employ expect payment expect payment prove allowed worker skip great skip level confidence follow payment worker report claim piece notation payment answer respect gold x worker payment compatibility form mechanism coincide define l payment worker answer evaluate worker select requirement compare vice versa
relate clear substitute kk bn self logistic completeness self self logistic derivative quantity achieve equivalently proof tail inequality random vector standard bernstein almost tail concern accuracy empirical moment bernstein random copy wish unbiased function aim sample absence minimizer commonly strategy desirable convergence erm minimize resource usage streaming regularity linear observe single super polynomially moreover quantify finite consider optimization euclidean minimizer sample sgd practice ease wish compute erm erm maximum certain regularity specification argument approximation
brief forest reader forest skip section forest represent direct tree internal leaf direct node hierarchy circle text em thick black minimum thick circle font draw black style level style cm parent tree b internal produce tree apply send repeat process node leaf make reach prediction component great sample phase construct greedy algorithmic construct arrive hand also one index optimize node split setting create splitting among child depth allow split leaf start training split make bootstrappe subsampling sample tree subsampling randomization reduce turn forest assign importance individual overview popular reason focus refer time entire tree value non relevant ranking threshold split researcher heuristic limitation extension record follow also underlie indeed frequency include feature theoretically provide informed however share drawback principled way determine beyond subsampling bootstrapping tree
tc point literature argument theory graphical lead close dc tc see dc kernel interesting happen dc entropy extend family stable spline stable spline exploit stability burden evaluation dc organize introduce gaussian via briefly review entropy completion property dc resp semidefinite denote order element invariant convenience white impulse response
vector manifold either suggest though bias make already improve recognition performance relu method triangle inference amount activation activation relu confirm cifar figure autoencoder hide unit train permutation cifar ie activation performance well invariant sigmoid relu detail light precede hide become small hide remove satisfy separate function active activation define
construction optimization path random natural function measure fx q algebra generate iteration write close numerically
plot benchmark associate euler univariate euler approximation euler posterior evaluate filter ng paper associate euler approximation reasonably inaccurate term location shape interesting consideration however dotted score key remarkably posterior replication rejection abc minute time case euler production still abc summary statistic euclidean reduction apply statistic produce parameter arguably provide exact despite slightly inaccurate reasonable estimate perform score produce reflect impose parameter provide quite poor marginal upon literature give typical mcmc note euler well exact effective use neither accurate linearity space confirm qualitative abc see produce rmse approximation indicate abc procedure statistic latter poor score abc accurate notably persistence parameter fp panel capture basic euclidean magnitude inaccurate ability consistent score euler dominate abc marginal rmse multiple score ss abc euclidean fp refer abc base marginal score euler benchmark top panel produce three run highlight
suffer global guarantee situation negligible remove project refer hard project non feasible projection efficiently low convex exception work demonstrate convex however able penalty scad mcp commonly use iterative thresholding htp pursuit sp however traditionally setting satisfy restrict isometry analysis sparse vector universal constant require rip wherein arbitrarily require perform number
complex reconstruct heuristic far classified genetic knowledge advance knowledge
latent continuous undirected model relationship compare four implicit sort model opposite figure discuss one prediction rating deal miss item certain take value user consider curve plot state pair predict rate roc plot class item rating rating rating play
trade computational estimator estimator statistical trade understanding phenomenon setting key come year section introduce notation paper ij u extract column let eigenvalue arrange decrease component length place eigenvalue eigenvector measurable unit angle loss change argument convenient bernstein directional condition turn particular size level whenever e u u every p pp n principal prove class distribution convenient level symmetric denote small element measurable sample bound subgaussian consider technique facilitate eigenvector minimax suppose restrictive mention introduction
window dependency field model rnn vanish gradient secondary prediction learn target cell gradient learn application secondary structure prediction feed neural concatenation
embedding unsupervise context paragraph achieve slightly paragraph version indicate add tuning well comprehensive favorable play role review positive exhibit significantly grain coarse grain svm movie review review sentence experimental protocol word document next compositional keeping obtain sentence representation recursive sentence sentence representation cross review baseline bag diagram quite much state
valid eigenvalue always indicate space expand determinant use verify association observe ii observe idea second use hypothesis follow positive infimum little extra capture complexity denote structure row resp mixture testing show calculation add see choose assumption estimation display imply complete section section remark grant dms dms gm part nsf grant dms award grant dms university nj ignore stability lead depend functional motivated difficulty functional correlation simple correlation rate exhibit phenomenon illustrate arise financial functional minimax matrix procedure component moreover characterize focus sparse
explore minimizer function purpose turn subtle structure observation formalize last section useful relatively include sdca accelerate ball characterization claim polynomial vice repeatedly apply produce chapter novel prove inversion sense carry low bind modulus upper root vast knowledge chapter develop tool root last lower precisely develop know strong requirement indeed chapter conjecture prove imply seek create accelerated descent root analytic theory polynomial use well gradient present believe future section notion important specification task sequence may randomly whose iterative increase denote possibly method random method draw previous mainly method explore property method nature sequence content elementary theory square pair denote root last spectrum likewise simple spectrum entry zero square note may size demonstrate matrix strictly exist sufficiently denote namely eigenvalue index plug yield convergence suffice norm zero derive aforementione
decay show generate propagate backward compute derivative near qualitative give intuitive task feed neural meaningful mean much architecture tree neural evidence dataset code website node evaluate code neighbor query sort symbol euclidean see seem meaningful reference break control moreover group table confirm cluster representation almost related mainly flow conjecture cluster group distribute vector representation symbol human program learn even though measure similarity program fail relationship different symbol metric e contrary aspect abstract benefit program compound union continue cast switch program interest feed representation base convolutional neural student student system run validity code code along cv cv
random univariate way avoid ill transformation table noisy create pdf pdf pdf pdf pdf pdf pdf configuration specify title vary axis bottom similarity euclidean noisy transformation truth noisy pairwise create column simulation row type transformation slope amount noise transformation information increase slow pdf pdf pdf pdf draw every run shape comprise
kernel som performance term seem section goal exception som aspect strategy som som represent give relational som dissimilarity som therefore equation part equation determination relational equation determination som equivalence variant equivalence relational som relational som som extend euclidean embed practice som som look relational kernel try address combinatorial generalize analyze differ use careful equation prototype neighborhood relational assignment rule value soft coefficient equation algorithm anneal implementation hilbert dissimilarity variant summarize variants som som relational give som htbp online som som batch som na som p annealing algorithm prototype batch som
lt lt lt lt bp ltb ltb ltb ltb ltb ltb ltb ltb l l l tw ltb ltb ltb traditional spin flip also evidence advantage increase arise result problem constant factor low spin flip gpu might generalise spin extra hard dynamically mention confident guess graph advantage considerably strongly subgraph would become restrict perform interesting describe markov field give motivate em p em use use value partial implement store slow would possible simple look table number instead store look arise slight arise small neighbourhood vertex convenient involve multiplication potential particular ideally able binary double bit processor
toeplitz realistic systematic naturally discussion section eventually procedure low suggest efficient parallelization whenever base probability relevant number include poorly informative provide clear contrast many subject expectation chose criterion stable setting intend retrieval namely preferred report protocol overall picture first give illustrate see error guarantee small covariate suggest error positive determine large bind false fp supplementary behaviour guarantee figure corollary coincide achieve value describe less figure positive positive determine notice false lasso latter include result finding positive factorial group toeplitz seem symmetry violate situation positive positive lie stability selection use loose probably room idea count outside scope vary disjoint false positive plot
show kl divergence expansion kl kl scale q continue bind multidimensional euclidean ball calculation ellipsoid claim two interpret eq mdp uncertain regret scaling demonstrate stationary control server queue mdp plan single queue customer queue bernoulli unknown mdp state action service resp service queue resp service service type action instant respectively hold queue whenever queue correspond total policy optimal policy policy monotonically range regard start estimate candidate q kl vector non degenerate mdp possess policy assumption theorem regret bandit policy arm completely mdp contrast huge compare summary state thus uncertain flat bandit large yield furthermore unable exploit state action exhibit scale expect return uncertainty scale recurrence cycle thompson completely flat force arm reward break cycle random
involve obtain density extreme surprisingly stein performance sample splitting difference I approximately microarray process accounting correlation effect size estimate extend correct bootstrap procedure purpose quite normally distribute future covariance additionally might explore connection correction discuss connection discovery provide fellowship grant dms solely author necessarily view manuscript extensive expect argument consider jensen b jk j left sign
state space comprise state assign value high process assignment denote want huge amount computing define separate module evaluate instead state denote concern future action propose actor carry deal huge likely problem turn actor algorithm add
study develop particularly since real like crucial comparison evaluate present taylor propose novel powerful covariate nan regression coefficient elegant thank simplicity comment issue condition
shift able leverage unsupervise selection robust mode wider evolve scale supplementary shift away calculation cluster cluster final converge right bandwidth isotropic respectively location plot correctly coherent mode water place smoothed fail detect mode conventional typical tendency segment boundary illustrative conventional level plot segment effectively local salient background exposition style kde px I np ic ik kde estimate set rise mm support satisfy regularity x normalize
protein odd make main collection protein present database contact graph available file refer graph contact consider atom connect protein attribute term attribute pattern literature aforementione physical label euclidean among characterization graph physical information regard protein protein remain note obvious ray physical analysis protein ds represent sequence thus subset ds consider effect viewpoint divide test obtain graph vertex explain real value vector aforementione denote graph algorithm ref accord matrix contain e contact edge atomic analyze eigenvector markovian walk however share eigenvector contact provide consistent protein principal walk far less split compatible ds protein ds ds sequence character usual dataset label describe protein vertex auxiliary ds direct ds g value hoc classification dissimilarity
score procedure budget differentially use score regularization combine parameter candidate parameter minimize denote h h l q r th entry generalize convex ensure perturbation differentially private perturbation noise noise noise procedure produce parameter estimate strong simple noise density xx need kb b
eq third adaboost specification span tree however structure natural tree maintain tree stand show chain bottom tree drastically interact adaboost mrf bp inference evaluate effectiveness hmm encode feature previous one differ original hmm extend partially state test whether improve datum level learn segmentation purpose macro average basis mrf vs limited newton suggest crf slow converge solution exact experiment forest cg per boost round meet inference stop message converge round final bp sensitive choice respectively appear stop converge adaboost mrf alternatives
future direction reliability cm work desirable important scalability combination expert independently learn expressive expert valid natural finally robust prediction theoretical combine obvious framework four I train
convnet great convnet compress kind remarkable large representation cnn compress marginally visually ranking compression pt dim dim storage I gb gb mb mb mb bin mb ms mb ms compression add encoding indicate gpu compression additional add time scenario exhibit compress hardware accelerate hamming feature work convnet cnn one amenable compression compression method ratio achieve instance test solely nonetheless representation retrieval dataset noisy across return image versus fix pool negative set result image datum scenario give pool query facilitate broad suggest diversity make convnet use correspond dense ii improve fisher encoding
positively neighbor scheme space make provide iteration eq positive determine adaptively alternative scheme maximum assumption survey fisher alternative score analytical naive gradient however name iterate derive artificial white noise component particle dynamic model ascent unstable carefully vector expectation popular applicable term argument characterize explicitly e path cost variance linearly computational use lag present path functional non vanish asymptotic improve asymptotic forward well filter smooth experimentally typically linearly result performance forward suggest admit mse dominate forward dominate understand mse sum particle forward confirm experimentally smooth limited path ml parameter accounting smoothing procedure approach applicable fast gradient prohibitive moreover run sequentially recursive variant ml justify ergodic ascent time increase upon ascent conditional new ascent except time form suitable evaluating time relie notation score filter use approximation fisher identity score property recursion e limit infinity study regularity algorithm assumption recursion possible originally propose space em
class replace bound rademacher say bind rank situation present run differentiable arbitrary subgradient property guarantee set batch arrive predictor side plug get optimistic terminology smoothness assume twice differentiable use express say usually
nmf role play nmf nonnegative create nonlinear norm matrix possibility must user nmf place rank singular svd two consequence naturally great factor nmf interpretation consequence processing document collection basis column nonnegative correspond term weight term assign document http www h familiar dataset nmf interpret instance heart basis similar factor topic sparse element strength document clearly nmf individual sparse I nmf nice interpretation individual structure svd create gain interpretability come perform equally reconstruct especially svd strength computation nmf unique nmf convex
decoder output style source source encoder pt encoder decoder transmission process square nx drop subscript distortion denote rate theorem encode mean infinity assume random independent minimize distortion store furthermore want possible bit communication incorporate variant coding encode r decoding call q write go define minimax base denote
minimize propose optimize order mn model complicate add gradient surrogate online see perceptron algorithms motivated exploit robust function surrogate cauchy confidence present presence function design traditional uncertainty guarantee classifier dataset conclude goal I many world consider corrupted classifier instead machine intuitive classifier decision risk df e fx minimizing
object voxel concern cardinality discretize coordinate grid reach point discretization ray discretization point represent concrete discretization correspond rotation angle rotation image discretize cube discretize rotation sense discretization value pixel way note discretized distortion phenomenon work element particular orthogonal checking assumption imply considerable coordinate ray sphere relationship hence span require subspace discretize sphere since permutation orthogonal employ grid transformation denote finite rx group coordinate view instance discuss power however impose strong grid graph behavior collect manifold build allow paper property fundamentally knowledge geometry familiar topic denote riemannian manifold embed bundle plane canonical ambient derivative laplace operator tangent connection spectrum l l eigen resp eigenfunction lf lf kf manifold inside f ambient lead indeed borel sigma absolutely volume associate abuse focus k case sense affinity connection mention decide dataset application context processing mahalanobis underlie state synchronization affinity
overlap exhibit variance systematically dispersion higher depict four result diagnostic plot color dark light symbol describe member class outli plot neither distinction influence correspond different majority space since outlier influence fit loading correctly identify outlier include four pilot laboratory production setting extract heterogeneous majority high ht depict spectra analysis valuable little fit presence many ht figure diagnostic scale enhance dark triangle examine use
affect change represent exposure people exposure change say reduction episode individual trait episode trait degree episode trait speak ratio causal exposure numerator denominator expectation however restriction difference expectation exposure paragraph matching subject additionally full match subject result match effect remain one match discard match variable control population discard discard parameter full incorporate individual effect individual fact control like conduct effect propose statistic statistic difference individual
sometimes form exhibit regret aggregation sequence possibly moment bound term bind term possibly remain bounded innovation slightly error I even quite work let start see recent extension assume predictor form large explain large page aggregation aggregation corollary moment aware introduce define denote denote interval older define extend differentiable rescaled sample locally stationary vary autoregressive observation represent unit variance say sample extend process time vary assume e see historical tend cope say sequence begin aggregation procedure situation result historical available allow infinitely past derive away vary condition ensure
factor model bayesian continue summary still open sufficient available statistic epidemic decrease abc material foundation grant ef work lee support office nf research utilize nsf grant grateful discussion datum lee consider biological environmental case stochastic candidate experience intractable dynamical suitable disease main hierarchical dynamical markov stochastic differential appear
adaptive use burn sampler benchmark iteration within show started fail adaptive part allow preliminary conclusion range general conclusion benchmark crucially start quite start maximum maximum pseudo start satisfactory yield variance start distant likelihood adaptive universal improve adaptive justified basic version
normalize vi consider arc six dataset model average method dimensional filtering ensure smc require smc neighborhood neighborhood intrinsic locate set vi smc illustrate superior compete robustness summary robust smc presence noise run propose tangent enable eliminate exhibit presence close smc appear dependence smc levels htb model ii appear contrary exhibit figure vi smc appear affinity spectral identify sufficient accuracy explanation outperform manifold computation neighborhood structure complexity scale datum affinity optimization principal eigenvector contribute operation perform fully since computation per neighborhood identify tangent entail calculation principal eigenvector ratio type manifold readily extra one smc ratio ambient vi unit sphere cost smc exceed ambient orthonormal unit sphere vi imaging methodology lie associate view orientation function nothing discretize pmf describe water pattern object image direction pmf image map map try modeling pixel segmentation modify similarity modification euclidean coordinate fit cubic cluster u randomly color cubic pass figure around spline cf region snr snr bar whose ten
mention pass state suggest camera pair static configuration ignore dimension parameter intrinsic validate simulated use basic experimental observe scene behave track multi tracking section strength space account use single advantageous amount newly object capability base two figure centre second translate along axis first around object locate axis distance model particle filter run particle ray move step ds pf ds pf ds pf estimator map map carlo result estimate truth cover particle notable dependent second space configuration camera similar one camera camera locate camera depth run algorithm approach cope linearity space limitation dealing depict propose propagate uncertainty move camera analyse distance camera camera angle successively update acquire initial solution
union vertex phenomenon fact understand fact optimization well return optimization continuous show easily f evaluate access oracle oracle oracle could construction prefer ise generally unless special vast traditional ascent approach statistic cover implement bottleneck quite np hard possible basic reconstruct
incoming activation unnormalized weight multiply w interpret model gradient compute derivative propagation plug numerator move continuous get value back propagate biased ignore fact bias relatively criterion hold usefulness choose serve importance sampling express dirac expansion normalize
propagation similar benchmark mrfs potential topology grid random edge present parameter field strength mixed interaction strength figure interaction bar interpret confidence interval fair otherwise parameter onto suggest large projection simple vertical horizontal chain randomly generate span cover use fix gibbs systematic scan variable maintain
estimating regression model influence early work misspecification extend notion linear predictor derive sparsity interpret even sparsity dimension maximize remainder organize background necessary divide agnostic inference result reader convenience j j submatrix row q order eigenvalue entry decomposition span gap pt eq column unique consider principal equivalent advance estimation require minimum well sense semidefinite equivalent row sparse assumption investigation gap principal correspond subspace sparsity basis assume
illustrate feasibility structured manner fully connect deep structured inference aim alternative encoding automatically determine auto aim explore efficacy scale image decomposition support science engineering innovation yu com interest connect graphical structured structured deep structured largely concept deal deep tractable structured structured connect field intermediate layer problem illustrate
assume small bound constant third select balance term order reasonable start small identity proof bind identity turn prove identity however stop time natural kt kt fs kt n kt kt obtain give state let definition index observe round period indexing round start sample hold fs v f furthermore proceed prove definition nonnegative notice stop stop kt turn hold eq calculation introduce hold appropriate last proof straightforward kt decay fast negligible handle lower bind introduce c keeping follow kullback sequence distribute independent distribute random clearly independent independent identically value even odd natural truncate parameter scale
far balance cross four work run default library svm model show obtain analogous outperform bad nine ten dataset support prevent worth even though svms build decision fold method outperform vice versa support claim different optimization computationally comparison svms perceptron h starting perceptron quite reasonable solution solution completely seem possible exploit already dimensional uci solved start random sample unit sphere hard valuable initialize approach last behave parameter control strength table score fit balanced h breast diabetes heart balanced result one superiority big interpretation error lagrange formula width part evaluation task chemical protein protein test compound ten ht fit
node preferable though propagation kernel proceed simply symmetric adjacency matrix kernel input green idea weighted transition obviously partially label graph marked accordingly learn involve graph attribute chemical annotate secondary measurement image inherently compose channel color way essentially similar advantageous per kernel attribute bin experiment normalize attribute set disadvantage ignore attribute graph database matrix metric initialization tw attribute combine attribute propagate attribute similarity attribute efficiently propagate continuous hash node associate attribute attribute graph edge challenge ensure compactly represent update matrix attribute gaussians center node share set calculate compactly spread attribute update derive attribute node attribute kernel initial equivalent correspond attribute distribution edge accord normalized transition associate attribute edge weight vector compare kernel ignore attribute vector exchangeable kernel reason associate space node exchangeability compact hash input performance accordingly blue green couple experiment iteration description
method often approximation perspective variational dynamical connect particle particle multiplier unnormalize approximation normalizing particle repeatedly iterate maximum variational free principle sophisticated mode coordinate practice maintain asymptotic importance converge combine advantage decrease kl correctness filter markov hmms hmms dependency filter history construct variable update markov select current set variable past product
primal decrease iteration line choice minimizer make parameter equally logarithmic decrease choice work cover theorem stop active motivated reach condition step yield eq q condition stop analogue problem community expect discrepancy always discrepancy satisfied solution resemble closely convergence separately proof strategy essentially monotonicity two evolution active elementary mutual upon let function identity imply denote characterize important monotonicity convergence arrive clearly
entry let fix model abuse result width yield state proof group similarity large size structured sparsity say zero encourage sparsity define penalty q element shorthand write norm group define structure constraint set form keep exposition work rest emphasize de emphasize almost need sake proving theorems literature know representation achieve objective relaxation constraint subsequently obtain observation number remark yield binary regularize group remain overlap group bind reduce ambient bind become combination structure regression structure know logistic group special efficient proximal recover elaborate detail bound sufficient ask interested constrain zero bind vector everything overlap logarithm number term pay price group recall penalty explore singleton sparsity lie group select
side probable correspond large quite monte carlo base dependent partial finite possible approach determine extensive law empirical density standard would theorem axiom claim theorem exercise proposition
value post literature propose correction usual controlling produce methodology computationally currently applicable correct invert kkt correct drawback interval restrict usually specific regularize estimator forward selection marginal screening follow marginal lasso orthogonal least primary model selection conditioning selection focus screen event limit marginal screening apply wide greedy
theorem show calibration without discrimination capability histogram method definition concentration transform histogram fraction positive calibrate probability notation define triangular obtain bin calibrate bin bins probability histogram calibration converge histogram proof proof state supplementary theorem show measure term histogram show base classifier measure definition classifier transform calibration calibration auc classifier auc third theorem bad due limitation theorem theorem calibration measure discrimination histogram classifier histogram mini
close request distribution skew trend toward balanced dataset could pa al pos aim aim pa pos corpus since label data pa process pa small step execute
yield superior learner base mean various scenario fairly dataset datum space difficulty generative method method apply try complex bp passive odd svm gmm naive scenario gmm naive bayes regression scenario naive regression gmm naive logistic gmm bayes svm gmm logistic regression naive scenario prominent combine batch illustrate comparative across multiple resource naive nature one present result winner b comparative unlike scenario employ arithmetic modify winner presence framework employ discriminative stream herein behaviour affect practical present scenario successful base criterion effort scenario stream address learn generate surveillance scenario
accuracy reduce kkt element vertice thresholded represent equivalently l l kkt block optimization precision fine thresholded admit statement thus induce component thresholded covariance resolution induce precision conclude equality labeling thresholded nest component partition thresholded nest prove give contain inside vertex precision l ij kn ij kp kk p lp stanford university rule discriminant naive bayes path
topic support addition hope gain deep understanding depend representation explanatory factor conventional nlp take word challenge one word representation semantic semantic index much representation deal semantic allocation lda word representation unfortunately quite train
measure draw positive drawing previously draw discrete useful cluster hdp hierarchical define group global concentration global vary control group control smoothing dp variable dp conditionally share extend multiple require hdp tf measure score query hdp topic document predefine e appropriate base hdp hdp base topic grow problem number topic reduce overfitte fix topic transform normal may glm linear capable relation covariate covariate generalise specify link relate response canonical link function family response dispersion dispersion generalise take q response choice binary trial canonical choice supervise extension topic learn control learnt act corpus regression coefficient response document generative document vocabulary topic proportion response draw ij iw ij response implement
request produce request active general request eq plug theorem agnostic theorem sometimes specifically compare maintain effectively replace replace state achieve algorithm mixture input budget request request produce er pf k j sign sign imply contain convergent subsequence j w kb b w kb sign I kb x x j surely sign x fx hx px gx px scenario recall vc get request constant since appropriate establish analogous axis align agnostic active discuss several specifically study thesis q b request q least return er constant request
mc receive nmf detector secondary set variate threshold true scatter set orthogonal diagonal scatter note estimate propose parameter detector employ distribute clutter sample clutter covariance shape towards zero gets tail nature estimate depict detector correspond curve remarkably huge desire especially clearly good maintain desire length well slightly would draw though performance underlie outlier recommend often heavy tailed shape whereas remain regularize secondary free clutter pd average detector set give study pd detector
fact transform algebraic link duality let tuple polynomial degree dimension polynomial polynomial ng nf ng express dx let let dx f f df consequence polynomial feature interpret decision kernel denote f analogue reproducing associate hilbert degree identify replace symbolic precede could also combine sec beyond
enable formulate formulate bayesian account available incorporate suitable mcmc asymptotically accord posterior turn approximate metropolis hasting admissible log computation inverse large image cf expansion adapt efficiently evaluate spectral effective small size assess representative construction cascade compound poisson cascade large pixel self construction outperform image value first patch pixel remainder main formalism framework underlying image result world benefit application analyze bound follow solution aim characterize call say belong small large process behave therefore fluctuation hausdorff dimension denote point take precise hausdorff increment wavelet formalism tailor orthonormal
suggest care correspond optimize first introduce component q moreover hessian x respectively since definite observe case convex propose gradient particularly suited introduce descent compute medium accuracy solution strategy problem close employ negative gradient definite iteration introduction scale choice describe respect norm induce definite positive step backtrack loop parameter fix parameter diagonal scaling projection backtrack fx go set accumulation sequence point iterate belong sequence admit assumption bound stepsize freedom choice exploit significantly practical main scaling stepsize behind stepsize approximate hessian objective minimization
processor keep vector processor continue standard condition asynchronous preserve convergence memory parallel coordinate descent randomization effective asynchronous decentralize communication failure optimization resource article dependent algorithmic synchronization tool continue discover ideally heterogeneity increase composite model map smooth big problem cope l obey certain say composite model whether fast yet work support european grant proof foundation grant schmidt support laboratory big review recent advance communication overview technique scalability survey parallel computation principle attain problem back area importance formulation dramatically last decade rise new successful vector machine wide process compressive sense medical imaging bioinformatic reason obvious
right core member university split optima division unsupervise learn overlap us division correct discuss elsewhere perform poisson community divide node contrast tailed degree network correct achieve overlap even emphasize analysis carry correct use interesting observe succeed value independent different parameter advance right learn known panel show move
van scheme group review computer education liu generative analysis online development social computer journal submodular maximization enable massive scale community conference tx framework online environment journal lee education plausible prediction bayesian computation membership journal research master university automatic coding act protocol international collaborative g support collaborative historical ed collapse journal multinomial mixture overlap community physical spatio stream journal rich scientific report pp mind university
sample expect perceptron recognize reward draw limit practical use exp probability explore previous algorithm advance adversary apply contextual propose estimate reward hypothesis separability reinforcement reward
k ci ic lie th row interpolation equivalently equation I hand c hold theorem sx subdifferential cost value assume w condition contradict conclude walk motion process brownian motion distribution bridge eq eq notice give fourth appear therefore q result say unfortunately opposite infinite come sub reasonably bernoulli cumulative cumulative sum zero nx absolute constant min max sum sub restrict last show minimum random attain
hash key sensitive along preprocesse call query hash uniformly satisfie note query transformation create let lsh transformation counter fact similarity failure lsh hash preprocesse sensitive explicit call without loss simplicity easily define concatenation qx q obtain thus suffice approximate neighbor transformation partially hash repository clarity
loading observe structured loading induce prior remove unnecessary factor load sparsity factor load iii neither gene disjoint may none iv possibly gene expression covariate unobserve systematically observation work induce jointly adapt loading zeros zero sparse dense use load dense mixture favorable gene number gene affect batch intractable enable possible subset sample search shrink zero markov mcmc expectation b gene entry response row generative loading select remain sim simulation ten dense loading factor component correspond dense vice simulation scenario residual simulation five method run set initialize warm final thresholded hoc tend recommend sim correct control
suppose si si si scheme obviously since verify numerical utilize establish imply compute compute partial take section corollary lemma g national foundation ed enyi institute mathematics email school institute technology national science ga email edu dr science foundation email nsf measure pair distance covariance correlation disadvantage compare fast distance computationally formula nice derive computing synthetic applicable much wide induction life aspect straightforwardly computational
express infinite moment follow lp orthonormal discrete variance admit jx representation direct fundamental random probability publish outline fact variance decomposition lp eq lp shape continuous derive scale interpret modify table numerical moment continuous first four moment quick marginal depict application normality tail lp orthonormal lp score obtain follow expansion moment decomposition k table list bivariate scientific question collect child aim chart help assess child normal comprehensive fisher child fisher properly tie recognize discrete tie surprising almost surely linear mid applicable discrete detailed investigation idea beyond scope elsewhere tool nonparametric
set imaging dimension vary million leverage space move towards high speed decay case exponent closely match leverage score set decay sharp decay present world would empirically usually decay law help theorem law decay prescribe algorithm orthonormal norm generation basis completion n kn k main km choose sort gaussian htb plot vertical correspond offer set row first equal perturbation avoid every
projection recover pose incoherent generate independently incoherence ensure mass spread fraction instance p pose serve warm exercise main bring analyze potential descent completion subset update iterate
sample practice make sgd solve expect network able fix exist observe rewrite follow bound bound
fdr c indicate procedure margin moderate proportion relative normal ise normal whereas increment set initial summarize figure similarly see incorporate testing improve dramatically voxel simulation lattice group mrfs simulation present multiple whereas procedure fdr automatically heterogeneity appropriately control indicate utilize dependency testing evident weak procedure slightly outperform globally procedure brain voxel median voxel goal identify voxel different rate nc nc procedure procedure distribution two cumulative procedure approximate three testing
recommendation mainly relation domain filtering relation comprehensive collaborative filtering information focus rich side information survey emphasize importance setting domain type auxiliary additional setting vertical focus study representative perspective setting interaction without overlap explicit overlap uncertain side tag two fm work recommendation scenario ii usually improve technique leverage auxiliary exploiting without difference iii goal recommendation efficiency aforementione survey research area recommendation worth exploration direction security
rewrite vector l stochastic angle block degree gradient add stochastic effect per difference quadratic recover directional curvature along stable technique descent rely gradient refer parameter expensive minibatch bias
consistent effectiveness ensemble far allow major classifier hand subspace reduce observed conjunction distance successfully three ensemble wide range base diversity technique mostly classifier robust variance subspace improve primarily decrease bias step technique ensemble employ subspace ensemble attribute important rsc investigate embed attribute promise preliminary rsc learner understand quick train ensemble accuracy comparable popular evaluate alternative classifier rsc rsc base nature rsc make ideal ensemble two ensemble tailor rsc classifier subspace rsc significantly ensemble bad demonstrate six subspace ensemble high attribute analyse source sphere rsc classifier rsc create classification
soon necessarily highlight exactly regard scenario ghz party numerically increase considerably increase reproduce quantum explore detailed dag input index parent equivalent demand q ac etc quantifie fulfil demand impose quadratic convex nevertheless minimization cast parameter illustration obtain projective identical inequality give correspondence inequality relaxation reproduce operational causal perspective independence locality quantum correlation causal model measurement quantum previously think consider scenario regard dependence finally quantify believe motivate importantly basic tool understand derive useful context randomness expansion possibility treatment characterization convex compatibility complex quantum support grant university research office w nf nf characterization verification support sake respectively norm causal model tool theoretically therein standard basis vector
sequence unstable bound fig box observe able unstable region highlight colored notice area correspond turn individual thus result salient motion similarly region sequence result global capture crowd motion robustness deal inconsistent subtle crowd synthetic potential region bridge note herein consistent interesting subtle motion discover employ similarity perform et sequence obvious region detect bottleneck able region addition
example likelihood approximate probability use iterative posterior bethe energy expansion rule general determine specify relationship model potential contain basis quantity scene labeling estimate objective vector interesting easily lead iterative log take hidden variable posterior variational inference belief nmf sequence simplicity ignore compute expect problem iteratively update step note step belief pass simple flexible descent iteration way consider algorithm unfold neural index node layer activation node tie different help fundamentally unfold allow course formulate derivative recursively sum intermediate derivative derivation give sigmoid obtain field markov mrfs level conventional unfold mrfs generalize change unfold mean field lead propagation deep architecture architecture power restrict mrfs mrfs high order factor easily mrfs create variable give formulation state
rule outcome target assign assign child c conjunction variable aggregate current split whether extract decision split tend happen top reduce extract metric rule rule popularity define incorrectly instance satisfy squared error value satisfy condition define frequency small preferred interpretable one accord combination pair rule tree include pair rule measure small error rule rule rule leave variable remove currently
eq hand bound cauchy schwarz inequality submatrix gram obtain remove column small eigenvalue investigate criterion write use correspond summation lower derive acceptance dictionary atom linear get low proof investigate summation eigenvalue distant dictionary lemma coherence dictionary coherence quadratic bound q q one hand second condition atom result acceptance criterion error approximate atom unit proof substitute term thank eigenvalue derive appendix conclude propose quadratic bind previously extend relevance onto subspace bind derive bound result
semi bandit online agent observe weight receive sum payoff close computationally combinatorial ucb like solve number tight factor tight choose subset ground item subject observe receive variant combinatorial combinatorial stochastic combinatorial practical application recommendation variant bandit access optimal cardinality set gap return suboptimal exist bandit variant call confidence bound chen recently contribution two bound significant improvement match factor consequence
important preserve linear block coordinate converge convergence also review choose choose q guarantee algorithm proximal operator ignore backtrack shift k I converge widely iterate another active pair decrease search algorithm start line paper algebra write notice observe union ensure finally coefficient asymptotic e note q u ni ty choose spam term regime error eq cccc report time step
var cast covariance available fundamental problem portfolio fmri study challenge grow natural ill medium approach obtain shrink specify autoregressive reduce view structural covariance attractive since provide covariance simulation show covariance proceed var integrate propose reduce fitting scenario var fit scenario ar var procedure reduce rank apply example concern stock return china derive reduced covariance estimator var modeling integrate reduce fitting var estimation apply dimensional latent latent independent replicate
rate difficulty control result problematic recursion simultaneously circular involve emphasize crucially keep penalty ensure would seek convexity strongly provide remove bind strongly loss strongly high put term extend case orthogonal genome streaming streaming regression average exploit theoretically rate try un exploit however add also streaming method goal competitive implement software experiment handle simulation create run linear instead tw c third method geometrically distance specifically random vector py w entry draw ccccc parameter regression linear time prediction measure aggregate realization slide window example addition online plot outperform correlate norm algorithm margin bad correlation rely strong perform difference particularly logistic incur achieve possible prediction recover optimal comparison treat streaming expect bring method around phenomenon fact desirable note term runtime fast run
size linear minimization option segment piecewise polynomial jump recursive recursion end provide parameterization calculate denote worst computation together complexity adapt present onto union subspace polynomial accelerate scale instead one ideally several update nearly natural maximal stopping course constant I estimate original modify impose continuity create binary program element change zero serve program continuity otherwise token thm corollary proposition thm thm cs il representation methodology variational strategy
estimator distribution perform enable divergence inference pair distribution experimentally theoretical use achievable nonparametric divergence nonparametric divergence consistency already mean divergence field generalize leibl enyi divergence rate probability distributional application information compression channel code mutual machine processing clustering entropy special distribution intrinsic estimation however beyond inference divergence detection hypothesis divergence e specify divergence establish
video forest choose million pixel require take tree train comparison forest forest test show accuracy forest huge cause high consuming maintain image forest colour f precision forest threshold quantitative analyse acceptable nonetheless able cope large complex tp noticed processing region return segmentation false face classification process region classify
tag provide estimate tag vary impulse third set quantile performance around evaluate accuracy sparse code matrix loss encode code example code dictionary dictionary negative typically solve fashion code update turn update fully frobenius may update perfectly loss penalty across consideration problem scheme requirement constraint specifically quadratic penalty recently broad statistical interpretation devise interior incorporate method code update make particularly model representation penalty discuss update scheme quadratic support reader exposition arbitrary nonempty matrix
environment simulation deviation guide provably construct prove impose guarantee principle sure respect denote action heuristic level deterministic path approximately reader manuscript result mention strong converge differential denote absolutely continuous strong deviation principle clear coefficient obtain recognize come deviation imply simplify qx bound almost unbiased via monte generate copy sample precise copy desire efficient jensen deviation q therefore actually opposite almost call importance notational convenience define wiener control sufficiently define
subsequent fully net remove interested identical allow big mnist version feature size initial decay momentum dropout reach fix comparison train class representation comparable artificial augmentation non augment conjecture
see article accurately relation word iii vary rating positive datum user examine recommendation movie show iii true movie unchanged new movie english american belong movie change get phenomenon observe number increase enough rule complexity representation user dimensionality computation update bias layer thus total matlab gpu acceleration second epoch dataset second epoch satisfactory large show scalable change pure significantly art jointly perform learn collaborative far first bridge state rs generalize propagation bag word representation alternative bayesian nature performance boost incorporate admit
consistent code word fairly reasoning therefore similarity number report kind insensitive sensitive sensitive insensitive publicly available embedding provide initialize denote good refer skip gram relation explain frequent initialization great job frequent bad showing skip gram greatly reasonable improve quality embedding context information note performance little embedding besides rare initialization word embedding embedding train recursive structure train less minute since update relation balance hierarchical knowledge rnn knowledge type actually knowledge basis leave word skip knowledge skip gram combination uninformative problematic inaccurate competitive noisy sample rare word embed embed cosine similarity five investigation accord combination task therefore skip combination skip gram relation besides four type process denote overview knowledge look actually job
identify extract salient demonstrate technique review scalable automatic extraction compare reference document create extract much preserve extraction convnet extraction extraction model acknowledgment would thank nlp early rgb united research present document computer vision extract scalable sentence avoid consume human symbolic researcher decade recent
effect mild learn outperform routine value reach reliably benchmark baseline analyze network insight challenge establish post becoming win correct development interpretable bayesian interaction interesting follow architecture domain uncertainty ideally offer balance real process benefit frequentist emphasis stay characterization conventional work experiment elaborate stationarity yield acknowledgement thank provide time google award amazon sciences institute advance
xu value lead conclusion additionally plot xu xx observe xu go generate xu xu relevant xu xu relevant compute xu xu xu xu hypothesis force x step collection suppose compute x x x I interpretation claim interpretation claim c claim unbiased report xu variability claim bias nature bias conservative compare deal support testing adjust numerically support multiple property microarray give gene white cell measurement basic h cccc cccc gene vs u w goal
center first center new analytic hereafter denote z z z global definite attempt close onto configuration fourth multiple argument global minimum challenging involve existence close local sufficient nonnegative definite encounter learn set empirical pc section c collect build rule classification train predict new select dimension sample whether discard considered selection space greatest z contain index group r modify statistic multivariate population statistic separable section choose principal inversion cause numerical exploit variance discard covariance accordingly permutation usefulness choose pc feature translate hypothesis test alternative calculate
parameter method mcmc source factor separately beta iteration report fm improvement fm dataset netflix dimension expressive rmse demonstrate fm topic use fm baseline experimental publicly implementation posterior training skip train netflix rmse baseline fm addition
split part surrogate f approximation moreover convex strongly also proximal gradient minimum often form f proximal proximal soft review proximal operator reader g appear dealing logarithm amount reweighte reference adapt logarithm replace consider real value function indeed convex note lead alternate proposition assumption surrogate useful instance regression huber huber smoothed loss represent associate problem formulate minimization x linear describe begin satisfied reweighte least huber inequality presentation smooth l l rate surrogate believe present jensen nevertheless instance procedure concept algorithm exploit concavity logarithm jensen
image scan disease assumption assumption positive novel standard assumption example instance belong suitable might problem describe bag classify anomalous advantageous classify cell classify decision influence generating application label face detector instance reasonable person oppose patch group bag recognition example image frame person image group annotation sense bag segment bag label belong background object song belong specie annotation annotate segment label costly weakly annotate foreground present bag fraction instance information output classifier get label name another spatially likely interest medical weakly annotate benefit bag patch
element wise restriction set simultaneously need adaptive measurement group restriction boolean gain cs obtaining therefore snr attempt evenly achieve least sublinear bit interesting measurement open question whether performance similar nsf corollary bind complexity sequentially low mutual information
corrupt conditionally conditionally mse attain limit ml noisy observation corrupt additive fc use estimator estimate sensor reach limit expect estimator therefore threshold htb relax conditionally observation conditionally sensor convenience fc observation derive optimality chen simplify system fc conditionally introduce variable chain hold conditionally independent n equivalence inference optimality optimality section derive manner pdf random positive early
generate feature episode twice represent link hand later extraction structure hierarchical preserve typically produce pool realize training aim linear risk parameterize weight loss follow convex order employ several ordinal loss negative likelihood outcome model lasso sparsity come instability theoretical intuition knowledge since independent clinical realize relation disease link ensure serve precision multivariate present transform temporal ed piece diagnostic exposition diagnosis version scheme applicable disease cover code letter digit digits head classify head medical contain represent clinical often episode episode visit end death health major contain event diagnosis admit home could come intervention assessment may list multiple problem historical transform sparse extraction technique precise instead exploit bank filter resemble filter sparse maximum history observation event discrete event diagnosis code duration parameterize th event convolution effect
correspond monotonicity positive value threshold put serve dag handle assign variable convention positive row argue monotonicity rather negative sample reflect monotonicity constraint strategy monotonicity penalize sample row large exponent specifically likelihood proportion row course include regularization combine define brevity asymptotically reconstruct correct correctly score monotonicity weight structure infinity play role overall stable choose develop score asymptotically especially enforce absence edge monotonic convergence size temporal see definition optimize
ordering count validate machine implementation count heuristic available package record count expensive none prohibitive apply heuristic give quantify output quantify single false might correct heuristic feature aspect express variable quite choose feature affect heuristic feature consider work h degree degree among degree occur polynomial proportion polynomial occurring occur place label heuristic could input defined polynomial feature feature across validation test section svms dimensional sigmoid
current psd practice project psd return mt task network common space mt treat follow enable transfer mt regularize learn qx rewrite incorporate constraint q ts ls q project psd q qx qx variable solve respect thing calculate partial subgradient metric update use triplet project psd hold regard dimensionality real world datum apply mt citation mt obtain article area wikipedia search also article article solely
em regard determined equation iterative mi tm substitute equation value sequence q limit since independent maximal correspond coefficient condition choose unique estimate close note solution penalty good initial quick super em tx procedure theoretical regularize large mx condition mild trivially standard chi trivially refer covariate consequence consistency condition j tend eq chi follow immediately chi recover mild oracle hold minimizer satisfy let tending theorem determine choose minimal adapt stability ss
hence fourth operation expensive decomposition typical identify dominate part right side approach exponential complexity assumption study uniqueness problem column affine independence factorization iff iff e affine figure uniqueness interpretation valid fail hold replace contain sequel play role solve extension negative factorization world factorization ask adapt solve column account negativity impose however return converse factorization principle feasible upper proposition bad vertex contain uniqueness uniqueness recently
provide first step understand brain network relaxed covariance independent normal whiten step preprocesse develop spatio possible instance study subject incorporate distance voxel assignment point choice voxel future research latter omit ignore irrelevant solve derivation equal derivative glasso like acknowledge national grant aa ai ns university research award institute brain pilot award university start em university american international publicly available brain attract fmri tool recover
tensor compute furthermore determine approximation np challenge rank become multilinear tucker unfold tensor multilinear rank completion multilinear multilinear problem open question class approach pass minimization though empirically achievable exist need substantially achievable formulation measurement focus find message pass need study tensor completion specific type oppose include specify show class pose state compute nonnegative pose study define risk function risk return question incoherence structure loss amenable loss partly justify interest approximation tensor compute loss approximation discuss soft attempt limitation move nonconvex combine hard soft issue design fractional factorial distinction incoherence differ factorial design write contingency combinatorial note combinatorial write many
bandit bernoulli bandit simulation allow illustrated robustness misspecification scale exploit replicate replicate regard tuning apply concentrated dynamically arm evolve understanding replicate favor work develop analytical acknowledgement work member facebook core team anonymous reference thompson bandit allocate arm thompson demand scale bandit dependent
minibatch minibatch method natural describe natural implement change convergence descent suggest improve rate poorly condition problem suggest method define outer job disk produce store iteration weight try work training datum number speed bfgs fisher average like weight stop aim optimize assume layer case would less optimize model average parameter duration find improve gaussians recognition use mixture gmm idea neural speech write mix dimension softmax number layer sum index class try class class group class evenly count class row old matrix plus term value modify normalize slightly result may truly effect rate describe note far conduct wu mixture regard mixture class multiple able improve result remove scale softmax proportional count average mention normalize zero mean affine transform accumulate multi discriminant class dimension fortunately lda actually space covariance desirable direction never type mention transform covariance singular decomposition singular motivation rarely encounter lead large transform rarely decide well establish improve get give improvement
monotone bound bound proximal possible difference prove solve low candidate exist solver broad nonconvex solver surrogate additional verify many surrogate table logarithm mcp laplace scad penalty bx bx gx kx identify give satisfy lie useful solver lie intersection bx supplementary intersection b satisfy xx bb denote
sparse component pursuit mention mild via involve nuclear norm recent many derive provable form proximal involve singular bilinear structure rbf low small trace error measurement robust robust account show call plus noisy component incomplete corrupt measurement calculate rbf scalable structure orthogonality convert scale problem linear constraint direction method solve linearization analyze remainder review background propose scalable develop efficient highly corrupt model denote subspace completion index result suppose incoherent r sign dimension haar measure probability measurement recover develop solve
synthetic dataset web collect amazon movie netflix diverse transaction click check etc ref ref predict preference behavior user pairwise build preference information novel rank generative comparison account essence approach ranking especially individual preference influence preference similar item iv comparison typically rank date aspect literature category optimally agree ref ref ref user preference category ranking population single ranking ranking ref ref heterogeneous preference preference behavior mixed membership capture multiple share ranking inconsistent preference mixture paper development efficiently consistently ranking topic corpus view probabilistic leverage recently topic modeling estimate approach run
artificial neuron reach human sense computer truly master visual e child consider core parallelization unfortunately despite year computer deep neural additionally fast ask high decade hardware evolve propose conditional deep positively correspondingly net
operator project one call amount compute perturbation state branching level insight proposition remark assumption infinite discount formalize markov variation policy conservative infinite recently policy per iteration error comparison particular attention highlight cost increase enjoy guarantee iteration contrary constant iteration problematic discount scheme confirm infinite decision mdp bound discount
extend dimensional extension dimensional chain integration dependent general formula explicitly q last sum denote set close eq I last unique point corner normal exist assumption b furthermore product therefore follow r dx last equality dx dx note plug standard ratio multiplying divide inside divide inside integral also function h h additional vanish old q therefore h know q draw assumption require perturbation eq r second
comment short test treat regard sdp methodology reconstruct incomplete noisy sdp lack bound take completely approach heavily distance semidefinite sdp lead convex nice importantly derive result social show treat work regard approach distance follow proof plan rank note j meanwhile since ta ta know directional jj j j thus jj conclude pt know desire show hold shall least completion r bad event interested shall x meanwhile rademacher know thus p finally exist apply bernstein bind old q x r proposition nm nm eq case know exist ex l thus random sub ex l ex l nc term dominate proof diag p continuously value
one want find marginalization direction encoder frobenius e generality regression minimize problem variance incorporate recover encoder replace x n unit supplementary material treatment achieved provide intuition equal join take choose poor trade prefer depend choice hidden solution highlight marginalization together capture principal axis non regularization cross optimal regularization hold trial record neuron form dimension marginalization hold repeat different train result clear yielded value argue axis encode decode axis try real material assigning variance argue toy axis stimulus direction vary correspondingly marginalization inside average arguably utility prefer projection full decoder therefore encoder essential equal figure marginalization matrix marginalization direction decompose stimulus decision bar plot figure decode necessarily variance work memory well average neuron explain simply axis explain decode axis stack row matrix standard sum eigenvalue covariance sum eigenvalue figure correction signal trial independent neuron random signal follow text assume variance therefore figure marginalization compute compute marginalization total angle star pair axis orthogonal sphere dot angle deviation quantify contribute activity
positive cone semidefinite cone simultaneously widely sparse signal variable group convex nuclear estimate lasso penalty encourage ensures interpret low aa throughout thing different statistical rather generalized especially sample take advantage see problem organization
perform via system implementation strongly assumption however physical addition ghz processors gb ram queue execution fix carry begin aim worker precisely worker perform computation complementary estimate kernel parameter dimension choice capability assess synthetic let mean consider addition range processor note experimental practical pass simulated dataset contain remain distribute acquire computing uniform bar
color correspond blue indicate green red triangle make individual individual use snp marker come train abc reference table subsample rate na I baye discriminant lda nn initial axis initial summary summarie axis axis neighbor initial summary weight neighbor regression neighbor implement I classifier number neighbors abc estimate minimize calibration error moderate I neighbor logistic regression minimize minimize use due calibration set heuristic summary use axis normalization use provide prior rate reference calibration table summary lda axes population albeit local bring expect optimal neighbor need quite large aspect local consume stress solely indicate forest calibration constitute I lda abc initial summary axis lda axes forest use summary summarie axis I bayes discriminant lda abc summarie lda axes local two axis random initial summary forest summary lda axis I discriminant standard lda lda axes initial summary summarie axis classification size reference solution average summary statistic lda axis contribute population meaningful discriminate important variable
often advance initial readily modify version dna fashion drawback error often propagate fine often employ extension network variant build stack rbms time top layer networks crf rao conditional random cubic standard inside algorithm quality respect early stage promise
se sp accuracy roughly specificity consequence misclassification kind make adopt science notation replace false positive false negative equally classify reverse formulation follow problem tractable false false negative use reason svm point section class feature traditional perform whereas measurement suffer would preferable new use fewer suitable combine feature preliminary version report training sample size repeat number choose formulation randomize vector threshold average zero originally determine use time assess classifier testing remainder performing value retain wherein mean statistically cpu comment comparable set half total application testing nonzero weight run another instead average run large adopt rank time randomized retain experience index retain iteration choose randomize right step training testing final specificity advantage many svm eliminate algorithm
weight defer carry believe situation evaluate compatibility factor evaluate restrict constant focus provide compatibility factor even correlated design belong set jump drastically result tv estimator ever lead importantly much piecewise vanish one bind first consider jump moreover work precede section fast bound two turn present suggest fast incorporate refine interest deduce minimax monotone example follow slow tuning lasso correlate classical logarithmic theorem rate set close span constant satisfie q euclidean span covariate fast effective number refine replace effective number correlation exhibit perfectly design belong span rate corollary differ fourth introduction finding comparison dependence corollary application
iii iv vi analyse vi conclude remark vector nf integer minimize convex positive definite denote fx policy within define policy usual asymptotically besides continuity lead characteristic tail convergent small consider contain origin exists control policy infinite exist admissible satisfie control bellman however bellman optimal curse look table neural term within approximated denote interest envelope valid state trajectory remain vi find vi initial guess iterate guess converge monotonically utilize converge reconstruct rarely problem parametric purpose function
kolmogorov cdf decrease practical reason rank weight calculate statistic monte carlo discretization observation expression sample level real low next point give gene inside observe vector test gene set sum return procedure k step function jump supremum test rely gene compute nan goodness fit theoretical cdf cdf
fact k correspond slide simplex condition course map k category tie sample mapping mapping crowd simple exact item favor expert vs crowd opinion entire scenario change g vector item category computation give assignment simplex basic affinity remain unlabeled satisfie harmonic unlabele respectively u linear affinity take train free eq clear laplacian laplacian effective I structure essentially set category follow principle harmonic lie maximum label lie constant thus assignment nonnegative sum predict assignment subject explicitly simplex simplify item belong category implement coding widely instead unnormalized latter variation square free intend extend replace divergence add term exactly meet play role propagate smooth item rely give
score preprocesse phrase provide author fold ten fold recently publish stanford sentiment include phrase sentence sentence sentiment label convert ordinal use structural phrase label partial sentence experiment employ ordinal multiclass setting nonlinearity softmax ordinal corpus experiments accuracy experiment randomly initialize word rand dimensional google b word reduce additionally initialize rand words acc word nb recurrent report bottom rnn
extract descriptor information response neighbor entire depend standard save hold dataset split aim adapt nn base cnn subset training error output cnn improve give algorithm accommodate centroid regular centroid solving solve test rate descriptor detail compare test training large large training curve intermediate compression ratio reduce nearly match nn marginally ratio superior baseline subsample cnn notable final
l noise omit clarity effectiveness cifar mnist image handwritten training experiment implement model multiplicative incoming weight mini rate factor epoch accelerate linearly momentum epoch reach performance result low validation train final good validation model random initialization outperform maxout achieve good
bfgs standardized estimate behaviour first correct regularizer bias structure meaningful rarely confident great research well calibrate equation collapse estimate show quantitative gray behaviour error relative estimate cg bfgs instantaneous linear leave projection px mb element q show blue stationary model figure stationary drastically figure show bfgs standardize prior outer albeit loose one course cg probabilistic interpretation exposition cg converge identify converge intuition one bfgs never explore block arbitrary elaborate course choose interesting way inversion like gauss direction text interpretation iterative solver derive posterior mean rank conjugate bfgs update rule cg apparent inference perspective bfgs well scale standardized sr lead correction form cg consistent definite cone bfgs cg possible rule
integer element intensity parameter density poisson likelihood evaluate give factorize close bm denote value marginal k observe define comprise initial treat mle nmf formulation want nonnegative one kl write q similarity generalise
low low excess differentially private strongly decomposable half decomposable minimizer terminology excess differentially eq give low every whose q minimizer case extra first universe choose entry change tight tight bound desire construction differentially private whose construction prove isotropic position reader deal general necessarily isotropic run distribution hypercube hypercube lipschitz sample property let j next walk reversible stationary walk fu mix walk distribution output step statement cell whose guarantee chain walk towards rapid chain space reversible markov set p observe eq lipschitz plug p completes bound output distribution close define standard trick weight attribute outside namely extension define function cube guarantee p position first namely variant
dictionary achieve fr truncate nn le de truncate le english en fr translation produce neural network system handle word highlight token novel achieve conceptually read translation appeal domain knowledge suit formulate network generalize phrase sentence store explicit phrase table conventional finally decoder unlike base despite advantage rare word vocabulary force vocabulary word sentence translate poorly sentence frequent word phrase base
dot represent series term dot represent dot pt repeat necessary diagram expansion suppose hamiltonian dominate deviation sufficiently series calibration response uncertainty show concept ref include uncertainty amount weak uncertainty reason clarity uncertainty ref taylor expand effective hamiltonian form strength contribution call new become justified correction diagram diagram identify pseudo accumulate uncertainty correction value e emphasize unity justification expansion sec small formal definition eq whereby formulate
bic n minimum cm white bic style mm cm thick white circle sep draw thick color bic black bic circle sep mm fill style sep white fill sep cm white fill color bic circle inner mm cm thick white color black style circle sep draw fill color black circle white style inner mm size draw thick fill text black bic style circle minimum cm thick text minimum cm thick black v v n v v v v v v v v v v v v v v bic bic bic bic bic bic bic bic bic v bic bic v v v v v v v v v v v v v v v v true include never choose htp n n v v v v v v v v v scale bic bic bic bic n bic bic n bic bic bic bic bic bic v v v v v v v v v v v v v v v v leave bic correspond high selection node never without htp v v v v v v v v v v v v bic bic
pre inspire predict automatically filter entity type object w evaluation table metric drop word modeling entity compositional phrase entity name promising greatly benefit train level relational scoring relational embed framework investigate bilinear also entity extra several interesting finding enable
lie lie strictly length least contradiction recursively construct definition real cube compact cube contain case subset assume diameter length recursion set vc existence construct property recursion say nice satisfie even odd addition every origin leave semi infinite semi final technical bring set help
consist define minimizer focus transformation sequence transformation representative algorithm consist minimization local whiten global structure view trade interested soon formally serve maximize introduce replace otherwise computation preserve iterative think modification include additive force cifar consist color partition contain comprise dataset
cl al deviation justify cl finitely elaborate cl considerably explain insight improve would true tree cl replace mutual estimate namely naturally small identifying however bad mutual information mutual theorem highly regime minimax worse require essential optimality latter cl intuition fix star independent give independent distribute entry probability normalization set overlap wrong edge set
select initial constant achieve early study fix boltzmann base axiom arm mean pick softmax select boltzmann mean randomness boltzmann act infinity pick uniformly decrease fix pursuit explain essentially pursuit maintain policy arm inform use version pursuit algorithm start arm actor problem pac form reinforcement pursuit maintain directly reward select increase decrease scheme design account case similar value maintain preference boltzmann play reward preference turn exist date family simpler elegant ucb planning prove go play program simplest maintain arm play pick fisher ucb bound ucb achieve multi armed tuned perform come without guarantee ucb variance arm pick maintain mean provide regret ucb ucb tune instance characterize aspect distribution affect relative performance surprisingly consider importance compare goal setup characteristic bandit affect arm type arm admit tune optimally setup learn curve
form structured selector dual suitable ds key aspect term original recently atomic estimation framework unlike consider norm atomic norm aspect norm selector primal homotopy linear immediately extend formulation alternate direction multiplier problem linearize prove ds inexact admm primal interestingly turn proximal update conjugate indicator decomposition suffice side interestingly set trivial operator efficiently support focus proximal setting provide error yield
analyze combinatorial sequentially construct count time new row previously unseen feature similarity argue flexibility ability count make suit wide variety world framework bayes random exist require predefine vocabulary share category beta binomial show outperform categorization need count arise text document term record many appear site record observe arise count relatively moreover major conceptual count row add sequentially row previously unseen feature word specie require count obvious row count unseen bayes classify predictive account feature ignore previously unseen issue investigate prior construct poisson gamma binomial lead count matrix time count exchangeable underlie arrive count take highlight certain evident rely novel
involve dimension cutoff subsample sensitive additionally possible merge easy user interpret cluster result potentially help overlap truly homogeneous cluster subsample sequential generally accelerate advanced method inferior convex cluster mechanism center iterative subsampling need number well noisy datum simplicity intuition magnitude present appendix n ari true estimated index respective indicate true respectively contingency ari base count c cccc sum use row row sum effect misclassification identify misclassifie noise estimate table count account number datum calculate year fellowship part nsf grant dms efficiency spc spc regularization distance cluster center capability recognize solution subsample order include mechanism ultimately tight cluster simulation ability handle dataset application gene class large sophisticated art
heterogeneous employ sparsity system minimize tr sn net aware introduce proportional heterogeneous region choose rest I network signal spatially input keep
inaccurate sample define event require additional since selection fs genetic sample genetic moreover
university usa group unique character certain people service become year study implement graph weight edge graph develop analyze characterize present dirichlet allocation lda lda predict generate representative group determine topic content author distribution topic preference website gibbs
nd x follow next lemma proof switch notation satisfied assume iii get calculus put contour picture g spectrum therefore order singular
distribution point nearby achieve parent marginalization tree tree label categorical normalize stable process parameter take discount variation prior ty p e consider modeling hence resort fast popular smoothing approximation straightforward instance refer reader detail use specialized general operation new split split update split exist child knowledge decision third unique tree follow walk input x j j proportional split leaf j stop leaf continue discuss version partition figure step h second third iteration partition though split gray rectangle new new lie extent
neuron item depend neuron let typical neuron neuron brain neighboring neuron fire action potential enough cause various neuron precede item strength neuron income time plausible firing fire threshold huge firing threshold small assume long consecutive step naturally comprise simultaneously fire direct neuron every discrete boolean whether fire another memory finitely keep consist another understand operation state influence potential fire status update q potential component neuron incoming notice function happen fire certain plausible description algorithm instead present state shall strength income next join operation perform exposition join operation operation basis least desire call fire nan state strength income total come fire strength strength incoming enter incoming
short backtracking variable short remove short negative even birth track integrate short confidence track style default vertex default default vertex default style vertex default style vertex vertex vertex default default style default default edge forward edge style edge draw forward style edge style forward forward style forward style forward style forward draw style blue forward blue draw blue forward draw edge red blue forward blue forward forward edge forward forward draw draw blue blue birth death flow red edge node pass successive difference dynamic programming shortest backward include pass dp array pass backward last frame frame original backward come one pass forward cyclic forward edge node short along variable iteration pass dp dp behave path pass go go track splitting track choose entirely track terminate flow quadratic eqn
nonnegative I directly neighbor comparison show without formally consider problem boundary outlier amenable theoretical analysis simplicity practice regression widely modification alternative regressor benefit include weight remain unchanged variance view much condition next interpretation prediction problem predictor predictor prediction preferable local develop locally conditional estimate cubic query consideration near neighbor recursively gain within predictive cart develop definition reduction heuristic split algorithm average split axis align split cart construct mode splitting leaf node intermediate align thorough base represent disjoint region correspond directly average point propose follow weight piecewise partition recommend decision constant problem recursive fully decision truly also cart interpretability interpretable tree interaction dataset simplest admit problem forest ensemble train component subsample forest one flexible tool ml thorough random extract forest propose weight combine idea aggregate cf forest weight extract tree ensemble practice rf flexible prediction algorithm perform predictive rf overall evidence rf efficacy absolute efficacy bias favor overfitte disjoint estimate black relative quantify overall universal scale determination absolute
tie tree propose locality allow multiple exponential quantitative set helpful left distribute preserve hierarchy activation ability subtree subtree induce tree feature
convolution xu wavelet section applicable hence general spatial rank determine throughout discuss n n easy var explicitly location apply decentralized filter posterior server server j central illustrate server central server move server computational moving plus central communication scalable massive matrix achieve assume compactly compactly parent case correlation j server treat contain know usually implement fashion extend vector dimension three parameter
evaluate perfect distance family appeal exist divergence chernoff divergence tight estimate come information theoretic bound kl divergence tv divergence inequality bind tv drawback inequality uninformative kl go tv refinement chernoff value minimize bc case bc motivation literature close form many chernoff measure bc beyond number exist author yield arbitrarily tight set bound empirically know bayes area david label new subset generalize author
k put iteration fix say clearly k x run b time fact well speaking possess dimensional tell solve influence good briefly discuss constitute suggest time find row partition less c taken solve leave heuristic ensure direction follow understanding say suggest heuristic idea dominant section example indeed parallel computing imply processor secondary device system computer equip storage device objective necessary additional resource solve usual row consideration readily preprocesse divide roughly equal feed parallel iteration main involve phase read memory modification already observe first phase multiplication operation concern across processor give processor secondary storage stream idea suggest parallelization secondary storage perhaps require hardware understand
achieve well denoise amp amp wavelet amp amp haar thresholding min haar choose well signal amp patch length search window parameter setting choose allow level amp original though domain capture effectively amp amp reconstruction algorithm amp bm bm amp filter omit result use illustrate correction considerably compressed test measurement matlab ghz processor wavelet optimally amp describe iteration amp run amp run iteration amp yield improvement code fail initial noiseless test iteration bm amp estimate final amp six processing house image present figure rescale restrict entire matrix store create amp store version signal begin recovery amp bm bm amp outperform vs dramatically outperform wavelet amp also comparison amp rmse bm clear amp bm amp majority denoise amp compete presence amp bm amp amp bm amp bm amp cs bm amp bm amp bm c amp bm amp bm amp amp cs amp amp bm amp c c cs amp amp bm amp realistic setting subject measurement sampling
show completion capable entry provide theoretical ad calibration increase incorporate property characteristic improve robustness accuracy extraction hoc organized pairwise distance coherence mathematical euclidean dedicate theoretical guarantee hoc array completion relate draw locate position circular room signal room locate room near base express distance represent sensor observe characteristic miss imply locality miss lead short distance miss underlie acoustic furthermore locate whose position euclidean transpose construct write hadamard square locate plane place circle rank exactly hence dependency low property pairwise distance introduce square distance kind miss distance structure miss noiseless recognize know square matrix noise matrix random
condition al orientation stand inverse inverse et al lda require cost release overhead right hand rank orientation present year reduction mining et et et
token member assume make restriction latent intersection label simplification model variable set disjoint restriction simplification length example shown illustrate labeling sum latent formulation crf crf see special employ calculate summing feature represent represent local consist parameter perform optimize objective term training reduce define analysis base
hypothesis parameter problem arise association set assume normality definite interest section restriction matrix shall composite density nan chi freedom coincide classical contiguous alternative n non central chi centrality contiguous centrality however contiguous htbp significance l examine test result nan statistic influence influence unbounde imply robustness corresponding statistic plot figure extend influence contamination point decrease increase test robustness type
method genomic bioinformatic brain kind neuron overall neuron differ chemical also type connectivity prominent processing movement make circuit challenge use e discovery aspect early property scale present understanding work repeat node represent repeat topology well commonly probabilistic connection rich cell neuron importance connection traditional arise genetic address challenge describe nonparametric automatically pattern location incorporate additional agree identification cell recently additionally compare discover human agreement future build probabilistic begin unobserved type cell connection cell nearby cell broad generative connect
generally something single science point general lie situation information exposure pre treatment situation unobserve exposure finally variable conclude simple information randomize study ann exposure potential individual randomization far suppose
quantity onto equivalently norm leave vector input outlier provide full truncate score well algorithmic date semidefinite interpolation rank leverage interpolation maximal ridge coherence also sum capacity provide notion statistical counterpart leverage well ready guarantee nystr sufficient number sample loss come sensitive ridge column construction find nystr om column kk matrix om induce provide scale effective
prove recognition acoustic dimension conventional dictionary dictionary new training svm classifier utilize voting class kernel version rbf bandwidth select via method exploit heterogeneous show modality employ main coefficient use sum coefficient vector final occur generalize aforementioned combination di category feature fusion fusion signal although sensor combine information sub optimality exist observation decision result misclassification aforementione verify test sensor different purpose nine sensor acoustic sensor combination process nine conduct six notice two acoustic sensor corrupt sensor sensor acoustic sensor utilize acoustic effectiveness sensor clean acoustic sensor nine sensor segment extract overlap segment dimension interference process sensor set coefficient minimal define six sensor validate efficiency method similar take accuracy single sensor set iv nine sensor interesting sensor three classification human h
obtain purely illustrate life worker collect additional care precision define integrate nest motivate present concluding remark mixed extend predictor consider response group distribute let known link ij ij unknown n q distribution say flat assumed parameter derive specification decide upon range mix interpretable
maximizer give start iteration facilitate chain run one could easily devise elaborate optimal rule open guess start use component final ingredient procedure quantification correction estimate intensity quantification empirical quantification straightforward example credible interpretation drive interval take regard frequentist property understand amount I bayes interval use bootstrap technique aim achieve sample standard arguably desirable actual physical quantifying technique frequentist approach result observe poisson regard one different resample empirical spline coefficient irrespective hyperparameter spline intensity result bootstrap procedure illustrate interval resample basic percentile interval clear superiority scheme usually difference basic interval percentile follow enable bootstrap probe large percentile poorly interval implicitly scheme bias conceptually percentile sufficient computational resource
cluster information exploit dynamically outside able high able successfully motivate examine already disjoint decompose separate seek combine denote combination random agent intermediate iterate iterate I k indexing estimate agent later small enough gaussian semi agent example difference matrix result infer indeed concentrate sufficiently agent hold probability observation agent determine member I I threshold agent test reach miss detection exponential therefore long agent successfully infer acquired dynamically adjust iteration accept hypothesis dynamically evolve neighborhood introduce diffusion iteration causality neighbor already summarize recursion update recursion key arrive theorem derive useful intermediate first recursion shall therefore examine evolution recursion influence study introduce error k use minimizer throughout network follow dt rewrite worth indexing definition combination possess diagonal
adopt recursively subset partition ask dyadic adaptive construct next question na dyadic dyadic policy simplify dyadic dyadic policy partition recall dyadic question f distribute take dyadic policy accord expect beneficial value dyadic actual term second converge martingale show sure asymptotic normality direct dyadic deterministic I random seed history include dyadic random binomial accord final cardinality dyadic policy nonempty product differential entropy denote term dyadic furthermore analyze term almost sure martingale convergence hold nz
efficiency optimal otherwise efficient separate inefficient fully always feasible model difference matrix exclude evaluation super inefficient never matter unchanged matter use super efficiency holds change diversity efficiency efficiency accord efficiency score indicate mi base redundancy perspective exist average label never accuracy finally reflect exclude negative express select link point natural handle redundancy label call separability follow feature always range classification form kullback leibler value assignment interpret program mi feature calculate label calculation process separability take weighted redundancy mi fig mm cc label cc cc ct cc cs mutual change calculation mi
bootstrap bootstrap shall selection regression consider maximize extension consider identically I estimate context selection shall criterion log correction try obtain bias adjustment use cross cv widely selection cv bootstrap base selection cv reduce introduce cv cv bias define follow thus cv correction direct let observed let estimate set replication define likelihood quasi cv distribution sure replication
select without low fdr good model lasso one unnecessary include plot whether look right lasso property oracle possible unnecessary oracle region inclusion offer validation would also phase tend validation specific figure plot cv sure poor nearly phase transition diagram replace make lasso probability cope independent screening consider cv carlo bic stein sure lasso sl
algorithm completion problem continuous choice list find recovery performance technique enhance dynamically decrease stop reach predefine initialize nonconvex function nonconvex experiment free rank r compare augment lagrange solve task alm evaluate recover relative regard recovery
computer store technology facilitate functional motivated various functional back extend detail extend relationship include functional logit functional polynomial nonparametric linear due popularity specific predictor predictor variable field functional density understand residual assumption cs symmetry residual density useful financial asset return hold wrong error density produce inaccurate asset unable risk estimate motivate response variable value continuous deal regressor power continuous scalar density kernel distance fit
odd behave predict conclude across message likelihood therefore connection turn question perhaps find way majority serious traditional word co carefully light finding help getting adopt responsible complicated think remain player careful consideration accounting wu economics business interest include hold management publish article international conference book candidate social
association correspond nm perspective devise mcmc two main mcmc design follow explore later move aim order new target mcmc nn nj mcmc move association algorithm explore extension da linear designing demand loop data step hmms alternative clutter hence convenient magnitude propose new exist modify link modify link observation variable four move move mcmc dimension birth extension move death reduction move move leave invariant self reversible pair essence move nz nj z z calculate move z move jump different birth death exist target sketch birth propose base change birth trajectory time proceed define logic behind candidate space observation locate randomly provide empty terminate
intuition behind view contain low monte mix carefully appropriately particle improve particle particle particle augmentation key particle process observation analytically particle allow less information particle believe augmentation particularly particle strong parameter process particle deal stochastic end marginal conditional function parameter process often depend approximate consist path full particle filter within
input autoencoder stack bottom top observing objective distribution arbitrary regularization ingredient lead autoencoder autoencoder objective autoencoder dirac since autoencoder joint mention autoencoder optimize reach objective reconstruct accurate intermediate aggregated loss goal furthermore representation vary continuously change focus reconstruct explore deep autoencoder nh layer enable regularizer single autoencoder autoencoder feed output feed layer reconstruct activation reconstruction input use layer gradient global activation autoencoder second autoencoder look optimize exactly unnecessary reconstruct back input parameter level input address drawback long
specify particularly work namely method correspondence way analyze parametric us framework transform regularize yield estimator shrinkage meanwhile substantially perturb lagrange q think make perturbation non yield desirable seek via feature closely connect aim network training dropout work maxout imagenet provide serve specify underlying estimator surprising isotropic gaussian equivalent shrinkage induce stable autoencoder solution autoencoder
suitable subsection update collect input agent intuitive reduce adopt ec explore toward visit everywhere central base update coverage simplify aspect agent movement particular markovian depend whole history function dynamic agent fully posteriori generic control phase coverage estimate control coverage review section variation movement another establish statistic vary past history
context content model inform information topic robust partially miss achieve cluster performance domain extensive scope topic counter part hdp word model cluster employ lda hdp extract proportion input lda hdp affinity human activity elegant discover first nonparametric atom achieve introduce base model document document corpus share throughout recently mix topic parametric fixing crucially attempt utilize context model brief account variant dp vast relate building constructive property stick break stick convention hereafter mix associate stick breaking use
real fast convex thus suggest robustness analyze special assumption recovery minimizer nevertheless important aware single outli magnitude relaxation fail normalizing center unit discussion compute subspace rigorous treatment guarantee result follow l l try minimize minimize whole iteratively least l rigorous explanation apply pca center ability attractive procedure outline first compute top scale point fast vector iterate
expectation numerator denominator value account auxiliary expansion relate f da upper easily verify spectral bound magnitude large recall contrast z fx recalling recalling equal last fact orthonormal plugging derive equal focus substitute write justify lower bound first least get derivation derive term plug back simplify upper place constant sufficiently recurrence epoch
digit choose aim extract linear database roughly worth point set challenge justify database individual subject near image pixel face database contain image pose expression subject image pose handwritten digit database contain handwritten digit use respectively image database normalize unit norm tb construction euclidean kernel configuration construct number set respectively line near construct essence toolbox lasso construct
proper convex true exist sequence convergence influence advantage fusion spatial response due cause scene mention response different variance response strict sensor response fusion relative spectral response use call response sensor try able approach estimate account regularizer discuss consideration possible reason hyperspectral normally full drawback subspace fusion onto subspace hyperspectral involve observation cyclic support column denote kernel denote j def bc def impose note constraint therefore even function minimize correspond noise remove neighboring horizontal difference adjust approximate without normalize dc unconstraine cover specify support estimate long actual concern deal overlap band constrain band denote hyperspectral band band band row contiguous hyperspectral band
ratio split reader depth discussion split significance assessment full raw online last ratio aforementione split assess significance employ sign sign test two repeat sample match order rank student use fold accuracy besides basis employ average accuracy common threshold significance additionally comparison apply post hoc method wise series furthermore optimize cross within combination grid specification give paper linearly spaced integer use spaced value linearly space possible distance low leave error keep test extra result measure outperform perfect accuracy e achieve good set perform fc perform ar count statistically thus
implementation streaming supplementary extension intend reflect detail across scale variant learning project learn frame spherical temporal layer technique abstraction occur towards stage pathway classification start file rate file overlap filter reduce environmental normalise root apply reduction useful median finding median band subtracting spectrum every spectral energy background cope simplicity across make onto learn spherical mean benefit model short variation derive order summary common alternatively reflect spectrum frame overall feature overall six six pool forest implementation issue datum pre release manually tune parameter label different decision label classification assume specie label match task potential potentially relevance classification forest full label situation difficult task large volume full situation model dataset comparison classifier multi relevance width fig contain long audio yet annotation time specie common format annotation annotation specifically automatic file make file decision
operate stream proposition simple defer distribution eq w w follow theorem stream adapt level exist active hyperplane bayes omit dependency show achieve previously without know label distribution characterize prove explicitly construct principle margin slight setting help adaptive provide sketch generalize concave neither differentiable minimize surrogate logistic loss surrogate passive learning achieve excess surrogate probability least nd minimizer generalization number I example theorem hinge c f excess working error many include hinge loss margin active parameter
ise ise use goodness biological organization interact ise arrange represent mathematical ise integer consist except associate edge adjacent associate quantity quantity count configuration represent configuration color white configuration read represent diagram tb ise configuration normalize sufficient sometimes consider temperature depend replace configuration get sufficient correspondence representation notation lattice nan configuration describe range homogeneity define distribution condition statistic computation give use chain mcmc ising grow lattice markov achieve way overcome markov chain formally basis limitation approach first note every uniquely nonnegative ise exist
slice give ica convert problem mixture exploit method precisely mixture polynomial certain non degeneracy incoherent dictionary minimization exact handle sparsity tensor program enough also handle level require signal expense complicated consider work overcomplete setting topic provide tucker tensor decomposition identifiable order observe decomposition decomposition technique empirical lead bad rademacher context spectral reduce bound tensor concentration trade rough cover dense require fine classification tight moreover general rip norm notice standard clarity asymptotic say real member vector may tensor I I pt convenience rest tensor instance mode refer column refer row fix index arrange tensor slice index slice rd multilinear tm mi u multilinear tensor similarly multilinear combination slice rd tensor rank vector generality said write closely multilinear form denote matrix since weight norm operator operator tensor rd section latent mixture analysis detailed decomposition efficient tensor decomposition latent variable provide tensor introduce exploit latent provide guarantee argue propose throughout simplicity high order view independent categorical kb define parameter cp decomposition rank hence third factor addition covariance denote noise
autoencoder learn instance word embedding goal place lexical offer strong performance easy word embed representation dominant unsupervise heuristic novel
control term vs account analyze recovery incoherence theoretical minimax regularization unfortunately practical problem tune additional cope practical tuning issue model cross easy give ideal could probabilistic problem efficiently rely heavily algorithmic formulation separation general rapidly evolve hence important highlight convex formulation constraint solution constraint also cover project accelerated gradient disadvantage formulation
consider th error bound approximation influence get function quantity get combine present follow influence propose quantity show influence satisfy parametric robustness propose statistic robustness get power substitute expression corollary whenever order approximation contiguous hypothesis contiguous contamination interpretation indicator derive put thus influence statistic contiguous subsection influence simplify univariate distribution thus contaminate contiguous chi square freedom centrality influence influence zero tc explore robustness stability contamination contiguous may ng restrict case make routine density nan
mid audio motivated improve speech convolutional network apply music spectral domain base feature mathematical stable via deep computing module invariance visual extend hierarchical additional invariant signal explicitly arbitrarily music sensitive small affect characteristic stable pooling smooth variability conventional wide band instability high keep pass component
intel ghz single improvement delay logistic regression simulate environment obtain main burden mixture jeffreys prior constitute set exactly logistic stem explain use benchmark paper indeed big mcmc researcher keep effective focus computing sampling classic scheme contribute attempt datum play computing control instance incorporate control proportional variant generic approach pick branch especially pick delay elliptical normal proposal runtime runtime stand delay hasting combination core delay acceptance version computational chain asymptotic delay acceptance classic metropolis colour sd mh highlight
twice almost surely poisson various characterization conditional model atomic measure sigma generalize formula classical kf functional fw w term apply denominator simplification q line completeness numerator yield undirected rest latent count rest hasting pl stepsize gradient momentum hamiltonian p exposure omit index simulate discretized q accept write pdf allow intractable hasting acceptance improper prior ij truncate efficient accept probability bipartite iterate distribution eq axiom theorem criterion exercise remark summary proof figure fill thick fill fc european intra european fellowship support fa fa modeling represent namely adjacency exchangeability apply necessarily empty rely certain choice underlie construction process degree distribution use derive representation network explore hamiltonian exploration range range facebook social circle political citation web include hundred thousand million class rapid availability importance drive behind attention build history os enyi fail world model recent conceptually
analogous detect relaxation statistic present computer simulation compare tractable experiment choose covariance assume nonzero choose varied entirely symmetric maximum canonical top eigenvalue mdp require power thing confirm maximum canonical eigenvalue well really top table eigenvalue competitive implement iid htbp vertical proportion though remain identity know oracle tight moment nan htbp cccc vertical leave interesting paper obtain rate whether computationally intensive even moment method analyze theoretical adaptation throughout shift hard unknown minimax low bound unknown hand easily lee way procedure accommodate scan procedure know canonical correction simultaneously variance nevertheless rely together concern moment mixture covariance paper population two alternative seem meaningful complex group however way population case identity alternative affine perturbation natural base top relate result investigate population population issue consideration researcher relaxation context detection covariance seem extend suboptimal test relax case prove bound line calculation versus hypothesis nan alternative sequel isotropic reduce hypothesis non zero implicit make problem versus bad case risk testing last risk lr versus low lr pearson lr simple cauchy goal lr reduce follow refer random red red
edge value conditional jump membership connection probability evolve propose membership rely extended try briefly think challenge heterogeneity method large recently cluster network handle possibly independence namely undirected group rely replace step low group small still room network second give social associate suffer obviously complex statistically valid challenge property model procedure asymptotic recently case theoretical inference definition corollary st fr present selective modeling heterogeneity extension development field application biology internet individual interact represent node individual interact relationship molecular interaction presence absence record huge graph reader e general appear method detect heterogeneity still cover quite past year present suppose complementary mention complementary focus review homogeneous vertex literature friend index
influence contain propose party gender year benefit likely associate influence relation adjacency common follow context however utilize challenge twitter probabilistic topic modeling extensive science apply lead analyze extremely big text statistical algorithmic see reference therein model apply tweet tweet tweet landscape medium assign topic entire preprocessing apply every use measure computing interval I th interest interval investigate capture active period limit concern user extend occur ten influential account financial time influential twitter media figure account mention
use relationship shape gain recent attention image digit image al human pose latter work train multiple part contrast powerful share mid feature part several analyze explain success suggest localization et convnet representation individual meaningful convnet sift comparison visual beyond correspondence perform architecture identical dataset publicly reference activation
clustering seek output shift trivial rise modal proportion modal modal jj ix spread modal manifold mixture density gaussian behave mixture like modal like regression mr variance depend datum base mode method term several component mode likelihood indeed np mode shift algorithm unlike algorithm np number method bandwidth mode mode kernel shift cluster mixture em modal simplicity eigenvector eigenvalue density span express modal set difference point modal align coincide ridge conditional mode locally ridge state saddle local condition axis modal
variance estimate identical hasting even dramatically ability analytically propagate easy influence graphical leverage applicability large experimentally long unobserved column specify belief yield approximate minimize factorization obey suppose natural family exponential write exponential log linear exponential equation derivative latter relationship eq vector
penalty lipschitz partial penalization keep constraint shall exact locally objective result penalty subsection eq nonempty assume lipschitz continuous suppose hold minimizer minimizer together relation fact immediately relation ii continuity yield minimizer local minimizer theorem omit continuous local subsection lipschitz derive cover lot minimizer lipschitz local exist minimizer indeed globally minimizer minimizer moreover minimizer corollary qx conclusion proof explicit modulus resp continuity modulus present penalty problem specific globally continuous modulus minimizer bridge locally
mid property steady diabetes l set testing training diabetes training htbp l diabetes diabetes include measure mid classification testing spend spend node feedforward extreme interesting cm plus ex ex plus generally lie selection algorithm select bias also bias avoid yield singular
inductive conclusion suitably inductive eq last rather exponential work subroutine r r prove iteration thus epoch drop ambiguity shorthand notice tangent angle obeys eq inductive hypothesis establish q along eq outer inductive conclusion base establish hold step run induction goal go deep algorithm noisy view special precisely also iterate analyze theorem follow trial similar requirement choice imply necessarily rely claim terminate final lemma noise add favorable conclusion g iteration noise requirement inductive inductive line lemma hypothesis hold proof inductive imply lemma particular requirement satisfy must suffice sufficiently large imply fix iteration union
bn consider facilitate comparison structure algorithm si reference discrete mutual bn sample bn file repository appropriate load alarm sim round skeleton dag backtrack false mi alpha skeleton sim backtrack hamming r hamming node whose unlikely algorithm power small hamming give dependence order backtrack note vary accuracy focus backtrack skeleton bn exception alarm ham great backtracking sample size hamming appear contrary increase get trend use backtrack range distance bn
add pixel confidence respectively fast derivative function everywhere sign adversary adversarial rotation perturbation process differentiable reaction adversary find perhaps adversarial well inconsistent adversarial yield apply layer activation unbounded activation usually original perturbation unbounded activation make comparison able maxout perturbation however additive adversarial training capacity adversarial universal apply layer sigmoid function apply final perturbation perturbation reason adversarial seem poor space live hundred another serve people think capacity different low exhibit rbf predict elsewhere default confidence layer get
graphic macro ltb lt lt lt lt ltb lt lt lt lt lt r r r r mm train propagation mlp neighbor na I rule forest produce fold induce use induce diverse hypothesis train training training diverse handle filter classifier vote train backpropagation forest uci repository attribute attribute pt c categorical mixed post breast breast anneal heart voting record heart tumor car evaluation census vs noise ten split randomly level sign test suggest classifier list classify nominal produce learn produce delta biased approach induce misclassifie weighting score instance induce
without constraint population extract nonnegative block dimensional localize task nmf notation q intuitive nmf vector product free multilinear multilinear widely exploit overcome discriminant later minimize element wise nonnegative define straightforwardly infimum optimization partial fixing optimize equivalent nonnegative square problem extensively accelerate proximal free extend gradient respectively equivalently base exist nmf develop gradient verify complexity space demand especially scale reduce consider decomposition perform reduce space less intuitive tensor simplify respect product I time matrix significantly reduce memory consumption
whole language multimodal language learn dense embed word semantic temporal recurrent image part deep multimodal connect language representation learn use detail parameter incorporate deep multimodal validate tc method significantly task image retrieval image extraction network potential sentence deep network field computer al margin recently al framework recurrent recognition learning describe retrieval
purpose first sufficiently hence stop width interval specify superior criterion propose since relative article sample property deviation modification high modification provide computer good knowledge attempt formally address long run stop rule hundred complicated fmri thousand diagnostic terminate deviation consider univariate weather collect united ii selection demonstrate deviation high illustrate rule automate provide confidence paper introduce relative modern illustrate hierarchical dataset discussion general target small restrict unfortunately setting analytically frequently basic construct ergodic invariant regularity
intensity function count dirichlet mixture gamma data technical instrumental n convenience follow dominate log likelihood denote kullback leibler th moment likelihood concentration aim usually first quantity stand number radius introduce kullback leibler eq posterior neighborhood define q prior express construct measurable fix sequence j enough complementary posterior concentration posterior bad n concentration converge global satisfy modify conclusion loss modification respect assumption posterior instance associate need construct transfer lie nu controlling become typically control density dominate dominating section dirichlet process dirichlet mixture use intensity context mixture conditionally cdf dirichlet stick break instance cover induce represent cumulative give radius study large exist ball test q eq
variety mobile pose behaviour promise overcome detection audio recognition usage mainly focus start insight user stress level mobile software state usage usage self longitudinal use phone mail collect user four day reach participant daily phone call discrimination mid term monitor student week period relate pattern social interaction phone call detect limitation comparison adequate daily people situation report recognition mobile sensor subject limit day major subject variety background usa china etc equip phone sense software collect mobile run manner phone device minute datum participant trait big report daily proximity minute enough resolution list phone participant ask daily
consensus show side show particularly encourage even gap eigenvalue cluster collection consensus cluster raw iterate consensus refine way element outside block iterate consensus heat consensus cluster dimension red pixel considerable block consensus heat consensus matrix refinement clustering diagonal show high iterate gap spurious relationship couple together h h demonstrate datum cluster consensus clustering step mi md remain stop agree stop solution matrix clustering determine repeat step ng common benchmark cluster web article evenly attribute
non default rf complexity parameter label expect nothing much would scope sd sd sd infer rate specific evaluate train test provide al gain simulated label study label quantify scope score normalise metric monte carlo varying statistically analyse careful refined methodology assessment iteratively grow method budget budget iteratively batches sense experimental resemble challenge realistic explore label experiment al plot experimental firstly sample pair
generality solution backward find point compute generalization matlab structure need matlab contain field return vector return way former matlab also matlab input return value matlab contain parameter describe fista method stand backward accelerate
useful network node connection come multiple example study membership association individual primarily facebook etc demonstrate machine algorithm quality heavily edge predict good source qualitatively subsequent graph difficult domain rigorous apparent big challenge connect aggregation underlie requirement locally aggregation incorporate good absence fashion inspire demonstrate community graph source community evaluate locally represent
right box recent deep require beyond random graph try individually definition hope mixture presence e change arise property suggest possibly seem quite plausible leave even though vision intend assume independent variable bernoulli relaxed concentration column denote denote ease exposition describe generalization nonnegative dictionary practical generality expect ax bipartite define magnitude j later effect every pairwise intersection among
solution significant constrain function problem parameter projection critical maximization correlation chain matrix critical scheme enough ensure implement copula estimate expression copula principle applicable elliptical compute gradient
hessian determination curvature bfgs low convergence argument complement characterization convergence sgd establish minimization function vary vary objective comparable sgd vary dimension problem exhibit degradation dimension vector svms points improvement numerical also compare non regularize bfgs fundamental bfgs observe separate hyperplane definition average function function convex strongly strongly convex descent motivate actual realization define give implement gradient require determination gradient average sense along intuitive formalize convergence hold control step problem small resort order evaluate suited newton whereby definite know select include bfgs e bfgs since practice approximated variation hessian tend
regardless sparsity contour specialized believe remarkable moreover recall almost line straight log observe function smooth novel primal problem primal sampling flexibility variant parallel variant sdca serial uniform serial importance nice speedup direct primal dual sdca sdca sdca accelerate leave sense involve pair mutually dual function convex conjugate follow pair concave belong interior last relaxed ex theorem theorem zhang two would acknowledge grant coordinate optimization penalized strongly propose primal analysis directly depend serial distribute bound match sdca predict speedup drive depend calculate speedup excellent efficient batch importance distribute distribute sdca pair optimization vary attract
identify exist identify regime measurement generic measurement generic regime perturbation measurement generic another irreducible k identifiability broad measurement identify theorem yield exactly apply infer theorem reverse implication hold prove another unitary n unitary identifying let single identify reconstruct phase invertible unitary identify generic proof analogous noting identifiability terminology determine motivate family irreducible denote small identify call small completely clear call write statement terminology allow retrieval state exclude formulation observable view element perturbation perturbation view identify signal perturbation generic perturbation except hausdorff hausdorff take furthermore statement hold replace measure subset closure subset closure already imply note statement contain generic make valid case analogue projection real vice threshold
noise variance accordance eqs compute hardware propagation across layer operation index sigmoid activation train sgd size objective momentum descent exponentially contain matrix vector initialize decay network sequence representation describe network corrupt control network precise entropy error train error show hardware suffer degradation compare control contrary slight presence incorrectly network prior shown improve neural network hardware virtue
generate try match slice slice quickly belief converge increase method entropy reliably explanation much present suggest develop strategy classical iterative additionally time latent prediction expense prediction might area rather suggest help investigate improvement investigate distinct describe useful optimize
index error develop fast solver proximal focus develop fast subject consider future analyze robustness whitening source try whiten good sdp fact whiten sdp whitening follow tight analysis whiten whitening extremely oppose pre whiten affect allow illustrate synthetic show whitening show well runtime analyze analyze reduce bound bound robustness since goal bind solution approximate w sufficiently interesting trivial unique column since see follow prove satisfie q optimal value
quantify method diversity dpp diverse without indicate significance sum gene sort radius indicate find diverse feature distinguish ii breast construct identify breast cancer comprehensive cancer pathway profile readily predict breast however poor feature gene pair protein protein network form gene belong similar community detection challenge avoid step collect genome breast gene protein top accord univariate regression respect tumor similaritie nature identify higher component specify gene approximately average impose dpp lead
visit remove repeat parent step leave parent child dag label hierarchy learn time million intel equip vocabulary major belong least major belong visualize hierarchy occurrence separate reasonably mesh vocabulary please well visualization rest subgraph care leaf representation predict child tend share
result sparse contribution computationally cluster dimension come complexity scale relevant inherent ambient spherical without notion feature handle organize formalize complexity high dimension generating point generate covariance cluster scope cluster e error
measure sign set sphere dictionary basis dictionary square expectation situation practical equally weight one measure random sequence component exist proof simple incorporate coefficient motivate requirement original dictionary signal probability base sparse product atom maximal freedom construct ascent need truly decay large study maximum near generate warm first incoherent noiseless draw unit exist c moreover sketch ideas svd include ensure consequently signal response response perturbation c still attain typical sign sequence permutation scale sign sequence original already optimum generating dictionary obvious special signal almost get ensure large arrive q find signal insight
evolve continuous neural define ideally equation dynamic eq frequently strictly mechanism behavior noise contamination equation recurrent indicate otherwise refer gene capture relationship node recurrent nonzero link interest time n u c I represent form dimensional time always point sde discrete observation challenge rarely analytical treatment approximate eq system depend separable x verify parameter minimize number limitation interest reality stock price influence stock market parsimonious interpretable statistically speak necessity penalty function
original future prior regular regularizer machine robust pca network share across resolve minima utilize induce sparsity newly outperform approximate optimisation mapping code extra one behind
later inner product obviously definite dense type core perhaps type expectation product definite figure accuracie kernel core perform kernel see core high lin lin bin mnist summarize see without tune compare abc mnist
reliability divide verification quantification verification concern assess quantification assess uncertainty model uncertainty mathematical reality predict decision verification difference mathematical concerned discrepancy verification sometimes practice largely technique control impact ensure compare source verification focus quantification uncertainty prediction unobserve validity model assess experimental comparison validity assessment closeness reliability uncertainty model assessment reliability unobserve discuss essence question justify validity really validity general produce valid quantity validity scientific strict e mechanic nonetheless valid sophisticated procedure validation engineering development validity make instance similar framework model representation framework observational insufficient validation predict observable strong decompose make discrepancy refinement quantify importance health economic introduce discrepancy within technique create discrepancy address prediction model highly calibration trained discrepancy unlike et al guide discrepancy systematic validation capability unique remain issue physics reliable physical couple reliable structure unique modeling entirely introduce discrepancy enter et unobserved validation address spatially vary elastic modulus model calibrate experimental consistent question compare always fit problem investigate match experimental account
interest q give meanwhile establish positive go term theorem asymptotically unbiased converge weakly correlate must hold establish uncorrelated maximize contain order complete prop conjecture prop prop prop definition prop comment prop remark grateful stanford fellowship forest prove area forest establish prediction forest paper forest asymptotically subsample number show asymptotic ensemble characterize treat black become popular box tool machine
independently use emphasis scalar scalar product ridge scenario experiment attribute decrease significant method lasso behaviour experiment popular mnist focus distinguish lasso scenario ridge considerably train check theory also number examine attribute offline utilize towards attribute htb perform slightly variability small examine attribute set predict forest dominant multi classification species address scenario data htb ridge similarly online examine attribute examine outperform attribute one appear perform well converge towards attribute small perform examine attribute grow budget ridge distribution excess art prove demonstrate even though quite partial direction algorithm expectation question arise
commonly multivariate multinomial distribution explore vector appropriate distribution random useful symmetric set poisson ip independently poisson dp prior suitably gamma conjugate cluster dp observation also dp restrict cache end temporal model temporal correlate count hmm dp incorporate hdp model case rise dp dp temporal come introduce define expensive dimension trace extend extend denote active symmetric otherwise emission across define rest dimension account case dp indicator denote natural distribution parameter bernoulli inactive case dp dimension hmm temporal instance hdp q hmm hmm describe hmm capture trace dependency exploit inference definition introduction dp
backward estimate obtain algorithm final look sampling bivariate full use quickly gibb low effective model gaussian variance particular exactly method much gibbs fact improve sampling widely hmc distribution space expand method dynamic hamiltonian update hamiltonian effective way posterior exact walk hmc invertible volume preserve similarly hamiltonian obtain stochastically likelihood bind
proper beta restriction give negative likelihood give atom weight atom atom location prior fix whose proper since assumption ordinary find form ordinary hyperparameter beta fact conjugate process posterior also posterior conjugacy approach bayesian nonparametric conjugacy still guess right conjugate likelihood automatically construct conjugate exponential give family family development bias representation marginal condition exponential measure lebesgue measure atomic mass conjugate eq natural bayes quantity eq belong family conjugate exponential start notion family location location atom unique location weight density density statistic share across atom ordinary component rate measure share atom unique automatic nonparametric prior automatic conjugacy accord exponential atom weight distribution fix accordance th atom sufficient atom ordinary weight rate measure conjugacy
minimize gradient useful ab q multiply side integrate eq divergence expectation density approximate average include analytically q e cross j size candidate sample except eq denote cardinality choose solution compute gradient descent euclidean give manifold ordinary tangent equip metric q denote
red black plane span principal vector subspace subspace let principal subspace preserve projection state state separate non remain help extend subspace multiple subspace ty separate margin projection vector respectively span use angle add dimension separate argument subspace preserve
explore soft margin use rbf range logarithmic rbf explore explore range produce tree bootstrappe varied boost fix tree range logarithmic ratio consider tree loss layer perceptron softmax minimize neuron value total dimensional space present generic different task crucial win either model ensemble run set good set winning frequency case
admm try fourth try calculate admm try admm adjust show admm completion effectiveness admm rank synthetic partial cosine operator compare rank take want illustrate noise compare admm show admm show fig run htbp propose noise namely approximately say robust corruption save illustrate effectiveness lr admm dct recovery illustrate advantage admm evaluate recovery compare reconstruction different sample fig addition compare generate run admm matrix lr easy increase achieve admm nuclear noise
match contain scale randomness operate lastly still hold partition approximate balanced cut energy property general multiclass unique balanced one sense balanced unique minimizer satisfie surface modify geometric substantially long theorem overall spirit role finally remark analogous remark convergence deterministic functional nevertheless random decide introduce space functional respect functional converge satisfying situation instead prove deduce precise converge enough deduce functional nonnegative functional boundedness functional property minimizer converge minimizer minimum make reason variational completeness benefit highlight work ultimately nonnegative functional statement cluster unique minimizer statement hold deduce arbitrary previous nx relatively know least inequality deduce imply prove dd kernel variation function rescale version sequence define surface weight define functional restrict functional characteristic function obtain convergence
order must strategy define stop determine base measurable control provide triple entirely setting strategy call natural draw advance almost choose recommendation fix budget strategy zero resp identification confidence resp fix setting follow confidence need average arm fix optimal require fix budget setting aim compare two complexity lower bind resp failure consistent sample resp failure lower present theoretic strategy round fashion sampling desirable armed bandit study regard hypothesis pair fix number budget rule determine sequential first confidence simple fully law know permutation pair hypothesis error small minimize sample type gain indeed gaussian armed bernoulli considerable interest introduction medical trial important perspective aim maximize reward equivalently expect complexity well understand bandit leibler
connection screen enable understand advantageous shot complex important hyperplane test among shot interesting study indicate significant sphere center spherical bind solve homotopy study wide range shot test select available successful application author constrain lasso concept screen lasso addition safe liu screening make variational sufficient screening compress select predictive outcome context apply screen music term codebook music xu et screening application dual give active change ai I nn lasso reject reject I e q w zero r yield reject nonempty hence diameter ds dr r r r lemma rt tt zero substitution ia substitution yield yield yield lemma empty nonempty reject partially nsf b electrical engineering university china receive ph electrical engineering currently
approach target latent bar intensity ground bar repetition selection gp rbf noise latter particularly adaptive trend datum book discussion model via already extract explicitly spike generate middle converge element hand converge element learn select inference gp cosine quickly ground truth provide quickly straight rbf ground select kernel
split coordinate child node inner facilitate formal version quantity tree begin expand otherwise height leave pt prediction predict feed observation predict h compute splitting x leave corresponding cm leaf bin storage size state depth store need store space order besides leaf need per tool sequence rectangle bb number initial jensen multiplicative want guarantee consider
text estimate extent political text note text text return probability extract hyper plane lie hyper anomalous encourage close meanwhile hyper towards text influence seem input alone gram alone yield suggest indicator author word perform poorly severe overfitting mention introduction ten agreement accept publish style death gram additional insight community author addition identify setup eight text membership identify close plane agree well preliminary seem p text non text correctly fact influence
symbol color three long represent red green choice make qualitatively wide typical give top lot similarity tend embed correspond dissimilarity identify rapid different score sequence whereas large sequence score bottom remove choice visually embedding embedding give correspond equal qualitatively htbp shrinkage multidimensional carry problem consider structure equip unique start extract exist form domain protein bank symbol contain coordinate coordinate case relatively compute dataset correspond evaluate
real movie sparse user rate bias towards rich user rate lot movie rating overall recover rating completeness present reader key study co algorithm well dataset recommend concern even study dataset limitation briefly user user continue well simple majority cluster subsection substantial recommend movie error rate rate htb show rate htb low c remark obtain obtain user ii movie conservative calculation rate netflix dataset movie movie select rating select user otherwise special computer svd ghz processor movie algorithms movie user error error rate htb evaluate rate among rate low particular error suggest robust independently cluster
strength nominal distribute alternative strength along nominal distribute evaluate compare test outperform test alternative strong right panel figure gain power become test propose test screening outperform agree screen substantially preferable whenever relatively simulation carry online supplementary material show particularly screen outperform covariance test maintain sample power sparse recommend biological know gene base focus extract biological insight increase identify associate biological state disease development research genome analysis produce name term category molecular mf gene group term statistically gene independent hypothesis model expression level gene gene mutation practice set set dimensionality set control wise procedure
appear good context mh algorithms reject single branch execution path evaluate core speedup core ignore circle font cm gray cm node state node state child state child child child child na I observe branch reject probable classic result scheduling proposal evaluate reject branch branch algorithm core useful reject proposal proposal root due core root computation speedup speedup core na I essentially advantage later consider special slow move evaluation likelihood slow
say different mirror image regard select score context globally discrimination optimize discrimination though probably give across especially behave indeed performance phone compare
differential object research rna seq relative assume come parameter dispersion individual shrink toward genome parametric relate dispersion notice estimation integrate frequentist paradigm place parameter thus able detection analysis typically start analysis gene de increase take cope investigate approach properly quantify uncertainty de
similarity yield much performance benchmark whole much well ensemble ill clustering add time base clustering method link graph link connect execution fig cost second pair method execution method two fast great partitioning ensemble crowd link crowd assess reliability individual exploit normalized crowd evaluate clustering unsupervise triple construct common reliability take consensus consensus term evidence accumulation partitioning conduct eight effectiveness method anonymous comment suggestion enhance science definition probably robust increase mainly aspect limitation exist firstly weight base ill clustering level ensemble fail integrate unify address limitation crowd agreement normalize crowd agreement quality
de en da fr en en es identify focus enable grain question similarity language determine similarity topology european language group triplets topology specify correct ordering fraction exclude subtree relationship language language similarity language nearby es ft triplet language remove ht cccc target es fr pt es es es fr fr choose four
denote homogeneous branching process root million old represent specie branch age branch simulate epoch give value binomial variation five lie temporal gap rejection simple apply require million simulator evaluation accelerate approach simulate due use generate estimate prior lead accept leave fit alternatively substitute less abc variance train noisy surface term approach successful result simulator evaluation rejection
node highlight central node centrality architecture network centrality connect scale centrality unity cluster centrality total independent band network free centrality dominate significantly method six condition exception recovery geodesic distance geodesic distance short path free geodesic scale network tend short feature small geodesic free lot band geodesic distribution band outperform control graph finally analyze synthetic band network connect component contain connect component band free size sample recover across figure nearly perfect graph inference construct module rare match assess accuracy infer interaction round final model threshold relevance infer node correct significance arrive gold independent test learn reference new network triangular distance vary interval ham repeatedly subsample disjoint repeating inference hamming edge disagreement bar disagreement highly visualization overlap color colored geometric relative american unify analyze reconstruct quantifying
identity lag walk forecast equation model parameterize possess hierarchical describe point decay range correlation reasonable lag lag hierarchical perturbation effect acquire kf enkf prediction walk dynamical thus solely term uncertainty quantification cost induce pixel inversion measure kf depend resolution resolution estimation error kf enkf produce ensemble realization accurate co day volume survey informative less improvement enkf represent simulate db snr instance day image plot function day co uncertainty quantification kf inside coincide ray coverage middle kf accurately enkf great kf plot three enkf solution
concentration model draw deviation table integrate ode htbp level range expression normalize normalization genetic example l error variance genetic l l pde forward pde boundary pde serve steady system head method function mesh endow gaussian squared length scale field eigenfunction integral endow prior parameter mode condition mode cover observational proposition proposition example mit monte carlo limited computational therein approximation hasting idea theory approximate inference typically sampler seek address characteristic local ergodicity distribution interest employ polynomial observation underlie regularity model evaluation greatly without carlo average multiple order forward ode pde inference computer experiment chain carlo computationally intensive markov monte scientific chemical often invoke constitute yield numerical cost forward quickly become prohibitive expensive cost recognize regularity interest characterize replace forward surrogate require forward evaluation chain reduce create global polynomial effort thus approximation accuracy expensive set approximate intractable significant improvement either induce conversely potential even delay eliminate need surrogate accept
svm aggregate train discriminate small instability sensitivity contamination unlabele bagging technique bag svm draw discriminate resample varied induce variability aggregation bootstrap bagging hold q positive unlabele choice bag classify instance high misclassification similar propose
self label euclidean fold fold describe angle translate seven rotation result c cm cm label seven source seven rotation angle high consider translation positive negative table remark change density robust provide performance necessity account disagreement label well result nn translation label nn labeling matching domain focus region remove match source close couple nice tackle target
potential first set item use translation map distribution setting space maxima vs plot original space hand map axis scale moreover matter map everything train correct increase cause map set spirit greatly simplify conjecture map consider appear similarity simple correct normalize item map length penalize correct rank straightforwardly implement put
response assess prediction unobserved response observe test response unobserve response sparfa range sparfa select control assess three metric receiver auc correspond percentage response predict py learner response classifier cf sparfa sparfa auc dataset auc trial sparfa achieve comparable performance cf sparfa outperform sparfa metric dataset emphasize sparfa
hence choice however require considerable htp htp solve investigate stepsize svm show evolution duality obtain stepsize mention previously stepsize theorem moreover computable comparable speed htp synthetic lasso generate average nonzero element maximal nonzero gb respect rate converge expensive compare angular
real multi modality datum label audio tune model baseline comparable modal dataset improve recognition video report result combination r v configuration range transfer deep fine audio semi bind achieve transfer bold chance performance video tune audio outperform audio datum show target specifically
patch patch patch difference final denoise result entire choice denoise patch case identify interpretation forward competition pass estimate scenario find balance generality cccc patch truth patch pool exhaustive patch adjust bm set show increase db increase db curve level propose mse recall patch clean distribution square define suggest optimal minimizer problem know never model prior patch difficult shape highly representative could draw center appropriately optimal diagonal patch measure respect improvement still linear mse define lemma patch eigen decomposition show noisy patch learn form datum eigen decomposition denoise assume subsection
half reliable systematically small magnitude real half predict correct qualitative sensitive contain shape frequency spurious predict obtain perfect impossible dot exact circle length dot denote green result representation denote systematically small coefficient magnitude ard result conclude learn v circle set exact dash denote length green learning learn time problematic green reconstruct give slice plus coefficient learn body b circle result dot red dash evolve set display look around half around applicable fraction representation set learn four fig dot dash learn
ccc ccc simulation lasso simulation c display b identical graphical correctly assume sparsity covariance overall little neighbourhood specialize intensive display adjacency matrix simulation c appear similar b simulation detect tune varied simulation panel panel correspond panel panel fraction set simulate datum examine
fact justification plug span td update parameter discount set instant td run length calculate td error value update follow td note bellman establish governed solution give vector rademacher component dimension unlike balanced one side primarily run come simulation side sf density integration fact chapter sf sf sf rademacher sf iterate define section necessary tuple saddle point sensitive around uniform rademacher eq similar eq expression operate finite sided gradient lagrangian recursion lagrangian enough sr discount actor sf sr objective rs sf recursion sr nominal sf algorithms sr optimization lagrange describe sr simulated sr lagrange trajectory parameter necessary sf td evaluate instant discount nominal perturb implementation would follow evident necessary base scheme simultaneous henceforth rs henceforth refer sf smoothed sf hessian confirm second inverting q update rademacher perturb update hessian estimate estimation technique chapter appendix expand taylor expansion observe vanish mean case update multipli hessian update ensure fast recursion see project symmetric projection ensure convergence hessian sequence matrix n eigen eigenvalue avoid incorporate newton update policy converge converge
nominal discuss great unfortunately share distribution high temperature rapidly start idea far intermediate traditional maintain high recent feature layer upper layer appear ever e ever play learn intermediate facilitate low hierarchy applicable energy equivalently rbms visible variable unit rbms much
model et al turn social approach analog grain infer latent influence variable validate compare several include movie model infer focus linguistic infer behavior combine model take linguistic possibility influence potential tool social brief description variant infer influence turn take et model pairwise action concentrate cluster influence scenario social nature relationship extremely valuable et al specify discussion correspond capture occur let pt calculate duration process
enhanced persistence characteristic illustrate validate correctness highlight visual persistent infinite hmms hyperparameter always informative often towards certain kind informative bias induce prior hmms remove indistinguishable hdp match explicit duration hmm poisson sample hmm construction hmm hdp hmm equivalently ensure persistence regime state encourage proceed monotonically encourage path skip duration emission emission duration emission give scale gamma poisson burn burn possible backward slice joint show score histogram explore duration posterior show duration fit datum distribute result hmm duration compare hdp hmm hdp state unit geometric normally implicit geometric fig poisson duration observation mean duration difference self observation difference ten dataset generate adjusted sequence state emission generate datum code audio utility duration correctness also
follow eq impulse response estimator estimator determine propose hyperparameter vector characterize impulse precisely jointly maximization likelihood integrate follow reason solution em method iterate converge global stop hyperparameter vector furthermore define
transmission measure image build transmission position across image replace detector array measure distribution angle frequency information smaller enable high resolution object need phase tractable redundant reconstruction often well convergence remain reader result see optical reproduce often practical refer convergence need global phase retrieval discussion phase recognize involve analyze projection descent metric focus iteratively enforce piece retrieval measure ap solution satisfy convex understand nonconvex still open question except quantify retrieval uniqueness proposition retrieval ap uniqueness show coincide factor result necessary sufficient condition ap unique solution factor ap algorithm become lead confusion throughout survey synchronization phase synchronization phase synchronization problem lead technique laplacian eigenvector initial guess accelerate speed large scale problem section show propose design convergence rate
curve figure learn adopt white notice region successful experiment coherence whenever show success sequence data matrix trajectory unobserve visual trajectory recovery corrupt recover art sim ssc std sim sim ssc ssc ssc algorithms trajectory correct dramatically ssc identity rate effectiveness dictionary environment sparse give might even typical structure multiple cluster go coherence overcome challenge arise coherent theoretically one capture structure produce suffer several practical simple lrr lrr furthermore utilize approximately still problem need domain document column new handle superior theory condition
solution hold instead b hold replace vx lemma relation fact rhs expectation high search expectation side show assumption u martingale martingale sequence lemma conclude convexity exponential markov view provide specific use stepsize policy corollary priori bound however need compact observe together part respectively definition conclude imply relation solution e point immediate exist stochastic b evaluation bound b knowledge time complexity establish slide generalize important cp specifically cp subsection nonsmooth give
toy class contain dictionary toy test joint ls group low eqs different toy region recover hyperspectral overall accuracy aa measure prior implement comparison whose fashion joint sparsity combine admm search among sort structured two prior prior row ht c c l
htb minimal quality decrease figure quite satisfactory active agree recommendation correct favorable stability time item recommend study item never failure account weak modification show offline evaluation factor recommendation recommendation user production offline contrary recommendation overcome
term limit two space correspond namely isometry relax quasi unity parameter dictionary atom say isometry entry preserve isometry constant isometry space isometry property dictionary investigate sparsity generalize atom kernel isometry property atom atom isometry dictionary gram large eigenvalue pair explore identify isometry measure besides expression theorem thank expression deal unit atom bit isometry dictionary r dictionary get bound divide yield isometry rescale atom proof theorem deal
dissimilarity tend prototype bag prototype bag reduce become consider rather bag concept instance originally approach ik radial however leave counterpart dissimilarity summarize representation dissimilarity information preserve per bag averaging dissimilarity preserve dissimilarity select relevant dissimilarity categorization problem bag image patch instance region include dissimilarity provide heavily redundant uninformative dissimilarity
intuition condition always strictly hold local neighborhood sufficiently concrete model equation function reasonable condition neighborhood confirm follow locally consequence sequence exhibit linear condition concavity side imply side bound combine original em repeatedly thereby give subset perform sample tolerance small tolerance require ball accordingly small scalar operator parameter ball belong enough iterate least size splitting figure illustration predict algorithm expect bind fix size suggest iteration particular focus least update increase choose remains fig conv part black ball logarithmic identical argument least union perform event suffice bind follow sum induction iteration whereas apply initialization bind argument complete gradient em separate addressing provide generate recursion analyze additional concavity assume pair intuition compare function update state strong concavity smoothness closeness requirement formalize condition verification gs positive regular hold oppose figs gradient whereas distant gradient condition radius triplet point gradient
character denote detail analytical conduct several efficacy algorithm draw note three choose index final lp lp lp problem analytical solution write explicitly objective inequality separately solution lp refer soft thresholding remark criterion speak whenever decrease slow term
worker disjoint summarize illustrate tradeoff along namely statistical efficiency epoch epoch ia ratio row wise zero cost writing read examine wise find execution depend hardware statistical efficiency row sparse read dense perform epoch combine linearly thus illustrate run converge error figure see number hardware efficiency change per epoch non subsample music detail ratio actual cost ratio row wise read wise method cost base optimizer execution read epoch svm row update zero scenario update estimate ratio read run long expensive reading cost access three similar traditional nothing architecture design machine difference model strategy exist simple leverage machine maintain version epoch share nothing art framework subtle worker epoch implement minibatch parallel calculate gradient implement dynamically requirement replica worker responsible update replica dominate minibatch implementation schedule way implementation overhead tolerance relative application paper implement implement force hardware deal coherence although reader converge single replica share core synchronization base epoch share approach consider fine sharing read
scheme accuracy small dictionary learn normalize vs binary classifier use regularization width validation dictionary match naturally sample previously svm train denote code comparative use supervise sample atom cluster close drop supervise previous rbf last texture task linear svm suggest classifier one accuracy well size moreover svm outperform test last rbf l coding coding last propose last compare different outperform code technique notable classification another mean soft consider task handwritten digits image training mnist compose image unit address class task use problem specifically separate classifier predict vs naturally feature different sgd dictionary nn code relu mnist describe previous sparse code addition building
similar predict protein protein know novel topological intrinsic develop second section annotation protein support machine spectral interior spirit make propagation obtain part semidefinite
detect sequence detect anomalous definition refer scale guarantee decay sample say exponentially consistent respect refer asymptotic regime go decay anomalous scaling develop anomalous I see anomalous sample draw anomalous capture level anomalous increase affect consistent mmd suppose distribution rkhs kernel refer mean hilbert reproduce clear map unique element associate distribution discrepancy mmd clear mmd equal embedding due base available paper scenario sequence start simple sequence case far anomalous detection anomalous case compute sequence anomalous constant naturally anomalous characterize anomaly generate apply kernel exponentially consistency desirable practice number small
complete target jump corollary definition em em height mcmc analyse move update particle filter application give estimate log proposal langevin investigate mala asymptotic dimension mala depends crucially accurately control sufficiently mala use proposal behave particle mala proposal compare walk furthermore acceptance particle roughly mala suggest monte well methodology year mcmc methodology tackle likelihood possible intractable monte replace estimate target work also particle refer mcmc particular metropolis hasting replace hasting sampler default herein walk rwm filter overhead proposal focus filter information region langevin
successful employ mainly probabilistic automate distance reflect mean language nlp ir use similarity semantic network base major classified base pointwise semantic term measure base semantic similarity direct graph arc partition dependency parent root one leave level third namely frequency define train consist path node believe explicitly scalable implementation improves implement outcome namely outcome probability v I px px fix identically outcome joint
drug chemical describe play compound represent discriminate document topic tailor domain similarity satisfy require instance learn typically problematic computationally expensive cubic likely overfitte especially rarely observe common dimensional pca reduce useful irrelevant similarity also discovery knowledge bilinear function dimensional mention sparsity parameterization frank wolfe learn incorporate pair providing ignore overfitte output
output identifie regard free recall mean simple numerical entry draw next randomly entry one plot trial range correspond estimation ix analyze reliable recovery output happen lasso strong convexity reflect change appropriately modify tail gaussian chi noise sub depend condition general recovery
increment denote cluster prototype assign empty centroid seed etc least one mean update empty cluster proceed point decrease decrease local converse suggest replace minima optima argue empty meet figure mean empty exception empty empty cluster create far copy however add seed decrease partial keep stage etc frequency
r combine get theorem execute quality privacy parameter hold differential call range therefore inequality computational quality suffice efficiently implement recursion operating easily call private recall show goal small error quasi concave algorithm label pt utility analysis let draw c j execute valid moreover execute execute quasi concave overall proper use axis align space think approximate private learner nc I align rectangle vc thus learn generic inefficient private learner give private learner efficiency however direct transformation begin sample draw component approximately likely fall proceed boundary use query probability positively place hypothesis th leave interval query private use transformation simultaneously could laplacian histogram straight approach learner overcome construct divide mass standard argument specifically interval class database b axis mention axis return database execute approximation roughly every execution theorem learner inefficient proper complexity fact learner must private concept class necessary database exist stand close database overcome tool approximate define domain approximately maximize optimization section database define function ii requirement could score increase neighbor moreover element private otherwise approximately gs gs choosing preserve fm output respectively hold two growth fact e proving
l e index transpose version call slow third runtime first evaluation call example sequential user parse restrict structure specialized solver inclusion decade solver solve alternate prox efficiently method like even problem spectral large sequential convex produce successful programming solution specialized branch name useful exceed care detect transform format subject extremely currently development parallelism capability leverage parallelism library simply implement software
concentrated verify variable mean event completely event claim turn convexity consideration contribute j q event difficult yield basic unsupervised family unknown parameter u goal natural useful subroutine various hand give recover maximize program relate limitation technique consider n reference coordinate feature first process sound wavelet useful primitive many state case recover nonzero generally entry approximation guarantee suppose arbitrary follow system satisfy equation solution degree satisfy moment cauchy schwarz sparse p motivate encode input subspace always close approximation ingredient subspace outline write subspace b gaussian
regression write regression k lk number serve effective facilitate selective noise component appropriately induce impose allow adapt ensure number interaction inclusion interaction scaling initialize se inclusion conjugate suitably section place component inclusion model inclusion c b inclusion default middle inclusion size vary fix inclusion prior inclusion depict configuration parametrize sample size large place alternate choice parameter play role interaction framework grouping within scaling realization parameter link crucially inclusion scale efficiency update discretization essential mcmc allow grid calculate enable vector grid ratio correspondingly length specify dirac delta standardize default sensitivity hyper though careful necessary allow nearby characterize inverse pattern surface scale characterize fine shape feature restrict smoothness grid range scale concentrate important variation response explain range ad pd active component across propose fast
assess rule choice perform extensive choice efficient method investigate seem yet criterion application real importance suggest numerical simple choice inefficient give unclear test consist end force second dimensional result experience two cg justification experience coefficient possibly scale condition cg cg however possibly conjugacy specific scheme furthermore proposal cg possibly efficiently aim effective tool context sequence conjugate context unconstraine detail study indeed choice might also
equip range expert convex corner gap cx corner robot find cc follow minimize therefore consist current linear velocity robot angular velocity expert always follow angular straight convex corner corner use controller decide version could analyze algorithm environment real characteristic environment length concave cx corner difficulty corner close front tb environment dim length cc cx home home office environment fig trace mark mark velocity environment grid map environment robot cm along environment robot real environment use obtain eqs universe rule fuzzy minimum implication universe variable min max distance velocity angular velocity three genetic fuzzy evolutionary fuzzy rule fuzzy generating genetic simplification genetic membership fuzzy base soft performance open software tool problem use perceptron hide bfgs number layer vary statistical software raw three group
base design operate pass thereby cluster find center select datum find center enable evolution technique divide task merge employ processor representative processor cluster cluster center cluster pairwise dissimilarity accurately cluster linearly capture develop som kernel full whose knowledge attempt scale reduce time memory mean reduce always produce matrix address challenge show cluster reproduce hilbert rkhs endowed follow cluster membership center domain matrix relax center find
follow p p f sec et al extend lemma directly adapt n hold km describe analogous e km km e proof operator k great second union partition union j p sp bound assumption corollary em address collective completion recover collection share partial noisy impose joint wherein across develop algebra represent collective collective tractable collective trivial
supplement exist hoeffding say almost nx imply directly theorems hand conclude corollary neighbor classifier theorem ni ok w ni ni corollary accord prior eq gain reduction relative gain logarithm improvement logarithm effective trading relatively research nsf award associate award dms stability concern conclusion population introduce measure instability capture variability plug classifier concrete classifier derive neighbor near trade stability possess minimax rate demonstrate near neighbor accuracy scientific scientific conclusion stability much many instability assess instability criterion dimensional selection stability tuning context purpose bag derive stability
item choose rmse yahoo provide rmse mf term criterion evaluation size also note give correspond default base user unable reasonable cs competitive less train version cs approach item expressive ability learn relevant item mf mf c yahoo na na quantitative cs present movie three last episode rating star episode seem rating movie half
run two reduction save nn classifier dataset computer package disease genetic heart apply technique dataset extremely snps space via multiplication dataset consider project table entire measure f area curve include process follow projection dimensional methodology testing describe projection dataset low projection course equivalent multiplication multiplication fix parallel roughly denote illustration projection ccc snps approach coordinate dimension projection realize take approximately month run comparative approach discrete contain snp divide testing see run observation select approximately day tree use test snps observation computational cccc total entire dataset observation snps testing accord training computation important snps random setting snps approach roc consider chapter genetic study projection follow forest conclude discussion apply projection nn dataset observation table result fold method reduce norm result value result roc result high value norm result area nn method validation meaning would two true area f area roc area roc feature forest observation apply entire regard result accuracy roc selection result table score three compare roc use snps area forest measure snps forest plot roc curve right consist experiment snps area roc snps area discuss table low nn successful predict disease dataset snps accuracy cross mention predictive bit genetic dataset hand snps observation able area curve curve snps previous score genetic
prior frequency run file partition classifying record record partition record one partition true partition measure proportion pair classify precision truly evaluate performance detection linkage big result present row correspond measure partition th percentile line refer recall gray solid average line show average average th depend amount identify contain file field naturally precision file general proportion false field file generally sensitive insensitive amount small poor mean truly detect file seven precision somewhat insensitive term precision panel easy truly prior indicate amount error potentially obtain specification performance trade recall prior wind prior indicate actually end simulation study file contain level intermediate list make publicly available thesis green utilize optical character technology transfer list list contain part current describe record database name name date death month article record file specify record believe field therefore truly confident name neighbor approach boundary two
basically estimator lasso isometry definition mse pack number application particular problem error projection contrast paper match error detail optimality estimate noise collection regression particular application among rank recommender system note write value motivate jx accordingly constrain square norm equivalently appendix packing pair see method match far sub optimality observe sketch vector abstract observe approach quadratic computational optimal accurate respect square solution set direction play important function unit norm important role sketch fashion sketch tolerance final sketch hold reference let matrix constant value estimation involve previously standard dimension sketch sketch
measure updating update must partial q distance assume topological space open question intelligence power otherwise distinction agent agent relationship mapping although satisfy exactly stage final minimized state indicate entropy mapping projection outcome total
candidate parameter ratio contaminate robust regularization grid parameter candidate need solve convex calculate present svm svm affect robust svm tends outperform svm choose straightforward grid search mean margin clear conduct another split test minimize prediction select computational svm necessarily classifier robust parameter carefully test outli svm train outli svm svm train outli svm svm svm present property compute robustness investigate show prediction classifier loss method another develop optimization although dc scalable deal massive
convex subset parameter convex entire show crowdsource function axiom axiom non convert aforementioned axiom applicable well exist inference crowdsource task axiom example two axiom section identify convexity would crowdsource satisfying axiom satisfy model easy objective answer non interval answer monotonicity worker ability exist decrease increase infer answer capturing
complexity capture behavior disagreement reciprocal survey summary behavior something maximize datum calculate combinatorial see section proof include kind star connection free growth literature large brevity may say star completeness nonempty star cardinality star equivalently data graph set node vc degree clear star guarantee exist bound gap generally infinite briefly calculation star classifier I least also contrast classifier I x align embedded star lie intermediate range x w h x hx x ready article upper low minimax abstract dependence logarithmic upper meaning reader logarithmic represent upper formal include mention comment regard theorem sketch underlying bound comparison passive aside refinement label mild case noise surprising primary prior root wide spread complexity active learning depend nearly passive complexity active know passive section thus hypothesis easy threshold classifier improvement passive classifier passive case literature label complexity exhibit spread literature star reflect passive problem passive class trend admit passive class passive come minimax complexity passive upper reveal sample passive improvement factor learn essentially logarithmic spread complexity long indeed vc dimension exhibit hypothesis classifier examine spread complexity increasingly complexity roughly increase strong improvement passive improvement dependence passive complexity learn naturally hypothesis complexity hypothesis class active complexity exponential though extent star roughly aside thus literature easy hypothesis passive reflect improvement consider regime hypothesis roughly aside make distinction easy hypothesis hypothesis always logarithmic label factor passive improvement nonetheless distinction easy begin hard dependence factor easy dependence dependence argue sometimes induce label gap low construct example class span gap instance sufficiently tight logarithmic suggest tight factor sx strong namely strong follow immediately fact refined loss generality introduce measure distinguish proof follow bound construction logarithmic sx tight bound embed variety machine maintain vc instance theorem tight another interesting implication separation classifier note result particular dependence reason separation interesting hx xy indicate achieve specific beyond discuss section complexity logarithmic refined linear unclear achieve separate generally hope bind improved match remain open extent restriction aside aside upper prove several bound
far corruption measurement additional signal smooth recovered signal interest graph assume require rank happen require sparse express bound formulate follow control frobenius quadratic nonzero recover use total reason computationally norm quadratic separate non graph form slight abuse reflect total recovered smooth low frequency rank force graph redundant minimize norm force magnitude coincide unfortunately hard replace sum sparsity minimization property practically alternate multiplier intend formulate augment iteratively update alternate leave summarize implementation measurement output signal stop criterion satisfy satisfied multiplier backtrack every element singular hermitian transpose stop consecutive cost completion review principal
look matrix summarize difference graph mainly pairwise random link topology central focus whose permit cast topology weight vertex respectively edge signal assign scalar unnormalized combinatorial graph laplacian graph particular column eigenvalue equal laplacian via generalization fouri graph extension tool detail smooth signal equip laplacian smoothness edge adjacent signal vertex weight consider strongly smooth supervise model statistical model try explain observation potentially unobserved q represents observe represent control signal adopt isotropic zero give classical key laplacian latent definition reflect link representation graph since many partition fourier
brevity htbp dictionary top dictionary element dictionary similar dependent parallel dictionary redundant layer propose dictionary convnet layer classification category infer send kernel cross model complicate layer imagenet sift training deep within bayesian map enjoy efficacy develop project high image accomplish mnist result near deep design jointly gpu scale deep novel probabilistic pooling operation integrate refinement
eq completes mention range consider random variable complete give method multiplier apply solve lagrange optimization monotonic far complete proof ready prove proof iterate achieve ready differentiable eq q conditions complete define follow kt iterate hoeffding satisfie meanwhile result kn arrive complete result deviation obtain kn k compose k n k kf kf z z difference ng similarly arrive least complete corollary engineering usa school science technology r china china
reach ergodic produce string bl note bl convex hull vertex produce string compute step derivative trace definition convex hull choose vertex generate associate string distribution state structure symbol find symbolic derivative define q terminate necessary connectivity output give asymptotic complexity essentially identical complexity input stream alphabet note corresponding take metric respectively stream generate probabilistic dependency process dependency learnable follow denote need result theorem coefficient dependence clear demonstrate directional process reveal possibly flow causality call stationary evolve need distinct process map imply encode inter dependency machine introduce notation infer machine string generate denote stream simplify coefficient give label strongly connect string run use establish stream run distribution current stream break j ik j dependence distribution stationary index equivalence minimal encoding strongly connect label g minimal converge stream state satisfie recall complete coefficient avoid composition correctness complexity inference complexity statement denominator appear ergodicity denominator surely infer bound surely refer assumption infer also see I cross establishes immediately establish small important bind symbol also imply relatively rare neuron network infer predict evolution evolve alphabet standard interested evolution notation accordance denote b
various hmms efficiency readily method seek active way possibly reduce try heuristic intuitive well practice exactly heuristic error largely theoretical result concern allow establish active strategy cost reduction maximum posteriori hmms bs brief active bs hmm study inference analytically essential hmms flexible tool bs insight demonstrate analytical map bs determine efficient scheme error remain unlabele allow examine active bs relate
ap fc yes baseline ap fc fc baseline fc fc sp yes baseline l method cnn acc kernel yes color bag yes ta yes pt table car car person person car person person person cat car map classifier activation local activation patch activation score train several confidence verify encode discriminative image patch discuss sec map utilize localization pyramid activation train several take characteristic activation consideration fisher uninformative however contribute invariance meet activation multiple reasonable equivalent filter enable pool multi
non distance close expression metric approximation indicate student test svm accuracy indicate score respective test c c dataset svm eq ten uci dataset learn former feature rkh respectively metric mahalanobis proximity learn distance mahalanobis cosine optimization computation finally compare classification multi multiclass svms psd computationally dataset low method set initialized identity initialize simplex satisfied default gaussian parameter parameter select
smoothed formulation variety weight need annotation latent annotation initialize approximately object manner effect initialization procedure correct feature box occur image short seek dense image find subgraph combinatorial encoding present signal old address description concept share information formalize flexible help box integrate cover positively versus generalize combination mode object appearance distribution image
emphasize solution denote optimal separate goal sample datum space q select unsupervised set full datum set main tool set score bss greedy name column v potential eigenvalue define iteration index include spectral rescale lemma theorem construct rescale top carefully e proportional norm singular select trial dominate time min rp
channel distinguish widely spaced artificial factorization gain source lee similar drawback stacking view instance spaced wide resource require resolution author point east west poor source issue vast majority extension audio allow structured decomposition dependency error etc paper could combine simplicity version nmf main
include several extensive datum represent network sequence denote adjacency direct general self e denote time respectively write quantity notation indicate node member node vector submatrix relation stacking index static consider snapshot time parameterize node relation node adjacency give random block dependent estimate rewrite number priori ratio estimation method sample switch spectral heuristic combinatorial possible class membership utilize adjacency membership temporal model call hidden extension conjugate initialize multidimensional approximation allow involve static namely blockmodel use extend kalman model decomposition
tackle limitation enhance software effort model present report rough extraction software project software software influence project effort fuzzy system enhance fuzzy enhance estimation artificial ann diagram help regression logic neural b boost effort promise comparative radial basis neural experiment carry dataset well regression genetic algorithm select optimize simultaneously improve effort al
convolution implement interpolation support input output transpose vertical zero input height width intuition summation reverse round operation weight combination element filter cycle right transpose filter store slice max channel patch operator sum detailed sect support convolutional equivalent extend datum zero boundary compute relu compute implement operator normalization independently location channel channel input dimension operator channel convolutional implementation implement batch somewhat whereas process image individually instance batch case array treat tensor additive map implement channel feature neighbourhood adjust must normalize section detail compute
work experiment recurrent ht module module neuron fully module module sequence modeling focus rnn long spirit rnn instead simplify introduce additional recurrent lag additional help bridge lag train difficult run slow term lstm architecture store error connect new gate network gate decay forget gate successful recently stack hierarchical hierarchy equip temporal
tell norm proceed induced atom norm follow immediately proportional know svd j sum symmetric non show psd span psd case would existence eigenvalue constant psd write psd matrix psd optimal decomposition positive might positive exist differently hull include cone nx ni generalize norm simple identity definition square replacing hand eq back finally give triangle bind denote sequence h fr fr ts inverse increase bound bound nonnegative scalar start prove tangent tangent closure inclusion prove k b duality corollary x x scale subdifferential deduce hull note tangent cone proof dimension part notation section appendix ab dimension normal cone subdifferential characterize cone introduce notation follow denote onto respectively form aa bb subdifferential write equal inclusion belong measurable characterization statistical give measurable provide cone fact freedom deduce simplify notation become belong sufficient g characterization prove inequality operator work know follow j I ij g j j v g g b u I derive
aic bic dimensional repetition examine weak pn interaction tn generate fit linear interaction typical example view ten form ten covariate oracle argue dimensionality prohibitive implement regularization fan candidate different oracle working benchmark select oracle criterion recover effect report portion portion simulation save positive negative report dimensionality tend criterion improve select small effect meanwhile interesting see correctly measure include supplementary several interesting performance specify except aic reasonably newly consistent selection misspecification interesting multiply oracle effect
trait control share evolutionary serious public health burden understand throughout evolution analyse human binary status assess trait trait pair correlation coefficient contain correlation genetic linkage trait present correlation trait correlation analysis reveal strong trait kb genomic among trait trait reveal history spend transition occur across specie binary trait pair definition correlation trait attract history question play investigate trait evolutionary trait population trait color orientation trait trait population evolutionary transition order latent include analysis use integrate strength traditionally alone draw trait matrix brownian
pt experimental rank normally sample zero normalize incoherence vary fix computational show scale increasingly dense fact hard intermediate intuition confirm show intermediate iterate frame time next foreground separation video form frame stack wise background static form foreground dynamic benchmark name restaurant dataset frame resolution extract several people near desirable
reward previously denote sect full portion maintain mean arm node directly observe q intuitively second account estimate value tight present identify along reach optimistic leaf node alg corresponding expand leaf arm maximum arm contain term add reward term point term become first mean uncertainty reward dominate reward amongst arm resolution need approach choose become occur sect supplement discussion expand two child lead accurate leaf create select big arm single episode optimistic outside optimistic remain unchanged node bound validity material begin need eq uncertainty resample internal optimistic fact second alg force optimistic become notice particularly critical choosing resample complexity theoretical feedback round upper iid iid find representative
snr sound model plane wave unknown arrival arrival isotropic spatial function sound component coherence aim estimate snr coherence sound field coherence first q estimate factor show coherent e coherent cdr signal bottom page omit cdr amount
length keep suffer challenge problem suggest length phase plan uncertainty margin appropriate regret solve provide explicit cost calculation paper control long arise classical parameterize sample start prior assumption positive semidefinite denote semidefinite main linearly parameterize dynamic choose kernel tw smoothly past know mx meet indeed latter markovian change example fit vector dynamic action select average loss e boundedness
feature large propose basic picking update instance feature norm adopt impose strong adjust primal convergence average follow let q tx execute factor make large robust unnormalized vector average describe method single coordinate update mini strongly lead omit technical considered optimization convex function develop dual convex appropriately obtain accelerate saddle form let batch update coordinate update update accelerate subtle primal auxiliary variable replace stay compare fact imply assumption batch equality bad become match case discussion order bad batch
combine feature exchange reversible implement name double reversible jump substantial gain double reversible jump enable model previously cost remain limit number propose direct efficient representation conditional elegant hasting bayes sampler cast bayes birth death set add remove associate birth event occur coincide provide substantial improvement status algorithm estimate connectivity
base localization improve initial notably degenerate class class remove point notable localization angular resolution increase manually position minute record minute propose h move scenario white move number red circle position correspond face video research world apply sbm method slide allow source direction fig frame number live manually select analysis center number high amount yield observed video speech activity detector adjust size segment sound source circle localization implementation face detector implementation cpu face detector annotation magnitude method standard histogram interestingly detect visible clearly may complementary visual face speak face even face ccccc wide field camera location location notice room impulse response circle find method face team research train system room room environment therefore likely capture room impulse response rely acoustic world testing position room impulse occur position moreover camera large right view room distance scenario online select improve use error average exclude large
english actually represent negative sentiment explore provide split involve classifier small sentiment explore effectiveness observe obtain fraction feature hope would guide future sentiment project number engineer university computer engineering engineering introduce sentiment date consist review rate star investigate property sentiment provide split validation testing rating unbalanced setting extend comprehensive classifier sentiment compound sentiment word explore available internet subject product book medium ever active sentiment among study classify piece opinion e either sentiment rating predict star
subject weight visual face multi modal refer comprehensive drawing trial faces filter raw hz hz ms stimulus trial channel classification decode regression show cross accuracy high leave pool trial use logistic penalization observe drop decode sg great table second trial
self normalize variance powerful alternative instance property consistent cite self subsample small necessary quantile estimator role study sensitive key ingredient finite product subsample asymptotically normalization nan directional statistic following distinguish brownian motion assumption pr dr multiple quantile depend process difficult implement nan simulated significance level percentile theorem local non p hold local alternative cross intermediate include reflect historical event ease rest dependency present lag quantile lag li k notice obtain analogue
take weight choose advance iteration store distribution spaced number grid value propose drive low computational trivial bayes approximated shape require implement efficiently perform hyperparameter stream derive update prior streaming update good covariance denote matrix wishart freedom parameter wishart joint lead expression conjugacy posterior assignment history
top identify identifie layer identify remain vertex cube specify vertex layer vertex identify contain identify identify main theorem review sect obvious always create maximum vc number dimension cube must use inverse move towards zero exist movement vc vertex shift cube anchor contain coordinate value notice preserve number fact neither must vertex easy number decrease offer subsequently sect existence projection sect vc embed vc iterate cube direction graph embed cube node face direction node contain correspond tb iterate every iterate reduction iterate color project coordinate correspond cube edge colour come cube complete cube iterate firstly maximum class reduction view time
term discrimination outperform focus development calibration probabilistic traditionally machine development improve discrimination improve prediction important make decision analysis probability outcome model method effective machine calibration modeling make may could lead addition affect calibration calibration parametric method parametric method model probability intend calibrate learn maximum likelihood distribution common non briefly associate estimate introduce method
power negative positive example argument language language trace rely positive large language language form language language build otherwise second family language language language language program intuition synthesis synthesis language language synthesis engine form language language synthesis technique access produce history recover easily program follow element far pick minimum see synthesis engine program return assumed iteratively discover eventually every positive singleton contain form large positive see previously candidate trace synthesis engine program language consider recognize language trace minimal observe z forget observe hence program class program fail program program synthesis
sa contain exactly solely learn call transaction factorization attribute attribute entity entity dimension attribute sa arrange tensor preference user item explicit preference rating typically sparse rating information entity occur cell negative preference assign real entity combination efficiently restrict training weight entity actual entity factorize entity dimension entity therefore actual although sufficiently generic weight leave exploration occurrence basic accurately predict entity one weight generalization square preference allow experiment usual framework vector consist hadamard product product linearity apply feature consist implicit method arise feedback possibility decompose computable part computation follow latter alternate usually accurate fast main square thus scale make conjugate cg square cg solve see linear generality base whether element reach column similarly simplify equation entity dimension weight sum difference part efficiently expand sum product argument feature vector rearrange feature scalar note change update solve conjugate high description
g curve give successful extraction source canonical correlation generalise noisy signal eliminate effectively blind implementation derive extract consider indirect preliminary research engineering university uk uk blind source establish
due induce high iteration good file despite fact file incur call observation accurate algorithm exploit popularity profile exploitation skewed popular file multiply also reflect period empirically content file cache cache replace file since cache applicable directly period file within period note history learn past period numerical mab section greedy terminal service average cache system otherwise cache capacity unit file user set file file size user uniformly skewness memory cache percentage refer content cache time algorithm switch greedy greedy algorithm mab plot lack theoretical practically steady switch figure mab mab switch period greedy counterpart greedy arm period reduction switch opposite ht mab inform popularity profile know advance cache horizon
integral case pathway variate density cm integral international journal journal pathway pathway h apply function pathway transform special pathway mathematic cm fractional integral perspective j york cm top bottom com mathematic university west cm mathematical science com outer united international cm
reward assume reward lie round apply martingale reward round decision holds obtain take wise optimal corollary bind regret end majority sr sr
differ proximal correspond hierarchical show proximal remarkably form write observation proximal solve exactly duality furthermore operation include special problem r method compete var modeling forecast dimensional fit square lag use aic per aic residual matrix lag selection follow square simple simply include var penalty lag lag lag lag pattern lag serve baseline unconditional sample ahead forecast form walk efficacy application evaluate method scenario component length describe
improve error medium sized modification produce strong test absolute unlikely grow bernstein sum variable empirical bernstein bound develop bernstein bound version variety bernstein small range variable random standard hoeffding bad increasingly likely binomial page bound advantage low q truncation inclusion refer depth depth multiply instead side bernstein binomial input randomly v
represented truncate normal tn distribution tn property mixture contamination scheme whereby amount proportion bad component g figure contaminate skew ignore write analogue respectively include fit contaminate schwarz good
estimation mse setting table display improvement significant mse improve primarily propose value similar unable abc currently bivariate analyze objective take recorded record refer ht ccccc represent customer propose bivariate beta binomial denote requirement joint beta distribute notation binomial introduce bivariate distribution gp b bf ep parameter determine valid beta form enable use bivariate propose kb beta beta furthermore
line line result performance recommender result plan investigate public mobile aware recommender model exploration exploitation exp current preference improve knowledge recommend user risk introduce name ucb user adaptively balance reveal exp feedback click near people become optimize mobile aware recommender learn may critical exploit appear frequently see select environment prevent maximize reward reward uncertain environment prevent formulate exploitation exp one solution exp arm hybrid combine confident ucb estimating interval reward algorithm document confidence essentially control difficult
completion completion compare well leverage perform repeat completion time attain weight completion procedure target output conduct effectiveness generate freedom covariance whose entry index observation noise unweighted collaborative collaborative filter datum unbalanced violate general experiment estimate score unweighted unless portion collaborative commonly unweighted compute solve repeat type report truth rank subsection test score coordinate synthetic descent vary sample set take row hinge loss coordinate respectively ht leverage result weight well intuition weight whether alternate weight procedure four ht round coherence perform procedure completion two set experiment
right dot node size right dot transform shape cm node right cm right b distance cm input node distance cm distance cm boltzmann hide bias hide denote input probability interaction visible differ bias share trivial regard wise normalize distribution namely yx px px approximate conditional represent conditional principle otherwise ambient polytope expect dimension triplet visible jacobian parametrization mild piece wise linearize version dimension rbm apply idea order denote cardinality whose small cardinality every hamming apart set implie joint know whenever conditional q proposition universal vanish imply conditional account follow divergence universal approximation analogue
shape three primitive record front fix isolated protocol consider illumination characteristic report recognition illumination canonical pm tensor standard pm characterize point product manifold high singular factorize represent modeling select approach perform poorly conjecture illumination hand tb use pm approach pm video traffic pattern light weather video record resolution range frame normalize version dataset normalization involve subtract mean normalizing intensity illumination traffic traffic fig select respectively method also compressive dynamical present approach obtain achieve worth et tb example traffic light heavy traffic video spatio compressive sc perform two simple property euclidean experiment realistic sample follow mapping back map create fix class variance problem medium hard give multiclass size map manifold propose sparse code approach task repeat ten tangent experiment sample turn recognition consider setup center projection sc fr characteristic prior approach
I word scope system collection use text specialize kp specialized specialized public public string double I I I else return j dp return use public count reverse public string return add dc else count return string solve string return improve collection use text dp vx xy long vx vx total total break vx scope write human understand human primary generative tailor base probabilistic extended variety baseline likelihood hold great deal human effort go develop develop tool development fast tool code problem outside machine public massive collect assignment thousand observe think code human human regularity combination work primarily
deep improvement unlabele minimal effort receive attention label train target improve performance deep bottom extend beyond structured output performance reconstruction objective prediction label bootstrappe prediction thereby structure may useful approach bootstrappe output proposal person agnostic region deep noisy multinomial softmax without noisy label addition log add encouraging multinomial feed forward posterior use softmax denote noise softmax follow learn
scene illustration independent human inference result mesh super impose image evaluate category internet significantly outperform dataset object internet superior current object inference transformation mesh challenge ask manually fit image mae score invariant surface mse much utilize approach attribute slight ground inherent object demonstrate b prior real naturally much part collect human person category internet comparison current art
rmse evaluation criterion order experiment result see vi design topology knowledge extract factor inclusion three model bad experiment layer eight verify offer design topology exploit properly knowledge stem good indicate nn network build fig omit modify magnitude negative large notice interpretation especially thick immediately influence neuron influence strongly neuron nd neurons layer affect moderately outcome quantify variable like valuable leave research try neural random forest achieve comparable necessary unsupervised incorporated try
situation variation miss simulation show simplified pooling yield short confidence mathematically trivial aware instance practical
weak tell efficient expect know perfectly predict expect size ensure tractable matter representation large etc statement fit hard possibility achieve unknown e agnostic edu demonstrate play central role multi feed forward argue analogy inductive induce sort encourage success expressive relative learning
extract pca percent improvement percent collect component ignore high precision optimization reduction localization position capability infeasible gps hardware gps service environment location location proximity localization measurement rely angle hybrid utilize various technique signal range technique provide movement location eliminate estimation surveillance object system divide define widely localization figure recognize location hardware manual reference transmission strength indicator reformulate discuss seminal summary column specify localization moderate aware moderate centralized localization centralize purpose soft localization moderate localization distribute localization distribute purpose localization svm localization fusion acoustic surveillance surveillance gp distribute spatial collective space som moderate localization som low centralize less distribute distribute less determination rl mobile localization correction method predict likelihood applicable system network thousand localization appeal investigate fu activity use phone music convenient way name ambient intelligence human home device automatic power management core classifier detect robust localization activity manually activitie limitation centralized system recommend investigate unsupervise automatic extraction localization scheme particular multi layer perceptron radial recurrent rbf resource requirement mlp likewise sensor node use anchor utilize system adopt node localization localization ability g alternative non probabilistic precision predict cost error device illustration mobile localization employ connectivity capability method movement movement detection localization design goal indicator though offer distribute effective outlier limit datum therefore idea sub start divide I predictor sub addition computational robustness preferred low tree develop target exact location target difference arrival tree also event
particular equation q langevin evolution fluctuation incorporate constant correlation root langevin equation convenience langevin easily arrive langevin concern markovian present mathematically markovian represent probability histogram ref concern moment average
x n f w encoder reconstruct conditional hold output describe symmetric model encoder decoder tie explore autoencoder denoise input autoencoder learn feature learn input train translate filter phase shift loss input entropy differentiable optimization momentum
one constant predict capture seem unbiased line metric popular regression tree package cart introduction forest longitudinal datum one subject computed entry cluster obvious accuracy cart forest tree forest subject cluster list close spline relation bayes forest besides distinction forest estimate test converge similar accuracy lastly focus area bic factor mechanism complicate necessity review attractive inclusion however
vpt def bl vpt fill vpt bl copy vpt arc fill bl copy vpt copy copy vpt arc fill vpt arc def bl copy copy vpt arc vpt arc bl copy vpt fill vpt arc def bl copy def bl vpt fill arc bl copy vpt arc fill vpt arc vpt arc def bl copy vpt arc fill copy vpt arc bl copy copy vpt arc fill vpt def bl copy vpt arc fill copy arc fill vpt arc bl copy vpt arc fill vpt arc c bl copy vpt arc vpt def roll exch def square vpt exch vpt vpt bl vpt bl def bl copy vpt square bl vpt exch vpt bl vpt exch vpt vpt def bl copy vpt sub vpt vpt square def bl copy vpt exch vpt sub exch vpt vpt fill def bl copy exch vpt vpt sub vpt vpt fill def bl vpt sub vpt vpt vpt copy vpt bl copy vpt sub vpt def bl vpt vpt vpt fill bl copy vpt sub fill copy exch vpt exch vpt square def bl copy vpt vpt fill copy exch vpt exch vpt fill bl copy vpt exch vpt vpt vpt def bl copy exch vpt vpt vpt vpt copy vpt bl copy vpt exch vpt vpt fill vpt exch vpt def bl copy fill def translate def stroke def translate stroke translate translate stroke def translate translate stroke translate stroke translate def translate stroke def translate stroke translate translate stroke translate stroke stroke vpt add vpt vpt vpt v def stroke exch vpt vpt vpt stroke def vpt mul mul vpt mul v stroke stroke vpt mul mul vpt stroke translate repeat stroke def arc vpt vpt vpt vpt vpt def exch exch vpt vpt stroke def stroke vpt mul vpt mul mul stroke def vpt mul sub vpt mul mul vpt stroke def stroke stroke arc stroke def exch exch exch exch add def def def def fill fill roll def get get get get translate mul mul def translate mul ne get add roll stroke ifelse true def def ifelse def def exch stroke stroke exch l fill exch def def l stroke exch stroke exch def def stroke pattern def pattern pattern landscape ifelse def landscape ifelse def landscape ifelse def landscape ifelse def ifelse def symbol length begin index def ifelse end begin def exch exch exch def roll exch def sub mul def mul sub sub def mod ifelse ifelse ifelse ifelse ifelse ifelse def constrain exch ifelse def add constrain roll mul exch mul constrain roll exch mul add constrain roll def rgb exch exch exch roll exch roll def copy mul add exch exch constrain roll copy mul exch mul exch mul roll mul mul exch add roll def ifelse ifelse ifelse true gidx gidx gidx gidx def loop def gidx sub def gidx gidx get mul def gidx gidx gidx mul def gidx get sub get gidx mul add gidx get le gidx get gidx def ifelse def def mul ifelse def pm gamma def stroke pm exch def stroke pm cf constrain cf constrain exch cf constrain ifelse pm pm ifelse ltb stroke ltb r ltb stroke ltb r stroke ltb stroke ltb stroke v ltb stroke ltb stroke ltb stroke v ltb stroke ltb stroke v v v ltb ltb ltb lt v v v v v v stroke lt v v v v v v v v v v v v lt v v stroke v v v v v ltb def exch exch mul roll exch mul mul def mul mul mod ifelse ifelse ifelse ifelse ifelse ifelse def constrain lt exch ifelse mul mul roll exch mul constrain roll mul exch mul roll exch sub roll exch roll def copy mul exch constrain mul exch mul roll mul add exch constrain roll def ifelse ifelse def gidx gidx gidx gidx add def def gidx get gidx gidx gidx gidx gidx mul def gidx gidx gidx sub mul add def gidx gidx get gidx mul add def gidx gidx def ifelse def def def pm ifelse pm def color stroke pm exch stroke pm constrain exch constrain exch constrain def ifelse stroke pm pm ifelse ltb stroke ltb stroke ltb stroke stroke ltb ltb stroke ltb stroke ltb stroke stroke ltb ltb v stroke ltb stroke v ltb v stroke v stroke v stroke ltb stroke ltb stroke ltb stroke stroke ltb stroke n stroke ltb ltb ltb v v v v stroke v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v stroke v v v v v v v v stroke v v v v v v v stroke v v v v v v v v v v v v v v v v v v v stroke v v v v v v stroke v v v v v v v stroke v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v
sensitive polytope convenient factor recommendation take actual one base decide implement strategy nontrivial perform experimental effectiveness impact design burden
corresponding argument observe round deterministic determine action round thus entropy relative equal relative recall determine pick adjacent go distribution coordinate node bernoulli l strategy plugging bind overall low least expression pick constitute round random neighborhood node need arbitrary expectation permutation selecting subset node term define arc consider j r randomness lemma outer remove conditioning conclude shorthand round outer expectation bind contribution overall quantity take bounding summing moreover action rearrange sum q I recall recall adaptively fall order overcome exp apply trick pseudo variant exp since guess run observe sum pay pay get take theoretic fairly analyze setting counterpart hold undirected direct let number induction start independence incoming arc obtain sequence small small sort node arc g state edge undirected refer arc recursively
x mnist original reconstruction largely sample heuristic multi heuristic well shrink shrink heuristic similar representative show note near optimal scale show efficacy heuristic pc elimination svm execute heuristic execution overall speedup comparison process original scale x speedup multi perform little execution approach spend reconstruction trend observe process dataset comparison set fair conclude pc extract shrink set shrink beneficial different set several hyperparameter fast elimination sample benefit shrink research shrink step shrink degradation shrink
might assign likewise test compare reward opponent pair test algorithm course reject ks rough ready merely mind interesting trend per opponent generator similar instance quite especially even block similarity show weak similar twice similar experimental metric study prominent us relationship converge consider equilibrium game relate payoff opponent finally tendency profile nash equilibria infinitely game difference could play static pure receive good static pure opponent play look regret record payoff respect significant thing ignore great reward learner always iteration approach infinity algorithm achieve lack regret achieve regret experiment algorithm positive achieve positive regret low gradient lowest achieve regret follow result practice come way see regret run negative slightly hand rarely run overall average regret converse attain take precisely wrong run involve around self outcome play claim break cycle improve per achieve low every distinct opponent well play converge pure opponent trend opponent regret dominate dominate gradient never dominate dominate opponent converge nash equilibrium self play nash equilibrium course play response hence dominate vice versa dominate dominate define another regret
factorize illustrate bilinear logistic bilinear logistic construct square become projection variational generate interpret mapping equivalent jk priors variational problem incorporate bilinear regression objective bilinear sparsity depend regularizers one explore future reason prior bilinear three bilinear ambiguity estimate estimate arbitrary rank unique ambiguity introduce logistic originally reveal spatial neural generating localize correspond reasonable sparsity third noise logarithmic concern regression intuition generalize show improve variational fix
lead quadrature quadrature intensity count component thorough study quadrature perform prove particularly realization localize section kernel estimation quadrature adapt inaccurate estimation mass spread study phenomenon financial issue frequency financial rather translate kernel second decay quickly small around much slow lag explain adapt clearly first step lag behavior precisely basically consist grid interpolation check approximation bring algorithm next last sensitive quadrature wiener adapt quadrature roughly kernel else precise estimation whole quadrature use point far able solve numerically consist point quadrature test simulate dimensional process simulation period second quadrature kernel kind multiscale kernel financial decade second
penalize need fix intercept control height dividing carry issue maxima challenging depend complexity consider quantification estimate perform latter avoid asymptotic number pointwise simultaneous bootstrap replication simultaneous confidence pointwise confidence contain fraction completely parametric hmms hold usually satisfied specific general condition zero guarantee domain sufficiently deal induce unobserved markov specific easier similarly distinct describe fit smoothing adequate datum drive leave infeasible generate
infection semi instance one could contact spline drawback involve cm school mathematical sciences ng rd uk mail uk occur methodology rely propose b illustrate show epidemic pick incorporate parametric disease prevent major high contingency planning past see mathematical disease understand disease nature disease fundamentally disease due ii little parametric assumption quantitie transmission infection period
draw black sep bl dot distance distance dot v circle width black inner sep cm fill bl dot dot h right h c distance visible layer cm rbm remainder comment present architecture extensively one feedforward hide unit understand composition unit tune input unit superposition linear unit see address accuracy feedforward maxout unit besides deterministic assignment give arbitrarily sufficiently stochastic see application understand development address impose subsequent connect restriction version approximation enough related minimal rbms nonetheless deep
method algorithm coupling e provide couple begin briefly present coupling diffusion suppose solve solution wiener univariate wiener finally orthogonal case need choice solution obtain stochastic indeed rotation however successful bridge close norm orthogonal onto vector geometric interpretation difficult imply wiener increment wiener increment increment minus increment increment increment direction increment wiener plane drive wiener meet treat define ensure really necessary bridge simulation rejection certainly ensure geometrically fast somewhat restrictive wiener initial path univariate wiener formulate review density function hence stationary diffusion solve stochastic differential condition ensure discuss lipschitz twice continuously transition partial
focus incorporate configuration detection within prescribed optimize auc positive trivial extend boost denote bold bold set letter entry tuple weak learner weak learner represent respectively predict output represent output learner training partial curve j learn learner score performance learner learner object practical successful slide main adaboost aggregated channel combine show outperform train bootstrapping train repeat bootstrappe classifier pool optimize roc efficient region proposal generation low descriptor spatial propose modification benchmark set pooling prove pooling pooling combine nearby statistic region preserve neighbourhood detail pooling learn lead state art object learn promise still time consume application low adopt simple operation enhance descriptor descriptor apply low provide relationship represent feature represent low derivative axis edge orientation orientation orientation feature redundant would select preliminary alone bad performance map image descriptor region store texture descriptor binary histogram version thresholde neighbourhood centre pixel
acc laplacian mode mode ex ex mode shift acc mode acc laplacian mode acc mode homotopy acc laplacian modes centroid interpretable centroid hyperparameter initialization centroid pattern surprisingly centroid object centroid branch manifold mode shape identity centroid overlap class expense class represent centroid receive centroid partition smoothness prevent centroid digit identity centroid move area homotopy assignment separate object achieve amount centroid mnist centroid valid image identity yield representative even mode produces shift nonconvex shift spectral find centroid pattern lie area individual
population baseline cognitive knowledge purpose eeg different baseline ec state subject room ask eeg data channel eeg signal subsequently apply anti pass restrict follow standard exclude retain subsequent analysis select available future condition ec eeg non epoch epoch consider observation state extract spectral assessment connectivity detail respectively eeg investigate purpose transform obvious physical eeg eeg signal extract epoch slide overlap psd psd ft characterize eeg namely cover eeg delta alpha
long remainder find negative baseline relative error indicate dataset hence claim include appendix average method performance large error extremely good run within dataset quite consider large see statistically efficient computational b appear computationally large implementation twice small primarily interested nonlinear expansion restrict dominate baseline dataset much essentially infeasible intuition adaptively effective sort try algorithm candidate rather parent inclusion also extremely base
change word change actual procedure call gets execute improvement thank pre central quantum phase artificial neural exploitation several investigate order perceptron method adjust weight though quantum equivalent classical quality purpose quantum perceptron onto compare perceptron able
case decision logistic sigmoid converse true return original predictive stock membership capable give stock membership incorporate far speak advantageous probabilistic sigmoid belong initial subsequent disadvantage poor relevant stock return address implement nn close neighbor within close neighbor may maximize quantity empirically speak case classifier single nn subset validation minimize large forecasting suffer bad intractable exist problem boost improve generate capable testing indicator evaluation use model ensemble behind return outcome subsequent classification prediction task meaningful future would application objective augmentation boost
classification validation cifar set consist colour validation size pixel colour channel represent divide validation dividing every get number choose epoch layer model pick da number training epoch da model time turn da combination parameter number step validation epoch da noise classifier pixel unit try outperform da error reduction achieve interestingly schedule detail supervise fine tuning yield low ever cifar error material test reconstruction error epoch hide unit worth
asymptotic present regard family statistic simulation carry comparison shall assume unknown estimating regard multipli lagrange multiplier n also test testing eq measure neighbourhood constant account fact complete assumption law consequently hold true restrict state rejection rule nan give alternative close term contiguous tend manner substitute n f obtain complete consider contiguous relax contiguous hypothesis variable obtain restriction able invariance empirical estimator inverse
bss task decentralize obtain multiplier denote derive auxiliary across bss w form lagrange multiplier three update lagrange multiplier quantity repeat bs neighbor via update decompose individual bss similarly plug simplify specifically become update perform locally iterate converge dynamic decentralize fashion centralize update keep essentially static mcp decentralize modification cut penalty cut bss copy bss cluster resource bss decentralized involve basic eigenvector subsequently execute implement decentralized fashion subsection bs initialize bb kt b start initial matrix decentralize equip th row localize let bs affinity b
hyperparameter eqn row order priori chi imply permutation variable propose modification convenience identifiable triangular loading distribution invariant diagonal entry truncate gibbs
find base hybrid outperform maintain start recommendation mf popular technique effectiveness netflix competition rating much matrix predict rating original previously mf represent activation mf value utilize phase weight simultaneously integrate description latent pure collaborative cf item involve separate weighted average combine depend technique content rating rating pass implementation address build description recommendation system input put backpropagation input weight backpropagation define sparse rating user matrix represent item
autoencoder variety mnist stochastic descent momentum training autoencoder manual search autoencoder decoder isotropic isotropic hide activation dropout additive multiplicative unless otherwise hide intel evaluate generalization autoencoder patch van image noise hide activation evaluate denoise equal average dropout vary activation hide effect learn autoencoder autoencoder structure learn digit
system reader reference article solver determinant contribution new direct solver semi separable matrix determinant separable determinant enable semi computational idea encounter computational autoregressive moving model sparse algebra
loss well equivalent w since agnostic ordering stream prune naive requirement arrive turn natural inclusion verify framework point loss l recover auc notion penalty function online interestingly exploit structural property penalty low framework follow style stress regularize batch problem let update require lipschitz continuous w loss decomposable function would would bound instead
feedback layer attention selective dynamically convolutional sequential allowing iteratively internal convolutional million dimensional space cifar cifar outperform deep convolutional cnns pooling recognition parse extensive stack feedforward bottom visual representation stage tend plausible detector detector object part train cnn evaluation discover feedforward pathway ask belong think answer imply feedforward perform
let get inequality new p c p r p experimentally verify minimization superior nuclear minimization establish observation sufficient condition quasi unique recovery quasi minimization theoretical
arm handle ucb arm confidence j optimistic reward express regret eq ucb comparison play strategy depend precise dominate influence ucb strategy explore arm arm good never stop optimal usually arm explore strength lie arm property exploration budget spread regret ucb play arm non ucb continuously avoid keep reasonable iii budget among promising arm automatically exploration grant ucb algorithm suffer greedy need knowledge reach regret contextual categorization profile core idea paper build
set stochastic sa briefly sa line fast pg sa aim analytical seminal employing govern solution cf sample plain propose try end episode shoot step size comparison policy recursion ode policy section likelihood ratio gradient ex markov single chain parameterize recurrent let sequence encounter optimize measure use simulate obtain sample gradient employ know estimate derivative
task connection establish market market period trading optimisation via market trading objective long market maker market multi period market pricing market maker aim optimisation trade maker quantity treat could apply translation invariance functional become eq pricing rule independent rhs substitute merge scheme converge lead despite behave agent contribute global goal trading sensible concern introduce requirement risk pricing objective optimisation problem risk pricing absolutely well functional essence pricing
input tail slight dispersion deep layer dnn observe inactive small unit input actually transform dnn depth dnn also code show trend activation subsequent hide true activation layer layer compare model model tend tb indicator unit behave hide focus unit number activation code layer tb length vary hidden information trend nearly deep observe unclear code capture enable versus redundancy hide unit length total trend depth reach trend evident parameter trend code length code deep task multi acoustic comprise address fundamental dnn design depth corpus improve overfitte utilize combine oppose regularization experiment suggest dnn architecture competitive specialized architecture architecture outperform architecture locally make meaningful property enable assumption specialized acoustic locally model work recognition task experience interference distortion train acoustic comprise evaluate date acoustic train optimization gain frame corpus reveal fairly dnn small begin encode information differently certainly layer fairly small certain point increase dnn depth yield gain dnn acoustic use optimization suggest dnn hide layer five reasonably strong modify acoustic procedure explore drive dnn acoustic stem acoustic train task dnn map acoustic input believe guide new dnn architecture train demonstrate fairly improve speech language understanding network component build acoustic design decision include offer investigation acoustic speech dnn final speech metric quantify factor task experiment benchmark hour compare network locally acoustic build systems corpus fisher corpora us performance
functional prove prune optimal extend pruning segment neighbourhood decompose pointwise cost condition pruning inequality generalise hold value pruning explain section pruning lead prune also functional pruning pruning demand pruning decide compare binary segmentation implement loss c assess well synthetic algorithm implement segmentation maximum search binary furthermore operation front segmentation point speed possible fast segmentation investigation occur database come different pac expect change profile run change execution algorithm benchmark array time
annotation assignment formalize assignment descriptor video annotation convenience element natural occur index action annotation background list actual illustrate interest regularizer f assigning regularizer scalar parameter jointly assignment follow assignment matrix one annotation put every obtain z indicator class classifier z z ta z ta assign row correspond em single replace skip replace descriptor notation want matrix assignment amount impose figure intuitive illustration column block contiguous occur
suppose bound norm tx similarity choose range wide close argument decrease large argument offset right plot curve big towards roughly big ideally small like big trend small hash maintain inner retrieve query conduct hash one reason popularity evaluate million increment rating netflix million movie movie rating integer form rating movie compute rating matrix appropriately rank call characteristic item row item product therefore rank method product outperform popular recommendation choice netflix proposal provable inner since hash
method tune bic elastic net smoothly bic scad discovery method simulation posterior concentrate posterior attain prior en bic fdr bic row correspond empirical bayes scad row empirical alpha beta pp pr contain pp pr fdr pp pr pr contain old simulation pp pr pr fdr sim pp exact sim pp pr pr contain fdr pp pr pp pr sim pp pp pr contain high dependent demonstrate posterior truth optimal minimax minimax via chain monte carlo simulation
show variance sgd enjoy propose reduce stochastic gradient dataset sgd indicate degenerate traditional prox sgd variance significantly variance gradient vary sdca summarize dual gap sdca uniform sdca converge standard sdca gap test sdca result sdca improve accelerate duality addition variance gradient sdca sdca enjoy sdca significantly might sdca paper study importance specifically sample stochastic sampling show depend norm loss relax smooth prox sdca importance show prox uniform sampling sampling improve rate optimization finally confirm appendix firstly bregman subdifferential inequality use cauchy hand result u divide conclude zhang department university
intensive experiment even practical often hold large outperform region provide valuable especially internet lead generation inherently dimensional data size long capacity dataset instance dimension reality web image word document power rarely document higher occur often presence absence binary representation locality hash lsh
close follow idea eq q keep magnitude drastically moreover also estimator optimal budget almost almost twice close pareto nest choose implementation law perfect hasting impact benchmark nested sampling estimation convergence ergodic dimensional vector stationary u reversible transition sequel suggest several final eventually sample keep theory practically speak approximately serve parameter need become generator explain combinatorial optimisation case reader compute walk simulation hence small implementation alternatively generate sequentially parallel recent sequential necessary
decentralize communication bit statistical instance theorem result binomial family communication linearly machine specify exhibit gap theoretic novel ingredient quantitative sharp upper recent subset current organize formalism distribute devote conclude variable abuse probability mass situation case density variable density divergence shorthand shorthand background minimax variant interactive protocol distribution consider quantile give expectation take risk define well bad via range paper minimax characterize problem sequel impose choice estimator minimax computer assume contain subset limited communication protocol machine local datum potentially past convenient model message central fusion denote collection message send protocol encode send correspond protocol communication variable length protocol distinguish two
irrelevant improve consider choose course pass possibility fold cross simulation illustrate grouping lasso use plus elastic publicly package fold elastic cv package noise correlation entry base average produce record htbp tp ms rmse rt elastic net rt elastic lasso rt elastic net rt lasso
context context element hypothesis correct context word pair lexical vector pair word call vector three use difference similarity reference similarity context vector word hypothesis similarity tendency learnable difference indicate lack similar reference similarity hand something colour thing label learn similarity affect lexical empirically algorithm three bad dataset lexical disagreement lexical issue examine present three detail present implication experiment limitation follow entail fulfil mean substitute occur sentence mean definition lexical present lexical entail able entail strong semantic entail fulfil typical semantic come semantic relation semantic relation outside semantic relation semantic relation relation typical relation agree whether definition lexical typical naturally suggest cat cat house cat word cat naturally mind cat sense cat house frequent sense lexical consider word sentence affect lexical decide lexical imagine definition lexical us relation connect condition relational determine entail entail cat semantic relation exclude correct cat cat imply lexical whether word implication depend semantic possible another implication follow relational lexical connection lexical relational case might cat relation cat house cat words house cat share cat house cat cat cat sense house cat agree cat cat relation say
efficient compute ordinal strength concept association discover association impose nuclear concept knowledge performance unobserve response collaborative interpretability relative tag ordinal sparfa pls provide feedback learner concept tag knowledge profile recommendation learner learner tag status benefit capability material support foundation grant air office scientific award science foundation grant pa f h k b r sec b n z cm sparfa com offer way wherein experience goal date factor sparfa novel framework base learner concept content estimate
distance node draw draw type rather construct tree tree count count action take local tree decentralize induce express flat shot difficult employ weight w otherwise guarantee develop priori widely machine however expect factor approximately information come factored mean show estimate converge plus policy dependent induce specify action global reward set cause value overlap component counting action value overlap expert recover action reasoning bound component assumption component overlap intersection profile profile expert return joint whose sufficiently optimize sequential ucb effective policy property bias interaction correlation
end include dyadic n l n note dyadic interval contain one outline provide theorem provide material find collect extension idea good since number interval help far q suppose anomalous interval efficient parametric basic dyadic introduce interval approximated property cardinality interval construct n pt p x successful interval outline proof supplementary material tp cause
ranking model user score decrease order transform collaborative ranking learn collaborative ranking correspond document observe train rank rank fu ndcg section extraction extraction propose tweet main feature part include rating user movie movie share rating twitter build tweet daily twitter original rating rating include extract feature feature tweet triple
filter individually focus orientation cause activation large try sensitive activate break scale small image small pick opposite column non scale object column work use fig map cifar size activation plot big response get activation filter drop gradually peak response peak size filter size compare activation weight however activate c htbp cifar drop cnn last drop
find cut cut frequency find node recover complementary note traditional reciprocal knowing automatically solution briefly refer graph non prove sampling normalize laplacian perfectly recover signal without identically belong lead clearly signal maximum bandwidth computing searching devise approximated minimizer numerically small eigen reduce complete control detail increase give hand due prove e subset signal converse question uniquely represent eigenvalue relax combinatorial need set hope reach cut understand relaxation define diagonal c hand side
object sequential learning also automatically order compare spirit attribute characterize different task apply variant allow capture task th eight annotation recognize specifie create task easy part create except part versus rest remain task image vs equal amount act negative different feature bag normalize act term repeat split across task mean value baseline semantic diversity marker vertical capture average rate background slice area reflect order purpose refer color number performance baseline treat prototype optimization
sparsity heart replace functional instance case reconstructed capture check regularizer smooth I vision image retrieval indeed compact signature anti propose quantification vector homotopy provide regularization quantization instance wireless optimization general complexity dictionary partly synthesis redundant directly j interpret regularize question solution problem give j minimal partly manifold manifold chain key equality know much complicated quite compact partly locally synthesis popular see comprehensive representation multiscale dictionary build wavelet isolate natural sparse piecewise dictionary make localized translate atom audio cope rich diverse dictionary research overview type regularizer terminology measure signal partly partly relative manifold popular example total semi piecewise image isotropic comprehensive review several generalized wavelet kind multi scale haar wavelet prior compose concatenation adjoint difference compose block signal define block write even block I e cope correlated type enforce unitary synthesis obviously overcomplete report compare distinction sparse theoretical development analysis regularization extension value low singular closed restrict smooth partly group build manifold absolutely nuclear j nuclear rank norm shown partly smooth around include low completion principal component retrieval instance collection relative manifold show also
involve clean eigenvector square associated eigenvector eigenvector eigenvector complete part eigenvector e degeneracy eigenvector column matrix leave equation complete proof distinct eigenvalue eigenvector last eigenvector back original know orthonormal normalize rewrite eigenvector arise dot j j last orthogonal kronecker delta term last state j equation two set arise equation release computationally explanatory examine version svd modern data analysis box sometimes poorly behind box focus build solid manuscript derive mathematic behind informally mathematic
recursion fourier pca interest thus maximum random tool try minimum gap maximum technique dependent process algorithmic learning mixture spherical ica gaussian extension detail improve complexity approach sample complexity tensor include I entry entry gap exploit approach tensor rhs copy recursive decompose projection random find pick eigenvector else ica jx kx jx lx maximum improve achieve apply fourier transform observe respect signal polynomial
efficient contour contour need contour computation simplify circular contour skewness respectively py py yy figure contour da b give f respect lot generalize skewness let dispersion scalar denote third stochastically matrix inverse denote density solve impose special special form write make canonical space depth contour depth family half univariate suggest contour elliptical approximate ellipsoid depth also illustrate contour panel vary
sequence quantify dna rna protein bind genome biology supplementary issue overview technology sequence review briefly double dna follow sequencing basis read end length basis end pair pair come double dna sequencing dna come plus read map genome position template read position read read minus plus read genome reference genome span read map length read pair derive minus read one read length every refer sequence detect copy read genome depend piece sample et simplify sequence read assume homogeneous intensity read genome call copy also content bias zhang process derive sequence jump number jump form symmetric walk increment jump indicate walk excess discuss except unnecessary since proximity ignore statistic effort make genomic reliable drop intensity interval reflect whether statistic involve maximization range perhaps obviously genome copy dna genome copy sometimes variation movement dna position genome site variant parameterize genome template read dna reference pair dna sequencing sometimes variant paired end figure region absence map span read pair apart span genome read overlap fail target genome distant reference read pair map
obviously solve score technique widely study accelerate base literature replacement score sufficient hold probability leverage score
establish asymptotic high setting parameter much assess develop paper aim mostly refer graphical lasso exist dimensional setting lie shall demonstrate carry covariance refer precision goal set candidate even constant exhibit impose consistent model impose restrict precision popular sparsity norm diagonal situation normally
mdps factor proceed policy episode produce proceed posterior factored choose confidence require graphical write return write return mdps remain treatment factored modification structure shorthand r obtain mdps accomplish impose artificial episode whenever factor strategy episode mdp simulation h encoding
come visual fourth discriminative cat cat visual appear visualize visual irrelevant look context among mi rs rs region localization dataset streaming method selection popular svm localization svm entire mi well visually region quantitative localization curve area correctness threshold detection arbitrary shape bounding mask yield localization pr value mi baseline also well linear usefulness
kind begin space type reflect spatial change feature reflect property dimension helpful strategy detect change system indicator eigenvalue slide line match report minimum indicate occur eigenvalue considerably significant occurring may detect subsequently overall instant day start recent weather environmental historical eigenvalue spatial baseline significance signal value cx b principal eigenvector principal c principal min p b mention type previous match setting criterion first fail contain effect fail recent solve typical usually perform construct inference environmental compare recent window current approach window ignore way baseline combination assume environmental setting environmental baseline tensor environmental frequent environmental main advantage fail deal
ii player iii maximal supremum player player inter hilbert open problem easily extend player game mean se determine player robust game determine mean payoff strong maximal supremum player could notation corollary play quantitative synthesis verification single average multidimensional vector boolean prove player finite strategy winner inter player game game partition vertex induce quantitative play specification graph objective multiple objective multidimensional outcome boolean formula atom well study payoff sequence infimum satisfy boolean
iteration one sample algorithm obtain consistently update irrelevant update objective number epoch amount especially possibly use gradient sample secondly training dimension different perform almost sg respect epoch outperform epoch value two ccc q underlie low standard figure recover low gaussian tensor middle slice recovered epoch sample solve shape slice along measurement thus objective figure objective see significantly outperform smooth local nonsmooth low cc tensor sample bilinear eeg apply format
maintain player player likewise payoff query record differ slightly public find private maintain private use instead argue privacy method minimal record versus trade since sample known hardness result release algorithm bad universe theoretically prior work guarantee practical benefit handle solver query data solver several heuristic improvement turn solver still good beyond maintain database let normalize largely routine later tune evaluation ingredient mechanism select output proportional sensitivity differentially privacy score round
unit anchor lda sure sparse show kalman filter separability weak instance column analyze uniqueness mild practice broadly number view j generates firstly draw draw view independently cluster cluster moment column linearity fall solve general matrix completion trick column sampling method perform multi randomly splitting obviously feature information rest capture assumption linearity actually general mixture hmm broadly use series depict chain state generate multi hold generalization emission consistency linearity convert hide current triple state triple build emission hull immediately recover give anchor extend
compute tractable challenge average variational maximum ise variational energy follow fig ik fc angular average energy derivative give element constrain momentum maximization nmf pair method choose cross performance assess
name bs anchor north anchor west delta bs c ex ex top plot rigorous surrogate error bind surrogate plot frequency plot report standard ex ex choose component dependent produce eight produce roughly rigorous perfect furthermore surrogate finally moderate around well surrogate rigorous surrogate e frequency fact compare detail surrogate recall figure surrogate overall converge surrogate characterize large interval compute align font legend column legend align style histogram anchor legend name title iii anchor surrogate gp kernel ex ex correspond curve depict report infer report frequency deviation variance gp accurately case align infer slightly realistic kernel optimistic interval predict correct discuss extremely confidence interval due legend style font legend cell align center label width center title anchor west legend name title ii mode bs north ii relevance infer affect measure surrogate method surrogate important recommend surrogate fidelity surrogate reduce basis decrease apply report figure plot order result surrogate legend legend legend align font anchor legend name title anchor north surrogate ex ex ex gp htp legend font
simulate realization surface observational observation difficulty latent non discuss appealing option spatial observational acknowledgement author university energy office provide valuable discussion approach especially provide thank university technology li special valuable htbp respectively set row show run htbp hyperparameter hyperparameter computationally property km variation model covariate observational grid scientific statistical wind follow spatial efficient mcmc sampler exploit construct vary quantile mcmc engineering sciences university mail result rare intensity public type
hypergraph mode briefly summarize color issue hypergraph count finding efficiently example contrast np optimization np exist appear generic interpret see algorithm polynomial number step need practically result example possibility sample exclude possibility polynomial good mcmc color care need study gibb c step metropolis distribution hasting mh way improve general structure simple bipartite blue edge red couple step exist propose eq add display move small allow move switch move mix choice property mh naive lead poor mix typically truncation edge weight certain zero achieve mixing without require factorization example w thus set follow differ balance improvement mix markov compare mh induce
regularizer regularizer involve connection regularizer rademacher denote mkl learn regularizer classifier I dependent rademacher upper complexity enhance flexibility optimize adopt updating utilize dual
notation permutation transpose inverse multiplying give row row effect multiply row permutation adjacency represent operate way characterize operator transform diffusion kernel follow similarity similarity correspond adjacency associate two graph adjacency matrix trace fortunately solution discount summation summation adjacency think series identify relate kind auto series model literature unfortunately consideration computing similarity expressive determinant kernel applicable lead dynamical
ordinary multiplication tensor tu w kk tensor establish return draw emphasize triplet expectation triplet equation take marginal distribution depend estimator moment estimator try function moment distribution moment result equation importance multi view ii ki ki phenomenon class moment multiple however structure mixture fitting circumstance rank know come allow difficulty try tensor store tensor option significant technical plausible individual argument orthogonal tensor explicitly online mixture interpretation potential
projection preserve randomly project row multiply matrix column suffice factor bound give communication algorithm improve reduction unconstraine application stream picture generalize approximation left row project row span small rank give strong strong line reduce cluster leverage specific specifically project dimension sketch sublinear offer interested know linear overview project top preserve sketch reduce projection result stream singular decomposition leave right singular kk frobenius remainder know multiplying scale imply multiply frobenius repeatedly dimension f semidefinite ni write multiply left project column rank amongst orthogonal projection formally set centroid mean cluster rely algebraic prior dimensionality assign ik construction disjoint rank rank possible projection find approximate indicator constrain either sketch certainly depend side tight constant preserve sketch preserve f f projection preserve final constant
benchmark identify test identify input show strong production weak production structure finding input application involve size experiment investigate size evident production efficient affected production exhibit somewhat except increase production improves increase influence experiment size second production method short execution execution table take c variation importance input production essential robustness result contribution output vary three production plot production process production process contribution inefficient input input
result vocabulary result construct dataset lstm rnns become architecture interpretation define could modify backpropagation validation set vanish forward gradient descent divide epoch moderately constrain achieve comparable lstm comparison forget hidden neuron favor neuron big contextual versus impose recurrent character affect dataset validation test cache lstm influence contextual corpus
pde pde require evaluation posterior amount pde occur optimization appear cost implicit accurate accurate without surrogate method significant neighborhood method around maximum sampling therefore efficient situation balance cost result surrogate near maximizer posterior maximizer maximum accuracy base moderate order scheme add use efficient mode cost
nine community database gap us air network contain network contain isolate isolate actually component large connectivity ns vi email email bind affinity electrical represent generator link network well ix energy www node locate cn table basic network divide treat probe move probe link probe consist link
basis sequence try incorporate cycle incorporate synthesis cycle flexibility incorporation rate utilize resolution consecutive zero incorporation case incorporate incorporate cycle sequence read analytical incorporation nucleotide unnecessary specification name kind permutation throughout sequence denote nucleotide incorporation kind template nucleotide incorporate cycle next next discuss special nucleotide incorporation eq
stochastic pairwise query coordinate optimization comparison algorithm convergence guarantee comparison tend regard descent consist determined search step along search therefore work contribution present block coordinate descent base comparison query show numerical explain devote proof
traditional singular fail bregman generalize dimensional occur logistic paper sequential property multivariate profile investigate receive attention wireless sensor huge amount streaming bring challenge storage overcome component find matrix explain portion pca contain naturally reliable largely bregman
configuration result mistake without adversarial mistake interpolation experiment span parameter parameter behave explore region maxout perhaps surprising induce run subspace span difficult cc cc explore coarse resolution structure run experiment dropout involve exponentially tend minima neural path two minima barrier random seed generator
sensitivity possible guarantee mechanism probability private cover guarantee mechanism find except sensitivity private unfold oracle eq probability eq provide include coverage cover guarantee unlike guarantee explicit solution output implicit describe pair approach interpret turn size database grow like continue solve feasibility private rescaled give approximate get kind multiplicative receive update aa normalize dense multiplicative maintain constraint two role solution useful define find least give ij oracle loss dual find may database vector pair think guarantee lp feasibility leave randomize scalar sensitivity slight offline building influential introduce express differentially private program throughout assume private neighboring database look oracle oracle private composition multiplicative private release exponential suppose score private sensitive output range oracle sensitivity private solver program mechanism scalar sensitivity find mechanism quality dual oracle guarantee universe constraint query privacy differ distinguish constraint differ constraint differ feasibility neighboring privacy want vector neighboring database think decrease
regard field conduct yahoo aim increase price work mechanism predictive intervention outcome effect causal estimation long effect work empirical mechanism desirable support analytical payoff payoff online adopt assume switch alternate design viewpoint make mechanism population mechanism agent mechanism agent receive round distribution select rule agent report mechanism deviation true type interest randomize experiment entire
elimination iteratively nested model pose magnitude regularization vanish regularization would regularization would dominate index gradient index manually challenge end adaptively fix advance fix relative procedure empirically performance encoder layer propagate threshold encoder train decoder reflect attain fix unit unit value retrieval demand code regularization stochastically minibatch correspond frobenius exploit structure allow long ability capacity decay across unit datum example tree balance completely representation however marginal representation bit encode construct code conduct retrieval bit b b correspond
however descriptor h lr adopt test blind transfer aggregate exponential class direct feature use additional attribute correlation give encourage restaurant restaurant scoring record overall multi task iii dimensional customer restaurant cover score set equally descriptor bit interpretation thus ignore outperform h go unified task core semantic descriptor variety categorical domain enable sharing novel shot
product stagewise keep unless reason stagewise zero unless reason keep stagewise lasso row distribution path stagewise middle stagewise path become stagewise ignore trend actually stagewise reveal behavior shrinkage factor stagewise algorithm yield effectively pattern capture path suffice stagewise scalability pure stagewise solution stagewise interesting study shrinkage algorithm say shrinkage applie shrinkage frank wolfe really strategy interpretation shrinkage stagewise history stagewise component implicitly place direction completion confirm stagewise tune even pure stagewise stagewise regularization parameter recursively inductive setup bind sake additional condition overcome inherent stagewise e exact path differentiable regularizer lipschitz constant stagewise limit number take stagewise reach parameter define effective lagrange exhibit weak constant stagewise satisfie need possibly nontrivial discuss remark technical hard interpret duality expansion difference fx explain stationarity z associate constrain lagrange along weak intuitively kind satisfied stagewise path exact display empirically replace decay condition furthermore decay topic htb stagewise figure ensure stagewise incremental stagewise constraint stagewise efficiently underlie exhibit stagewise estimate stagewise offer apart ability rigorous characterization stagewise future work throughout mark understanding stagewise capability work attempt explain stagewise teacher thank grateful frank wolfe lastly review constructive compare stagewise frank wolfe problem begin frank wolfe iteratively loss successively small minimizer frank wolfe gap could stop iterate duality sufficiently face frank wolfe stagewise similar former iterate whereas iteration substantial informative setting run frank wolfe run frank wolfe warm newly guess frank wolfe mind may compare something wolfe frank careful stagewise step frank wolfe actually quite make comparison direct wolfe regularization parameter step frank wolfe initial frank wolfe stagewise linearization around frank wolfe stagewise constraint point gx feasible point value frank wolfe new estimate frank wolfe opposed continue iterate repeatedly minimize stagewise consider construct logic finding maximally align frank wolfe stagewise strategy
benchmark model robust perturb input bag part distribution boost great deep leverage noise model relate classic model formalize pseudo ensemble collection child parent process define ensemble relationship pseudo ensemble ensemble create sec develop regularizer input state dropout reproduce fully supervise extend supervise produce state real world dataset recursive pseudo ensemble generate ensemble process parameter latent empirically type
generally examine improve benefit hyper optimization high maximize sense none validation provide perspective improvement result could analyze determine filter misclassifie ensemble search algorithm result high validation accuracy entire add hyper hyper give amount constraint show perform hyper optimization default classification increase filter increase filter demonstrate benefit motivation datum notation training instance equally machine learn concern input set generally
create process uncertain associate classification denote label recent numerous method management however none address collective possibility generate collective classification use final label major disadvantage collective label collective information algorithm collective account network encode link sometimes link successively link optimum contribution collective provide formal introduce labeling incorporate uncertainty edge enable across accuracy evaluate technique serve practitioner collective mining uncertain collective uncertain collective time present experimental conclusion classification especially context propagation refer class improve tend collective al propagation perform collective email speech al exploit et integrate leverage label collective
bellman bottom visit end algorithm algorithm fig respect instead solve path maintain close obeys weighted rule dags dag instance dag substitute add contribution show enhance contribution decay constant provide
data asymptotic journal american let hessian hessian ball expand maximizer interior x conclude hold mb hessian interior recall exact taylor modes bx kp assumption approximate case hx gs gx sx st x flow apply
represent pixel image utility indicate strength want ise self equation determine strength ahead compute common apply selection field notice recursive formula present selection design bs ds choose
super strength training explore master testing dataset take result one kind another relatively test standard dataset super list top decomposition subscript test super subscript indicate approach usual sense super target multiply successfully successfully domain view testing successfully approach likewise standard provide super train dataset success cross actual composition label show interpret composition estimate value value column prediction vary value show super composition column give may calculate match cd percent percent top percent percent candidate testing calculate cross domain super decomposition dataset need search consider matrix row considerably full full search super estimate able search pt column super column performance super candidate expect column decomposition expect use testing seem decomposition cb ce percent percent candidate source cc calculate search apply super decomposition super train interpret approach extend interpret b performance standard observe column exactly cb cd evaluation percent top percent percent percent source calculate super dataset unsupervise training tuning dataset unlikely much impact training improve
additional work unit integration minimal write short probability short running discuss mixture bayesian analogue mean methodology inference way isotropic covariance wishart cluster center speed finite chinese restaurant merge proposal two modification new testing implementation keep sufficient efficiently entirely standard term unit thorough
acceleration region ct scan figure image reconstruction reconstruct demonstrate os behave exceed image see algorithm light
stochastically dominate simultaneously write q fact equivalent numerator rhs satisfied one generative fact marginal orthogonal axis invariant first claim eq variable induction random base product iid unit independent square product joint induction assume joint sufficient joint j equal v new axis add term j k k k jk projection onto since j norm vector draw random define orthonormal row equality follow chi square freedom chi tail bind chi variable freedom
section focus click accurately usually compute observe click predict actual user click position alone differ click similarly predict click query unable user click therefore follow question well predict click click evaluation model correctly even make mistake actual sequence actual click rank term top click word click sequence click accurately ad predict click click predict reverse click intuitively high query easy low understand strength report across access query label human look rank score time post click relevance measure vs recall consistently particular able relevant document auc able achieve htbp recall xlabel ylabel ylabel legend pos north recall roc roc pm
although rd superior performance layer convolution benefit convolution term grow find performance always grow become increase size train either ratio per class suffer overfitte poor result sample class comparison around cycle train convolution layer choose comparison significant overfitte poor generate video dataset million respectively identical storage diversity produce recognition experiment examine approach video recognition entire video video frame video frame extract video video extract video result video aggregated video kernel compare learn nearly kernel indicates describe learnable initialization train pattern natural video intermediate simultaneously domain parameter update conv layer restrict convolution reduce learnable avoid improve depth initialization policy conv fc conv yahoo fc conv fc conv yahoo fc conv fc yahoo fc fc conv fc fc fc conv fc conv yahoo conv fc
r step everything know beneficial follow show alg algorithm purely term computable begin perfect momentum simplify c note matching noise place benefit inaccurate estimation example choice size dominate governed information alg assume allow arbitrary fisher per iteration update dominate gaussian covariance g diagonal large note parameter setting inaccurate decay mh likewise proposal increasingly close bias explore error supplementary momentum gradient sgd momentum formalize rule alg become sgd momentum reduce momentum setting sgd momentum guide e sophisticated involve use momentum scheduling elaborate select supplementary material without naive variant replace gradient use
cascade ensemble cascade procedure cascade yield place private incorporate cascade leave team competition separate challenge assessment reveal
understand effective convexity perspective characterize establish oracle lead class extend evolve seminal showing rate agnostic square loss function losse effectively pass risk minimizer bad way precisely heart minimizer bad learn empirical good bound unique minimizer sort converse problem addition stochastically class perspective fail look perspective separate minimizer agnostic bernstein include complexity first minimization necessarily excess good work complementary previous work come form set yet
smoothness goodness previously set gps se ard implement unsupervised method pca follow ard implement sample figure consider sigmoid x negative log predictive per function sigmoid capture rmse se ard identity gp ard intensity learn learn feature sigmoid presence frequency focus spectrum space compact compose layer neuron per performance outperform result confidence figure implicit appropriately
chance mass normalization reasonable note initialization cost reduce multinomial routine metropolis step proposal simulate true proposal term initialization long use sample hide coefficient lda superiority real root root node node node node grid thick thick rectangle thick thick innovation also collapse bayes hyper document word representation hyper vocabulary illustration gray correspond occurrence highlight ij associate realize vector update occurrence read store gradient algorithm completion lda update require modify update huge room parallelization exploit efficient asynchronous case lda change first dependency weak element summation negligible exist yahoo server server retrieve recent update certainly decentralized asynchronous
reader schema alignment schema encode formal language describe piece reality e diagram schema source schema entity call schema element schema relationship specify connection schema schema schema instance relational coincide table schema process often equivalently design find often call formal matching matching give semantic matching tuple schema confidence measure range state strength relationship iv specification name mean student point attempt schema match introduce schema level source schema schema data source body activity schema allow type infer semantic look source amount relate largely attract many researcher graphical overview idea main approach document dynamic distance idea coincide call transform former latter document direct document path root document leaf node associate base approach suggest distance vector real scenario along schema filter similarity among element degree instance adopt amount therefore computational hybrid map basis let kind find approach find group extract instance could element address could element address match return example system capable handle complex see explore schema form account domain schema model describe vocabulary variety linguistic neighbor develop extensively design deal language carry entity symbol character digit detect token article filter root exist rely basis example basis lexical university group english schema match system background knowledge element approach use knowledge wikipedia attract researcher investigate management level wikipedia mean wikipedia sense wikipedia couple yet contain million entity million fact whereas fact link entity fact coverage page fact entity therefore purpose provide rich consist thousand concept non create concept entity fact significantly quantity fact add individual specify semantic new schema handle fashion without promise aim explore
explicit construction intuition construction relation sp experimental sp bp allow max I message col color none color sum give sp combination bp message equation care valid contribution sp message factor product message assume false false however valid message basically message message cluster control contribution estimate sp refer contribute either former case sp sp correspond sp boolean fix options sp col sp clear advantage sp cluster choose difference reflect comparative success col see usually fix sp uniform particular assignment remain happen phase efficiently assignment sp switch local soon sp update need combination take minus update consider incoming exact substantially
see robust robustness see increase significant efficiency value recommend near fair know practice censor aspect complete driven choice might censor solve hope research present power censor robust applicability theoretical censor asymptotic scope censor pc pc conjecture random censor apply science medical planning etc non estimator censor covariate respect presence examine study censor covariate far censor robust power regression failure analyze science survival analysis observe model censor mathematically I observation censor sample observation portion censor event aim derive see limit maximum function presence
mlp exponential smoothing change associate player divided two evaluate price historical price predict forecast propose al ann south analogy fair week select bt component less exclude mae price forecasting hybrid ann hybrid demonstrate ann forecasting ann svm fair display ann svm paper problem forecasting system machine preprocesse technology use bt rf
suitable minimax low fix
scalability approach see field offer uncertain however learn human cluster reliable compare pairwise query additionally sake phase least cluster learn reliable option explore initially acquire may rare encountered framework cluster contrast active allow modification certain ground truth experiment specific translate selection query pairwise negative consist similarity similarity manner set constraint constraint indicate semantic certain set initialize select change indicate assume propose detail constraint empty raw reduction pairwise query choose assign create aside ability collect maximally one
consist update major analogously reference informally sum loop mod view sgd update less leave maximum begin whose randomness communication since characterize descent converge exponential mild bind variable define assumption load machine computation decrease invertible optimum bound var capture randomness update argue var monotonically decrease sgd close optima see variance much tuning term bound induce main weak sgd update step need directly rely threshold ever mail certain mail later day regularity reliable become ml drawback size carefully either knowledge sophisticated may
short walk code reduce size stochastic gradient descent line derivative propagation sgd begin decrease vertex far vs want say quickly reduce future walk vertex word long therefore affect nature asynchronous sparse scale show run may interest approach variant walk pass directly code instead learn worth application softmax tree leave still code decrease graph sequence page website stream walk graph sample also variant sentence view walk design language like streaming evolve
determine uniform dropout eventually optimize dropout dropout difference visible get close informative dropout regularization effect hyperparameter input unit require vast large likelihood mask always choose contain proportional discrepancy error map estimation use prior posterior geometry behavior singular increase necessary number explain big
neural baseline work project dimension net rbf svm std try rbf find critical differ order magnitude mmd
x x I I follow correctness dominate much way retain count generate use training instance random frequency neighbor nn weight rejection thus act choose balance variability nb localization separately emphasize preferred retain imbalance choose real number close search nn instance eventually classifier full multinomial model
gaussian express hessian identity derivative expression integral two back discussion rule back express main rule integral part line
accurate small component frequently expect regularize sparse quadratic computation previous iii occur solve optimization lipschitz term domain problem generally continuous penalty extend difficult optimize show algorithm also complexity nesterov require respect advance lipschitz regularization constant compute appear dominate change feasibility rescale independent care tackle arise always inside lipschitz moreover estimate dominate iii issue grow sample norm trace nuclear smooth
pass segment total validate list replicate replicate result successful replicate consistency replicate methodology region include marginal segment derive calculate posterior show sample accurate posterior way scale simulation method pass benchmark perform segment quantification focus change mean baseline specifies normal place segment bottleneck likelihood parameter parameter intensive
identical unified template follow package support importance constant carried easily rand rand definition co co kl knn co member member name
newton step majority computation spend al newton parallel library matrix operation gpu matlab gpu matlab worth interpretation datum elastic support vector feature select net equivalent recover special svm numerical treat call hard line algorithm p minus running spend result input advantageous implementation mode mode run dual well assume vector elastic feature keep
vector component acknowledge training image prefer use pixel sample unbalanced training pixel region eventually serious mapping handle base collect image graph homogeneous patch call region frequency detail color return accuracy image cost local close centroid within manually intensive adjust attribute region adjust representative training image reduce make strong necessary reasonable coverage select representative learn codebook descriptor training descriptor close codebook quantization bag descriptor furth histogram codebook histogram bins histogram initial collection accumulate histogram summation representative entropy evenly image vice versa encourage coverage feature essentially combinatorial seek add maximize cross entropy expand subset illustrate initialize propose suit highly nonlinear spatially rely content local sense operation could complex previous g design address challenge help contextual strong capability hundred report conduct technique learn apply material show visually numerically mit evaluate far conduct study positive throughout fix hide color confirm color transform reproduce color
equality follow conditional px I fourth equality rule definition n x eq third equality n follow formula mean eq exploration localization tn let obtain observe x ct tn tn access gp strength localization determine signal widely incorporate besides signal camera also bayes omit contribution gp filter resolve work claim obtain measurement give every call label datum pose estimating map offline obtain optimization localization bf filter highlight give trajectory divide slice slice slice calculate update mean
di contact test correction summary contact list contact remove gap reweighte count file estimate compute di symmetric entry score average exclude adjusted pair interaction predict rr protein rr pair rely efficiency introduce definition base existence unify concatenation interact protein encode protein estimate whereas family domain co decide interact horizontal concatenation ratio depend sequence come normalization intuitively score coherent assume rr sequence odd ratio conditional probability cf people action european fp grant agreement cb acknowledge european grant direct model quantify interact interact scene describe denote interact repeat analysis position correspond main interact p x
dnn formulation dnn character dependency dnn dnn output distribution enable temporal use weight hide distinction activation since depend activation work find nonlinearity select activation prevent set maximum act dnn like use backpropagation always gradient subgradient recurrent connection reflect perhaps powerful sequence maintain state
market circumstance able expect increase impact family see family tie make scoring score concerned scoring rule expectation infinite characterize also special characterization elegant rule score possibility seminal market sequentially score scoring rule space market introduce generalization continuous outcome market extend infinite outcome market well practice sound price agent neutral imply move belief extreme market information agent however fundamental mathematical finance portfolio notion convex measure indeed axiom draw finance market single financial market connection machine consist agent potential absolutely lebesgue outcome measure discrete throughout let denote interested variable device expectation score statistic scoring affine arbitrary value term implicitly affine transformation version strict avoid score density rather must large summary note place agent reporting expectation latter complete recover rule statistic exactly probability function point scoring case outcome generalize family generalize log scoring
k k combinatorial interpret matter extreme indicator complexity shape shape determine number space projection view require addition compatible multiplication minimize identity construction note enjoy definite keep randomly input k u input large choice application discuss solve exactly solver contain projection report across trial representative c angle norm minor major high minor c minor major major minor contain input give residual normalize angle mx solution normalize major minor path major
bound see mean p representation every functional exist df existence lemma f h ps h h l j j l j jx dx j allow integrable respect w equality dy jx dy dy e jx b third hermitian dx dy elsewhere last reproduce f expression proposition derive b w dy compute derivative simple computation notice h h obviously constant rest second hence integrate p equality operation fourth rw correspond symmetry partial derivative next summation product simply apply derivative combine theorem conjecture
intra inter vertex unify discriminative hypergraph partitioning formulate cluster hypergraph hypergraph hypergraph eq similarity perform solve optimization newton solution propose consider separability intra aim intra inter formulate denote vertex intra separable cluster partitioning maximize vertex membership concatenation membership p p xx conclusion diagonal form argument vector element matrix optimization sum sum ratio n reformulate trace optimization utilize newton three large well repeat step choose eigenvector singular solving obtain follow value hypergraph partitioning satisfy requirement may
though remain available moderately must meet since must bound word continuous continuous q old old cx consequence concrete cm cm continuity h also continuity metric continuous measurable compact guarantee schmidt k compact operator separability separability general section consider hilbert section due boundedness satisfied boundedness somewhat condition special output simplify boundedness eq map l kernel older suitable evaluate I k k nonlinear see natural exponential cauchy kernel continuous canonical see continuity continuity h present consistency embed theorem result concrete rate cope turn idea demand section
bc vs bc bc conduct cc nmf nmf vs relate domain em give vs em conduct performance recommendation user item combine different share propose novel filter probabilistic account knowledge across domain hand rating discriminative space rating indeed rate cross recommendation extension
gradient derivative iterative number condition example convergence matrix thought bring close would require solve cluster empty subset partition overlap cluster ie dataset cluster extensively field bayesian random partition chinese combinatorial define density description cumulative density moment generate program way alternatively program
collection grouping need proceeding shall set db rd r c k converge exponent suitably almost summation suitably adjust calculation exponent reverse order summation db value discrepancy original need sum term occur induce extra discrepancy geometrically complete universe number
nmf kullback leibler parametrize translate noise gaussian offer fit heavily coefficient update decrease update employ nmf solution proper update next heuristic scheme prove update turn e previous nonnegative preserve result per iteration involve nmf divergence purpose extend current step bind produce descent algorithm indeed I auxiliary rely concave decompose dx dx yy give dx c inequality lk kp lp kp lp use convexity
compute point point lemma e addition k therefore find appendix consequently subsection template c either terminate update update either rate discuss later alternate advantage nonempty norm dual iteration smooth analytical algorithm primal bregman smooth use use analytical achieve choice feasibility gap lead due know method describe unfortunately avoid expense section specify solving lipschitz lipschitz accordingly similarly f strongly rule bad analytical primal depend prox long absolute residual aim algorithm state variant limited technical assumption objective eigenvalue positive bregman cd pd pd c pg define satisfie pg td pd solution attain require fulfil strongly follow strongly concave result scheme corollary generate show certain problem priori augment lagrangian smoothing subproblem reasonable addition convex nesterov accelerate satisfy imply addition inexact inexact proximal operator start compute inexact whose analogously scheme omit initial bad sense primal feasibility sufficiently practically relatively reduce burden explicit gradient admm fast idea smoothing technique function augment lead nesterov smoothed dual either augment bregman
derive throughout focus sampling seem sample attribute natural p obtain technique pick mix strong direction zero coordinate divergence matrix eq budget iw analyze begin follow assume equal expectation eq let law eq divide observe eq side proof bandit dependence since low detailed idea sp loss
version consider nan enyi alternative increase difficulty resample among high enyi subgraph difficult choice plant dense make edge independently plant dense subgraph entire alternative arbitrary dense model first plant distribute hardness detection financial edge nan hypergraph big notation mean trial tv divergence base ask small detect plant dense subgraph question investigate statistical sharp regime treat regime sparse regime separately focus interested absolute asymptotic treat regime unified manner limit deterministic hypothesis result ok subgraph reliably statistic scan correspond whole subgraph counterpart minimax test test test q implication parametrization
update environment explain trajectory relationship relationship work plan planning learn tv human tv crucial planning show heat human center tv might move planning activity different function instance activity cost activity human object proximity around vary distance object angle certain angular activity spread wide fig human angular preference activity center angular preference parameter define human activity fig human project length prefer robot vision pass add center parameterized variance product von robot careful pass behind move back preference vary human cost capture activity along human user prefer robot cross whereas activity preference symmetric normalize human normalize relation object mostly commonly planning reason human human book display read bar
density pairwise intersect dp prescribe experimentally selection mixture lx n description length etc retrieve deviation complete dynamic seminal outline bellman paper contiguous element incorporate cluster size solve mean median center report tailor application contiguous bregman mean
grid plug back scale symmetric reason normality assumption plot furthermore individual contribution different calculation suggest play minor part compare violate implicitly phenomena world short tend model exponential homogeneity add flexible heterogeneity available otherwise
finite issue address ica subject degree restrictive permutation ica second rescale infer source assumption trick simultaneous goal scale covariance simplify remove prevent give challenge ica solution highlight benefit exist ica primary advantage fairly highlight ica identify basis maximize ica make apparent quite salient variance fail direction world lack orthogonal merely statement orthogonality factor recovered source uniquely source middle lie identify underlie source appropriately statement justify statement lead ica source permutation recover flip recover rescale recover dimensional ultimately heavily whiten begin examine assume factorial concrete arise invertible factorial ica optimally factorial independent distribution ica principal
manifold tangent fully alternative perform demonstrate easier able separate far good stack accurate previously generative among good mnist generative structure datum firstly demonstrate separation latent mnist latent nearby space correspond writing write right approach pass network generative class figure far house vary way feed significantly inefficient able perform
accomplish atomic similar require sparse overlap order ensure feature description report speedup parallelism cd svm cm cm avg cc couple asynchronous version basic problem consideration high interesting direction function obtain partly nsf comment expectation kronecker ready divide side statement I e l objective take expectation relationship substitute simplify theorem bind follow equation q take side give inequality fx step satisfy substitute require reader exposition analyze unconstrained similar manner recall iterate l fx fx x derivation schwarz inequality lipschitz continuous variable step inequality
behavior color estimate extract frame represent yield choose test average random depict accuracy baseline bit pl kernel reach hash outperform reach overall high tb b kernel pl body formulate atom depend compute dictionary atom well dictionary label comprise body static dataset video environment region represent video order yield point size b outperform baseline static maximum static art manifold tb accuracy sparse code introduce positive
compose unstable precisely map condition unstable stability absolute summarize p randomly hand symmetric dynamical equation write actually map essentially performance method sequential averaging momentum list asynchronous supplement scheme master unclear pseudo supplement center follow master center master move set dataset cifar imagenet convolutional deferred supplement gpu node gpu local gpu processor variable master store update denote color channel fully operator max denote operator softmax linearity inner experiment neural p bias initialize master break also mini
normal x position obtain adjacency embedding position identically multivariate position propose wherein incorporate adjacent vertex tendency present position instead dot gibbs position adjacency employ mixture corollary quantify suggest role bayes latent indicator identifiability constraint blockmodel eq embed gmm within present investigate utility blockmodel hierarchy name primary benchmark vector assume know gold prior position theoretical limit covariance distributional convention adjacency finally absence adjacency embed theory prior give rise spectral might increasingly see section corollary
give cdf length vary q cdf eq imply sphere fig moment formula htb ccc htb ccc sphere length mean contrary moment stand moment property
reduce ordinary singular I k symmetric problem coincide sign precisely identically note define symmetric form nk normalization q finally vector recover flip suggest constraint loss establish problem matrix coincide large turn result follow background exist eq evaluate large asymptotic digit indicate large tensor bind incur maximum likelihood proof real likelihood prove lemma expect sharp suitably constant appendix provide background random study physics spin particular rigorous replica spin theory confirm rigorously prediction maxima
application section panel first motivate time additive case intercept consumption price consumption individual min slope z pr intercept l type unobserved error visualization affect remain correspond extraction factor yield right htbp estimate panel individual existence check interpret effect testing describe less within estimator inconsistent least ph p consistent residual variance choose suppose nan hypothesis exclude true factor favorable violate iterate inconsistent recommend user follow argument specify argument test statistic message dim consumption dim level assumption fulfil unobserve true probably assume inconsistent alternative nan discuss way check whether classical sufficient reasonable decision additive section
analogously uniformly sphere thus interested variance projection operator must case assume careful detailed appendix say complement compressive tight qualitatively project subspace use per demonstrate figure predict quickly average formalize simulation plot thing simulation demonstrate
perfect promise possibility success practical audio speech recognition audio clean sound input situation
manuscript devise dynamic setting decision point observational collect decision disease extract medical record datum national survey assess health status children united states diabetes third wave control pressure manuscript simulate patient diabetes one challenge avoid result backward induction experimental observational start decision find option optimize decision point history regime nest key notion outcome regime outcome propose tool use regime time introduce subject censoring create
imply multivariate finding respective provide view main feasible solution match kkt become replace weighted benefit extra even vector condition design provide benefit scale restricted isometry property sparse condition restrict condition weak prediction cone group group cone restrict eigenvalue compatibility also introduce notion cone sign cone cone define sign follow cauchy eq eigenvalue imply scale
explicit mark x td txt thick color solid td txt thick mark black option plot xlabel ylabel ndcg pos north east color blue mark thick cd mark td txt thick color option table header td thick mark color minor title yahoo learning ylabel ndcg legend pos east color thick error cd mark color mark option header plot txt thick mark color black option solid minor title yahoo xlabel ylabel ndcg legend pos south east mark plot txt thick mark mark option solid index header txt black option solid header minor title xlabel ylabel legend pos bar explicit color red mark plot thick mark solid header true plot scale xlabel ylabel ndcg pos east color blue mark thick table txt mark mark solid header true header title xlabel ylabel ndcg legend pos south mark thick bar cd error x error txt mark color option solid index mark color mark header minor
list regularizer huber nonnegative unit regularizers advantage parallelism yet initial datum assign process column compute portion objective complete begin criterion add automatically function detect type appropriate implement method loss ignore frame fit function fit aspect day include value measurement miss encode pick encode use feature embed separate axis embed despite automatically course embedding regularizer embed embed set two code line across divided core restriction fit leave server possible hardware local many core avoid core process partition input copy column gradient subgradient signature implement square entry similarly implement proximal operation experiment several recommender netflix matrix dimension per row sampling location finally location uniform amazon ec master machine cpu core gb ram run matrix regularize capability depend merely illustrate c ambient table iteration useful iteration author grateful de sa art david price taylor van comment insight create support foundation fellowship stanford fellowship fellowship extend pca arbitrary categorical ordinal factorization pca mean maximum matrix heterogeneous miss simultaneously interpretation parallel generalize implementation send stanford mine large collection datum row represent column sometimes type record patient survey yes sometimes test question finance know asset class science record record customer difficult understand complex visualize example correlate feature identify anomalous enable low dimensional plot anomalous value well embed principal analysis pca find minimize sense extension handle miss extend pca square appropriate extension beyond pca add regularization factor impose encourage structure factor refer problem dimensional consist factor datum still familiar optimization formulation completion pca svd margin problem relaxation tight globally however involve
growth distribute challenging topic real world machine read order leverage hardware significant communication worker read popular version stochastic sdca propose distribute allow trade adapt diverse low core framework support objective model leverage avoid simultaneously employ step master perform data method round scheme naive communication theoretical reduction come moderate increase order sdca ascent optimizer smooth geometric platform propose variety show magnitude gain mini
hold mle belong function great mle study condition surface unimodal unique mode concave let symmetric concave real symmetric problem study impose characterize course seem asymptotically study concave uniquely characterize cdf stay integral empirical logarithm change slope modify cdf mass word nx nx n nf z dark circle knot concave mle mle nn main establish mle carefully density concave boundedness weak class compact start state reader b dt prove ft real sequence converge concave let class log concave class measure fix log hx mle mix g yield behind introduce avoid issue fact previous zero stay bound generalization result mle combine unimodal claim unimodal arbitrarily support approximate compact dropping towards quickly point convolution density center deviation choice problem exclude log concave log convolution importantly condition unable integral appropriate consistency proof two concave goal definition refine respectively location shift ensure stay uniform
regardless time additionally regular mean covariance matrix definite toeplitz solve fast current slice smooth interpolation ft tb pdf pdf pdf pdf finding laboratory exploring drive purely domain clinical stream prior intensity distinction intensity smoothing use discretized intensity rejection share intensity loss piecewise
problem work space call community problem issue flat sum need introduce basic construct volume probability partition disjoint subspace point uniform partition bin product estimate iterative generalized simulate markov bin reveal landscape mapping bin variant record belong probability among estimate suppose mcmc decrease schedule adaptively proposal move intuitively landscape improve method simulate annealing process sample visit bin equal step project identify local especially flat merge minima identify descent two minima barrier straight line minima energy dotted level descent
diagram reflect second diagram sample see experiment dominate variation persistence sufficiently instance near persistence thus allow distinguish class b differ persistence vision persistent homology valuable purpose outperform art problem would rather topological advantage shape benchmark consist shape synthetic human child pose human pose resolution classification synthetic real texture benchmark pixel benchmark predefine training image training shape retrieval ten choice piecewise mesh discuss compute persistence diagram texture classification descriptor report rotation operator
finally norm define norm shorthand notation pp ph turn start discussion problem square absolute favor model often take address overfitte measure take square absolute shrinkage employ sparse nonzero alternatively examine attempt language uncertainty set g via procedure procedure key rise interpretability n denote uncertainty norm fix regularization coincide extend setting begin identically triangle final follow definition dual homogeneity triangle completes result corollary likewise recover know equivalent follow conjugate observe square arise uncertain uncertainty induce
unified power within hierarchy vary nonlinearity generalize version complexity grow begin overfitte fluctuation aside decrease stop velocity connect vary consist release som infer dynamical supervision adaptive qualitatively produce elliptical monotonic trajectory color line show dash leave single variable use single recovered som biological hierarchy infer crucially distance exactly within system elliptical trajectory zero radius finite completely specify dynamic object som display object dynamic dynamical motion statistical fit trajectory would different understand automate adaptive inference sir software find perfectly biological phenomenon may interact able useful site yet site arrange affect site produce imagine setup evolution total site treat site due site measure corrupt scale decrease datum simple amount although expect likelihood dark
ex false em mu size th mu mu end align end align environment define style define array construct allow false tag true receive fusion fc via limited capacity cloud access communication receive
n stand determinant recursion condition natural initial satisfy rewrite nj ed arrive eq approximation argument finally stability derive state mse representation stack rewrite respectively kronecker stability
exponential hyper experiment schedule decay depth depth potentially small key aspect penalize learn decay get gets test standard depth deep rate important although study beyond gradient descent sgd minibatch gradient varied learning result roughly display value distinguish lower left except error combination nonlinearity result agreement affect accord walk initialization
solve multimodal get seem optimum issue former contain hard global one depend case cb seek quantify region common cavity powerful considerable supervise forward explain improvement datum show effect learn benefit play important expect annotation account training guide retrieval ec rely meaningful ignore ec also hierarchical ec number ranking support summarize rank obtain high indicate color higher unsupervised show correspond correspondence distinguish population within correspond similarity illustrate discrimination similar query overlapping supervise unlike match ec reason supervise value similarity relationship derive indirect cavity similarity division indirect show correspondence infer protein protein ranking supervise rank indirect expect indirect detecting entry noisy one indicate performance degree influence rank clear distinction roc
compute tv distinguish follow eq deduce imply distinguish case start two interest case imply mass apply schwarz norm assign bound translate case recall q take substitute distribution equation translate show plug claim square distance make constant choosing employ square distance distinguish show sampling sample distribute instead variance induce among number concentration sample denote symbol independently define
bias bias expense mse although use stop qualitatively case adjustment already adjust sample accurate adjustment ba produce size estimator part bootstrap increase bias adjustment bootstrap coverage close inferential estimator long integrate bootstrap filter preliminary parametric parameter simulation bias estimator via encourage suggest correction yield long encounter practice estimator result subtract triangular e less integral yield consider hz fact operation substitute sequentially recurrence h x neighbourhood origin small n extracting term ol regressor regression discussion nr j array weakly dependent e conclude ok ba analytically adjust
argument consequently omit proof extension easy leave note na follow inequality valid event large case let cart clearly rewrite argument thus collect argument property fix moreover k k supremum side grid k q finally hereafter tuple one level leave define fix inequality rest inequality enough confusion cut countable cut clearly result prove finite subset infimum modulus observe plug conclude prove fix exist exist satisfy p k fix conclude cut
projection node reduction project hyperplane span centrality well high component highly gender project central amount project variance allow decay classification accuracy heuristic extracting compare association provide association generation face fit avoid explicitly compute matrix association unweighte direct graph domain edge union set label subgraph note intersection edge ignore reason accuracy cost seem topological vector describe fast currently know carlo incremental reach get limit
ensemble detection unlike classifier propose detector preprocesse competition rank database publicly auc absence ensemble process dr serious disease diabetes common early prevent affected dr mass diabetes highly desire manual resource demand effort establish reliable computer system color promise automatic screening close clinical recognize fold first sign dr precise highly detection detector remarkable ensure reliability detector prove efficient aim value provide coordinate
collection positive location write density xy multivariate notation follow zero gaussian marginal recover process margin estimate bound rather mean problematic depend distribution asymmetric variance provide order input entry distribution cdf median median copula height x bottom axis line table toy txt mark densely mark black toy txt index toy dataset txt
principled research showing indicate positive element underlie link choose optimization problem relationship predict edge e real world co network ca test done relax solve link method svd train svd svd approximation omit link candidate computing false snapshot show accurately achieve large relax general overall link completion highly descent whereas eigenvector second semi supervise cluster inductive completion
slot execute eqn indicate action represent current learning know generality cr queue cr queue instant form represent total possibility value policy execute queue cr availability threshold capacity belong fig primary slot decrease queue hence slot figure demonstrate secondary queue secondary increase secondary service furthermore certain service share cr decide sense action
jensen mass false I exact false loose imply care worst bad absolute perfect suffice quickly like substantially convenience expectation become convenience order q
dash mark style fill red mark col sep comma fix utility col sep comma size xlabel utility legend style draw legend pos south grid style col comma expert thick black square comma utility thick mark table col comma utility ib xlabel ylabel none legend south major axis height sep comma expert smooth thick mark style red comma smooth thick mark sep comma size utility ib classifier measure reward contain discount property assess single return discount correspond traditional discount medical diagnosis category discount visit discount visit substantial risk apply discount accuracy score instance accord discount discount utility consist discount
fitting hyper penalization explore considerably markov mcmc lack investigation fitting prior hamiltonian monte short test critical scale real microarray confirm finding add tailed mcmc fully throughput genomic identify level disease classification diabetes response hereafter verify biological diagnosis disease collect relevant gene disease challenge analogy commonly univariate relationship ignore redundant gene include meanwhile weakly attempt take rather capture greater often maximize prior use tail propose prior hyper prior regression differently normal laplace laplace scad bayesian gamma generalize pareto unify review addition broad class commonly high bayesian popular mixture see express belief enough feature irrelevant totally sensitivity theoretically empirically jeffreys nan point markov mcmc lasso still lack probably difficulty likelihood hyper multimodal coefficient ordinary heavy tail cauchy
solve book numerical area medical imaging arise algebraic specific would mention sparse also topic network mostly spectral huge matrix algebraic seek complete scenario tractable size desirable scenario region resolution scenario recommender recommendation item recommendation theoretical practical kind optimally run many assumption work intrinsic ingredient dependencie circuit circuit kernel circuit restrict small circuit least costly ingredient inversion circuit circuit size circuit locality trading cost circuit capture algebraic
mixture analyze cluster laplacian operator compact weight belong interior probability simplex specify via weight mixture drawing formalize recover latent observe unlabeled course generally recovery become difficult overlap increase easy formalize overlap follow background normalize embed symmetric say kernel kx semidefinite throughout semidefinite function cluster relatively decay apart normalize embed associate laplacian rescaling kernel since symmetric construction large eigenvalue eigenvector typical cluster vector apply cluster embed reveal formalize normalize function conjunction suitable orthonormal result component section
operator generalize mild regularity regularity random joint function px lemma review first function regularity basically r integration ready vector respectively density parametric mild formal regularity stein parametric identity mainly differ elaborate see form closely know covariance parametric density identity function relation e since px score property stein property higher establish notation section formula product notation thus score function reason score enable stein order derivative yield differential statement mild regularity function operator variable formal description regularity polynomial satisfy orthogonality property polynomial know mostly involve thus order coincide polynomial instance convenient polynomial orthogonal w interpretation need provide order similar previous construct score exploit parametric stein identity parametric respectively parametric formal result regularity apply
approximate graph limited edge question spectral cluster assume distinct work obtain beyond beyond graph approximation approximation consider one important cut cut original graph machine etc approximation wish laplacian algorithms laplacian connection spectral analysis need cut focus behavior nd eigenvector laplacian crucially generic framework sampling costly full eigen experimentally appear range theoretical somewhat access g course edge sparse technique guarantee consider weighted graph graph
outperform td alpha bind iterative linear singular regularization notable recently temporal td keep varied estimate td error regard adaptive step alpha heuristic td error sign variation
notion utility diversity modular submodular differ entity maximal relevance order fs recommend item implicit penalize instance yu measure multiple objective consider tradeoff relevance diversity axiom distance implicit metric capture movie instance fs pe satisfy diverse propose objective recommendation offline document find case belong category diversity address et exploit user preference study diversity grow recommendation argue solely author similarity intra compute pairwise similarity item high diversity topic balance diversity zhang et problem find address maximization list recommend maintain item hybrid target weight require tradeoff consider idea marginal submodular approximation paper prior recommendation optimal target concern maximize diversity elaborate behind greedy online evaluation efficient contribute diversity motivating formal description user movie ccccc utility name x movie depend choose recommend movie match satisfied utility
analogously expensive smc use intermediate carlo stochastic approximation em replace simulated smoothing intermediate next approximation enjoy maximizer reasonably weak simulate joint smoothing structure rao start rao model smooth rao find apply mcmc base smooth gain jump separate infer jump compute
highly get additional change contrast x pure underlying solution implicit augment x square imagine panel ellipsoid
q measurement allow nonzero convenience bregman continuously differentiable assume inclusion subgradient norm addition piece compute linearize bregman govern exponentially bregman linearize bregman iteration image sense change variable iteration iteration augment linearize euler simplify wise motivated inclusion path always reach see convenience replace aside obvious piece wise though obey subdifferential lasso consider identity linearize let least solution oracle following evaluate hold estimator selection consistency also path refer existence select lasso provide consistency reach nonetheless bias
convex let compact define simplicity maximum possible optimality justify non estimator outperform r hausdorff hand considerably see enough necessary boundary ball interior meet intuitive sort limit see convexity et al compact convex reciprocal restriction converge desirable case account see prove proposition prove sufficient convexity convexity account pt pt general h compact relationship different geometric ready
significance p serious year du exact hypothesis possibly great tr et approximate normality sp normality sufficiently poorly demonstrate decrease test close comparative test central possess property propose
converge suffice p w go use inside supremum identically sm note possible c inside supremum converge far diagonal use bind sm sm sm sm e conditionally variance x cx x cx x sm find converge converge sufficiently sm x sm n x n sm c w sm cdf far obviously converge zero zero turn sequence inequality hand eq use lem k combine proposition corollary condition hence independent distribute distribute freedom convention possibly r r prop coverage corollary bind respectively instead use x satisfie maintain section far clearly denote orthogonal complement column triangle probability display p probability far display depend form square degree infinity finite turning term let denote matrix p converge definite converge x supremum precede prop lemma prop prop prop prop remark prop consider interval quantity minimal
search everything optimal r plug direction quasi bfgs update solve let maximal decrease overall repeat choose improve mc focus mc strategy fitting entire surface although mc penalty place certain subspace simple form produce notable scaling set expand coordinate standard descent step step converge simultaneously mathematically actually optimal appendix attractive non function appendix selective correspond put limit restrict search implement
affect macro optimal threshold theorem describe uninformative maximize base widely recognize particular prediction maximize expectation example prediction depend apply quantify optimal threshold dependence make difficult relate thresholded show perfect optimally thresholded calibrate get always macro argue actually weight rare label study consider article mesh control rate rare lose cause macro rare application desirable binary output label dimension probability column dc false negative negative
chapter ac uk sophisticated parallelization gpu acceleration parallelization naturally modular gpu million demonstrate dataset source integrate gps parametric reduction formulation responsible limitation output collect plus th
shape rotation fig contour contour plot large eigenvalue sign behavior kl change condition sign develop algorithm theory ml merge expectation divergence decide mixture merge elliptical gamma ml solution apply follow change helpful turn eq gamma em maximization step current weight k l l b ga step update maximize p I explain two modify case likelihood one sequentially step step maximize log
x filter x x x architecture use four layer eight feature map two model size allow translation invariance conv conv conv full x x pool also try three decrease factor validation stop momentum mini batch importantly turn layer normalization gpu training pixel preserve aspect computed randomly sample leave right global sample corner right visualize
cycle duration west mm fig south mm percentage west south mm transition west mm occurrence universit du financial existence learn learn perform unsupervised induce knowledge development dynamical theory focus broadly traditionally imply assessment post process long term continuous assessment behavioral
bound proof approximate coherence kernel since coherent side therefore complete strictly form root polynomial root diversity low rkh yield previously embed borel measure topological detail dp algorithmic one expression radial function radial function radial base kernel feature radial decrease namely decrease low distance diversity atom proof decrease building corollary bound entropy shannon generalize r measure measure generalize enyi entropy low window shannon entropy enyi order overcomplete provided examine linear dictionary examine show dictionary low
give intuition sequential allow follow since event observe achieve capture restriction first time final opposite versus intersection occurrence know presence access rough effect spin spin neighbor update time determine interval compare time start configuration output remainder towards runtime gives state runtime homogeneous let event none aside
low limit n discount figure bind number require amount grow require iteration simple impractical discount factor behaviour complex single mdp stepsize rule study special program extension consider value iteration derive formula l l n require quantity optimal minimum squared estimate constraint unconstraine objective produce satisfie manner e e observe e e minimize objective equal yield assume stepsize expectation decomposition write term expression complete positivity numerator complete numerator covariance stepsize explicitly account observation furthermore closed bias balance close showing behave correctly case collect simply add discount stepsize rule stepsize stepsize long easily induction reduce stochastic approximation algorithm convergent establish low proof requirement observation recent weak condition convergence paper condition common
x skew balance get x stationarity determine element eq want minimal equilibrium rate lead adjust skew ultimately db violate converge cover scale diffusion fig improve stationary apply transition make counter transition bias continue jump replica visit walk introduce site e copy auxiliary gb maintain markov chain site toward walk enter likely edge walk direction towards continue circle likely
train student thing teacher ambiguity mistake teacher make thought teacher help classify digit compare field create sphere gradient descent minibatch mnist difference sgd sgd stop sgd gd reach bottom gd mnist sgd noisy nature capable minima high gd
dependence keep notation cause neither confusion characteristic sampling lot accept accept stage overall operate characteristic stage operate thus control overall operating determine error equality want overall acceptance sampling give stage appropriately treat impose remain guarantee overall one impose obtain valid acceptance plan select give put hold wise risk procedure stage second put risk symmetric example yield acceptance plan measurement decision procedure recall additional usually different location identically distribute time instant inspection degradation effect relate determine degradation know work degradation yet enough knowledge degradation module justify degradation act power degradation practical
use plug generate subsample prediction simulation slightly limit sufficient necessary simulation find ensemble large approximately begin increasingly figure bootstrap limit estimate bootstrap build ensemble replacement estimate distribution normal move interval quantile form confidence limit estimate calculate prediction generate tree assess coverage true ensemble mean ensemble take ensemble close mean contain interval horizontal probability limit estimate high likely underlie external repeat internal remarkably external estimation ensemble build testing hypothesis assess feature training depend value additional feature sample uniformly interval independently look distribution point interested tree build notation run simulation consist estimate covariance estimate ensemble result figure hypercube predict feature total histogram statistic
eigenvalue zero diagonal quadratic since always small aspect stability regularization parameter network independent topology range go indeed one clearly step regularization topology low upper example become hand increase connect improve increase necessary distribute conclude step range small stability agent consider verify know diffusion stable indeed verify conclude example noise node consensus observe fully stable strategy al steady issue network leave corner al shorter increase fast vs good color auxiliary matrix select network approximation definite sufficiently simplify see assume follow assume positive definite connect surprising improve strategy act independently agent
correspond goal learn involve moment representation exact moment available carry throughout otherwise third depend score exist go support mixture input instead extend follow cross glm stein identity form moment separately message contain glm moment glm mixture glm section set appropriate continuous identity state satisfy mild regularity condition glm bias weight ix moment suffice let look moment expectation I iw subspace span however moreover bias vanish mirror trick span recover appropriately obtain cp
arm always arm fair mean value useful strategy least concentrate metric discount formal time skip relate safe exploration aspect sometimes lead outcome take account consequence htb parametric term discount return b average step forward per run concentration small mixture little arm multi bandit uncertain grow safe become slow prevent kind generalization policy skip environment justify explore cost translate generalization despite cost avoid ts suffer discount return intuition past value thus discover inform time success incorrectly pick positive arm exploration similar rely sampling yet challenge agent uninformative hyper infer data hyperparameter agent conservative explore robustness increase
bind bernstein v px iii expectation integral second combine inequality nc assertion rejection satisfy one instead define bind except obtain chi square freedom tail assertion small bound almost manner argument assertion apply old inequality recursively assertion assertion paper occur many collaborative filter spatio temporal convergence rate accuracy optimal rate cp tucker rank model high relation
probable viewpoint fc orientation train imagenet annotation ap r r c car train avg I aim angle instance far circle c c car leave right jointly classifier c c b database explore find yet stability issue conduct performance cnn baseline surprising error detection viewpoint baseline show effectively b detection detector cnn baseline score candidate square observe cnn really baseline investigate jointly optimize pose test variant probably lack improvement detection
special france paris en france universit e paris online aggregation prediction new regret version average loss instantaneous algorithm demonstrate expert interval use excess loss loss simple expert make choose nonnegative every incur k cumulative k k good improve possible possible advantage form expert loss unfortunately explain depend monotonic consequence tuning example trick issue sequential fashion suffer
indicate decay rate cross kronecker consider temporal q satisfy drop dependence belongs dependence memory let show weak decrease integer imply na thresholding dependence stationary process consideration together lyapunov apply n nn restrictive homogeneous decaying series factor result brain voxel discard brain grid place regular mm throughout voxel voxel center
segmentation coherence distributional count distributional advance contrast approach distributional drive supervision annotation parse task representation perform considerably generic propose recurrent parse idea argue introduction expressive generally distributional view recent literature distributional semantic entity take purely distributional logical representation try add amount formalism purely logical distributional obtain type relation logical semantic distributional compositional combination logical distributional semantic sentence generalize sentence argue distributional sentence element recent sentence text semantic word modeling factorization text nlp task sentiment political similar information call belief recursive bp
value penalty extend condition method ode norm method learn highlight classic ode realistic conclude loss classic rkhs scalar value kernel propose literature spline seminal kernel ridge choose positive definite smooth solve gram observe calculate scalar hilbert theory apply empty hilbert space adjoint
dpp np hard extract diverse run inference standard dpp achieve margin significant improvement performance evaluation metric f w broad applicability diverse frame open pixel historical summary independently validation frame trivially redundant low visual task public frames compute precision extract frame histogram sift extract intra compute standard quantile visual similarity neighbor compare select frame tradeoff precision multiple parameterization base flexibility point hamming advantage fine tradeoff adjust baseline statistically measure tradeoff valuable know want free summary recall preferable user take third party see video drop frame detailed analysis video material offer powerful status maximum avoid specification demonstrate dataset real conduct inference dpp call
theoretical rf rf computation estimation bias integrate addition choose r build tree use leave subsection toy bl situation fit scatter plot plot behavior toy rate section forest right rate indeed forest rf rate tree forest section particular forest tree rf result ht toy rf bl situation fit scatter act compare forest reach toy rf reach forest behavior hold model r denote scatter linear graph forest hold reach framework moreover suffer divide increase rf divide forest rf gain forest extra extra final risk improve risk plot scale rf slope fit regression five depend slightly bad forest decrease explain reason presence five well forest well obtain informative
constraint analogous r orthonormal basis ii theorems notation ai reduce aa coordinate basis principle distribution little definition use tail probability lasso simplify description dimensional write respectively estimate precisely value solve distribution expectation note trial distribution target direct routine weight trial calculate na weight sample sum routine estimate sample tune hypothesis coefficient consequently become distribution nonzero choose accordingly increase wide spread increase variance uniform procedure n n routine minimum trial dominate target value application give level expression factor potentially rejection trial routine lk lar algorithm direct sampler sampling lar comparable direct trial simulate effectiveness diagonal predictor vector lar along active turn design test aim calculate e routine choose dataset routine routine value estimation deviation estimate quantify estimate deviation across result probability coefficient
combine surface identification alternatively challenge relation stanford berkeley syntactic relation annotate argument sentence syntactic span branch multiclass identification entity distributional improvement show semantic
define formulation support zero reader number time parallelization year discard coefficient coefficient test rule discard screening rule overlap group propose screen discard screening nonzero use screen rule nonzero consider overlap perform screening test perform screening whereas contribution overlap lasso ol screening rule overlap
end monotone tv next say spread seed sketch coupling select edge live otherwise live live edge verify live graph subgraph live graph node must also could spread budget maximization influence spread maximized find influence greedy maximization mi v monte simulation estimate al exact spread marginal monotonicity seed estimation ps cascade mixture item use item topic mixture direct social influence spread exactly ic give influence probability topic aware influence seed e study topic aware include datum study due constraint section describe complete supplement dataset american movie discover movie learn movie direct rate movie rate movie later obtain topic mixture service search edge contain direct topic
equation compatible versus normalized distance dd give original find concavity turn reference boltzmann minimal nature binary perceptron classifying pattern weight force phenomena easy part potential vanish dp constant entropy curve fig dd dpp support convex tendency become constrain constraint explain simple local increase minimal grow rapidly consequence algorithm
iteration algorithm attract interest admit obtain rate optimization primal applicability high dimensional separately exploit explicit backward combine forward computation proximity operator computation backward manner deep justification terminology maximally monotone operator go rather view backward admm recursion converge fix mapping view extension perform respect variable solve saddle relaxation factor involve adjusted profile iterative thresholding rescale previous exist extension also symmetry primal dual obtain see encounter literature admm eq convergence guarantee condition g g converge primal generalization author primal dual hybrid accelerate convergence rate primal approach problem eq condition size restrictive convergence solution primal converge solution also propose specific primal interesting deal operator indeed differentiable proximity operator g gauss likelihood handle adjoint main inverse primal base forward primal proximal inspire seminal work extend convergence guarantee cm admit primal often enjoy advantage operator scale version addition satisfy condition respect slow dual algorithm rely tucker solution f terminate generate
subsample synthetic set respectively resample number like control negative alternative due incorporation selection structural precision discriminative fraction retrieve recall fraction relevant retrieve discriminative snapshot many fmri brain weight thresholded visualization purpose zero zero determine algorithm vision experience also threshold voxel voxel brain adopt scheme situation reach point prediction acceptable beyond validation kind fdr control main feature sensitivity specificity probably positive extra reliable least portion voxel truly achieve corresponding region accurate scheme permutation order positive finally voxel control voxel interest patient five voxel voxel element cluster three pattern I linear group cluster spatially cluster brain region voxel discriminative voxel last mean false figure demonstrate retrieve relevant also sensitivity relevant retrieve keeping control together randomized discover almost discriminative notice work well identify discriminative voxel different voxel right discover computationally visualize recall curve figure illustration specificity voxel aim chinese study history fmri subject kind
divide worker consider update column finally parallelization update independent fix versa mf h matrix frame schedule scheduling counter counter else return counter frame single empty list counter else col col frame counter k else x col counter mod support static take consider parallelization coefficient fail converge presence dependencie lasso challenge avoid simultaneous dimension highly contribute formally regularization loss standardize intercept loss coordinate cd soft thresholding schedule strategy pick dynamically observe j substantially
describe angle width b width width angle angle width set part snp available individual slope day measure datum al trait trait kernel linear use genetic define correlation phenotype set genome display depth region detail multiple building follow divide box multiple replication display correspondingly trait association previously trait width b datum chemical relate health property carry record environment year marker
perform due objective function perfect bind minimize trace consistent state need approach minimize corollary note centre france de langevin paris paris investigate simultaneously enforce structure
denote simply useful comprise gps coordinate respective q measurement nf assigning update move previous however reliability weight notion indicate user trajectory reason estimate expect visit assign measurement close proceeding follow pixel weight consider performance exclude pixel also provide novel analytical justification scheme choose estimation challenge briefly automatically detail scalar equivalently nj update dictionary end thus take measurement notational column similarly minimize cost scalar fit turn positive responsible second discard irrelevant datum dictionary change design provide feature help new hope scheme close optimization already measurement available hope track reason backward low function differentiable minimizer nonempty
mean tangent measure tangent work recently stein bregman matrix bregman divergence asymmetric undesirable jensen shannon divergence barrier cone obtain stein bregman stein divergence transformation computationally relate establish bind point matrix classify similarity vector aid stein hence convert similarity euclidean riemannian discrete dirac
speech low comparison return non evaluate system test file evaluate synthesis rnn hour train clean clean train noisy clean clean combine ai speech prediction system dataset clean number several inspire previous approach early replace effort classification loss rnns speech similarly simple activation rnn et multiple enhance scalability focus scalability simple lstm scalability dl instrumental dl reveal significant gain gpu hardware locally optimize library connection cluster inspire scalable utilize try potential train set large label set feed
previous result graph summarize acyclic decomposable hold reverse form add reverse add reverse operation neighborhood change change previous step denote node denote acyclic operation lead cycle add reverse remain cycle remain cyclic operation lead cycles acyclic add reverse otherwise cyclic form reverse operation follow cycle add lead lead otherwise acyclic lead previous need remain proposition search operation status score acyclic status assess implementation greatly graph sample second published replicate gb detail appendix bag aggregated prediction rule base version prediction build effective improve idea learn structure model dag highly drastically perturbation propose aggregation dag greatly dag bootstrap dag aggregate ensemble aggregation dag nontrivial straightforward
simplex axes de close plane hausdorff essentially orthogonal contain w kk w volume q ik multiplying side therefore return fact endow inner decomposable element entry odd note orthonormal square sum square inequality corresponding basis hand equal finite space easily cauchy branch easy substituting say infinitely jacobian either local direction imply say strictly indice jacobian matrix exist direction path w everywhere since prove isometry imply volume circle follow hausdorff euclidean cube isometry image jacobian r metric ready volume measurable exist possibly jacobian since constant prove singular
probability prove kn algorithm smooth k rewrite unconstrained apply summarize need compute operator soft thresholding operator sparse illustrate regularization edge experimental network world social transmission transmission transmission set cascade cascade infection infection illustrate consequence use cascade precision network present fraction infer network cascade successfully income polynomially neighborhood infer incoming hierarchical degree different super cascade pa summarize cascade neighborhood study predict value lead line regularization cascade large cascade may satisfy cascade hierarchical transmission transmission model outperform find competitive first score cascade contribute establish problem
interesting flexible spatio temporal separable acknowledgement award amazon research conditioning express give set posterior pf give predictive curve spaced grid predictive package mat ern determine refer distribution beta kernel epoch decay along place prior constant hyperparameter exceed bound name develop rapidly setting machine decide training start new previously machine
bayesian strong role much framework recent attempt indirect induce structure bayesian tool matrix good suffer attempt low introduce variance singular generalize sparse
almost attempt represent generative manifold theorem asymmetric kernel original limit work latter describe operator limit direct embed algorithm limit novel direct spectral apply directed graph density directional flow source separate directional geometry manifold hoc consequence principled model generative recover idea geometry result respect asymmetric kernel everything expression symmetric asymmetric kernel go elegant four able directional work direct graph manifold relate generative direct embed embed analog like undirected present asymmetric express
regression function rather accounting relate choose likelihood value observe differential variable independent distribute clustering assume multinomial additive function thus penalize maximized algorithm control complexity penalize establish closeness fit fit tend complex however discuss coefficient maximize robust curve training iteratively em cluster step penalize see initialization stop step compute log curve computation posterior maximize respect show maximization perform separate mix subject proportion obtain subject use lagrange multiplier eq role competitive discard update cluster proportion entropy stable enhance less finally penalization set describe competition enough discard cluster discard proportion sufficiently partition robust stand prevent large
rich label generate user tag human different vocabulary describe concept conceptual end multi net multi modal vector conditional imagenet dataset output image representation concatenation tag description skip gram appearing time
h nmf call tweet create conclude tweet nmf topic topic collection tweet prove fast provide tweet visualization datum nmf algorithm prove valuable since text well explore visualization aspect singular value consensus run reliable exploration would evolve national science foundation project nc mathematics
modern day record east united census want part country collect database record often single database database record corrupt quantity merge database return apply paradigm database corruption process provide uncertainty quantification generative unique record relationship database mcmc approximation
q distribution gaussian radial basis whose constrain dimensional interpret centroid manner model choose probabilistic gaussian pose gp euclidean I assumption straight line euclidean track recently pose prior application space metric space less efficient dimensional may reliable way deal geometry provide metric able smooth manifold gaussian model
eigenvalue alternatively employ initialization choose value pick initialize initialization initialization simulation stop value however progress likelihood stop determine basically acceleration decision make algorithm reach log value rand index true ari agreement rand ari perfect ari well chance alone although incorporate parsimonious family structure remain model
operator epoch initialize iteration ir I inexact sparse refer epoch epoch within ball radius center inexact employ efficiently level perform processor regularizer simplify form soft step update extremely carry average propose note convexity strong convexity curvature function relate variable continuously differentiable positive definite feasible regime matrix solution notion strong work f q bind hold eq q dual assumption epoch least last epoch improvement result depend bound propose problem multiple variable property bind constraint allow dimensional rate consist error gradient impose term convergence
attention distinction indirect cause application finance portfolio causality portfolio risk causality measure field information theory experiment perform well nonlinear causal structure well system set direction window economic identify range wide application frequency work separate briefly relevance causality measure largely ultimately researcher confidence possible relationship rather measure analyse causality still causality economic model gain even wide little nonlinear causality finance economic hand many could filter dealing parameter primal equation get dual weight depend parameter functional analysis space follow generalise modification metric fundamental concept space rest paper standard notational convention following prove continuous later operator use mean cross operate important functional hilbert functional trick explanation describe point future loss exist satisfy equality present definition hilbert schmidt criterion schmidt induce schmidt u rkh space define field joint expectation ensure schmidt schmidt schmidt norm hilbert separable tensor notational first denote application multiplication hilbert schmidt cross schmidt operator eq early element measure element rkhs respective bound denote rkhs strictly kernel k yx introduce early schmidt hilbert schmidt follow kernel copies eq cross eq normalise cross covariance operator normalise covariance gram equal u u uk
demonstrate feasibility base tight complexity result value worth utilize proportion bound cover infinity cover lemma class lipschitz cover w generation cover definition growth class dd vc lead hx hx pure bag correctly instance f probability least thus correctly bag draw distribution denote copy instance bag classify immediately z nr h bag bag select last instance bag treat tx I iy find two first trivially let constant see later find space three define equivalent ex x tt relation relation derive
drawback lack view derive reverse end start globally consistent first wish trajectory deterministic thus solution seek procedure point require class recursively assume define covariance iterate apply alternative future window window replace move retain computed implicit solver plot window xt
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public knowledge connection n I objective cell immediately protocol community detection maximize polynomial algorithm ease want infer protocol detect community execution protocol close limit protocol sequence objective accuracy maximization privacy max abstract protocol protocol symbol size challenge challenge successful accuracy privacy define denote exclude adversary get know
signature procedure yield first implement classical sampler virtual virtual order value rank become log quantity quantify goodness namely correspond value benchmark discriminate describe identify contain spectra involved optimally figure display histogram virtual log bold cancer group select detail optimize discrimination vs display virtual log value compute length bold vertical black true spectra binary two function clique restriction since value observation quantify two characterize fix precisely number determine equivalent pearson proposition symmetry concrete discrimination know independent consider parameter simple rule vector sign error precede situation matrix mean zero asymptotic variance supplementary material power decision affine length two parametrize fix two probability recall arbitrary pair material two parameterize configuration resp compute estimator correct resp number supplementary mass acquire patient patient moderate inferior spectra discrimination
low reduce original group classical lot attention approach discriminant propose centroid naive thresholding combine penalty discriminant vector scoring essentially reduce discriminant group direct avoid misclassification rate desirable unfortunately approach include one versus classification assignment usually multi canonical constraint undesirable viewpoint well canonical burden pose challenge guarantee bridge superior performance estimation group goal develop novel multi observation canonical form affect neither rule
bilinear model frame training extract try predict bilinear way transformation relational well relation transformation end frame frames bilinear infer transformation derivative temporal series way individually frame bottom subsequently tune parameter back may system equation stack account demonstrate surprisingly image multiplicative filter relation multiplicative recurrent neural work interaction gate state separate consequently work interaction transformation
associate transformation influence temperature etc measure variation computational transformation generally expert instance software decision evolution measurement instant record example pattern reproduce indicator normal behavior learn statistically model g measurement distribution failure signature describe indicator see expert generally specialize mainly diagnostic diagnostic take incoming come situation early sign failure could perfectly provide level trade false alarm general operator role sign failure recommendation identify failure monitor reach automate precise optimally visit
observe correlation fundamental range economic practical trial na correlation variable causal former latter seek likely case gender cause induce direct influence distinguish possibility leave trial carefully assignment drug common cause consequently correlation must causal fact common record correlation preferred ability complete solution variable cause strength precise causal whether correlation direct act possibility theory place restriction measure nonetheless show early constitute bipartite determine state device markovian implement type experimentally causal quantum signature causal word type correlation causal purely cause framework place common cause extension devise impossible scenario experimentally passive problem quantum causal quantum relate nonetheless causal form influence common cause causal mechanism simultaneously possibility depict acyclic specify causal parent specify circuit depict specify circuit maximally subsequently unknown quantum operation parameter swap pure pure swap common connection middle hybrid causal pdf aim discriminate causal relation
train block size block cifar block c decision tree much well svm perform function step art loss motivate employ support codebook example increase improve retrieval dramatically binary encoding sample outperform speed precision decision hash encode suit plot support outperform also high codebook feature codebook dimension time map train cifar employ two block report cg show spectral
call turn stationarity bethe consider bethe studied point hessian one somewhat call point expression bethe rescaled eigenvector multiplicative correspond transition ise start identifiable give cluster motivate bethe hessian like graph sbm analytically limit know spin graph community situation interestingly spin pass hidden take place informative remain fig bethe hessian analytically
cover call basis pursuit sparse recall criterion suited value non interval overlap consecutive active cf note group tu tu tight x random datum report average figure recovery criterion use include overall sparsity suggest try solution show error group envelope ball tu text envelope tu envelope note denote bipartite linearization trick employ reduce program j focus translate determinant submatrix contradict tu def tu surrogate proposition intersection surrogate text
eigenvector equally good machine weight covariance equation nevertheless extreme order flexibility section principal new encountered dr eigenvectors set constitute eigenvector allow straightforwardly substitute start typically towards eigenvector eigenvector encounter start iteration near within amplitude number drawback retrieve remove find component filter thank regard norm along function highly dependent commonly produce orthogonality manually check obviously case example classical require operation solve matrix build follow compute algorithmic various hereafter solution algorithm mainly multiplication refine eigenvector give one spend build
ball density function two pn dnn l dl assume h hilbert rkhs h fy detail integrable stand transform large introduce necessity measure borel algebra finite geometric median rather condition hilbert see x another univariate median metric define generality distance compute median ellipsoid important transform collection independent significantly strong collection constant collection concentration median true fast preserve estimator disjoint parameter contain arbitrary amongst affected family distance goal admit assume measure special case include f x f wasserstein infimum take simply write
norm stay rmse plot assimilation window norm reduction middle reduction cubic figure analysis tend background one assimilation norm approach corresponding residual norm consequence residual reduction correspond panel residual norm panel certain norm magnitude consequently ensemble residual norm converge optima slightly spike adopt conjunction cubic fig close simply instead adopt sophisticated parameter equip divergence end element I observation adaptive appear panel assimilation analysis norm appear assimilation nonlinearity exponential operator well stability comparison algorithm observation understand view type suggest start linearization remain roughly valid regard appear flexible make step small large guarantee focus examine experimental adopt g assimilation unless default experimental follow
quality bootstrapping enable notation denote mean assessment standard confidence analyze set highlight limitation overcome acquire online action datum decision probability ask action reveal account discard acquire quite allow acquire cost evaluate bandit close exist adapt recommend cite know output mind thing may data unbiased dynamic could indeed argue everything change item news news hour two systematically reason believe contextual bandit
cycle reach separately first go back switch affect likelihood bic section switch trial flip switch bic switch compute switch perform require likelihood v j md obviously constrain document zero document trial update compute datum likelihood note topic switch unlike compute evaluating update minima use select occur document initialize likelihood decision assign topic use document topic frequency count topic potentially however impose complexity involve iterative loop topic flip require scalar part experimentally update estimate proportion trial switch loop active experimentally time compare choose bic range order sensible use order way range order bottom fashion initialize specify predefine remove least plausible minimize bic
measurement spectral project organize mathematical problem formulation discuss various conclusion discuss vi system equation represent however analytical find explicitly unconstraine b ax whose analytical ta tb available wavelet expensive instead analytical solution prefer initial another previous project expect algorithm select row select version
chebyshev lemma choice lemma hence upper meanwhile piece plug give w bound user neighbor type rate via user rate exactly user neighbor crucially joint exploration choose item item u iy iy replacement item apply current scenario incoherence item item represent entire preference yield argument lemma hold user jointly item explore bound different write reproduce lem begin rate neighborhood user neighbor rate stochastically low good user bind every item suppose neighborhood hold neighbor condition km stochastically dominate bad variant bad rating good neighborhood inequality choice finally eq item union bounding provide must exist item time
kernel deterministic simplicity ei use informative present background area regret cumulative bound enable rapidly converge sub formally say sub laplace example zero uncertainty mutual play central role note
development belief network decade network default option machine learn success handwritten digit problem hard classification cnn notable good respective mnist report recent achieve learn algorithm superior cnn achieve significantly low complexity publication cnn whether architecture popular choice step result point obtain past machine feedforward conventional hide neuron neuron classification similarly sized begin classifier input distinct matrix mm contain test size activation class prediction vector unit test contain projection unit include b column dimension element always unity leave description
network approximately network indeed compare sparse easy practice point transition autoregressive mechanism bring dependence g rule condition fx ax n definite imply place penalty via cb follow joint exist care graphical computation union node screen dimension reduction facilitate remove unnecessary edge small brain connectivity reveal datum divide region economic divide decompose finally helps improve overall identification estimation accuracy base literature evolve order introduce undirected either represent give toy share structure cluster two node integrate picture network
operator relaxation f interest different namely simple guarantee search alternate direction multiplier admm
paper take proportional norm project onto system q alternatively interpret stepsize expect pick respect coordinate minimize let coordinate equal aforementione univariate stepsize inverse lipschitz gradient convergence detail section examine behavior algorithm bring opposite behavior algorithm represent system happen rank call
mutation population frequency depth interval confidence allow pseudo second associate affect describe use frequency copy broken population frequency equation allele frequency distinguish copy assign furthermore assume copy replace lie affect possibility relationship occur occur branch site relationship cell copy cell allele numerator copy cell denominator average number copy numerator numerator case still infinite affected lie branch cell average affect expect affect observe affect variant allele case affect possibility likelihood copy decompose possible situation distinguish example condition unable circumstance branch consider tumor come add contain read variant allele frequency half variant frequency copy genome position copy read allele method incorporate incorrectly incorporate alone tumor recover tumor recover
concept seminal building work give uniform approximation symmetric subsequent approximate degree agnostic approximated combination contrast strong limitation polynomial hypercube theory limitation hold arbitrarily high class rely type approximation generalization classical degree polynomial interval inequality function weight strong statement origin powerful jump must quality full multivariate generalization rich agnostic agnostic hardness weakly even boolean hypercube line give independent hardness assumption show pac threshold intersection imply e et al work prove unconditional
inference factor random respectively correspond unnormalized joint iff maximize map message formulate sum function represent part represent select entity mean entity match least entity match since entity equal without match similarity weight constraint entity entity connect constraint connect definition show variable total similarity constraint entity variable relate evaluate variable plane plane evaluate infinity serve enforce matching entity entity sum entity sum max weight maximize objective pass message factor pass define message reverse direction message pass stand source entity message update constraint number node factor node message calculate affinity propagation intuition pass either message sum really variable concrete matching follow c c message message except similarly since ia x ia except jk subtract formula equal update rule match pass number
representation coefficient dictionary polynomial update accord real behavior synthetic localize place performance collect experiment graph wavelet transform ii purely numerical dictionary svd treat iii graph kernel otherwise fix toolbox always orthogonal normalize apply thresholding would careful stepsize code synthetic unit set edge thresholded ensure experiment set synthetic training localize dictionary capture component signal random learn expect dictionary collect infeasible training phase lead impractical training signal allow flexibility polynomial fig fig able recover dictionary learn snr sm signal training testing set compare performance run test approximation polynomial well
rank ccc netflix collaborative interested unseen remove movie agreement compute movie entropy proportion movie movie randomly choose rest testing keep report result fig show netflix moderate pmf seem information state c c pmf mf pmf closely ai social focus preference preference express preference ai limit complete order set close normalise normalise previously rank involve former additional offer explore ease readily capacity similarity assignment
convolution iterate heat smoothing diffusion eigenvalue eigenfunction trivially eigenfunction eigenfunction literature eigenvalue eigenfunction laplace medical imaging vision eigenvalue shape al topological al shape detection et al second surface eigenfunction scope paper issue eigenfunction heat heat heat kernel smoothing take estimate signal truncation automatically determine heat avoid discretization wavelet wavelet complicated study offer simple unified consider wavelet generalize euclidean difficulty arise one try wavelet surface unclear translation try modify exist wavelet immediately grid surface diffusion wavelet bivariate wavelet scale eigenvalue eigenfunction heat translation bivariate heat rewrite truncation wavelet framework wavelet therefore diffusion heat surprisingly wavelet
direction allow divergence change motivation physics sm entropy rise partition solve efficiently formula sm practically multivariate interested form formula sm frank close sm analytic gradient closed similar produce adopt output kl regression minimize sm relative entropy analysis begin new point similarity vector vector apply rbf kernel form k xx exp k overfitte noise otherwise firstly kullback leibler marginal gp focus result pose analytical define eq index kernel use optimizer search gradient gradient complexity store gradient present gradient detail sm matrix algebra k xx k yy hence could rewrite definite xx x k eq worth introduce large calculus logarithm derivation follow eq computational cubic due solve system
randomization nonlinearity first approach match set approximation pilot show decrease piece affect large number show accuracy piece wise fit tolerance distribution sample marginal sec coincide give different stationary generating process input variance monotonic fig transform black line grey dash fisher confidence expect cubic transform display thick line spread spread expect correlation indicate fig monotonic gaussian gaussian correlation var determine transform turn close correlation differ decrease na I denote expect
cascade object star model filter part filter coefficient specify placing detector slide window output pyramid specify position level scale pyramid possible position tuple pyramid detector gradient anchor root part score write kx k perspective part labeling argue necessary correctly treat score label receive stop incorrectly formally score denote letter order response negative make assumption response star root true
variate selection x x variance distribution representation worth representation advantage parameterization reflect skewness parameterization implementation em generate far consider conditionally random truncate namely pdf aa pp limit satisfy rule group rule function rewrite discriminant vector normal discriminant rule e group region
conjunction learn representation reflect layer supervise contrast classification body finally optimal maximize activation neuron resemble human face visualize neuron heterogeneous two type pose aim predict window human g use body bound box contain human summarize location joint normalize bound box use truth part window body part train part separate detector appearance contextual corner annotate body window window body portion inside
theorem cm cm proposition mathematic university department mathematics mechanic state pr university school economics department mathematics mechanics university abstract family test characterization statistic family integral degenerate reasonably efficiency alternative efficiency simulated type recommend practitioner local favorable fully
fashion claim exist drop negative supremum clearly choice nh v bounded right denote sequential tree later except minor fact affect proof cauchy fix one tree well everything upper conclude end sequential cover close eq follow claim provide fix tree along recall hold cover put everything together lemma eq simplify far conclude since take case value particular choice definition value necessarily except sake root
discuss cross theory use domain base graph represent adjacency n eq subject avoid constraint formula rewrite minimization subject extra specify matrix equivalent maximize stability introduce
substitution movie describe interest consider dependency illustrate consist correspond learner template share base training identically share object classic ignore alternative discard clearly suboptimal sample providing achieve
dimensionality image mention vision concern variation recent work neural network reconstruct view face randomly generate face approach allow viewpoint relie find probable match viewpoint procedure sample model fully face show variability expect model applicable task cm cm first generate high description formally n camera transformation target mask parameter increase amount variation reduce overfitte augmentation cnn color variation apply artificial target mask plane change
manner fast exploit ensure vanish assume normalize inner monte carlo gaussian gaussian rbf combination admit expansion datum recursively recursion dense multiplication subsequent achieve reservoir place ensure permutation effectively uncorrelated iid independent draw distribution change kernel remain adjust rather without flexible adjust spectrum way range invariant accomplish since term frequency frequency moreover spectral frequency translation frequency mass basis undesirable optima enforce smoothness therefore spread expansion fast expressive parametrize section already efficient framework learn formalism introduction expansion clarity likelihood innovation process represent demonstrate extended process particularly suit expressive kernel choice objective instance list set gaussian finite process equivalently parametrize mixture integrate away express solely term matrix parametrize independent additive likelihood negative likelihood covariance close inspection expression simplify greatly perform storing require regardless efficiently computation approximation even memory use preferable gaussian
describe decomposition row wang zhang state take theorem column prove actually take author define new eq round integer I p follow argument replace point difference moreover weak upper make little difference purpose fact linearity equality variance e remainder strong weak whereas let pairwise hash need trial pick correspond overall back state omit deterministic operation apply result transpose switching n summarize decomposition bss version extension original bss spectral frobenius randomly subspace see ns nr I I I I construct small lemma simultaneously apply times column result version discuss version randomize use take respect randomness immediate markov careful implementation routine
outli impulse model vector give stable spline encode stability tailed density student identification procedure mixture gaussians variable propose posteriori make derive identification optimization propose advantage new compare worth datum popular huber contribution noise find year description describe approach derive impulse response kernel particular subproblem connect system admit computationally especially compare alternative propagation fully organization introduce map describe time causal transfer system drive output corrupt noise sake output measurement
linearity nonlinearity curve meta rank curve strict rank fig principal translation strict perform know size interpretability rank principal curve call monotone give cloud explicitly parameter rule however lack monotone fig requirement monotonicity new principal model ranking five meta formulate point b place particularly complex bring problem simple represent possible monotonic curve rank cubic meta rule nonlinear end determinant shape curve point scale translation facilitate calculation control derivative order exist cubic type strict monotonicity point nonlinearity cubic coordinate interior hypercube monotone parameterize cubic monotone group existence fail exist monotone eq appendix control optimal curve
ks table deviation percentile standardized version ks statistic adjust variance rv choose testing computation quality normal table asymptotic normality hold well stable rv sample inspection normalize show clear significant level precise eventually conservative small practice actually constant reference therein relevant
extract small eigenvalue consequence nc nc subsequently eigenvalue next nc nc subsequently eq hand q q inequality due therefore combine mm general though method convex globally almost theoretically analyse method experimentally promise back age theory hyperplane case apply outli theoretic principle learn hypothesis finite equivalence observe hypothesis label
special player attention th matrix notation matrix row discount expect reward initial eq denote stage markov property addition state state eq transition matrix eq write notation transition value express note introduce use relationship stochastic rational playing player seek minimax player change game value favor start minimax simultaneously action player receive determined minimax person exist player reward player strategy play player express reward receive value minimax nash equilibrium minimax discount satisfie game remainder playing
formal intuition behind robust update procedure short iterate explicit sgd iterate explicit widely sgd motivate implicit introduce still maintain quadratic proof present appendix formula variance sgd implicit unbiased rate derive close statistical comparative estimator stable implicit hessian iteration although quantity explicit implicit implicit update perform approximation na fisher general analytic equally define generalized implement sgd essentially univariate interval generally sgd evidence performance goal tangent analysis procedure principled purpose identify seem viewpoint score learning rate add insight issue explicit help identify loss sgd estimator whether one show low mild condition sgd information sgd information become apparent represent amount else learn receive minimal despite much jacobian curvature show I refer explicit sgd parameter e way sgd use order hessian jacobian effectively iterative procedure thus uniformity practical arise normalize aforementioned proceed sampling efficiency quantify theorem consider explicit update simple
show present appendix expectation follow behavior large follow mle appendix simple bias substitution integral mle v rewrite q obtain
approach recover lose performance uncertainty human st switching complexity annotation variation synthetic approach template question typically spatially relate frame argue evaluate threshold order predict accuracy architecture nd segmentation pair class whole test yield total automatic rd table switch human automatic class propose approach across baseline th human ask answer question collect answer question answer class single world binomial check preference
quantify illustrate method performance equation mathematically phenomena heat structural pde law law practical application real simplification tractable ignore physics certain condition pose difficulty refer pde knowledge pde accuracy predict compare pde model acceptable current pde define model alternative assimilation broadly speak assimilation incorporate reflect inherent mathematical assimilation share estimation assimilation calibrate assimilation define possible good involve parametrize reconstruct experimental obtaining assimilation physical square however mean quantify recent pose
classify strongly ground combine category classifiers cnn imagenet object close coarse confident coarse belong category category rank nd overall cnn place demonstrate need category classifier impact coarse train classifier coarse top error overlap execute trade classification large lead execute category enable hyperparameter merely minor speedup time build net parameter imagenet size hyperparameter compression factor compression decrease mb mb class net execute fine category error hyperparameter imagenet dataset building net independently obtain error building block coarse category method top top imagenet net build net layer imagenet layer total layer
perform class extract class dataset e relationship dataset predefine unseen class fine cat cat cat herein attribute label confusion though method predict achieve lead well contain drop situation apply share public achieve conventional art cifar conduct experiment low possibility similar order employ aware semantic visually likely similarity electrical device pick relate nonetheless work coarse class achieve property theorem manually
examine nature choice familiar law model algebra fine grain become refinement think initial triple degenerate triple singleton way account broken choice measure mapping implement value swap measure together form specify particular mathematical decomposition purely usage probability implicit somewhat outcome outcome choose outcome assume outcome belief type distinguish belief impose restriction make restriction probably highlight illustrative cascade mechanism sub experiment implement conclude affect odd pick know stage increase odd hence affect belief future mean causal aspect concerned scope potential stage stage part namely sense separately exclusive historical pick nothing pick give know execution sequential experiment situation fail example precisely text htbp clear belief choice respect boundary separate operation parallel namely represent right collection experiment sense operation transformation put choose illustrate figure htbp recall situation choose know first discussion amount stage measure subscript similarly table list table experiment swap colour black leave black white yes white yes black plausibility q choice provide stage draw right plausibility obtain intervention evidence support hypothesis past intervention deterministic moment intervention subject external recall
orientation orientation velocity connectivity presentation source fig histogram noise velocity connectivity stimulus make long range finally group reduce kind error subsection consider perform spatial visual predict stimulus trajectory group apparent motion move coherent trajectory one temporal offset one explanation specialized motion contour orient study suggest rule drive orientation role play specialized evidence trajectory drive connectivity come population response change direction show low angular provide sort spatio temporal interpolation mechanism velocity already motion movement extend stimulus present area asymmetric column orientation role play horizontal suggest biased motion recent show segment support two mechanism spatio temporal moving shape twice bar dataset depict create segment move frame curvature bar opposite bar orientation embed consist uniform path direction stimulus frame full spatio surface stimulus rise visual grouping spatial level identify visual grouping spatio recognize object movement reasonable two interaction point integration trajectory cluster spatio unit hand grouping law contour different implement structure experimentally confirm composition mechanism result summation argument cluster sum affinity construct matrix additional coordinate
bias improve much bad proportion miss disk indicate near contiguous composite distant conditioning determine design lattice low incomplete approach compete month spectral stationary observation region south illustration unknown mat ern function effect miss total posterior minus isotropic process mat ern q smoothness modify find bayesian mcmc non square lattice lattice maximum original pixel approach radius lattice choice lattice lead eigenvalue sampler discard constrained yield embedding describe run burn period three draw unobserve closely smooth draw region figure
case loo conditionally parameter evaluate importance separately correction correction produce also illustration finally dataset put scale interpretation firstly sometimes secondly calibrate interpret pseudo calibration error loo quadrature loo loo really completely fail loo useful la loo fix whole la tp l loo loo na na g g la l loo significantly ep loo loo tp ep loo loo na na na loo ep na ep quadrature loo significantly well satisfactory loo bias correction therefore force truncate loo loo cm add well bad tp loo loo loo la la table loo data loo quadrature latent ep result loo ep already good correction worse
area may community community community goal community detection citation author undirected author paper replace direct author b easy analyze research discuss separately rather split community large community dimension reduction interpret community design study htb citation identify score stanford bayes citation large parametric network network connectivity sophisticated identify community research propose score community undirected community follow datum respectively investigate cluster al seem largely section discussion b citation citation largely detection method community multiple testing spatial statistic present figure convenience road map section connect also section therein trend citation trend finding limit period paper proportion decrease distant figure become collaborative competitive degree bipartite phenomenon frequently obtain quality hard true quality resource google try portion challenge author overcome primarily home attention except recognize country bioinformatic primarily period statistical area period seem serve meaningful community collect long
counterpart mnist exception establish evaluation variational report softmax perform well score lda softmax competitive dataset hide record dim news training intractable direct sigmoid belief demonstrate emphasis possible model inference architecture considerable gain expressive continuous latent promise latent appropriate conditional latent would require make inference would make model make powerful direct latent acknowledgement thank comment provide gradient outline computation
walk proposal inefficient explore offer use improve mix hamiltonian monte proposal would full whereas optimization objective surface apply simulation feedback determine next location function idea gps abc cost abc institute world demand challenging computation standard tool handle likelihood problem two sampling algorithm hasting adaptively gps simulate challenging realistic biological illustrate potential algorithm biological birth star weather rely naturally observe hypothesis generation evolve critical match observation hypothesis cycle trivial phenomenon inefficient potentially interact program scientific play
improve change strategy alone equilibrium one play could strategy game payoff player check alone nash equilibrium games game incomplete intuition know opponent playing set consist player denote player though extension game transition
disadvantage problem movie rating average user rate movie rate practice uniformity far netflix user vary quality dramatically order suggest nuclear norm column marginal improvement unweighted nuclear pattern bound low rank column possess incorporate completion form regularization netflix assume movie rating dimensional weighted represent define column denote imply laplacian column column row term add frobenius nuclear outer non propose outer
model target try risk rare influential event concern notation parameter term roughly estimate period crucial system capable handling include year extreme heat loss minimal translate book opinion incorporate alternatively calculation identically distribution practice drawback tail put second degenerate maxima sequence exist degenerate member q shape correspond fr support understand I mean maxima increase would relax limit certain unit fr q helpful consider member transform unit fr margins unit margin transformation assume unknown may taken extreme transform location fr site generality one assume unit fr extend multivariate let maxima noting unless occurrence block coincide spatial maxima location degenerate sequence multivariate distribution one
transpose jacobian relate row jacobian stack hessian hessian equation follow form definition trajectory derivation hessian parametrize assumption hessian give parametrize policy trajectory hoeffding inequality section conduct domain linear water
correlation output plane require storage testing memory tolerance template take elsewhere filter correlation filter design formulate form expression vector notation equivalent channel signal formulation due circular circular replace filter introduce combine localization base svms traditionally cf design g template equality constraint relaxed constraint margin negative slack variable image wrong margin svm class negative typically express obtain add problem formulation signal determine dft template q inverse dft wish constrain find considerably computational numerically outer cm minimum width height cm fill blue thick width cm fill illustration require computational consideration discuss challenge arise filter computationally intensive memory overcome design typically great conventional counterpart usually train usually image kn kn reduce explore several way explore portion template observation portion nearly dft tail circular
stationary stationarity hold stationarity constraint derivative lagrangian multiplier constrain substitute set stationarity burden lie th order fact prove two center straightforward characterize denote two critical vector much strictly half coordinate similarly second half u claim q inequality x x kn quantity inside kn ki stationarity equation choose first claim k true low precisely p k reach stationarity u ki positivity dual since positivity always define express stationarity multiply side since choose contradiction follow item lagrangian apply kkt condition sufficient symbol represent abuse constant actual vary theorem convenience suppose nk k fix use concentration sampling replacement convex gaussian union put constant bounding hoeffding inequality term union bind eq apply q union second inequality use part third claim let overlap segment segment take union
run prescribed weight tree alternatively run approach demand homogeneity possible study cause difficult small branch length random gamma log branch trait describe time view possibility selection procedure search rate change laplace motion brownian given brownian drift gamma increment interval increment gamma define
layer kernel extract pooling layer use bandwidth time show net jointly achieve net net two layer contrast normalization max pooling test extract top net dimension rbf times median without center result drop phase gradually achieve net net imagenet million color image randomly random horizontal jointly neural net dimension net pooling net rbf set comparison max voting variation color neural net produce rate neural much speed test ridge table dataset representation convert base randomly break
go blind dr even though computer field create automatic system manual effort screening raise financial issue study dr specificity screen sensitivity specificity combine novel screening feature make close meet sensitivity automatic early recognition serve essential dr screening system dr investigate exclude direction dr recognition framework extend component automatic dr center novel assessment feature exclude image
probe degree structure I denote entity probe similarity transpose structure basis stand bipartite learn edge total bipartite utilize programming solve solution return illustrate come shot shot ambiguity cross word occurrence slice pixel pixel operation single locality appearance change ambiguity view codebook carry information visual large codebook map codebook preserve appearance look distribution associate spatial collection visual word embed use visual make spatially locally distribution account appearance change similarity way embed hilbert rkhs universal preserve inner similarity two two end image occurrence appearance visual word rkh entry generative discriminative view insight activation view location joint j p eq inequality rearrange efficiency simplicity idea handle specific bind eq spatial fig whole latent spatial appearance slice
able extract descriptor binary picture form dimensional canonical h hence binary accuracy linear svm feature semi side validate alone statistically respect pair svm scalable training autoencoder nonlinear reconstruct regressor mnist cifar compressed mnist cifar full anonymous comment discussion mark van la method principal component pca cca able reveal relationship datum nonlinear variant cca prohibitive large scale randomize
extensive separation important annealing nmf decrease entry high amplitude indeed descent decrease iteration refine final threshold independently estimate level solution threshold choose amount keep last also q aim minimization use descent minimize subproblem differentiable must additional indeed thank constraint hence fidelity term domain permit prove rule subproblem line proximal able continuous low q close wide interest function proximal operator update account negativity soft solve proximal operator constrain diversity greatly improve enforce make separation complex see aim explore extension tackle impose transform noticed contrast prior negativity challenge focus formulation sparse transform
restrict constraint dictionary minimize datum contamination objective atom recovery compare svd al condition sparsity atom dimension k apply denoise image corrupt additive slightly peak noise structural similarity indicate efficacy metric technique gain community survey
change objective soon main evaluating criterion form candidate optimize degradation see value fall threshold could coefficient discard atom atom removal successively update stay basis iterate basis coefficient implementation allow atom add spurious atom add later phenomenon atom form iterate would form iterate computational comparable approach require calculation vector ht iterate column reduce conclude discuss practical guarantee converge decrease termination property proof appear sublinear convex generate true optimum standard omit formal solution boundary atomic place
context replace span relaxation plant recover provide essentially exceed provably semidefinite programming sdp q possible say relaxation naturally intrinsic price pay similarity efficient theoretic simply guarantee sdp relaxation order recently introduce plant recover perhaps surprising runtime sample size interest raise practical provably
neither true source subtree pg pg ml proposition definition claim protocol usa anonymous medium share anonymous message initially sensitive consider observe snapshot spread advance exist protocol adversary introduce protocol call message fast perfect network message nearly experiment sample facebook effectively source cycle core aspect internet propagate message text video platform link message rely post brevity interface party communication communication anonymous enable message friend message author identity message exist service store company access solution store node know never party distribute monitor architecture simple anonymous protocol anonymous adversary recent advance building protocol truly anonymous platform contact strong adversarial condition consider contact graph identification fast connection internet principle receive equally likely key real scenario message spread share filter inherent happen message standard delay message spread time spread continuous anonymous popular decade anonymous communication receiver
fig contrast analogous hierarchical able detect significant specificity glm ols perform poorly even corner expect rapid column significantly thresholded map ol well perhaps detect sensitivity specificity fig replication simulation show portion glm ols approach give significant brain result able consistently detect truly activate avoid finding glm perform time canonical computing brain significantly canonical voxel correspond closely voxel fig believe diagnostic assess glm analysis ability recover peak left show hand column estimate replication width occur shift result single slice slice illustration secondary work involve activation think discriminant voxel level shape group shape also another response delay reach peak roughly apparent slice problematic canonical basis glm show
parameterization enable candidate induce convexity partition efficient proximal svms initialization generalization partition global candidate increase fitting indicate candidate achieve practice competitive art locally relate category assume specific interpretability assign flexibility indicate clearly relationship input model predictor probabilistic framework partitioning sp sp utilize region predictor highly advantage sp vc dimension classifier develop utilize predictor uniqueness specific account test formulation sparsity induce relate motivation test kernel
angle triangle retain discrepancy map integration triangle important graphic infinite carlo vanish integrable triangular van estimate merely integrable discussion conclude give volume cube ball simplex generate space sphere cube simplex simplex cube cube lie spherical triangle mapping identity equivalent input differ corner singular variation present cardinality point hybrid computation triangle point degenerate triangle via corner convenient equal quadrature right triangle borel
skip gram embedding gram starting suggest gram embedding retain retain compressed specific lexical acknowledgement centre france centre word particular lexical relation
proper bottleneck reason may redundant computational might lead bottleneck bottleneck rarely address bottleneck choose bottleneck bottleneck layer metric refer train autoencoder show bottleneck layer pattern hence determine critical bottleneck notice slope clearly within bottleneck bottleneck change critical percentage connect bottleneck curve line percentage difference give large enable automatically
q easily notice equation equality independent complete section thm partially support rf grant nsf grant nsf dms city wiener detect minimum observation couple strength differ adopt approach minimize mean alarm partial correlation sum stop mean without bind emphasis decentralize centralize arise engineering detection analytical consideration concern observation delay formulation time constant interesting variation assume depend treat yet formulation far either one stream regard post stream couple assume wiener different
project people project project average project segment project frequent project display frequency display project distribution feature skewed update goal mm project comment available project mm analyze behavior resort type type e project q counting project project project project hypothesis project less support one ready hypothesis project comment increase level dedicate web site matter extent previous pearson management activity correlation seem decide project depend receive dedicated site overall project depend goal goal frequent likely high project support project confirm recommender project goal
resolution obtain coherent take player position predict coherence respect process path coherent interpretability kernel simple coherent estimator require long substantially realistic convolution manual normalize branch exponentially state continuous previous homogeneous depth transition kernel convolution formulation reduce integration state simple play expectation taken provide useful summary unfortunately appropriate tradeoff homogeneous define average together irrelevant begin pass currently markov would estimate
wireless channel network adversary finite enable static adaptive receiver discrete asymptotic static pair propose novel even armed bandit work armed armed bandit regret sublinear sublinear reward adversarial measure minimize knowledge pair achieve linearly strategy bound strategy wireless detail minimal early rest adaptive learning conclude paper receiver receiver amplitude phase pass value symbol normalize unity pass channel signal represent see receiver assume receiver perfect synchronization match filtering sampling kt k signal level duration word signal level power send probability analysis receiver receive along offset
computationally kl model vocabulary word content may topic model might aspect different similarity multiple possibly source compare similarity similarity significance apply
supervised classifier good four supervise rate well unlabele good improvement classifier offer improvement picture different supervise optimize option among supervise provide far observation implicit constraint lot consistently likelihood achieve beyond deep insight issue cast light safe attain offer improvement test perform
likelihood average stochastic drawing uniformly definition involve missing involve pick layer flexible dark input fix value add add deep boltzmann structure mask vector indicate component multiplication simple illustration sigmoid index eq evolve reconstruction eq
concern argue reasonable geometry insight dimensional method infinite dimensional measure space tendency disjoint mcmc define straight forward proposal different note though still slow geometry incorporate also geometry ensure review langevin langevin metropolis hasting proposal though facilitate focus appropriate manifold chain manifold set rotation directional manifold problem appropriate need area geometric problem discuss geometric mala cope target distribution heavy tail dimension mala geometrically tail scenario geometric ergodicity fail mala acceptance scale related picture show metropolis target identically symmetry regularity shape tailor scenario change serve practitioner requirement acceptance rate rwm discuss focus langevin exposition riemannian geometry full derivation langevin diffusion manifold highlight question geometric hope field award suggestion field smooth
right h node target structure sampler empirical center generate gibbs sampler strongly identical simulate simulation moderate particle emphasize kernel respective good drop considerably sampler autocorrelation strongly however autocorrelation comes set hand form fully sampler magnitude computationally involve partially sampler autocorrelation suitable convention recursively simulate definition document remain index implicitly artificial particle particle assignment eq ratio unnormalize explicit set explicitly particle factor depend interest
translation testing dataset summary noisy dataset technique address wikipedia annotated software language technique statistical machine translation evaluation human annotate work supervise procedure generate wikipedia discuss statistical machine use nlp preprocessing wikipedia summarize table previous language specific preprocessing stage parallel language pose bottleneck language cover contrast rely agnostic compare sufficient replacement dependent entity token entity classify word problem task tag depend neighbor competitive sentence dependency
versus list bind especially efficiently threshold desire figure integration play role determine b subset expression correspond small theory lb theory c c simulate compare three numerical determine threshold estimate delay scenario delay threshold mix community mixture know mix inside b demonstrate
encounter practice value asymptotic bias would trial baseline trial randomize patient illustration outcome treatment baseline dl gm dl base achieve normality dispersion five covariate patient estimate logistic treatment successively log correlation table ap ap fit value correct reduce reduce far mark
computable hessian precise subsection stem structural strictly eventually lead barrier move along locally convergent scheme assume compute clear much progress path need fast convergence rate attain idea complexity need define function precisely large compare newton optimize together standard initialize subsection euclidean matrix obtain start definite newton idea behavior start close enough strict local fast convergence assume hessian fy x eq newton well follow formula formula obtains write hessian note insight newton look start show word affine share affine invariance sis inner would without third differential operator idea fix issue replace euclidean function observe one use always definition self barrier keep mind self describe region convergence newton self important eq without standard newton iterate barrier plan plan since q immediately initialize remain iterate obtain infinity need control fx uniformly fx fx fx safe penalization concave arrive barrier self logarithmic barrier difficult barrier universal key follow generally ensure central generalize identity close central formally basic self barrier small describe subsection x k scheme describe show iterate remain central indeed theorem obtain thus obtain finally yield explain central path trick one path small enough central path iterate equation start k analysis correspond obtain path roughly self barrier om computing newton direction compute invert thus barrier nice barrier barrier come
mix marginal usual effect spatial section penalty scad interpret obtain sparse induce around zero subdifferential net scad example notation net pattern convex subdifferential origin interpret score penalty scad practice penalize necessary determine pattern convex penalty optimum respect use algorithm sparsity penalty leave depend yield meaningful instance regression meaningful large perhaps simple estimating nuisance penalize acceptable context bias incur score thought test useful manuscript focus score effect modification section instead effect group appropriate straightforward
good sentence begin repeating song detect choose sentence certain presence document frequency size sentence weight c c raw binary parameter music context future include feature type gaussian help markov model leibler frequency relevance singular value term document frequency letter generic successfully text speech
link direct centrality proximity distance short centrality quantify node centrality another indicate way divide subgraph great within modular theory shannon contain entropy
belong inter level document label experiment connection inter inter decomposition control seem appropriate interpretable question whether use representation dense one representation space primarily regard interesting benchmark modification flexible applicable semi science computational intelligence system x entropy ie x p eq iterative alternating rule iterative rule instrumental analytical
response causality locality require superposition rise cause event background precede formally attribute background rate augment product poisson density causal iii cause impulse single parent process unweighted indicate direct edge node whose indicate reflect recently framework conceptually class graph exchangeable relate graph trivial endowed characteristic convert representation transform many construct nonparametric fall leverage formalism combine impulse eq binary negative respectively capture interaction parameterize support us structure interaction empty background process recover make event allow strength interaction suggest
experiment except node initial choose identity evolve leave tp plot standardized factor predict still membership estimate checking mean calculate state tail experimentally deviation size factor class encourage priori cause asymptotically true class filter assume student posteriori set figure comparison accuracy adjust rand index snapshot sbm surprising probability follow accurate
modify sharp bound additional show excess hold excess risk exp unknown instance regression respectively domain classifier aim generalize unseen interested problem function logistic loss classification portfolio management arbitrary e minimize exp concavity
multi task figure achieve small recovery short demonstrate particular observe outperform test noise exceed solution involve sparse test therefore choice real world I high collect name english letter alphabet twice group subset refer task correspond feature response letter experiment letter treat different training evaluate multi average square whose test commonly task
tangent axiom positivity symmetry triangle inequality simplicity geodesic derivative geodesic view distance follow end field call field characterize gradient field manifold hold exponential cut cut locally integral curve distance function pass note condition pde simple example second appendix distance fig field everywhere characterize distance distance v require red color indicate value blue indicate riemannian embed space pf ready heat flow schmidt learn set illustrate justification whole heat gradient obtain geodesic function analysis control approximation rely cut query cut distance fail since field cut would embed undirected graph neighbourhood near worth note
wish dynamical drive external parametric neuron goal insight analog second nonlinear state spike count bin poisson relate second smoothing lag particle smooth training figure sample addition prediction clearly frequency datum capture show cycle tb trajectory trajectory smoothing sample posterior trajectory simulate trajectory inside cycle attract tractable dynamical suited principled learn expressive
partially input denote miss refer observe union realistic g human nevertheless model within straightforwardly model jointly contrast input take account uncertainty location empirical experiment well q observation model fully partially location input define incorporate partially principle specific predictive train incorporate auto case confident prediction outlier due together bad gp equivalent miss gp semi supervise use traditionally encounter know approach accord model label incorporate specific follow achieve miss confident label build instance framework adapt tackle miss appear treat difference input associate uncertainty concern method self also train portion however hence uncertainty measure discard contribute make self semi gp simulate give input correspond world datum body formulate regression portion input miss extend train gp handle miss straightforwardly reconstruct fully input size motion plot mse compete method vary percentage miss plot seed supervise gp make portion miss converge gp miss identically gp latent automatically constitute generic dynamical modelling able capture complex rigorous bind range world give research extension become feasible propagation gaussian allow filtering application variational system linearly unobserved state non past latent propagation lower also promise research place application formalism process function five learn complex
exist q permutation many annotation rank rank empirical rank rank key aggregation statistic list copula rank list rank rank rank mid retain definition define informative domain call rank miss expert rank number expert rank object expert axiom e consistent notation axiom monotonic axiom furthermore call extension tie map strictly important rank imputation whereby list rather
researcher relaxation qp seminal evaluate various qp show dominate sdp relaxation aim obtain tight add high interaction marginalization group practical open decide attempt semidefinite primarily state semidefinite redundant linear develop proceeding notation throughout conjugate operator denote nonnegative frobenius state configuration discrete consider parameterized potential mrf maximize energy probable assignment estimation hard combinatorial problem cast mx estimation equivalent follow encode hardness arise aspect binary ii motivate relax semidefinite relaxation step
safe test appropriate dynamic screening reduce radius region sphere improve test effect test may approximately induce regularizer separability problem sense assume advance weight group induce soft thresholding eq dual n optima link small publish extension except contain thank please detail safe extend screen st define capacity screen readily thank concrete propose safe static screening implementation algorithm usual appear line line backtrack dedicated safe l safe state successive use screen test lemma screening compose line
vanish perform embed define rx proof exploit embed distance towards first sum assumption bernstein term sum bound bernstein resp resp fraction nice neighborhood coupling generalize random node everything else attribute parent attribute child label attribute equivalently label random accord everything attribute parent one attribute denote node tree couple g r attribute leaf independent consider eq lemma lemma give symmetry
microsoft early widely proper noun coincide w shift occur analyze shift amazon review content short book corpus table amazon review twitter dataset like acquire sense movie game application change acquire east usa shift meaning release book capability method book aware web request intend evaluate quantitative linguistic shift corpus copy wikipedia corpus wikipedia corpora speech tag introduce word shift finally mean word pair exclude functional word occurrence word occurrence illustrate two type perturbation frequent tag might car frequency method observe degree consistently word category perturbation quality perturbation annotation
report community detector label four unsupervised anomaly report traffic detector anomalous traffic technique multi resolution method report traffic distant use useful kullback leibler detect prominent traffic report anomaly tuple address traffic select alarm avoid miss build retrieve
square essential present satisfie instead restrictive arbitrary excess loss give straightforward taylor around end minimizer obtain identify level loss parameter z play thus result may role require identify cardinality proportional version find theorem assumption question risk rapidly decay rather concentration argument small thus heavy tailed heavy target scenario bound empirical minimization precise space almost close close loss distance average distribute goal least minimizer unlike select describe price cost like minimizer partial note problem deal predictive reflect good exposition estimation prediction select element minimize
limit posterior obtain process relative n support I distribution variance square jk ki hold w jk w formula matrix check posterior select impose n observation bb reliable value way eq always nonetheless tie population function tie become represent expression verify symbol use indicate go sum ease equivalent sum converge vanish follow procedure prove equivalent statistic consistent conversely instance different significance return powerful conversely alternative asymptotically experiment range facilitate test traditional never outcome call issue chance stress
propose ng jj ng shrinkage method assign get sparse elastic net covariate elastic tuning estimator reduce lasso select cross reduce package coordinate reduce comparison package
measurement imbalance momentum direction particle number event black angle imbalance momentum angle observe projection little discriminate production production capture derive sum observe p p tp tp angular mass mass visible simulation first low level variable benchmark text neural
effort design progress current benchmark upon descriptor decade show arguably claim traditional hand make different discriminative robustness numerous complicated stage optimize module pre encode transformation careful individual module intensive ensure whole module individually present unified easy framework face basis convolutional neural pyramid face pixel
fx cx giving obtain since exploit ei replace readily familiar g analytic ei monte predictive equation simply average f j observed generally error suffice arise explore ei surface whole numerically whereby outer loop yield yield ei search improvement al composite cause little deep consider constraint slight composite removing case ei new composite notation ei substitution integration otherwise rearrange shrink besides point analytically ei idea avoid boost efficiency ei drop lead way carlo search know feasible lie drop large region towards boundary restrictive consideration modeling extend much constraint surrogate summing extreme former using could monotone include active discuss motivate implementation provide material
something wrong something effect seem isolated appear datum table nan test quality jump step trend behavior practitioner desirable smooth jump one smoothing bagging adopt choose large cumulative size bootstrap estimate size pt value median regard mean seem value step total view
definition lemma statement denote portion lemma choice next problem integer follow claim suppose begin follow z argument proof replace follow fix arbitrarily every three follow repeat throughout arbitrarily nonempty hold satisfy exist prove repeat argument utility establish cyclic arise context logistic regression estimation cyclic minimization subject symmetric negative number observe variate definite variate inverse minimize concentration model gain popularity particularly develop tackle setting necessarily inverse covariance provide cyclic minimization objective function example convergence guarantee however selection version
computation volume page world page page extraction author analysis international conference page organization title supervise principal journal journal american volume year supervise title principal via advance system page year extension author zhang yu reduce title reduce journal page american title feature scale le chen title motion motion author journal pattern intelligence transaction year title author journal page year title comparison
reasonable convolutional neural work architecture architecture gpu expect particular architecture sort restrict connectivity layer current might connect activation effective parallelism layer column dependent varied scale modern convolutional short new descent sgd present perfectly sgd approximation perfectly nonetheless neural way train parallelism parallelism parallelism
metropolis take keep gibbs dirichlet topic prior ip variable parameter text token stanford speech extract gram tag simple pattern phrase obtain vocabulary phrase summarize htp case token r phrase token brief phrase brief manually neither lexical identify section template classifier introduction statement conclusion evaluate classifier split tune set range limit evaluation posterior great obtain classify support side precision topic phrase control plan plan plan company plan health right control rational r country vice member limitation act security rule security exchange act act communication act vote vote political political free speech
independent define index I zero rearrange since claim
general family term surrogate option calibrate surrogate sensitive classification interest extend surrogate multiclass early work zhang multiclass framework multiclass use surrogate surrogate lee particular multiclass lee al calibrate multiclass multiclass surrogate calibrate multiclass study consistency involve space prominent problem consistency calibration contain model rank document relevance query losse recent year zhang subset discount cumulative simple calibrate et normalize ndcg consider focused subset disagreement pd popular surrogate r loss et show surrogate calibrate pd precision reciprocal subset calibrate surrogate loss standardize show ndcg loss standardize standardize finally consider al use surrogate regret certain surrogate multiclass loss et calibrate surrogate note develop consistency multiclass calibrate multiclass introduce measure certain algebraic geometric result existence calibrate surrogate type subset surrogate main conference paper
linearity upper need precisely go follow absolute eq gr adapt argument go end equivalently rewrite standard extend odd define uniformly balanced algorithm query achieve overall wrong exactly chernoff target fail stability approximate title corollary lem theorem prop conjecture example observation definition acknowledgement acknowledgement university institute technology cl l grant li monotone monotone boolean circuit paper boolean circuit
median maximal point base point adversarial outli always fraction straightforward see outli base output estimate machine deviation ground truth estimate point low bind number overall fraction outlier outli adversarial distribution make appealing average consider machine outlier outli fraction break aggregation severe robustness base outlier preserve besides favorable alternative randomly permutation uniformly take face arrive respective perform central often outli long randomization division average communication follow section
line sbm see appendix claim set succeed throughout learn maximization loop boundary phase instability dot black line overlap retrieval parameter sbm additional component group degree sbm regime find spin phase see appendix ground vary appear determine classic modularity exceed principled method network agree perfectly find modularity overlap grind google network common novel find retrieval state world google truth statistically height c social book political google appear community look structure work group
observe difference consequence interest future ga ga li li li ga respect rank determined procedure become ga ga li li li precisely response bring canonical inverting
suggest upon literature number constant state space domain e potential maintain dependent location stem e galaxy basic physics galaxy mass geometry galaxy constant universal isotropic wang respect angle invariant also recall isotropic remark thus motion dynamic isotropic isotropic attempt use ensure freedom keep ease space addition per cross vector q isotropic vector include isotropic aim model help demand space use ready express mass embed within cast bin min equation unknown rhs mass density plug rhs compute discuss domain space e invoke functional relationship angular momentum e v circular distinguished circular speed mutually speed parallel simplify thing coordinate rotation previous v refer
type unify similar intuitively naive pick near else construct neighbor convert pairwise trick query inspire boost coarse fine human human perform coarse pruning recognize distance nearest fine distance query sample query convert trick utilize global query htb aspect residual obtain dimensionality identity dimensionality rp b database randomly select per person person totally database comparison global focus dimensionality recognition person totally average residual rp graph see five approach achieve similar indicate reconstruction residual table identity rp graph recognition discrimination ability difference high rate pt identity rp htb direction gd gd dimension interval gaussian zero vary gd perfectly classify poorly gd datum case unknown conventional subsection evaluate public database reliable experiment htb contain digit experimental setting
bivariate employ sn employ indicate limit density diagram identify sign classical sn systematic log link another inferential simple possible case diagonal factorization translate product likelihood sn hence stationarity implication diagonal involve project qualitative front qualitatively family discuss match skew replace one student examine skew dimensional student result skew density degree version function univariate freedom version dimensional qualitatively correspond sn tail via
van ac uk inference desirable property uncertainty estimate way tune hyper scalability dataset remain variational inference efficient exploiting give induce formulate inference preserve load scaling amount variable mnist gps big show complicated fitting variable dimensionality use model big dataset limited scalability dataset natural ignore amount datum
pool list image easy sophisticated sift descriptor importantly magnitude sift feature map see superiority feature map map show average c sift feature focus object extent soft convolution sparse intersection large codebook atom complicated et type dataset performance purely level level mechanism therefore base task drive specific parameter model include ns layer also study ar database face accuracy sensitive stable range reveal bring performance gain vs neuron gender sufficient gender peak neuron say ns serve efficient mid hand propose produce argue much propose layer neuron mid feed support result perform order achieve comparable effectiveness propose gain
noise section visualize organization cluster encode hierarchy connect dendrogram compute sample recover topological study code generate construct evaluate library expand grid measure smoothed detail near among point follow knn kde kde h h visualize graphic package code kde plot col h band estimator
box core marginal polytope minimize project gradient computationally hard approximate backward argument constrain polytope evaluate polytope hard nevertheless knowledge polytope normally ellipsoid barrier polytope inherent keep inside polytope enforce constraint consequence property state proposition reduction approximate partition tractable polytope surrogate gradient entropy polytope relate polytope reduction hardness obtain polytope hardness marginal polytope manuscript learn independently high level differ establish hardness marginal specific core collection
replace take justification question arise might argue modification thing critical thing arise trust region classical trust region curvature constraint pick suitable minimum trust trust approximation answer discrepancy second expansion impose partial accept take curvature square would difficult solve difficulty lemma matrix measure upper generalize replace inequality minimum always jump trust multiplier solution constrain step cifar algorithm newton curvature information fundamentally shape curvature method unlike newton saddle unlike rapid even requires approach dimensional subspace iteration span
arbitrary boundary rule machine excess loss condition excess margin correspond convergence convergence risk excess constant eq generalize margin da provide limit neither depend explicitly tight may achieve motivating intuition formalize soft appropriate hinge loss corollary result surrogate linear condition use theorem show suffice minimizer loss size separately lipschitz modulus logistic separately surrogate section lipschitz loss modulus satisfy close radius cover b iy I I following make theorem provide bind consistent modulus state combine follow satisfie generalize constant obtain non loss surrogate modulus condition surrogate iy satisfy constant obtain piecewise theorem theorem rate
dirichlet topic length guarantee assumption test corpora recover align bipartite column accordance matching topic recovery across il il report recover kl mean document length result ng small sample recover show error topic dataset green quantitative measure algorithm lda word exp dm real hold consist document document dataset comparable moderately length coherence semantic quality approximate experience quality top word coherence iw evaluate tc top recover topic topic topic top match pair topic gibb recover
uniqueness freedom j j unbiased freedom agree p knot small provide freedom total knot fit replicate set freedom spam spam j j spam spam propose freedom compare replicate data freedom calculate axis overlap colored degree spam axis axis plot solid vary spam black dotted indicate consider solution sparse completely pg completely never corollary
filter feature appear set fair also compare f nf dataset state htb top top concern art precision increment alignment distant supervision generate incomplete label discuss tackle noisy truly effective information extremely poor gradually imply approach principal recover low furthermore discuss feature program show consistently
heart rate pair height weight heart height know cause heart age measure person biological body uci learn repository concern systematic regard compressive function addition coarse water making incorporate material chemical show water influence concrete compressive age measure per concrete pair pair scatter compressive pair compressive strength pair compressive compressive pair compressive coarse compressive pair compressive pair water aggregate aggregate compressive strength compressive strength simultaneously component water decrease measure cubic nevertheless uci repository medical thought arise consumption instance constitute record daily consumption number gamma available h pair scatter consumption consumption consumption daily consumption cause individual abuse drug lead mean reverse chemical whose concentration test daily consumption exclude evidence consumption pair consumption volume average consumption cell consumption cell great amount disease level disease highly disease consumption primarily rarely condition uci repository collect national institute diabetes disease usa diabetes risk criterion year age select instance nonzero seem encode yield cccc pair pair plot body pair pressure age body define height obviously mass age cause hour measure standard tolerance exclude cause age bias pressure pressure causal direction alternative age measure cause could way may weather locate include temperature height datum evaluation summarie weather plot day temperature day year temperature temperature month consist variable day year year year drop pm count human twice useful heuristic day cause angular position around true infeasible change incidence pair relation temperature year area day surface temperature average daily process interval assume cause air easy artificial enough decrease air temperature long also environment play daily average national research laboratory website observation weather date national centers environmental national realistic grid four cell air pair pressure pressure pair relative day day across north across cccc pair plot pair temperature surface pressure level pressure daily area km km propagate
select illustrate effective region choose bad anomaly pearson correlation coefficient possible performance strongly favor select effective model simulation time train train hundred implication possible necessary implication design rapid train student student select initial scoring quickly start score student receive feedback receive refine collection pool subsequently maximally effort ensure
return solution return incorporate rule see worker round round certain monotonicity regret minimum optimization greedy eliminate worker monotonicity avoid exploration elimination worker follow give approximate denote follow loss index ms mb I k l l k k l pick return allocation rule monotone worker task task cost element similarly big call cost worker decrease worker remain appear worker nothing change worker worker already remain worker worker never big element cost worker element follow change bid big worker become big big suitable set thus exists discard perfectly equal contain run meet big true candidate therefore candidate hence drop c r rule worker eliminate elimination se confidence bind elimination se transform mechanism compare efficacy propose simulation solve greedy compare four ns solve explore
token map list representation turn rather token sample sequential token classic search index bag word worker record eliminate addition sparsity first reason great efficiency change within incremental document frequently word every word e maintain token fractional along make note dense fractional mass partition however state require disadvantage sampling parallelism salient parallel yahoo available notable similar token throughput yahoo lda roughly yahoo lda per compute core per medium machine sample throughput yahoo yahoo show throughput yahoo converge fast per careful synchronization fail
function minimizer point essentially manifold function manifold function perturbation partly indicator constrain refer constrain convex compact follow definite imply bound set equivalently state let accumulation k classical whose solution constrain partly smooth nearby soon enough fact understand operator imply enough partly smooth partly
hold need pick choose linear map strongly specifically hold pick next give sufficient admm existence boundedness proximal admm exist either sequence generate ii l x plug hold boundedness boundedness third next boundedness boundedness finally boundedness relation complete proximal admm choose theorem shall follow linear include hold sequence generate admm bound conclusion hold end recall comment condition shall fairly large twice differentiable twice lipschitz modulus reader inequality modulus differentiable whole generate admm proximal admm algebraic rely kl kl property analysis like
technology division contract term inequality decomposition extension apply derivation substitute side q eq apply eq difference involve line solution single e surrogate decrease adjust trust update newton trust newton consider within trust poor trust region leave axis decrease huber penalty approach dominate reweighte image ray specification ms filter strongly filter continuous create spectrum fill divide integrate computed accounting detector detector pixel angle resolution david convergent alternate image reconstruction transmission extend determination law object voxel additional latent voxel importantly image comparable penalize exploit parallel line search voxel considerably smooth penalty employ allow positivity inherent demonstrate algorithm ray compute minimization automatic determination transmission ray ct integral along ill pose sense solution consistent desirable measure represent inside well know example ray must increase physical transform filter back analytic formula incorporate mention prominent type iterative latter incorporation knowledge
angle begin interpret generalization pca ultimately nonetheless emphasize employ entry entirely underlying belief validity unlike exist turn theoretical condition whereby substitution nuclear nonetheless guarantee produce rank dimension nuclear limitation fold measurement force certainly importantly provable recovery place strong restriction singular ever hold realistic condition check theoretically measurement violate guarantee support empirically distributional adopt enforce next denote aggregate definite symmetric define accommodate case follow convenience abuse notation come negative wise denote necessarily per se primarily desirable give define operator expression clear overall require common bayesian regard likelihood respect definite transformation standard convolution integration minimize employ standard bound em bound simplicity experiment herein currently explore add independent progress even low obtain optimal closed form optimize dependent iteratively importantly
independent essentially discretization invariant posterior forward model mesh certain regularity discretization organized introduce informed reduction approximation present construct likelihood inform pde inverse efficiency property discretization invariance technique conclude overview random appear probability product describe commonly additive data section posterior prior advantageous evaluation expensive pose equivalently compact decay thus project onto covariance good involve balance decomposition define together yield optimal minimize positive covariance linear reader detailed range inverse basis maximize direction informative represent mode parameter space output insensitive particularly inform conversely smooth variance relatively inform case direction
state assumption long represent processor section encode development probability associate indicate otherwise write hadamard hadamard matrix notation identity elementary give one semidefinite identity particular matrix elementary element right hand side follow already rely play identity identity fundamental identity identity far illustration identity hadamard linear operation h follow hold set I deduce formulae restriction combination formalize probability scalar j matrix intersection equal
structural result low coherence fast solve massive problem rely building block narrow shorter examine avoid entirely state proportional unfortunately leverage score regression place simply random work could contain row remove preserve product drop include mix avoid storage runtime elegant sampling score need row mind straightforward sampling cm label sample small row repeat reduce small previously prove small come routine analyze ultimately require mix something avoid second possibly condition rely primitive possibly
important approach consist transform couple item transform implementation item relevant link strictly provide generally primarily aim rank stress relative state improve scalable link graph call ranking close prediction ranking supervise network section article social node kind interaction phone dedicated metric evaluate unsupervise consideration improve aggregate unsupervised implement selection parameter suit investigate record european phone service month interaction user phone filter social link activate interaction phone link call paper
design participant participant per assume per year combine population duration trial design rate equal reflect reality likely slow assume implie amount regardless take amount item per n sc participant stage stage ss participant item boundary constant calculate boundary boundary design calculate section design boundary sc item design ss boundaries h ss treatment range treatment set effectively basic great treatment set output panel right design section software link full design sc ss ad efficacy boundary participant early duration design input design design fourth together adaptive number participant stage ad break stage table efficacy k ad stop boundary k stage inf efficacy construction see research continue stop plot page display efficacy boundary stage design sc
master order frequency controller automatically synchronization structure fig structure add mechanism enable master master neuron master pass output master neuron inactive cutting connection master pass consequence equation detail frequency automatically combination frequency neuron satisfy activity synchronization neuron active inactive master note enable adapt period anneal feasible suitable discrete search robot record direction step angle last initial angle deviation fig green forward angle otherwise occur automatically synchronization stochastically process value combination record new period variation combination period depict calculated deviation rate indicate trial deviation select period assign compare particular period assign angle measure stop loop
throughput sequence tumor broadly simple consist copy variation widely throughput sequence technology generate read rarely variant partially tumor reconstruct evolutionary history tumor evolution tree assign node representation tree chinese restaurant assign leaf knowledge assign e pre specification apply stick breaking
convergent admm inexact fitting term method admm admm penalty select verify fit two admm suboptimal organize method inexact concrete mathematically show convergence two regularize discussion admm situation support inexact split admm multiplier al stack equality compactly
arm arbitrary reward let decrease purpose epoch q impose non maker evolution form expect reward continuously change variation correspond budget ht continuous spend first measurable abuse notation action mapping together class policy past history difference epoch oracle performance policy take worst guarantee policy quantity establish refer constant regret formulation reward adversarial manner budget lower achievable performance assume reward bernoulli reward stationary well least embed stochastic non mab stationary increase variation budget regret run optimality imply regret reward evolve brownian motion achievable relative stationary
discuss idea extend overcome drawback achieve tradeoff among orthogonality balance loading truncation research encounter signal process computer vision etc area pca approximate represent orthonormal loading represent component project product loading loading decrease lead maximal variance lie mainly vary direction loading enough obtain achieve physical financial biological specific asset dense principal make difficult loading zero principal combination principal component far physical loading dimension aim sparse basis interpretable represent tradeoff statistical fidelity interpretability past decade method consider tradeoff sparsity three yet orthogonality loading loading global advantage loading orthogonal pca satisfy orthogonality highly indicate mean loading loading orthogonality support
parameter pc window bit core processor set calculate classification gain popularity area open resemble basically
example convolution periodic reflect condition extend development assume list orient respective sum frobenius norm ff group horizontal slice norm like problem tend inside enough driven hope tensor size orient seek point ideally matrix efficient generating build tensor problem give algorithm whole translate give I chance intuition zero multiplication fourier equivalent solve multiplication next build norm affinity
dt bias rmse rmse sl sl sl dt sl bias rmse logistic nb nb regression spatially aggregated count cc cc sl dt sl sl dt bias sd rmse c sl dt sl sl dt sd cc cc sl sl dt sl dt sl dt bias rmse sl dt sl sl dt sl dt bias rmse simulation study lattice spatially aggregated approach bias standard sd mean scale sl car nb nb dt skew skewed calculate informative bias conclusion follow sl method bias dt almost good sl discretization univariate
summary frequency estimation little intervention chemical presence absence transform robustness low noise peak analytic acknowledgement author thank microsoft connection grant dark microsoft nuclear exploit atomic chemical environment model weight function free decay challenge estimate general model decay noise offer practical solution conventional using experimentally acquire robust low snr overlap peak enable snr conventional nuclear advanced understand molecular analytical nuclear behave spin axis generate later use share develop comprehensive lead study protein specie conventional lead ever
start consider innovation component unit thus kernel entry fig show reconstruction different observe verify phase transition signal particular section use mean gaussian reconstruction conditional ik ik ik ik ik ik ik ik ik notation reconstruction side without decoder notice match numerical ht section compressive sense resolution pixel signal vector patch picture represent patch extract divide portion section e test ik ik moreover extract patch notice introduce substantial distortion relative value side train covariance imply almost extract image reconstruct test image feature represent image input side image I reconstruction number linear also reconstruction phase transition obtain match mathematical phase occur regime namely develop significant approximately natural exactly matrix term account component way extraction full covariance mathematically equivalent low covariance noise zero entry zero unit noise variance correspond noise equal offer principled effectively task error compressive hyperspectral imaging example image subject presence measurement collect snapshot scene easily obtain expensive hyperspectral device rgb improve camera image code represent patch hyperspectral image system analyze vector hyperspectral image perfectly characterize run time close camera ht htbp side c region real camera projection implementation imaging belong side image compress single meaning
plot intuition dropout expect dropout plus weight mark minimizer dropout criterion generalize perfectly distribution strongly separate section whenever dropout separate define optimal classifier denote region green section figure regularization bayes strongly dropout regularization predictor dropout create predictor error number draw otherwise uniformly achieve despite first vote correlate ensemble discrimination feature put bayes placing accuracy find optimal dropout even conjecture source generate equally accurate optimal predict small require dropout make enough even p number distribution start theorem call strongly interpretation regularizer interesting dropout regularizer dropout minimize plus property dropout penalty verify dropout penalty dropout behave dropout loss plus convex weight penalty training weight vector go
row fourier autocorrelation fig sparse use autocorrelation slightly support plain versus phase approach linearize convex real case circular shift retrieval problem break derive invariance another greedy algorithm group valuable universit france electrical engineering sciences university california berkeley usa electrical engineering retrieval measurement nonzero relaxation norm result circular shift complex recover typical
hyper principled complicated conjugacy lose similarly certain actor logistic normal additional inferential unclear class sbm dataset potentially space may fundamentally assess hold employ possibility primary view equally know give general ij vice versa conditionally independent presence diagram since calculate ij ij ij I similarly multinomial since give parametric e ij log intractable low ij ig ng estimate newton step iteratively simpler note ip np np np w nr np nr little nr np j nr np np actor manner due political may
applicability second semi label nb theoretically decrease nb document presentation community plus add iteration ideal learning third trying minimize competition former determine fitness conduct facebook dataset community exclude finish combination network display training year bar network probe item perform item find community year attribute large true attribute reason see suitably year attribute suitable attribute also explain item network infer behavior item combine multiple good discover community worth community item k expand method hierarchical agglomerative agglomerative cluster partition explore collective edge algorithm merely cluster second community community reason partly poor
I valid produce valid singleton belief plausibility plausibility specifically plausibility simplify plausibility frequentist coverage consequence variation challenging framework singleton plausibility plot plausibility fairly true plausibility interval particularly plausibility seem plausibility I plausibility region I around magnitude I arguably give method probabilistic fundamentally experience scientific especially live failure probabilistic argument
arrive q two satisfie e dy mx combine properly choose fix due enough take eq straightforwardly study q integral decompose decomposition q arc proof natural hence desire fix denote n sequence k intervals v v constant z pn
heuristic approach large class variety model partially observable stochastic refer usually usually begin usually mean mmse state greatly measurement nonlinear filter possess recursive filter state gauss representation desire sense chain follow comprise almost compactly state measurement constitute vary known employ change appropriate summarize ii result converge define sense compact probability unity nevertheless deal almost continuous result time fact counterpart treat mode stochastic convergence compare regard approximation paper heuristic approximate recursive approach wide variety limit recursive sufficient condition operator highlight
contrast factorization furthermore motivated problem entirely topic numerous represent often meanwhile rank form able inherent help interpretation intensity amplitude count value nmf lee parts
extremely reverse lstm ask proximity fairly fact easy sgd output transformation greatly lstm method english mt translate input list translation visualize sentence representation english dataset train consist english choose specific subset public availability typical language frequent target every vocabulary special involve maximize complete translation search translation decoder maintain hypothesis discard soon remove add hypothesis decoder
close linear ref table first calculate approximation increase reach become negligible respective value cc cc exact curve normal mean variance eqs nucleotide incorporation fix cycle agreement exact skew left complete incorporation flow cycle nucleotide composition probability nucleotide delay synthesis incomplete incorporation sequence leave work school technology next sequencing sequence length flow cycle nucleotide probabilistic incomplete sequencing sequence generation sequencing generate huge amount
velocity discriminate galaxy datum several optimal obtain combined htb galaxy dataset dataset hyperparameter group test paper mixture complete respect rather depend hyperparameter sensitive choice sense use update however informative hyperparameter prior frequentist cluster configuration obtain little advantage categorical situation posterior optimization routine stem iterate extend present improve adapt context greedy property negligible mixture special consider avoid dependence hyperparameter hyperparameter differently possibility exact formula several framework real various simulate meaningful however strong solution indeed finite mixture pure force use informative
ordinal logistic job research innovation use useful datum international economic operation development institute statistic become datum simultaneous data matrix reduce dimension row individual column measure method principal factor fa combination regression essentially square algorithm normal exponential family describe never never alternate perform iterative extend sparse none logistic procedure matrix score calculate external procedure successfully nominal region nominal variable nominal adequate categorical principal item response ordinal row along
c bring picture consistent half inconsistent correlation function satisfy correlation fc measure either code able force inconsistent one interaction code good need show sum concentrate standard unable apply instead proof verify concentration moment generate prove concentration novel lemma dropping cauchy k k three random suppose obtain score large correlation arbitrary deterministic may able round power round receive apply firstly however prove score score nj ii thus bound imply completeness suppose sake n eq q contradiction thus rearrange equation give substitute eq q contradiction construction analysis interactive code interactive code robust fraction failure false pair construction analysis interactive interactive code interactive user fraction false answer adaptive working answer statistical query choose adaptively hold value answer iid answer moreover choose formally ht dx
decade view currently consistency graph model estimation meet heavily furthermore standard especially validation classical criterion aic bic stability different
observable active develop shaped stochastic understanding variety e continue et et publish parameter formula r specifie machine stimulus possibly internal external source stepsize machine additional call assume identically observe stimulus adaptive specifie optimal behavior learn value training stimulus x x absolutely notation sum integral stimulus discrete absolutely expect loss equivalently choose parameter characterize paper consider machine stimulus
cloud dimension algorithm depth proper variant tie prove keyword combinatorial complement suggest propose centrality cloud call depth location motivated notion decade depth depth cloud point assigns degree centrality later depth notion possess affine invariance monotonicity upper finite point lie
technique procedure guarantee normality condition write term brevity influence derive influence function py q derive alternatively definition construct density function sample denote without taylor precisely remainder segment first term cross g mention boundedness density give need condition formally normal finally construct analysis form estimate estimate estimate estimator proof corollary normality begin influence q jensen last boundedness density function boundedness fourth compare shannon follow fourth fig dimension dimensional comparison shannon entropy partitioning power representation hellinger rather wish problem category sample
serve naive normality necessary high grow definite grow function completely member covariance matrix covariance discuss asymptotic sometimes sufficiently ratio eigenvalue consider show asymptotically critical mostly goal turn attention parallel loss estimation frobenius parallel begin interpret tend clear part asymptotically dominate might convenience pick eigenvalue property specific alternatively result could correction look simultaneously asymptotically form thought correction discussion regularity strong statement appear
assume particle equal particle time identical relatively element brownian motion model diagonal sample speed accommodate object module appearance conceptual tuple origin basis extract consider top corner locate appearance affine track specifically result track time bag use appearance particle specifically obtain singular choose dominant affine object along able reflect appearance change accordingly propose keep model object track bag model use
member department electrical engineering national university vision research group dr areas vision hundred paper range topic google citation index associate transaction receive paper winner task winner winner mention associate award research award award award minimization involve smooth reweighted updating solve affine mixed minimization essential formulation concrete norm regularize theoretically previous one depend
multivariate mm department york mail current framework measure risk involve risk cf frequently model copula tool financial cf reference cf g two risk leave rise upper concerned phenomena unify copula survival index dependence survival copula duality dependence low tail al reference restrict consideration behaviour copula instead
report environmental service md pp mm weather small poisson b c york journal p stochastic day axiom theorem conclusion theorem conjecture example exercise summary section test study case fitting researcher generally central chi check validity asymptotic chi square nan variance goodness spirit example refer
basis window apply search optimize correct frame involve update order efficient practice computation code description stream detector seed detector frame initial detector start negative pool alg case rank approach publicly benchmark long video object wang independent unsupervised set scenario detector adapt stream compare wang art video also rely train detector confident positive wang consist ranking scoring
keep around one small subsample much replacement si usual si assign inclusion illustrate variability variance fraction impossible scheme explain si design represent equal si extremely contribution contribution much include replacement equation suppose sampling probability constant sampling weight approximately evident likelihood way replacement ps obtain deviation vs ps marginally efficient p many si example probability generally impossible construct model construct weight normal proxy goal time common computationally costly taylor aspect subsampling refer consume aspect numerically quadrature
svm weight specific express base lagrange assign support stack generalization assign performance correspond combine meta binary multiclass environmental narrow approximated similar band white noise noise precede front full band adaptation feature account vary contamination colored normalization method additive noise furthermore stack generalization tune classification similar distortion range instance meta train score score mixture clean stack limited amount require optimal meta stack offer individual problem useful major counterpart front si diverse sx compact sentence core test sentence consist sentence train test set meta stop certain group class fu lee classifiers multiclass initially decide use overhead impact degree slack training sentence energy add sentence hence level snr ratio
estimation conclude statistic perfectly formal mapping jointly statistically independent dependence extent know standard relationship note imagine scenario reason exposition back motivate pdf horizontal interpretable interval colored sampling reliable interval straightforward restrict attention ask much let close interval proxy proxy exists size attention guarantee produce illustration correspond differently closely distribution measure dependence close contain interpretable proxy interpretable know guess relationship noise omit reliable interpretable reliability interpretability resp diameter resp reliability resp interpretability call approximate distinguish perfect imagine way reliability dependence dependence interpretable resp interpretable resp reliable interpretable reliability resp imagine belief various interpretability reliability interpretability give terminology correlation perfectly proxy bivariate perfectly interpretable perfectly proxy simply restrict perfect second sufficient interpretability equitability amount measure suitably reflect particular might equitability measure case versus model state precisely term relationship call noisy write functional placed gaussian way arbitrary necessarily functional property equitability relationship noisy relationship measure resp average proxy equitability uniform instead evenly clarity opt produce analogously also distribute difficult say follow ideally equitability behave model somewhat narrow might add coordinate modification paper equitability interpretability differently formal
annotate however si design inferior abundance correct biased account variance size third hybrid design achieve error wide array si htbp detail cost hybrid cover abundance perform structure variance commonly annotation investigate mining determine hybrid second simulation assumption applicability quantify monitoring survey collect survey site year image hour annotate minute annotation sampling identify randomly annotation use percent
delta period worth note refer eq importantly likelihood concave maxima maximum sp spike spike ascent g stimulus fit roughly spike generalize intensity notational simplicity q stimulus though stimulus number
discovery cell material thereby address energy generalize extend prior specification combinatorial specify priori include traditional negativity dependency come chemical system factorization call program show outperform art material discovery scale large knowledge direction namely concern formalism combinatorial science nsf award grant grant nsf nsf energy department office office energy sciences award research energy nsf national national institute award joint center innovation office energy de stanford national laboratory support u office science office
error square dominate gain descent optimize gain quantify limit quantify asymptotically expansion let matrix multiplication notation pair drop implicitly eq distinction operate usual application sense symmetric large stable take frobenius look precisely left operator denote operator denote vector setup small eigenvalue exact moment bias would model lead contribution tr extra supplementary material specify actual matrix term generalization eigenvalue supremum supremum necessarily follow proof definite dimension
compare participant randomize switch decision covariate feature gender status history care feature intermediate medical degree burden consider covariate treatment decision quick rate final record covariate outcome switch randomly assign participant treat score select cut reason include good response reasonable categorical information response lot indicate potentially include moderate important covariate decision regime patient treatment among variable answer diagnostic screening sr score change level patient rate rest select covariate baseline relate patient give treatment decision comprehensive patient situation baseline examine optimal regime ig ia ia method get subject treat regime
tune stage function open gram precision translation mean estimate softmax deeply hide softmax figure detail extend purpose major layer softmax costly solution short list class adaptation scheme equally adaptation scheme back selection model practice mean adaptation process train several g generic usually take substantial hour combine translation adaptation train new method adaptation outline weight update
achieves general assume define return finally achieve begin guarantee assume cluster exponent exponent nd n similarly establish simple choice recover show candidate independent exponent furthermore fix constant addition constant finally recover rate exponent unfortunately simple selection correspond unclear strategy component present select proof fx nx integer hx fx measurable integrable lebesgue absolutely thus former lebesgue monotonicity show case persistence iv imply c part lemma hand connect prop db find return notation return become write v v ia w b find repeat assertion small let check remain conclude assumption check obviously exponent exponent
cluster research unify hardware cluster area innovation valuable capability capacity template question investigate school mathematic american center xu acknowledge technology additionally national foundation support
part gradient sufficient straightforward implement memory computer distribute key store hash worker support entry server store resolve access update suboptimal address max tradeoff made return immediately server fast worker get fast server propose reduce improvement include filter support add remove server learn topic lda million token yahoo lda server specifically bayesian fast software model yahoo correlated topic parameter partition machine computation copy globally size large challenge model address challenge partition store part e update part carry parallelism flexibility worker fast converge delayed parallelism storage automatically compute together user worker apply automatic primitive user partition vocabulary assign full cycle complete data lda use read fast parallelism replacement complement parallelism show parallelism consist several parallelism asynchronous distribute linear algebra library intel mkl could library matlab server basic algebra vector matrix add multiplication algebra package include equation factorization implementation intel mkl implementation aware cache aware also claim always find level library build upon efficient architecture algorithm effectively linear easily base support choose toolbox topic model hierarchical review include boost bag linear combination tree iteratively appropriate prior tailor recent advance massive control important factor parsimonious regularize estimate truly optimize averaged take inferring include spike shrinkage heavily tail prior e laplace reader selection aim expensive planning recommendation big become huge tuning prohibitive progress method select multi core stochastic method space
add table cut complementary available htb cccc lemma number add lemma improve come complementary proof lemma theory newly improve effect however since try high clear prediction category consequence vector fact add close theorem work lambda function eliminate try find corpus idea logical dependency atomic element conceptual organization ai topic semantic corpus serve automate reasoning core automate develop improvement due introduce evaluation proof exactly measure contribution ai whole corpus work make incremental quality method style detailed level inference graph intermediate prove hard run help ji many discussion extract par lead library consist million step give rise statement lemma analogously mathematical statement formal criterion usefulness furth criterion graph library add good automate mathematical library intelligence decade corpora mml
tree project leaf compute result exploit standard make reach leaf retrieve sample see appendix b material split score approximate project subsample split construction balance small generate significantly original grow may without predictive idea exploit context building project grow pair jx jx exploit projection kind diversity among share output projection output projection output combine randomization grow ensemble tree already extra discuss empirically look single small suggest variance subsection jx jx n ensemble computational difference output projection ensemble tree
patient treatment day day another treatment treatment use day patient base addition year survival estimate regime simple treatment estimate regime two confidence interval bootstrap base run bootstrap confidence stay compare year indicate optimal treatment regime improve survival regime iii interval approximation stay contain conclusion bootstrap confidence propose various estimator follow treatment smoothing regime regime year survival work treatment decision generalize decision define multiple year clinical treatment regime take du restrict survival regime treatment regime interesting topic investigation establish regularity recall work assume parameter proportional model survival tw nu specify satisfy
encoder encoder intermediate obtain rnn recurrent auto encoder auto mapping decoder latent stochastic introduce year similarity auto rnn partly
excess interested derive sure result exist surely almost surely fairly output satisfied omit clarity know integrable xx general depend well linearly quantify r w define g minimizer nonempty
exploit indexing multiplication fully leverage engine multiplication complete indexing require division modulus division modulus operation shift reduce indexing convolution auxiliary require convnet implementation layer configuration average throughput convnet importantly show well l layer layer illustrate architecture build architecture peak precision throughput build use architecture peak single throughput range peak provide gpu gpu deep
description vc depend vc dimension rademacher straightforwardly dataset bind error erm intend large small say generalization since erm limit collection radial create placing limit magnitude function treat control equation must solve equation choose turn bound policy performance decision tuple primarily done facilitate analysis equivalent rl maximum episode without everywhere straightforwardly v reward
cp cp bs bs nan ccccc ccccc cp sd bs sd bs ccccc ccccc cp sd bs sd nan choice ccccc ccccc cp cp sd bs sd bs compare sd table show sd large comparable sd worse compare sd test dispersion diagonal favorable sd show large turn project search large benchmark power describe power power
decomposition multiplicative literature machine investigate propose conventional special kernel wise division hadamard nmf worth identity half polynomial kernel balance impact high common polynomial update rule gaussian kernel multiplicative nonnegative since q division wise nmf include follow impose typically motivate different impose improve estimate turn constrain function interested solution variation less exploit smoothness minimize space term cost get reconstruction smoothness estimate yield additive method multiplicative turn unconstraine denominator nmf multiplicative rule unconstraine add denominator easy
l good fit htbp parameter aic statistic e pl customer service bank et data observe exponential good fit among consider result distribution figure htbp r pl stress play important
duality equal orthogonal support word imply bind approximation indeed solution duality gap notion value value space contain satisfy eq version unfortunately nuclear calculation subdifferential nuclear giving give yet
incoherence allow vanishing go infinity applicable likelihood define incoherence two structural si relation rsc restrict fisher structural si guarantee si marginal satisfied lemma rsc si prove remainder negative gs ks define next family partition derivative show author q coincide norm definition relation hold due rsc precision inequality omit derivation incoherence si condition remainder incoherence due verify definition curvature theorem statement error sample bind bind eq verify guarantee specifically require bound first wise choosing
eq state inequality matrix help c eq lemma perturb ab consider span orthogonal contradiction omit ik positive semidefinite ta b I run time reduce albeit pre modify complexity pair nz py pz tie arbitrarily reduce half pair replace run winner sketch df complexity complexity add happen close therefore run union generalize spherical covariance along kl divergence spherical distance mean consider consider subset code distribution subset differ coordinate show first
role event possible enforce ensemble take annotate use annotate plus none case q paper score classifier suggest weight explore arc perform composition triplet link
traffic management newly specification fit value derive error assess meaningful furthermore provide tight probabilistic reach avoid fit iteration approximation assess process reach avoid concern evolve endowed control investigate theory quantify dynamical known reach avoid specification formal verification algorithm advanced process endow past evolve deal theoretical add markov interest endow state work engineering life science controller synthesis selection order maximize figure go investigate system theory deal tracking focus specification know formal verification field verification deal simple chain development computationally deal task synthesis instead specification model continuous endowed avoid deal likelihood within horizon trajectory avoid equivalently property express goal state reach invariance specification core modal verification temporal computational controller becomes either allow reach path
particle ess tune I na j nm w bring two extra make sample quantity smc sampler use gold simulation slow turn ep framework derive target distribution constant pseudo rewrite ep generate case approximate un lead type exponential family see update site site
along preference ordering tell quickly word budget large allocation lp lp cp price established take maximize bundle modify maximize undesirable modify price increase price still good either thus price derive bundle despite xu c fractional instance time xu np xu cr ir n order budget uniquely remain unless budget know bundle remain minimize remain min wherein bundle first induce slightly show like bundle rather bundle order desire cause item prefer item price preferred price fully fractional fractional price
outperform outperform remain technique rate angle gap still fast fast dynamic adaptation reach angle gap dynamic topology distribute grid dynamic topology adaptation employ result mse
formula power derivation yield power explicit component bivariate role use information nan positively substantially biology economic finance thousand hypothesis investigate asymptotic theorem present parametric long bivariate normality maximum applicable theoretical lemma appendix prove almost surely reveal propose conservative incorporating drive asymptotically f ft pointwise sequence integer carry simulation study aspect control various bivariate unless otherwise simply proposition replication compare procedure dynamically select scheme determine preliminary multiple filter proportion hypothesis share spirit serve example come mean normal right sided testing versus hypothesis test bivariate nan overall notational convenience rt value nonparametric role value e respectively case control conventional alone affect power thus combine detection significantly compare conventional procedure scenario take value compare optimal estimator close except phenomenon bivariate choice provide control combination panel
individual never population time individual correctly desire discrete prior heterogeneous link formulate probit normal cumulative individual latent version share similarity recent probit detect specie level effect probit behavioral w denote capture give record h assign close population abundance modify effect remove effect binary basis propose sampling distribution enable gibbs include feasible readily available assume frequency correspond zero frequency create none history individual history full tn update one update conditional update bernoulli full hasting integer discrete distribution integer tuning frequency
without component consider thing namely find unimodal elliptical away core clustering estimation researcher mind core look implication outlier truly belong degree freedom core identification whether separate one interpret consider artificial dataset distinct point start cluster generate red generate region mass reconstruct membership region figure reference expect measure imply generate although non base robust define noise definition component assign cluster outlier mean assign classification cluster necessarily cluster really gaussian covariance determinant scatter existence wide measure correct achieve equal covariance proportion could assign set define th ellipsoid region define union membership g membership need outli outlier
european please project small project life easy boundedness signal local reference title signal j appendix r f manifold vice versa dictionary b establish consider parameter basic proposition f hold identify assumption imply kp kp show also lemma remain leverage yield kp kp use need h eq signal surely lipschitz frobenius metric bind appear equation column hence far consider condition program existence uniqueness invertible algebraic plug along span approximation exploit conclude unique lemma quantity exploit desire observe surely shorthand r fm apply apply triangle j j j appendix technical
gene responsible rna seq whole four bl j treat rna seq end bp rna seq read supplementary material experiment differential usage bl seq read genome fdr identify gene differential usage differential gene annotation respectively gain insight gene list category gene annotation significantly associate relevant projection supplementary figure imply rather bl list identify annotation functional imply read depth without annotation manual recommend sample one one evaluate comparison annotation functional category gene identify supplementary gene identify target study involve could effect disease potentially candidate treatment well rna two incorporate snps genome rna seq separately genetic variant annotate rna seq identify much less gene bl fdr gene gene identify six involve bind great bl follow behavioral phenotype bl bl might induce allele usage rna seq allele map allele seq genome wide allele seq supplementary allele specific rna seq assess usage cutoff differential treat
category comment later order iterate given order detail need translate definition acyclic graph income e formalize consist ordinary edge incoming set together map incoming acyclic acyclic definition direct acyclic graph equip type understood tensor assign graph diagram tensor way assign satisfy contain node equip union carry intuitively place diagram interpret value requirement refer composition order carry assign coincide half edge compose obvious composite carry armed label joint evaluate diagram number correlation generalize see regard claim circuit thick minimum leave quantum application partial start follow use contain edge lie boundary whose operation need marginalization operation take new induced apply result coincide claim simply likewise put guarantee induction gives depict show far follow linearity interpret hand side diagram object label leave side latter exactly element iw ix e go composite system figure regard composition part label coincide rule tell income rule diagram coincide take complete rectangle thick draw fill minimum mm thick black minimum e rectangle mm e e smooth first compute value empty binary correlation equation subgraph disjoint operation choose still independent choice modify change permutation nothing exchange neighboring node among among correlation original new respect original order swap thank
function separately function derivative derivative concern provide derivative derivative mean function integrate reconstruct addition even linear find gauss prove carry precisely ols reconstruct variance unbiased compute explicitly dimension nevertheless practitioner represent expansion truncate ol blue moreover ol practically projection consequence rigorous generalization present advantage
n constrain satisfy constraint ball j project generate project descent gradient burden per size stochastic project stochastic compute involve suitable f slow key improve project inspire reduce strong efficiently solve estimate estimation construct reduce refer algorithm lemma technical contribution require gradient choose rate initialize mi
observation able reasonably anonymous constructive suggestion improve acknowledge financial research model production provide procedure auxiliary k k problem interest pdf instant kp recursive probabilistic recursively conditional condition incorporate evolve obtain time instant consist pdf pdfs conditional pdfs eqs obtain criterion maximum lead derive filter largely integral eqs intractable approximation situation approximate posterior pdfs kf extension extended sigma filter ensemble enkf assimilation gmm bank nonlinear g nonlinear kalman current work adopt prior particle filter pf applicable assimilation problem th notational convenience start particle instant together filter incoming particle remain unchanged update
case twice average return hold sec sequence minimize risk online batch datum batch obtain bind become solution immediate result average hence rate zero infimum advantage ball always term exactly price pay know advance binary lemma also misclassification risk trick suboptimal reduce applicability show tune assume guarantee otherwise regime sample big see misclassification worth adversarial small deviation theoretical
grid gradually modify frequency iterative refinement maximization method typically carry guarantee global optimality dense sufficiently variation discrete atomic version dense noiseless frequency recover atomic norm stochastic presence noiseless study atomic exact prove atomic convex programming include incomplete atomic measurement encounter method line estimation demonstrate base note method incomplete exist atomic norm possibly theoretical result miss spectral framework estimation consist improved estimation explore atomic
development try build evolve process see upon stick hmms emission resort strategy model evolve family rise dimensional value hide simple infinitely evolve continuous infinite distribution give signal space evolve former whereas latter evolve support treat evolve doubly intensity specification process control correlation infinite model distribution intensity bayesian statistic application popularity bayesian tractable family consider filtering
letter column letter g entry utilize norm dr large singular nuclear moreover entry failure mt theorem recall well establish lack parameter argument sphere union
thick cm circle pt look generality c looking contribution follow heat nearly heat heat kernel laplacian heat heat formally heat define similar form relate geometric continuous walk refer heat heat embed vertex heat embed eq q embed point heat embed show range definition heat q proof heat approximate condition contribution coordinate give contribution remark heat weighted reason heat embed spectral eigenvector linear algorithm approximate nx matrix nonzero algorithm compute correspond output follow almost
smooth beneficial surrogate motivate develop binary take main show condition smooth smoothed hinge well use hinge convex r ff b excess loss f learn empirical source error bound excess since error emphasize surrogate complexity affect excess bounded excess error affect aim investigate small parameter result approximation hand small understanding
pick school friend explain pick school would remain school small prefer school explain school represent friend trade explanation little must characteristic network facebook suggest mutually exclusive setting user membership keyword page membership college college aid college easily create node create member infer label explain college influence college college membership feature next first membership age difference infer handle modify modification previously intuition support type label graph run consist user profile current city school college five friend lift respect increase recall
simultaneously relationship field make physical model physical system specify inferential bias gp simulator output nature pair inference allow source contribute gp predictor know desirable burden datum burden framework calibration consequently methodology moderate large number computer increasingly experiment accommodate also recognize exploit relative fitting simulator argue argue mesh acting quickly describe application goal detail section canonical calibration limitation include gp design meet goal project together modern demonstrate proof concept variation discuss limitation motivate equip conclude brief discussion section team interested study matter complex evolutionary arise phenomenon super temperature simulator novel setting experiment disk front end wave energy material b physical observation pre experimental input list range field column three specify disk fill pressure image geometry perform circular
slope expression affect way example consider conclusion display cumulative correctly clearly difference axis switch simulated dataset display datum reject surprising might consider involve discrepancy highlight lr interesting hypothesis test classical parameter space characterization frequentist way test ie decided reject pearson derive bayesian pearson lemma proposition symmetry break adopting paradigm reciprocal alarm detection probability measure exist proper conditioning eventually depend type way error add frequentist integral underlying maximize lr equation pd x x contrary frequentist hypothesis pearson deterministic stand vs composite compare interpretable couple compare unlike bf improper unlike crucial fundamental natural alternative bf many aspect likelihood whereas bf compare likelihood composite
gibbs sampling exhibit slow tractable although approximation exponentially apply introduce multinomial reasonable apply raise maintain dependency critical count question prove asymptotically show correctly able preserve expectation highlight property property original inference arise propagation ep allow speed evaluate define set random variable assume domain configuration second show exponential set clique marginal node marginal rely existence relevant henceforth add edge
solve fix update standard lagrangian duality term region locally normalize normalize constant message essentially variant depend experiment use message region et al greedy sufficient converge traditionally conditional map feature margin easy conditional generalize way rather margin take secondly vector learn aside
transform interpretation generalise divergence begin treat poisson outline countable subset integral may course interpret sum distribution equal domain follow state treat expand preserved likelihood iid observation write estimate introduce follow alternative additive likelihood intensity estimation equivalent remark along incur treat maxima interval confidence obtain poisson addition maxima family even preserve yet compute integral possibility approximate via noise divergence iid
laplace distribution c logistic laplace pareto mm consider instead score function regressor generate table empirical variance base laplace pareto van simulation consider regressor measurement quite similar previous confirm relatively finite fisher r square estimate heavy acknowledgement grant research nsf dms grant support section remark section recently regressor
minimizer pseudo likelihood concern definition consider familiar cover argument uniformly metric entropy hold identical identical throughout rf constant proportional eq concentration assume rf iv knowledge onto preliminary essential remain ss result stage eq possibly grow unit cube eq cover element clear diameter gradient theorem constant evaluate derivative gram deviation define bounding term term chain eq control notice large lie within arbitrary choice element lemma eq eq hence mean hence nj long
exist learn exist sample define note pac randomness great dc sm unlabeled times construct take chernoff unlabele differentially concept either least pa om show boost combinatorial allow private move boost back private private boost algorithmic representation arbitrary probabilistic pair c h class every end choose probabilistic class concept notation align denote h h denote show probabilistic randomly tc think leave fail good hypothesis tx tx cx tc tc probability think experiment h h tr cx dt tr tr h dr characterize private learning concept
discretization space tail deterministic fashion matter never distribution particular consequence discretization serious produce third threshold proceed direction severe appear extract law pdfs cutoff cutoff behave dy py kk suppose instead practical importance situation typical generation prescribe object construct node ingredient consists compose priori
centroid generate create vector scheme centroid centroid close remain however feature mapping centroid feature competition refer triangle gaussian data mixture distribution feature map membership learn em nonlinear scan input micro unit multi contain unit activation map image output feature detector remove irrelevant robustness clutter think pool apart window center location window act overlap window map difficult commonly pool
order reach accuracy far rank speed conventional refine obtain apply fail find large lead gradient reduce instability cp issue scope future show superior architecture character net scheme accuracy b second drop model dashed tune solid fine third layer fourth ratio number point greedy cp bad degradation cnn fine try layer
jeffreys motivated asymptotic prior nearly boundary space weight uniformly spaced mass weight without negativity long magnitude absolute become grow non representation mass point suffice degree freedom equation mixture straightforward algebra hold implement minimize point log ratio except avoid initialized prior
source separation foreground vision extraction separability nmf closely numerical therein pixel hyperspectral long problem see therein convex nmf root comprehensive overview separable scope provably new show efficiently noise provably noiseless extreme span normalize discard identify vertex hull combination combination presence near separable separable nx w nr near separable classified geometric noiseless case separable formulate reconstruct minimize column reconstruct separable particular denote constraint reformulate relaxation problem relate relaxation exact involve system incoherence nmf highly correlate hence extend sensing model relaxation correspond potentially remove successfully
rao expectation construct rao low variance work expectation control variate consider finite eq index whose taking set estimate covariance maintain control depend easily see expectation variate scaling scale follow control basis control reduce black compute gradient maximize memory modification compute small set computing average require sample computation tb field family initialize randomly variational n p ix lt less extend
dr zhang provide code dr dr provide datum science education project electrical engineering university hyperspectral high characterize complementary characteristic image formulate contain data edge preserve datum quadratic regularizer align reconstruct hyperspectral operator infer augment admm convenient splitting exploiting
use agglomerative context cluster natural think average mean thresholding cluster ignore cluster cluster plot belong fourth plot curve dendrogram large specifie total obtain addition stable evy poisson measure mathematically apparent stable simulation marginal conditional advantage memory requirement store well ess conditional general purpose bayesian prior add toolbox could block research ia thanks european european research european framework fp
linearly alphabet size factor learn generalization shannon eq seminal enyi shannon namely order shannon entropy requirement entropy symmetry normalization grouping restrict many replace shannon interest diverse adaptation enyi entropy generators determine bit physical randomness help read dna motivate estimator enyi entropy physics study enyi differ shannon vary depend alphabet size show need fall grow linearly interestingly order grow
condition mean moderate node percentile percentile eliminate remain outlier usually significantly synthetic detection far implement mean plot detection mean show graphical datum correspondingly spectral cluster adjacency average rate percent cost second apply cluster equivalent treat outlier group graph adjacency misclassification correspondingly consequently let sensitivity community plot unless close major cluster cluster possibly fact synthetic large form second two solution figure theorem community detection test modify convex analyze collect compose political division conservative ignore direction component totally skew variability political membership manually evaluate efficacy adjacency political degree derive sbm perform well instead degree correct sbm spectral back modification heterogeneity early choose mean variability
minimum big explore ground onto hill repeat entire slope allow sufficiently routine may momentum add advantage preserve persistent velocity update momentum value learn partial accelerate momentum formalize parameter update trick input inspire gaussian behind matrix income layer second close decrease possibility vanish delay transfer draw pre output train involve calculate pseudo hessian setting though instance clear vanish radius restrict small free train deep network random initialization deep previously due concern
gmm exhibit structural exploit include derivation study behavior mmse draw gmm linear whether mmse converge zero heuristic sense gmm heuristic pick component greedy greedy approach use greedy error utilize component ensure mix signal x p power noise snr approximated indicate measurement covariance ia ia vector covariance gmm contain vector far acquire c distribution correctness initialize c eigenvector solve update parameter signal estimate application find add constraint greedy allow integer outer outer cutting cut optimization problem solve feasible region g integer software cut problem original repeating sensing standard deviation covariance n xx zero tolerance dash
model optimization linearly vary scenario optimize separability exploit another complement realize normal sufficient success es multidimensional marginal apply new distributional copula copulas evolution es free suit black optimize context normality however study call evolution exploit separability multimodal several modelling generate normally distribute
perform nb sophisticated error almost unchanged size nb allow road nb fair svm newly expression cancer separate vector along standardized mean road nb fair fail road nb fair inequality density copy jx jx later q linear empirical feature eq excess base procedure expect fix development label theory carry penalize notation regard couple sample bandwidth marginal oracle p define technical condition consist transform jx nj compatibility compatibility exist compatibility jx compatibility restrict signal risk density penalize
hold quantity eq agree definition body failure event time either failure occur rational rational initial value initialize lemma rank lemma success finally use definition outline bound occur theorem prove success event occur markov angular radial instance quantity order produce low numerator denominator next condition logic eq apply desire neither occur result p verify choose substitute arrive substitute desire probability event occur success stopping failure occur neither success stop failure event solve desire stop time run condition failure occur logarithm jensen therefore stop eq eq solve substitute finally angular theorem first notice failure happen event neither failure union markov part since
switch pixel probability set transition occur precisely adjacency model likewise noise parametrize conditionally change site least favorable distribution draw color latent noise interval change tuned stay range characterize model distribution field experiment rule yy statistic capture discriminate compete fall depend front substantially joint lie outside family sufficient bring concrete geometric colored connect component undirecte coincide site induce site share color component geometry vertex connect vertex form partition rely summary worth note component help search empirical represent abc dimensionality simulate statistic sufficient edge induce graph trial success component connect fig picture notation number component experiment apply approach base two pixel
reconstruct geometric thesis cloud characteristic forecasting purpose paper define classification consider partially amount iterate use data value column incomplete miss transmission topological comprise sense focus geometric construct extension directly geometry analyst choose determine particular integral semidefinite care
assume psd eigen decomposition replace column nystr embed psd one basis expand normalization explicit subset feature eq nystr point svm solver combine similarity measure svm solve eq regularizer scale scenario reduce preferred regularizer straightforward svm evaluate basis share measure lack psd argue measure likely particularly vision consider fix instance therefore similarity kernel differ similarity computer vision kernel pairwise measure position object image define similarity measure potentially pd representation latent variable possible similar mrf position pick variable
main advantage come mcp fast term runtime mcp roughly material runtime unlike previous long estimate take respectively spectrum network reconstruct analyze measurement continuous cell underlie experiment biology hereafter assess infer observational experimental presented test apply logarithm produce close analyze discretized measurement transform nonnegative correspond magnitude preserve interesting reason represent misspecification develop nothing prevent three proceed clean much make use unable final leading formally modify metric count true skeleton successfully orient pc treat dataset likelihood able dag log score compute run behaviour split choose comparable possible mcp figure consistently many edge positive small across develop mind discrete mcp hc fp log mcp pc fp dag skeleton largely negligible take complete processor fluctuation low report observe improved performance consistently fast bayesian along computer design test limit approach accurately handle node compatible high accuracy regime applicability domain exploit nonconvex nature progress break barrier one since around edge poorly combinatorial nonconvex affect already indicate improve even change core algorithm coordinate descent sophisticated nonconvex rapidly develop merely technique network acknowledgement comment van de suggestion dms dms conceptually simple would dag topology coherent carefully outline dag eq define map matrix match confusion write mathematically dags ambient wish cholesky recall definite permutation cholesky triangular cholesky take compatible triangular take deduce dag compatible proof original similar topological sort permutation order
prove c j apply conclude c jx b n jx p hoeffding know z c since measure condition illustrate figure horizontal rt jx eq cluster total index represent square distance dx I I intend diagonal indicator intend want construct show complementary tell us vector q mean give dual primal ensure linear combination get uniqueness result define intend rewrite equivalently note subspace span expression simplify euclidean distance row norm coordinate semidefinite state satisfy sdp unique intend mean sdp coincide intend condition satisfied next statement precise probabilistic model cluster assume section let measure center continuous translation center consider disjoint euclidean ball center center separation search attain minimum attain desire rhs coordinate constitute distributional identically constant row recall span point row j quantitative spectra
rapidly convolutional translation purely neural recently translation deep neural consist encoder decoder encoder variable target translation model mb contrast system appeal unlike conventional every component maximize translation much vocabulary affect performance translation crucial approach strength neural machine integrate machine translation neural rnn decoder replace encoder rnn recursive convolutional english sentence vocabulary nonetheless model newly supervision kind syntactic neural able process recursive ht
example positive objective creates naturally tend box figure illustrate visualize expand near set production trivial find baseline always class varied fine mean meaningful order weighting produce possible consider interpretable cart produce trivial box split option imbalance imbalance split data cart c fast box corner corner breast breast box fast baseline handle box ghz gb cache core box minute optimality minute instance provable repeat able show exact box box font summarize
precise really something com software gain lot state embedding research paper paper presentation obvious neural language
leave identity consider euclidean dataset competitive write rather synthetic embed triplet embedding leave near neighbor category label neighbor within recover thing triplet triplet unseen leave reveal embed hidden impact embed experiment triplet acquire converge triplet triplet figure music dataset converge triplets triplet time triplet grid triplet several triplet bottom grid triplet converge far quality triplet wrong metric triplet wrong idea consider inferior triplet triplet triplet individually grid individual triplet human decrease
agnostic state b except along degenerate throughout algebraic moreover immediately characteristic mode cf remain characteristic turn term exchange decay manner symmetry organization fig figure line space line correspond tm qualitatively different predict note doubly sum sum associate unity unity eigenvector unity stationary throughout unity eigenvalue h question direct towards answer eigenvector normalization denote easily check via obtain draw useful break mention early critical phase transition along eq eigenvalue case operator projection remain note obtain operator addition term turn contribution nan specifically process make
outperform alone observe accuracy observe gain mae prediction accuracy observe mae analysis post hoc reject difference across mae rmse regression chart magnitude interpret across individual across versus examine relatively note derivative mae window monotonically across consider descriptor set combine observe performance relative consider determine audio similarity specificity content retrieval task rating song evaluate utility descriptor quantify descriptor descriptor descriptor confirm propose choose domain specificity content retrieval descriptor track moment descriptor potentially content complexity descriptor suggest base temporal advantageous employ representation yield small frame domain advantageous explanation abundance observation choose similarity base short characteristic structure future aim close feature towards track chart entry control audio production song audio production chart acknowledge audio respective outcome future suitable
dimension consider class r tie breast cancer vs diabetes r tie heart patient ed cancer vs data column diabetes identify cut mahalanobis region quantile take outli observe regard outlier firstly approach cube robust depth depth employ dd suffer vanish performance obviously portion portion lie outside class leave inside least table portion vary sect sample relate train classify finally tie knn classifier tie point pool determined remove pairwise construct depth moment approximated depth depth choose depth treat number sect sect polynomially space polynomial fold cross validation complexity characterize need sect classify knn traditional knn neighbor perform wide save satisfactory max mahalanobis depth calculate estimate simplify classifier slight modification traditional knn
great surveillance possibility surveillance translation wikipedia know happen forecasting often motivate model wikipedia forecasting test horizon day preliminary important likely work article current try manual process yield candidate article feasibility comprehensive evaluate thousand million plausible article also facilitate predictive disease incidence proxy inherently fine country level traffic foundation wikipedia relate project ip address aggregated datum public preserve privacy traffic sophisticated try single wikipedia traffic disease incidence non wikipedia variety wikipedia share internet highly traffic cause news cause difficulty preliminary use english article roughly article month repeat inter lead page title none author google translate notice title article traffic day new period article change exclude article maintain aware tried latter manually sum discard history article transformation full available comment article would facilitate change health article often map international simply model testing recognize inherent internet source strongly internet access technology age gender education role quantify survey sometimes study limitation bias wikipedia
move list patient hyperparameter list likelihood posterior list list parameterized space boolean clause give advance fed determine node risk associate plus default patient monotonicity give early prior list consist clause frequent boolean output presence diabetes presence boolean return diabetes clause decision monotonicity constraint posterior log real constrain great
newton usually drop inside r evaluate alternative newton simply linear center replace g gauss semi generalize newton method parameter compute hessian free approximately minimize compute explicitly typically generalization method newton backpropagation implement point generalize previously alternative classical gauss situation arbitrary substitute formally maintain unlike linearly approximate capture think linearity compute many simple happen section psd positive estimate arguably conservative reason accord reflect version obtain whole contrast
create th highlight black increment bic concern apart perturbation former select fix sort bic always factor observation observation c rr misclassifie correspond misclassification perturbed lead ht c bad obtain perturbation number perturb recall increase contaminate contribution involve gaussian mixture contaminate cf sense view generalization analysis observation
describe integrated neighborhood adjacency list call along spectral guarantee recover via approach expectation suffer local three leave triplet hidden learn denote x triplet joint provably recover learn parameter triplet tree triplet employ tensor procedure triplet next decomposition require label structure grouping estimation triplet integrate bring merging step sub parameter learn triplet tensor decomposition recursive recursive node recursively group parent continue carry decomposition triplet find close decomposition carry learn moment continue active v v v decompose divide parameter estimation compare popular optima stable guarantee since avoid quantity
empirical propose able produce baseline search important mean internet drive pricing ad affect engine research design attract attention area intelligence require bid ad search ad one correspond g ad price use ad bid bid price maintain current rank role engine group theoretic symmetric nash optimal require information hold involve situation limit therefore game theoretic analysis group conventional machine assumption
layer pm grateful grateful comment work support foundation award modeling system partially grant gm learn broad learn directly yield break difficult language despite success little successful feature compression show deep successful theoretical physics coarse extraction relevant operator physical examine introduce deep architecture boltzmann rbms illustrate near neighbor ise employ generalize learn central modern directly promise successful learn representation training datum utilize achieve record
sequence graph fix order element law eq common testing item graph general dynamic address natural corresponding notion view present statistical parametric model issue result occur natural definition involve implement network exhibit become standard finance application development diagnostic base network particular cognitive fmri eeg state conclude propose similar em canonical introduce setup find fix minimum distance part center foundation central graph stand adjacency
ask could quantify art panel panel institute similarity red especially st panel stand reason work area restrict panel faces check particular extent separate method thin parallel panel possibly interesting
accounting category representation activation ensure contain row matrix contain notation column compound draw multinomial integrate vector interaction count interaction exist gaussian represent well ibp ibp chinese restaurant crp discrete comparison crp prior crp assign case latent name chinese process stem follow chinese restaurant infinite customer choose assignment node act crp
outer cluster decrease color speedup distance metric fine tuned color fine imagenet training variety svd institute york university facebook ai edu design million make internet cluster problematic dominate layer exploit approximation significantly demonstrate convolutional layer keep accuracy within vision test problematic mobile compute life spectrum scale electrical neural take extensive effort devote relatively effort aim improve test
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long satisfie turn borel axiom conclusion definition exercise euclidean metric set scale coarse curvature almost coarse underlie laplacian cloud data laplace manifold paper together demonstrate curvature wu develop compute vector point compute form coarse take euclidean discuss choice uniform motivation stem space coarse speak deal infer manifold give cloud
either follow assumption vc vc fact appendix minimizer minimizer reasoning roughly next sparsity literature select follow notation conditionally x x ij nx exp exception interpret expert feature influence difficulty classify sample influence assumption feature expert simple many situation situation similar product likelihood feature joint response explanatory indicate model log
e due good scenario fig comparison expect failure year good actual failure failure case reduce use early replacement take roughly period decrease time directly cost period slow optimal skewed towards expensive part study integrate source leverage interaction refine joint use combine expression similarly protein protein protein type base present diagnostic service dependency data
partition superposition partition use exist learn literature take boltzmann unknown label obey pr pr multiplicative assignment obey convention et term q consistency function exploit classification exact polynomial long expansion optimum mention semi community use min derive cox exist cox place additional maintain necessary seem motivate mrf use map hamming discuss function improvement mrfs use consideration section pairwise mean possible kolmogorov minimization represented cut exact minimum work result kolmogorov value input expansion
aic good simplify assumption fit however law necessary computation scale removal low kolmogorov minimize gram language family attempt establish classification actually power law subsequently gram database depict international similarity list item list stability
practical usefulness highlight application access example build machine predict interval stay short goal fact reliably next nothing aim make give computer similar study situation observe capture vary embed vector dynamic task learn adapt situation already one create sample serve sample blue dot point curve thick mean dynamic observe thereby prediction unobserve blue desire thin dash dot
increase become distance tight comparison stable analogous decrease slowly varie unless difference important gap suggest concern c occur fraction dna li normally distribute constant know variance increment align exhibit speak interval various evidence positive error candidate conjunction multiple comparison involve length actual problem often hundred relatively simulation compare modify modify follow comparative similar contain change high repetition simulation theoretically calculate false hc hc though conservative inclusion continue procedure signal high
ica fmri smooth ica versus component represent smoothed ica reconstruction error represent limit ica conduct fmri realization specification verification validate artificial fmri difference log box count simulate fmri blue versus curve fit sigmoid recover slope curve central log parametric recover fmri box counting realization simulate result complete estimate ds instead slice fmri employ fmri e voxel grid brain volume evolve ds blue blue sigmoid slope accord plot point sigmoid ds smooth sm box count ds analyze subject present estimation ds estimation ds confidence significance plain exclude large value table instance ds ds estimation sm plain exclude small complete cell correspond estimation range sm sm ds smoothing sm small x range significance range sm sm ica model activate brain area standard cognitive activation brain basic fmri modality eeg moderate regard see nature inherent constraint source perfectly statistical satisfied minimized fmri ica reconstruct
lagrange hmm generalization expectation notational expression inside
nsf fa materials generative likelihood generative log likelihood different category whereas summation want assign score observation discriminative want weak constraint generative continue refine satisfy example node final fully connect wu california model convolutional contribution include construct cnns form reference cnns importance
bellman instrumental bellman equation regressor deterministic typically inconsistent instrumental regressor regressor e method instrumental previously context reinforcement provide instrumental instrumental comment idea project bellman minimization call bellman space span basis function minimize project right hand onto span euclidean mapping well respect close typically equation least project bellman consistent equation instrumental appendix instrumental variable instead easy estimator equation policy row I instrumental regressor full ij il l assumption necessary converge mean present equation follow assumption consistent xy show every mean observation explanatory next instrumental uncorrelated variable assumption trivially law see therefore proceed instrumental bellman square bellman instrumental project bellman minimization equivalent start recall
risk current situation compute aggregate set line risk situation empirically environment conduct randomly split five mobile application recommendation software get document recommend system impact exploration fa ts risk recommendation fa ts fix ts document fa ts consider recommend
surrogate prediction directly relevance score relevance score surrogate develop perceptron ndcg rank instance perceptron sufficient query ranking surrogate rank instance consist list vector relevant query relevance formally input list document vector supervision space relevance supervision multiple ranking rank query technique learn score sort score scoring quality learn query compare rank relevance normalize discount ndcg induce gr popular eq number indicate place position rd position rank actually intend maximize ndcg function ndcg performance surrogate loss lipschitz relevance incur irrelevant score document induce score sec write convex follow q scaling wise
focus gain overhead possible fix fix processor convergence irrespective splitting introduce algorithm optimization problem optimization rigorously argue optimization increase admm among expect strong scaling thus split accuracy strong experiment show summarize partition memory consumption gb vary gb minimum node speedup nearly speedup high node parallel node increase cause learn experiment number column speedup parallel apparent imply processor per toward scaling stay processor natural contrary accuracy resource figure example per node increase time improvement improvement gain show effect requirement graph
scheme exactly simplicity number empty greater extra complicated change final argument prove hold new change web reduce overall variance theorem precisely empty increase empty improve compare large hash experiment scheme compare near lsh empirically theoretical effect many practical extract similarity web vector presence word united review mean
v conclude I I side v third inequality take full side c summation cg cg definition divide side surprisingly limit theorem choice choice optimize work likewise seem dependence
average encoder much simple manifold small generating challenge back find find optimize gradually transform easily machine multiple representation abstraction together conceptual challenge deep good question unsupervised setup whereas constitute good obvious follow unfold stack autoencoder rbms appear yield manifold decode experimentally probable like close rise input immediately recognize simple addition
mb mb rest descriptor local estimate entropy term three linearly rf step cost remark interested predict two force entire typically constraint entire parallel assess experiment publish address infer causal datum size continuous dependency parent model relationship lx fx lx three medium number sample number dag structure dataset remove introduce unobserved descriptor return denote existence dag dags descriptor positive direct cause negative cause forest train balanced test independent simulate dags dag configuration configuration select negative pair
question number go sequence library treatment firstly equal ratio define useful ratio nucleotide difference statistic library secondly deviation library around mathematical software library calculate library library program server derive
color practically either training higher explain scale translate activation example contain translate type invariance autocorrelation measure g translate autocorrelation wide main supplementary invariance towards much mod cifar mod whereas supplementary invariance good dot neuron reflect neuron blue horizontal axis much model analyze material notable neuron mod whereas invariant vast neuron invariance neuron use test mod stay invariant layer increase material invariant neuron especially mod nonlinearity unit convex side tie concave
kt tt kt uniformly iy tx update estimate mini loop comment let x h max mb ms gd consider nontrivial bound prox mini batch strategy fix simplicity
face align subject illumination condition pose sample pixel subject control experiment focus subject top pca light leave informative region automatically pca adjusted r ccc ccc time pixel high dimensionality interested recall pixel another pixel total reduce similar adjust pixel conclusion cause simultaneous prove sharp rate pure reduction deal sophisticated reduce term vanish sf issue pca strongly nonconvex challenge situation initialize case strategy study terminate analysis property big universal occurrence denote orthogonal onto projection orthogonal complement scale write ambiguity satisfy lemma stochastic combine statistical
proposal core propose merge move propose thus prior influence move probability create k merge move adjust merge pp ap denote constraint row lie simplex solve proposal need conservative recurrence correspond likelihood automatically document propose topic give graph sparse lda interpretable world art nonparametric focus sparse propose iteration factor procedure hold first toy arrange tree uniformly concept include initialization problem document comprise test likelihood hold skewed somewhat importantly graph
slice result outperform regular mcmc carlo proposal operator exact variable dark choose requirement likelihood must recognize obtain problem produce inference monte benefit black believe example mh proposal global mcmc mix carlo much empirically slow mixing offset fast computational hope chain continue mix closely challenging variable bernoulli become unbiased
rest ensemble expert mistake evaluate ensemble conclusion consist first learner learner assign learner label exploit randomize know predict weather think expert expert yes question opinion weather weather predict opinion since nearly opinion guarantee bad consist stage receive correct answer majority predict majority whenever true factor wrong label among describe pseudo code majority correct label
stack usually map apply pool adjacent map output output average value reversible pass pooling effectively high computational convolutional wise boltzmann contain convolutional intuitively layer wise pooling provide intermediate store recurrent feed convolutional spatial alternate pass feature map position path pass architecture top connect convolutional depth grow convolutional protein figure channel channel supervise reconstruct compute activation layer correctly predict secondary
discriminative row far appear novel one discrimination class generative generative describe distinguish view generalize discriminative pick manifold distinguish alg point point vs class features basis discriminative stress alg alg sufficiently generative learn manifold yield take instead cluster correspond manifold know motivate discriminative circle ht circle set basis small noisy experiment radius center origin space proportional generator illustrative purpose important black handwritten digit describe minimizer
eq q continuous q first statistical second impose restriction verify proof difficult cdf uniformly mean segment r u r r u uniformly gradient uniformly vector eq hold stationary ergodic unconditional appear around reflect effect hence admit ml square minor allow ny function cr c nr follow limit suppose asymptotic continuous nn nf ty uniform iid close drift tf conditioning away elliptical condition family drift ar ny ty ny ny fix u true
user final respectively map reach please model dataset name capture preference bad item co variant inferior achieve ensemble cf user preference factor rate extensive art cf algorithm evaluate predictor popular name name occur name criterion also try optimize map rank suggest approach factor apply desire reach predictor
paper corpora hierarchical seek rich form additional successful perfectly regime hierarchical direction relax structure document topic nonparametric effort include chinese restaurant corpus provide tb describe posterior corpus category proportion topic generalize lda base parameterized approximation distribution latent variable fully factorize accurate approximation multinomial figure approximation leibler eq corpus concavity jensen compactly variational far factorize approximation proportion document category prior express rise intractable average use dirichlet dirichlet intractable average distribution although degenerate work practice first error novel average arise average note
rs c method meet meet meet se se rs na I cover three replicate classifier effectively literature hold advantage explicit theoretical target vote somewhat less approximate pool compare outperform except fact naive implication suggest form resample future quality approach al explore behaviour generate new optimal selection al method suboptimal choice dependence target label behaviour primarily theoretical address application experimental compare estimation algorithm across range motivate address practice ground aspect weight straightforward estimator section describe function calculation brief exploration calculation loss see loss analytic calculation density give class give x j distribution respectively sample nc expectation term z nz nf nz eq turn np f nz nf nz illustrate complicated analytic calculation give express
introduce equal p inequality typical I batch index replacement gd update reduction compute every sample sgd minibatch practice minibatch reduce employ instead uniform separate k ts gd expectation update provide minibatch
lead parametric evaluation complexity unnecessary rather expect indeed gap approximation spirit gaussian strategy confidence budget use non bandit gain uniform gaussian bandit variance improve pac stop follow two provide draw kullback key increment spirit iterate logarithm structure distribution low bound uniform bandit model include illustration performance algorithm practical budget element leibler divergence q bandit call identifiable identifiable class provide armed bandit consistent algorithm confidence proceed
hx get e pr complete initialize integer synchronization stre sx string occurrence synchronization string negligible uncertainty symbolic entropy iii estimation satisfy e font south alphabet title yshift legend align legend style xshift draw white gray axis thick corner height grid background top color gray xlabel length xshift ylabel ylabel style yshift xlabel yshift scale format format figure draw black inner sep axis cs begin yshift xshift plot xshift font south title title style yshift cell align legend xshift fill gray fill text style black thick corner grid dash gray width background style color xlabel length symbol style ylabel ylabel xlabel fix format sep table figure circle fill none black inner sep axis current yshift text font current confidence letter alphabet uncertainty
statement state theorem map pca map r l map must use complex triangle write j prove connection correlate square
generalize e cd moment provide fit match thus near moment may arbitrary fy derivative substitute cause yield note q minimize e take recognize moment combine close fall observe minimize gmm moment justify alternate tractable analytically cd moment equation intuition work
produce performance compare table reflect ability completion achieve demonstrate outperform especially lr n cp naturally incomplete tensor employ bayesian significant automatic determination moreover prevent advantage amount pixel validate discover ground extremely synthesis superiority due interest attractive receive engineering department science engineering university china laboratory brain interest learn zhang degree china computer science china research computational theory brain compute statistical publish paper international receive dr degrees electrical engineering team laboratory advanced brain processing science technical associate journal transaction figure proposition zhang cp powerful completion capture multilinear cp algorithm
concept visualization call appropriate along mode tensor mapping interested unfold unfold rewrite operator another unfold kronecker represent concatenation introduce operation able multidimensional record separable parameter serve weight entry thing second minimization constraint trivial role
miss try neuron input north count node count dim output align align center gm encoder deep encoder low layer decoder consist reverse reconstructed version set encoder train simultaneously autoencoder since provide dimensionality gp laplacian locally mapping way deep encoder free parameter decoder train cost sum reconstruction find normally feature processing model training function minimize minimize respect compute efficiently minimize consider q
make mass policy support random always obtain context policy word action context obtain reward zero observe take policy policy experiment evaluation initialize sensitive control uniform policy ta tr ta ia step minimization rather doubly reward function train contextual learner repeatedly observe choose access classification guarantee call round number policy policy obtain contextual amongst excellent performance online contextual reward action take round previous round feedback observe contextual clinical supervise necessary exploration action result achieve round probability set policy reward cumulative collect logarithmic dependence likely reward use
rf center specify constraint candidate comprise specify reciprocal distances around intra balance mild maximize respectively submodular induce collection greedy gain speed greedy construct marginal gain element gain one merely gain idea increase return property naive method element gain element lead practice investigate rf rf essentially nontrivial mid pool candidate generate due sparse code metric pyramid grid grid please pyramid instead totally different pool grid represent whole extracted sift rf measure rf grid rf l x p grid grid please also
diversity dirichlet bias general whole diversity index distribution predictive species discovery estimation posterior flexible limitation diversity shannon diversity diversity health classify group produce index specie specie belong population actual distribution species shannon far widely measure biological diversity identify measure mean value relative sensitivity rare rare specie difference specie easily recover shannon unique year c introduce physics generalization shannon
compact observe invariance enable significantly accuracy descriptor l vs intensity classify texture descriptor adopt supervise support standard purpose denote class column distinguish classification separate assess coefficient pick thus coordinate pick texture image fig randomly sized gray scale patch texture patch texture patch use descriptor near neighbor assign patch texture report coefficient accuracy comparable long feature obtain accuracy descriptor magnitude short descriptor different
specific segmentation segmentation generally identify discard room cross audio segmentation overview note literature often type datum contain news result audio explicit segment
hilbert advance hamiltonian position show careful assimilation model prior dynamical kalman enkf ensemble assimilation gain wide ease practical enkf algorithm restrictive applied filter posteriori assume posterior design observation operator degree nonlinearity handle propose assimilation name obtain monte carlo hmc non probability distribution carry level water linear highly nonlinear enkf assimilation variational filter monte datum assimilation information uncertainty system family ensemble filter application variational root costly development tangent model assimilation scheme root considerable development enkf formulation class stochastic member update perturb lead root filter transformation apply correct enkf observation enkf accommodate linearization spirit extend kalman alternative handle non linearity use operator instead linearize mathematical pose subspace minimize depend operator jacobian nonlinear addition inherently posteriori multimodal distribution advance feasible promise towards sequential monte particle density
design essentially provide turn recent compressed signal signal technique counting entry maximally skewed stable develop compress compressed compressed linear
par safe safe safe safe safe par var safe define lambda safe safe var par safe program lambda stack safe safe safe safe safe safe observational respectively histogram right program e code efficiently many dimensional conduct ability automatically discover procedure encourage text common perform abc space pre text learn representative histogram summary statistic figure discovery sampler program sampling mechanism exhaustive computation tb lambda stack par lambda par
perfect achieve lk achieve linearly linearly place centroid regular margin grow place centroid maximally however margin separability maximum experimentally dr uniformly keep projection fig separable first column apart center big achieve perfect basis suffice projection enough role dimension number geometric projection nonparametric fixing projection find perfect vs svms different improve drastically critical separate degree experiment lie simplex boundary increase ideal corner simplex centroid compare experimentally simplex quality random c ideal reduction dataset boundary black achieve boundary
jacobian remain submatrix make independent row column continuous identifie level framework incidence generalization framework framework molecular conjecture body constraint system frame affine introduce hypergraph property useful hypergraph hypergraph combinatorial literature concept vertex identify tail connect admit orientation exactly nash tight map compose edge disjoint tight ready theorem combinatorial characterization finitely subspace incidence frame outline replace copy row determinant identically apply tight hypergraph show matrix identically long avoid main call behaved notice modify hypergraph copy expand hypergraph let replace last incidence row expand expand hypergraph incidence framework underlie expand
seq shrink letter reverse infinite sequence seq sequence seq study nucleotide incorporation interested cccc cccc flow cycle nucleotide flow seq self contain mathematical algorithm software implication threshold basis intensity variation threshold base drawback sequence availability positive potentially direction bootstrapping threshold call basis iteratively threshold test independence target
concentration specie assume observable system diffusion observe specie case steady specie regularity form fp ss patterns b concentration steady large fine ss reaction diffusion finite initial I p desire observation guarantee perform two design develop search require pattern descriptor goal logic superposition treat problem superposition tree descriptor reduce find logic specify machine observation step synthesis correspond reaction diffusion formula end semantic assign superposition tree quantitative
omit obtain continuous time consider purpose update policy offline linear remark establish introduce policy optimal use converge residual reinforcement learn collect increase collect learn offline employ real effectiveness algorithm verify free policy reinforcement method weighted powerful reinforcement rl great optimally simplicity rather policy
main additional observation seminal completion incoherent recover nuclear later incoherent subsequently incoherent bernstein inequalitie significantly simple proof recover incoherent svd manifold restrictive entry uniformly sampling power law universal robust devise matrix scheme universal scheme dependent furthermore important perform know problem recovery bit mostly rip
vary label regard er model label occur probability labeling namely edge probability edge graph label regard mode consequently case multi view anomalous less empty consider within community edge graph unlikely multi regard possible yet model characterize inter intra community give capability detector inter intra community edge degree external er multi anomaly detector probability anomaly subgraph possible internal internal unbounded degree excess sufficiently negligible detector subgraph graph fit statistic value degree give order use adjacency node spectral modulus computing observe three lastly newly statistic anomalous normal base
spline satisfactory basis regressor generate unknown implementation match regressor introduce regressor differentiable conventional basis well basis regressor regressor strong without significantly performance linear define incremental decision regressor approach performance initial adaptively dependent incremental tree complexity regularity meet length paper detail section performance nearly piecewise incremental algorithm twice observe datum start simulation remark nonlinear regressor regressor I ti regressor past regressor vector transpose xx piecewise divide regressor region regressor linear regressor regressor vx assign independently method mean square regressor process internal parameter e limit regressor desire signal piecewise partitioning regressor regressor achieve good partitioning
algorithm behave like wang table quantitative study fix depend still see average square confirm validity wang rate seem value assessment number provide magnitude accordingly vary varied slope c slope conclude numerical wang look behave wang table value times one increase retrieve asymptotic observe confirm since time illustrate observe close bit temperature wang choice times parameter wang histogram visit indicate vertical well visit wang section corollary expect behave regime wang stepsize prediction previous consistently state behavior visit contain initial condition drastically change approximately know wang stepsize discussion advantage increase exponential rate one
distinction involve similarity hc ica variance empirical variance estimate em modify I covariate voxel calculate statistic estimator determine covariate voxel level map apply testing rate false fdr effect map specific sd sd hc dual hc ica n medium sd hc ica hc ica dual medium n c em em hc fmri ten fit hc ica exact em approximate em comparable exact temporal domain population major advantage exact em signal time exact ic em exact em time
body near position inside body intersection know surface ball cone cone volume graphical illustration vertical transformation bring isotropic position scalar keep keep consequence hyperplane construct case event least oracle call achieve closeness variation quickly
estimate scalar least estimate imbalance appear lemma error even rate plug estimator asymptotic error appear regard issue covariance matrix tensor present imbalance restrict independence classifier mixture two product address contrast totally tensor decomposition imbalance imbalance suitable maxima b attain imbalance fx denote vector instance likelihood imbalance expression average instance numerically unstable close true imbalance imbalance justify constructive b law delta method prove
sparse select base know law viewpoint carlo suggest discuss previous paper viewpoint case extent height still inside illustrated figure region union discussion interval stream hypothesis test construct rejection specialized require base independent reference side large empirical expectation manner cutoff small test bit principle solve efficiently describe computationally apply away several effective sample get small carlo possible truncate multivariate gaussian hamiltonian carlo efficient multivariate distribution constraint selective cite sampling propose conditioning well sign fit event consist single polytope sign variable exclude conditioning selection selective validity power put nearly price acceptable quantify tradeoff far work selective add quadratic sample instead sphere section gaussian setting selective binomial scan statistic generally selective simple selective clinical binomial experiment similar clinical trial heart patient correspond patient heart trial efficacy treatment th order construct odd tie select law fix remain selection heart attack heart attack margin conditioning give right conditioning conditionally extreme selective fisher aside otherwise family interval observe poisson constant intensity unknown maximize confidence
isometry appendix far condition success iteration combine get recovery sparse matrix rip isometry ensure index clearly guarantee condition justify former k summary index condition I actually k totally induce exceed recover l projection terminate recovery reduce conventional recover
implementation validity property claim propose I model add plausibility think ok agree validity property wise error familiar paper selection paper right select unknown seek desirable dependence structure design know efficient xu zhang framework valid worked propose inferential I special variable regression interest develop optimal set product I validity I sub select I approach valid post I drive base example arguably one widely tool scientific possible planning subset useful variation inclusion explanatory explanatory variable fundamental go selection central pursuit general solve arise comparison variable always suggest variable overcome limitation add kind variable include method schwarz severe allow candidate naturally etc despite certain property ranking inferential meaning aic bic conclude correct plausible model criterion meaningful measure correct variant elastic net summarize meaningful significance test lasso resolve concern parameter search specification posterior computation remain furthermore
project figure inference answer appropriately involve incorrect free nothing put splitting lead power ordinary try inference parametric invoke strong focus weak like completely sample predictive seem property
fig modularity trend simple increment monotonic worth note early entropy fitness subtle problem pattern narrow target good auc consider eight misclassifie represent report degree see auc high membership hard decision zero please data may still picture additionally yield insight recognition among optimize towards relevant property denote metric would find perform role entropy modularity situation pattern leave corner green pattern fact red fig assign trend modularity peak I trend blue pattern belong blue one correctly misclassifie respectively assign depict axis perform accordingly contain isolated herein uci miss retrieve comparison version uci normalize ensure unit uci usually principal ref
operation exceed ssc apply title width legend style east legend pos north mark none dash face none dot index time face mark none black none name anchor west title legend legend style anchor east pos north x mark solid face none dot black
evolutionary evolutionary process possible set normal function consider function default protein think accordingly posterior use suppose move say proposal value possible density model distance keep gap setting prior distance various first analyze sequence method identity align identity suggest sequence prior distribution top leave panel evolutionary modal acceptance proposal modal even approximately contain dominate modal protein possess structural similarity top show unimodal suggest long evolutionary two previous mode mean large move towards around expect bottom model evolutionary alignment consider pair multimodal posterior distance mode consider unimodal mode setting little sensitive parameter discuss influence keep give comparable match difference
one paired weakly pair training precision precision recall experiment one retrieve document rank document retrieval precision retrieve document denote retrieve score precision display scope curve visualization retrieve precision present user pca cca pls method text pls texts database wikipedia database dataset split multi contain select training word frequency
simplicity tool problem eigenvector especially dimensional ad hoc absolute take fit original approach desirable motivate research especially pca et sparse nesterov sdp exploit connection singular svd extract principal component pc approximation et optimization involve although differently except phase base conditional gradient variety efficient multiplication extremely large algorithm suit identity direct deal substitute difference require qp every intensive amenable restrict identity special multiplication need shown adopt approach develop eigenvalue unified method maximize objective function consider quadratic reweighted square turn eigenvalue sequence problem efficient ascent lead generalize spirit solving type often suffer get systematic inspire minimization nonsmooth
knowledge choose noise subset know specific case unconstraine consider quality sparsity viewpoint maximum relate bernoulli dedicate greedy explore iteration atom current subset gradually improve pursuit mp orthogonal omp square ols refer match iterative hard subspace pursuit category resolution series consecutive error increase lead extension forward subset support element error exceed square inclusion removal induce decrease design forward backward search either atom backward elimination omp single spirit extension ol early wrong backward ol omp low complexity search dedicate regularize result nonconvex relaxation convex pursuit lead homotopy whose omp homotopy closely connect angle lar simple forward lar importantly homotopy solve dedicated penalize ss font lb lb lb lb lb lb lb lb lb lb concave piece regularization compose support instance minimizer piecewise appendix understand consider concave envelope curve curve support solution vertex advantage suboptimal greedy value good replacement repeatedly minimize decrease descent complex maintain improve decrease approximation error
dependent variable influence regressor covariate variable commonly proportion interval common line transform long transform kind usual thus model datum practitioner standard specifically tailor regression underlie distribute flexible rate proportion shape parameter mean precision mean covariate function fashion formulation incorrectly take estimate density maximum likelihood estimate slope covariate vary simulation precision incorrectly properly model large dispersion identify source variability dispersion model beta relate
r glm default perform cross report assess statistically metric evaluate pearson correlation bold encoding signal cross sign obtain significantly irrespective across study relative averaged sort rank induce separate oppose free glm inherent glm model purpose design follow glm basis sign subject use figure voxel encode score first code peak difference level voxel encoding basis glm voxel metric pearson correspond exhibit score glm result investigate encode valuable peak width characterize mis reference voxel voxel exhibit method commonly software define www fail voxel score encode score plot peak code peak even single volume second coordinate encode score canonical axis rank black glm previously estimate voxel canonical glm significant use exhibit sufficiently suggest computational
I independent grow increasingly face curse translate scale hyper cross hyper support feasible mp train different fold pair sample sign star demonstrate solution reflect rmse toy curse dimension factor curse gp automatically factor basis rmse cc cc concrete red datum sparse rmse fold statistically indistinguishable except concrete compressive considerably standard require factored regression
autoencoder autoencoder stochastic gradient online minibatch epoch autoencoder decoder epoch reach weight leaf autoencoder epoch stochastic gradient mnist report scale word relative mnist attain error reduce autoencoder perceptron ten tree dimensionality tree might autoencoder news less architecture autoencoder autoencoder reduce ten
vanish obviously source introduce framework negativity source partially non negativity combination positive source generate seek simplex volume spike source sparse spike verify support illustration leave mathematical perspective characterize column mix translate spectra different spatial consequence mathematically emission emission etc physical density temperature make detail panel report take correlation wavelet highlight partial chance finite sample realization theoretically uncorrelated likely available negligible large htb limitation standard source negative generally negativity negativity source assumption approximately transform vanish entry present correlate source build blind source separation sparse correlate source noisy make assumption non negativity range separation problem especially image organized review limitation correlate introduce report performance standard finally illustrate diversity q first fidelity discussion choice penalty appealing make
consist per semantic differ weight often assign graph language relation resp represent total equal probability model everywhere axiom relation incidence incidence every element axiom axiom relation identification treat relation axiom retrieve domain function uniquely graph logic axiom edge go opposite axiom axiom graph edge complete edge connect collective weight equality fix analogous cycle weight size construction item statement subgraph working logic weak statement logic emphasis axiom universal sentence presence weight difficulty investigate implication logic artificial technique recognition ann direct graph edge weight connect neuron connection say write activation neuron neural neuron activate previous language binary ann introduce new predicate represent neuron activate satisfie threshold update ann neuron begin neuron activate accord digital activate states neuron neuron iff imagine property phrase logic answer validity q conjunction axiom axiom unlike weighted artificial theorem finite particular order language finite unary predicate use predicate ann almost finite predicate language issue e still expression limitation many case logic mention quite logic locality reader logic calculus model countable moreover subsection countable regardless shorthand shorthand n yy number depend free finitely
close back euler characteristic physical underlie decade considerable cut minimize normalize modularity cluster edge removal among progress researcher spectral majority reader comprehensive review stochastic respectively little previously community input specify quantity address select wise edge bic criterion variational approach highly researcher blockmodel undesirable restrictive bernoulli observation apply misspecification stochastic blockmodel variant motivation conditional restrict exchangeable graph composite bic community blockmodel exchangeable simulated show outperform background cluster methodology simulation discussion adjacency diagonal zero random
google frames assign frame correspond nf logistic unknown index correct sgd method axis epoch improve find decrease version almost batch illustrated show improves measure marginally right effective obtain curvature figure memory mark degradation great yield focus datum dot train remain error measure percent correctly classify latter account yield suggest occur set monotonically quasi newton fairly large sgd since set compare cost batch explore efficiency quasi report performance product order line vs dotted vs sgd significant margin cost dot scale crucial quasi allow acceptable cost
prove lemma definition ac il show modification bayes work proximity near grow appropriately regularize enjoy considerable bound principle speed encourage empirical nearest nn continue popular practitioner despite numerous continue yield paper near since amount effective analysis vote neighbor guarantee consistency
shift modal signature signature may signature mean partition evidence contain signature may signature signature shift signature signature verification competition signature curve signature dataset priori group signature sufficiently separate shift modal curve analysis represent unsupervised nonparametric signature evident signature signature polynomial get polynomial smoothing estimate acceleration signature normalize norm depict apply asymmetric gaussian smooth acceleration distance norm acceleration depict figure figure acceleration figure show algorithm mean shift signature fourth original signature original signature blue middle signature cluster display functional mean shift homogeneous summarize signature green signature adaptive ascent scalar counterpart dimensional shift design dimensional functional applicability mean shift correspond ascent practitioner establish
nf propose impose constraint introduce write diagonal account influence code introduction function black dash function get optimization bfgs et kernel regression consider locally constant weighted equivalence prove appear distance hellinger hellinger analytical calculation hellinger
fundamental object yet greatly covariance decade additional covariance key tractable lie toward order spectra constitute observe variable order assume variable distance series assumption allow depend depend assumption process likewise model use whose apart weakly correlate situation specify unknown depend matrix I n norm measure use estimator toeplitz form allow frobenius relative member matrix decay diagonal estimation establish frobenius minimax utilize practice estimator positive motivated estimator partition block zero adaptive class technique norm construction obtain heavily decay away far regard covariance diagonal arbitrarily situation decay estimator thresholde guarantee positive estimator cone would norm notably
natural extension relative partially support dms thank international conference great author express thank united national title guarantee mean bernoulli random widely success
mean answer question answer task never environment task quantify evaluate answer answer question quite understanding involve concept hide ideal manually every individually since infeasible ambiguity interested inherently bias frame grain categorization interpretation coverage automatic answer consideration member attempt issue similarity score lexical
tail large tail particular type note marginal fr xx tx yy ty tx ty ty cf bivariate characterize useful example copula see popular symmetric joint tail correspond bivariate coefficient freedom tail extreme tail asymmetric structure return structure chen exhibit tail exp exhibit estimation main technique univariate heavy independent index standardize marginal fr pareto observe let minimum fr
validate scheme provide predictive compare fix predictive interval model regression conventional interval conv purpose upon interval hard interpret mis interval measure big dataset compare precision width conventional observation contrast assumption error occur manner estimate increase able quantile one hypothesis hence provide table display explain conv k dataset var value find minimize fold compare display constraint sign fail sign put pass mis distinguish two consecutive annotate bold proper hyper illustrate row look proportion var conv conv reliability constraint situation stay var conv nine look almost everywhere large wide obtain band eight list sake compare mis mis chart chart display mis mis figure chart mis reliable mis find reliable envelope interval chart small mis ratio chart chart predictive fix look see mis conventional conv ls testing stay var conversely fail predictive interval nine conventional conv l wide envelope scenario var envelope error look figure fix decrease model note conventional small nan accept model observe interval neither reliable summarize display row summarize reliability quality band fourth display efficient ignore value normalize c efficiency var var var var var conv concrete fix svm l svm conv var k predictive goal detailed manner reliability purpose choose
surely norm surely expectation amount affect column k treat rewrite independent ia bernstein theory equal hadamard product hand hadamard independence q dt matrix bernstein know use normalize normalization happen algorithm part contraction power iteration tensor contraction argument argument update overall guarantee convergence similarly argue local convergence update define notice constant contraction perturbation suppose hold suppose h asymptotic include rate algorithm actually quadratic contraction quadratic beginning involve quadratic observe quadratic appropriate initialization rapidly sake clarity propose convergence convergence lemma explicit contraction observe denominator numerator contraction result tensor perturbation tensor w contraction define addition imply iteratively tensor power provide proof notation incoherence notation th column remove decompose unnormalized unnormalized c follow derivation repeatedly assumption use inequality exploit last exploiting term eq inequality exploit j distance expand notice argument provide incorporate inequality definition denominator use coordinate
tree subtree subtree leaf node uniquely define suffice mistake subtree subtree lem prove show tree bind one quantum communication express concept communication numerous low sample pac vc parallel dd aware counting restrict ball b define see already separation quantum way communication bound coin simple randomized public coin prove differential privacy differential privacy somewhat stronger bind upper since communication asymptotically learner zero privacy sample sample subset margin optimal achieve efficiently vc
order objective replace mask th mask multiplication variable concatenation copy mask function training criterion user corruption process sample dirac delta mask take corrupted version q autoencoder effectively mask capacity encoder copy px assign input logarithm essence maximize eqs every chain sampler although train agnostic sampling propose deep select generate select dimensionality hide unit layer use
report reporting apply might preprocesse simulate infected calculate neighboring infect internet inter consider free note infected node near neighbor infect infection value preprocesse report large theorem infected internet inter compose maximal relevant various epidemic occur report epidemic specificity epidemic available test standard plot carlo scenario body paper graph perform report ball radius ball radius standard show distinguish follow describe false positive fig infection size report positive type assume likelihood epidemic reporting plot obtain facebook level extremely infection setting succeed negative show succeed presence information contact zero hundred spread across epidemic begin experience epidemic process trend public across common
generation know k continuous parametrize term parameter intuitive also assumption interesting take follow binomial c check straightforwardly viewpoint mechanism individual birth next remove subsequent could population take follow condition check I either expect another process growth e z whenever depend unity classification likelihood shall entire z lk k lk li li n lk represent individual exactly intuitively accumulate generalize likelihood use parametrization shall worth mle knowledge thus address obtain ii intuitively rise parent also order investigate estimator necessary power series assume establish verify establish preliminary omit remark I iv
initial effectively controller training simulation switching preference mode next control initial investigate capability controller initial capability simulate initial plot initially change controller control different loop moderate robustness uniformly act depict system loop open mode lead instability system degradation steady demonstrate apply switching admit incorporation assign different investigate go solve cost go current develop burden many value south school technology city edu nonlinear investigate adjust switch feedback development ability switch switch lead deviation mode preference
contain cause identifiable soon period provide insufficient occur equivalently sparsity degree obviously condition necessary much hard eigenvector study suggest condition irreducible everywhere identifiability unable counter critical assume markov assume I introduce let arbitrary affine fashion function convenience denote small letter g p lie work operator q view preliminary obtain arguably estimator obtain q derive minimizer
detailed discussion comparison propose analyze frobenius sample see good bound various completion recover observe minimization uniform incoherent want compute rank popular frobenius norm consider scenario guarantee frobenius present element rank comment ingredient number leverage tm neither leverage exactly leverage account given optimize factored sample element routine matrix give objective function follow sub routine set span prevent heavy provide main rank show I constant specified completion
proved reason analysis thompson resemble sampling carry modular particular set bandit whether idea generalize maximize modular variant problem adversarial base solution key polytope polytope hyperplane combination prove first exist vertex exist therefore contradiction prove contradiction polytope express vertex greedy I ie contradiction ne e se least event hoeffde event claim get q last fact note conclude span solve optimally greedy work unknown learn interact repeatedly bandit setting formalize know computationally favorable dependency efficient massive drive greedy sequential learn combinatorial item number potential huge
structure histogram image derive finally trend scene intuitive method consideration complex acoustic forest poor second might valuable finally exploratory section audio baseline existence fail performance significantly suggest effect modelling describe one drive development normalise compression dissimilarity investigation design individual public avoid addition choose human testing phase evaluate experience experimental choice human rigorous comparison believe reflect employ qualitative capability interesting significance test human protocol cross clearly algorithmic human result figure achieve similar human suggest median benchmark secondly misclassifie acoustic aggregating take acoustic correctly misclassifie encounter music retrieval whereby always misclassifie moreover unlike human challenging acquire human experience comparison confusion present reveal misclassifie human find misclassification acoustic observation contain sound event even semantic environment universe mutually exclusive exhaustive word include application ensure category important suggest far first consider design learn acoustic give mobile service place resource intensive processing line need signal application real instead
serial consider nesterov special convergence rate prove set result obtain accelerate proper vector z fy kx z result algorithm satisfy specialized serial q unconstrained case serial nesterov prove accelerated utilize restrict indeed simplify improve uniquely suppose block lipschitz gradient serial q useful confirm block pick update often mention algorithm dimensional operation unless suitable implementation assumption gradient focus algorithm variable note present latter require address k therefore set block
location full location visit sequence instant previous image mixture sift histogram sake extract descriptor within analyze subset pixel fair spirit life grid combine image together investigate separate grid visual input approach image pixel comparison place bag map bag agree fair complexity counting grid g train window scene illustrative purpose label label realistic train image likely field view simulate generalize scene possible count scene close question neighbor comparison none fig approach bag feature simple bag particular example reason window top window bottom sometimes track train infer track existence train track furthermore proportion carry layer previously elsewhere organization retain reasoning iterate eqs count window scene take bag low right window compute count appropriate matching reconstruct
present analysis approach stage bag bag bag define basis take analytical typical gaussians exponential log method reproduce kernel reference therein divergence gaussians compute product straightforward divergence consistency number include construction overlap concentrate dispersion type negative performance remain also similarity enyi index prior address estimation assume response consider covariate nonparametric form assume regressor reproduce use classical regressor construct continuous meta generate handle dataset estimation learn bag finite algorithm ensemble convolution case similarity establish introduction rate bag
frank com building answer question intelligence promise progress recently achieve learn logical database cost either amount human label define tailor practitioner question answer schema without fine tuning supervision resource empirically demonstrate meaningful supervision similar label challenging answering answer topic bring huge building way development store huge organize database connect entity answer define entity give express simplify issue collect search e answer open answering remain challenge triple million difficulty machine language problem semantic convert logical subsequently scale hand schema parse negligible intervention might generic database schema broad english
rw model hamiltonian hmc favorable hmc favorable scaling guide posterior neural use hmc traditional feed back propagation though hmc explore manner rw mh expensive large drawback outline challenge outline variable influence assess relative background avoid testing model genetic examine possible combination hmc mcmc method rw greatly massive real assess network popular method advance technique clear use basic net parametric method classification sense smooth precision need relationship net appeal include phenotype complex capable classical natural consequence strength fit occur start represent include noise may method exist method another net black
figure goal versus movie review train bag word example exhibit behavior simulation run size result learn good suboptimal error relative generalization suffer bias show behave link favorable tradeoff dropout implicitly assumption label original dropout naive naive generative recent survey bag document dropout state art accuracy suggest discriminative strength generalization interesting assumption explain helpful topic make training dropout erm
determination deep refer reasonable activation even supervise kernel model model propose unbounde layer connection adjacent introduce draw amenable series fix encouraging layer analyze recurrent network back make gradient hide stable know systematic deep composition polynomial application construct kernel kernel correspond infinitely deep attractive study first show view neural architecture capacity decrease degree propose connect input examine obtained finally obtain gives define
assume prove assume bad regardless satisfie cd kp leave side go sufficiently contradict q claim easy fp fp p approximate cut see undirected exponential number subgraph cf property calculation possible sufficiently minimal generate input decay positive fact noise bad instead independent chernoff hence interestingly accommodate adversary cf nature node label adversary input arbitrary bad case bad inconsistent semi well purely effectively set contribute adversary maximize adversary grid attain consider addition marginal mention optimal recovery procedure generally per instead resort mode hard relaxation locally marginal lp relaxation tight field interest cycle relaxation edge noise another local significantly baseline latter likely map
amongst much become unstable relate proximity policy policy reliably realistic setup reason believe ideally suited framework turn aim learn task neural ensemble diverse error part rl diversity aspect experience diversity diversity diversity experience high assume multi diversity issue unless sound diversity mdps generalize train mdps aspect diversity discuss express think multiple heuristic situation
conclude discussion contaminate gaussian random degree contamination bad common maintain distribution advantage establish either otherwise robust contaminate distribution unfortunately relation covariate covariate contaminate lead covariate property contaminate replace contaminate contaminate contaminate contaminate distribution mixture conditional gaussian linear also convenient square concern distribution model regression marginal integrate contaminated contaminate depend mixture contaminate nest family contaminate mixture contaminate particular contaminate provide maximum two
social medium root expand aggregation variant meta share similarity system cluster exploit social algorithm unfortunately assume twitter streaming scenario expensive obtain popular user mention driven similarity social medium author e title tag date location propose cluster label combine similarity reveal group medium event twitter incorporate module temporal interactive location micro event benefit variation mean spatio dense topic term message assign close centroid work deal media stream twitter pre aim token tweet strategy processing particularly efficient suited streaming scenario aggregate computed pattern stream observe build cluster carry track least comprise twitter systematically baseline tweet assume knowledge social network system cluster paper simplicity
method handle network competitive compare relatively tb fp graph come domain penalize logit separately individual factor utilize encourage solve quadratic constraint impose network descent procedure free world latter real demonstrate pc procedure data dag candidate causal network demand size nature logit room cd moreover since nonconvex introduce global future consistency node grows investigate review topological sort topological topological sort acyclic every sort sort
standard ball grow edge boundary decrease cut edge budget upper contain get control remove boundary maximize solve connect capacity capacity every source lie cut easy check minimize give detail remove remove edge edge decrease control procedure size control statement I trivial budget remove increase budget total budget budget budget execution control remove edge possibly assume argue great hence remove control boundary decrease apply contradiction violate upper trivial bind equal piece solution nd sdp mapping sdp sdp coincide connect active second condition property sdp solution ball radius around ball graph vertex first sum option remove cut iteration go length cut cut step none cut let go know cut increase budget lie initially budget step change
theorem tell twice exactly compressive cs general reduce signal directly paper brief conventional compressive approach compress choose suffice reconstruct
send barrier barrier overcome develop successfully deep seminal propose hundred million conduct extensive recognition automatic recognition kernel deep net method appeal cost training reduce two competitive question deep application automatic arguably instrumental huge million adopt drop parallelism new trade add hundred excellent various consideration dependency effective sample early reflect extend million svms solve svms approximately matrix kernel million million sample time publication none compare reduce feature feature dimensional optimization approximate dimensional recognize promising observation product approximate spectral weight major random random learning training report speech recognition vision context automatic recognition example task
affect student measurement student teacher teacher time corresponding year teacher contain current teacher effect subsequent score student expect diagonal teacher intra student student year block student observation year refer gp indicate intra student great flexibility future year teacher effect student correlation assume independent analyze year year correlate moderately effect average teacher single future year teacher aspect persistence require scale measurement refer current teacher teacher contain student year alternative autoregressive specification persistence structure teacher effect correlation intercept implement alternative except teacher g define exception teacher diagonal set corresponding year new definition student express include teacher effect fit without student compound block student intercept formulation
eq result batch prove integrate case involve boundedness apply cumulative risk boundedness adapt recursive apply end application recursive argument discuss optimally observable develop parsimonious strategy aggregate dictionary complexity reduce generality e learner empirical iid interesting deterministic easy equation sense generality remarkable dependence risk natural optimal rate general stochastic ms restrict strongly iid observation iid copy assume jx preferable convert learner average jensen inequality give optimal regularity satisfy
transpose side obtain system linear subscript find store consider regression determine secondly dictionary due selection criterion however find solution relaxation np certain incoherence basically dictionary co linear typical viewpoint incoherence requirement bayesian treat concave add prior typically approximate yield eq
majority review diversity pathway generate key convergent trait benefit effort acquisition genetic specie expand molecular toolbox trait thank n conduct analysis experiment model test extract rna j figure paper study supervise discuss manuscript source correspondence publication view website material c seven specie characteristic record intermediate c majority abundance bundle bs study employ methodology comparable cross quantitative partition score cluster em assign g presence absence allow component two complete linkage agglomerative partitioning euclidean common em quantitative study specie abundance band abundance band qualitative presence score assign appear bs cell represent string string trait absence trait presence trait miss pca fundamental transition consist denote label binary phenotype meaning allow transition transition change towards string possibility involve simultaneous constitute order influence evolutionary dynamic
proposal combination namely error design independent regularize cast computed norm rate mu selector embed zero mean order advantage convergence propose analyze acknowledge gain rate additional regularization main constant property procedure auxiliary rates convergence stochastic notation integer cardinality say sub gaussian variance sub
sum parse parse short message step message definition message part technique message current form pattern pattern consistent cm material step compute step take turn vertical message passing imply short correctly show make nonnegative add connect incoming assign since acyclic source path take compute checked rescale r rescale short path initial rescale suggest vertex edge equal iteration thus bad complexity cubic practice small reach good
write permutation sign stems combine conclude differentiable frank wolfe problem recently interest improvement become particularly atomic ball compute euclidean proximity frank wolfe gradient regularizer regularizers fuse elastic net en glasso variant rely depend namely linear usually arbitrary regularizer permutation neither outperform consist pairwise sparsity equality estimate proximity efficiently accelerate proximal fista refer regularizer object order weight
map relu three third fully relu final send logistic stochastic momentum epoch tune decay network run file architecture source code scale convolutional neural science university college md david department college md show excellent visual exception carefully object align naturally feature get increase appearance recent imagenet discriminative use learn simple feature manner mnist build convolutional neural network achieve excellent handwritten digits sign category imagenet come learn pattern increasingly layer
projection take input simplicity give computation degradation l projection bilinear entry bernoulli binary eq kk section apply sign carry preprocesse drop embed fast fourier matrix clearly need datum efficiently compute operator convolution fourier transformation dft dft dimensional dft conjugate transpose transformation convolution original hadamard therefore bit bit dft efficiently
platform pac bound analyse performance classifier adopt classifier centre prior bound vector another ingredient logarithmic determinant inequality lie evaluate source view offset explore pac view consider promise applicability content video audio view feature improvement split
derive set properly pdfs directly block r p k j j dt dt I find intercept r dt dt r dt proof appendix material information drop subscript convenience know imply bf k r dt rhs rest nonzero bayes drive bf k k bf k exponent bf true entity denote index recall block index model result lemma slightly b denote j tn n bi ib ir r proof attain tt apply two satisfied block portion material bf k I maximizer lemma equivalently large bf om om q r b b bf om f positive structure limit rate first term go
anchor south west box west east box south box north east lag ar aic k time specific result ar goal lag series lag divide series second half estimate order lasso aic freedom fit apply adaptively estimate non freedom
shall denote ds ds denote integral entry proceed namely martingale properly lemma large eigenvalue random trace ie side notion calculus let whose integrable quadratic martingale quadratic r quadratic martingale identically r twice self adjoint matrix follow twice continuously application act importance result trace martingale td entail q x
next training repeat process repeat sequentially numerical optimization time need short initial result jump large poor prediction less approximate exponential particle show predictive provide c dataset log method compare rank task whether significant analyze comparison summarize across dataset differ performance level statistically superior addition predictive overfitte dominate sign p pairwise comparison give functional log return
purpose test bayesian prediction extra exist end section consequence split input output look determine model believe consider global choose commonly use covariate give satisfactory try aic supervised version bias situation see practical usefulness need experiment consider determine bias aic focus perform experiment promise derivation start equivalent aic hierarchical greatly applicability acknowledgement thank comment regularity item wish use explain explain select likelihood validation variant estimate vary assume correction see aic adapt input bayesian averaging case also
space maxout artificial hide call popular success maxout neural become I e hide layer product unit deep sum compute certain extend kind analysis discuss estimation artificial feedforward borel lead choose weight behave activation precede time overall look give compute activation precede consider activation hide set vector activation layer subset map onto subset map common recursively disjoint neighborhood whose function compute layer recursive formula count along branch root space linear region linear activation r r construct region input neighborhood distinct maxout discuss identify fold coincide absolute fold twice
gene test complete carefully elsewhere factor specific gene co expression uniquely loading b regularize control loading x estimate w calculate zero suggest gene gene precision matrix know gaussian network use observation construct connect great combine single keep edge run network co expression observation run short pass node project datum snp multi trait nucleotide snps snps represent copy frequent genetic variant total quantile although snp perform perform gene single snp across perform association snps biological meaning build snps snps conversely refer trait gene trait association interpretation snps minor frequency snps let gene gene gene apply repeat initialization estimate sparse active snp snp trait association demonstrate model table l one average represent zero snp dense individual individual among dense next analyze sparse snp observation observation active gene level gene level snp zero
processor gb ht marker respectively disk represent interested robot previous grey represent cell robot cell marker step mean action become know terminate satisfy specification horizon converge close second become policy second output optimal state three observe maximal error loose bind correctness apply robot planning north west correct cell robot arrive adjacent intend cell north ne cell fig robot maximize
rr depend omit follow mean eq easily convergence effectively whereby appendix situation density find noise wavelet integral due integral quadrature rule q know obtain whereby mm mathematical ex promise usual hide specify new dependence wavelet variance equal provide flexible inference become edge detection image auto laplace statistical dependency coefficient wavelet take value application hide determine wavelet
frobenius relative spectral miss cccc miss e miss health survey approximately low certain specific human data combination compressive sense offer currently first normal successful define classification ratio number trial rate contain
hyperspectral motivate particular gaussian entry take assume normalize sum pass column separable column probability index translate stream implementation section index nmf upper r r I factorization nonnegative factorization work singular show scalable serial residual e error residual choose datum near nmf extreme solve matrix fortunately complete remarkably pass suffice achieve use ever matrix thin
example label goal available training datum develop assume link involve network node accurately pac apply big technique verify match however available match organize expectation sample replacement section present validate match refer match node match validation extend match algorithm rate precision subsample combine conclude research later average estimate example use pair finite estimate replacement estimate similar
sample belong label addition unlabele observation class predict future minimization lipschitz hinge supervise act come represent transpose operation especially powerful arise setting must link default label ji c constrain j capture label lead j semi supervise survey link equivalence I j e index unlabele nonetheless proxy encourage instead partial show constraint j ta ellipsoid ball restrict unlabele label unlabele constraint necessarily quite yet nonetheless verify c upper bind energy x mahalanobis act estimate follow constraint variation smoothly encourage example neighbor predict difference operator ill square matrix like also encode label constraint diagonal ij e twice label node encourage
cite role one compute another penalty regardless apply call wide strategy describe strategy make order qr block zero take case skip cover triangular wide strategy minimizer qr decomposition minimizer change boundary empty square old initially differ column qr square detect begin compute qr special permutation orthogonal rotation special qr refer qr exploit qr boundary set differ qr appropriate operation operation meanwhile naive simply encounter magnitude quite favorable operation operation drastically near strategy naive naive primary naive one summary total qr give detail latter implementation complexity filter derivative operator define trend produce piecewise fit favorable argue trend quickly via sophisticated trend filtering actually key row make boundary tr problem two system polynomial implementation time coordinate list cholesky practical I least qr practice though yield necessarily importantly qr operate k reason preferable qr package use qr cholesky qr utilize maintain iteration successful sort efficiency
guess permutation time text create together implementation create gram retrieve contribution letter long v multiply calculate letter new letter complexity letter roughly language project plane implement create text corpora sentence language easily vector
choose risk risk arm good goal identify measure correspond pseudo regret bandit directly correspond stochastic try empirical similar even switch good arm undesirable usual rx focus respectively kf rich lot stochastic armed problem risk measure
langevin h drift satisfy discretization proof c position establish constant produce control two langevin langevin discretize choose precise assessment vanish step go thus goal step get lead reasonable trade computational error complement lemma satisfy second langevin diffusion path formulae achieve initial draw view convexity divergence get h function point total variation triangle hand call error time hand side error tv formula desire result satisfy level time output step th addition h mt applicability claim
codebook iterate start classifier rf codebook training decision bag codebook learn quantization codebook decision construct replacement specific node patch respectively recursively right compare gain class serve codebook learn element number image accordingly specific likelihood soft independently patch object class region background patch label patch produce label conventional cause codebook assign soft estimate model
chain step unbiased sample cd perform chain visible alternatively calculation persistent cd assume efficiently mrf produce unbiased learn perturb energy perturbation perturb
perform introduce logic adaboost solve propose operation respectively demonstrate layer greatly improvement significant case traditional dataset vision though decision usage algorithm complexity variety vision support nsf nsf efficient key problem cart operation feature weak classifier combine implement dataset repository convenience tree boost dataset thousand million cart remain mostly logic incorporate algorithm
observe aa one ii variable weak accumulation x n tight apply weakly bound every belong rejection nothing assume would continuity interior rejection continuous part proposition remain element otherwise b b clearly construction symmetric definite nonnegative square root c choice continuity remain invariance assumption yy belong conjunction precede equivalent cumulative distribution evaluate lemma turn equivalent impossible kb first assumption claim turn directly dt assumption claim obvious remain claim obviously invariant w additional condition rejection complement second claim bt obvious otherwise imply kb kb kb absolutely satisfy c iii equivalent obvious relation iv symmetric orthogonal complete definite must nd dd du immediately last xx xu z l spherical symmetry clearly prove proof analogous represent suppose inspection arrive would inspection equivalently furthermore multiply q I dimensional equal prove claim satisfie radius conclude arbitrary every obtain since coincide eigenvalue make eigenvalue q convergent necessarily equal inverse eigenvalue view invertible showing see since w maintain establish since I prop bind phenomenon cite intuition main serious part theorem incorrect proof part particular concentration effect present concentrate already observe somewhat development distributional include weak well allow absolutely theory invariance test convert precise advantage weak argument avoid tool class test treat iii much distributional theory build test autocorrelation characterization situation invariant correlation help phenomenon organize specialized test appropriate test boundary belong complement closure interior rejection region constitute proof
name label language label optimize objective simple optimize objective embedding separately alignment consist project embedding word onto embedding obtain approach learn extend target embedding canonical correlation conceptual relationship word accurate pair word ignore sense natural language two align representation align due train expensive apply objective everything jointly tp approach al formulation train less train resemble feature disadvantage slow stem objective
article hour run svm rbf svm convex l train test train cifar image choose category cifar cifar versus original pixel per hill easily less train train test cifar bit train cifar bit k c train cifar bit remark novel powerful propose circuit classifier enable efficient operation extremely obtain framework compare conventional circuit circuit present circuit vector note gate bits output gate gate look bit boolean
decide hidden layer neural adjust since major safe range ever bad dropout separate number net hide neural net allow hidden net hide net unit except constrain delay fraction iteration anneal parameterization though program round initial minibatch allow neural net hide task neural net select type annealing mode discrete learn straight annealing start final stop anneal anneal ensure stop anneal final believe precise long momentum weight cost except single net two hide unit either unit force deviation initial use weight subsequent control natural bottom adjustment scale weight bayesian optimization large matrix notion
becomes find lead remarkable automatic configuration cdf limitation optimisation base tree understand determine compare recursive feature threshold reduce uncertainty measure mean average could gps find uncertainty follow reduction uncertainty objective cart splitting convenient child gap gp cover unknown unknown variance place exactly
final rank propose algorithm recursive principal subspace reach much requirement algorithm distribute numerical simulation system zero rank signal generate ki ki ki ki ki ki ki variance
divide possibility combine idea enkf variant inspire current status assimilation thus point conduct enkf couple assimilation two extension trade efficiency consider assimilation aspect physics biology flow name assimilation couple state treat dynamical conventional assimilation enkf ensemble kalman update conventional consist mean I forecast observation construct q similarly decompose ti decompose joint filter step kalman construct lead center matrix element reader center formula front dimension one obtain assimilation divide express root kalman respect divide mathematically counterpart formulae divide estimation give formulae formulae framework diag h therefore transform leading eigenvalue correspond eigenvector square root formulae centering previously discuss accordingly I n system background
form noun roughly constrain syntactic require simply subject rather fine needed date expression include address ordinal proper name construction addition usual linguistic indicate clause comparable learn strict connect type considerably english fairly rigorously answer several linkage definition type apply corpus parse sentence broad enable parse word pressure apply much thing lexical entry word noun leave least hundred lexical learn lexical entry common lexical grouping word lexical splitting differently song group together lexical apart lexical observed sentence complexity distinction purely syntactic relation content syntactic applicability fine scan corpus lexical mistake place belong likewise merely subject pressure fine appear syntactic extent place heavily input corpus contain suggest book perhaps semantic appear generalize relationship semantic semantic subgraphs subgraph may syntactic order need phrase syntactic level different subgraph semantic subgraph may syntactic semantic relationship category early stage word become many x entity physical person physical category understand set include set phrase learn expression whose challenge complex construction simple content phrase ground none usually manually construct dictionary contain lexical construction essence construction treat single already accomplish automate phrase lin existence syntactic dependency rather author attempt rather pre existence attack sense markov take abstract distinct term markovian express internal category theory distribute entropy lagrange I np function maxima hill evenly algorithm slow commonly successful network assign naive bayes fast naive immediately independence word independently nearly impossible english viewpoint really count neither entirely satisfactory keeping fundamental outline relation relation view form constraint
differentiable smooth causal calculus probability calculate observational causal effect form observational reliably create challenge imputation nonparametric transform complete estimate demonstrate nonlinear effect conclusion give straightforward parametric causal model unknown calculate turn implement causal could utilize nonparametric causal simple structural causal operator intervention calculus
equation together equation x give result orthonormal unchanged calculation u v v convert orthonormal column thresholde ordinary solution solution result generalize design design select remove
metric large paradigm boolean task take form qualitative difficulty order observe human value predict human literature date b paradigm iii iv vi application task depict definition subsection discuss mention predict general aggregate indicate relationship type calculate classification aggregate indicate outcome participant task look rule neutral individual aggregate reflect type may subject focus fix focus fix learn agent focus rule heavily suggest like vary also display level neither alone capture general order sum quantitative fit metric paradigm block complexity metric find human ccc ccc ccc
ab g g pe business feasibility aim primal dual classifier understand condition determine difficulty aid establish generalization theorem use end von relevant deep margin convergence apart vast topic represent concerned zero specifically exist generalization theorem affine applicable one characterize feasibility problem inequality bind leave right side later theorem affine margin explicitly familiar quantity sec argue subtle especially margin
specialized choice significantly feature identify body something take computer pose prominent influential recovery plan similar pose body mesh inf mh ground mesh term mesh person retrieve visualize deviation edge higher beneficial experimental help viewpoint record mesh sampling predict body measurement application observation distribution height fortunately original training corpus per relate parameter many regression regularize regression good posterior show figure measurement dash line correspond value recover corresponding ground advantage generative ability miss perform depth code parametrization non account inf mh compute show reconstruction expect work propose incorporate discriminative enable diverse computer analyse behaviour baseline inform perform applicable frequently main inference many tailor
central tool process functional h first show l r standard statistic mark intensity inference framework cd continuous r pd pd locate locate outside additionally absolutely assume negative measurable l provide explicit construct section section spatio light univariate property mark stationary uniquely turn distribution mark impose eq independent mark marginal g e f satisfie mark independent retrieve independent weak obtain eq f measure p ff functional mark choice looking mark wiener latter idea indicate measure see martingale see poisson marks setup fairly may filter construction pre l connection reference measure create mark since diffusion setup choose wiener reference brownian motion wiener assign sample brownian q ask adequate obtain explicit density n conditionally diffusion process explicit expression f fm ic c apply apply underlying extension discuss l sometimes mathematically explicit purpose process auxiliary mark l fm b distribution although analogous conditionally thereby mark absolutely case may functional mark markov g mark mark filter ti nm pm ia pm density become turn reference locally appropriately probabilitie
user majority friend friend denote respectively denote stand alignment conduct experiment base rating accuracy user class comprehensive recall map trust trust value e list friend option friend ndcg friend alignment user relation rating user process similar increase rating explicit neighbor compare table align ndcg type explanation application example may specific inaccurate information implicit trust distinguish utilized conjecture metric side distinguish friend lead alignment implicit consistency social relation align opinion trust relation user trust conduct randomly social relation social friend majority social people justify correlation set include review product user user private alignment friend accuracy mae l c mae rmse rmse mae mae rmse mae rmse mae rmse investigate social recommendation experimentally incorporate enhance trust recommendation different pure factorization factorization matrix factorization propose dimension early amount create four different increasingly evaluate mae rmse mae rmse four sample set performance surprising algorithm exploit pure factorization algorithm case indicate incorporate recommendation indicate relationship social rich source recommendation need incorporate carefully nature totally huge relation relation remarkable likely influence friend make trust private nature quality trust deep investigation verify question investigation art end subsection rating second utilize neighborhood crucial make propagation direct neighbor consider propagation trust propagation long propagation level propagation far away user affect recommendation significantly result trust propagation trust neighborhood sense perform propagation constitute user user less neighbor recommendation I
classify exploit crucial specifically since code pls guide learn much figure consistently consider always na initially na behavior almost class visual dimensionality lower exploit lda simply apply pca localization domain adaptation target present split extract sa split art classification setting present result classification theoretical david integrate subspace target subspace optimize extensive experimental principled combination evaluation domain adaptation part work demonstrate convolutional
moment match show gain gain category concrete likely ten em model corresponding differ replace support wireless color video establish em runtime em discuss practice moderate average explore parametric novel algorithm recommendation task naive modeling item similarly advantage acknowledgment work support part grant give give briefly nature condition formally dpp assign regular dpp dpp constant dpp elementary compute term marginalization unconstraine hold conditional recalling express
lie choose auxiliary range payoff round sublinear worst need exist polynomially weight front instance exponentially tune advance block increase length version fix block regret action response discrepancy vector payoff block round payoff component propose length denote let bound range difference remark proof sequence payoff possibly denote integer payoff last block comment work original recent form aim quite demand calibration strategy grouping finitely payoff quantity involve round minimize consist represent loss cumulative expansion control impossible example control block prove ensure hold terminology computationally strategy indeed existence axiom choice constructive explain reference know translate case straightforward noting try quantity towards
rating well alphabet whereas multinomial insensitive summarize new penalize provide estimation bit completion mild value formulation directly grateful en l des big support discussion lemma control consequently control distinguish need leibl point control e I score diagonal eq
eqn proportional q likelihood last eqn discuss eqn model factorize attribute attribute imply probability pl product probability correlation detection auxiliary eqn variance th corresponding respect dimension reject highly product error difference measure decrease avoid eqn eqn obtain eqn contain six simplicity remove regularization simply ignore minimized parameter although jointly term highly nonlinear respect cnn stochastic search demonstrate reasonably decay filter second eqn regard negative logarithm third dynamic update fix current value last thus write loss decay term decay eqn combine least loss
trace norm give large iid mp b express complexity bound theorem exist sphere even likely margin order incur lipschitz eigenvalue lipschitz ambient logarithm potentially curse grant ep international high kernel width mail replace paragraph please hour height ne b skip h ne
variance pp assume development year year year effect satisfy follow estimation quantile formulation procedure model predictive distribution density specification section likelihood present along associate formulate need present al gb structure pp adopt relatively uninformative reflect magnitude instance skewness regression select shape gamma distribution discussion choice al gb prior combine result likelihood model ensure proper set constraint parameter restriction derivation mcmc posterior metropolis hasting rejection parameter value fail slowly mcmc replace allow simple design tune mcmc significant gb intractable et metropolis hasting popular mcmc technique reader suggest mcmc implement available request iteration discard initial burn convergence also carefully check autocorrelation plot posterior predict cell involve quantification predict mean al alternatively central measure adjustment base calculate require parameter integrate triangle include mode predictive interested loss random distribution convolution state feature sum loss low note low tail vary long precise value regular
hypothesis survival length tree significance survival fail detail test test permutation importantly change existence heterogeneity goodness effort article appropriately inferential structured data variant distributional connection develop asymptotic statistical one functional popular albeit see reference therein detail functional know functional tree appear promise weak convergence paradigm wherein ground development worth availability combinatorial tree prevent idea validity result simulation conduct contain branch length partly wherein branch hierarchical approximated normal partial sum lead current binary hierarchical ultrametric use connection exchangeable develop fit informally ultrametric arise sense leave tree choose vertex step tool euclidean space dependence tree structure rarely compare straightforward account attempt model approach use representation path explore tree regression tree structure branch length model investigate brain couple use detect record represent rooted tree secondary structure rna sequence order trees rna describe suitable flexibility method depend choice develop inferential straightforward determining interest support simulation tree dataset contrast probability tree consider wherein topological branch
report report status yes phone status yes mobile phone status yes car car yes status report yes cart report cart yes status report report report status yes material main nominal respectively asset status survey data difference cluster scientific interest aim appropriately asset care project survey economic valuable input serve study structure present mixed idea mixed modeling employ joint mix categorical categorical recently miss latent factor context analytic base gaussian copula cluster early variable none capability model binary ordinal present factor model nominal establish root expansion include extension partial approach model latent lie interval model trait response binary response multinomial probit fitting response treat nominal multidimensional continuous depend covariate item response variable advantageous
constraint namely negligible pyramid train train dnn directly second layer layer wavelet modulus haar dnn unit leave chance coefficient linearity report neural create alternative impose filter architecture relatively approximated softmax mse reconstruct evaluation discriminative setting gender generic per gender include adopt division contain testing compare mixed sir bss metric symmetric discriminative nmf frame overlap feature
explain interval recommend interval bootstrap error percentile interval natural derivation asymptotic interval formula come different sample time ultimately indeed say effectiveness also report side picture biased confidence interval I side test says call short interval trade coverage reduce length statistical interval intervals confidence side cover cover test confidence interval side sided nominal test percentile error skewness adjust statement regularity statistic e moment estimate detail many bootstrap interval early day develop accurate accurate handle skewness transformation sample accurate little thing thing effect percentile poor order order accurate rejection probability differ procedure skewness behave circumstance vertical top bootstrap distribution side top bottom interval bias vertical line normally sample middle correct middle range include truncate center bootstrap percentile interval coverage include right bootstrap scale interval happen bias thing simple center statistic bootstrap percentile coincide exactly side acceleration vertical line truncate middle distribution normal bias coverage text wrong unbiased skewness worse leave simple bias r positively bootstrap biased symmetric end copy bias enough percentile bad copy correct percentile interval would percentile side happen middle percentile bold error avoid miss bootstrap percentile even interval bad skewness percentile skewed skew percentile toward counter intuition intuition may much confidence must especially interval reach right conversely average error parameter roughly normal bias give bootstrap bias bootstrap percentile correct interval whether leave correct whether cause bias skewness second accurate bootstrap information implicitly percentile skewness interval asymmetric cause transformation confidence differ interval absence quick early percentile interval use
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replace side leave hand per repeat scalar summation lead removed compare remove vi state use vi sum upper decrease converge positive assume analysis trajectory enforce stage side become moreover analysis present regard I x generate process implement optimal control cost system establish approach perfect e function play derive result use nonlinear presence problematic phenomenon complete analyze vi open publish include good cost solely valid computer science matter discount infinite analysis easily compare example approximation denote possible write boundedness include value function besides stability rigorous consequence error great practitioner reason great tool control practice vi online control pi start
trial activitie activity etc show ds length ap deal dissimilarity compute similarity nearest ap sc relationships spectral graph partitioning time run make follow conclusion come trajectory choosing point try dissimilarity model challenge sc ap yet still come effective obtain due different target result representative datum ds dissimilarity dissimilarity class ds length model trial change value local activity contain good underlying activity pairwise dissimilarity source consider efficiently encode set propose row problem solve grouping find reveal set parameter representative implementation admm hence reduce two categorization representative model motion representative model currently statistic theoretical theoretical make prove order need notice
attribute perfect attribute classifier score add km km draw mean keep follow noise specific level plot axis test pick market chemical art school yu share conjecture principle shoot possible specify category generic attribute classifier category property possess even provide standard zero shot suffer novel image random forest explicitly account attribute obtain robust discriminative unseen class devise extension shoot bias mid attribute develop signature category operate characteristic idea association shoot number demonstrate three show clear advantage suggest valuable attribute play object inherent reliably shoot two stage approach give novel attribute predict attribute unseen object
require depend version bound prove statement uniformity tight bind finding reduce attempt tight factor regime bound instead interpret regime match strong statement prove associate game parameterize set support random I correctly otherwise sample failure convert oracle run algorithm obtain member game occur output correct answer triangle inequality fashion construct sphere pack code constructive merely win game intuitively probability win look like imply constant win p nn nn mention examine another al equality distribution distance uniformity metric extend utilize uniformity hold translate coordinate sample exactly number focus small prefer tell require learn versus uniformity use author knowledge order regime open uniformity chi arise closeness establish question may seem author match come biased coin distinguish coin bound perhaps give result convert
spline locally allow suggest fitting bandwidth insufficient smoothness fitting occur measure degree select smoothing monte explain sec thus stable datum heavy tailed mass however cause curvature heavy tailed worth em realization shape I exponential combination realization event simulation algorithm exploit branching process formulation estimate use em choose bad start procedure parameter uniform random branching ratio choose branching ratio fix point expect process bad another introduce overlap clear branching systematic study mle em overlap decrease synthetic realization c address process simulate nan model generate aic ratio level ks discuss option test type test alternative hypothesis
pca share filter part find violate half run bottleneck grow template contain candidate location bottleneck qp line qp solver ratio bottleneck second method discover seed train seed candidate top part well despite use heuristic slow comment correspondence long independent multiple experimental well pool part note take extract apply transformation see definition optimization visual group visual school science university uk representations useful base view collection informative discover use response part randomly select good subset namely intend correlated part discover look correlation computer make reason part invertible nuisance factor generation include break visible reason part variant object scene often object cat faces discover contribution unified framework
correction field isotropic task fmri analyze first tool temporal cutoff number algorithm reader deep area test statistic conclusion robust arbitrarily strong statistic fdr use differ conceptually regression recently propose paper relate local fdr smoothing rather covariate fdr differ fdr ordinary smoothing differ ordinary build testing assume statistic mix describe report appeal offer may discovery discovery recover know estimate parameter important call either control adapt unknown involve conceptually simple use spatially adaptive discovery defer fmri encode three change site site odd log odd let assume odd regularizer impose penalize define odd lasso differ mathematically differ conceptually size orient adjacency th
wang li al among establish depend two objective full section residual sum describe screen rule minimize separation exist p strongly separate large whereas large fairly separation imply screen n borel denote assumption hold separate n n nx n h nb nb nx nb nb h h ax complete assumption note restriction satisfy hold asymptotic literature de conduct simulation verify fitting well result multivariate whose vary angle fan independence screen sis well produce sis screening sis cr c sis solution optimization solution whose objective simple framework obvious effective theorem follow way separation basic include subsample robust statistic besides new subsample selection distance method well
state reversible jump crucial well want accommodate mixed support circular circular measurement circular modelling mix circular visit propose problematic particular known identifiability hyper addition mixture hyper density difficult introduce rely project normal distribution study behaviour association recover basis hide wind period km organize review discuss specification circular project provide large application real conclude circular direction circular challenge meaningful directional many circular book overview onto circle
express filter mechanism might odd goal also attractive algorithmic synthesis henceforth refer np ignore large penalty work mean lar package seek minimize approximate approximation penalty activation exponentially distribute mostly omp also problem simple feedforward hide pick encoding must update also nonnegative
propose serve tune double distribution linear updating equivalently aggregation add additional acceptance set propose calculate ratio corollary proposition remark section ensemble outperform linear approach motivated regression list adapt receive open truth convex linear tends concentrate good truth illustrate simulation dirichlet aggregation learn minimax misspecification many pick suitable model set problematic hence substantial practical aggregate combine obtain aggregated potentially single one towards attention search convex combination focus select combination aggregation function iid aggregation randomly primary interest adopt fix translate context basis orthonormal overcomplete weak learner special map average place updating
property contraction von distribution error typically derive robust misspecification slightly slow contraction assumption research lead european grant support grant bs e full high continuous compatibility contract sparse study credible quantification class regression possibly close selection coefficient priori prior usual reality select zero product coordinate crucial one finding weight decrease exponential performance theoretical paper identity see model share case must take account oppose factorization axis entropy refine prior laplace distributional insight nonzero coordinate contraction spike example routine dimension dimension result computation number develop cope consider recent various feasible value hundred thousand increase come clearly short dimension programming approach cope truly model present time surprisingly overcome see sharp
flip tf fs flip tf fs flip tf flip tf fs flip tf fs flip tf fs flip tf fs flip fs flip tf flip fs flip tf fs flip tf fs tf fs tf fs tf fs tf fs tf fs tf fs tf fs conv conv filter convolutional geometrically filter reconstruct output method sect horizontal flip flip rotation representation abstract mapping sect one look demonstrate representation layer manner fig transformation learn empirically sect importantly transformation sect new able property allow invariance build equivalence look whether information million optimisation may question whether apparent network equivalence obtain cnns sect sect
noise rbm term equivalent benefit bernoulli rbm benefit rbm positivity condition rbm put benefit condition benefit predict improvement deep learn careful contain cell need dna cell rna outside protein synthesis occur code gene cell gene cell adapt receive expression code dna control turn factor gene bind site locate basic bind specific complicated bind site fuzzy inexact give experimentally verify admissible variation bind specific single bind area miss exact dna model dna sequences discrete take learn sequence explicit em length mm generalization success human genome bind method success version supervise method strongly principled noise exponential em likelihood log reduce simplifie follow condition benefit replace art year idea emission advance advance gibbs hybrid frequency detail generalization accommodate maximization figure mr image segmentation mr brain voxel voxel grey matter effort automate difference water mr mr mr water unlike medical modality water mr within distribution current water segmentation post process automate annotation identify characteristic accurate water assess risk diabetes disease relate segmentation intensity mixture gmm pixel classification spatially aware hmms localize localize traditional algorithm causal involve show image segmentation also prior show apply mr image expensive speed apply noisy fast show improve classification examine fuzzy bayesian problem conjugacy rule rule bayesian idea inference convergence sampler pt chapter chapter school california partial requirement electrical grateful development work much growth year thank feedback closely grateful many friend support I like thank teacher united states control cause converge basic scheme neural fuzzy general derive theorem consist short theorem chapter broad em em variation algorithms review discuss expectation intuition derive give noise speed chapter derivation end enhance beneficial sparse artificial heavily mean algorithms backpropagation proof theorem bayesian chapter approach technique approach imply apply analyze effect approximation approximation bayesian uniform fuzzy approximation validate use fuzzy inference via chapter discuss extend show noise sample substantially maximization iterative corrupted speed benefit present derivation sufficient benefit demonstrate speed processing include backpropagation feedforward artificial analysis corruption general framework main uniform via multidimensional expectation maximization algorithm discussion chapter schema converge slowly dimensional incomplete problem expectation maximization chapter describe simple corollary cluster model benefit indeed theorem secondary expand function pdfs bayesian closed function function uniform lead maximization em lead map continuous property exactly replace equality define recursion solve condition general generate let closed convergent stop sequence infinite convergent subsequence monotone converge subsequence subsequence original advance derive convergent subsequence closure assumption contradict compact sequence apply interior current estimate element eq large compact generalization parameter general satisfying condition condition imply iterate point map map theory nash equilibria fix point give map apply always close map close incomplete base exponential pdfs application censor gamma condition partial derivative close map may restrict maxima instead non singleton flat sequence kind kind easy detect wise estimate sequence information carry approximately measure show information prefer measure standard fisher quantify complete complete datum miss datum observe expect fisher decomposition principle parameter estimate al em current em ratio neighborhood high slow number variation step forward variant root hard another multidimensional replace single conditional may perform multidimensional idea iteratively medical approach rate proportional amount embed iterative em may iteration e reconstruction space every pixel voxel liu liu
appearance correspond exchangeable log upper outer leave part show negative part show region exchangeable ce crucial exchange provide upper provide improve span optimize appearance exchangeable consistency constraint probabilistic work area demonstrate possibility highly connect arise frequently program language
forward propagation correspondingly problematic computation output output prohibitive since neural network output vocabulary become increasingly propose attempt address difficulty fall fall category impose define initial kind present actually gradient implicitly computationally manner actually compute
encode string report thing token pattern sort search store carry check contain dictionary extract match object size dictionary number pattern distance dictionary extract intersection dictionary concatenation drawback limited table totally token type constitute text contain plain english mean additional
expert importantly quantity expectation randomness signal conclude rule explicit form underlie intuition omit provide investigate exchange time reach connectivity play w knowledge markov invoke technical give connectivity w establish convergence rate govern characteristic generate true assumption strong agent lemma converge delta state let borel interested let proceed cost function round numerous expert adversarial belief bound identifiability strong connectivity q explicitly calculate apply primal dual bind centralized identity take right
minimization associated number completeness admissible triangle use observation hand find put estimate conclude subgaussian define assertion subspace subgaussian set recall main wish recover represent measurement terminology subgaussian probability measurement projective quantitative statement theorem entry assume dimension respect note statement subgaussian idea expect argument base approach riemannian induced notation terminology tangent associated tangent piecewise curve geodesic subgaussian map sufficient preserve
lt age shift lt college college sometimes le le sometimes heart abuse abuse degree include mml htbp mml htbp lift none size narrow lift lift class lift lift none rule lift color lift size narrow none none color color surface narrow narrow surface eq narrow eq leave error mml use fold cv experiment train long minute ip mn rule oppose ip thus mn rule involve compare result long classifier path plot mn train accuracy tailor tool diagnosis lastly numerical accurate interpretable many use ill literature interpretability particular predictive many relate comprehensive review popular assess interpretability distinguish visual popular linear address work table interpretability improve sparsity ensure monotonic relationship predict restrict popular neural support machine ensemble interpretability box mainly auxiliary extract prototype relationship predict useful practitioner accuracy interpretability practice interpretable require control multiple require address control multiple interpretability relate practitioner model monotonic exist way measure surrogate interpretability measure heuristic procedure round poor interpretability robust match non restrictive condition issue aside poor interpretability extensive adjust available parameter claim understand method design interpretable market produce interpretable domain study often predictive interpretable domain produce use highlight agreement outcome hinge discrete round ip classification discrete practitioner flexibility previously lee many train classification minimize early attempt minimization improve modify heuristic design specialize recent feature framework linear accuracy addition reproduce interpretable literature often accuracy pt control interpretability mention ability incorporate need mention address database address
discuss syntactic condition multilinear decomposable validity condition theorem degenerate strongly decomposable complete multilinear transformed degeneracy hypothese validity condition easy verify c section restrictive condition limit expressive ask efficiently criterion weak question prove provide extend np hardness property checking goal advance expressive ultimately focus well circuit efficiently logic circuit overhead polynomial easily constrain monotone arithmetic input circuit obviously simulation boolean logic circuit would monotone arithmetic circuit impossible circuit negative value monotone seem much insight general multilinear multilinear circuit moreover make arguably theoretical place conventional deep meanwhile ensure validity aware also condition validity seem arithmetic circuit constant kind structure kind perform something compute become impossible kind efficiently despite apparent efficiently example prevent basic operation work design formulae couple computational show simulate system close like vector multiply whole product give formally fix permutation constant input maintain crucially process limitation read function dimensional distribute possess limitation lie representation linear transformation positive entry simulate compute fact suppose one affine special multilinear branching polynomial require exponentially function computable polynomially section circuit decomposable univariate proposition convert decomposable size provide indeed polynomial require exponential prove fully characterize capability compute efficiently existence similarly even efficiently simulate work initialize state process fix stage transform real permutation sense state decode map g machine state size subject restriction input ahead allow large enough see compute would grow exponentially bit number bit combine ability combinatorial high capacity state establish size dimension proposition state unlike readily easy construct wish
assume shift goal x target domain indicate come source target example training time want feed predict decompose map vector feed label mapping finally map domain aim minimize annotated part optimize minimize ensure overall want want xt covariate shift assumption make prediction domain source measure dissimilarity change dissimilarity classifier provide train discriminate feature idea domain feature seek possible minimize label predictor multinomial domain evaluate saddle minus predictor prediction maximize
contrast behave fix difficult create incoming new overcome development systematic learner structure ensemble naive learner drastically ensemble component desire highlight increase sophisticated procedure construction practice subset allow acceptable accuracy generalization context sophisticated dimensionality play crucial role uniformly
rate quantity suppose estimator rate dimension demonstrate mix objective gaussian might hypothesis like attain bias seem population shall biased examine power alternative hypothesis increase might suggest power conversely power let emphasis gaussian dimension mean origin dimension need decide suggestion possibly unclear choice verify communication slowly affect author former power fig make justification
shorter noiseless fig determine unstable take prediction also instability instability sense analysis positive compressed sparsity show demonstrate gaussian matrix predict recovery dimension theorem conjecture theorem theorem question remarkably decade signal small synthesis measurement logarithmic factor signal signal use
column range true scenario see example value fit change interval aic tend change sublinear optimal
disjoint separate sufficient conditional practically check hard hand read kernel induce graphical important understanding thm thm corollary elegant graphical proposition
decade mention deep chain converge chain numerical correlation chain need posterior independently start collect chain take chain parallelization approach parallelization exploit example come target execute decomposition remain markovian technology generate single draw method gradient tool component function posterior model gibbs sampler implement small require strategy conditionally conjugate might possible augmentation transformation method action conditionally sale example hamiltonian hmc software posterior benchmark complicate assess method parameter think estimate reasonable illustrate inherent mcmc package example heterogeneous unit deviation place conditionally gibbs describe confidence indeed iteration get need proposal mac minute draw collect take minute collect sample processing median ten show population unnormalize tb example sale sale store week shape price volume week covariance inverse wishart conditionally
algorithm far variational similar nonparametric modeling variety nonparametric principled nontrivial literature use sampler sampling suffer poor scalability involve beta successful standard derive atomic random measure stick break yield construction integrate mean inference instance stick break seminal inference dirichlet stick break naturally lead model variational framework stick breaking process derivation measure
illumination illumination illumination cluster cluster matching compare balance hull near propose affine hull adapt show local distant cluster restrict middle variation enforce query constrain convex reduce convex treat constrain cluster adaptive adaptively distance reference take indicate distance fig conjunction comparison object method adaptively cluster base continue follow briefly affine hull technique describe evaluation comparison research represent affine hull distance near
base code link depend desire back programming facilitate particularly extensive aggregation scheme insufficient extensively broad overview development library mind major lower
mesh motivation voxel analyze voxel intensity neighbourhood instant possess illustrate intensity instant axis correspond separate cognitive discriminate voxel classify voxel slight squared intensity value voxel carry intensity instant vector voxel neighbourhood system high power voxel intensity mesh voxel employ neighbourhood fully activation brain spatially distant neuron neighbourhood model euclidean cognitive additionally spatially voxel strongly couple cognitive defining may redundant mesh arc improvement accomplished usage functional voxel neighbourhood feature vector whose consecutive connectivity voxel voxel example pair connectivity spatially distant inter analysis connectivity brain node correspond voxel connectivity study connectivity
diagnosis beneficial science voxel approach development may enhance interpretability even identify voxel biological etc classify human inspection motivated classifier attribute similarity superiority technique strongly suggest capture subject class dark knowledge within classifier study classifier effort make visualize
preference evident supplementary quasi restrict quasi ultrametric state include partition quasi dendrogram represent resolution singleton method formation asymmetric fig resolution form singleton cluster influence highly asymmetric resolution california every depict leave california explain fact california large area country california influence partial resolution permit california force minimum height leave thick bend leave thick bend bend edge bend bend edge bend bend thick bend right bend bend bend right importance resolution california important ordering form three precede happen already hierarchical quasi application asymmetric undesirable generalization asymmetric formalize notion introduce axiom framework rise new admissible axiom furthermore equivalence quasi invariance property application support nsf fa dms nsf dms map ultrametric dendrogram dendrogram quasi partition partition right continuity quasi ultrametric attain non attain negative clear identity imply boundary since conversely identity strong triangle inequality node ensures equivalence hierarchy property x partition definition eq q definition substitute consequently satisfy triangle ultrametric define converse result show
branch sdp cut plane evaluate experimentally suppose mrf characterize undirected node index unary potential mrf unary without generality lp polytope space q definition singleton node globally joint vertex extreme correspond problem difficulty optimize polytope outer l example vertex cycle fractional lie outside relaxation guarantee lp relaxation map cluster consistency constraint ed consider symmetric semidefinite discard rank convex outer generally boundarie polytope semi outer mutually neither dominate marginal polytope semidefinite outer work solve result general lp integer problem round lp sdp feasible round sdp relaxation base singleton pseudo would done label cut hyperplane round sdp relaxation expect objective paper simple rounding solver overall b subproblem branching present briefly optimum feasible relaxation feasible separate branch proposition branch global branch terminate lower sufficiently show selection branching branching subproblem selection strategy find investigate rule branching selection rely improve initialization bound queue subproblem
observe build fraction direction derivative direction effectiveness sampling manifold ref ml approximate quantum dft highly self energy principle dft test gaussian low cauchy cross test prediction hyperparameter generalization set construction grid calculation far explain origin search euler effectively project introduce pca constrain project work prototype yield highly constrain energy thank nsf kb correspondence li li rgb rgb rgb li li non interact functional explore highly energy accurate modify euler lagrange project derive effort research review dft functional reader dft typically fall derive principle tend work error treat certain semi
gene biological provide good usefulness rank set patient ac uk patient moderately severe patient severe measure intensity value indicator eventually protein serve discriminative outcome estimator though could favor framework filter step yield store patient iv demonstrate rank predictor version estimator fold lasso predictor predictor response average iterative sis
affect noisy happen far journal sharp observe topic journal decrease star system cell much vocabulary document document proportional edge likelihood document assignment fact use data compute mean never scale much see fig dataset example corpus computer minute minute english wikipedia decide link unique less million result decide total graph word roughly hour extremely take hour filter iteration take hour iteration run word interestingly change guess significantly lda synthetic likelihood explain topic merge small variation inference function topic heavily broad use fair comparison indeed high see degeneracy landscape provide well web topic area perform well lda splitting cell american economic comparable comparison likelihood model english sec english first group equivalent english fall binomial ingredient fit go back call english average q hold q word fall english entropy variable
spc performance scenario perform slightly tight initial categorization without include cluster clearly demonstrate sensitive pre specification noise recognize cluster group mean mean recognize resample show ari score well leave clustered might challenge become order reasonably correspond neighbor mention qualitative comparison lead previous demonstrate converged figure case take iteration solution table run fast spc tight calculate length include tight mean search prediction resample spc currently implement cluster write r spc good especially ht ccccc separate separate overlap noise performance spherical dimension group normal high correlation around one show true gene expression spc relatively size big green plot spc perform recognize small assign end normality data spc compete apply cluster dimension convexity ari generate present spc high score perform cluster usefulness spc high spherical comparison non comment path tuning parameter could
solution bx bx bx nonconvex approach consider nice dc decompositions dc program original sequence sx sides ii assume kk kx contradiction subsequence proving theorem replace zero approximation optimal tight approximate contain solve several context suppose concave define bound z carry resp concave bound resp general carry much result nonconvex motivate local minimizer describe problem I local optimum kx x implication obvious backward exist neighbourhood case case fx problem state equivalently solution problem dc dc program necessary local condition express equivalent fx able set eq q hx g gx fx ki b ik b kx k k prove mention approximations first concave context smoothly logarithmic apply concave increase verify particular dc general dc investigate
illustrate evaluate completion quantitative evaluate firstly incomplete complete error rate method subspace cluster sim ssc handle h preprocesse fail motion object produce miss error sim ssc segmentation miss possibly exist trajectory entry cluster improve dramatically advanced example verify realistic sim sim sim max std min std ssc algorithm ssc std pointed could call
deep feasible severe lack training medical imaging cnns generally number use image patch purpose effective amount apply classification improve view filter name max layer layer final layer recently publish call early dropout allow open source gpu
constructive repository proof algebra necessary clear distinction theorem proper dependency ultimately explain learn well need dependency among prediction theory corpus discuss future dependency extract development describe overview base formula encode logical essential bind environment
variety galaxy shape human shape code allow dimensional space shape statistical reduce
description involve additive predictive addition additive heavily penalize interpretability version accuracy predict divide small outperform interpretability construction great outperform mkl despite poorly dataset explain show outlier square linear regression occur centre supplementary material divide result supplementary material open detailed describe capture demonstrate discover describe pattern interpolation building technique non thank helpful part google code experiment scalar input
negligible simulation gaussian probability p denote gaussian terminal linearly joint notice depend value correlation expression mmse source noise denoise message pass correlate bernoulli measurement random see match se successfully recover run signal rate two contour rate distortion source dash line boundary case independent region contrary private obviously independence individual rate enough gradually result
preserve preserve embed canonical cca pls linear suffer input vector may bag similarity instead representation linear handle handle version end solve central choice make automatic combination application multiple initially receive significant attention excellent mkl extensively relatively explore mkl instance ratio optimization impose general lin extend cover ratio trace formulation refer reader trace ratio mkl result optimization iterative guarantee similar mkl lda trace equation lda cca pls optimum
active inactive treat free base early lin start pick index solve sub optimally newton express equivalently define onto hyperplane one convenience represent simplex denote interior simplex problem homogeneous collect elementary work optimum limit dirac peak insight actual regularity description scale consequence application amount operator scalar chain onto projective space line e equivalence projective identify sphere equivalence canonical step end denote distribution restrict union hyperplane superposition least progress rate eventually progress define nz define w tw tw trivial consequence invariance existence lift depend homogeneity homogeneity technical converge exclude cycle homogeneous coefficient progress capture define progress note coordinate everywhere set multiplicative ergodic corollary limit denote norm immediate consequence chain distribution obvious rate coordinate adaptation maximize function dependency problem aim towards cd converge coordinate arbitrarily boundary remain continuity maximizer simplex identification problem distribution goal adaptation could
vocabulary practice numerator normalize instead opt specifically probabilistic root node word associate leave branching path path branch observe bag use full decoder word bag thus decoder share internal decoder compute assignment leave well choose decoder eq
datum hypercube block contain partition block location next subsection convex solve precision matrix calculate distance block form always matrix assume magnitude diagonal attain relatively use multiplier proper close n similarly map efficiently auxiliary equivalently augment dual multipli linear constraint primal admm iterative primal tolerance condition experiment converge acceptable admm iterate guarantee constant minimizer penalty admm admm iterate linearly particular q c opt opt z rp opt return method block fit framework assumption hand selecting obtain
q assume strongly recover q minimized recovery sufficient mn mn definition since
distribute weak learner local base communication cost network graph depict star improved select parent receive child root back node work agree tree far operate neighborhood agree span communication construct simplex atom restrictive since proof design optimization amount communication atom node atom evenly two one instance select atom atom need show minimize f deterministic lead select atom bound precision communication algebraic approximately vector near base orthogonality near length precision near simplicity
break arbitrary form satisfie x lemma use u p v p positive homogeneity I vector increase norm homogeneity trivially convexity lemma combine
comparison passive risk default definition might achievable optimistic sequence regularity probability achieve challenge relatively extra active predictor therefore label predictor equality label budget ki kt list alg operate stage stage
rnn achieve capability especially long help accuracy click prediction unfold rnn algorithm unfold incorporate unfold necessary effect rnn unfold understand accord increase unfold good auc achieve unfolding drop check discover error vanish unfold explain unfold span several lead unfold dependency implicitly accumulate affected event explicit run history background intrinsic click click traditional click temporal recurrent structure art continue several aspect build user user pair even whole go
design set I usually lead excess bound independent distribution free excess parameter latter excess deal hypothesis large large factor close upper excess free
hold random euler gamma scaling sequel valid claim next mean et r proof q asymptotic vary suppose depending imply asymptotic j u
achieve relative improvement htb bt unlabeled bi svm center possible load hold vector gram news corpus word repeat experiment loading sentiment stanford rate grain grain task improve attention retrieval task length return engine content web page web page query test triplet result query three paragraph distance representation close paragraph paragraph paragraph call call report american paragraph call share information paragraph health patient pay pay triplet split
mf three continue rate test three layer usually quite usually gradient essentially sum layer try momentum lot mrfs edge energy sum potential eq crf make explicit simplicity apply conditional model mrfs image denoise optical flow field find factorial
use nonlinear transform reach stack reason degradation layer dimensionality layer improve greatly good pose view information view largely cca pt cca pls probe varied fix varied conduct view sketch view utilize benefit fold criterion gradually narrow gap two view cross image superiority method art world view endow capture reveal object image face modality condition pose computer
summarize statistic attribute ks test numerical attributes hellinger univariate define maximal numerical ks test size get picture hellinger discrete percentage nan hypothesis attribute average strict important similarity base rand set overlap u n u uv clustering base counting agree category contingency value indicate rand rand rand take value clustering clustering adjust rand contingency q ari ari ari instance instance datum cluster similarity near cluster get clustering instance union generate clustering base illustrate assign exist partitioning besides output use cluster distance instance attribute dissimilarity nominal attribute dissimilarity attribute cluster ari compute probably good substitute largely dependent classification classification performance indicator original outlier capture comparable generate show
open study examine exploit heavily topological effort precisely general fr metric satisfied verification increasingly geometry complicate various geometry challenge allow pose answer analysis propose remain additional fully establish capability limitation recently brain one tool order manner call leave determinant want ball continue first zero must thus symmetric determine block determinant block matrix entire perturbation preserve still conversely symmetric parametrize satisfy alone cone intersect open component positive zero conversely stay open prove manifold convexity statement graph rank show lie affine none hull point plus determine affine convex hull dimension convex choice since connect let euclidean segment neighborhood exist lie thus corollary proof condition basic manifold corner corner coordinate move top left corner take corner chart chart corner corner convexity affine follow nonzero determinant corner
search direction rapid construct intermediate direction variable general algorithm proximal proximal gradient proximity operator descent proximity proximal rapid involved gradient alg equivalent eq current efficient constructing follow guarantee comparison illustrate difference auxiliary variable rapid auxiliary
imply hadamard margin complexity triangular imply probability max combine select large enough include norm well use lsh fortunately require point generality database inside sphere indeed establish existence lsh refer pair query simple universal lsh need hashing database database two review threshold compare recently suggest hash quality movie parameter value follow pair mapping combine q
crucial step fully resample weight intuitively marginal assign specifically fully accord simulate optimal simulate point equally weight approximate monte carlo approximation primarily normalize partition turn provide estimator unbiased due channel
noting lastly encode direct cycle encode acyclic topological cycle admit topological represent zero encode positive well encode encode ensure cycle cycle encode cycle penalty represent direct graph inconsistent neither feature weight ji positive inner arc represent arcs network head spike arc bit variable fully arc bit hamiltonian row represent bit problem color logical set arc arcs bit total enforce slack bit parent bit hamiltonian indicate bit together hamiltonian encodes dag local term full
experimentally convnet whiten fig accuracy template template believe template improvement weight run convnet beyond gpu management issue similarity template carry weight template similarity template convnet achieve accuracy unweighted similarity hand similar convnet layer super linearly number word accuracy would template reach network template considerably accuracy template despite fact latter overall size worth performance unweighted convnet confirm hypothesis analyze shape decision region unweighte essentially hypothesis unweighted abstraction far defer accuracy problem network account variant eqn good code high absolute art observation whiten architecture whitening template use facilitate fair modify template svm sgd modify template reach high level author turn triangle improve accuracy phenomenon reveal local minima lead successful find report template code reproduce display template template template peak peak template almost fig importance sec template zero mean unit variance accordance patch
model solution tuning solution algorithm base occurrence leave bottom leave co occurrence might introduce employ algorithm specify linkage misclassifie bi f stand mutual standard real uci repository figure consensus four consensus solution contain breast hard optimize solution balance cluster cluster employ p lr lr lr cf c sc sc sc sc sc c c sc c ground b propose consensus propose mutual parameter involve dimensional ambient lie dimensional subspace aim subspace represent matrix case define parameter apply normalize node affinity wise cluster segmentation video object frame video object treat yet group instance propose bi consensus instance value consensus subspace sensitive wrong number analyze result range enable consensus low misclassification feature lr lr b b green computing cluster instance mark dotted line competitive misclassification consensus solution without video project pca base subspace misclassification consensus bi cluster cluster discard miss help understand need bi lr lr one size extract bi bi must row column cluster compare size bi return discard drop
success variable internal node put prior easily proper posterior make smc basic note conditionally value leave normal value propose node internal analytically pass therefore exactly two leave propose weight involve density compute message pass implementation perform qualitative agreement economic indicator five high child school distribution tp internal comparison method method within variance standard intermediate distribution forest post internal implementation implementation hamiltonian algorithm marginalization kalman inference limit baseline experimental setup measure efficiency effective sample ess convergence ess implement std non standard smc measure intel e processor ghz experiment ten begin ess run particle ess std perform mcmc concern ess min deviation inferior explain section sample impose ess smc mcmc investigate indeed challenge remain marginalization normal baseline std slight additional support normalize little k respectively reasonable assumption c outperform computational roughly std accuracy implication particle mcmc light recommend estimator particle deviation close particle case deviation std number experiment distribute environment idea particle description implementation implementation running particle second less single sampler
covariance matrix diagonal prevent issue power exact adjacency probability great property lemma nz I center gaussian take high bi unbalanced matter chernoff choose left degree may graph hence deduce draw let thresholded recover whose regularity bernoulli regular particularly lemma bernoulli q input important impact uncertainty aspect
amount approximately iteration conjugate gradient length present gradient via adjoint depend make recall direction free discretization I particular involve structure inexact newton references newton termination criterion cycle backtrack regularization warm section gradient hessian action cg express adjoint quantity action pt algorithmic scalability result component maintain core ingredient overall scalability exhibit algorithmic number increase indeed outer newton cg mesh hessian operator hessian exhibit mesh cg computation gradient fr lagrangian formulation calculus strong velocity pressure adjoint velocity pressure problem account pointwise adjoint stress fourth identity adjoint notable velocity nonlinear adjoint adjoint velocity pressure characterize linearize tensor act adjoint jacobian newton forward consequence self mesh solver problem newton step forward discretization field velocity pressure pair adjoint pair surface boundary hessian gradient evaluation expression form linearize solve hessian presentation expression defer hessian play role solution performance inversion section scale observational ice use newton slide decrease inexact characterize tie mesh slide velocity treat field make tractable inversion characterize linearize must core underlie objective vector product computation dominate cost remain newton algorithmic performance fine
choose back issue decide second notable go result mix explain filter denoise signal child heart
interested happen experiment experiment leave impose future annotation query match return excellent platform manual fit researcher find phenomenon take relation possible alone term take literature obvious limitation capturing demonstrate capture retrieval imply natural good additionally bring experiment miss feature computing model model include query however approach implicitly capable somewhat contradict retrieval experiment different small approach model
nonetheless hand great cccc model pre naive latent logic friend friend fan fashion fan fan friend live live device like friend devote automatic profile social major level public extraction study examine interest political year propose highly extraction education job base training distant supervision google treat use base supervision return predicate job education fundamental assume share attribute chance friend medium friend share adopt like attribute extraction extraction seed learn additional pattern distant supervision another methodology source supervision raw logic reasoning logic order logic representation day ai variety reasoning relational logic language capability type logic logic log linear incorporate relational dependency network combine bayes jointly consider reasoning logical reasoning language understanding web cluster propose
six fista algorithm fast support recover compressive consist proximity art array fast accurate algorithm recover sense array case ill pose address regularization seek eq form
identity replica bx ab yield expression average follow statistically independent finite q ab ab saddle expression operation lead restrict candidate dominant saddle point analytic restrict replica replica cavity utilizing provide analytic n df n account become lead particularly eqs dominant include complicated handle significantly simplified allow st replica compose z p ny ensures involve change saddle summarize equation evolution notably bethe usual replica factorization section evolution derive mmse matrix factorization bayes optimal datum evolution arise analysis derive section pca blind definition basically interest blind trivially take equation discussion present generate eqs obtain explicitly eq eqs channel additive eqs evolution greatly simplification equation evolution explicitly dictionary mmse evolution mmse find phase transition uninformative initialization limit correspond fix instability symmetry limit initialize plant constant entropy phase simplicity uncertainty completely unknown take learn blind calibration pca blind source separation problem eqs number channel distribution computation large obtain q point locally large bayes mmse remarkable implie set identifiable large give count bind next question whether mmse tractable answer uninformative behavior state expansion lead uninformative evolution mmse achievable g approximate message analysis see blind side let compress particular sense identifiable recover static phase amp match mmse sample take right transition finite value see phase dash calibration achieve amp surface plot mmse signal sharp replace smooth mmse link dictionary blind try possible plot negligible value less know matrix amp mse
shape satisfactory measure year fix previously repeat one converge upon hyperparameter eq likelihood numerically laplace expand allow partial brevity order hyperparameter require computationally arise search space direction occur invariance latent let unitary consequently invariant unitary transformation depend via unitary undesirable consequence firstly partial positive secondly representation infinite rotation latent eliminate
always noise error relative yield well representation next repeat paris city range pca art ssc one report result relative reconstruction thereby learn subspace capability mc report denoise handwritten digit demonstrate nonlinear denoise unlike mc belong compare kernel kernel eigenvector fix number te x te te noisy eigenvector oracle different subspace mc subspace equal mc training level te te te denote pre rely cope lie high always embed improve also help communication storage requirement geometry high dimensional utilize application denoise segmentation broadly fall two category learn latter drive geometric outperform lie near hilbert transform application nonlinear generalization linear remain computationally investigate decade popular generalization manifold also model mapping
performance phrase rnn encoder decoder contribution rnn encoder correlate expect try word neural network I shorthand opt penalty able achieve ht come analyze pair score encoder decoder translation phrase corpus score phrase sec expect without phrase rather linguistic occurrence corpus focus pair phrase phrase phrase target phrase translation rnn encoder decoder perform phrase list target phrase phrase either translation encoder phrase phrase encoder decoder short phrase pair rnn pair different fig arise rnn encoder rnn encoder learn simply frequency phrase pair
case case localization w return dp also desire runtime w claim w part tp hence cc large enough conclude lemma exist px let establish xx xx indeed dt te n know check polynomial proof
rgb new statistic fit well network observe take spectrum diverse algorithmic generative network statistic assess gap role inference dyadic covariate arise rule diversity research relatively develop assess goodness describe assess type community substantially gap leverage spectrum statistic percent make goodness provide observed exist assess roughly group simulate fit function assess fit structural actor orient know framework algorithmic subgraph approach fitting comparison simulate ask draw simulate network unlikely generate well specify readily example theoretical
ni consensus iteration distribute power satisfie mix mix u state exponential case tell estimate due iteration look block r k r help characterize cloud svd centralize perturbation recall definition sec sec express I mix along make arbitrarily long perform appendix rely usefulness cloud use demonstrate cloud svd representation deviation cloud centralize help simulation mnist cloud svd benefit distribute site enyi synthetic individual column distribute site site randomly follow noise carry average weight describe update power method cloud oppose canonical centralize canonical svd cloud svd see carlo trial centralize local cloud illustrate power cloud svd consensus consensus iteration denote eigenvector distribute power iteration monte trial power distribute fundamentally consensus highlight cloud centralized still first experiment experiment method iteration collaborative centralized error site monte
spread influence social influence medium compete information nature content recently diffusion work information diffusion diffusion property correlate instance event formation edge work focus arrival follow even behavior dynamic steady explore result like behavior user receive edge event cause information diffusion event lastly accuracy spike provide insight cause occurrence spike spike well examine aspect compatibility create content lead insight additionally tweet foundation grant yu fellowship microsoft like twitter give social consume content create underlie social individually study past much know user interact evolution sharing examine twitter post create connection structure rate cascade post create significantly discover cascade phenomenon way drop connection source refer create connection refer follow user increase cause discover interest diffusion connect exist similarity
rate feature sc na normality misclassification sc misclassification sc overlap entry valid layer layer entry sc mean variance layer layer na ht c identification misclassification sc misclassification spherical rate misclassification simulation primary identification datum identify sc var joint participant study participant show return contain patient individual follow exclude survival analysis section measurement pressure heart quantitative variable sensitivity experimental description contain third contain overlap select identify structure type exist computing case auto sc min var var layer df df df value membership risk subject exclude analysis third second neither sc fail ht c statistic df
seed force seed replicate mc windows program number single stream reproduce platform serial execution window explicitly create stream may key aspect latent trait response variable produce possible pattern variable parameter via monte analogously function justification focus example point quadrature grow error monte monte integration high dimension univariate integral evaluate integration remain dimension value general sense conceptual payoff mathematical simplification estimator typically suitable weight serial aspect primarily issue turn parallelization case code develop similarity solve two different include supplementary evaluate frame label double array
truncate capture retain separate norm posterior scale sensor error quantify average plot show reduce order model least rate reduce build prior reduce versus signal ratio amount order adjust examine concentration basis gradually increase record vector quantify posterior define marginal dimension noise amount reduce concentration complement ex ex ex ex ex ex ex ex ex ex high employ field point eq eigenvector preserve avoid inverse figure pressure head measurement sensor case simulate full discard burn mcmc approximate simulation mcmc burn evaluation reduce cpu ess speedup threshold mean standard field reference full approximate produce generate record provide maintain steady medium drive achieve classical approach order challenge modeling convert observational process play quantify framework model characterize chain carlo
universal method combine walk discuss main multiplication long reach start difficulty cluster refine cluster vertex vertex singleton mini accuracy news even trivial refinement achieve refinement fine scale slightly version initialize scale modification random walk fast start refinement proceed let hierarchy refinement perform level return seed termination denote seed initially level
respect quality excess excess q rd internal randomness risk mechanism preserve differential risk minimum achievable define min differentially private mechanism work fairly loss contain square give strong go beyond constraint paragraph privacy replace well quantity convex never large similarly bound assumption function bound analyze noisy mirror simple result precise tailor gaussian kn show interpolation norm view one assume satisfy appendix get perturbation width dependent bound width bound noisy mirror mirror descent often assume e entry lipschitz polynomial loose beneficial lipschitz issue bound differentially version wolfe constant precise
sketch algorithm present extensive report def layer variable gamma bernoulli two datum pt improvement collaborative corpus lead deeply sparse gamma overall corpora k baseline art observation generate def collapse def gamma sigmoid sigmoid sigmoid poisson poisson def def layer netflix netflix ndcg ndcg mf layer double def double double document hold conditional multinomial additionally percent document percent form document completion document fix correspond evaluate ten document differ latter difficult
write indicator overlap demand drop subscript effect solution must lagrangian self derivation remarkably number marginal marginal eq define follow sum value value constraint rough sec relax accordingly uniformly ix implement soft solution limit smooth choice sec tb py li calculate update iterate find optimum bound pseudo detail mini batch size estimate marginal representation consider use variable objective achieve mean set solution representation different tight bound ability begin synthetic
fit precision indicate time furthermore datum effect rather novel sufficiently regularization good inherent training objective dramatically worse sensitive
require e need loss report daily average performance performance entire daily prevent number entirely dominate instead assign b reporting performance slice allow circumstance improvement fig per total upper show performance click average daily apparent figure score click significant daily also evaluate score click test ignore entirely click use find period auc plot difference leave daily box circle mark outlier interval quantify order summarize relative day performance click set
parallel another picture evolve go represent dot iterate stop effort computational resource hence process describe greatly need network fig cascade architecture report predict confirm novel schema couple feed use separately predict similar cascade feed output stage connected nn cascade confirm effectiveness evaluation phase show nn architecture
svd approximation
rely nmf refinement heuristic tolerance assume soon relative error predefine stop conservative value numerical arrange display possible five initialization along paper local perform example really point numerical relative threshold choose high improve additional matlab precision open nonnegative g semidefinite see detail section matrix linear initializations refinement refinement idea behind obtain avoid initialization iteration algorithm pair small give state generate obtain fair different generator matlab execute outer execute far reduce time numerical stop soon check every ten display find run find exact ten display nmf find bold detail nmf algorithm use intel core
various sequence suppose assume sequence strictly nonnegative function constant verify definition verify sufficiently sufficiently control study balance contraction smooth collection spline wavelet fourier commonly hellinger w h p usual distance density prior sequence positive positive constant sufficiently contraction verify describe define way approximation imply rate polynomial series spline wavelet series wavelet j incorporate setup scale normal model suitable whose metric entropy control complement exponentially concrete theorem contraction crucial choose equation divergence square different choice
combine reduce part template chance turn resolution global attain bottom e value randomly initialize tendency yield situation max minus min composition parameter move opinion stage however gradient em beneficial initialization part care exist minus shift rotation part template version transformation allow transform template configuration probable procedure matching pursuit start add increase improvement material yield opinion
v intra similarly st inter pairwise solve regularize connection form generalize intra neighbor q obtain true block guarantee crf constant type intra inter satisfy eq hessian node universal constant solution recover neighborhood recover inter neighborhood node recover union extend selection evaluate numerical example ensure parameter meet discuss parameter simulate appendix sampler node conditional via gradient descent elementary sample operator roc recovery replicate top panel recovery instance variable panel panel decomposition x product specify bottom highlight advantage mix mrfs mrf crf crf poisson mrf represent mrf mrfs truncate variant mrf instead introduce simulation simulation selection recovery distribution homogeneous ordering sampler connect neighbor block curve recovery binary mrfs poisson mrfs mrfs able even extremely model set mrf recovery dag much treatment question beyond scope section brief demonstrate achievable via node neighborhood figure plot roc curves structure crf crf mrf good increase
implicitly bias value calculated get bias discrete minimize size describe sample take sample average risk mc bm misclassification additive constraint substitution plot curve figure thick grey us task suppose concave attain hence
ie draw therefore branch likelihood branch minimum accord exp convenience respectively work understand theoretical specie tree multiple focused tree reality tree molecular sequence recent understanding tree take fact gene generation associate assign move root change random leave length time site specie write mutation adjust branch xy disagreement character p xy goal learn character easy presentation realistic reversible molecular hold
aspect general focus modular focus maximization combinatorial triple item modular maximum short bipartite matching hard problem computationally similarly agent approximation randomize combinatorial recommender typically item value expect rating never perfectly refine repeatedly recommender typically write structure assume formalize bandit semi triple ground arm sum weight arm observe return combinatorial structure like stress expectation episode adaptively choose weight episode gain observe weight te learn semi goal maximize return randomization
layer gray convolutional consist size neuron pass layer project feed softmax produce bad leave tie except pool architecture network gray rgb neurons input choose well match cost network negative run implementation
human capability order become automatically variable variability obtain vote joint voting validity classify unknown light curve vote rare voting detect bayesian joint explore offline test algorithm million curve anomalous divide outlier air bad calibration instrumental remove outli list select object pass post stage match identify blue type outlier follow deep analysis important examine effect star nearby star star candidate visual discriminate remove systematically object run step repeat obtain apparent outlier execute purpose group interpretation finally match aim know belong followed analyze identity rf rf extensively use start rf construct bn purpose extract classification final outli method anomaly machine theory perform real training elimination analysis conclusion vast publish relation anomaly detection classify class unsupervise learner unlabele consequently training class turn partition anomaly method detect anomaly generality
cnn preserve internal input straightforward treat pixel cope word example vocabulary computation unit region rest image vector convolution learn call net convolution representation seq cnn seq keep potential seq cnn however rgb vocabulary size weight dimensional vector handle vocabulary desirable provide bag vector vector convert convolution word seq convolution size image fix sized output convolution also give layer pooling would variable sized output pass output require connect alternatively top receive sized fix dynamically pool datum cover overlap max pool datum whole
process develop tractable real work tune hyper automatically adapt degree poisson intensity gaussian advance term well nonparametric method carefully tune hyper reference therein challenge nonparametric devise procedure avoid overfitte intensity intensity location
well small eigenvalue criterion newton invariant conditioning issue filter discrete extremely ill reasonable trend filter gradient method perform primal dual descent admm simulation setup give appendix variant descent standard admm compute trend method drastically reach thousand latter technique admm suffer less issue regime accelerate accelerate form iteration specialize admm perfect descent filtering cover model specialized algorithm conclude describe specialized admm trend filtering algorithm differ admm require operation iteration lagrangian write implement cholesky update wise multiplication full admm time specialized begin augment yield update q small update trend fuse specialized solver
empirical kx dx f minimize upon achieve standard become get stein improve upon gain reduce incorporate adopt shrinkage estimator knowledge close provide knowledge interest investigate without require assumption dirac ensure case positive suppose non thereby yield characteristic function characteristic definite xx stein zero risk decompose part behave estimator easy great bias decrease hand become increase mention show shrinkage unfortunately fact know require oracle measurable bind uniformly virtue dirac kernel characteristic uniformity bind norm characteristic empirical upper transform ix proposition kx regard shrinkage dominate empirical estimator lebesgue
color green marks solid forget sep crcr blue mark option forget plot crcr blue mark mark option forget sep crcr symbol classification handwritten digits computation depict less solve toward set showing pair draw training equally test analogously misclassifie pair multilinear form infer relation equivalence relation misclassifie objective training height unbounde xlabel ylabel relative title test format cd fix cd white forget crcr nan nan nan nan nan red forget plot crcr nan green dash crcr nan dash forget sep crcr nan color forget plot sep crcr nan green solid forget sep color dash
justification contaminate gx minimize mse regularity condition asymptotic component pilot estimator recommend use pilot estimate general framework divergence different include divergence efficiency property explore estimator explore robustness performance analysis prove link robustness al family recent elsewhere beyond discrete distribution estimator family bandwidth smoothed density divergence equivalence practical implication well support real example anonymous suggestion lead version manuscript institute inference divergence technique divergence name family discrete develop continuous smoothing unlike avoid usefulness extensively et base quantification minimize include pearson chi pearson kullback leibler kullback hellinger bregman rao
pass slide communication device could human co operate power effectively lose control machine intelligence improve initial look loop artificial study investigate compare electrical load learn forecast load robot despite precision feedback increase purely feedback user control term lead great assimilation device predictive copy operate expectation remain verify life cm thank insight need individual one promising end interface supplement learner control sense live interaction part internal copy move cause desire
involve laplace cauchy boundedness even problematic dealing noise infinite far tailor situation reference section dedicate assume predict place take tail ensure set small notation assumption behavior satisfy tail investigate near family laplace distribution integration gamma satisfy immediate pareto regardless gaussian law tail univariate law satisfy inspection q spectrum justify influence consistency first minimal necessary obtain rule consistent discrimination classification assumption minimal mass non compactly density classifier discuss curse support real compactly support though show reference therein second mass compactly support proof mass major former family case excess risk strictly long neighbor occur neighborhood contribute near hold precisely assume hold sake convenience adapt tail note recover compactly density compact case compact
single outside leave unchanged observe together element exact remain r note remain even strict broken mean strict extend comparison rank index index j score get tie score observation comparison index shift row remain comparison intersection column corrupt generality get eq apply intersection row corrupt shift observe remain matrix strict except element row similar column tie break use mean strict matrix proposition corrupt induce score case pair adjacent break arbitrary corrupt exact rank comparison estimate corrupted maintain true corrupt set pair guarantee admissible pair random replacement denote pair sample admissible contribute pn notice provide small fraction
q serve pareto network specifically reasonable condition gradient theorem agent limit denote agent symbol gradient process past iterate agent verify whether agent dominate effect broadly consist flow illustrate particular agent equal limited sub sequel sub illustration figure strongly receive sub self loop indicate figure run diffusion mean sense third bottom network towards bottom third influence feed behavior third influence network still exhibit independent implication discover sub end sub limit sub regardless stand connected sub run independently network agent group use letter refer denote equal agent across would strongly truly
indicate denote product represent reflect simulation verify high robustness statistic decade extensively especially deal dataset fitting accelerate robust like name author fast inexact alm publish later outperform efficiency competitive robustness dense normal operation mathematical observation
selector inaccurate fail identify vector provide selector selector short mu selector subset discuss selector apply random mean finite admit drive estimator bernoulli value rewrite therefore model easily shown admit good available mu contain induce whose expectation vanish idea thank lead call selector entry constant choose estimator selector study design one interest selector enable error
advantage particularly efficient simulate pose perspective table clearly exercise draw burn auto mode middle simulation mean simulation setting leave zero perform simulation growing consider variance covariate variance pairwise absolute via simulate gibbs iteration burn death move probability posterior draw hyper prior posterior sum grow difference tend grow lasso ratio figure horizontal l ccc select predictor validation bit ghz processor ram assess experiment gene potentially show cancer patient predict datum gene predictor unit material assess scad method bp predictor moderately expect related observe prior remain parsimonious validate roughly remain small predictor drop observe predictor various finding effective detect parsimonious dimensional time order competitive implementation code follow storing likelihood memory focus mass burden tend
kt ft proxy equation enjoy conjugacy mask factorize separately correspond variational obtain initialize variational kt kt kt kt lt variational inference example check well separation bss comparison exact potentially complete conditional
convex modify ridge hold validation let fold tune relax kkt solve constraint fold avg intuition behind solution path curve relaxation compare use
assignment subsample mix schedule quickly empty pick assign remove assign little code effort implement linear subsample schedule initialize empty assign gradually latent schedule perform henceforth choose real consider iteration differ sampler subsample anneal hyperparameter hyperparameter inference per cycle early relatively inference subsample toy intuition asymptotic speedup subsample time simple two energy barrier way subsample anneal simulate anneal
membership partition community overlap overlap offer community measure vertex divide community attract relative benchmark event unstable suboptimal modularity yield modularity introduce new section define let community degree node modularity modularity value degree belong community e contribution w w simplify verify modularity calculation accordingly assume calculation perform vertex move difference step compute c z z md list individually compute r may infer follow c c simplify computation provide study semantic without discussion suffice say follow vertex apply equilibrium ne community assign achieve goal benefit player access strategy execute derive profile
different relaxation case feasible one relax remain since make enforce constraint easy think control surrogate cardinality enforce outlier li sdp model justify fairly chen xu belong sdp also generalize inequality advantage chen know number outlier natural restrict n balanced balanced relax note n kx xx x apart nonnegative remove generalization sdp pca relaxation lead generalize possibility leverage sdp c consistency sdp call sdp consistency mm note q block consistency differently eq plant kp qp follow eq show order let sdp sdp sdp sdp piece argument symmetric triangle establish sdp balanced plant rescale namely precisely similarly let c sdp consistent c sbm block point
jt q kt eq multiplying noting since real relate invertible also know optimization find proportional negative current scalar q gradient descent q denote increment step use jt kt update rule scalar cr calculus make derivative make derivative
mean namely similar also filter first detect change second datum detect use random analysis later publication scheme remove present example conclude trend derive iterate integrated trend
descent gradient choice vector call fill height width em coordinate thin distance cm right cm xshift block sim text ex sim update theoretical optimization albeit unlike noise component establish establish regard difficulty evaluation adapt carlo horizon discount set transition truncate trajectory approximation simulate want discount cost truncate artificial outline horizon set horizon truncate trajectory define trajectory limit induce discount current one il j c estimation algorithm provide algorithm free transition algorithm complexity respect
quantity similar matrix expand partial derivative valid two taylor independent variable independent width prior error reduce propagation variable use straightforwardly generalize generalise fisher fisher cast http www fisher toolbox cast grateful physics ann united states school mathematics university south sciences department
average recently player player strategy reinforcement response recently broad setting underlie consideration player payoff payoff specifically rank strategy score rigorous tune player response x assign strategy high multi value interior history theory refer discussion refer reader purpose seen x finally rate cumulative allow explore accordingly throughout reinforcement behind seminal bandit discrete set variant continuous systematic account two characteristic example reinforcement logit good response prototype unit simplex gibbs negative standard simply equation thorough therein consider penalty hx function project response support change game evolution play games reinforcement sort noise access cumulative assumption rarely
graph realize modal reduce distribution sample dotted sampling distribution capture go beyond initial concentrated estimate serve heuristic degree aspect seem property degree whereas thm thm condition interpretable structure statistic formalism core statistical consideration social specialize os construct serve estimation network concerned node record count familiar framework associate network degree comprehensive review base fail connectivity connected seem additional
language corpus involve discriminative usually rich primarily linear unit particularly behave deep impact difficulty leverage benefit piecewise linear propose adversarial net adversary learn whether team discriminative competition game article kind article pass perceptron adversarial net train use
ability difficulty function infer maximization likelihood sn set response expand student positive pair take share hamiltonian ise introduce several introduction ability exceed share extra adjacent could account reasonable room site situation demand contact range ability
offer auc really poor know subject performance significantly performance offer across notice subject calibration calibration quality point short minute acceptable game record generalization across dataset ii record realistic p p pz sample hz classifier subject increase calibration average subject ht trend outperform reference also despite subject cross performance show cross difficulty high result good adaptation propose necessary dataset iii record life live hold laboratory th
steady difference state steady unique invariant concentrate expectation much sum steady expect phase reveal phase expect cost strategy proceeding lemma throughout proof lemma heavily uniform continuity every unique invariant mx evident specify regret bind present reduction steady steady cumulative cost steady cost policy fix consider loss therefore cost steady complete phase e decompose nonnegative hypothesis strategy state decompose cost phase q steady get everything ahead last inequality due situation everything ahead result steady actually everything ahead q value repeat combine next bound quantity phase moreover matter routine calculation choice complete demonstrate tracking move undirecte probability dynamic topology neighbor simulation vertex passive random
free svm gram yield expression center propose center singular gram singular author consider optimize center centering model nest center either partially center comprehensive centering center beyond scope investigate investigate non update connect center gram perform matrix provide center machine subscript recognize center oppose center versus singular singular eigenvalue variance deal sake inner matrix associate outer machine present center set available inner essence focus gram non decomposition gram data decomposition realize financial economic coherent order non moment center center centroid decomposition rewrite mean contribute center center entry I n central rise
property term modification argument give apply rest lyapunov basically l convexity adaptive size round k unconditional drop convexity summation inside simplicity gradient element avoid illustrate correctly np np perform double lag lag double long amount lag lag amount dense double double double v double epoch long double store format double index double double np double track lag np
choose sure payment effort dominant eq dominant equilibrium proof course could arrive mechanism computationally overhead execute complexity ode statistic ode find certain criterion g unlike ode ode optimal ode solve approximate unique feasible minimization feasible preserve reduction minimization translate box manner special minimization regression restrict minimization behave separable minimization polynomial behave iff expectation regression bipartite ie convex function cost polynomial cost wants find minimized matching cost find min min optimal regression vector finding take well define hope
upon identify green blue respectively collapse ask clean fig manual piece together fig hmm difference uninformative freedom depict grey reveal critical axis motion loop display low axis show fig interpretation experimental reveal protein family role cycle md atomic effect protein biology gold provide know protein suit approach protein family whose lead role cell anti prominent member size complexity md task show water accurate inactive take adequate
consistency posterior assign precisely entry denote completion summarize survey reduce reasonable assume rank make consider e dirac mass large consider close lead variant consider iid conjugate among author give intuition seem computationally prohibitive
temporal classification layer backward recently acoustic deep rnn stack rnn predict phone show get art phone recognition database language gram phone lstm networks phone database result paper obtain art vocabulary recognition modify architecture lstm network address computational efficiency standard architecture lstm output layer lstm layer forget cell connect network parameter lstm
arguably category score htbp hard negative add positive region around foreground image overlap score intersection union foreground less precision improvement negative region automatically discover negative region green box discriminative part box fit configuration car cat tv svm negative neighboring negative discover negative supervise method frequent configuration discriminative visual show discover configuration provide coverage way useful hard negative together challenge berkeley due
fc layer scale ram ram scale ram ram ram fc layer fc layer ram scale ram scale ram test successful policy experiment patch enough experiment also test ability ram patch input also feedforward two unit test additional improve ram training full digit successfully combine information classify task translate mnist generate place mnist digit patch case effective size multiplied contain random translate several model translate ram network convolutional filter nonlinearity
require provide symbolic believe science division berkeley berkeley computer division berkeley berkeley lemma corollary remark em minus gibb several augmentation map gibbs allow expense slow parallelism excellent modern hardware graphical factor give throughput competitive fast symbolic fast report validate method applicable general distinguish represent miss generally want optimize value
filter right filter histogram model integration solution dense simulated particle filter particle visualization filter particle measure novel wise provide particle filter present filter
n substitute problem objective easily partial derivative regard ranking code complex use sparse code consider reduce solve sign rank code consider irrelevant turn sparse coding solve
claim lemma technique randomize tangent cone technique warm show obtain cover put piece z system prove theorem j width inequality calculation suggest maxima apply difference consider cover see appendix subsection result apply tangent cone guarantee u restrict piece q pair condition result state turn inclusion consequently apply embed result suffice least guarantee z rescaling throughout make convenient shorthand equality write conclude q ensure early bind claim dimension regularization nuclear contain analogy suppose optimum sketch eq must hold proof inclusion trivial q notation
exist definition reverse triangle n f continuous schwarz yield hold limit bind hand remove specify projection powerful corollary matter satisfie possess smooth hold consequently indicate validity cf remark since completeness sketch less elementary argument apply exist neighborhood orthogonal small limit taylor zero space mn euclidean equip inner norm save space prefer representation inner orthonormal writing onto subspace orthonormal basis convergence algebraic matrix see continuously fact assume analytic g dense relatively open hence real task tangent point tangent cone arise
nonparametric layer filter unlabeled task thresholded dot simplicity recognition natural computer solve face recognition task think require alignment able operate totally unconstraine face hierarchical architecture memory thus propose take advantage principle greatly simplify frame system architecture module abstraction cell input module considerably abstract represent input cell dot cell invariant concern architecture construct signature layer layer temporal association associate template training accomplish
laplace smoothness induce laplace model logistic highly resemble gaussian among cauchy advantage suitable second tail highly capable substituting scale framework cauchy q cauchy pca extend entry observe otherwise maximize introduce explain locate set estimator influence contamination estimation supremum take worst desirable estimator unbounded explain pca noise another neighboring stable possess cauchy estimator unbounde sensitive
figure reader dark gray background figure take legend title inside serve sequentially place graphic center figure wide figure place column environment format require input initialize ix center title title unless leave flexible breast meta table material figure material draw row table column place page please format regardless file include name name place otherwise separate author publication person blind please style leave subsequent reference give journal conference book author precede reference author year publication sure reference page strongly encourage publication version whenever camera ready copy reveal instead anonymous review acknowledgment acknowledgement version blind review camera acknowledgement thank give financial help student report student raw source involve crowd engineering extract student non model art examine model characteristic generally predict difficult attain massive open leverage advanced topic hundred develop time markov present make stack hmms finding contribution end raw advance methodology scalable interested fall extract combine student usage leverage collective brain crowd create temporal feature predictive comprehensive art technique
confidence coverage method work well asymptotic cat significance power cat point cat almost cat assertion test consistent sample consider home team call team time goal home team bivariate
uniqueness policy set bayesian hmm field health technology political business research patient com computer site percent company online analyst example instance lasso application speed approach optimization approximate sketch preserve computation lasso property solve root fast solving equivalent
recall bivariate partial odd model ml asymptotic penalize literature whereas patient
logistic describe requirement fast lead posterior technique often group g write usual glm notation datum form logarithm become extended derive posterior normalize logarithm inequality approximation equality apply logarithm tx tx tx bound proportional tx computed approximate
auc gaussian event background event baseline layer network whose hyperparameter optimize architecture top hide overfitte unit performance dropout method dnn relu relu relu unit effect benchmark activation complicate try initialize position activation keep
popular regularizer general xlabel ylabel avg legend style true legend north header x dataset server update processor main drawback update make guarantee framework although recent make coordinate observation gpu collapse sampler independently parallel good saddle parallelism contribution towards could epoch machine predict measure average epoch per machine value scale iteration ylabel font legend north title true serial txt
evaluation protocol keep sentence total group cnn semantic dependency tree rnn match global rank fair cnn multiple representation object cross evaluate prevent hyperparameter devise modification cnn discuss rnn global rank cccc search al devise model low good test cccc et al devise alignment sentence good quantitative table significantly rnn competitive objective suggest bring cost directly minimize global scene
frame convenience denote peak parent multinomial random peak subsequent th peak intensity value respectively center gaussians axis oppose peak consider rather currently consider gaussian bad far otherwise alignment enforce frame e peak score e alignment log random score peak likely somewhat simplified previously constraint variable utilize alignment subsequent current simplify come observe spectra resolution spectra assume state amongst return require among differently spectra comparable another essentially score score target let dd return score linearly calibrate order score percentile interpolation score largely target identifiable typically decrease
moment distribution section fractional procedure carry five full characterize parameter use robustness case case two density reconstruction criteria namely test start compound loss business density marginal diagram empirical use visual idea density cumulative whose figure observe difference fluctuation indicator good reliability diagram diagram measure probability beginning seem closeness norm distance mae detailed plausibility display mae rmse distance reasonably mae reconstruction correct consist transform distribute deviation uniformity poor display different power histogram seem robustness method perform reconstruction routine carry reconstruction
status physical consider ny memory bellman calculate respective scheduling conduct sensor side hence controller utilize transmission identical instant lose whether controller new aim controller side hence communication period end possibly measurement receive conduct k state variable update system physical system network reason dependency control controller new information derivation change sensor controller controller need use protocol fulfilled present simply communication delay extended delay loss utilize rl handle stochastic process delay losse bellman available besides mention behavior schedule capable handle moderate include delay losse final give weight backward dynamic concern lead unbounded control admissible control within
basic confidence round choose policy predict maximize implement construct set contain least confidence translate confidence transition select simultaneously optimization efficiently perform employ eq solve specify interval time ts confidence
almost choose ii aa significantly outperform iii unlike sophisticated convolutional aa helpful cost drive per aa sc visualization regard image internet video public methodology present example visualize request paris accord dense descriptor interestingly request paris city reduce influence outlier heuristic choose point even learn fisher point classical figure
linear gaussian analogy dictionary section compact cf exploit part enhance extend pruning dictionary strictly prune emphasize single approach contain multiple multiple compactly efficacy propose toy example red color color curve blue light color employ section linear adaptive filtering adopt basically filter kernel input sequence draw output corrupt trial mean average respectively complexity summarize curve reference include dictionary see outperform counterpart
b fourth last equality last definition expression fourth definition b u th machine core block band core compute respective block fig block main block band e follow block compute recursive step mr I block band nr r resolve transpose e recursive block ir u nm thus albeit black red likelihood curve jump area blue gps jump boundary yu chen department mit technology usa edu thereby perform region furthermore accurately scale locality
triple q check rate predictor guarantee x x rate distribution agnostic case algorithm logarithmic applying theorem number get agnostic isotropic log treat linear classifier complexity recall apply simplified computer science california la study generate main high requirement fairly major challenge achieve provide rate prediction rate study generate access large request learn pre label target aim active survey main challenge consistency maintain generalization binary search inconsistent agnostic algorithm agnostic minimizer maintain exist agnostic label sphere unclear major
error iteration pass predictor consensus message indicate red belief iteration message red significantly slope indicate bar also regressor capacity fig well comparable bottom message pass mm noise gaussian gate gate pr gate gate gate pc l south south circle pt red l circle south south consistently problem make low challenging devise inaccurate center circle square might wish reason colour foreground background make hard image perform sequential schedule marginal latent additionally place message lead pass fig highly inaccurate marginal center inference see quantitative fig predictor red consensus message message come pixel foreground internal design test two position
particle serial continue core theoretical additional computational overhead initialize run alternate trace new particle addition retain operating trace late branch arbitrarily achieve every process barrier compute weight terminate away branch loop new child process barrier albeit barrier execution program execution retain branch reach particle retain select signal retain branch process retain except retain trace retain branch
express c infinitely variable evy e find separate absolutely atom positive count dependent membership observe atom joint assignment size fix label atom joint likelihood r lead pmf order group membership size q lemma directly bayes rule augmentation marginalization laplace transform parameter analytic concentration use sample discrete distribution grid gibbs hdp thm corollary thm section integer stochastic employ count
equality ef lead appendix cover characterize bind number normalize n cover radius cover covering fx xx x variable complexity closely cover exist average uniformly f h lipschitz constant apply inequality I I sample h nh proposition eq h e vx u n yx yx j yx nx v vx h similarly e x yx
continuous theorem rv independent motion index pass coincide aforementioned let copy th skew covariance copy give stand gamma case coincide recent also excellent calculation extreme value normalize constant linear normalization supremum converge stationary limit weakly minimum process
maximize minimize bic bic parameter value component coverage model function bic bic number component configuration correspond diagonal dark wide wide classifier cluster simulation free measure observe uncertainty mass simulation randomly deviation scatter noisy decision thus draw contain training distribute uncertainty use classifier cluster inside repeating generate realization contain uncertainty draw make previous density cluster sample object statistically target gmm make straightforward method outlier target nature
learnable agent dark remarkable start track observe agent passive observer stationary condition occurrence situation observe finite let still admit decomposition process theorem obvious omit tail induce decomposition learnable stationary tail sketch ergodic theory partition atom refinement dynamical system seminal follow algebra every measurable q natural tuple stationary I I admit decomposition primary object interest outcome day history outcome previous relate prediction decision observer
q decrease accordingly proposition imply sure backward describe mdps action initial path merge common couple decrease path couple first chain time begin common state get couple chain couple couple overcome chain infinity simulate accord continue chain k define thus couple couple rhs picture henceforth q precede clearly proof k equation g provide bind developed subsection also early notable sure iterate however confirm simulation result
cc decay however suggest speed shrink construct confidence residual satisfy choose discussion stand introduce define bootstrap xx remark reason first develop suggest effective goal achieve coverage accuracy motivate lead representation also term bootstrap propose quantile regression iterative less effective term accuracy bootstrap analogue serve standardized section process approximate xx section ec regression replace limit ec asymptotic confidence bootstrap two coverage discuss bootstrap quantile affect quantile depend bandwidth hence location choose negligible slightly optimal order nonparametric eq validation gaussian study select rule package
redundant leave near neighbor decision gain decision split choose split attribute create accuracy return competitive meta decrease et identify instance correctly determine misclassifie instance difficult classify correctly hardness measure instance overlap instance instance measure cover produce feature subset tree percentage instance total kolmogorov instance depth tree provide instance belong calculated attribute feature skewness instance instance target
form good principal tool computation provable building recommender prediction customer base netflix entry rating customer movie dataset entry customer rate movie make assume rank customer movie highly correlate rating bound form customer turn recover often completion corrupt person identically sum loss least sensitive outlier classical example reconstruct would matrix presence become study call new representative basically approach minimize rank two factor rank call factorization rest review solve section factorization review finally conclude discussion recover rank minimize estimated rank combinatorial convex popular rank norm singular value two fold nuclear convex feasible global optima relaxed secondly prove mean operator analogy recovery norm model summarize nonconvex method discuss problem matrix element entry outside equality coincide exist early replace nuclear tractable work solve eq prove
describe label receive empirical label constraint propose rademacher global label analysis generalization global rademacher converge complexity set fast learn svd unitary value assume complexity consider orthogonality follow eq eq local rademacher erm label determine
reformulate exact penalty motivated structure zero problem consist problem propose partial deal method yield space numerical exact well quadratic past decade relaxation norm brief historical account signal reweighte due rank relaxation follow surrogate tend nonconvex study demonstrate reweighte nonconvex norm zero minimization mathematical constraint variational induce add e coincide penalty parameter though exact penalty solve special structure decomposition reweighte consist show favorable locally globally weight software subproblem bfgs newton section penalty method solve bfgs state code good solution feasibility
survival evaluate instant moment exist uniquely distribution moment density two entropy system polynomial turn convenient fast uniquely need survival moreover instability polynomial broad among densitie compact support contrary polynomial like prefer semi infinite transformation coincide approximate polynomial every recurrence relation moreover normalize uniquely decompose basis raw moment coincide sum give procedure make stress polynomial expansion might fail positive resort importance proportional normalizing require support support natural
interpretation intend since fa loadings fa explain loading fa factor result singular order convenience generate data sparsity form fa fa papers generative use tool interpretable combination variable consider perspective variable variable approach propose conceptually exist pca lar analyze arise variable affect strict subset experiment non elsewhere analyze scale accurately infer graph vertex define
consider discard burn compute running root compute modify smooth hence expect see five accurate filter simulator particle backward trajectory rmse show thick gray serious classical particle unknown already outperform particle gray line correspond smooth see sampler level order use system toolbox fix truncation nd system posterior discard rmse plot increase truncation level th consider year million severe hundred cause severe public ability predict disease activity sir environmental fluctuation specify discretized euler yield infected death rate r google infer epidemic proportion query relationship odd relative odd proportion infect count day month mh sampler sir model though observation base particle augment sir applying
whether constant achievable equivalence et almost question furthermore identity perform conditional demonstrate intrinsic interpret show fundamental distinction usual low support large al give doubly factor support adaptive support query prove low uniformity test conditional answering conjecture et al non probability conclude bind query estimation constant exist adaptive access zero perform query domain query indicate adaptive support lead low set construction carry tight query significantly provide low equivalence deal conceptual long quantify thing much restrict property namely invariant permutation show simple core distinguish work instance polynomially domain probability deterministic guess separate query
inequality n fact n rs complete lie complexity experiment decrease intuitively prevent encourage contribution feature phase try theoretical former leave introduce concentration real value eq write easy function regression network base n type neural random draw distribution random inner product
problem introduce source short compose vector short database superior group selection select dictionary term search efficiency compact experimental sift netflix demonstrate superiority area attract interest popularity compact code design inner inner product many scale cone design product fact exact naive scan observe similarity thus inner category code locality sensitive base hash iterative compression show superior performance little acceptable category specific adaptation quantization constrain research hyperplane hyperplane relate product address query hash maximum
u u tu n u u u u u r nu complete eq solution eq q complete rewrite u science chinese university chinese edu high tensor become scientific social mining challenge propose parallel trace formulate require factor mode optimization observe trace tensor core cast multiple trace alternate multiplier verify outlier term array
density kullback leibler respect optimal pdfs expectation unnecessary since normalise pdfs depend distribution vb expectation bar current variable necessary compute calculate h tx ty c wise hold verify write square linearity similarly horizontal covariance compute gamma independent derivation go tv case q case compute moment value update estimate one extract mode parameter density lr parameter gaussian isotropic briefly prior difference model next briefly distribution pdf bivariate distribution dimensional bivariate origin encourage periodic boundary
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formulate inductive empirical replacement various application fundamental replacement employ om technique concentration practical implementation replacement generalize bernstein function illustrative hold fx prove finite continuous technical x take let n get coordinate moment provide chernoff obtain trivial bind supremum calculus bind moment chernoff lemma sum obtain result derive presentation cube point thus uniform distribution define distribute thus thus concentration trivial quantity function modify assume step theorem
em guarantee need propose experiment carry mixture variable sample true mean mse true ex ex ex ex c c comparison approach ari ari standard ari small dimensional sample ari occur sample size good projection formulation widely vote house type vote yes party list htbp issue item water project share education right anti aid south binary code yes yes fit
infer assign thin observation solid line model infer panel gene infer subsequent function replicate cluster show supplementary material allow similar signal cluster slightly david tool differently cell whose arrive intrinsic show cell comparative purpose without hierarchical spirit maintain model vary microarray well variation gene activate temporal pattern vary correlate hierarchical cluster hierarchical dramatically hierarchical account significant model replicate variance discover supplementary version profile inference vb describe gradient find subsequently merge set sensible value optimization variational enable optimize fast vb biological time application capture consistency biological
unsupervise dynamic scaling affine operation correspond shift follow deviation compute standard training instance dataset time instance feature vector equation deviation define dynamic stream label vector feature th instance stream real feature e scale map range note irrespective original whereas relative enable appropriate range resemble typical affine deviation respectively frequently special belong follow write symbol overfitte
dirichlet collection assume document generative follow token phrase clique introduce lda assign clique token depict figure utilize encode distribution hyper simplicity conjugacy multinomial integrate lda assumption completely ignore infer great proximity phrase motivate strong graph relation bayesian association relation direct causal propose un direct dependence among near word latent topic un show clique clique phrase introduce express high clique define normalize joint collapse sampling variable ideally influence phrase clique topic state normalize get choose specific potential constrain merging significance clique possess develop efficient gibbs choice collapse assignment integrate configuration variable value take value pz w optimization lda experiment propose phrase visualization phrase label model attempt external wikipedia candidate parameterized allow user control provide phrase easily generalize probable phrase adopt clique guarantee topic phrase token iteration tf topic assign indicator instance document definition visualize sort relax assumption lda category phrase phrase incorporate use dirichlet gram latent variable word
go review four sampling explain usage popular cart automatically classification cover involve area improve difficult structure even offer array scan array characteristic raw return gps translate ground compute neighborhood systematic point
filter connect filter pattern match pixel predict resolution patch prediction fed image fit column location bottom add track digit move train connect image fully image experiment publish pixel patch resolution image give size search consider batch fully certain physical task vision decade application however primitive ability conjecture good human difference vision vision process around human integrate human currently instead extract location across different
limitation poor improve apply singular employ solve conjugate gradient ill unclear convergence regularize optimization whiten speed reveal propose closely whiten thank cluster computer overhead minimal computation employ compute present regularize loss introduce ii reduce condition boost convergence validate facilitate analysis henceforth loss respect gradient similarly lr scalar
path connect assumption connect function constant identifiability joint structural px px kx ix full connect function proposition none example prove density necessary intersection connect intersection call intersection direct causal additive assumption mu bx argument
interpretability way naturally dendrogram fit evolve combine change application perhaps social interpretability crucial interaction detect shift occur mean structure characterize norm distribution identify contiguous period relative allow subsequently process find detection change fact anomaly identify traditional point focus shift norm qualitative like parametric family slide one window represent nan hypothesis window conduct choose change past convert network scalar value traditional introduce novel define parametric graph compactly scale interpretable
proof various bernstein able build let concentration measure result coherence establish strong complexity completion use provide yield suppose tu complement identically statement note use coherence fact conclude notice ever fully live thus probability invertible th one thing either current happen ensure reconstruct live estimated strictly estimate event invertible latter via factor operator behave algorithm space fully invertible write invertible find q bound cost involve reconstruction projection multiplication ignore inversion take since run finally run gram schmidt take necessary argument specifically theoretic dominate minimax randomized say consider verify fact line inner minimize simplex deterministic suffice deterministic minimax guess
disease diagnosis develop medical mind question medical run available reconstruct diagnosis disease knowledge twitter could serious economic detection accuracy health expert surveillance traditionally report practitioner control surveillance costly slow public novel surveillance compare traditional system diagnosis automatic term self
method label image box exclude modification loss fig simultaneous term framework structural annotation specific semantic segmentation weak function level bounding box object seed annotation usage annotate annotation background give object bounding box segmentation quality annotation effort microsoft structure obtain try present instance supervision unified technique infer quantify annotation demonstrate effectiveness challenging annotation popular semantic segmentation compose generate pixel annotation bound label specify annotation dramatically improve compare classifier burden structured thousand scalar annotation human
crowd total cost crowd phase stage historical labeling result worker homogeneous definition worker worker provide label decision maker crowdsource service instead maker need infer aggregate size mutual overlap natural question ask allocation concrete example motivate c instance current toy worker noiseless aggregation vote put confident optimal lead policy value compute expect table assume table improvement intuition choose ask frequentist accurately limit accuracy instead optimally budget crowd discuss policy beta rich variety shape domain instead fix begin budget equally label procedure uniform distribution simulate section reasonably unless skewed term uninformative prior jeffreys adopt uninformative prior discuss fact bernoulli update th write current easy next choose current budget sequence formal collect historical labeling section allocation process phase phase since allocation phase aggregate aggregation process terminate stage maximize conditioning minimize inside binary write depend historical define q determined rule follow I proposition completeness place plug hand simplify estimate label confident
provide acyclic model definition say child causal divide class parent parent call discussion key regard principle identification causal fig cc ccc specify assignment x dx v give two variable unity accordingly reflect check conditional appendix accordingly determine imply way truth define identification structural causal advance contain finite unknown permutation identify boolean estimate enable conditional various boolean problem binary list compute list propose
uncertainty inherent underlie simulation uncertainty metric objective design uncertainty rise broadly design decision however probable approximation form value mean function taylor hypercube typically tail probability uncertain quadrature alternative single moment definition function uncertainty desire seek point match target density estimate pdf formulate suited software design improvement discretize pdfs histogram difficult optimization detail design pose elaborate essential heuristic efficacy seek pdf uncertainty target objective resemble target response physical
inequality show indeed rate proposition handle slightly submatrix q note divide block row block element word block diagonal strictly triangular figure form block estimator circle outside truncate know row
type product result applicable variety specialized high space decision involve form need decision leave interpretation completely computation aggregate vector also approximate architecture concept resemble speed assumption normalization derive constrain assumption vector normalize value positive none agnostic normalization provide conservative bind validity approximate allow algebra library demonstrate library significant
exactly laplace conjugate however mb mb model mixture distribution common rkh mean mixture x mb demonstrate note provide upper input pl w pl I j x j x fourth equality employ pl prove statement estimator iii pl consistency pn l consistency iii n function include upper sect denote mb sect mb f g ij derive equality mean q w mb estimator I I te j te f fourth equality explicit inner operation sect sect eq h q mb derivation thus omit present definition elliptical condition generate elliptical begin invariant rotation orthogonal matrix spherical characteristic scalar describe generator
form comprehensive comparison synthetic tensor promise tensor completion outli detection acknowledgment national china zhang national china grant china grant cb ph department computer science engineering china laboratory brain processing brain science institute institute technology interest tensor vision brain interface publish international ph china china currently laboratory advanced brain processing brain research ph china china current interest visual inference paper international receive ph dr degrees electrical engineering team laboratory brain translate chinese processing received work university brain institute institute international award award award proposition model outlier
description set first super website game round team attack serve gain opposite proportion point win attack point serve proportion opposite team remain part paper organize section definition compositional regression study methodology super remark transformation introduce
affect know coefficient importance recover probability sparse may correlation also error compare practical assume jointly observation correlate setup necessary snr note let variable mean iid zero conditionally analyze require correlation decay comparable limit fundamentally correlation f characterize achievable error set w bind recovery vs fig degenerate since possible necessity sufficient exact model along us recovery figure snr cutoff regardless measurement cutoff relation snr low theorem asymptotically identical bind incorporate show increase relative noise consider observe describe exhibit nonlinear noisy upper model mainly extra term denominator term reduce
break segment power whose fix exchangeable incorporate law objective power law simple iterative algorithm optimize datum compete baseline edu cut objective normalize cut cut become year widely simple computation cluster cut still suffer several require advance tend uniform case cluster importantly objective cut equal produce consider application normalize follow segment law frequently power cluster census fail cluster phenomenon intensity
collection globally ii maps ground truth map detailed consistency across map across truth relaxation tailor input investigate encode specifically encode binary I shall doubly block encode partial map I n nm diagonal notational input map encode constraint useful j component methodology novel sub start discuss ground map universe object element object truth shall connect point associate speak underlie universe clear clique positive psd explore collection obtain reliable universe motivate develop relaxation lift tight outli pruning turn constraint remarkably improve theoretical two matching procedure tracking spectrum bias effect represent candidate provide small degree exceed pre estimate h output decomposition estimate method output position guarantee maximize correspondence additionally map inherently natural add encourage intractable propose relax constraint semidefinite refer regularization compatibility sensitive default formulation
nonempty must contain computable use computable take example effective formula classical calculus expression effective examine finitely hyperplane computable half coefficient computable point boundary convex effective reference concept feature effective always behave concept class learnable effective david countable point behave
image percentage annotation annotation correctly predict annotation annotation train document representation rbf svm parameter rate weight rbf choose annotation word image base solely visual quantitative comparison feed visual word l c annotation separately modality figure exploit position usually beneficial exception grid slightly topic superior perform publicly separately accuracy less report highlight modality jointly modality treat separately pyramid adapt image classification representation vocabulary near knn prediction annotation prediction random split annotation figure confusion topic explore semantic associated class label unit visual annotation large connection visualize patch extract learn association window person word window deep large challenge multimodal art baseline multiple tag annotation among image class car etc
parametric test use assignment satisfy parametric assignment consistent fail parametric take general decision fail hypothesis significance classical model set model modelling statistical kolmogorov consider decision represent mean random identify refer determine less student hypothesis decision assignment reject identify hypothesis generate note equivalent enforce leave strategy direction variate
rather distribution take graphical consideration identifiable graph distribution find access access unbiased oracle access take access method statistical call fx mass definition statistical average dimension fix search pair consist set valid possibly solve call unbiased oracle probability unbiased graphical node degree computational cardinality assign accord soft let equal contribution choose
divide rank various hash perform lsh top rank hashing choose publicly ep news except mnist representation mnist consist handwritten mnist commonly generate two big partition create hash refer training statistic dataset dim std mnist news evaluation asymmetric section hash hashing describe symmetric hash base asymmetric lsh asymmetric lsh asymmetric hash hash measure task gold standard element hash indicator subscript distinguish generate sort function hash underlie follow compute gold rank list rank suppose gold increment count move recall balance obtain rank relevant average summarize clearly well hash hash confirm product hash different sign well lsh ht
image person report field well improve start author adapt negative pair domain experiment network illumination pose change highly nonlinear exact section architecture neural network output predict handwritten digit person identification subject generally style network apply include send probe figure compose convolutional cnn cnns connection exist share bias could share deal task person call mode view figure pooling connect channel convolutional pooling dimension pooling include channel filter filter neuron capture body train cnn person three train independently
variable derive subsequently related latent parent
exploit fisher consider score lead various score auto encoder efficiently estimate close tune score fit g autoencoder obtain estimation employ feedforward feed mild degeneracy assumption weight tractable output moment propagation procedure empirically improve dimension reduce term weight layer feedforward learn hide however incorporate label consider set learn correctly present
closely trend success field divergence increase understand scale function obey graph synthetic algorithm exact issue compute weight visible unit allow scale divergence rbms give scale arise deviation weight drop rapidly provide add decay success qualitatively necessary scale constant typical ml scale error obstacle apply natural generalization success state decrease case also adjust process risk produce address pose large wherein mass investigate superior result substantial improvement achieve perfect inferior fact need still later strategy divergence restrict boltzmann question whether substantial difference divergence find ml cd layer rbms visible hide cd training optima cd optima differ optima quite cd vector bias difference percent limit noise suggest optimization substantially difference tend unit model bernoulli tend distance arise quality find bernoulli determine relative error tend error machine trend significant location cd optima quantum closely current art reasonably suffice wherein cd ml base layer boltzmann machines quality optima two learn graph algorithm train field polynomially overlap main model exhibit superior train boltzmann slowly optima visible range consider important bfgs single visible four hide approximately machine unit unit although
homology vary work imagine side also cloud choice dash persistence diagram dot death indicate birth examine gradually neighbor note version point cloud union ball radius around cloud intersection increase infinity evolution homology persistence diagram persistence work space almost merge quickly merge grow merge birth death pair diagram persistence diagram three hausdorff z unit sphere thing restrict theorem close
provide user another sp behave sp stop tc name publicly past publicly available corpora corpus spam corpus perform split document category ten category tc tc dataset occur publicly surface feature word name word employ base model subsection select close example batch large great result method sp sc l spam fold avg cat avg avg dna fold avg cat avg avg avg avg fold avg macro avg
intensity neighborhood comparison package contrast take sparsity mark exhibit least exploit auc subsection top besides outlier furth outcome bias subset drop contaminate numerical experimental denote collect paired comparison live attractive property pair balanced live include video video database correspond round pair top vs blue circle c baseline rectangle baseline base rectangle pt line indicate line treat outli pair comparison pc open circle circle table base rectangle baseline base rectangle simplicity take reference illustrative video detect pair comparison pair comparison agree opposite occur video video arrange accord score calculate pick corner outlier preference order l addition corner confirm top pointing
region multiple interval ideally imputation corruption node varie opt clarity empirically digit response anomaly alarm depth leave pixel alarm vary detection alarm realization indicate statistical parent cf choose anomaly corruption alarm model show label local partitioning consider acyclic accurate mainly happen conditional explained distance pattern although label parent euclidean distance assign corruption nevertheless possible order reasonable euclidean contrary case rank euclidean rate detection detection corruption localization detection empirical frequency truly instance attribute alarm attribute localization corrupt experimentally localization small corruption widely spread localization corruption decrease since euclidean fraction attribute deviation algorithm local anomaly hence corruption stop lead distance emphasize alarm localization operate single phase alarm euclidean make determination reference imputation affect correlation imputation wise imputation imputation wise imputation test quality distortion corruption imputation define versus alarm large imputation even imputation case euclidean produce superiority unlike mmse replacement corruption imputation fair mmse imputation visually regard gradient naturally
incorporate additional pair specific bivariate incorporate constraint fourth problem superior recently undirected realization vector population relate feature specifically conditionally covariance inverse induce sparsity regularize x unbiased problem formulate covariance backward iterate always maintain dual estimate early dual use solve analytically close expression optimality rewrite soft thresholding apply entry observe yield x thresholding via identity convergence detail minimization sparse algorithm covariance tolerance backtrack step k x k covariance duality gap size bb step equation backtrack line conduct iterate satisfie k backtracking criterion safe take give write smooth
consider nominal feature mod task feature discretization either round real take allow accuracy many value oppose working mod metric despite issue frequency accuracy experiment synthetic uci value near portion create behave voting portion cause neighborhood correct output currently explore little value neighborhood give output use fold cross validation na I neural layer
stop dataset comprise evaluate performance system label tuned learn grid unit number use sample randomly dropout shot computer core gb ram want near neighbor share confirm network semantic h restaurant plane near display capture semantic well network special train deep layer dimension allow datum visualize network embed movie figure leave dnn comparison
like segmentation label may come maintain ambiguity propose encourage separate predict hide joint build map average interest optimize output significant improvement jointly example extraction semantic natural unfortunately graphical marginal hard structured efficient popular generally implement slow especially smooth general framework transform iteratively widely particularly compare consider output characterize unobserved suppose h
stop class text radial feedforward primary difference activation sigmoid statistically significant selection transfer kind activation probabilistic preserve neuron weight input pass input output threshold modify function arbitrary layer determine function capable g equivalent case create entire category capable radial basis variable act inference subspace topology fisher replace sigmoid activation function neural generates predict target mean score
machine learn categorization popular learning thus exist spectral mainly find method select successive regressor appropriate select output identically pair either categorical classification consider vector vector responsible predict lasso efficient feature optimize k regularizer lasso irrelevant lasso number sample dependency limitation wise linear introduce transform instance term instance use select kx nn
yu wang xu department university com com mail li old email cs edu aim dimensional redundant feature attract interest feature nevertheless serious framework neither intuitive framework lead guarantee consistency attempt drawback develop sparse cluster concept partition previously intuitively explain principle cluster realization extremely implement interpret exhibit well feature important high sample size problem example feature penalize framework analyze regularization analysis high report really rigorously feature comparable difficulty partition statistic set intuitive definition interpret property organize notion formalize framework k mean section develop brain
denote multivariate usual unit order construct information mutual first introduce correlation information multivariate mutual kl correspond terminology total correlation px extent explain correlation reduce argument mutual common exact explain also independence relation therefore construct model appear interpretation contribute successively tight constitute px conditional table representation quantify search maximally definition representation rbms auto describe generate grain hold negativity information representation thm next repeatedly invoke replace
manifold deep linguistic want require embedding relational schema hope tackle task similarity work center contract reproduce conclusion recommendation reflect work lexical representation map dimensional instead advantage capture uncertainty product cosine enable expressive boundary distribute embedding benchmark embedding asymmetric novel task retrieval relation extraction semantic model map goal low embed object map nearby embed approach prove single space embed represent estimate concept typically
neighborhood close point know kl algebraic case satisfy property continuous continuity modulus throughout compute find engineering application sparse take regularizer convex adaptation split termination criterion meet go optimality subdifferential convergent pass see point establish dr heavily dr motivate call envelope convex moreover alternatively relation elementary relation relation complete exist furthermore cluster rl moreover must simple middle comment split proximal mapping strongly modulus behavior splitting notice last next use minimizer relation equality
task difficulty acquisition address acquisition follow step task evaluate search criterion framework depend location extend solution seek reduce relative uninformative uniform intuitively maximize space differential affect entropy condition integrate gp hyperparameter full outcome select constraint hyperparameter several turn spirit discretization expect must approximate drawing find objective function constraint gps satisfied incorporate simply cost per become q illustrate minima disk indicate evaluation online topic
weight max set triple correspondence numerical specify assignment accord dag represent independence give bayesian structure challenge conditionally test stochastic alternatively structural pose relate avoid commonly include equivalent criterion dirichlet uniform depend local score function store base network learn seek dag parent e computation take unless retrieve constant despite optimization certainly introduce direct cycle say cycle undirected graph undirecte cycle four clique one undirected graph minimum undirected graph connect child drop network dag node order pairwise admit elimination elimination elimination elimination perfect elimination reason free
illustrate graphical represent circle model section propose priori across convenience generic build transform cumulative gaussian denote gaussian multidimensional methodology present care define field covariate distribution new apply marginal obtain beta idea include transform within stick break previous article copula process fully function covariance write normalize square rational quadratic choice apart shorter adopt deal gamma rescale bandwidth minimax contraction rate indicate correctly path smoothness function study behaviour covariate deal process multivariate gram write estimate
intrinsic dimension open supplement offer insight norm scope need challenge supervise asymptotically sufficient practically beneficial evaluate overall variable appropriately acknowledgment partially support content represent view national science foundation acknowledge idea height em bandwidth algorithm laplacian exploit connection manifold riemannian operator set preserve experiment tool semi model parameter neighborhood formally put common unsupervised obtaining optimize construct empirically geometry estimator manifold interest asymptotic
translation result significant translation effort machine phrase system recently entirely network promise system exist phrase decoder length encode sentence sentence sentence long long corpora sentence additionally nature sentence representation neural fail translate clause translate improve long translate segmentation encoder model encoder decoder independently consist rnns act encoder maintain hidden update decision
retrieval specifically query find problem query across search engine length utilize probability pareto log suggest density evidence indicate underlie typically concave pareto database automatically assign keyword annotate middle middle pareto transform label class label major annotation find organize discuss front property pareto pareto method result pareto visualize partially order retrieval decade image query provide user measure texture propose sift extraction vision research query query relevance feedback issue individually retrieve visual attention retrieval bm vector link manifold effective database rank anchor ranking assign pareto front wide pareto machine complex objective find pareto front use
number advantage regressor sgd accuracy conclude remark denote trace determinant semidefinite stand expectation sample sample convex strongly gradient descent size iteration identity reduce hessian quasi whereby attempt matrix method bfgs work since bfgs curvature finite define select tend completely resolve bfgs possible hessian differential close permit update hessian implementation avoid formula stay definite arithmetic operation step newton cost total computational bfgs reduce scale likewise bfgs storage motivate memory objective limit describe curvature curvature proceed previous curvature information idea restrict use early current iterate expect precise pick positive definite form curvature perform refine approximation cf plus inverse yield complete update recursion although expect recursive product information reduce storage operation yield iteration context compute gradient gradient gradient realization gradient curvature descent function past gradient gradient simplify discussion gradient associate subsequent cf purpose natural stochastic see need stochastic version identity quantity eq update matrix recursive except hessian approximation
model extreme regression special dimensional vector diag tr write exclude aforementioned approximation hereafter linearity long require likelihood ratio find parameterization parameterization interest note systematic specification dispersion sub sign obtain diag diag diag diag diag diag diag diag diag gamma sign replace diag wu adjust sign statistic q q exclude
collect approximate inner element wise approximate derive commonly kernel square distribution fact random kernel limit shift invariant instance relu invariant cosine angle compose alternate follow connected layer top perceptron much connected layer account often extremely redundant greatly layer replace new perceptron illustration despite simplicity feature storage reduce require name reduce constructing introduce eq diagonal
structure parsimonious rank decomposition see comprehensive application subspace track history representative recently term subspace observation incremental subspace converge rate incremental second seminal handle miss minima see lack unstable behavior especially amount accordingly data paper imputation offer provable performance stationary flexible accommodate model leverage subspace estimator exponentially similar put anomaly focus incomplete measurement upon separable nuclear amenable subspace track complementary strength convergence simplify technical assumption propose algorithm provably attain nuclear optimality complement claim present algorithm decomposition accurately simply reconstruct cube model assumption leverage stochastic case entail fitting criterion frobenius decomposition propose online offer solve large tensor massive main simulated internet traffic traffic anomaly superior efficacy conclusion draw bold letter denote letters hadamard cardinality magnitude denote n additive signal stand index correspond sampling entry
construct generate binary supervise encode semantic thus metric merge forest powerful semi random construction two contribution previous nonlinear point collect relax every available node focus address tree situation incoherent thereby data algorithm robust additionally propose tractable neighbor major limitation tree remainder paper describe detail hierarchy forest metric max learning section near near neighbor complexity compare forest element forest tree independently segment distance conceptually represent rather distinction
covariate principal involve reconstruct deconvolution impulse non somewhat similar bias traditional make deconvolution approach deconvolution dynamic voxel apply deconvolution substantial million spatial voxel temporal actual moreover focus study integral impulse voxel biological assumption particular assume moderately inspire deconvolution multiply assess moderately realistic image slice homogeneous take provide quantification discuss voxel eq input impulse response voxel reality contaminate version decay voxel ij j largely decay nature estimate deconvolution truncate could situation difficulty function reconstruct handle curve voxel voxel spatial spatially voxel voxel three signal dimensional smooth may available voxel feasible k
ica ht similar calculate mixing coefficient compare order consider total activate reach representation activate hide complete case prominent far complete frequency orientation filter see achieved use spatial orientation difficult propose success training patch train preprocesse datum especially
message pass indicator central essentially pass computer pass coefficient average negligible htb estimate median justification wide number coefficient c criterion q consistency term model note almost sufficient sign relaxed concern condition mean square aggregated choose theorem boost rate heavy tail addition preserve almost known significantly load demonstrate
replication per size star dense setting already indicate section difficult star next analysis ask score association biological concern behaviour range perfect datum strength center goal graphical representation estimation diagonal interpretable impose constraints negative discuss estimation lead negative entry graphical lasso impose usefulness describe initial quantify validate analogously regard ten fold value except report loss sign level come rather feature induce association specie link evolutionary model flexible alternative keep mind validate liu tree support latter extra relative validate impose sign concern spirit one consideration base help answer concern attribute answer clearly amazon simplicity treat minor modification reduce five fold cross different tuning entry square block contour present section modeling contain face dataset employ comprise collect cf panel
oppose mapping object localization recognition place categorization environment another therefore large database vast place recognition develop visual distinguish use global feature computed model feature cluster visual quantization discretized lose generally speak
identical present solve proximal optimisation operator regularizer complicated structure point use write make symmetric dual algorithm h constant initialize round dual main row feasible essentially decompose original solve operation complete sort ordering minimize plus differentiable subdifferential one subgradient hence actually ascent since fix sufficiently convergence use
parameter dedicate learn test section vs enable model one model force generation model ten discriminate related specialized high vs assign example assign trajectory posterior assign great specifie specify format allow file value file file load sep specify file load weather specify column file load default file incremental number column trajectory possible trajectory file allow specify file specify column except consider specify name model validation partitioning file section txt txt class probability txt true test predict txt predict txt predict file identify specify load txt txt txt test txt txt test txt txt file cv allow remove name file reason txt ex txt test txt txt txt ex txt ex automatically require definition file reduce trajectory percentage datum line force trajectory original default specify trajectory trajectory time transformation specify file file file file file force soon datum training section file file file file test file specifie store
length note red marked amount variation popular vision circle fashion trivial perform example instead insight pixel classifier setup expression make remain aligned translation example hard dataset perform second interaction pixel perform prior normalization add capacity locality go like error classifier fall improve attribute solely
dependency get draw testing distinguish suggest ratio test helpful two observe problem always since hardness stay roughly problem lead difference correlation random variable convention chi degree freedom deviation strength correlation estimator maximal risk use c universal
circumstance resample though dominate coefficient take account via calculate uncertainty uncertainty suggested discuss monte accounting uncertainty return coefficient
label counter simultaneously learn representation confident infer multi annotation make inference image competition infer class investigate idea carry preliminary end jointly fully instance
end factorize eliminate elimination operation marginal gm similarly priori would gm group gm linear achievable grouping might simple tree correctly represent grouping clique resort gm become exist gm remain far possess maximal clique vertex clique contain form connect arise illustration implication take complete three
require year asymptotic contraction rate regression et al stick obtain high van categorical consider allow exponentially bayesian finite spline construct show smoothness agree logarithmic use smoothness density conditional allow devise typically reversible vary give group dirichlet coefficient conjugacy structure utilize base direct univariate computing method close posterior discuss tensor spline contraction section uniquely closure indicator denote inequality pack symbol stand generic
state art mb mb interesting implication cnn times efficient embed device onto embed trend mobile bandwidth allow send typically applicability platform order storage requirement consider cnn eight layer three parameter art widely heavily parameter explore interested parameter reduce dense layer run convolutional layer network early work cnn publish explore method test researcher show
good distribution parameter respective ad aic p statistic approximately chi ad von give fit distribution plot cdf survival function fig distribution comparison ratio lr likelihood ratio freedom df reject significance significance significance conclude h freedom exp u ax b therefore bx
shorthand position define vector intersection hyperplane illustration characterize independent row algebra state clear linearly row move discuss position zero entry use arbitrary contradict one simple extremely zero non contradiction form zero contradiction discussion determine mind dependent identify give characterization notice represent theorem document iff row say play crucial lemma iff proper subset fact verify trivial would nonzero entry linearly first accordingly generality zero write immediately statement suffice bb linearly independent contradiction reduce q know nevertheless last contradiction row position position row linearly assume loss row loss generality may scale construction otherwise accordingly linearly particular algebra since transpose entry back common polynomial zero f entry theory algebraic zero ready course still subspace example contain vector vector incomplete
segmentation theoretical option relatively measure color let location color distinct radius enforce sparse greatly computation decay small use spectral hand label label number cluster human median clustering complete place specification appendix theorem hold idea sphere give basis system unit accordance p right origin maximum optima fully optima relative sphere body satisfy strictly constant mh local specify hx x hx e ii theorem ht h xu optima outside consequence necessity demonstrate necessary maxima valid assume twice differentiable twice maximum strictly g g conclusion enough convex convexity convexity
tailor speed difference root able generate candidate metropolis satisfie exist help sufficient location mode location divide distinct modal density design assign region determine likelihood know evidence candidate note pick requirement region decide candidate current locate normalise take center
htb dot curve cccc word symbol represent heavy tail convenient method fractional detail sphere characteristic stable graphical sg acyclic bayesian acyclic ordering order fact triangular diagonal determinant triangular equal jacobian determinant transformation z j hence bayesian sg multivariate definition sg imply concentrated number unit sphere sg represent lemma every use order unique use index base true assumption dimensional stable multivariate form forward variable verify bayesian criterion bic minimum score select acyclic major special represent gaussian
dynamically observe network observable represent table observable hypergraph edge characterize hypergraph basis move move edge summarize term adopt r negativity table move applicable ensures move task suffice goodness fit purpose large applicable discusse construct applicable metropolis walk output nf probability construct fu tu k symmetric periodic hasting ideally structure hypergraph employ crucial usual rely basis markov basis basis due rejection usual metropolis draw full markov impact contingency entry entry independence replace produce outside interesting similarly suppose independence hypergraph walk primitive closed correspondence primitive primitive walk correspondence perform edge say walk degree move depict hypergraph represent edge highlight table highlight seminal dyadic relational summarize
detect technique principle real access exceed accuracy specialize nearly discovery rely ability automatic discover system compare record classify odd rapid process quantitative knowing look search specify distinguish principle remove traditional key include correctness dependencie digits symbol acoustic intensity traffic road anti stream opposite stream produce inverting sequence appear digit common anti stream vice anti stream statistical relate stream character symbol simplest map symbol generate string select stream generate anti stream anti stream stream flat stream mine heavily prescribe encode stream result stream leading cascade application involve decision rgb rgb rgb rgb rgb distance font none fill text white none major axis axis none width height gray thick axis none axis width dash gray none none width major style dash gray line axis none auto distance font draw bend leave yshift b loop yshift bend auto font bend yshift b bend leave node yshift bend yshift c bend bend yshift xshift bend yshift xshift yshift align font hide xshift yshift center gray yshift align font space text black font align align center distance generator capture statistical symbolic stream model construct require either exact alphabet admit wherein trivial usual operation context key group observe alone synchronization reference generator imply sum unique hence generator make stream anti unique zero characterize arbitrary alphabet size minimal symbol realization variable uniform symbol alphabet entropy among sequence generate identical distance thus exploit observe alone without require give stream intuitively observe symbol alphabet deviation historical ensure contribution address occurrence system identify eeg hz font font thick style gray bottom font false axis scale style format fix cs cs axis cs none inner axis cs fill white sep none fill sep axis cs ia ia nature hide fail heart ex file sample letter font font background gray bottom font width scale x font format scale ts axis none inner sep none sep axis c ia art art ex series letter font font thick top bottom font axis axis false scale false style font cs none fill fill text inner draw sep ia accuracy achieve specific hand optimize classification ia eeg visually potential subject variate trial quantization letter style font font name axis top color gray style axis width height scale false format txt cs axis cs none axis none fill white inner sep cs knn ia achieve consideration database series letter font font thick axis style gray style height scale style font number format fix axis cs none
design investigate let select play take time play play refer machine play machine early central distribution new carry follow h playing describe function qx upper quantifie
cascade maxout last layer feed directional rnn hybrid architecture combine ability maxout transform nearby precede gate account term train observe condition add decay score phone basic treat error recognize convert sa testing model evaluate auxiliary development gmm dnn gram network use similarly result testing validation adapt gmm build frame feature per gmm force align library
translation impractical mechanism comparison summation applicable solely logical limitation formal logical reason complex language group arithmetic contain logical predicate mix rational constraint boolean value problem logical formulae researcher maximize formulae maximize amount formulae strictly solver
indexing search many retrieval key practical development locality hashing lsh rely projection retrieve near grow database long employ derive section work visual near neighbor asymmetric excellent performance hash kernel classical hashing variant nystr om able angle show angle provide theoretical issue around lsh pca demonstrate tradeoff classic bias variance tradeoff lastly importantly potential boost validate technique retrieval recall query implicit reproduce interested find database query lsh function result hamming note
advanced scientific technical result statement make previous section auxiliary approximation important semi matrix seek perturbation identity rank frobenius column identity end write symmetric gradient functional gradient iff condition equivalent iw nonzero eigenvalue solution critical diagonal either monotonicity subsequence prove distinct satisfie importance next negative look prior matrix root covariance update semidefinite update rank converse identity semidefinite update main approximation covariance minimize generalized belong also specify finding minimize invariance w lead identity unique th corresponding leibler divergence rank eq belong approximation follow argument hellinger hellinger dimension minimize turn q belong h proof turn optimality posterior root possibly root equation furthermore lead result svd use root square root w norm entry back yield ig theorem optimal give pseudo reveal q v g dd di dd di I expression recognize optimal mit edu inverse problem characterize approximation lead large posterior chain
compute power square polynomial let fix exist right equation p proof equation approximate factor polynomial extend parallel inverse eq fact approximate factorization polynomial apply task one construct sparse matrix om depth number simply maintain non achieve present preserve p refinement behavior analogous
model representation largely representation semantic phrase evident early attempt distribute multiplication representation representation model relational order contraction adopt mean composition observe compositional approach sentence recursively network vary semantics share take token
ph interest statistical sm sc engineering work research member electrical engineering digital communication mathematic paper maximum equivalent look parameter meaningful interpretation context synthetic coherent illumination call interference ratio describe scale moment quantity negligible substantial moderate thus correct bias used image propose correction ml estimator profile
ordinal input covariate cumulative know ratio ordinal good choose review development attract attention past decade add margin statistical multivariate ordinal study decade latent generate rbms interaction advance massive develop specification well procedure ordinal reach rbms wide generate response recommender review indicator current rbm hand ignore nature treat drawback category interpretable ordinal modelling ordinal adapt ordinal utility along
datum since typically expensive besides handle basically relation nlp base domain adaptation attention domain adaptation aim target consider domain adaptation validate assume underlie unknown domain fact difficult manually divide discovery discover domain discover testing benefit include object vision multi recognition detection part former vision latter rarely address viewpoint domain however introduce interesting challenge deal unbalanced vs background illustrate assume target traditionally target pool domain tree apply information capture datum regularization fact adaptation svm give rise concept sect adaptation sect develop multiple sect source objective adaptation target
dynamic certainly machine generate infer bottleneck machine future develop rather extend graphical compressed arbitrarily redundant despite restriction prop distortion suggest distortion property optimize predictive process symbolic dynamic dynamical distortion analysis provide tool identify stochastic phenomena key feature principle approximately train ref science relate kind organization encourage complicate quite fully automate author helpful member material upon support part laboratory office contract nf sm foundation fellowship berkeley fellowship proof elementary markov highlight statement prove repeat formal solution ref come anneal anneal objective find equivalence distortion straightforwardly consider process associate codebook sample unnecessary distortion codebook denote unnecessary distortion implicitly specify via instance codebook codebook realization codebook codebook codebook process distortion measure eq quantify probability distribution shannon jensen divergence q codebook consider codebook contradiction codebook random expect distortion codebook code incorrect code desire use early complete sketch next consider class state realization code information distinguish new rate thing proposition main prop relationship
three article fast symmetric symmetric factorization symmetric factorization identity hierarchical structure scale validate scaling encounter rank diagonal particle scale within group nest matrix hierarchical extension fast symmetric definite hierarchical factorization factorization scale allow generation factorization algorithm depend first arise dense arise dense fill radial density kalman filter efficiently recursively tree matrix represent
video record view record paper multi metric video combine advantage margin minimization ability find good datum meanwhile learn systematic video knowledge multi view video viewpoint metric effectiveness usually highly kind encounter processing
project layer denote transfer define sign derivative denote n f z ii training backpropagation make playing play put proceed one ann rule show clear index variable lf define l w l computing let play normally implication analogy usually backpropagation w ij l base lead computing ann b ann bias
j metropolis former latter draw df conjugate admit ss admits demonstrate df replace obtain approximate online distribution stationary ij l conditional j conditional close expression say base income suggest identify interested conditional inference nuisance df partition l j j update g approximate example c df choice partition df simple serve intuition parameterize sufficient computational benefit main assess sophisticated consider probit df augmentation probit df model update variational admit online derive approximate df conditional see df conditional distribution admit surrogate metropolis applicable conditional approximate c df measure define large plot observe time appropriate arrive sequentially horizon approximate accuracy addition square replication error appear table plot show representative parameter simple predictor proceed
try training well expect consequence easier generalize dl compare identical except time dl task complexity pattern able generalize pattern reservoir dynamical computation dl memory computationally capacity create dl place reservoir mechanism study reservoir usually place requirement reservoir understand effect reservoir architecture task varied requirement nonlinearity greatly outperform generalize novel increase reservoir dl result overfitte lead generalization
machine problem infer ode infer ode classic purpose well regression attract return instead solver way particularly candidate probabilistic solver deal uncertain definition gp ode share construct ode family first third yield combine strength gp ode solver interpretation classic question fit fit gaussian linear member family fix statistical ode find ordinary differential hold solution simple treat base extensive
dramatically resource variance first employ analytic expect quantity parameter stochastic expensive rely discrepancy mean measurement exceed threshold trajectory exceed threshold involve reasonable number deterministic behave related trajectory trajectory use eqn fall outside abc immediately reject determine check whether calculate exclude protocol implement rejection threshold perturbation obtain new meet describe theoretical birth death process replicate rate illustrate birth process trajectory record randomly time
performance graphic accelerate pass cnn extensive implementation cnn computation processing accelerate separability network filter difficulty optimization penalty consecutive recent speed cnns convolutional rank scale train minimized accuracy demonstrate speedup convolutional keep cnn connection layer structure state art cnns fully connection training output connection cnn also connectivity investigate open far apply structural conventional feedforward acceleration cnns constrain path difficulty optimization successfully learn
nonnegative since collection dx grid case author grateful valuable manuscript valuable point author grateful know elegant book valuable grateful dms support center nonlinear grant dms support fact lipschitz sequence argument conclude enough theorem find sequence bound constant bound sequence du exist x u nx dx axiom claim conclusion theorem conjecture criterion definition lemma remark pt measure euclidean domain cloud weight based develop available learn good address question scale connect hold metric allow suitably functional goal develop mathematical rigorously problem go infinity application establish algorithm task largely determine minimizer converge increase functional set setup cloud define sufficiently weight computational task represent cloud minimize graph functional cut balanced cut variation range variation term total dirichlet
dimensional topology induce hellinger distance expand haar volume support wavelet derive rate upper density easier successively introduce notation support rectangle edge power assume support edge fact support equality plug normalize support require support front support rectangle tensor haar rectangle number solving reach f application variable advance haar achievable traditional achievable convergence tuple nonnegative integer differential define differentiable density achievable major density affect decrease quickly cite straightforwardly advantage fact quickly restrict attention
new alternative discriminative efficient spectral matrix weight fast theoretically discriminative extract feature feed quickly classification moment label raw e guarantee contrast score feature show score provable semi set unlabeled handle processing framework operation also scenario input source instance crowdsource application different feature input
considerable information state detect background intrinsic physics express brevity remain experimental match normal true model rarely know applicable adopt k deviation measure trace discrepancy drawback error matrix method ideal ideal deviation
regard pa several test investigate mean et al exact difference log
effectiveness illustrative simulation two inherent graph also boundary fraction vertex match via increase fraction increase vertex utilize dissimilarity plot dissimilarity figure figure dissimilarity run begin simultaneously serve highlight strength cut edge pair new perturbation follow probability probability graph independent level increment increment dissimilarity dissimilarity dissimilarity datum drive though point future dissimilarity flip parameter seed increase indeed performance os enyi highly
set begin choice inequality communication engineering apply mathematic hold institute school science currently ph electrical supervision image process computer b ph vision framework employ dimensionality base exploit fact enable geometry formalize operate potentially use column algorithmic utilize separate convex denote nuclear column identification subspace span low component inference overall depict correct low rank component transform overall identification sense carefully column know facilitate compressive matrix column bernoulli set second approach reduction identify outlier nonzero subspace span low rank simplify significantly high ability identify outli detection column sample bernoulli regularization orthogonal sense acquire exploit devise reconstruct object sequential numerous recent area cs summary article therein subsample inherent strategy parallelization utilize one partition effort utilize formalize rank plus effort early robust seek sparse extension entire otherwise low analysis direct seek
mostly perhaps significant problem event city edge section time interest pm road count graph block day week eight measurement count note tune substantial observe description news st traffic could ground truth trend laplacian two panel compare laplacian term smoothing sparsity impose already spike able laplacian far localize event display throughout distant decrease flexibility increase display node truth asymptotic trend focus broadly throughout concrete statement graph defer similar argument basic inequality square q denote connect quite general yield sharp trend filter trend trend essentially simple inequality large k k small loose graph control bound converge nd stronger bound trend recall chain operator fact operator link trend tight univariate trend filter minimax optimal prove connection trend locally adaptive spline sharp rate adaptive spline
isotropic change deviation away approximation isotropic tight logarithmic perturbation let suppose q show group moment desire want sum p kx I pm tx call block block rescale give lemma relate gaussians gaussian eigenvalue project note symmetry triangle suffice variation dimension rotation invariance scaling assume term symmetric term term case option two eigenvector single dimension empirical transformation I getting let arbitrary probability concentrate univariate degree lead factor get eq proceed induction formal induction condition univariate get result reliable verify symbolic multiplying expression standard code explanation formally variable mu sigma sigma beta mu moment sigma sigma sigma sigma sigma mu mu sigma mu sigma sigma convert mixture mu sigma sigma sigma p sigma mu sigma sigma sigma claim excess moment alpha alpha gamma alpha alpha alpha gamma alpha alpha alpha beta gamma gamma alpha alpha alpha beta alpha beta beta match alpha mu beta mu
either hyper bf eq incorporate redundancy flexibility illustrate call redundancy scenario predictor association essentially specification receive
description region methodology offer computational economic couple generate concept financial volume factor five nominal past year form international web trade investigation economic growth flow economic readily project economic make cross country trade flow clearly united china macro point picture near assume weight node united states china within capability unite year china steady score global trade measure might node flow flow contain one element zero I act gate working show stream enter go stream flow letter one evaluate
movie arbitrarily movie select prediction rmse separately movie run evident qualitatively omit baseline arbitrarily movie baseline denote subset least since outperform base estimator user rmse metric baseline ensure optimal design base outperform baseline account identically distribute practically indistinguishable utilize rating conjecture clustering seem perform space result reader train state percent start try tackle item answer give rating collaborative filter considerable rating
novel gap free low match bind episode choose th pair suboptimal optimal difficult discriminate item item index cardinality motivated basis result ta ia ta item pair indicator function function event episode inequality gap basis episode regret episode indicator instead item episode kt kt follow every suboptimal match remarkable aspect regret decomposition rest gap expect cumulative regret regret episode choose
perform choose solution axis robot alternatively wish see vs height expensive behavior store difficult visualize refer define space behavioral characteristic behavioral dimension variation height descriptor controller genome location create behavior simulate space produce controller dimension behavioral robot weight height map beneficial performing solution behavioral search solution many way perform solution extended search perform search design effective efficient modify robot design begin generate solution evaluate record g space record robot performance current map behavior type keep beneficial reason newly candidate location initialization evolutionary improve generation pick solution mean chance randomly change behavioral determined keep parent meet g time stop map iteration number behavioral adaptation accomplish via k search unknown measuring model rigorous create use select update next acquire prior choice particularly cost nee several f f normal therefore variate covariance relate nearby value correlate via distant influence distribution correlate put differently distant almost nearby correlate kernel noise user specify optimization maximum select next improve explore part predict acquisition section classic likely concept expect behavior real objective incorporated process mean function equation previous replace prediction start code supplementary well store performance map pf function controller physical estimation test nearby test fig square mat ern kernel variant curve mat ern square become parameter fig function mat ern stein interpolation error select extensive acquisition finding optimization reality exact
ts ls l l drug protein drug channel drug protein drug drug protein interaction base chemical unified cv trees svm classifiers particular global approach relate specific choice paper base ensemble drug interaction besides local global include among completion method base base systematically investigate exploitation supervise biological formalize pair homogeneous global local interaction unseen node discuss highlight term interpretability unsupervise bi experiment predictive result family competitive advantage approach structured pair protocol local relate several biological conclude discusse find generality connect
yield least compare candidate rank dendrogram poor subset feature fail enough candidate subset follow average value imply subset may likely contain among candidate discard high rank subset increase rank increase incorporate large mean prefer smaller easily apply default cluster dendrogram previous candidate candidate unknown subset different idea candidate high average subset prefer detail idea list g candidate pt argument default size global candidate every size
element draw close iid influence hence row ensure row adjust suitably relate rbf thus radial normal analyze moreover consequently rescale entry gaussian sign retain checking albeit feature rbf hold considerably block retain change radial straightforward approximately length rescaling direction normalization radial part operator kernel key conventional requirement well need rather discuss determine first kind fourier function ball fourier transform iid use scale appear entirely also surprising rbf dimensional concentrate fix characteristic latter dot draw correspond direction draw operation address gaussian initialization implicitly dt equality homogeneous polynomial second reason stability access
similarity formulate paper good implementing outperform trivial one range learn meaningful conclude perform nevertheless closely probably receive increase online reinforcement customer simulation implement confirm reinforcement use middle plug hybrid home extra electrical deviation
discriminant model training observation class class conditional observation multivariate fit optimal set know covariance structure allow set call gmm matrix structure initial class probability optimal conditional probability bic gmm high solution maximum estimate calculation apply test high insufficient overfitting solution former risk fit information
propagation base graph field tractable quickly converge form slightly informally strength neighbor ise attempt model basic take mixing project different divergence show project iteratively decomposition secondly experiment project use avoid drop kl experimentally perform divergence evaluate respect approximate gibbs divergence approximation expand fully factorize mix extensive presentation wish draw obtain
condition criterion drift absolutely stationary suffice rescaling empirical converge weak combine lemma substantially define knowledge isotropic place integral scaling graph entirely underlying scale near upon generalization truncate connectivity decrease exponentially connect limit corollary isotropic graph vanish vanish bound tail ensure
thus change decay prior direct integral decay exponentially datum illustrate utility classic gaussian second classification gaussian data uci machine repository subspace normal plane comprise subspace plane challenge infer contrast mixture normal level gaussian subspace follow five truncate parameter conditional ten plane table range assignment ten perform poorly decrease temperature acceptance achieve affine precision normal uci repository breast heart breast objective classify tumor ten three multinomial logit mixture infer period theorem remark
geometric designing perform evaluation criterion task answer retrieval feature cost handle tie weight grow establish technique thus picture aim rather traditional setting probe functional application answer question site web retrieval another quality answer greatly question answer question rich contain linguistic often link page relevance search difference suggest object modality represent separately likewise rich content structure document retrieval relevance indicator well decade pair functional capture specification functional rich object first usefulness quadratic overfitte reflect particular place list metric
aforementioned effectiveness synthetic np hard moreover show nmf ill pose exist nmf case rank nonnegative constrain variant semi semi nmf sense discussion detail initialization thank mm nonnegative factorization semi look nonnegative approximation property nmf heuristic error nmf unconstraine singular svd approximation algorithm svd certain initialization extremely well optimal semi nmf decomposition situation np hardness notice paper available treat semi theorem contribution go prove orient algorithmic algorithm provably matrix prove hardness ill nmf imply usual nmf lem let eq geometrically affine triangle big sm tight matrix
mean impose seem return odd finding reduce one return interval involve large point plot return recent level estimate access multivariate quantity mean angular four variate bivariate version angular simplex horizontal dependent angular threshold six pair location excess another location tail outside location fr transform comprise dependence extreme independent explicit expression incomplete beta vary one strong dependence I six plot datum display easy marginal threshold condition threshold return period take excess value point theory increase horizontal estimate dirichlet limit
simulate recall simulate achieve strong integrate way importantly contribute overall independently contribute fraction pathway hypothesis support addition particular indicate location possible fix priori validity issue address future node domain diverse evidence perhaps category prior might evident result domain posterior notice perturb presence pathway however hope divide pathway gene dividing pathway various pathway heterogeneous include indicate
multivariate copula multivariate dependence base copulas quantile quantile work model drive force beta employ rewrite contour positive relationship negative axis remain copulas degree respectively copula possible extend property financial correlation require extra copula copula upper tail copula copula class copula follow new inequality hold numerically careful discover illustrative eq harmonic special dependence copula efficient proposal copula represent copula tail dependence describe parameter copula parameter give relate copula substitute attractive invariant monotonic correlation dependence correlation
assign cox proportional cox regression constant relationship ny py patient period right censor observation maximize partial driving weak towards zero cause instability instability choose highly correlate
markov combine get eq line get discuss subgradient point recall x n assign online example transform choose online lp differently explicitly appear follow dual variable rewrite ty kp mb x ip tx tn tt online toward subgradient size normalization examine four
applicability doubly run dependence question estimation influence zero generalize complicated require work unobserved know help generalization likely generalize alphabet complicate uniformly setting modify graph unbounded degree acknowledgement grateful david graphical topic year I thank comment proposition theorem reconstruct ise physics direct towards various model model neighborhood greedy allow assumption necessary allow ise maximum notation doubly exponentially dependence exponentially doubly probably suboptimal implication learn ise learn ise neighbor
bayesian behave fine grain easy processing aim successively cluster agglomerative dissimilarity new cluster merge artificial highlight interesting power consumption home year highlight multimodal balanced dendrogram concave pareto chart emphasize focus difference able work plan method distribution consider cluster universit paris paris functional sample evaluation pair arise weather hardware quantity sampling exploratory large consumption monitoring reduce small set segment part cognitive ask complexity representation unfortunately induce additionally adjust parametric make sometimes implicit density
know smoothing definite rkh smooth kernel size affect initial step online dependent kernel cv free plug plug computationally intensive suitable widely although multimodal distribution also topic relate goal kernel base normally avoid complexity mention online adaptive mode sample specify priori final practically real work treat propose paradigm adaptation filter size sequentially square incorporated quantization compact rest kernel section
agent pick free agent pick draw join round minimum cm rectangle em green policy update converge limit system simplify ode asymptotically ode sg sp result stochastic quite crucial stable give ne discount single game pure never show ne quick per review section use problem solve game sub present sg sp condition equilibrium counterpart single game stick conclude remark approach nash general discount game minimax sum meaningful learn convergence nash restrict equilibria solving equilibrium traditional nash equilibria propose reference model free prove converge self play however state game objective strategy definition play game infinite sum singleton mdps numerically equilibrium share aforementione offline complete game iteration converge ne design program involve find equilibrium solve equilibria nash strict homotopy tractable infeasible game model free cardinality nash equilibria rational agent combine ne setting
bayes regressor expert predict iteration execute policy regret compare additionally action rl I rl n n tn square go iteration sample collect loss ij ij estimate j ij ij ij km jk bound hoeffding nm empirical regressor rl nm nm nm obtain nm reduction alternate present denote prove lemma eq iteration policy time define guarantee online sensitive j policy I I
exceed provide relation weight simplicity bias processing consider branch integration concentration simulation interval plot want surface change neither close effectively perturbation concentration output surface genetic responsible determining species specie solid line stand specie row previous adaptation output incorrect analog replace adaptation weight sample vector define integration implement precisely rather qualitatively provide time
kk fact lead increase distinguish slow increase ht provide empirical spurious secondly high topic great topic utilize show see solve validate experiment average experiment keep fix norm large match theoretical serve upper red go structure vary keep parameter fix vary fix
top since low stage low stand eigen particularly matrix multiplication accomplish helpful large combination alg show picture stage epoch active triplet sgd recover dimensional extract activation convolutional network imagenet fully feature learn predict conventional reference cluster compute query distance center assign distance performance conventional smoothed number compute explicitly base use refer empirically original recommend fully exploit
less negligible complexity obtain pruning possesse comparable demonstrate relevance acknowledgment partially usa pt cosine require introduce assess compression show implementation nm dct degradation dct play signal image video stage dct dct stage
exp exp persistent homology topology relatively attract great deal allow object persistent resolution evolve identification cluster discover type interest notion explain depend adopt ideal modal yet background new risk similarity clustering applicable identify modal cluster estimator necessary type object formulation clustering comprise partition population measurable content define permutation differ probability detail distinguish concept essential r clustering procedure induce henceforth immediately assign get theoretical focus different clustering ideal population cluster well establish induce precise minimize denote usual assign group ideal corresponding mixture clustering
present combine multidimensional stress least square try preserve pairwise mapping put paper note solution eigenvector diag visualization constrain stage mode point mode mode scale factor stage individually work th jx j visualize overlap remove visualization treat mode improve visualization accounting connectivity measure connectivity connect straight connectivity adjust panel much summarize smoothing size rest parameter free bandwidth cluster bandwidth merge large reduction j experiment parameter bandwidth conduct merge visualize step pairwise high kernel
error ht cv cv error cv error cv cv rbf variant svm superior neural dataset achieve svm evolutionary quadratic evaluating employ carry future multiple hyperplane place kernel evaluate performance hyperplane may reduce particle method technique deal determine hyperplane boundary problem weight use programming constraint minimize neural svm quadratic
check contract observable effort ask worker level pay constraint payment increase back maximization problem effort show contract contract contract contract conceptual contribution adaptive contract design conclusion area version contract round basic agent set prior worker contract specify agent set effort outcome payment cost contract outcome contract maximize minus payment employ contract heavily influence work build line lipschitz continuity discretization require immediately natural shape arm reconstruct approach virtual width similarity information result mab pricing crowdsource market essentially round either reject offer call worker directly thorough problem describe capture extension static multiple worker pricing simplify classic bandit occur complete worker specify worker complete contract derive complete normalization minus payment worker observe contract effort set result complete choose derive contract offer emphasize agent effort randomize line crowdsource effort worker make error worker utility task payment minus cost contract worker effort expect utility crucially effort observable choice task model effort value nan level mapping call mapping effort call worker completely production traditional agent worker observable extension round worker worker unknown contract type effort outcome worker choose effort utility production observe specify contract worker reveal adjust contract round total utility round assumption relax
surface recover transformation shape treat transformation limitation model mesh interact exploit reconstruction enforce consistency surface volume hard surface sensitivity paper representation propose automatic spectral segmentation move coherent infer body action voxel rely body compute accounting technique g rely enforce handle enforce space propagate technique favor desirable segmentation virtue isometry reduce constraint relate body shape affect initialization issue technique pose multiple volume sequence body recovery pose method recover intrinsic shape embed able automatically term consistently spectral see stick recognize perform successful favor extract instant instant explicitly motion graphical hide extract technique reconstruction availability approach promise body consistently propose easily exploit preprocesse stage example segment feed hmms identify understand coherent physical interpretation segmentation initially propose extensive quantitative evaluation real comparison compete move voxel domain embed
hz original coefficient I fit curve original coefficient coefficient whose absolute show equation fig figure spline almost base fit well fit original curve curve
improve error batch elimination eq limitation analysis rely investigate might always functional particular function refine theorem corollary functional three specific functional decision mean risk problem formulation application finance reinforcement learn give unbiased unbiased good sample eliminate round imply mean case upper
accord adopt estimate initialize regard seven procedure em hierarchical initialization hard second hierarchy initialize high probability log high acceleration use estimate decide reach close asymptotic li respectively asymptotic log likelihood q cf em converge q adopt replication ic also definition ic definition aic aic mm definition illustration attention focus accord
see test visual similarity semantic representative semantic engine indirect semantic base advantageous direct verify early base semantic inter develop shot show bipartite computational class set dataset label take first support provide belong see element row test although relationship see class relationship employ word representation vector regardless unseen correspond connect find construct semantic node see I
bounding box aware publicly suited annotation reason many image people action target annotate evaluation action detector extent many often make localization action database class class slightly specialize keep trivial contain perform action annotate maximize set region define either space consist segment order box grind produce map represent human table furthermore show human match computational spatial learn ground truth bound annotation alternative annotation consist
use fact span without right similarly value vertex edge define complement sum unnormalize let laplacian denote nonempty subset index lemma show ap initialize like subspace origin ap nonzero result satisfy generate sequence ap attain attain imply ap nonzero write equal change large q eigenvalue suffice examine derivation therefore da e red theoretical low sequence plot blue successive ap initialization
equip eq monotone function submodular function shift apply greedy specific construction broad envelope diversity greedy
result notation inspire invertible penalty parameter suppose correspond slight abuse treat active variable element subset element assign know argue induction induce active select inductive
start convex convex operate rank large decide decrease value value operate goal operating take six hour gb relative package big gain per leave netflix fit implement right give early option reach package available implement number missing correspondingly center make prediction fit package compute large column center find package author upon factored split many machine transpose split row across current hold zero cpu core restriction memory machine copy cpu copy though core act core copy ram core exploit low signature method four distribute consist row full spread across method distribute multiply similarly
metric versus svm sl ne due formulation sl sl ne scalability computation class linear sl ne constant quadratic sl follow sl ne sl de sized middle large sl examine optimization sl ne sampling strategy selective lot initial attain similar retrieval sl ne shown see sl image class sl ensemble range sl de project single two ensemble observe learn sl sl subspace dimension performance validate several datum
encoder relu activation superior previous thorough analyze regularization augmentation observe indeed pre provide gain unlabele label art performance enough training label form pre unsupervised work help deep unsupervise zero encoding train unsupervised take improvement convolutional show randomly cnns boost technique popular supervise cifar unsupervise additional pre training unsupervise
fidelity non overview mainly latter mention algorithm originally variant applicability since fidelity appropriate split perturb variation technique field cf many apply image decade follow reconstruction receive ray ct encounter ct limited view image cf edge material inherently split receive strong ability regularizer efficiently classical ray ct ct image mainly invert case fast protocol cf fouri fourier reconstruction sense find exploit application velocity encode accurately operator emission nuclear classical imaging poisson available half life water suffer inherently splitting emission see modern allow imaging reach suffer live imaging imaging scale deconvolution address appropriate poisson quite beneficial apply achieve broad imaging highly perturb probably proper modeling essential g weak choice since marker decay detect datum model process ray cf ray transform coincide transform model describe property acquisition algorithm discuss split highlight strength carry image total variation tv algorithm backward metric expectation step emission tv adopt modify efficient weighted square problem within proximal problem stop
rx square define accurate equation restriction indicator k lead pearson indicator plug rearrange drop external selection subset challenge difficult intensive exceed often genomic bayesian penalize regularize computing ise show rapidly compute present result accuracy applicability regression gene broadly frequently hundred year challenge genome study potential feature regression statistic classical square turn quite difficulty feature especially general phenotype body seek nucleotide snps trait height mass trait represent absence snps highly explanatory trait thousand thousand million presence absence snps tend linkage
good expect rapidly region expect gradient vanish concern pursuit overall ultimately output experimental architecture one additional feedback think analogous produce local main act kind although error train result training focus raw input notational sample consider independently goal net convolutional neural layer error output weight weight filter layer layer map layer refer filter response map classifier softmax
record complicated background clutter object object rotation illumination variation motion find material extraction brief descriptor task task throughout consider estimate tracking criterion score I frame regard frame average success successfully frame sequence track accumulate accumulate detect frame compare approach boost approach
variance economic financial market management heavy tail problem kk solve constraint determinant setup model fortunately identification solve fix proposition modify var justification find approximate decay centre distribution apply addition involve choose threshold discard significant modelling distribution distribution credible interval ci return period extreme large estimate hierarchical framework section model discuss briefly illustrate temporal result prediction section finally discussion west reliable day region insufficient surface water rare duration important public planning propose weather resource management region
fortunately avoid exception convexity instead penaltie simplicity penalty add convert compete generate model n act reflect correspond difference another nan run repeat loading repetition predictor penalty repetition pick hard achieve start find result much coefficient seem much perfect bad first add job recover unclear exactly applicability run biological
space hmc satisfy stationarity balance sec hmc non must joint express hamiltonian invariant ode system invariant continuity term eqs reversible obtain apply simplify need aim ode explore rapidly reversible large size ref split split energy idea express integrate reversible discretization splitting product contain dirac delta law splitting law eliminate freedom exact value field f follow determinant facilitate sec
immediate generalization nonzero column I sort otherwise state infinitely subspace match give hold row form state column condition condition prohibitive state pattern assume notice typical compatible
extraction tool gain lastly use image grey image image refer isotropic voxel result volume voxel provide image measure scan lag slice right normalizing slice grey
rewrite bag write hence affect instance log set maximum estimation maximize log form efficient therefore approach maximization hide propose logistic lr hide framework surrogate function r specifically step pt p decrease observe begin derivation log rule likelihood logarithm surrogate expectation instance probability py kb px em surrogate equivalent force keep q obtain substitute b b bi bn b use summation instance bag overcome prior computational probability introduce programming approach efficient challenge bag associate instance convert figure allow grouping complexity chain dynamically compute finally figure algorithm b p bn equation incremental illustrate e probability exclude computed recursion recursion bag compute bag p py b py py replace obtain store store store illustrate bag label box currently b p b
function strictly root thus since root notion behavior successfully power element algorithm error stability uniformly realization copy x z follow kx hold reproduce hypothesis hold stability theorem concern proof involve generalize newton sum eq symmetry symmetric hold let generalize implie pose conduct address explicitly section conduct experiment world dataset uci repository attribute attribute instance
htp discrimination illustrate newly publicly cancer publish newly roc accordingly cancer patient cancer control observe poor normality figure closeness monotonic outperform roc auc obtain htp htp htp outline methodology approach utilize
ep prefer approximate namely n ep approximate non joint un normalise bi variate q ep belong product division show variate normal ep try fix factor probability convergence n z ng leibler divergence previous kl expectation derivative close mean constant find particular gradient prior investigate approach aim correspond
calculate value nan alternative population problem model right estimate curve surprising relatively power pick quickly birth biological indicator birth affect classic study large state associate birth weight factor population medical black white status yes history history yes yes birth birth gram set relationship birth show ols black vs vs history history positively birth perhaps surprisingly age check predictor age standardize residual adequate indicate relationship age birth could expand specify ol age fit value age
hierarchy release mesh link mapping disease produce disease th imbalance gene disease gp gp pmf identity gene generalize disease full disease disease gp pmf pmf gene gene difficulty reflect table extreme find approach trace baseline pmf outperform baseline model rank constraint fig highlight propose top unable require store size implementation reduce initial indicate inferior disease correspondingly achieve dataset gp baseline pmf perform disease level recall result disease disease describe association gp pmf gp utility compare approach gp suggest helpful essential recommender information service matrix effective recommender recommender covariance may rating pmf side measure metric recommender typically concern
framework laplace draw zero spherical ensemble times error take mse analysis experiment aim reconstruct regard amplitude vary zero show amplitude error error tail b laplace fail amplitude reasonable amplitude range fast performance measure reconstruction
aggregate update however fully aggregate distance serve kl divergence bregman core traditional generalise generalised aggregate use prediction expert update distribution aggregate via log loss kl divergence bregman divergence notion apply update
place tuple weight assign use utility empty location opponent player hand al present possible include network position capable position function line player standard heuristic exception position randomization move player long play correlated ability play player consist game play aggregate scalar count average score constitute paper eight time advantage symmetric eight symmetric expansion tuple return perform recognition game al advantage tuple network
detail svm physics randomly problem tn still solve condition life recently call constitute towards begin method gradient cg remarkable property descent rely linear quadratic depend hessian index lipschitz gradient present relate expand cg minimize span direction propose expand minimize function propagation gradient cg method increase gradient version cg neighborhood extend direction smooth convex equivalent
quality process detect false observation tolerance change belong good reconstruct quality control ba propose nh nh denote probability alarm specifically q propose new smoothing dimensional bandwidth asymptotic given plug assumption
modify smoothed negligible standard assumption asymptotically close unconditional validity whether assume make object relax distribute main result iid gaussian imply observation iid assumption relaxed interval surely upper end quantile moment prove discuss symmetric denote covariance e hand significantly significant obtain rather rely iid tail generate illustrate line line asymptotic
stream supervision question statement statement segmentation know label segment slightly know statement statement statement contain e office case write feature directly model directly add triple loop keep win winner value whether old old old support memory compute old old old support memory feature find first thing memory train write eqs loss match term final word remain rnn eqs triple term argument side every rather compute task answer neural train predict read stream encode accurately fact past compressed dense rnn perform task sequence term speech incoming signal term movie answer modularity level
subgraph mining investigation eight popular dataset propose always return art magnitude exploit dependence test increase power frequent subgraph mining correction object massive chemical pathway structure web mining mining graph database two distinct discover subgraph statistically drug discovery activity similar fashion protein require event subgraph candidate subgraph extremely check subgraph exponentially cause huge significant represent number subgraph mistake science subgraph far severe resource significance level control positive subgraph common highly conservative test
kernel x x taylor correctness mention lemma repeat element define stand
readily calibration availability problem solution associate well proper probabilistic setting density enable computation static challenge sophisticated content raw image ignore process emphasis mechanic denote infer datum also example outlier calibration generally observe attribute model term datum conditional bayesian framework specify discussion dimensionality aspect employ improper prior naturally available priori mention primary work low could forward specification fluctuation turned assign nonlinear map different amount different would mle equation extreme variation direction even would cause huge drop variation whereas dominate identify effort drive argument approximation algebraic objective bayesian employ motivation behind intuitive resemble component pca vector along span decomposition attention field signal image audio attempt encode possible
require keep bound resample particle branching particle branch event mutually observation order particle order keep arrive online relative particle posterior allow online asynchronous particle weight particle process average particle process far number child weight design particle need careful appropriately informally weight alternatively approach scheme could poisson integer particle number particle start particle particle recursively would
advantage spectral utilize structure notational convenience term element equal covariate categorical beneficial center simplify section form robust node edge also brain compute use tuning procedure describe next alternative classical canonical analysis algorithm inherently reduction run spectral specific tuning block information practice contain tuning parameter balance value approximately interestingly lead continuous stable continuous range consider eigenvalue within minimize find define appear covariate vice static static remain slightly perturb transition value occur square th tune cluster estimate propose model estimator assume covariate bernoulli variable membership addition support depend
layer denote multimodal tangent force lead rnn rnn softmax probability next adopt sentence sentence length denote sentence generating layer likelihood sentence calculate sentence generate differentiable train rnn generation retrieval image retrieval sentence image sign start word pick model sign calculate generating treat affinity image retrieval query rank retrieve rank equivalent retrieval sentence retrieval consist appear problem sentence condition across training normalize original image sample ignore lead much performance use task architecture word language initialize layer pre imagenet detection combine treat experiment well
friend child parent movie dataset originally sentiment total movie review remain label negative train label set break map sentence train sentence experiment logistic eq sgd mini document iteration total epoch different minute evaluation particularly contain identify correctly part sentence sentiment rate extract sentiment entire
sample proportion crp cluster assignment construction k analogy stick break stick portion break stick remainder proportion first break stick infinite proportion cluster place relation dp place set namely cluster first correspond specific domain correspond list item entry item record l j crp prior indicate strength typical assignment object relation matrix object index discover case binary domain describe j eqs domain object point explain imagine user mutually act user reach first domain user single reach domain applicable node limited applicability domain whole discussion old one drawback cluster assignment membership blockmodel multiple assignment change edge ibp handle attribute flexible many view model model superior interpretation somewhat counter intuitive simple advanced probabilistic attract pure limited nest chinese restaurant membership representation base possible connection three connect link separate combine triangular representation simple generative scalable limit cardinality triangle binary network consist million node consider cross collapse
free network htp color free network extend accommodate node extend accommodate optimization ai onto per complexity two penalize package penalize adjacency describe finally standardize standard deviation display simulate correctly fine obtain varied obtain figure display compare use appendix htp colored line proposal figure surprising explicitly estimate ise refer network refer discussion graphical model density ensure sum one network penalize log likelihood due difficulty several neighborhood solve separately penalize pseudo q proposal involve maximize sparse network admm solve step
sentiment want distinguish label review product movie book domain achieve transfer unlabeled review book hypothesis example hypothesis true suggest even generally classifier approximate example example sample train approximate equation example eq subset show empirical combine result similar vc tell risk tell find small fix minimize risk divergence strategy domain indistinguishable source risk target present
closed form know factorization spline exploit burden computational scheme spline spline recently become mainly due extension expression order spline factorization diagonal interestingly share closed term spline highlight spline error method model adequate
yy local ds ds show exist increase decrease section lemma going result lemma deduce correctness log concave convex characterize dirac exist measure functional clear dirac hold constraint iii either give eq q integrating useful continuously theorem try point
annotation combinatorial harmonic compute image reach database sample al pose question toward acquire strong supervision multidimensional pair landscape still landscape grey factor element semantic representation people ordinary texture proper color texture highly color analysis investigate great et reason make meaningful feature fully automatic method edge base arrange create van style help distinguish conduct van collection small state rather van test statistical et al ordering embed one tried apply unsupervise laplacian feature fast unfold capturing contain database name last name title style image detail primarily
overcomplete observe spherical iteration learn significantly large signal noise previous efficient desire optima overcomplete model mixture achieve gain speech computer unsupervise challenge task label extensive topic gaussian decomposition order usual tensor tensor method blind detection current decomposition degenerate complexity requirement exceed often paper power make progress repeat tensor main intuition characterize dependency component randomness argument idea inspire algorithm analysis aspect pass typically infinity handle factor amp polynomially idea vector initialize core state follow gaussian initial universal tensor multilinear form w correlation j statement dynamics rd mixture show initialization condition initialize initialization combine overcomplete
need enable parallelization streaming later build adaptive sampling idea branch technique back center focus whole cluster encourage understanding result provable presence base family study book survey aspect area suggest elegant primal area problem often present semi number simpler tight nevertheless online
g problem assess efficacy presentation illustrative use namely toolbox locate conditioning represent along straight ray rectangular situation sample problem insufficient domain take full lead right side take represent hand sample l curve know rate lead problem solution deviation copy level obtain matlab problem apply preserve find tolerance white variance approximately matrix white color error detailed table level order randomly illustrative rate noise
prove return enjoy normalize generate subset
network field researcher experimental specialized specialized compare possible classify contain high physics matter physics physics cover scientific author author undirected author network extract along physics physics matter physics individual neighbor undirecte summarize belong task belong physics matter physics medium differ physics cm classify physics classify matter vs cm classify domain know network behave network nod formation random graph real discriminate social closure degree twitter twitter make information adjacency undirected graph generate random node number add edge twitter binary consist similarity run classification base five
change vote translation score later potential label class formally identify true predict invariant gap aggregated score serve score item invariant quantity gap aggregate across item labeling meanwhile bind rate relate assign labeling besides interest introduce notation score gap aggregated error change amongst aggregated score translation label high bound error kullback divergence high model worker notation define require condition aggregate basically predict item predict label bind bounding solve solve analytically thus bound serve condition present notation entropy bernoulli variable notation formulate theorem practice might evaluate conduct take estimator thus general setting compose generally dominant inside min rate recall tighter behave tight proof defer apply various labeling correspondingly apply binary majority voting majority voting scenario cover case multiclass labeling weight voting since hyperplane rule later posteriori vote weighted majority
bundle bundle admissible bundle vector outline argument h see reveal accord need utility bundle map kkt bundle linear follow sort run allocate namely admissible bundle admissible cost allocate bundle bundle p bundle admissible design admissible bundle j j j admissible obtain bundle amount amount increase crucially good well bundle exist clearly good else case transfer bundle bundle latter amount increase contrary suppose allocate get otherwise case admissible contradiction well admissible bundle check next help lemma clearly bundle second partially bundle case increase come check namely second need mapping utility h time mapping sample preference particular class know segment see
prior sensor parameter many let paragraph note let bound boundary positive self adjoint trace respective unbounded know note choose assertion proposition summarize derivation employ lagrangian pde lagrange multipli function ik ik ik n ik ik ik space counterpart derivative state adjoint vanish weight basis vector derivative gradient provide appropriate adjoint adjoint vanish variable define require vanish notice adjoint rearrange system ik hand side reveal follow thus adjoint eliminate right side element adjoint letter discretize describe computation gradient q note rely gaussian trace justification procedure compute I adjoint variable point accomplish follow differential appear discretization pde block eliminate solve ik right expression discretization eq discretize email edu edu email edu email optimal differential seek infer parameter field govern small moderate aim inverse moderate parameter author forward compute design gain
rate constant case td derive require mix handle assumption td handle asymptotic broad reasonable mdp second inherent iterate reason carefully variance iterate average large succeed necessity mix stochastic optimization variant td use iterate contribution summarize bound probability td variant incorporate center easily approximate iteration scheme show finite obtain expectation eigenvalue
requirement tune extra information precise topological without frame distance stop agent move status rw meet stop correspond occur observation mode set metric cloud dimension cloud persistence agent make improvement metric incorporate static rw status nod percentage stop node event I configuration agent cloud space proper persistence diagram hybrid static existence environment persistent distant outlier world importance cause appearance
similarly obtain existence moment drop throughout calculation
request unitary multinomial purpose classical square contrary logit recover however estimate cdf logistic significantly low rating avoid summarize outperform h kullback leibler divergence remain split rating range binomial rating rating l suffice prove tensor get f control apply ensure plug eq polynomial relation
send en en tr cr cr could rely successful coarse sentence even able outperform ever language language language obvious act unlabeled datum prove useful nlp representation syntactic unlabele usually small datum exploit unlabeled generalization allow vocabulary thereby partly sparsity concentrated start look document representation align language align language develop english document train english learn represent document resource language project annotated datum another projection resource translation
fx f f I actual constant big large section nesterov prevent without fall bind uniform actually
evaluate element sense principle converge optimal available generalize space application third define convex crucially remainder respective near state desire precise representative fix order lemma appendix let assume basically representative partition relative correspond sum adjust lemma result guarantee regardless specific respective representative proposition small kernel fix impose possible flip side statement appropriately rather guide sound limit illustrate simple world capable contain move difficult task classical double balance drug schedule domain action discount algorithm derive decision evaluate task appendix capable two reach goal avoid along except discount evaluate state collect policy action random case define group mean place varied report pick maximum return use convention instance coincide effect representative improve surprising magnitude indicate contain depend reduction consumption resource replace approximately operation policy figure fix performance improve however linearly explain huge time evaluate compare reinforcement task iteration popularity build transition good balancing long unstable problem apply cart track cart balance hard compare single implement reader carry transition cluster version fair varied policy iteration figure call hard variant decision policy probably achieve rate balancing figure contrast decision able balance attempt interval balance difficult reinforcement result opposite conservative estimate run computer use policy minute compare evaluation step involve require contrast operation fit iteration fit conceptually also build transition adopt choose
x ever transform lagrangian lx ta respect wise ty ty substitute write n ty ta ty ty equivalently ty constant composite ty ty ty hessian boundary pseudo parallel extensive literature crucial gap proximal gradient method fista regularize yield payoff partial outside scalable rigorously demonstrate derive
scale try hyperparameter prediction show comparison three setting term shape hyperparameter factorization variational vb vb prediction receiver operate auc measure prediction extreme imbalance less motivate auc since auc view imbalance link independent term variational cp tucker factorization model randomly unobserve incomplete factorize cp extract factor full missing performance
positive fewer certain accuracy level might come slack formulation margin slack actual object interest classify hyperplane slack training datum proxy slack th slack scalar dimension grow number optimization qp medium hundred solver suffice qp suitable algorithm tackle general qp package fast specialized qp propose call solve solver adaptation setup example look example check example easy classify hyperplane subsequently know classify learn improve train result classify hyperplane easy classify even indicate hard classify train svm enforce margin small influence compare inequality value correctly possible weak optimization overfitte original tolerance
nature impose inter dissimilarity latent dissimilarity isolate need central spline priori dynamical evolution occur euclidean curvature sensor may geometry space euclidean euclidean measure vector incorrect simplification bregman measure bregman positivity bregman
study recommendation situation experimental result validate energy mrf image minimization play role computer vision early vision follow pixel define pairwise dependency graph minimization accuracy accuracy thorough comparative study dense three closely hardness figure hardness situation perform validate combination exist reasonable unary computer vision combinatorial graph minimization decade hard exist general problem characterization mrfs state art mrf include cut solve graph partial labeling
baseline projection multimodal clearly numerical require majority handle cluster original I replace present flexible capture content unit randomly country remove project posterior people china interesting north east south south task fill answer response answer ignore create randomly portion answer remain answer generative answer predict person answer multimodal collaborative filtering subset ccccc categorical continuous category six representative problem country country origin size country iv
attribute invariance desirable vision low level pose abstraction detail label spatial invariance signal incur combination max layer employ originally dense substantially relate invariance inherently boost fine employ fully crf semantic score interaction pixel increase crf efficient ability fine largely improve performance boost pixel classifier state couple pixel system virtue fully ii challenge good margin system compose cascade fairly technique make system potential
success logistic give interpretable ht grow tree predictor predictor role categorical include candidate categorical predictor fitting effect capture split split among regressor candidate logistic ordinary determine effect response grow wider able estimate logistic logistic logistic numerical regressor criterion regressor node terminal node stop fit degree minus turn terminal node otherwise select tree chi ordinary satisfactory divide interval construct table pearson chi result chi dependence also select predictor splitting interpretation cm variable take consideration depend fit split simple node good regressor conduct chi split fit candidate regard count failure expect instance success observation category cell thus chi square cell proportional count hence degree freedom numerical numerous contingency distinct count pearson number cell theory violate fit find additional contingency logistic numerical predictor chi nonlinear adjust compare group divide fit sample cutoff point table chi way pearson chi roughly approximated chi variable otherwise two lack summarize run logistic model construct contingency category cutoff
parameter intractable inference technique variational fast problem handle multi problem evidence estimate parameter inference technique approximate tractable approximate variational kullback kl maximize non act logarithm evidence apply low estimate maximize slow recent advance approach gps use variational variational write since diagonal diagonal diag variational variational bind log integral jensen bind write maximization positive covariance diagonal diag diag concave respect inverse determinant concave ascent parameter
map z z patch patch square point possibly locally shift parameter cm cm rectangle rectangle aa node aa node every c right leave yshift style rectangle node cc middle cm every grid yshift every aa ab ad ec ed aa aa ab ac ac ad ad ec ec ed ed rectangle xshift mm yshift patch right style bad ba bb bc
forest type know datum allow test manual calibration visual assessment parameterization adjust pattern typical create virtual output rate option model know priori sufficient virtual practical understanding process interaction posterior width posterior create synthetic supplement median panel visualize refer display low fit black correlation model tree specie seven detail availability distribution virtual aggregation output concentrate v brief detailed density parameterization table represent assign parameter retrieve order plot mean display correspond uncertainty substantially subsection represent sample along respect
train add layer connect exist stack head subset label cluster train augment pre minus hold pre make capacity rather adapt new tune perform fine method
minimize variation transform transformation form signature signature transformation transform signature denote robustness cluster art test parameter mean rely information cluster spherical split
whether extend paradigm instance methodology measure instance identify examine set learn weight filter especially useful affect backpropagation result induce generalizing however beneficial induce instance even datum label correctly model misclassifie remove training solution beneficial multiple misclassification repeat benefit beneficial significant preferable generally remainder organize review relate handling motivate weighting conclude c package conjunction terminal explanation use load package graphic terminal
red circle positive variable toeplitz assumption replicate circle positive influential predictor scatter plot smooth region definition fs fs fs height depth pt mid pt fs fs keyword modern set observation variable selection important flexible variable call resample procedure add control boost simulation insight minimize observation descent boost intrinsic one begin compute residual call base learner good percentage iterate final fit learner learner spline square boost iteration iteration early
follow illustrate potential policy iteration policy approximate generate extra exactly except induce represent decision represent incorporate action gap policy tree partitioning define extra obtain build fix represent corresponding decision vary tb describe space day high load bar impact performance compute well restrict efficient expect trend policy influence difficulty number adjust complexity specific greedy policy restriction benefit term general truncation policy tight vs vc gap property easy report order magnitude tree show conservative may thus performance performance algorithm stop monotonicity still upper actually distribution policy iteration converge actor evaluation implementation ac algorithms actor use gradient actor policy space would ac tailor continuous share form actor algorithm problem vs finite remark allow space slower reason complexity supremum advance localize rademacher modulus empirical hand relaxed gap regularity goal decide efficiently informative sample
discretization refine posterior along inform maintain mh proposal inform subspace dominate prior prior dominate adapt proposal adapt without target proposal proposal replace symmetric regularize sample independence absolute continuity condition comparison produce size direction influence autocorrelation mh improvement operator proposal several limitation art first representative structure non problem also broadly local target form great part limitation present describe simplify mala stochastic newton langevin sde method hessian simplify mala use derive proposal inverse hessian mala newton related function expect newton data operation expensive mcmc sample mala generally local riemannian metric insight direction curvature insight distribution particular
include major education conference number color topic actual topic uk actor computer computer operate system system label record music record label company production company company cc cc gold natural world ten discover assign characterize basic foundation consider study describe research year execute corpus number iteration similarity experimentally approximately node set candidate significance topic nsf grant ten see label two evaluate label show six mostly topic inter lda topic grant label rank return present comprehensive visualization reveal scientific subtle effort education conference display towards indicate color discover characterize
observation occurrence context pair may explicitly co occurrence local bias relevant context weight
able surprisingly benchmark deep architecture boltzmann auto encoder size adopt map otherwise map size really performance curve dataset besides aforementioned method local atom randomly fine tune sift increase consistent encoding encoding dictionary variation image opinion atom relatively advantage unsupervise mixture learn increase redundancy decrease efficiency hence desirable combine sift base performance table sift dictionary increase begin slightly k however c effect conduct original image extraction block map pool yield generally layer mean size improve translation add sift pooling effectively detailed fig representation size field pool representation well representation complementary individual stage encode sift representation voting
reach regardless add sum entail ultimately estimate discuss asymptotically case r enyi model hold pn n n enyi know nk nk nk nk nk na vanish instead claim consider parameter notice non degree os belong easy denote k proposition nb n
q one evaluate q aggregating observe across lemma construct packing every satisfie packing bounding desire issue boundedness final relate likelihood hessian q making allow must define minimize n must virtue coordinate sub substituting get everything final set guarantee binary code hamming code e ed vector pair satisfy j third vector l j step make value decomposition u pt minus minus r star empty name display tag display display tag macro name cr cr cr cr
constrain domain inequality distribution close impose categorical logit distribution rank supplement p z z read k tractable approximations box field employ q method applicable persistent per explore explore entire fit idea cd create discard e collaborative filtering maintain moderate parallel helpful boundary collect mcmc eq describe three namely handwritten digit collaborative survey supplement difficulty end progress reason map nature underlie norm rescale hide unit deal another posterior add evidence leibl posterior posterior give simple integration constrain
learnable supervise may arbitrarily long dimension space linearly c give well define actually calculate take early proof class tight dimension let dimension generalize number follow standard optimistic dimension confidence set function square l tf growth empirical c distinguished lead function set scheme argument essentially statement value derivation deviation
acknowledgment acknowledge helpful lin thank berkeley national laboratory energy office office advance compute research contract ac berkeley national laboratory york lin support national grant dms dms work office office advanced mathematic contract ac foundation dms lin sampler variate assimilation laplace asymptotic expansion implicit sampler regime improve confirm use assimilation direct sampler give variance weight determine independently literature introduce noise smooth density
distinct graph order graph node hence theoretical definition loop multiple simple assignment node contain edge walk repeat path cycle node cycle graph edge path edge path edge say node list
reaction nucleotide template determine light intensity detect axis determine nucleotide flow light nucleotide flow excess nucleotide ideally incorporate reality light incorporation identical pose inherent difficulty accurately specification kind permutation target denote nucleotide distinguish nucleotide cycle cycle show sequence particular sequence nucleotide need flow cycle sequence
stock portfolio specific portfolio need stock portfolio correspondingly examine investigate financial fast transform filter technique publicly financial rapidly hardware enable eventually stock automate machine finance frequency focus company reduce easy stock combine capability support portfolio suggest
likelihood constant linearly however condition von posterior concentrate around reduce see plot remainder linear multiple need maximum yield construct cauchy guarantee refer take outside inside linear line exact sample challenge address first mean way stochastic define generalize perturbation work require infinitely draw perturbation break dirichlet insight rejection lead likelihood behaviour illustrative proportional independent unnormalized perturb location basic ib consequence choice axiom moreover independent generalize continuous maximizing perturb analogous max adapt sigma finite disjoint consistency requirement requirement ensure consistent ensure requirement restrict
improve various parameter reduce prediction provide experiment dataset e neuron describe rank pair neurons association rank ground participant recall curve show sensible especially density small truth manually association roc pr curves network case partial correlation filtering function pca case increase really improve various minor last show tune good performance
usual hour work indicator term three child status mi width confidence property challenge high nonetheless offer coverage age figure regression improve method tend coverage particularly interaction toward effect expect imputation coverage drop coefficient hour work lack appear effect effective similar age ci examine quality estimating cell plausible display proportion child home perform rate drop coverage never drop every good coverage arise misspecification tends somewhat extent probably due large cell standard error evaluate field construct production brevity subset panel subset wave subsample size census researcher estimate production record however variable
high linear regression observe estimation regularize selector therein biased lasso estimator normality example base measurement wide recent attention observe measurement recover nuclear minimization noiseless sign recovery compare estimation theoretical confidence increase bias correction de wise constrain quadratic procedure high inverse structure pose proper relaxation atom structure solution efficiently geometric intrinsic difficulty tangent play role develop general induce answer inferential question local rademacher complexity studied capture difficulty pack volume paper quantity computationally difficulty main summarize unified cone formal ratio define establish normality linear compare consistency remark beyond intuitively present fold unified estimation feasible program provide collection provide feasibility guarantee
scale nuclear relaxation one
mis obvious ij convergent moreover I j var almost surely eq argument c nz c nz remainder decompose integral equal follow conclude precede analysis respect substitute nj cauchy n thus complete establish show deterministic chen q gradient examine recognition ever approximate fit rise response advance mis significant issue mis move memory chen incorrectly integrate model study demonstrate mis obtain inferential specific dependent precise degree mis mis specify regularity true pseudo pseudo limit proceed behaviour incorrectly dependent equal specific integrated move average distance extent mis several firstly derive commonly estimator namely
water essentially table use boost boost boost boost circular simple increase move emission boost variation emission boost variation illustrate uncertainty observe well star circular variation marginally include imply calculate turn well example biology us protein atomic datum determine mechanic force field calculation force field determination typically contribution force clear realistic field comprise weighted force field freedom atom angle inferential determination determination appropriate weight field good force normalize assess observed prediction field incorporate boltzmann partition prior inverse temperature force explain realistic temperature identical unknown force van link drop atom van force attractive van interaction force attractive
give remark take fusion derive fusion diversity classifier could investigate additionally interesting study performance video retrieval assess trade rank diversity retrieval perspective take classifier indeed pac obtain majority vote vote belief kullback vote problem part european fp agreement author corollary st de st france universit france past
author would like thank grateful family department introduce convenient strategy predict distribute random size implementation optimal moment condition string moreover string alphabet poisson universal law alphabet compression understanding outcome individual quickly shape small example chinese character chinese character frequent choose redundancy count whole n n minimax familiar universal pointwise p cs nm instead slightly suboptimal independent indeed preferable size advance count conditional sum count close relative unconditional subject expectation
I I lm ij lx lm lm lm lx ij lm lm lm previous example lp qr tp walk metropolis hasting mh slice alg tb dimension proposal tb tr pseudo handling scale slice reject reject lower decision slice accept need new inside slice whether interval slice upper bind outside low low slice make tb prior tb z tr l let inequality nonnegative gm inequality symmetric trivially use
hyperparameter broad uninformative gamma accuracy approximate entropy compare es compare rejection rs objective process fix data hyperparameter know ground rejection scheme work discretize grid mean variance evaluation entropy use rs measurement collect display x particular measurement display leave figure approximation ground truth es see discrepancy rs discretization draw ground truth rejection method plot objective similar rs ground performance optimum box reproduce comparison optimize dimensional unit first gp
video code good energy closely decade devise dct prominent work dct et different nevertheless multiplication year see advance new dct exception arithmetic cosine background recently yet scenario
complex correspond respectively phase formula critical fig transition threshold match leave gaussian random column predict obtain formula I case predict predict independence property theorem temporal perform simulation window monte trial observable htb analyze propose complex value correlation synthetic stationary mean white series band pass add ik ik impulse band pass stationary series stationary impulse band band ideal band finite impulse chebyshev part see sample show magnitude finance focus detect series correlation detection lead reduce computational cost wireless sensor network useful remove study consistently finance financial correlation become e scalar naive sample dimensional random completely correlation consider instant time
hand show pseudo herein operation compression technique us prototype element ds compressed prototype dm compression compress prototype set order execute retrieve prototype interval effectively proof narrow size extend evolves show tr set iteration herein expansion replace discriminate prototype perform threshold graph maximally prototype analysis outline evaluate dm two execution rs initialization size relate single fitness variant initialization detail fitness generation dm rs initialization compression operation distance characterize operation cardinality operation entropy cost cost relate embed dm cost expand deduce version sec implement operation notably rs synthesis depend tune compression seek ms procedure consider element different heterogeneous mode ms filter characterize neighborhood define neighborhood size weight characterize reason initialization synthesis systematically value neighborhood cascade compression computational fitness involve algorithm
markov since correlation limitation normalise compression define correlation string string short string common long string lag string string motivate consider inner cross involve lag string rely determine bit encode perform illustrate observe alphabet may interpret require optimal thus quantify bit require represent accounting dependency finite define give base increasingly term cross entropy quantify expect source code estimate iterate describe term source quantity denominator serve analogous denominator entropy observation prediction quantity b b mx nx prediction evaluating vector precede r parameter predict base determine embed space illustrate depict far future motivate
multiplier kkt satisfie let l c rl j j j sum cauchy schwarz oracle choose say moreover plug dd j p j coefficient integrate logical among g two main hierarchy interaction associate e jk turn achieve see benefit involve parsimonious inherently hierarchy traditional hierarchical turn multi ad shrinkage purpose vanish guarantee optimality slow large vanish main enforce magnitude coefficient desire apply introduce part constraint drop study decade
gradient short dependency dominant researcher try devise simple gradient descent use growth derivative guarantee interested sophisticated follow element wise nonlinearity unit attempt result recurrent rnn recurrent task capture long task speech et al evaluate recurrent lstm empirical describe
author far least text must author estimator adjacency text text text text proximity word sequence index word symbol symbol require direct proximity discount sentence direct proximity word word word word primarily include exclude gender type text sentence illustrate common symbol window worth load worth index accuracy unknown text correctly attribute denote write accuracy classifier construct direct network word edge represent edge text compose word measure pair select node discuss calculate sentence way discount factor apart similarity sum find position apart direct word eq similarity proximity word every amount text undesirable want normalize author positively node divide word normalization matrix inherent serve know text interpret mc probability word every
reject reject sample vector approach reduce post hoc difference due dimension eigenvector eigenvalue importance hoc significant clique reject disjoint post hoc find clique retain level post hoc correct experiment show univariate show multivariate seven discriminant discriminant knn neighbor random tree svm svm bioinformatics ec fold multivariate normality normality precision normality compare use algorithm
write derivation unfortunately proof essentially instance whose metric derive formulation svm segment letter tune validation misclassification average along standard across competitive kernel svm remark derive global algorithm advantage exist way learn analyze approach generalization evaluate superiority scalability measuring instance distance euclidean lead grow past year survey mahalanobis constraint close typically standard unable multimodal boundary overcome limitation work metric per locally simple metric perform integrate burden severe overfitte method
block belief layer distribution rbm logistic unnormalize low recognition p efficiently intractable case concern markov mrf rbm denote potential f latent p pair want compute test place mathematically outline closely applicable addition unlikely reason mrf situation appear denominator log partition translate log problematic inaccurate dramatically lead researcher direct performance fact sigmoid network cite
desire hold l sf nf ix bound thus uniform statement yy unseen belong specify unseen q ms mi give feasibility sf eq apply statement probabilistic statement inequality least contraction inequality bind yy side yy nr c yy nr c lemma yy nr yy n nr imply low bound yy yy statement unseen realization low simultaneous unseen realization problem sm j equation feasible every change appropriate eq add third value replace max term hand due concentration quantify get probability around maximum change follow statement right sl events ne ne sl n event happen e
architecture fundamental propose score datum fit great difference tuple circumstance know nature find poor local initialization multiple begin kind stop experiment step maximum
explicit hoeffding concentration inequality extend argument upper bound union bind long technique naturally case contain detail lead suboptimal optimality proof tight proof generalize appendix detail hoeffding type martingale like except bound explore emphasis work prove slightly stop martingale theory exploit favorable convergence nonnegative nonnegative possibly argument begin construction hoeffding
sensitivity specificity g svm evaluate benchmark uci rna graph library framework matlab evaluate implementation objective certainly implementation implementation normalize zero demonstrate data sn accuracy letter clean eeg proportional weighting study experimental imbalance ratio detail ann run deviation small bold acc sn letter rna acc sn mean
acquisition restrict minimization recommendation portfolio compute entropy detail rest visualization objective dotted plot acquisition ei blue pi ts red select candidate histogram blue dash draw blue green triangle vector bin minimizer depict utility first monte carlo integration e step sample quasi carlo entropy analytic simple impractical intractable compute replace desirable propose expect close target
software author read final manuscript acknowledgement helpful continue user additional program useful suggestion plot frequency leave variant chi table small summation use impact impact accuracy rapidly frequencie table consider skip improve problem increase plot space rs phase e likelihood total historical extreme value finish due monotonicity dot horizontal mac mac mac mac snp mac mac mac mac calculation dataset mac mac mac mac snp mac mac basic cluster mac mac mac k mac mac mac k snp mac mac k k
moreover change posterior bias varied plot appear value situation although bias appear variance precise dependence scale appear marginal standard via square instead concern long plot fitting fitting exactly hold reasonably consistency likelihood base estimator proportional impact improper twice repeat change difference well practically improper likelihood consider present table rr std ci exact process via approximate run via exact use red respective correlate interval slight shift exact imply less excellent frequentist level exhibit behaviour check method entire analyze result residual present rr std slightly close large small vary analyse similar vary posterior deviation result posterior plot log ccc three proportional reliable conclude posterior uninformative default ability explore
nuclear term step k alm reweighte norm like e lagrangian descent optimize fix alm rather update inexact alm projection soft threshold singular minimum follow toeplitz determinant heuristic weighted need solve nuclear subproblem analytic add definition constraint lagrangian augment lagrangian follow alm strategy weight nuclear completion minimization take gradient formulation update look reweighte derivative equation arise lyapunov alm reweighte nuclear section
initially negative parameter increase threshold true threshold event comparison stable partly term filter addition stable peak wider well noisy scenario time sect curve performance drop peak cc close parameter setting investigate temporal tweet sect b spatial tweet fig decrease hence especially temporal interval tweet generally noise finally influence relevant tweet increase tweet observe gain match intuition high signal detect event concentrate correct temporal spatial threshold comparison employ filter synthetic generally suggest detect different absence simultaneous spatial also lead event two highlight multiscale dedicated detection scalability public tweets york bounding box leave gps pair stream public tweet twitter stream request retrieval tweet predefine bounding box tweet tweet contain check implement daily detect day vector implement toolbox remove provide toolbox http minute eq term appear tweet evaluate standardized term valid minute number scale method post
ex ex school department berkeley pa berkeley dual establish selection even nonconvex guarantee rigorous nonconvex nonconvex vanish recovery may require incoherence condition corollary wide method composite loss square modify conclude study prediction body relaxation nonconvex arise see broad identify among candidate encode entry vector optimization hard much slightly problem constraint norm well condition relaxation estimate parameter therein encourage sparsity aspect norm value linear regularize sample setting nonconvex smoothly absolute concave regularizers become regularizer cause numerous empirical variable nonconvex paper penalty assume continuous allow apply one exist open may take extra meaning minor abuse notation appear side univariate act readily regularizer restrict homogeneous estimate function goal consistent consequently feasible study say symmetric differentiable amenable vi regularizer vi amenable notion past regularizers vi goal conclusion vi everywhere differentiable furthermore appendix useful amenable regularizer many popular regularizer amenable amenable example illustrate take form zhang take form eq amenable finally consider example amenable mention penalty amenable
potential structural biology determination interpretable protein enough particle particle intensity collect ray sample fact copy collect treat particle ray orientation pattern modulus particle pattern different slice complete recover pattern compression experimentally artificial coherent light source hz correspond hour european become operational hz alignment demand high volume achieve balance object light parallel keep experience heterogeneous computer fully implementation scale cluster hundred
one unconstraine bound singular approach find nuclear convex relaxation extremely find recently since problem indeed demonstrate extensive recover vector g reweighte minimization variant ji problem arise network localization al reweighte solve addition extend problem square produce method either processing solution apply expect reweighte capable yield aid solve variant study stationary minimization minimizer first order minimizer
small anomaly detect expensive enyi bottom outperforms gap stanford analysis project http stanford edu co record amazon represent product frequently represent internet take undirected link eigenvector large residual subgraph align amazon com modularity residual shown leave frequent co isolated set small large subgraph possible edge subgraph internal neither income compare take million comparable average internal none among internal great however primarily outside span anomalous consider graph represent norm get large index highlight deviation eigenvector align subgraph align vertex primarily external degree subgraph take sample size vertex share vertex dense external vertex dense eigenvector extremely external degree anomalous background residual principal algorithm simulation utility identification detection network subgraph show anomalous background demonstrate anomalous processing recent focus extend attribute particular dense subtract edge rather add
sensor switch dual past decade range environmental air increase decentralized scheme reduce rate ensure presence failure relevant centralized literature distribute adaptation interest communication range dynamically sub share contribution dual average distribute method allow adaptively diffusion network uncertainty third weight uncertainty reliable statistic derive regret highlight link organization background graph review optimization network weight topology distribute application operating environment online subsequently online demonstrate conclude future utilize system
somewhat behaviour mode move mode phenomenon high gp tp consistent close elliptical tp task show gps computational home message tp gps modelling flexibility extra suggest useful gps almost application tp orthogonal expressive kernel proof lemma corollary paper student analytic marginal enhance explicitly depend verify student situation change structure covariance good come additional rich bayesian non simple exact
minute nearly minute figure safe greatly superior pca develop algorithm problem norm convert constrain online stationary moreover empirically propose recently online nuclear suggest hard corrupt relaxation stochastic bound exist exist bound solution q examine uniform note uniformly prove constant eigenvalue equivalent basis produce norm upper consider uniform trivial feasible upper uniformly bound make solution boundedness surrogate lipschitz uniform bind uniform tf subgradient theorem measurable subset borel measurable nf suffice hypothesis corollary particular set uniformly proposition family pf ff p
ks score quantify lead text unsupervise term component quantify gene activity sometimes component characteristic set likely use collapse gibbs lda lda hyperparameter collapse gibbs yield em laplace select predictive repeat well fold prefer expressive fold separately optimal
fit label semi supervise good principle semi datum generate make like process unlabeled text text empirically task good well cluster unsupervised model many care serious b average datum explore supervise natural focus study paper book explore possibility semi supervise thorough carry explain limitation language large intelligence scope text organization present overview semi technique follow semi consideration conclusion label train little label generate label model label working principle value
reconstruction error depend vary component component error log balance bad relevant digits quantization numerically might want possible change change relative scaling theoretic homogeneous intuitively compression viewpoint divide error move bit per gain may actual error usual problem might prefer conclusion minimize reconstruction provide term certain regularization add reconstruct backpropagation noise namely provide noise differently translate feature auto criterion neural need include variance parameter well modify reconstruction involve square quantization still
want distribution maximize newton read entire propose modification require dataset substantially decrease analyze theoretically open dirichlet distribution long prior analytically bayesian maximum data recommendation engine rating user rate inaccurate prevent estimate opinion week represent week start update originally similar dirichlet biological core level
decision process markov robust amenable decision maker scale dynamic programming use bad leave adapt black available problem seem relaxed theorem corollary claim ie il ie ac il il ac il framework belong min solution realization type case uncertainty lead semidefinite paper solve stage program solve number call robust fundamental perform parameter convex concave optimization
otherwise size within fact clearly go well state sc construction special polynomially relate hold ready first transform find see every induce cover covered equality would conversely induce norm cover put contain exact cover conversely cover sensor problem choice sp solution p set
public invoke review role causality reasoning broadly concept fitness begin causal difficult persistence right context cause competitive training physical fitness account failed meet bar contract demand role causal origin lead mechanism connect character fitness might concern role student character access causal pathway argue fitness use decision origin fact
large correspond consider eq choose set minimize square think consist nonparametric tail effectively use number thus consist output mapping use linear mapping rbf dd project respectively note I I set q nf tn output rp compute multiplication vector include metric every input space complexity need store basis contrast space lastly evaluate
sequence rbm plan introduce multidimensional rbm drive motion right plan simulate free motion conditioning satisfy previously another namely continuous uniform simulation technique tolerance piecewise least complicated wavelet localization tracking jointly maxima minima dyadic ultimately dyadic reason preserve easy piecewise input gradient since computable lipschitz combine approximation fact strictly eventually construction indicate rise
regime covariance compare gender cover body g oppose spatial spatio temporal appearance highly gender especially heavily low result maintain low e avoid curse dimensionality paper organize dc feature
act na image unitary rotation act feature without cost iv canonical fundamental property readily apply implement sign simulation obtain sign separation ease applicability trick invariance induce transformation possible focused sign sign finally experimental result come class form one sign cluster sign namely obtain eigenvector norm thus interpret learn representation increase compare ambient try projection g scalar heuristic lead fluctuation scenario compress different combine kernel learn
experiment dataset randomly class filter filter testing error average show table outperform improvement follow classifier good method histogram object cifar image test cifar significantly scale object explore simple database face digit roughly align accommodate database spirit verify q mat k spatial pyramid pooling aim database help object experiment face digit train cifar filter overlap region bin pool pyramid pool histogram pool reduce learn descriptor paragraph combine dimension descriptor table gain degradation art method augmentation extremely encouraging triangle convnet convnet conv maxout combine arguably simplest unsupervised convolutional deep pca binary block convnet network layer train extremely efficient involve build comprise follow output alternative yet could facilitate couple extension image classification hand write digit experimental show outperform par convnet comparable hand feature task face recognition robustness
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task need ideal operation reduce first point equation minor coordinate lie generic form certainly avoid difference ideal symmetry restrict affine chart x assume normalize vanish looking apply yield intersect equal n symmetric easily compute determinant express determinant vanish statement lie lemma following hold proof apply describe u obtain replace th sign u jx h h combination polynomial comparison strategy ml precisely action symmetric group look ideal polynomial equivalent requirement
accurate answer mechanism exist super smoothness mechanism accuracy much inherent private mechanism answer direction exist algorithm synthetic database accurate smooth query specify immediate question output database preserve natural contain proof section query notation proof discretize dataset please step similarly laplace noise finally notation mechanism preserve privacy straightforward output private immediate specify later good approximation also follow equality correspond error error round error respectively equation five type separately derivative discretization eq let depend satisfies calculation yield chernoff q eq q calculation run arithmetic operation
privacy mutual information perfect protocol minimal vs theoretic privacy preserve mining investigate namely analyst privacy trade study consider analyst multiple clear analyst perform statistical operation operation publicly party statistical operation study set include social recommendation analyst enable user perform aggregate privacy private linear constraint amount analyze privacy recommender system recommendation focus make output application private recommendation differentially private privacy consider recommendation rating differentially recommendation user might link party adversary rating manner private none work study privacy perfect independence work privacy necessary amongst main theoretic develop asymptotic datum grow arbitrarily asymptotic private problem impossible practice privacy approach distortion privacy mechanism assume general private develop quantify privacy privacy design mapping inference information information private hard involve paper private datum privacy simple also limit statistical keep mutual review highlight challenge factorization address prediction analyst possible news article etc user item pair rating rating rating attempt comprise rating particular term da kb ij profile typically minimize
theory hold assume neither symbol meaning estimate unbiased unbiased mean observation symbol symbol stand data moment fourth observation shrinkage shrinkage shrinkage quality target contain target limit behaviour integer overview shrinkage combination estimator optimize manuscript generalize optimize turn linear arbitrarily rescale quadratic program equivalent optimize eq govern quantify target estimator unbiased estimator adjust target element contain target correlation entry variance estimator element target optimal shrinkage quadratic follow estimator relate limit consistency sequence index kl assumption estimator behaviour behaviour none target identical relative go limit property combination
performance texture highlight dataset create step datum construct discard particular class c sample exclude combination average hyperplane exclude datum train exclude still face well result euclidean sr locality sr code locality preserve relational sr synthetic datum texture recognition necessary improve texture sr code locality preserve seq seq plus accumulation locality projection divergence
evaluate matrix multiplication generative obtain px generate computed rest inference probability propose factor performance modify metropolis hasting infer hide dimension factor integer within draw modify choice positive integer run iteration expectation plot influence initialization great infer value
form wolfe frank wolfe reformulate value term slowly issue fw essentially wolfe cost problem feasible apply frank wolfe exploit step idea auxiliary summarize g however frank wolfe optimal convex set obtain one solution objective value imply bound feasible modification apply wolfe obtain hence produce straightforward sparse slow diameter frank method well performance frank wolfe generate iterate l expression gradient solve linearize subproblem subproblem homogeneous l easily verify fairly wolfe direct lemma algorithm k l v
star introduce neural able mean low preserve sentence semantic build document representation compositional embedding sentence sentence task sentiment nlp network entirely feed manner recurrent notable exception convolutional neural convnet short fundamental building network nlp fundamentally symbolic operate necessary create symbolic text word embedding receive several excellent mapping neural embedding relationship serve excellent neural document compositional word embedding sentence combine embedding inspire convolution great model
net state update step accept reject transition go change equation however incoming transition make transition modification transition state precede repeat transition etc two time flip accept step update fashion eq transition transition state exceed direction exceed rate direction alternate physic generalize poor behavior detailed balance remain assign momentum flip fashion take hmc rough rough hmc
various effectiveness extend dictionary propose approach sparse collaborative name dl aware kind extensive experiment superior art term robustness background present state conclusion propose version handle superiority suppose discrimination without well job lead result agree superiority dimensionality correspondence dimensionality discrimination zhang arbitrarily quite dimensionality dimension still effectiveness besides drawback zhang experiment limited face though variation collaborative another third party dictionary shot version extend coefficient face valuable party bring
past year effect sort choice wavelet single choosing basis function assess user assess publicly method contain computational implement al et al al software software start package software method response predictor general represent effect predictor position goal curve deviation frequently assume iid whose structure covariance curve deviation error primary curve ia covariance imply st enter model accounting potential later regularization interpretability potentially estimation interpolation curve predictor truncation penalty choose inclusion within correlation induce affect account weight lin wu et proper analysis correlate idea far affected function especially joint inference lee exist back method scope covariance aspect focus inferential capability fine common approach correlation handle highlight sample moderate sized although design sized iid classic represent rao pcs represent case determine pc mix fit form mean curve curve deviation frequently orthogonal estimate curve ar bandwidth could autocorrelation demonstrate mean penalty parameterize decomposition involve regularization truncation spline separate penalty deviation spline truncation inherent pc decomposition spline white parametric effect model spline basis curve perform curve mean curve spline location represent curve model matrix pc decomposition penalty curve deviation reversible jump bayesian average number wu spline knot deviation truncation inherent zhang polynomial represent mean effect smoothed subtract white level function pc decomposition adaptively free truncate spline sparsity spline diagonal fit thompson level spline candidate wishart function process partition sum weight kernel white al specific gaussian bernoulli spline determine four scalar green wavelet space regularize induce wavelet adjust like et fine grid flexible represent introduce mean curve parametric shape
nearby trade operational consideration search initialize restrict grid subsection c sequence supplementary separate solid dot image show criterion circle current supplementary high red undesirable greedy row choose circle span column focus first four eventually possibly third big show design choice impact choice red close corner third behavior surface create fourth column eventually fan behavior first column fourth search initialize use nearly zero follow arc first arc creating near final row start empirical local design robust parameter number orientation work regularity accommodate select along suggest strategy leveraging discover different structure partition place along exhaustive amongst quickly numerical optimizer
var appendix scale purpose vector suppose also experience expert opinion background word prior let transform obvious way uncertain prior uncertain p base statistic component appendix posterior follow f posterior integral box gamma completion square gm define thus density proportional g q g thus proportional eq f e
team team player team player team effect effect expression probability team mean team equation maximize accurately situation team team team comprise team team probability team margin solely eigenvector centrality maximize historical game occur maximize historical formally historical occur year algebra edge network calculus algebra three genetic search global team implement would maximize occur initialize player vector genetic run carry fit certain genetic take converge therefore decide genetic attempt optimization iterate randomly vector simplex
outperform accelerate speed method reduce accelerate solver achieve acceleration observation acceleration author like suggestion besides support project natural foundation china important proposition institute chinese ia ac ia ac cn massive increasingly various challenge generally speed selection subspace specifically rank subset suboptimal property find globally optimum category assume center center select
verification accuracy thank exploit performance face verification adapt approximation anchor graph introduce speed take addition application need dimension memory time speak far gpu large inversion issue online intuitive representation acknowledgement chen work partially vision project lc ie edu remain illumination single source insufficient variation propose principled process variable name comparison additional multiple verification unknown target adapt automatically therefore must determine since due use end propose name face verification source advantage improve constraint asymmetric task constraint maximize multiple gaussian gps parametric also
introduce define infection remove equally infected individual improper initial infection augment likelihood respect measure st j denote parameter period formulation common model although also densitie datum introduce allocation yield simple gibbs st st dt nm infection infected say infection sample may accept sir period datum shape rarely bayes epidemic scenario b respectively table summary respectively factor behave model scenario
determine set tree amenable projection diagnostic exploratory distinguish characteristic evolutionary brief overview concept tree acoustic study investigate evolutionary evolutionary tree traditionally object specie field formally graph graph network thus analyse assumption process oppose tree constraint unique connect interior node white circle leaf think language interior version attention interior node adjacent leave express three connected leave tree fundamental circle fill sep circle draw sep relationship case expand recently attract considerable example advance author geometry space model see constraint implicit binary understanding beyond active semi algebraic assess whether tree language acoustic set model applicable random specific order constraint yet exploit purpose investigate tree precision latent matrix univariate relate covariance result result covariance calculate necessary margin positivity strictly triple positivity impose moment derivation considerably emphasis give apparent focus solely fundamental tree constraint linguistic
language semantic acknowledgment citation page stanford computer science neural network mean open support demand logical evaluating whether plain neural learn contradiction representation first artificial relational recursive structure quantification natural simulate logical natural structured neural successful array sophisticated language sentiment detection encourage ability representation question whether learn achieve fidelity algebraic whether match ability core semantic phenomenon like quantification contradiction algebraic yield framework language representation yield effective representation typical reasoning
lp qp qp lp qp lead dual sparse svm solver separable p implicitly middle panel figure define cost k k k primal soft comparison qp rest derivation standard result qp equivalent standard svm solve solver r svm learn function quickly let kernel ranking function become learned summarize svm qp tune kernel tune hold cc pair pair compare pair inequality accurately test two
readily close minimizer soft thresholding multiplier update price ti summarize topology unchanged available period reality power grid frequently expect thousand condition cope challenge recovery enforce square show nh value absolute cost cope batch price online vector suitable tailor processing algorithm aim form regularizer leverage grid recovery setup problem whereas equivalent regularizer online coincide regard attain sublinear building introduce copy soon admm update upon complete reformulate linearly
individually fraction item pooling include multiple linear allow item detection first operator extremely due commonly amp since property match recent use amp optimal one fact however bp situation efficiency involve nan
exceed note technique usage dp adjacent dp show memory need disk store disk scalable large usage scheme recursively problem subproblem subproblem split space order space although scheme practical limited increase ai ability solve intensive serial much several already develop bn mention author pairwise scheme parallelization processor solve compare run parallel scheme subproblem variable parallelization overhead become parallelization present take subproblem control complexity algorithm redundant calculation dp solve day processor processor processor decrease processor processor pose barrier large parallelization dp efficiency dp lattice equivalent powerful topology computer exchange adjacent
validate offline live engine use evaluation subroutine offline organize describe capture interactive search describe offline discuss solution issue search engine section discuss finally section contextual bandit formalism classic armed information interaction useful important present content recommendation formally contextual learner contextual reward reveal learner contextual optimize action interaction environment convenience q run action round reward almost increase search engine include vertical news vertical web click variant vertical optimize give reward another
improper begin fisher mean multivariate follow discrete discrete dimensional sample white denote vector slightly abuse notation let bayesian assign conjugate covariance hierarchical mean covariance normal wishart distribution version introduce remain hierarchical bayesian conventional wishart hyper whose posterior belong simplify form gamma ax e bx hierarchical constant normalization incorporate prior state posteriori follow order derive calculus formulae ax linear I xx information thus require initial b iy ty I I converge surely
rank list large relatively dataset essentially plot confirm previous essentially well offset need ii target recall well relatively recall g iv retrieve top optimal fraction retrieve panel retrieve value code plot retrieve code panel retrieve target recall retrieve code panel retrieve value
restriction continuous exploit ol natural krige positive definite gauss krige mean banach space continuity ensure kernel ols krige restriction strictly krige maximal ols krige unbiased predictor diagram eq banach exactly ol infinite banach ol generalize krige hilbert certain array manifold krige function smooth prediction effect compactly consist co functional array array functional array compactly support value b ci evaluation array integral scalar array scalar space compactly support measure array co let co array functional measurable mean arbitrary quantity ai product co array assume throughout co array support automatically array satisfy e mi array array dirac mass weight dirac act evaluation embed explicitly structure integral operator ki ci operator closure existence justify diagram summarize necessary integral operator machine represent hilbert important continuous ci ci ct ci maximally almost continuity assume correspond array almost discover certain satisfied proof arbitrary compact cover entropy index
adapt system predictor outperform predictor electrical engineering centre ac electrical engineering centre wireless technology technology ac wireless try serve reliably use selection feedback feedback change set rate wherein build like prediction partial frequency prediction problem complexity provide upper information theoretic simulation loss throughput abstract evolution adaptation exploit variation wireless rate bit per suited channel condition support rate would optimally sequence joint user propose source encoding symbol study encode practically algorithm modification source encoding namely build converge depth asymptotically practical may stationary long period short furthermore difficult requirement use frequency depth tree implication discuss analyse sequence use extensive upper tree depth optimally pick reflect propose use order
core security system modality define structure datum project template maintain classification paper desire property preserve random good formal structure far main subspace say exist subspace formally state independent tell definition margin subspace separate q geometrically say dot subspace dot subspace angle subspace sample linear structure preserve
hyperspectral contain snr pure pixel run observation abundance different able rmse favorable additive snr converge equal reason mean row abundance matrix large value converge db db db repeat time large predefine threshold estimate detecting
bootstrap amenable parallelization store project high space access file minimal memory storage access requirement compute furthermore even necessary summary bootstrap distribution translate summary quickly bootstrap calculate bootstrap dimensional confidence region pc construct solely similar complexity available arithmetic pc e statistic calculate bootstrap procedure eeg procedure later pc pc purpose functional pc easily demonstrate feasibility health study design analyze relationship metric health patient entire use place use stage primary pattern eeg among quantify uncertainty variability reflect subsample history eeg hour eeg subject eeg eeg consist two measurement raw eeg subject windows proportion hz recorded proportion preprocesse raw eeg smooth subject result hour hour window panel example function across subject figure first primary way course four correspond pattern fairly smooth pca eigenfunction course five pc pc explain variation pc material
completeness adjoint matrix main difficulty side coincide hand rewrite analogously mean adjoint denote mahalanobis dm put implie since put direct consequence fact feature concept generalize mention approach reduce calculation rbf usage invertible invertible replace consequently retain geometry preserve rbf one restriction class lead restrict possible adjoint
sometimes fix huge amount annotation subset namely car face social form manually annotate positive evaluate precision rank dataset new indicate region spee berkeley vision deep clean architecture rapid file switch gpu gpu architecture convolutional neural visual sentiment concept mostly eight main conv fc five convolutional fully fed softmax class correct multinomial
place bipartite partition sbm perfectly odd vertex also bipartite whenever identical partition illustrate sbm force break symmetry find partition provide prefer demonstrate moderate find well fast solve small likelihood sbm explain quantity evaluate correct thus find vertex separate search search search community roughly fact large space degree correct sbm gene sec iteration well rarely find pure eight replicate find converge sbm take second replicate eqs difference sbm random initializations mixed solution exhibit qualitatively optima additional flexibility suggest sbm optima sbm infer replicate easy correct sbm perform small perform reliably partition sbm fail projection modularity moderately inconsistent corrected sbm nearby section partition
filter within ranking user item community collaborative rank paper pairwise simultaneous item less candidate far rather rank differ million preference item c important focusing assume user successive wise matter namely effectively distribution way introduce parameter community belong enable rich modelling community ranking generation process parameter unseen second approach stage model
predict computation correct implement event thus fundamental sensitivity demonstrated present herein prior table fig insufficient explanation support random dimension average strength incorporate parameter narrow prior parameter expect uniform exclude value likelihood similar impose prior often exclude ideal recommend look impose mixture low limit look divergence time uniform place uncertainty suggest divergence constraint divergence must fall state allow place majority believe still detect divergence reduce support cluster implement less cause prior unlikely exclude true prior likelihood rather causes exclude divergence show narrow weighted support model less space carefully bayesian strongly probability unlikely region support wrong evolutionary history inference historical event analysis treat cluster fortunately uncertainty avoid region improve inference
validation subject cluster two safe place immediately affect reinforcement function pass input value safe observable configuration op place action encode implicitly three demonstrate robot action probability use belief human subject execute actual robot predefine execution assume preference subject demonstrate task assumption later response post experimental human participant robot perform action phase reveal evaluate act human human indicate although exactly human enough validate leave cross validation demonstrate weighted assignment likelihood cluster compare manual label code type expert safe expert safe type robot place person alternate finish algorithm perform partition
visual nf training test feature error observational entirely spurious predict nevertheless may accurate observational completely inaccurate recall causal observational train learn hypothesis could section pt net mp pick representative class tell observational principle ignore issue observational observational causal class question represent consist laboratory causal observational input class causal acquire observational step predict learn mean metric experiment metric induced metric space induce distance propose way approximate requirement output random observational choose causal loop search image neural net general set output desire causal mean agree accurate want minimal variant break desire perform feature observational confirm learn
random posterior classifier expert solution marker p generation visualization fourth plus marker concentrate abc statistic let covariate classification expert allow drive insufficient specify classifier expert namely proportion individual infect posterior marker comparison expert summary marker affect suboptimal choice expert select expert additional consequently posterior curve triangle p expert pdfs reduce affected posterior marker abc cause triangle red uniform convergence three condition condition equation classification accuracy classification classification rule validation denote set validation set disjoint point datum disjoint splitting fold possible equation validation rule classification probable measure unknown rule obtain q unlike sum make weak stationarity belong expectation occur classification bayes step frequency law rule meaningful analogy sum consider sum logarithm opposite unnormalized probable
cox enable mention cox close baseline estimate flexibility incorporate recurrent event form censor time assign slot correspond binary takes happen censor observation period recurrent period start recurrent refer full event censoring express let value cox cox dependent logarithm transformation ease baseline hazard assume smooth process write intercept shift relative baseline hazard order accommodate
vector weight ki k output economic pursuit mp storage mp mp pursuit prove useful accord eq x x show number rank residual decrease observe hold uniquely stop nonzero hold singular r conclusion eq contradict consecutive residual back complete ready definition property property view extend low f completion study recall map define express rewrite ad mn mn mn mn ready tackle apply singular least square kk converge procedure present detail hold approximate set able rank recall
movie low anomaly score movie movie rate rate user group action star series specific lr lr title full vi mr generative multi view anomaly find inconsistent confirm detect view anomaly anomaly several relax assumption would application cca annotation indicate cca wide nonparametric probabilistic anomaly inconsistent view cluster view cluster view view
limit relatively minimize fitting design noisy huber loss usually original deal effectively motivation roc space outlier exist consist load tr pc low r pc q nature term matter robust satisfy vanish regardless loss outlier estimation outlier motivate mean throughout give project onto decompose part stand term describe entry finally sub problem challenge increase dimensionality subspace outlier outlier regularization refer observation sparsity project necessarily roc pca estimate pc identify rank reduction roc way enforce e take convex p suffer biased inconsistent fusion penalty scad hard ridge also apply row type roc row arise contamination reader roc conventional modify frobenius estimator subsection estimator build roc generalized estimator begin definition n pm tm estimator derivative
tm use q super search classification describe range allow framework admit fitness evaluation focus algorithm search drastically algorithm assume error iteration independent consist spike consist incorporate algorithm super greatly never design hand case find random guess polynomial necessary notice learnable necessary search computation algorithm study evolutionary domain domain analyze last datum whether acknowledgement national foundation china evolutionary purpose inspire phenomena pa subroutine
determine include parameter one definite redundant parameter well unseen position order accuracy task divergence generate give task prediction practical situation give make formulate latent observable gray estimation target figure observable estimation latent circle latent target estimation estimation panel show probability target
toy use challenge power hierarchical benchmark infeasible dimension usually within strategy useful much computational however gibbs sampler slowly scheme limit hmc novel separability hmc semi hierarchical vector hamiltonian example result hmc large restriction enable restrict importantly restrict g
k method take algorithmic prior requirement demand feature select possibly solve need easily incorporate exist meet requirement individually enforce sparse regression hand regularization presence correlate essentially correlation successfully regression incorporate knowledge process demonstrate alternate admm net quick solution converge slowly interior svm able svm qp solve quickly hybrid simultaneously combine novel knowledge propose able exploit knowledge feature
f r strict inequality characterization american european binomial suitable adversarial upper american black price calculation characterization always exercise merely payoff payoff calculation resort multinomial recursive tx gx discretized uncertainty length analyze approximation american give arguably american nature round execute trade decision discuss address connect adversary bag one price either decide underlie factor figure movement pick total payoff movement option high option worth option asset case free gain go back asset price replace movement binomial low rise strategy conclusion uncertainty game binomial bind price also equilibrium bound binomial movement movement neutral particular upper option price risk neutral neutral measure use pricing risk neutral measure analytical correspondence play keep mind price backward programming suppose entail movement write solve iteratively low match illustrate price option apply maximal next instead final grow make device pricing c bag bag adversary standard model price european uncertainty round payoff option black start analyze version write r formulation follow definition minimax represent upper
likely block presentation auxiliary definition column index instance return straightforward exercise finally never query result require quantify insensitive deterministic formal technical intuition kernel expectation choose uniformly point quantifie hold uniform sampling block sub query block member value member quantity query discover query use case k note belong discover neither occur term account one previous contribute discover contradict logic least eq query bind query pair r j tr
rank recovery apply mathematics nj department nj department recovery mle convex relaxation likelihood
improve fine significantly complementary property expect conclusion per fine tuning raise performance pixel cost improve support pr c ap token fields se stream convolutional contribute third mid detection provide boost feature stream contour performance contour task fusion contour performance sketch token information structure stream use convolutional layer achieve contour fine tuning
circular eqn circular q substituting eqn eqn eq demonstrate mmd circular discrepancy contour distribution fourier transform kernel generate sample unit circle calculate circular discrepancy mmd circular discrepancy construct implicitly furthermore note norm norm investigation begin mmd sampling spherical ratio large eigenvalue match mmd result fig bias mmd quantify absolute exhibit see increase quickly mmd unbiased indistinguishable method mmd prefer superiority mmd synthesis rectangle locate within circular mmd kernel family compose isotropic bandwidth multiplicative exact
arrive apply inequality present property prox eq regularization depend publicly dimension source list process choice machine benchmark illustrate characteristic prox splitting uniform function choose prox number effective pass effective pass evaluate component full pass appear curve prox big remark iterate dataset vary varying period leave right objective gradient
adaptive generalize simulation train median sequentially median hour process adaptive resampling would hour occur resample rgb rgb manuscript resample scheme value tuning parameter efficacy procedure sub parallel gain procedure explore possibility produce identical tune maximum depth prune another spline prediction identical fitness procedure never unnecessary resample risk use remove identical choice need performance might conservative could confidence discard setting adaptive procedure rgb end rgb rgb tune parameter efficacy
assumption empirically also package effect illustrate confusion large make illustrate left training therefore start decay away start base accurate however case clean parameter fix parameter cost increase want prediction identity belong additional outlier enable apply describe noise base sample outli across unfortunately exact add outli make become eqn outlier principle experimentally sensitive network without first label clean dataset flip
aim payment compatible generalized axiom eq length payment axiom payment worker expect payment assumption present section utility case describe experiment amazon crowdsource platform would emphasize mechanism research amount typical crowdsourcing task expect worker understand mechanism act instance expect significantly upon modification amount compatibility propose mechanism standard platform amount high worker researcher design game compatibility prevent check experiment reduction comment worker suggest work theory amazon crowdsource platform call worker exchange pre payment payment part task nine task range annotation speech answer skip mechanism requirement worker amount total task mechanism constraint attempt worker complete least task respective certain worker execute depicts worker baseline skip mechanism nine question incorrectly average payment iv break answer fraction plot gold question prevent result choice time time put payment mechanism self mechanism receive comment worker receive increase complete job complete thank interface task obtain solution website author gate bridge three fix gold compare baseline gold skip confidence worker number worker baseline correct gold standard base mechanism result worker convert upper remove answer error match true worker identify ten depict gold baseline gold skip confidence classify depicted amount
rule q twice apply cauchy schwarz arbitrary desire calculation inductive complete inductive relate bound use complete substitute use shall eq statement recursion second step ps ps ps induction hypothesis conclude necessarily aim seek algorithm regularity quickly usage certain generalize smoothed huber estimation algorithm achieve erm every consider quantify implement space algorithm decrease standard super polynomial trivially see handling importantly quantify obtain comparable erm trivial guarantee size number large divide order address question low erm rgb initial
nature subsampling bootstrapping formulate subsampling explicitly subsampling first subsampling feature formulate bootstrappe forest subsampling subsample explicitly apply scheme use various scheme forest scheme slightly popular accordingly make applie subsample focus approximate detail first mathematical framework key node equation randomness build mathematical support construct space naturally subsample us feature let selection situation use easy index feature determine word focus feature probability get equally likely number possibility hypothesis regardless stem limitation infinite detect label feature choice purely realistic many dimension case reality spurious relationship label feature probability select usual tackle spurious relationship bag bootstrapping solve problem get still useful relevant feature important make correlation say imagine nonetheless complex correlation derive assumption tool determine threshold odd feature surprising statement somewhat sensitive strength insensitive study affect change commonly
correlate tc spline ss problem pattern undirecte edge cycle great connecting specify band along complete fundamental result discuss assign become definite large possible view particular band notation convex extension admit entry definite call let gaussian independent word sample covariance equivalence
preprocesse whiten whiten patch expect performance patch suffer get activation suffer mlp relu initialize train hide logistic yield supervised fine increase performance slightly dropout regularization yield permutation cifar slightly whiten extra none video transform dot whose random image subsequent transformation video frame intrinsic frame frame model interpretation
generally good satisfactory measure put emphasis expense measure propose opposite endow g lebesgue remark
parameter produce overall sn sn ratio dot line correspond described box sn marginal horizontal dotted correspond box pdf sn ratio horizontal dotted decrease sn qualitative result sn impact sn summary statistic certainly tendency closeness sn volatility asset pricing portfolio management segment three decade demonstrate volatility motion asset price inconsistent volatility empirical see option pricing notably volatility also view evidence price geometric motion black price review response many time volatility stochastic augment jump prominent option price become option pricing mcmc filtering assess adopt simple square root logarithmic asset period day wiener restriction positivity previous steady gamma density eq kind function view discretized version diffusion return observe retain diffusion discretization step abc occur exactly composition match return daily realize namely stock reference inclusion volatility would necessity financial abc invoke euler approximation detailed outline sigma deterministic specification define respectively filter particular step sigma
limitation infinitely none analysis square recent attempt made extend differentiable objective continue restrict constant remain style method hold large satisfy rsc particular statistical result work show relax optimal rsc respectively offer employ greedy gaussian family satisfy rsc convergence solve hard thresholding prox provide projection crucial unified descent confirm prediction
distribution additional advance unfold histogram treat histogram distribution come demonstrate regularization similar reconstruct check
say rating ignore observe adjust bias perform movie rbm gradient eq use divergence cd much produce log minimize kullback divergence minimize usually learn use new unit activate add rbm could extra conditional unit distribution become hide affected activate
efficiency factor minimax write algorithm input attractive adaptive sparsity level adaptive sparse zero inequality provable ask computationally statistically tight occurrence indeed otherwise adapt plant formal definition discussion price efficiency pay estimator study essentially rate computable order pair countable distinct clique clique plant problem suggest variant classical associate word pick vertex plant plant input graph sample random nature plant clique possibly algorithm know plant clique clique point asymptotically vertex plant clique
go go forward backward normalize layer stack lstm introduce feed responsible network single softmax secondly expand feed forward recurrent recurrent similar idea explore show
zero task minor classification gold part parameterize dataset example optimize gradient efficiently scalar possibility project lie parent illustration sentence far continuous standard backpropagation understand standard implement representation sentence experiment binary label sentence constitute resource supervision setting fair mini batch fold implement embedding treat parameter implement
draw draw divergence constant testing see imply infimum class covariance matrix complete observe prescribed lemma imply desire lower trivially observe treat separately pt combine case define integer yield ii simply invertible invariance eq q schwarz yield observe result list illustrative comprehensive detection however literature focus various performance functional covariance covariance attention originally gaussian white density paper study estimation regularity estimation nonparametric become phenomenon regularity decade high sparsity unknown parameter parameter interpretability simply sub particular covariance zero uncorrelate
lot describe dynamic algorithms analytic polynomial framework analyze optimization algorithm formulate popular utility iteration principle able term iterative dynamic process spectrum able generate sequel sake brevity quadratic order define typically prescribe constant function cast chapter one way generalize sdca nature randomly reason inversion become mapping mx clear convention hold inversion matrix execution algorithm induce test assume sequence close minimizer initialization interest algorithm take matrix satisfy identical matrix method add matrix treatment examine otherwise choose dependence true see readily naturally ask range characterize speak consideration technique instead answer specification l gradient method see accelerate rewrite stochastic gradient sgd straightforward extension go x sgd satisfy appropriately choose stochastic coordinate repeatedly minimize coordinate denote th row matter sag like sdca sag closely sag method derivation result slightly order sag framework straight implication optimization strictly come q apply
choose report optimization neural network exploit field program exhibit rbms autoencoder speech like nlp word datum like gray value become accordingly pixel word meaningful unlike nlp feed index equation representation representation etc anonymous value reflect degree satisfied word fed work word modeling probability linguistic word conditional introduce energy maximize word calculate researcher realize normalization essential merely feature word consideration local pattern word recurrent neural rnn dependency rnn would either vanish propagation nature rnn treat language language rich structural solve problem detail representation node abstract formalize learn objective subsection answer fundamental previous section vector representation real symbol character token level character
observe transform transform globally consistency denoise manner close apply correspondence main contribution present consistency pairwise transformation programming permutation globally special matrix present invertible similarity euclidean transformation interest shape multi view transformation introduce perfect information straightforward extension handle noisy finally type transformation zero invertible transformation
generalize median dissimilarity generate difficult solution optimal som principle median som iterate well determine update generalized notice prototype variant solve unit neighborhood neighborhood dissimilarity natural know som problem batch som request unfortunately som algorithm rather cost careful cost actual per avoid introduce arguably drawback median som intrinsic restrict massive som sub need determination propose subtle restriction unit empty apart prototype prototype usefulness visual representation matrix avoid point form generic lift prototype restriction som intrinsic som prototype representation dissimilarity euclidean q som mean define solve eq
appendix tw strategy gs review respectively measure handle tw monte move respectively spin immediate carlo section big roughly term powerful search state big approximately equivalent configuration repeatedly subgraph subgraph update threshold back go step subgraph update run collection state way try long find energy find ground oracle know solver terminate target successively value look something let repeat em terminate run result interval generator energy obtain occurrence discard minimum serve purpose estimate accurate statistically also second reasonably confident energy energy landscape behave ground reasonable energy energy ensure chance course rigorous
ensure multiplicative front large choose simplify integer choose expect false decay choose large forget subsample close large probability close rank base contain maintain relevant set increase reduction quantify discrimination occur available trade extensively see theoretical intend goal goal truly strict positive provide relevance false improve recommend indirect assessment method scalability recommend computing capability take covariate selection probability effect subsample section depend iterative base see limit procedure result generally strongly previous already highlight subsampling procedure follow covariate signal covariate case expect amplitude noise subsample omit procedure output randomize return similarly consider uninformative score denote procedure pd covariate deterministic analyse quantity circumstance concern determined estimation true word grow
lemma henceforth consequence event posterior epoch boundary stop vector put epoch choose define convert precisely represent eliminate suboptimal time epoch distinction eliminate eliminate break resolve eq particle modification action eliminate early lc cc lc suboptimal define bound value arbitrarily assume region positive quantity contrary eliminate parameter epoch sample eliminate display follow nonnegative suboptimal policy choose let epoch step post respective elimination write calculation write iid bernoulli assertion large enough thank coupling iid bernstein n estimate fashion probability application give enough sum give complete lemma conclusion occur remain suboptimal th standard schwarz c hence ct boundary stop write regard state lemma epoch kl whenever write obtain thank infimum set finitely expression probability assumption observe complete epoch see factor
oracle oracle htp negative stein oracle outperform estimator unbiased instead statistic rmse generate equal note perform normally small large test statistic average replication table c freedom stein oracle model unclear version right wrong incorrectly also method show empirically model preferable wrong inaccurate group feature standardized control accord standard belong observation group belong wrong share pool covariance wrong pool covariance rmse small
remarkably scientific people different rational revealed preference expand upon despite process interaction agent loose make algorithms process viewpoint problem far namely process evolutionary cognitive decision brain neural information cognitive make novel take
focus range underlie derive computable high dimension misspecification covariate define covariance net contribution covariate mean variance ingredient inclusion covariate alignment parameter account contribution covariate
partition boundary depend bandwidth roughly capture take scale size near mode mode reasonable converge mode one reflect local bandwidth utilize usage shift restrictive utilize cover fix shift heuristic density smoothing fashion isotropic track evolve primarily aim result somewhat offline supervise process neighboring stability estimate recently automatic bandwidth gradient offline focus gradient error bandwidth computation domain shift create partition multiple domain mutually segmentation utilize
allow interpretation high loading high loading turn loading allow specie column structural analogous follow base solution consider classification compute three specie case obtained refer svm look sound substitution space mathematically translate correlation physical maximize combination magnitude heterogeneous inside inter field correlation statistically significant first canonical variate give explain table table among substitution canonical correlation transfer physical drive act second variate link minor discrimination scale significantly variate matrix report expense substitution imply change relative substitution significant canonical correlation substitution discrimination logic root move e substituting opposite moreover move negative loading aggregation move different move svm basis explain svm svm result cost majority component collapse general express without lack relevant aggregation point relation physical canonical report
snp various performing screening concentrate second differentially private screening paper procedure elastic regularization differentially private step accurately end differentially regression interaction elastic penalty extent penalty exclude absolute experiment high score snps convexity impose sensitivity privacy show often recover interaction term factor middle bar
also thus new domain h update learner requirement meet set tree guarantee carry incomplete loss move functional functional may optimisation author point decrease subject h translate select large boost proceed likely accumulate boost add ensemble control average mrf boost learner fail correctly instance mrf training particularly desirable formal tree selection direction condition search direction small scalar continuity differentiable constant find experiment satisfy mrf linearly take whole network example grid g successfully tree purpose tree subsection belong
capture multi explore gaussian depend rather valid filter individual expert overall parallelization property combine power allow subsequent property combine poor product expert achieve scalability result comparison mixture probability expert generally
entire dimensionality nonetheless common category retrieve pre consume intensive grow differently generalise category runtime attempt address however attempt limit produce necessary small stage complete inherent nothing relatively compare ac ac uk overcome world computer vision carefully imagenet category action video offer develop scale purpose category retrieval operate millions second system reach product amazon current system typically bootstrappe search source learn third video rank retrieve contain category aim stage happen matter second retrieval performance trade high severe penalty severe high costly training rank excellent compression quantization page entire aside gpu gpu
time typically iterate comparable experimentally discuss translate property provide use kalman grid black example regard iterate gradient em smooth exposition far fix either computation benchmark kalman shape scale subsequent example run particle normalize kalman bottom average true bottom variance n mcmc put model consideration problem numerical program page author first long statistic n nx ny result omit top panel show mean evolve ratio kalman exhibit bottom scale computed filter grid estimate decrease support approximate provide method portion confirm mcmc fail space false security scale run method experimentally estimate instead linearly mix suggest increase remain degeneracy problem particle convergence rigorously red versus truth one short know estimate ny single gibbs addition figure posterior independent method seem consistently inaccurate negligible increase methodology performance run additionally compare result estimate particle display improve particle mcmc increase computational particle gibbs display parameter prior favorable particle black dotted versus
query relevance get detail exist map document convert row abstraction consist iid document often vector rank sort scoring contain bind norm accordingly similarly natural counterpart ingredient function set negative vector score differentiable smoothness depend twice norm
h approximate replace alternate constrained define scalar scalar scalar penalty assign measure user enforce easy hoc negativity practical nmf dense use another solve e element solve thereby execute far solve fast available truncate comparative nmf factor incorporate nmf believe call reason well nearly multiplicative become iterative element remain mean improve toward continue provide flexibility path towards poor prove saddle converge local ad hoc drastically saddle suffer nmf guarantee prove otherwise superior alternative hoc linear whereby
carefully follow decomposition loss characterize codebook loss stein estimator simulation result show empirical compare estimator randomly draw rate comparison stein see estimator towards shrinkage rate stein choose signal ratio observation code method compute repeat combination recorded error theoretic low error convergence sharp bind estimator nonparametric euclidean similar ellipsoid principal ellipsoid great
generalization since df df r f consistency conditional presence asymmetric rate provide conditional accurately trade surrogate next estimate x show dx asymmetric x p rate estimate probability sample low universal guarantee n nx r kx separately guarantee mapping universal induced reproduce kernel hilbert slow preferable density hard density reduce curse accurately variable therefore distribution
power logarithm power turn level method perform mostly equation clean image relate manner replace sample natural kind practice analyst potentially indeed invariance general modification modify free noise datum result see put large rest spectrum eigenvalue proposition notation call I similarly since hold asymptotic assume assumption conclude come unit modify work instead diagonal matrix q large asymptotic eq clear spectrum spectrum try regime hence neighbor near robust avoid near intuitive context systematic future impact kernel started compare level much partly result focus laplacian map create difficulty rotation impact additive broadly proposition collection dissimilarity consider regime call noisy asymptotic though way change version clean version suppose consequence furthermore gaussian like robust recognize essential move situation elliptical scale model largely noise I modify instance situation approximate approximately spectral free suggest original noise past practical theoretical refer interested reader aforementione paper detail
method default yield evaluation criterion elliptical distribution covariance stand stand contaminate contaminate model fix rank contain equivalently outlier separate observation spatial configuration rotation vector greatly reduce scenario need contamination belong complement concentrate simulation subspace span eigenvector whereas setting online resource belong robustness contaminate contaminate simulation data facilitate parametrization drop measure indicator observation come location find separate outli configuration frequently
quadratic denote deviation assignment th odd receive unit odd match affect equation vector unobserve lower since p nan indicate reach value reject hypothesis quantify sharp nan note statistic simplify I use eq sharp nan hypothesis test sensitivity refer refer odd equation outcome change inference furthermore q let follow third moment chapter pg normal theorem page combine fact effect lead finally normal case strength identical look quantity
grid theorem face bind estimation loss random estimator thus require nontrivial rely intermediate leibl divergence construct certain spread pseudo consider define hypercube eq fractional definition expect check choose need appendix proof depend certain sufficient choice rely depend lemma apply constant simulated picking randomly smoothly bound obtain realization innovation sequence center display online predictor explain corresponding specification overall iterate aggregate aggregated equally shift loss right line predictor one predictor accordance theorem aggregate first outperform predictor achieve original aggregated loss original td ta derive predictor improve quality maximal
model description transmission role deterministic typically ad hoc preference systematic approach considerable environmental aware work illustrate example consider deterministic base show calculate abc smc popular choose dynamical modification develop option factor previously dynamical become approach keyword selection ordinary stochastic differential dynamical choice dynamical expert knowledge hoc function objective
infer need deduce conclusion square root two use synthetic gibbs subroutine newton mc log nr approximating follow algorithm fix instrumental produce basic version propose examine assumption ty produce stand sampler gs importance step likelihood subsection computation use nr
lemma imply q affinity matrix connectivity solely gx rx covering ball graph riemannian determination adaptation riemannian use geodesic cx neighborhood point cx imply tangent proof projection onto easily difficulty arise logarithm ball arbitrarily uniquely among minimizer inequality follow restrict chart two subspace chart mt rt follow sample technical bx htb rest proof follow appendix see equality combination h tc eigenvalue h remark inequality derive simplify strong requirement q immediately follow graph function constant imply bind conclude exist constant estimate connect origin proof geometric let respect chart projection thus combine furthermore fact minimizer appendix exist depend manifold consequently conclude claim carefully constant riemannian manifold letting immediately respectively threshold satisfy gx word proposition point conclude noiseless geodesic due replace parameter satisfy requirement requirement I next explain satisfy requirement sufficiently satisfy rh conclude portion fact sample multi analysis multi geodesic tangent without noise difference level precise trivial claim robustness noise experiment study space solve propose
represent actual objective though low show explain linearity make axis actually extend camera estimating assume object detect view discard fitting represent green straightforwardly camera space use statistically track object false method camera throughout move accord velocity augment respectively object often size additionally population construct motivate remainder reader within object object object object might probability either detect detect dl spatially distribute image plane object know association empty restriction function association denote density object conditional update costly density density possible propagate multi density cardinality set first density set gm object previous framework mixture require survival assumption propagate gaussian assumption strong consider might significantly differ necessary relax detail adopt birth detailed scene restriction
efficiently p f definition sx joint entropy last let emphasize appendix control additional bind approximation bound intermediate bound outline theorem intractable sample q empty soon even approximate adapt technical aspect mention achieve ratio state fact check return recall
properly could give capacity especially face outperform deterministic able reconstruction c c c bias na na na hybrid na noise weight interpret probability away particle show estimate final substantial challenge reasonably reasonably separate p like promise address issue latter explain neuron feedforward
specify sub thus principle turn expensive smoothness equality add extra quadratic arbitrarily closeness yield eq like subgradient slow general course different euclidean projection operator descent minimization place result convergence ij ij jj l ii indicator otherwise constraint solve like gradient solve quadratic analytically h
utilize elastic net concave interesting unique prove play role basic denote basic eigenvector principal maximizer j uniquely proof satisfy sample cover selection easier relevant regression analogous order q technical limited contain condition special pca imply relevant select thresholding thresholding intuition illustrate difference toy select relevant sufficient condition assume block negative control sample provide satisfie satisfy unique addition part positive part additional individually negative exact part
water variation deep table noise visually fig start able truth unary gmm feasibility structure computationally purpose structure structure connect sparse encoding layer inference interactive perform manner tractable manner solve field structure state structure wide application natural language use mrf limitation utilize unary potential neighborhood smoothed boundary range state boundary
sampler cost cost report reflect particularly cost k allocation training perhaps achieve indicate line estimator underlie allocation combine weighting still considerably variance especially observe consequently allocation thompson outperform static practice tune automate adaptation benefit simply new strategy monte pool estimator produce cost map correspond problem bandit take area finitely infinitely estimator sampler bandit study variance family thorough investigation acknowledgement microsoft greatly decision subsequence chapter iii need second tt return denote begin round recall round cumulative likewise definition estimate mean may give history n kn I sequence share k identity cf kn get mse prove easily
threshold monotonically move close center classifier sphere maximize generalization margin separate margin method truly aim search minimize show svm deal cauchy schwarz divergence projection prove constrain reduce instability apply maximization project big circle point express procedure rule window width ascent gradient formula omit gradient computationally issue practice maxima start sphere run also possible start solution model perceptron reasonable short investigate section operation construct tx tx qx qx k qx projection classification point decrease search iteration build however access expensive change parameterization width window width factor replace equation analogously evaluation beneficial tend tradeoff big lead simple
label propagate along edge appropriate accommodate tend induce induced propagation achieve competitive avoid pair exploit bin node new feed common scheme construct preserve show hash hellinger use classical chemical sized cloud propagation originally addition expand kernel introduce propagation numerous central huge graph represent video care propagation kernel easily kernel scene domain kernel begin introduce propagation walk propagation kernel section example information scheme propagation well respect choice graph bioinformatics real application image classification object kernel develop mining establish relational graph kernel class walk size subgraph kernel subtree base exist often slow attribute slow kernel subgraph subtree introduce compute signature set label compression feature fed base kernel iteration although kernel usually competitive runtime design label label propagation mark unique symbol however propagate run label observation motivate avoid termination similarity neighborhood recently become popular
entire variable truncate replacement induce choice closely probable variable assignment particle approximation connect compare first sometimes particle fail three dirichlet hmm state differ particle filter total particle ess sequence filter run illustration sequence particle filter conditionally degeneracy without time conditionally unlikely particle resample size fall ess ess place
htp accurate primal penalize square arise compressive sense technique incoherence property sense extensive competitive state recovery without exact sparsity dms partially national science foundation china section remark primal dual frequently compressed strategy identify dual variable square define update certain sense incoherence property restrict isometry noise global extensive present illustrate strategy global convergence ten sense amongst broad application formulate follow sense vector denote component vector
raw unlike generalize discover group pattern suggest voxel subject individual individual raw align across reduce validation relative glasso find align expert carry train fold voxel shift voxel dimension misclassification show small successfully cross subject location allow fitting note pre perform report train expert pre select assess select priori involved picture classification voxel significantly lasso overlap elastic net overlap return cognitive elastic net make interest force across individual mean subject drawback interest succeed voxel task glasso make figure overlap group lasso ill overlap force force undesirable select elastic treat independently across voxel also pick voxel interpret lasso elastic leverage inter subject similarity solution voxel allow task correlate voxel lead cross indicate inferior predictor like elastic
depend contribute specific coincide global method illustration plot component versus histogram figure eps know indeed bayes density em method classify show entropy
develop without distribution constrain affine b characterize truncate deriving conditioning follow q picture give independent decompose dimensional clear condition univariate fall q truncate lie plot truncate note width quantity cdf cdf f b v yy gaussian agreement summarize confidence interval
calibration capability base classifier assumption loss measure plus rank counter develop processing separable dataset calibration base learner discrimination auc acc perform rmse retain discrimination obtained prior post processing improve histogram size experiment fix testing method capture calibration size experiment see steady datum real uci repository information people decide letter cost whether person among include build two predict person letter people expect return letter concerned choice
develop svms cost corpus imbalance imbalance discuss experimental setup present evaluate hyperplane svms close positive pos recall suffer present pl many time tune training overhead subject max part quantify reduce
descriptor processing hence feature explanatory depict stream ccccc diversity appearance scenario identify literature experiment order study behaviour option choose popular member design classifier constitute quite situation unary strategy completely gmm use unary least batch slot combine slot little effort accuracy label batch refer misclassified batch formulate period label batch available effort knowledge stream non stream three baseline approach passive batch label obtain train stream online learning need complete second expect odd batch odd keep buffer time train buffer classify batch partly set passive however odd original
email spam relative occur mark choose split spam long tail community community fold yield number community correspond lr spam positively direct conference internet business email mail receive report address quadratic discriminant path graphical naive interaction single linkage cluster community procedure applicable diagonal sub problem show community sparse encourage matrix sparse produce interpretable
dim dim dim dim dim dim cause entail benchmark research word embedding microsoft research relate word describe select build potential scale recently rapid learning researcher start model amount text nlp notion relationship deep approach
model vocabulary integrate iteration chinese restaurant collapse gibbs posterior section describe general conjugate auxiliary alg word allocate hdp term conditional document denote allocate exclude word give word allocate topic empty topic weight new new atom range topic indicator remove unsupervised replace document sample assignment difference topic update allocate gaussian response though response rewrite cluster residual empirical finite count prior parameter remove document average use calculate regression glm sigmoid close coefficient method converge number instead sample common laplace unnormalized memory bfgs map estimate parameter sample document model simplicity also consider estimate find benefit sample word allocate topic randomly document old km parameter binomial value use l dataset classification financial direction price stock
reasoning law jx ix j reasoning event union event k ij hx x hx ij x hx k ba jx ij jx jx j j b j j prove differently reasoning implication combine implication label disagreement combine yield bind improve publish reduce marginal product space axis result simplify complexity achievable active request great maximum classifier value budget exist marginal active binary classification establish low p b k k thus aforementione establish second hx prove target allow request great return classifier ix xx h xx jx
b appendix shrinkage estimator scale interest covariance matrix scatter regularize estimator I condition automatically sample estimator recursive converge convergence mean convergence penalize respective shape regularize estimator shrinkage mse due general estimating problem scatter scale estimating question estimator shape select e derive estimator shrinkage estimator alternatively scatter close scale copy w hereafter seek minimizer mse q oracle oracle plug verify employ value matrix component uncorrelated toeplitz note identity rank measure
sample vector alone kernel duality kernel cut pca basis manifold necessity subsample ht compute cut component matrix dx kk compute truncate reduction principal agree pca right component onto datum ht feature like feature like entry coordinate manifold entry orthogonal apply right vector compute vector
j n c short connect parametrize positive numerical reason conditioning covariance limited dependence covariance log formulation block form work wavelet function convenience stack use independence q must admissible quantify admissible practitioner application additional minimum mmse bayesian complicated posterior determinant parameter situation common markov chain mcmc accord turn sampler successively accord conditional associate sample instrumental propertie reader tc p c r instrumental c r tt burn define n mmse subsection require inversion computationally prohibitive numerically due grow replace exact assumption realization regular lattice additive
tc stepsize matrix variable great equal problem multiple experimentally fraction instance discuss stop iterate improvement tolerance maximum impose scaling gradient scale gp whole scale scale method cl backtrack differentiable limit degenerate local experiment plot iterate fast decrease function observe impulse impulse obtain relative scale method scale importance versus performance ht ccc result test number nf average second whole run server dual intel processor mb cache gb ram parameter matlab interior trust region ip tr table option suit
variant computer software framework language communication datum provide tolerance communication central communication bottleneck method principle communication substantial develop version simply parallel term processor need coordinate need change serial furthermore classical convergent require step serial variant recent coordinate surprisingly requirement decomposable minor modification modify fact communication degradation decomposable compute contrast robust happen asynchronous show highlight model system globally ability prove useful formulation useful mathematical place pressure accommodate increasingly dimension internet text long progress utility interior go discuss locally algebra cholesky multiplication take prohibitive solution big inexact describe big review non term
transition recover plant hard hard transition occur transition transition bethe energy cross exhaustive search regime lie transition plane meet order arise introduction bp plant along height reach critical possible ground label separate realistic use maximization minimize discuss parameter namely political label
relatively proportion people people another final project sub group south massive infer potential explore influence enable practitioner connection meet global online finally note flexibility specification robust generative capable represent label validate benchmark make candidate towards learner group low future like run service course student learn feature reveal massive new factorization computationally efficient cluster topic local explore underlie user sub level statistically significant
art contextual vector sequence context reward advance adversary round reward regret hypothesis purpose contextual minimize cumulative one hide layer learn action know many
dominated exponent ultimately determine rate change change presence phenomenon force close therefore observe distinguish piece assume notice likelihood follow order piece character inspire estimator drop subscript kkt coincide formally establish follow originally prove fused coincide applie segmentation covariance scope huber instead convex filtering piece wise approach piece arise consecutive decrease however double constraint hence integrate walk z analyze detection fuse true piece
theoretical runtime note lsh scheme approximate threshold vary norm scenario approximate query change parameter also costly preprocessing query correlation norm keep inner product upper norm word transformation sign thing work norm transformation know precision recall well l parameter recommend correlation cosine note
skewed type configuration loading ss sample loading factor dense loading dd capture gene co er differential network batch ds gene variation dd variation affect variation gene population calculated percentage ss sd ds mean order run orthogonality component explain quantification assume normalize across wise calculated sum total gene contain loading component sd ds dd category ss ds dd number component total select fit identify differentially across identify er across er component er differential er component panel dd component ss ds component er er er panel component er er sd panels range minimum across precision matrix subset test
algorithm utilize second sort adopt sort via algorithm argue dyadic define close interval q evident time cite numerous disadvantage adopt correlation idea idea root idea utilize computing extend suitable also detail effectiveness evident size become availability detail publish study rest paper covariance correlation relevant property unbiased additional capability finally conclude remark make proof correlation co several pearson pearson dependent
shape peak flat shape end excellent compare finding fig substantially computationally intensive discrete datum lp framework discrete free goodness discrete develop continuous make test whether procedure em compute goodness test statistic justify h equality goodness chi square probable compare chi fit equal square measure interpret lp drive statistic interpret chi statistic powerful utilize universal orthonormal construct polynomial recurrence relation often complicated example investigate parametric fit select shape base model skew lp skew exponential estimate u plausible heavy tail performance statistic alternative detect lack contamination component traditional ad er von level nan cutoff level set
right singular vector top singular running appear appear completeness description construct factorization column k lemma construct k rank replace consider step arithmetic output modify imply bind hold orthonormal accord conclude improvement spectral lemma spectral aware deterministic randomized factorization method construct matrix k take randomized sampling modify let k imply square eq expectation respect proof mention analog decay leverage
observe fraction recover version pose machine task analysis several practical np hardness show problem broad constraint trace sum memory time prohibitive iterative call optimize hold
index choose g sgd architecture try break mistake classifier run iteration word iteration loss monotonically
fdr procedure develop three image field dependency generalization replace markov markov spatial image human brain assume grid follow two parameter ise voxel hypothesis normal extension hmc major difficulty normalize likelihood product tractable thompson may field mechanic intractable turn sensitive zhang algorithm et maximum despite incorporate name se restrict belong exponential family search mle prevent mle average adapt backtracking increase lead generalize et fdr motivate image subject patient ad subject voxel cubic voxel line posterior ac line wang
incorporate auxiliary design designs integrate feedback auxiliary item raw rating rating rating user movie measure actor extract wikipedia engineering restrict case fm description rich interaction fm capture recommendation integrate auxiliary rule transfer constrain vector tag inform incorporate obtain introduce basic knowledge near neighbor feature trust introduce different friend include user social knowledge friend feature similar term friend term one tv
noting assign current mechanism detect tends adapt equation grain reduction eq reduction technique cover batch advantage tune trade thresholding bound thresholded weight bias weight normalization layer convergence sgd condition size guarantee
v vs image rsc vs vs vs rsc vs reduction comparison due increase similar experimental bias use bootstrappe eight datum bootstrap replacement bias set show attribute four observe increase biased determined case reduce comparison single unbiased forward reduction complex structure pruning systematically reduce bias average rsc make result low rsc commonly net lie ensemble bagging rsc overcome instance rsc attribute new reduction rsc compression compression general keep large number fall knn explore rsc examine cardinality cover probabilistic sphere degradation possible heavily prune strong candidate create rsc control retain instance
observe efficiently derive analytical normalization drop without loss generality precisely combination solve evaluating denote standard vertex type q iii condition use similarly third vanish correspond vertex arrive c c take point write operator point condition instance inequality bind simulate illustrate box maximally achieve full distribution numerically impose plus normalization see arrive distribution achieve quite simulated moreover result another nice much bi full minimize value almost display fig decompose measure influence particular simulating drastically move region finitely analogously feasible polytope define compatibility region show main xy pa violate state plane measure angle obtain exceed compare even causal
capture velocity consecutive capture particle position velocity cubic velocity jacobian separation spatially instance jacobian stability root offset satisfie query point unstable vice stability field result time scale vice versa frame transform comprise act unstable point scene denote height frame collective crowd simple grouping velocity similarity group respect change crowd phase shift
unfold recognize sigmoid structure please layer frame weight relax constraint conventional sigmoid network note feed sigmoid start layer perform forward update structure schedule illustrate general mrf feed sigmoid interpret either interpret mf mrf may interpret deep unfold compact structure either one inference variant neural model belief propagation probability apply loop bethe approximated well investigate explore unfold without parameter bp bp mf update marginal posterior know mf update message contrast mf belief normalize belief output mf mf versus incoming message prevent ensure incorporate belief yield marginal mrfs cycle long prevent feedback true problem message pass field formalize two unfold layer implement update accomplish message optimize message schedule bp instead message mf schedule
add already calculate ordinary forest global score regularize gain concept particularly represent base maximum control long thus addition add importance follow normalize rf control importance also select score one rf calculate frequent tree association association association transaction transaction association rule transaction contain rule condition side measure follow define proportion transaction transaction outcome transaction contain item contain length consist use side
essence criterion approximation regression process principal operate current discard belonging include dictionary residual obtain project span coefficient cost current function eq removal budget concept discard nonetheless removal process issue motivate complexity dictionary computational expensive complexity scale dictionary operation moreover condition express multiplication instant computation affine reduce linearly dictionary instant error atom atom moreover discard combination atom possess duality discard bridge criterion purpose derive approximate atom obtain coherence criterion
gap small event eq suboptimal depend argument event happen time event remain choose monotonicity satisfy moreover q bind numerically upper bind event item specific associate q item observe sufficiently often event item gap far bound definition bound decompose regret part gap proof gap inequality conclude proof follow definition fact green green definition near whenever offline
linear allow feature response discussion support nsf grant finding conclusion recommendation express necessarily reflect view descent section exploit property optimization approach completeness proximal use solution new extension proximal framework subproblem block compute step size avoid proximal solve efficiently coordinate project backtrack search summarize coordinate cycle proper backtracking exploit quadratic basis matrix coordinate subproblem formalize problem minimizer eq operator one thus completely track search qr block gram
autoregressive pa model many science science finance aspect var explore var estimate covariance coefficient compute var forecasting estimator end residual condition underlie uk u show ensure identifiability variable first dependence account variability individual structure encode separate dependence pattern motivation variable relate unobserved provide motivation similar see relation assume identifiability orthogonal equivalent model model latent variable covariance latent variable isotropic identifiability become estimator observation ignore proposition exist analytical reduce reduce reduce link widely
batch loose respect minimax rate much modify slightly yield discard extra obtain generalization error besides direct estimation dealing feature batch modify replace replace regularizer q correspondingly large new new attention allow latter prove generalization online verify batch lipschitz idea section generalization present error decay early minimax long problem extremely bind loose end probably problematic weight streaming weak good analysis guarantee penalization sparse result run probability output direct expect condition regression streaming setting achieve prediction contribution iterate convergence obtaining convex still similarly work generate isometry latter regime restrict isometry enough usage imagine finally design orient increasingly important streaming analyze procedure computational property favorable use believe combination examine proof check obtain emphasize random also regularizer mirror sequence
one use sense denoise focus mainly extension simple believe scheme believe art segmentation denoise processing lead work texture optical thank provide code author helpful constructive greatly jump representation additive polynomial segment approximate parameterization polynomial temporal jump jump residual approximate equivalence method find close jump subsection criterion algorithm constant option optimally solve minimization index sub recently sparse optical flow denoise vary solve traditional framework recover piecewise
realization case hold certain eq statement condition exponent side argument neither one hold covariance become om decay however fact pr l om term third apply eqs om lemma w standard ann mi k k divergence machine information statistic particular normality propose estimator example divergence plug histogram scheme assume none
rate variety quality level illustrate detect face detect conclusion draw qualitatively sample fail image environment high comprise human secondly robustness illumination image camera characteristic colour qualitatively notice dataset illumination approach discrimination stage effective opposed column combination notice quantitative analyse dataset result comparable colour high true prove human visual colour
scalar equivalent newton bfgs line converge iteration huber rapidly convergent scheme problem figure fast huber quantile behaved b b loss study demonstrate importance adopt framework sophisticated emphasis briefly discuss possible benefit paper form corruption simplicity corruption additive happen sense transmission scenario ignore image improve loss function discover corrupted randomly occur image kind noise therefore huber penalty thereby loss infer tends spread dictionary image extract patch stack specific randomly replace percentage
assess monte multiscale environments investigation simulation associate issue partially science dms condition infimum condition generality hx assume lipschitz lose due notational convenience notational omit use control rearrange lipschitz continuity definition jensen sup old eq omit distinguish denote explicitly mention put sufficiently inequality conjecture question proposition proposition proposition corollary proposition conjecture proposition proposition stochastic equation scale equation ergodic provably sample rare event expectation functional perform simulation presence randomness construct measure prove asymptotically numerical
feature extract deep despite importance often side task notable variant representation distinguish example label representation par include also sensitive calibration person person object provide person
social behavior hand recommendation tb topic image internet community facebook friend recommendation contextual planning article search ranking annotation via topic online internet community share friend tie query tag search tag feedback event wikipedia indicator indirect article group formation social membership growth measurement crowd social tie recommendation collective social capital self site longitudinal increase human computer interaction topic flow cloud code lattice galaxy mobile wireless traffic fluctuation centrality heat article user top flow cloud boundary galaxy galaxy dynamical trust mobile wireless monitoring article galaxy formation
learn word embedding also use rare distribution word tail frequency compose word furthermore word appear thus make capability new know top close calculate representation score ht type compute definition public split implement tool test base type build specifically four set vote balance word use suppose vocabulary frequency reach feed similar frequency approximately total word coefficient similarity rare discussion balance knowledge character wikipedia totally million text process digit replace word replace occur discard result ignore compare baseline baseline use feature skip input vector length projection baseline skip gram update word design baseline coherent human cognitive skip gram employ type update note fix share fix context window
less outperform several example create see review description brief sentence since express opinion movie review ignore sentence consistently convnet documents architecture structure technique network language domain language model translation answer sentiment language task entity language us encode semantic language geometry idea nearly network
search already complete consist example handwritten demonstrate search speed search dataset consist example train show transfer see optimization task benefit search logistic get minimum able consistently evaluation averaging configuration toward high conversely low agree less roughly agree offset epoch minibatch increase model mnist solid line indicate mnist line develop process especially well suited approach
decision depend mid example reject mid retain decision retain verify mid retain conservative turn mid example valuable information formalize motivate propose hybrid abstract could whenever could lead decision occur choice opt mid affect testing simultaneously likely microarray reject utilize usual mid value natural mid section reject study section describe operate characteristic proceed abstract method computationally adjust abstract randomized motivate new adjust mid mid mathematically randomized remark method define relevant relationship value test countable variable specifie goal reject retain
web supplementary material set feature selection york supplementary include optimal section scale factor web section web table matlab file optimal classifier classification model introduce propose section hierarchical write classification k simplicity operator summation relate correspond work determine majority vote individual vote optimal lda poor performance simulate majority voting strategy suggest enable rule predict look angle case class use concentrated point theoretical observation classification thought building space remain question good high assess support motivation thousand observation investigation thus observation emphasize let ip rip pc pc empirical space sequence
history view word latent item get name fm model obtained skip skip maximize pi item feature fm item train order user item relevance handle fm model evaluate movie rating collaborative filter movie
may unconstraine minimizer replace introduce n tn nf inspire convergence sdca duality gap proceed sequence use e represent n e g use ng add twice eq come immediately e obtain rate use inequality prove become inequality replace sdca low play role sdca rate sag apply conservative heuristic section use size present update relate sdca strongly result composite one well surrogate even though suggest rely surrogate equal inspire part l randomize available warm surrogate composite minimizer solve address small value empirically warm could would proximal surrogate convexity even lipschitz
si bag weak proportion si mi bag contain test object independently underlie class learning general introduction refer train instance train classify individually classical dependency field mrfs popular originally describe labeling assign part speech tag account sentence label sentence achieve si task mi mi task si si observe label exist advantageous bag provide bag derive train correlate although convert si si mi instance correlation instance call trained feature label instance relational repeat encode learning mrfs simultaneously label bag object bag name scenario bag directly distance kernel bag space bag
compare necessary achievable relaxed imply bind similarly measurement achievable therefore contrast bit cs might room measurement sublinear performance improvement individually power information sequence maximize observation restriction measurement formula group testing bit testing bit cs
result low observation low resource correlation affect case distribution coefficient fisher contain observation go approach limit correlation sensor bit sensor sensor observation also make decentralized htb certain present optimal estimation independent noise noise local sensor locally process fc fc design previous section one sensor fc observation stochastic employ possible rao fisher conditionally probability minimize maximize optimization
assessment patient one assessment successive within week patient least event assessment patient collect gender country birth status status among consider age assessment broadly classify health refer moderate class risk assign look diagnosis convention event period risk code event open would typically complete rare month period attempt moderate r horizon day filter bank sec event primitive code map high level correspond severe episode hierarchy list reduce rare code diagnosis disease health diagnosis table alternative view health likewise frequent economic robustness rare event risk give good specificity intervention detailed convolution bank compound filter min medical risk suggest max similarly item suggest serious surveillance assessment moderate list max ii max item time item mean rating item rating l order network ease interpretation divide non distant encode belief old
edge recover place detailed figure realistic material area significantly bic relation bad trend relatively opposed finally randomly run average incorrect find non low mean almost twice result supplementary material demonstrate efficacy filter filter approximately possible hypothesis considerably less mistake drop score bic across level realistic training variable great surface leave bottom box demonstrate performance knowledge learn ten bad random measure positive true negative prominent fusion event persistent theory
variable output heuristic al know study alternative likely heuristic thesis upon problem relationship heuristic artificial intelligence infer make machine variable elimination quantify average heuristic heuristic first machine formulation algebra follow recent background learn section describe conclusion work free formula problem produce free formulae reduce prove algebraic formulae polynomial method elementary doubly complexity survey remain work produce polynomial eliminate polynomial decomposition space form real root
st observation suggest boundary mean test social circle statistic circle overlap ratio less compare exploit get inferior mt operate learn metric task final preserve structure network mt article citation circle well reasonable analysis importance lie task perform exploit mt convergence source author support award fa multi task learn number jointly network priori structure provide network come attribute associate variation relational entity often source performance work
parameter maximal mse choose bias em indicate mse well mse result correlation structure suggest large select right induce mse deal high structure report ccc r sf mse sf ss deviation sf mse square compare margin correlation among feature identify true feature mixed mse addition indicate tend independent regression htb panel increase go zero extremely parameter force estimate irrelevant go feature specific simulation three selection regularize without consume especially pick aic sf
distinguish specify advance typically factorization moreover column choice contaminate additive distribute achieve reduction reduce discard arise case identify one instead solve factorization row candidate subsequently pick alternatively pool backward successively apart k k alternate coordinate descent propose g along feasible perform exhaustive search possibility impractical stand alone initialize latter block step help dataset entry probability simplex size setup report average follow hamming denote box five
hundred alternate dataset fmri treat fmri bold connectivity bold spatial fmri insufficient thus connection simply treat direct hypothesis conjecture nearby voxel explain range group close brain nearby grouping provide group interesting biological develop precision assignment difficult network dimensionality fmri hundred thousand challenge applicability hierarchical fmri brain interact introduction interpretable enable infer hundred thousand develop update compute simulate go model tool relationship
estimate characteristic rank denote nuclear th estimation exploit method consistency thresholding np notational drop bound question ask whether could assumption result case keep sufficiently large contain large check extension technical continuously multiplicative specify measurement value change alternative focus iid one relax assumption unbounded direction appeal interesting sub gamma consideration insight simplicity boundedness unbounde numerical rescale condition iid measure alternatively tensor notation multidimensional array call clear low rank specifically upper substantially encode structure reason well low tensor ill pose might bound tensor converge fortunately
dependent observation person motivate computational need thompson become large require feasible get extremely good real problem online hard complexity scale straightforward computation core different bootstrap replicate later replicate probability empirical simulation bootstrap heuristic solve bandit thompson replace thompson
define expand substitute gpu equation convenience mathematically diagonal later due small assume expand never diagonal eq never instead directly note seem like factor term gpu gpu gpu scaling multiply gpu notice invariance lose proper fix great symmetric computed gpu cpu element nothing cholesky compute extremely rarely bad already disk minibatch initialize top top eigenvector let ii quantity compute fact square enforce per minibatch use inner product gpu factorization trace follow expand update need multiply quantity enforce towards gpu nature computation bottleneck try fail apply inverse fisher fisher expect unit normally experiment nonlinearity maxout like typical matrix something rank input matrix matrix sort greatest scale input value leave unchanged last like take scaling set unlikely sgd learning rate direction track change tune minibatch specify interpret minibatch reason believe speed update every update sgd summarize matrix fit picture ignore explanation instance correspond separate copy describe typical configuration
solver c l singular singular proximal separately apply proximal use subroutine nonconvex experimental show outperform previous small nonconvex minimization extend nonconvex affine alm acknowledgement national international centre office lin china china grant cb collaborative fellowship department electrical engineering school technology science technology laboratory university edu sg com com edu development differentiable bx give denote
svd bilinear factorization rbf product desire uv observe missing solution complement incomplete corrupt product lemma impose much exist solution exist note case entry observe alternate multiplier admm solver non algorithm produce different estimation guarantee theorem compare convex common impractical problem rbf complexity scale understand rbf relate method special desire qr update modern parallel architecture regard section algorithm run fast accurate bilinear hadamard product miss propose alternate multiplier admm extend analysis exploit admm assume simply zeros uv uv sd uv
isotropic assume distribution step inequality element wise conclude direction p ip j suffice union bound ij similarly result require proposition improvement eq could since isotropic lead overall projection let consistently recover permutation direction spherical normalize hull state discuss post computation propose preference develop generative account population user inconsistent natural ranking statistically model leverage advance latent ranking provable consistency computational complexity empirically art approach preference provably go comparison demonstrate competitive performance collaborative metric demonstrate effectively variability real world estimation partial extensively last decade various prominent user ranking homogeneous center truth
observing exploit unit bit bit give rate consist dimensional input parametrize bias single propose layer parameter computation consist parametrize basic bit weight indicator input may choose possibility predefine make sample
repeatedly compute able call apply value white amplitude project onto span quadratic state action mapping discount receive well know fix bellman similarly optimality point policy iff equivalently implement greedy input practice achieve call direct suggest approach hand shall policy scheme section comparative performance go describe error
without subtract supplementary large sequel use descent gradient theorem guarantee furthermore convergence sample var estimation reduction sampling mc sampling suitably modify unbiased incorporate fairly material provide detail significantly well problem detail proceed realization variable distribution apply obtain I smooth facilitate iterate common enough optima projection rarely occur return keep consider denote equilibrium suitable technical surely application hold differentiable surely discussion slow requirement algorithm rl involve decision
artificial wave ml example try pose light high input pose importance connect method successfully result simplicity nonzero furthermore percentage interesting observe treat three equally eigenvalue model well efficiency digits mnist set digit pixel two major two represent capture capture percentage variance consider image mnist mixed top obtain indicate projection two capture major image successfully represent digits face image gray pixel represent dimension representative face randomly eigenvalue return eigenvector matlab experiment mac os bit intel core ghz cpu gb memory numerical experiments matlab software package semidefinite sdp efficient sdp solve sdp network termination tolerance train inexact proximal employ terminate primal meet respectively tolerance choose train report eigenvalue h cccc ccc cccc problem e e e digit term
exclude mistake error correct additional rr frequency lead work spatial task correct stimulus arrange exclude trial position discrimination name distinct six one mixture two proportion uniquely water group right figure exclude trial categorization previous exclude either qualitatively exclude reward trial decision protocol combination trial cutoff neuron exclude neuron fire neuron fire fire rate hz exclude neuron minor transformation square transformation qualitatively neuron work neuron work discrimination neuron categorization preprocesse spike train filter kernel ms smoothed stimulus histogram neuron work direction fire directional show neuron prefer sort separately neuron prefer preferred dataset trial trial fire separately trial interest figure illustration water align four reward take trial water across discrimination categorization trial describe alignment error along time piecewise linear align introduce cut trial similar pooling datum reveal qualitatively marginalization neuron stimulus trial trial filter train vary stimulus trial fire rate thought condition dimensional collect column mean single neuron decompose part vary stimulus decision averaging
population estimation sequel constant play role result decomposition eigenvector eigenvalue f obtain j eq desire commonly give regard rate assumption small apply choose sufficiently suppose n assumption reduce minimax rate attain determinant barrier zero barrier
operate resource necessity tolerance continue sensor wireless obtain information environment growing acquire sequentially efficiently avoid communication fusion sequential decentralized communication central unnecessary designing analyze amount mathematical challenge question ask elaborate asynchronous computation develop deterministic stochastic asynchronous online property entity processor hereafter perform estimate meanwhile processor receive update form computation regression goal integrable assess classical offline setup batch prototype
nan distance fm across distance proportion two nm proportion ap population variance diversity allele var across mean diversity two allele two index dm coefficient population universit de france france france universit paris uk range purely motivation likelihood propose selection forest complex cover algorithmic output indicate probability correspond forest recommendation sparse forest severe table performance computation methodology illustrate approximate near bagging subsampling introduction approximate abc ever increase cover calibration always still critical implementation partly explain widely accept specifically major quantify vector practice finite lead summary produce summary crucial role provide inconsistent answer exact bayesian factor probability summary pool summary statistic simply avoid select random forest therein probabilitie approximation well posteriori try build lead compare dimension predict solely probability toward loss estimator assess reliability select posterior select wrong simulating tool random perform suggest tailor implementation support argument favor rely forest forest expense production forest value direct possibly large collection summary statistic
run divergence htb cc belief networks boltzmann share use stack simple block capture dependency correlation building field restrict machine dna building inherently dna distribute rich carry drawback inference rbm stacking field representation rao difference dna modelling purpose specifically design
say sparse sparse norm whenever disjoint subset g adapt argument norm sl decomposable notion say constant constant definition prove respectively ny ax decomposable q replace interpret state minimize decomposable near ideal problem group generality constitute future objective discriminant present context merely find discriminant relatively discriminant whenever third great choose discriminant devote amongst give label linearly weight discriminant separable weight r linearly situation dot dark dot clear circle circle circle circle circle pt circle pt determine feasibility lie hyperplane linearly way instance power behind high hyperplane separate many hyperplane support separate hyperplane point hyperplane formulation hyperplane illustrate concept separate hyperplane circle dash circle pt pt pt circle circle pt circle
become replace argument denote lead precision contain compatibility factor c well permit comparison decay invoke usual isometry restrict slow design consequence substantially compatibility concept evaluate application new weighted compatibility may apply one hand example poorly prediction moderately covariate develop optimally gram matrix define penalization lasso selector recommendation explore consequence result nonparametric least proportional th derivative study lead notation denote proof tucker infer subtract relation may appear somewhat c equivalently write display divide side accord eq replace introduce vector display combine jj hand negative n classical subset identical left reader I
hence may continuity approximation difference approximation former use iteration continuity function hx u directional equal select belong boundary follow continuity forming follow continuity contradiction side contain away margin contradict close reason versus lead function arbitrarily disjoint hence finally continuity sequence vi value function upper recursive relation initialize ix ix bound bind lemma mathematical induction x x ix comparing ix ix ix mathematical induction part lead tight see dynamic consider condition converge
carlo primary genomic treatment review set typically experiment may consist correlation noise repository gene cancer etc list molecular signature significantly weight size genome identify choose appear smoothly contiguous assume include gene practice subset interval say nan replacement gene fact set identify uniformly distribute tuple identically basic object inside
assignment exist author paper write regression given expect assignment intuition give solution derive conference paper category soft assignment category suitably define typically pair category category negative dissimilarity allow want optimally one nm qp row column column likewise index determine case assignment item category type encourage term encourage item although could fix mean dissimilarity laplacian separable problem couple certain solution extreme program lp separate k category tell category correspond assignment category quadratic mn e tell differ generic unique close n k nk similarity assignment large laplacian dominate follow assignment category similarity sparse category similarity point similarity qp minima since semidefinite multiple characterize give sufficient minimum assume
make prediction spatio temporal set type compute transformation input layer formally type memory output nonlinearity wise sigmoid nonlinearity softmax layer among unit connect intermediate score sentence discard output length sequence incur supervision whenever step since dense weighted transform mean interact learn interaction layer task sentiment detection consider come might model multiplicative multiplicative recurrent sentiment retain rnn
nn training discard find drastically storage requirement facilitate stochastic neighborhood visualization compression compression devise efficient cholesky change variable carefully world baseline error set lead order nn case dimensional vector input compute descriptor matrix via metric euclidean mahalanobis approximation along matrix affine geodesic definite rank matrix f accurately describe intractable moderately sized roughly burden distance similar call jensen q near neighbor classification nearly asymptotically
improve network see capacity early stop decay training involve proportional objective weight incoming addition effective equivalent regularization denoise autoencoder additive signal reconstruct denoise criterion permit overcomplete reconstruction regularizer idea fitness species co likewise dropout performance complex co feed procedure function define l
focus inverse update rule ph point direct rule exchange ph section inverse update bfgs cg arise ph equation note estimate computation projection however goal sub explicitly singular n nu matrix mt designing solver step identify span ideally regularity equivalence cg offer elegant way additional cost recall covariance span semidefinite property eq normalise positive matrix span covariance bfgs scaling prefer use standardized prior prior cg hybrid construct problem conjugate run store conjugate gradient multiplication cg estimate cost minor construct gradient I problem yy e estimate rule propose cost standardize sr corollary quite sr use cg use give overhead posterior algorithm alone multiplication cg store mean require relatively problem crucially external distribute eigenvalue external
compute sequence choice guess sequentially receive online rule subscript law compute sequentially use detail step execute add estimator discuss implement exactly implementation general implementation recall initial density since close suppose approximation dirac
convexity set dr hence excess condition brevity sign last inequality follow fact p tt exp give construction output arbitrarily close multiplicative interested pure differentially efficient multiplicative fact interested total could however achieve sample distribution well explicitly work mainly highlight construction construction private isotropic position eq choose write since bound term measure distance denote derivative point membership efficiently suffice oracle polynomial run highly isotropic isotropic particular however convex efficient place isotropic position apply transformation take fit inside finally transformation isotropic diameter put op norm lipschitz walk building done define walk input cube output close respect argue
nlp group helpful early lastly valuable feedback stanford google v le google google google york university comparable approach significant conventional correctly translate rare tend symbol vocabulary implement system later utilize post processing step translate every use hand suffer extent phrase allow extremely word strength phrase address rare problem corpus explicit enable corresponding sentence utilize translate use experiment english translation winner task translation map sentence target
include influence shape order reduce diagram flow boundary positive arise signal gain turn effect calibration sec numerical example reconstruction color structure calibration calibration respectively panel low exhibit mcmc panel mcmc explicit absolute measurement local tensor pseudo instance principle field optimally calibration start calibration basic idea infer vice versa fix reach estimator hamiltonian equation must iterate base mark new infer determine usage hessian
draw thick fill color bic circle minimum thick bic circle mm size color text bic style circle sep mm size cm white circle inner sep text rgb rgb rgb rgb rgb rgb style circle thick black style cm thick color circle sep minimum cm draw fill style circle sep mm thick black fill black style circle inner mm thick fill style circle inner white style inner sep draw thick black circle sep black color circle inner sep draw fill black minimum draw black fill style sep mm minimum black inner thick black fill style sep size thick style sep mm black minimum thick style mm color style sep mm minimum black style sep mm thick fill color black style cm color n style sep thick style mm cm thick fill text circle sep minimum draw white circle mm size draw thick white fill text draw white sep mm cm fill text black style sep thick fill style circle sep mm draw white color black size draw white white text inner thick fill color circle inner size cm color circle mm size draw white color black rgb rgb rgb rgb rgb style cm black fill black circle sep mm minimum draw thick
bilinear slice layer bilinear bilinear diag special bilinear matrix c linear bilinear bilinear diag result observe worst overfitte publish achieve discrepancy sgd vs bilinear consistently comparable much require parametrization relation expressive simple bilinear bilinear diag baseline comparable bilinear note bilinear
since odd every origin contain accomplished interval semi origin replace replace left interval infinite lie ball vc consideration family obtain let scale arise primarily represent accord collection vc recall final subset hull rest connect hull vc hull rest pair convex denote lie convex hull lie
connect filtering spectral insight particular stop filter introduce whitening demonstrate fast standard collect sample processing aim classification transform transformation unsupervise simplicity scalability three processing unsupervise encoding abundance image interact crucial ensure efficient paper study contribution filtering benefit stop performance
location model mle stein show towards uniformly low underlying prove statistical development shrinkage compressed shrinkage estimate dimensional small variance mle bias risk wherein dominate exist analogous x mle mle stein example good prefer uniform estimate dominating recall family bind mle perform parameter mle design handle save generally address solution idea shrinkage reduce methodology
strength excellent rapidly arm valuable choose specific problem empirical guide heuristic bandit consider clinical trial vary effectiveness unknown identify successfully heuristic assign patient extreme variability done subset bandit characteristic affect arm reward implication work regret reward et consider moment impact performance precisely identify accurately evaluate apparent tune bandit example evaluate easy bandit strategy could effort towards turn algorithm clinical trial whether bandit suit clinical answering question implement dropout identify good treatment confidence bandit trial offer trial term patient simulate clinical strategy randomization criterion successfully treat patient clinical trial conduct treat patient early treatment particularly suited context initially patient study patient receive day test patient treat provide patient individual assign treatment patient achieve success patient condition course test patient result mark strength indicate indicate publicly unclear
inexact simple inspire simple subproblem proximal proper define correspond c admm optimal initialize k work need efficiently proximal stem subroutine one function convex pt mm enable negligible flexibility algorithm proximal efficiently simply admm term spectral radius accord theorem error measure gaussian error span far result entry probability choice notion gaussian gaussian norm standard ds eq
conditioning thus truncate distribution involve kind fortunately likelihood augmentation negative generalize replace binomial share binomial dispersion row exploit binomial beta binomial jk express absolutely evy laplace express eq laplace transform express sum independent compound pmf pmf binomial pmf may truncate q laplace augmentation concentration step gibbs unstable calculate number rapidly allow precision machine numerically thm proposition probability matrix potentially unbounde three derive negative binomial binomial binomial lead natural count although wise distribute fact use derive explicit drawing column map random order certain derive random random count framework construct naive text require predefined account unseen analyze completely suggest propose poisson multinomial laplace
go compare low result spherical gene usually cluster splitting big cluster price discover tight cluster fortunately split detect merge reasonably number decrease apply cluster four preserve pattern four amount create splitting extract cluster biological small small dataset demonstrate spc iterative subsampling cutoff discover interesting pattern approach relatively trade large reasonable either preprocessing could remove valuable dataset filter coefficient cutoff remove carry profile function lose without knowledge number able decade rapid rich ever problem size computational intensity effort far create consist essential exploratory simple model cluster dataset numerous modify mean computational modification include random subsampling sophisticated review summarize abundance outlier
sparsity aware rest assumption give box aware doubly coefficient sparsity aware diag I fact simply express towards ii I I possible prove
lastly environmental gibbs control additional explore snps rest snps effect snps study
find author department engineering create account account receive research process lda appropriate format thus list similarity list likely package word word filter count tf word list project properly part five section generative detail describe estimation
present special interesting construction state spectral adjoint eigenvalue eigenvalue lipschitz equal numerical hand unlikely give satisfy metric space say property ball r lipschitz maintain
bad rf implementation optimize implementation test five exclude simple scale take range subtract common forest pass rf hyper default initialization rf vs training rf batch version evaluate split version candidate split split every recommend training increase realistic streaming setup store multiple pass vs time marker pass training batch rf mini mf streaming setup new significantly fast batch version tree balance mf rf remarkable label competitive similar test comparable rf world irrelevant method independent rf well mf attribute amongst attribute
identify total operational strength conclude four simplify join miss create cm operational operational state operational double strength operation elaborate join input join soon one firing firing come parent henceforth item fire fire accomplish parent state henceforth predict go back operational operational item neuron fire show fig diagram create operate algorithm transformation assume join implement fold fourth former subset consist strength argue operate diagram easy transition operational state loop operate join self loop parent come item transition event fire consecutive shall briefly actually implement operate algorithm require need besides fire neuron fire necessary plausibility fire soon primitive plausible pattern mean parameter visual one possibility special pattern item sensor remain unchanged period presentation presentation
maximal carry series round find round track unary include effect interaction activate track overlap commonly occur element direct acyclic short start path absence term correspond dp approximation solution quadratic track go could otherwise accelerate interestingly greedy update pass dp quadratic penalty pass find pass dp learn tracking potential feature depend extract video spatio relation candidate track parameterization sign convention maximization represent appearance template represent pairwise track temporal transition represent birth feature location make detector appearance consist detector allow motion connect later time window window overlap window lower flow transition vary track appear move thus single birth death geometric object spatial context bins window window location additional intersection box area set corresponding feature ratio video tw vector encodes geometric object w object way intra
turn internet movie database rt rt website review media user rt show meta datum rating vote actor review sale b entity extract content tv original release rating user tv actor rate pg rt score rt aggregate rating rt voting item american office provide scatter plot attribute week user rate title sale title report vote public engine office reasonably informative box office access american european quantify understand turn google trend volume google search give search shoot relative establish query volume region search volume neither else volume report would relative search world european country country community normalize query interpret fraction google search week search approximately common scale engine panel scale sale release search engine attention look close region search north correspondingly sale release panel engine attention sale engine advance particularly successful search engine sale sale suggest search engine local sale volume instance predictive consist cumulative new backward release release indicator vector country vector identity location search engine title release
fold cross discriminant tree tune rate pair sum report node tree distribute note almost leaf node percentage term four remain
water system ground gps operational environmental integrated system feature error vary server operational water national environmental service send national service movement water condition operational create lead boundary goal spatio low coherent uncertainty without transfer processor h c period hour sensor unite top sensor product filtering inference spatio parameterize isotropic mat ern intercept walk neither cost elaborate parameterization parameter also walk initial
confirm design evaluating task rate divergence divergence estimator divergence extensively process involve segmentation separation function divergence perhaps widely signal process family distance kullback leibler generally indirect class divergence measure useful dimensional two class assume restriction prove useful application knowledge underlie divergence measure parametric parametric introduce also estimate measure investigate unlike divergence fisher utility focus classification section iv come provide vi contain discussion future divergence without fr
match partition block come algorithm update step convergence rate strongly ht partition use block p remark method store row update translate run update method run choose strategy pick ensure row block store separate secondary storage convenience mention crucially difficult rate hence exist arbitrary effort line viewpoint fix strategy round easy evaluate alone find pick block iteration randomize pick row per pick randomized pick proportional use block approach effort correspond worth one mention selection algorithm focus greedy adaptive deterministic new estimate block hence greedy pick amongst candidate choice appropriate strategy end time briefly greedy projection block possible pick fact refer bs pick iteration emphasize b
correction carlo efficacy amp framework evaluate imaging amp recovery briefly review amp later demonstrate within reconstruct denoise denoise amp behavior simulation kernel implement name suggest apply filter whose gaussian noisy image gaussian pixel violate image convolution implement efficiently remove noise filter regression pixel neighbor close compute addition spatial proximity window noise dark prove amount extend average pixel neighborhood take neighboring neighborhood pixel however edge opposite side edge usually neighborhood wavelet transform basis coefficient transform hence inverse sort thresholde soft thresholding performance unfortunately image wavelet thresholding least overcomplete wavelet coefficient square neighborhood coefficient prior expect noiseless bayesian square remove noise coefficient bm filtering begin grouping perform transform dct haar transform bi haar amount group perform transform pixel thresholding second wiener filtering estimate outperform compete additionally author bm optimize complicate quite combine bm patch filtering help group patch mean dct haar bm retain performance bm unfortunately provide among filter non maximize closely signal use denoise rescale automatically
correct equivalently bound square aa respectively assume circular therefore hence small independent lipschitz base next rank orthogonal show denote therefore incoherence incoherence property enough prove j define correct bound incoherent right n array thorough evaluation conduct theoretical insight hoc classic mathematical matrix projection completion achieve array calibration position semi program stress discuss reader detail low space preserve matrix relative position first center distance position real scenario map compute shortest consideration sum minimize approximate miss distance short classical programming show reliability space semidefinite increase gradient method calibration topology hoc optimize reliability control state importance distance incorporation simple classical assume distance
triangular orthonormal identity c al theorem al nan eqs notice give characterization suppose contradiction nonzero q orthogonal x full
latent jj ix I I position respectively labeling definition follow inference analysis base extend construct valid clique length e setting score path contain horizontal lattice layer valid non node valid least pass correspond path node valid layer layer pass position connect position valid become remain
previous independent set section contaminate density hypothesis give distribution define g gd call information section type show belong g hypothesis statistic g follow nan type linear combination nan context chi unity unity useful robustness asymptotic test exactly illustrate consider fix contaminate proportion origin contamination influence function boundedness quantity towards explicit index statistic theorem
anti correlate ai electrical engineering university california berkeley department institute department department university produce massive new discover biological discover neuron crucial neuron type traditionally connectivity manner show neuron enable reveal structural enable automatically derive connectivity massive far circuit computational method impact high throughput sequence connectivity fig cell profile cell probabilistic cell belong type connectivity vary typical profile historical classify base give start stochastic block cluster connection salient logistic link additionally body validate perform simulation accurately cell simulate compare estimated job recover correct fig extent exist infinite assume
introduce potential potential value potential intervention sequel particular response pair ab rs rs none causal mechanism mathematically mutual independence ann experimental randomize property serve validity observe
n rgb definition main statistical sketch previous algorithmic result time number depend kernel adapt kernel lastly capture fine difficulty semidefinite emphasis obtain guarantee primarily work area focus issue guarantee inferential low result recent excellent combine obtain fast kernel improve relative state recall importance perspective input bad decomposition work leverage randomize positive rank parameter projection approximation recent qualitatively bad
solve multi sensor simplify method optimization constraint common tackle splitting multiplier splitting break close minimization together introduce efficiently optimize splitting rely burden dictionary yet augment smoothness fidelity variable keep updating present weight converge update involve subproblem intermediate subproblem update element separable mean operation equal summation constitute simplify solve svd determine soft thresholding q subproblem unfortunately close difficulty come regularization row group operation dictionary multiple modality alone restrict sparse regularization arrive sparse modeling normally require iteration achieve converge tackle exact third function taylor expansion achieve expansion last line I separable property simplify separately solve utilize approximation utility yet use whose sketch j penalty combination domain nonlinear empirically multi fact extensively validate become linearly onto
specify assess influence unit company expect vary political regard eight include potentially logarithmic width size medium random effect slope covariate beta logit beta complete specify remark flat improper component precision parameter indexing random wishart assume reduce define intercept covariate add related intercept large effect categorical
tail order account normalization mean decay rate mass invariant datum discretize bin width narrow central correspond pair event reference order dataset draw bin histogram marginally mutually independent belong unfold belong cb estimation cb invariant mass maximum indicate high cross check find agreement carry bin event outside extended side place knot result unknown sampler condition number hyperparameter initialize iteration hasting burn observation size extend square confidence interval replication core confirm converge iteration little variation roughly autocorrelation whole plot mcmc verify bias pointwise bootstrap compare unfold reasonably overall shape figure histogram count divide unfold reconstruct peak smoothness boundary intensity reasonably tail intensity invariant mass sample figure na I confidence figure
sufficiently concentrate around r k k ahead examine analysis eigen decomposition orthonormal far follow substitute element obvious independent hypothesis weight random square zero gamma distribution complicate pdf instead rely construction sequel error test type alarm rejection right tail ii correspond tail ii bound chernoff pdf origin skew long tail need subset chi degree chernoff central euler number I exponentially mutually function k degree centrality substitute c k h rhs gaussian last step chernoff enough exponentially pdf form pdf chi freedom central degrees centrality pdf pdf scale chi f plot pdf red curve mass around close curve red shape agree make evaluate mis rhs demonstrate assess error alarm type incorrect decision become proceed operate steady recursion cluster decision
discrete able modify policy set description policy make ability ask simultaneously good figure substantial benchmark dyadic benchmark contrast dyadic question nearly benchmark well dyadic object location dyadic policy dyadic question remarkable compare little lose hard compute policy dyadic much go dyadic computed quickly pre compute ask question setting compute dyadic compute provide least dyadic sometimes dyadic right dyadic low turn dyadic policy dyadic explicit analysis dyadic conclude lower expect first notation
consider assignment contain total estimate proposition straightforward similarly two sorting rank sort merge complexity apply conduct pair already process sort cs grow subset computational cs straightforward thus assignment order therefore case estimation cs refer programming interior polynomial bi thus f interior necessarily ten choose benchmark result discuss summarize uci dataset domain range within characteristic discretize prior selection lr kp discrete dna discrete discrete continuous evaluation uci namely support c near neighbor na I influential mining algorithm classification empirically three rank representative
model dispersion regressor specify selection criterion regressor dispersion present highlight propose criterion yield easily identifiable stand version parametric scenario compare performance criterion size performance become parametric weak criterion evident competition outperform criterion least relative regressor jointly weakly identifiable also weak clearly instance aic replication whereas regressor mean dispersion interest must mean identifiable display criterion outperform
select kind predict measure risk relevant covariate part discuss select coefficient measure technique criterion prominent one high denominator fdr lead achieve low fdr compressed noiseless thresholding identify thank situation estimate regularization good estimation compatibility order relevance strong bounding inequality key role screening find asymptotic true rate stein unbiased concerned discuss prediction selection
nonsmooth base reweighte nuclear solve compute proximal operator nonnegative decrease monotonically point guarantee note nonconvex stationary last nonconvex synthetic algorithm nonsmooth need extension nonsmooth nonsmooth concave differentiable nonsmooth inequality call denote versa concave versa subgradient useful explore monotone
infinite aic optimally gives forecast particularly attractive ease explore typical credible account variability plug bandwidth cauchy inf rgb rgb bandwidth nonparametric error density lin nonparametric value recent error density admit bayesian estimate simulation apply nonparametric type regressor chain recent advance bandwidth residual establish rate estimator framework recently propose bayesian simultaneously kernel error bayesian mixed regressor nonparametric regressor essential idea functional scalar
medium discuss implication google differently social key evident finding google controlling message length day exhibit incidence large indicate increase lead message increase decrease due sample result allow turn across picture discuss effect variability effect seven variability follow differ measure also clearly find indicate message successfully effect finding would contribute confirm
measurement observer observation assume mean covariance absence example set make uninformative posterior direction get k k posterior calculate method matlab processor linear evolves need mcmc marginal calculate continuous length performance da observation clutter loop move update jointly window loop move contain move alone da da particle line slow especially report time mcmc da cost resp particle include cost particle mcmc association respectively efficiency propose move particle da da region include inside target plot connected star connect measurement target clutter good non replace kalman kalman target particle per window parameter density z compare joint nz nz converge log evaluated truth apparent gap ground truth alg track iteration
single particle number calculate weight nj ip py ij extend stop particle particle iteration propagate particle particle first ii simulate particle proportional likelihood output particle choose particle proportional iteration monte estimate product unbiased implementation output arguably simple particle filter liu implement calculate depend latent
confirm would three autoencoder joint epoch autoencoder representation test model performance table report test classification computation layer table joint achieve error extraction unsupervised contradict performance regard good model regard helpful good generative necessarily translate good discriminative p l cccc j mnist rand h detail cccc cccc rand apart perform initialization train train joint beneficial perform another weight autoencoder train autoencoder table use performance improvement joint correspond irrespective initialization joint addition clear scheme scheme suggest use role autoencoder
eigenvalue recover linear autoencoder generic model computationally demand shrinkage note equal isotropic noise autoencoder discuss previous difficult heuristic selection problem iterate scheme low high unconstrained solution procedure describe update encode update show small cone ordering reason scheme subspace show eigenvector cutoff find isotropic case algorithm admit close give particular never isotropic iterative
centralized find also non plan provide distribute accounting phase trade coverage final certain proposition ensure probability relax belong kernel hilbert induce covariance see key question phase location markovian process adaptation happen infinitely converge se answer rkhs sup location location instant relationship estimation field rkhs q contain location think borel probability stochastic mechanism
distribution yet structure formalism effect hdp strength across group share make precisely cluster precise cluster share introduce dp modification definition group number exchangeable collection observation group level dp cluster pair product base dp draw base measure realization repeatedly within specifically base observation conjugate respectively stick break stick countable atom eq stick stick break form ij stick collapsed step refer chinese crp second conjugacy l ji z v integrate dirichlet conjugacy exclude ji ji z pl ji
plot ambient variance ambient varied runtime randomly generate mirror descent large require ambient ambient dimension gb ram intel cpu demonstrate also require synthetic accuracy draw outlier draw change variance draw plot scaling change dataset perfect scale offer complexity lead well demonstrating release digital survey database spectral resample gap correction use find center subtract spectra value pca reduce
generalize option tw tw tw iw iterate small therefore use iterative algorithm option recover singular via recover recover j l extend need form disadvantage straightforward runtime epoch implementation maintain scalar scalar appear sec eigenvalue sec sufficiently epoch practically relevant regime success algorithm succeed high
neighborhood characteristic informative researcher enforce graph sr derive representation select graph explore powerful discriminative sr noise base sr recent representation sample capture global drawback corrupt clean sr base capture structure liu representation graph graph jointly constraint capture global subspace mild lrr correctly preserve membership sample belong dense undesirable lrr interpretation negativity visual performance nn draw ks direct basis coefficient many
correspond bottom corner illustration perform matlab implementation intel cpu gb ram work pixel pixel pixel sort ground one similar due color band nine fig dataset term fusion obtain resolution problem closely relate challenge large normally image normally spectral fusion program solve split augment multiplier estimation sensor formulate intrinsic dimensionality hyperspectral space image define adequate splitting effective publish simulated life detail optimization form k k k q computation fouri include inverse advance involve q separate minimization solve th convention dominate perform multiplier acknowledge dr providing acknowledge zhang providing provide gs pca bt process corner fill blue style thick width minimum height text style gray es universit
capability early classification economic communication time author accuracy euclidean distance wang claim near experiment believe carefully assess extensive broader consider exploit world distance raw fourier transform dft notice parameter dft coefficient notice equivalent raw time usually take close sometimes effect rapidly relevant accuracy accelerated reduce extremely literature aforementioned acceleration capability et al financial use reasoning al use medical wavelet coefficient instance system diagnosis provide advantage generally series base piece wise aggregate coefficient polynomial computing similarity extract employ model computing use chi cluster coefficient stationary series chain discover
fold strict mm mm heuristic measure disk write store take setting reduction determine fig code save load state disk process classify disk time extraction take two pooling learn high dimensionality automatic recognition volume across potential specie realistic variation demonstrate case unsupervise learn operate knowledge training find large volume benefit apparent indicate lack feature create feature lead order magnitude order specie audio datum substantial attain raw transformation preserve implicit perform input often hold back spectra common low result use audio volume availability annotation crucial small label item increase annotation boost due single individual audio cause problem dataset come directly annotation format portion auc classifier suggest label yet automatic outcome importance annotate intend public collection highly specie automate audio width mm plot nr indicate scale width
label differ construction adopt threshold learner suppose every bound bounding standard robust introduce underlie coefficient unit pass origin disagreement coefficient result improvement term rate also considerably simple low bound close q say present unknown connection multi convex strongly tie together propose adaptive base datum furthermore parameter appropriate conjecture existence algorithms adapt notion convexity introduce mention direction unknown query target instead handle possible algorithm improvement convergence learn prove corollary convergence rate surrogate view
use basis move would alone hasting randomly preserve chain lead inaccurate value markov guarantee intractable following modify irreducible computing consist move expand sufficient prove chain form canonical lattice pair paper configuration simple swap construction configuration simple propose uniformly remain unfortunately irreducible unable leave swap connect sufficient htb ise configuration connect project result result reversible holding present maximize adjacent exposition subgraph maximize connect prove configuration ise unique configuration connect unique singleton configuration singleton configuration component unique max singleton exist q reversible
show number either rip claim detail partitioning concentration fine partition several similar concentration hold exploit trick spherical modify tensor exploit draw show median ica tensor modify observe dense case subgaussian nonzero empirical bound ica set ica ica bernoulli random variable subgaussian sample remark detail counter intuitive dependency actually expect require idea net argument addition ica subgaussian subgaussian see argue claim ica exploit assumption incorporate conclude section state organization propose section exploit learn model alternate asymmetric update update alternate mode view alternate least multilinear mode intuition tensor orthogonal tensor suppose correct tensor expand point power update incoherent power initialization successful recover algorithm moment learn provide initialization supervise exploit technique power multilinear l cluster member cluster tuple alternate output center tuple tensor draw compute vector initialize tensor perform many need expensive carry procedure lead model decompose moment tensor view hadamard thus matrix operation initialization bound distance distance ambiguity issue recover tensor unchanged one sign two section tensor k factor asymmetric always adjust appropriately simplicity dimensional overcomplete precisely assume highly organization section appropriate tensor concentration employ latent model mixture sparse semi supervised information
approach reduction time machine slightly although representation domain seek auxiliary prediction approach denoise provide exploit feature nlp processing
pursuit decompose signal sparse develop variational accelerate practical superposition surprisingly look different principal decompose sparse principal seek image scene mathematically state ex solve specifically relationship cite remark fast ahead greatly cross restrict adapt generalize smooth penalty frobenius include huber penalty besides proceed section cast general product regularization enable use discuss computationally accelerate project quasi formulation section demonstrate efficacy new
solely get actual unstable hand large value empirical stable almost region power satisfactory value size nominal pure even size generally finding indicate preferable would empirically preferable nominal level power test maintain usually actual unstable use worth note concept simple composite testing would scope proposal paper significance consider section proper coincide nan idea divergence restricted subspace restrict divergence base testing coincide robustness property easily minor change impose nan decide consider extension excellent robustness already empirically remain
set signature g k projection onto template implement histogram signature impractical require version property gx instead transform version neuron store transform version template template allow signature transformation visual template visual adjacent correspond memory audio observe similar template transformation computation
dominate relevant perspective fit metropolis nx ratio prior overall acceptance metropolis hasting beta posterior product bernoulli acceptance acceptance centre sequence application variate illustrate likelihood prior step often converse apply costly prior ratio eliminate unlikely normal confirm histogram result term individual evaluation potential rejection costly decomposition bring execution binomial replace sequence adequate sequence fall acceptance integrable therefore improper original note irrelevant sense try far successful value b high though requirement ergodicity
borel let nm n old inequality let q square integrable theorem surely kronecker value almost surely almost surely without evy monotone relate evy intensity tail evy intensity eq slowly vary satisfy part monotonic monotone infinity additionally infinity chebyshev inequality variable hierarchical node edge graph total function decrease work concave low markov eq q chebyshev type obtain conditionally poisson poisson go invoke gamma direct graph income count notion exchangeability represent variable indicate restrict generic space infinite jointly exchangeable infinite exchangeability representation involve transformation uniform constructive array study represent exchangeable either terminology theorem conclusion exchangeability sense sparse real network exchangeability obtain alternatively rescale network size lead sparse finitely rescale distribution node auto right node right node node benefit law aside array structure adjacency different notion exchangeability network link otherwise exchangeability exchangeable small unlikely fall lead intuition implication exchangeability
cauchy schwarz inequality moment cauchy schwarz triangle schwarz use schwarz moment limit supremum reasoning n combine eq q rely straightforward provide deviation necessarily quadratic random deviation four deviation obtain q back taylor k imply standard variable x ax power decompose product ax k fact nonnegative office la bs calibration rgb rgb theorem gaussian concentrate detection procedure top projection able keyword detection mixture sparse eigenvalue test fundamental aspect number number literature proposal convex relaxation greedy multiple use dimensional accord appropriate choose selection hoc performing preprocesse necessarily useful propose penalize method suggest crucial estimate propose method two psd stand semidefinite specifically sparse nonzero integer belong sequel parameter arbitrary note identity dimension read q exposition mild approach setting treat would burden vs otherwise limit adopt quantify test bad problem manuscript minimax way versus nr nr noise pseudo nr alternative hypothesis bad versus error maximize introduction alternative hypothesis maximum far enough infimum test formally speak hypothesis index correspondingly also limit well powerful separation rate satisfy
bayesian presence covariate generalize poisson q variational heterogeneity inference residual review case integration actor application consider sbm likelihood force actor sbm wise block rectangular height state sbm sub interval fall receive attention direct suffer intrinsic identifiability problem preserve paper cite interpretability result show subgraph characterize evolutionary network value vary consist typically time keep connect sbm suppose aim detect share connection stress node cluster latter former group never p fill white circle present particular performance review focus sbm constitute broad network popular social account social induce heterogeneity identification unobserve simplicity decide discuss model review general cover section assume cluster latent controlling
account account baseline general baseline large topic media stop discuss account account account model reflect activity covariate l require member without follow obtain estimate treat nuisance full form hazard pl get leverage measure total hazard change bring hazard algebra obtain hazard change bring value express scale next newton logarithm q smoothness employ use hessian given initialize optimum maximize update constant newton analogously entry proper book keep obtain entire matrix hence complexity iteration
replace diversity layer locality column pick cosine similarity instead average center entire neighbor color center near image together expand average conjecture pool mid hoc fashion use convnet style sift retrieve neighbor coarse dense impose smoothness image alignment neighbor alignment convnet location convnet response formulate grid feature image target source image edge
subtle determine membership total entirely trivial principle perfectly every determine clearly thus output shift simple determine output manifold modal differentiable derivative clear context manifold curve cover support vc fourth derivative derivative make sure represent manifold critical point mode saddle assume usual variance k condition regression integrate error proof pointwise kde kernel nx pointwise hausdorff q fix curvature density nx nx z conditional derivative uniform error estimating mode closely see uniform q pointwise rate additional pay square nx analogously mode yy yy yy nx confidence
section define equation fix inspire boltzmann propagation normal perturb often derive interpretation derivation j define stack substitute evaluating covariance impractical variational two example constitute often estimate go likelihood denote univariate mean precision th wish precede generative experiment approach covariance hasting mh
consequently iii also assumption suitably assumption bound accumulation point ii accumulation subsequence take side theorem arbitrarily f k go end virtue sequence establish penalty method accumulation feasible subsequence f penalty method fact accumulation subsequence point problem denote vector form sufficiently together sufficiently unbounded closeness finite follow contradiction pass convergent subsequence finitely q kkt ax kt ax contradiction unbounde contrary generality divide side contradict assumption subsequence take limit side along subsequence finitely see last relation together loss generality
n em w em I n n b n dx surely computation generalize invertible matrix dx dx ii denote sample algorithm train choose propose extra spend bias view part simulation ghz intel cpu fx create distribute normalize range long obvious quality learn
condition completion frank wolfe convergence typically require accurately use entry denote throughout change resp resp asymptotic span denote onto index proof involve integer mark gap spectrum decompose decompose j eventually stop concatenation matrix column point index definition bound quantify whether end q present f satisfy gap tb nb tu l kx tx matrix call result imagine thing obtain distribution independent split precisely explicitly present least procedure analyze subroutine subroutine control intermediate arise matrix tight coherence description iteration way top vector arise use reason ap see randomly main return good condition n element return guarantee
variable marginally conditionally rest choice local multinomial univariate variable frequently combination interest variable parent configuration parent advantage possible task regardless set separate uniquely parent child redundant inference bn implement consist encode algorithm seminal graphical ic bn test graphical information student correlation score optimisation technique candidate goodness algorithm attempt arise independence dag latter pc gs incremental hill search min parent semi pc implement across several package extension
nearly approach computer vision adversarial positive primarily linearity space extreme need yield rbf presence phenomenon sample fed assign probability maxout softmax mistake change top drop confidence mistake cifar sample convolutional maxout experiment suggest need imagenet image encode search uniquely able focused softmax confidence mistake rbf behave find error confidence mistake hard problem belong cifar skewed class maxout network none classify likewise network classify classify introduce propose succeed randomized runtime cifar success step ten step class image gradient method member dataset activation
lt bp r package conjunction terminal explanation load package package graphic explanation graphic macro ltb lt lt lt lt lt lt ltb lt lt lt lt ltb ltb r ltb ltb ltb ltb ltb ltb ltb ltb ltb handle significant high determine handle add effectiveness artificial figure graph reduction handle percent reduction percentage handling compare handling handling handling axis handling handle decrease handle increase exception classification noise broad application beneficial task weight filter algorithm diverse suggest produce result filter weighting great estimate handle examine forest generally high inherent instance
gaussian snr population nmf positive uniqueness uniquely end nmf rank unique describe case tensor nonnegative factor know uniquely recovered idea reduction propose order perform rd tensor obtain follow product rd tensor unfold original uniqueness close nmf sparsity moreover directly reflect zero focus core tensor break curse improve uniqueness ideal nonnegative essentially unique mild sparse suggest quite sparsity core tensor impose sparsity matrix pp db able approach improve q approach
extract sentence dependency recursive feature feature feature sentence generate generally two method category assume sentence attribute field kind sentence multimodal input e sentence probability sentence give affinity retrieval fall close build bilinear whereas store architecture allow arbitrary htb word image architecture illustration rnn share frame simple recurrent neural network widely language task speech type time
specify terminate simulation validity validity ensure terminate interval coverage rule difficult high dimensional magnitude component far unlikely suitable idea terminate simulation ess sufficiently bit variance asymptotic variance note general easy estimator standard terminate interval magnitude simulation terminate poor reasonable default minimum specify reflect analytical condition asymptotic validity relative stop three theorem strongly setting direct practitioner automate criterion applicable first one need relative knowledge magnitude suffice excellent however adjust balance effort exceed criterion
b theorem dependent step assess case super smooth case consist finitely fouri transform xt integrate tail relate exponent analytic law vanish case prove satisfying condition monotonicity condition respectively finite support strong bayes posterior condition hold satisfy mixing via case mixture gaussian derive rate wasserstein assess empirical deconvolution ordinary recall borel probability moment wasserstein metric recover investigate correspond symbol scale b mixture selection deconvolution smooth error optimal measure herein consider rate error dirichlet hyper mix super recently investigate frequentist rate deconvolution model density measurement convolution p yy transform density density iid mixing derive rate either ordinary smooth super ordinary super smooth minimax super corollary inversion inequality relate density density yield deconvolution follow
iii iv phone call incoming call receive diversity incoming unique call send entropy ratio behavior percent call call percent regularity average inter event inter event consist incoming total follow call call receive percentage ii percentage iv percentage text text receive user hour text characterization diversity apply online measure variance inter call inter quantity measure evenly distribute address three feature unique total entropy explain bin function fx np apply deal bias entropy filter case filter proximity feature table general proximity accounting see time diversity interaction regularity two event variance inter time extract general iii correction calculation formulate
explicitly beneficial great consensus superior add paper consensus herein matrix treat input reach initialize randomly consensus final consensus consensus run mean consensus via consensus final membership method variety mean ensemble pair unclear potential datum essentially round collect herein popular consideration refer resource spherical mean low objective matrix cut accord normalize cut ng choice membership algorithm mistake assume rarely algorithm mistake introduce consensus matrix proportion must agree keep vote initial consensus form
maximum compare pool largely available score rise degree autocorrelation contrast expect experimentally autocorrelation score contrast compare performance al al common autocorrelation trajectory al small would lead method trajectory dominate rs budget would present picture al much al performance expect comparison show would score resolve score autocorrelation show show examine base way compare benchmark rs stage first seek
five start structure represent former matlab take input strictly matlab contain field list help default default solver minimize gradient determine toolbox find contain cm cm prox proximal solve prox proximal proximal operator prox
goal offer represent observed network inspire boost handle noisy overlap signal noise plan thorough framework cluster potential corollary observation pt significant relational give challenge general boost inspire weak entity similarity graph suitable community detection real demonstrating measurement consideration local structure community real absence ground quality proxy learn measure contribution aggregation framework learn application heuristic measure operate incorporate make
indeed expand signature rough individually inverse polynomially correlate set empirical define simplify main explain correlate expand find signature rough solution get polynomially signature imply dictionary entry whether ability individually k j assumption large entry ns using argument shall dictionary formally write unknown expand large ki negative outline quantity exactly additionally deviation signature require additional standard deviation general signature
distribution show present strong package show thank valuable suggestion correction estimation student copulas I use univariate compute estimate matrix initial copula current covariance ii derivative
represent differential entropy matrix differential semidefinite close minimize differential among semidefinite condition strongly variation explicitly see positive definite conclude stay definite subsequent still eigenvalue zero mean arbitrarily bfgs implementation challenge introduce enforce matrix proximity exceed show restrictive solve regularize correct explicitly regularize bfgs replacement variation correct variation term bfgs regularize curvature iterate think gradient iterate stand hessian compute bfgs batch determine add positive differ account relative regularize regularize differ gradient observe add curvature necessary variation discuss hessian stochastic gradient satisfie explicitly eq guarantee variable cf cf approximation core comprise require
dataset ip importance convergence ip show almost sdca ip practical sdca u somewhat prox sdca explanation primal update q conservative indeed satisfy large primal variable confirm test change result display clear primal convergence prox sdca prox sdca less hence close difference rule prox sdca speedup predictor speedup focus hinge factor specialize nice dataset several value practical theoretical prediction large roughly speedup reach lot year smooth primal machine incur predictor regularizer especially interested big million much big let n nm mi separable list mind name know simultaneously solve primal however apart block choose like method dual perform primal primal update
mutually exclusive exhaustive mutual algebraic elementary logic equivalence triple triple follow signal identifiable identifiable converse identifiable equivalence assertion proposition identifiable open equivalence establish third triple generic measurement generic identify measurement identify generic measurement regime identify generic identify furthermore dense open either three formulation mutually exclusive exhaustive analogous retrieval treat verify generic rank enable signal reconstruction measurement signal recent furthermore algebraic algebraic term non intensity measurement image ray magnitude sample phase optical task reconstruct finitely rank algorithm involve projection fit nonconvex optimization semidefinite programming success grow algebraic formula derive require measurement scale dimension jointly algebraic semidefinite successfully reconstruct generic identifiability signal signal open enable phase retrieval signal phase rank investigate mention identifiability
expensive computation hardware corrupt descent true stochastic hardware shown avoid additional present proportional differ descent gradient descent proposition proof directly applicable enforce monotonically satisfy preferred fitting relaxed understand hardware induce successive discover hardware computational analysis lipschitz iterate eq batch gradient bit sequence converge variable expectation allow decay I hardware accurate
maximize bayesian design minimize leibler eq random additionally mix ct slice select entropy progress accuracy overall posteriori fig measure truth gp depend subset dimension learn seven possibility truth correspond fig significantly method uncertainty design perform poorly
continue standard generative hyperspectral full dirichlet without loss generality n go particular parameter characterize pixel simplify symmetric follow phenomenon uniformly decrease concentrate around vertex concentrated contain pixel separability implicitly assumption ii exist statistical assumption thereby implication pre formulate whitening q large small ba deduce whitening assumption n tw whiten aforementioned bind conditioning plug corollary provable spread play get consider fix natural concentration interesting specifically
dpp space space say mass specifically subset determinant contain column semidefinite matrix speak representation ensemble similarity dpp subset square tend volume co machine appeal marginalization model dpp preference six except ensemble group green rank condition dpp sample interaction probability proportional rest tb dpp toy except select although big block block standard model often make binary random specifie include assume bernoulli covariance spike
exploit hierarchical learn representation co occurrence qualitatively among hierarchy occurrence highlight facilitate reasoning space experimentally illustrate dependency multi classification area machine contrast instance goal model structure space occurrence attempt structure configuration label classifier handle classifier efficiently effort maintain update generate ground
result supplement assume constant setting probability depend let proposition satisfy define zero proposition establish result v result contrast provably identify method computationally primarily dimension however goal mixture two spherical
argument behaves simplify special noiseless signal order maximum within dictionary conversely expect dictionary incoherent dictionary take account need least incoherent imply reach small choosing translate mean maxima conduct input lagrange multiplier n arrive scaling ensure giving generating bad atom signal sign iteration determine multiplication comparison thresholde instead omp procedure algorithm local refinement furthermore version new old signal process learn accord algorithm conduct experiment htb generating signal far give decay factor uniformly vector sphere choose permutation compare error basis number perturbation correspond approximately noiseless signal
ridge thresholding network ij j fitting square j construct jk perform hard ridge j cf threshold medium large link must maintain solve single solution latter ridge impose application allow medium th exclude maintain link show implementation th integer decrease costly iteration simple directly network topology relatively fine either dynamical possibly nonlinear approach reason observation perturbation dynamical exponentially fast important investigate lyapunov representation degradation
transfer code much also decision result problem come gradient optimize code train create iterate create initial robust coding produce latent present evaluate motivation robust pca
demonstrate effectiveness core core basically q achieve mention package modification pre substantial store moderate issue summarize expensive kernel half total pairwise store realistic pc merely example computational demand another serious issue issue
use purpose predictive support source validation assessment phase inform methodology illustrative involve validation bayesian advance hardware advance fidelity enable physical phenomenon system complexity capability heavily nearly complex nuclear use area inaccurate inform decision could response critical reliability systematically characterize reliability science reliability computational necessarily computational thereby regard computational compete design meet operational decide evolve informed prediction quantitie important feature use observational scenario available would course engineer currently assess unobserved address propose call predictive broad define specify enable test predictive physical reliable theory law whose highly must less reliable reliable embed various modeling approximation empirical interpolation mechanic embed might molecular fidelity fidelity embed composite build highly enabling reliable though specifically require reliable restriction composite specific ingredient representation representation observational model provide uncertainty development model predictive approach discrepancy model unobserved advance accomplish embed physical directly uncertainty bayesian model condition observational uncertainty unobserved composite provide physics subject validation integrate assess reliability prediction describe calibration predictive regard uncertainty infer
formula appear subset etc uncorrelated follow result follow lemma desire combined hypothesis suffice definition way limit suffice infinity never condition hold replacement limit distribution forest example consistently forest prediction remain largely answer subsample training randomize predictor prediction forest averaging explain formally resample random forest main base prediction provide consistently thus bring forest bootstrap forest remarkably estimator surprising property
calculate size lemma k proof appendix enough know always bind dm however prior knowledge prove analyze second know able algorithm calculate prove eq phase examine expect bind expect bind km improvement attribute reason highly dependent require hard attribute eq efficient scenario call efficient present develop estimate operation update update result project ball yield build unbiased gradient modification p r tp ti slightly let idea unbiased estimator therefore use analysis full proof present notation author algorithm ridge scenario analyze tell
mt media file mt user file mt file mt monitor mt project file mt file trace cache simulator build cache service request maintain cache request simulator cache block cache record else record add cache cache limited cache trace cache simulator cache exist policy keep track previous cache sparse hmm baseline trace trace htp r r trace sparse hmm without dp mt mt mt mt mt mt portion dependency train operational mark real world set require vary periodic require explore version setting htbp minus pt pt pt though
useful specification additional mcmc adjust achieve even quite well solution small hamiltonian dynamic effectively result improve posterior step hamiltonian black handwritten digit handwritten digit benchmark consist spherical gaussian parameterize connect output distribution pixel consist datum necessity individual map jointly unbiased construct replace chain hamiltonian describe vary number auxiliary choose deterministic
demonstrate biased nonparametric prior odd bernoulli discover gamma process couple heavily description work particularly value atom interpret trait atom proof essentially fix ordinary translate bayes full moreover entirely process reasonable carry tuple need real believe broad acknowledgement project university berkeley fellowship automatic prior far generate bernoulli location atom accordance bernoulli rewrite conjugate fix location atom beta hyperparameter normalization ordinary component proper hyperparameter ensure measure must must represent improper beta either distribution integral hyperparameter restriction recover bernoulli parameterization condition q pick biased beta point beta condition pair particular construction location atom weight atom location recover posterior nonparametric prior sequence finite represent full bayesian via finite motivated prior exponential likelihood construction
none b comparable uci benchmark training measure choose summarize describe select significantly stock tend bold c reduction kde auto forest energy stock kde kde auto red forest concrete energy evaluate propose simulator body part roll roll roll value angle velocity
zero forward see finally real world even slight independence tackle corruption heavy reconstruct belong subspace coefficient reconstruct corrupted assign unconstrained quadratic program run loop gradient conjugate straight projection total matrix identity program technique scalable synthetic detail empirical face individual illumination face
successful describe agnostic agnostic successfully construct ensemble predictor constrain grid generalize agnostic attempt directly infer accord contrast bayesian implicitly data likely member irrespective ensemble approach risk hyperparameter loss rule posterior estimate repeatedly sample could vector predictor low rule repeatedly predictor obtain stand obtain sample replacement empirical risk
completion recovery conclusion accord aim paper effectiveness explanation experiment window core ghz gb conduct representative completion partial dct effectiveness admm admm admm former value performance quality admm nuclear norm recovery lr conduct visual situation dct conduct admm lr situation dct conduct compare admm rank dimensional dct zero deviation matlab generate randomly completion experiment randomly x terminate admm terminate criterion f empirically test parameter admm matlab code use peak ratio
construction lipschitz boundary disjoint eq disjoint collection satisfy together modification variation concrete visualization minimizer cut eq appropriate take datum distribution two rectangular eq easily domain appropriately characteristic analogously partition base cut problem optimal cut n nk k red line indicate cut utilize nearest eq near descent cut initialize ground truth algorithm three consecutive graph partition return quantify partition simply misclassifie I sequence satisfy last inequality piecewise way convergence context nn measure graph ratio maximal average realization degree compute become become increasingly relate perform exhaustive experiment three domain consider correspond distinct fall surely connect also rise rather structural increasingly graph see figure scale connectivity geometric graph vertex leave connectivity alone responsible balanced scaling serve benchmark context provide balanced cut test fail outline pose balance consistency practical difficulty may
abstraction prove useful statistic whereas well introduce generic notion consider confidence low theoretic specific armed bandit derive refined bound confidence particular fix set familiar behavior alternative addition improve sequential time proof result deviation lemma good arm exploration divergence sequential testing paper find arm armed arm arm option receive expectation agent goal identify index arm expectation tuple expectation sort decrease depend several analysis include kl ucb thompson without try observation study name identification advance consider confidence introduce identification successively discard example bandit arm sampling model subgaussian propose algorithm subgaussian upper imply rather comparable gap recent bound sample dependency term gap go remain exhibit pac algorithms exist improve upper work bandit follow relaxation consider literature tolerance optimal compare
iteration recommend pass execute break terminate loop early suitable input determine solve shot screen external set additional examine algorithm table generate rand mnist write digits sampling randomly subject face pick word uci repository represent occurrence document remove leave first solver solver give percentage reject speedup divide sum solve reduce lasso speedup use value select problem r feature dim rand mnist average shoot st dt spherical conservative shot screen dataset sphere dictionary also oracle sphere center provide bound shot test fig salient default shot indicate potential spherical gap indicate worth default spherical except value default dt x dt effect mnist speedup plot test dataset iteration confirm low sequential screening scheme salient mnist robust yield rejection speedup compare one shot giving screening use successfully solve
function posterior via regression function simple spike behave accordance expectation obtain performance improve performance since great ridge via development regard stochastic inference acceleration part terminal cell cell acknowledge foundation fast number alternate relevant state whereas computationally infeasible entirely approach problem selection variety inference challenge case hierarchical
unknown start sequence rate predict form number start weight forecaster start exponential wise loss strategy perform step sequence negative algorithm rule rate forecaster defer article expert advance final explain let increase sequence rate regret lemma lemma dividing grow last main derive stochastic assume formally learner ask knowledge observation strategy consequence show
character demonstrate recognize detect close admissible evaluate length admissible occur character character contain model bernoulli hoeffde document decrease exponentially acceptable proportion necessary achieve desire probabilistic right side proportion unlikely exceed become impose allow percent opt evenly throughout maximally construction character document criterion meet case order among application quantitative produce programming principle implements object oriented programming implement represent analysis let seek represent
thank addition amount pairwise distance geometrically shrinkage project euclidean encourage effect analytically give principle leave entry minimal eigenvalue principle q eigenvalue decomposition light minimum dimension equivalent assume eigenvalue apply amount shrinkage embed circle circle right circle circle pt embed characterization shrinkage estimator characterization clear convenience projection derive oracle stand close euclidean frobenius norm large tuning parameter embed explicit error general true distance eq light
word rich rate average assume across item item receive rating receive item item heterogeneous rich word item receive receive assume rich item model rich assume user item rich otherwise analysis relaxed comment modification proof condition form satisfied fashion matrix furthermore noisy channel two likely upper size size constant require recover rating develop scale asymptotically occur rate rating e function require recover cluster satisfy separability condition hold exist completion observe rating fewer accurately proof present sufficiently hold rating least item recover rating find present recover cluster key modification information rich clustering compare select user say prove rich algorithm select accord user majority
hereafter hereafter test via monte simulation propose value size test therefore empirical critical reasonably maintain weak signal loss take alternative take magnitude take follow identically distribute k block identically dependence e range possess robustness propose autoregressive distribute generate distribute variate move beta consider p covariance simulation test sparse sampling take significance compute simulation hc summarize figure test empirical test reasonably nominal maintain fail long range structure hc procedure fail maintain strong ex ex hc ex hc hc nominal diagonal autoregressive move consider level signal covariance signal strength matrix empirical took propose maintain nominal test screen
variate determined iteration time examine schedule expect speedup true quantity tree predictor predictor predictor immediate transition always choose available acceptance improve increase worker move sequence affect scheduling branch promise branch ultimately step chain compute completion distribution apply monte posterior unnormalized modeling correspond model decompose independent logarithm term mh eq form normal separately variance together expand perturbation concrete subsampling subsample construct term subsample empirically deviation multiply estimate finite correction eq form
work show side metric dnn phone embedding extend multi gold derive term feedforward neural input stack contain hide layer sigmoid activation embedding dimension type
technology quantify expression throughput possibility rna seq yet appropriate statistical assess statistical bayesian treatment empirical approach parameter quasi account tackle lead highly couple probable differential expression rna sequence uncertainty theoretically parameter gene yet small across log fold ratio continuous exhibit difference cause random variability
cl sl ensemble accumulation similarity connect triple time consume fig infeasible dataset method perform benchmark agglomerative cl sl baseline cl sl cl sl cl sl performance baseline dataset consensus dataset method stable pair good dataset overall method ill clustering heavily clustering pool ill pool clustering pool clustering clustering partition mean randomly cluster select heavily clustering ratio clustering experiment base clustering ill clustering link pair wise clustering clustering accordance common reliability collect clustering consensus accumulation partition gp respectively eight result effectiveness robustness propose ensemble consensus accumulation partitioning link fundamental purpose partition unlabele homogeneous
project axis distribution axis consistent discard might size powerful regardless compare asymptotic efficiency relative great size tend equivalent value relative efficiency relative follow theorem small equation case scale imply error conservative xy dependent numerator denominator test high test test random rotation assume size kernel associate uniquely respective reproduce hilbert xy
gps gps wave unlikely poor sequential size importantly gp achieve interest wave improve gp wide advance number wave sufficiently detailed whether gp become increasingly problem support model wave cope suitably say accurately predict posterior metropolis hasting proposal acceptance gps parameter likelihood mh decide simulator onto prior wave report volume dimension act
du univariate correlation generate marginal binomial via generate margin topology original element monotone slight normal sample poisson close great appendix practice package cdf several pre abundance zero account sequence normalize sequence multiply desirable depth preferable filter sequencing depth normalize serve count fit use target zero binomial good account count newton candidate superior normality assumption pattern adjacency undirecte graph topology generate step undirected adjacency ii convert assign diagonal convert focus representative structure degree recovery thus topologie maximum band model disjoint association across scale biology network serve comprise specie many specie connect sparsity control start adjacency type network chain connect neighbor edge fill cluster comprise divide approximately randomly assign scale network law specie connection adjacency matrix add adjacency adjacency precision
procedure kf use approximate mean kf q explicitly enkf ensemble bottleneck enkf describe converge small size degree require version enkf adjustment ensemble transform kalman subsection kalman take couple initial product second cross reduce kf gray operation state covariance cost nm tn essence kf store adjacent cost form covariance prior cross already kalman equation product equation walk fix variation derive formulation assumption store update assimilation step therefore kf covariance initial assume available model covariance spatial section kernel dense oppose provide sparse engineering application
ergodic converge rate gives point select log likelihood happen exactly quadratic match proposal move discussion three point mt mt mt always situation substantially detail setup genetic circuit response concentration chemical figure vertical correspond algebraic switch z datum six scale around nominal endowed uniform nominal observe low average without posterior remain seem useful towards quite add difficulty heuristic heuristic rigorous give expansion designing grid respect induce convergent density inefficient whenever illustrate overall approach provable convergence surrogate attempt resolve propose hasting local process use sequential experimental exploration design reflect quality local random refinement local quality practical permit wherein enable quickly smooth simpler asymptotically exact walk couple local approximation demonstrate posterior refined walk metropolis broadly adapt metropolis hasting time propagate broadly theoretical complement experimental several order involve ordinary equation partial equation approximation remainder organize describe exact defer several emphasize present representative therefore discuss several future metropolis infinitely refine problem forward
justify bag contaminate resample contaminate set contamination explain section contamination enable create diverse base resample bag exploit resample tradeoff place variability increase stability model influence svm quantify distinguished instance ii iii incorrectly classify margin bound
property return set every share notation associate firstly risk classifier subsample self hence close minimize risk deviation disagreement rewrite confidence majority keeping performance confirm label close source risk apply optimize disagreement risk label however decrease guarantee gibbs decrease avoid design tune question concern da reverse circular reverse perform intuition source label seen previously validate analysis make sample show
list originally english word obtain return translation one devise training distance negative margin negative hold descent tune task improve least propose cross modal mapping vice versa contain representation wikipedia gram mode image represent train train labeling set task label entire distinct return cat chance observe differently case cosine domain improve standard setting chance
logit lead inverse probit sparfa next give response response minimize observed response constraint use practice nuclear rank one constraint via validation emphasize sparfa regular sparfa negative logit fista start iteration perform aim follow projection make give size inverse logit link bin boundary measurement
htp red solid accelerate accelerate suffer slow amount overcome issue one strategy fy k evolution dash iteration dot one clearly already well solution help htp htp lasso htp physical average coordinate htp
modality normalize temporal original frame derivative compute temporal delta element vector calculate way delta static audio video dimension whitening frame result audio build network structure two letter deep machine final rbm unit unit letter size space embed although preliminary one embed
angle face second put clean patch image database adaptive denoise apply localize new exist bm bm bm new patch procedure determine text face demonstrate superior amount available generation give eq drop orthogonality take later lagrangian multipli setting pair must correspond observe trivial satisfied take constraint denoise column become sum lagrange take eigenvector orthonormal q recall simplify first difficult note substitute eq hold therefore standard exist give remark fact procedure propose denoise image denoise patch noisy generic new patch database denoise filter problem denoise filter solve sparsity generalize denoise offer systematic enhance second determine denoise localize bayesian localize intensive computation
approach generate expansion abstract approximate term give subscript representation example calculate cost multiplier introduce assume component separately separately column contain length specific kernel identity calculation assume multipli component may equation need obtain exponential radius process avoid square cholesky intel core ghz point define fraction ground way solution call remain apart training test accurate result use predict detail use sec interpolation shift parameter prediction example ii either literature general study quality assess mean predict predict different ard
human exist infeasible screening gaussian possesse screen gain popularity context possess go approach generalize generalized screen partial screening response select screen recover call motivated fact precision obtain feature onto neighbourhood sure show exceed threshold grows establish surprising exist procedure unsupervise conceptually implement estimate
away unstable equilibrium future discount reward actor convergence fix lagrange td td lagrange multipli constant recursion set equilibria ode ensure evolution ode stay recursion convergence limit policy recursion unique eq bellman transition state similar manner stable govern almost let unbiased vanish vanish martingale cf see actor recursion asymptotically track point ode eq bias converge uniformly follow discount show recursion saddle ode define surely proof manner discount claim use use convergence rs g ode satisfie context application delay motivation behind variation road infinite setting traffic formulate briefly recall queue time road since turned belong road factor traffic discount implement green simulator order neutral parameter follow td unlike rs neutral multipli neutral use smoothed sf technique neutral n j q rest symbol rs sf neutral counterpart sf hessian actor update form actor rs g attempt accord sf variant sf hessian update sf sf counterpart rs consider rs underlie boltzmann approximation road approximately order increase experiment phase nominal iteration policy simulation converge average snapshot road simulator traffic frequency specify traffic proportion horizontal fig set weight
proposal neighboring constitute single cause mix eventually low point mcmc move manner configuration accurately rbms typically train stack greedy wise deep belief layer latent feature another rbm mix well configuration draw top level gibb lead mixed well result appear consider layer model even phenomenon receive attention learn texture
make consequently cause al shape kernel learn non parametric fashion control person person decay person characterize decay enforce value instantaneous reflect material new dynamic bayesian specifie occur people person interval token word type token upon multivariate process token person categorical vector draw person specific base discrete vector characterize inherent usage person person self enforce make token type person end time person characterize decay parameter
duration delta duration counter always state hdp possible transition hdp restrict ps equal ps draw dependent hdp bayesian hmm setup hdp hmm replace auxiliary dp instead hdp create hmm reversible hmms cardinality death operator incorporate duration might would complex finite hmms emission slice sampling emission mixture responsible observation subset mixture relate identity state describe could arrive inference perform dynamic exactly compute map implicitly cardinality kind guarantee auxiliary duration emission sampling hdp jointly employ filter backward infinite duration hmms duration associate time log likelihood quickly backward slice sampler move trajectory infinite backward hmms state time backward difficulty overcome use slice auxiliary result sampling context auxiliary variable introduce variable
bayes relate optimization efficiently novel iterative maximization performance problem drive access identification specific namely interested wide engineering reconstruction science communication hundred method impossible thorough literature generally pose structure impossible retrieve partially intrinsic
ray foundation rna protein solution nevertheless phase mechanism year experimental enable potentially resolution chemical specificity ray enable combine high precision resolution imaging advance detector world develop hundred help ever material device study life macro molecular machine picture recover atomic detector small focus numerical exhibit encouraging conclusion organize introduce setup show sufficient ap section relationship ap objective ap synchronization propose accurate also design achieve field set non negative phase would set symbol form scalar hilbert lk ii f pi li I wise entry resp resp notation aa ba ij ij form hadamard matrix express format stack column embed j I record wise fourier magnitude relationship coherent pattern discretize camera
select detail bernoulli variable one cardinality presentation assume sign nonzero sign ij globally place elimination scheme prove distribution noisy follow tw like column equal fairly hold denote difference first coherence finish follow ie ie u ie ie ie ie norm norm sense u ie ie give ie imply u next sign entry bernoulli sup entry distribute u long ac u ie u ie see sum u ie randomness easy ie ie ie f vary ij u
enhanced also replace subproblem easier approach q iteration bound point subgradient function application problem gradient method latter new namely slide gs evaluation approximately solve subproblem accelerate method gs need compute pair use place output show subgradient subroutine nk ps prox slide procedure let parameter given observe slide compute approximate clearly problem since affine skip gradient differ remark gs firstly occur increment gs algorithm update secondly solve consist solution relatively notational convenience procedure conceptual yet
joint force pixel matrix pixel atom group equation sign inherent since compose several e dictionary classify pixel measure reasonable enforce atom accomplish encourage group active inactive group dominate classification inherently mixed optimization represent weight collaborative collaborative regularizer define refer coefficient lasso joint
vice versa item modification weighting scheme weight weighted item leverage one allow time limit bias induce equation thus optimisation amount dissimilarity reduce influence rare important calculation gradient system limit modification precisely subset large calculation descent notice point costly associate statistical
sense consider coherence function coherence initially propose cosine angle become j construct coherent candidate coherence cosine functions control nan computationally atom perspective coherence deal unit explore analogy norm gram eq atom exceed follow include j view extension coherence approximation extension dictionary eigenvalue span sparse dictionary sparsity low low bound investigate condition cf proceed bring mind gram dictionary unit norm atom every lie provide gram sparse dictionary
bag classifier evaluation undesirable attractive base strength ensemble classifier two introduce prototype bag form dissimilarity prototype bag stand correspond size stand offer default alternative well alternative dissimilarity set split data dissimilarity change expect choice affect well single dissimilarity ccc problem drug image dataset scene formulate concerned categorization song dataset audio feature zero
satisfy concavity operator previously subset size sequence use sample size contraction coefficient initial sample bind q probability guarantee dimensional instead analyze discussion state theorem algorithm term addition initialization iteration estimate illustrate iteration size figs em eps decay rate iteration number em fast cc fig eps fig eps regression em decay geometrically decay geometrically devote linear covariate vector miss ratio give choice define guarantee miss missing covariate miss satisfy ball previous corollary somewhat seem counter intuitive minimax covariate show gradient algorithm formalize amount information confirm show eventually decrease grow fig plot mixture algorithm optima splitting em em operator usual full subset contraction em iterate probability perform large constant logarithmic sample achieve analyze particular iteration sample give stochastic gradient eq appendix fig mis eps decay ccc fig em mis eps figs mis eps missing covariate
e achieve oracle property section equal follow reason variance advance accordingly even know consequence fact variance fact subsequent analysis notation l analyze lp discuss capability recover result l l definition lemma previous l ne
space tradeoff show access use allow well lp qp choose factor expected distribute overhead however svm outperform wise advantage incur node request contrast lr l primarily tradeoff descent number converge fast factor speed overhead break scheduling second use second scheduling inside remove epoch hold implementation tradeoff space difference qp row wise access wise fast primarily issue support claim interesting choose throughput compare throughput different system simple sum model core throughput figure high throughput system incur cache write single copy copy node cache single fast overhead dynamically schedule maintain overhead language scheduling computation fast impact modern architecture row wise mean ec validate access dominate strategy row wise force correspond report measure take achieve side segment epoch reach wise access least phenomenon column access slow simply read preferable coordinate descent wise amazon google lr qp lp dominate write capture describe impact factor wise strategy figure ratio row column amazon machine increase ratio wise become slow wise hardware
adopt paper logistic loss training keeps prevent simplify exposition straightforwardly vector learn nonlinearity jointly significantly learn classifier however mapping raise generality set rest fixing norm essentially permit adapt advantage set general focus go class general original cast dc program last classifier rewrite optimization rewrite eqs hold restrict component vector hyperplane remove small component positive class knowledge atom desire half encode prior balanced estimate assume proportion atom sample solid smooth qx parameter approximation become soft play important training easier soft latter handle essentially relaxed optimal variable still nonconvex type write dc solution dc function
local ratio cut define one choose cut correspond weighted think vertex divide unlabele wish assign assign represent vertex similarity vertex cluster give order
anomalous unknown priori test apply consistent stack sequence case anomalous exponentially comment scenario lack knowledge scenario study anomalous sample characterize sparsity anomalous consistent sp substitute theoretical mmd apply reference choose I choose anomalous laplace respectively change normalize horizontal axis clear converge theoretical furthermore curve drop mmd run number anomalous plot converge consistent threshold increase anomalous priori choose distribution mixture distribution figure converge confirm theorem test scenario reference sequence gaussian sequence see reference probability agree drop increase importantly error zero theorem comment variance variance different
proposition similar mala mala know mala rwm study asymptotic regime rwm mala asymptotic mala rwm natural whether mala rwm extend within mala log property particle mala mala depend crucially behaviour accuracy log decay mala behave mala asymptotic rwm mala rwm scale furthermore explicit posterior particle mala mala though implementation particle mala mcmc theory extend beyond model introduction sequential carlo filter filter guide introduce mala result asymptotic size improvement rwm discussion measurable density represent arbitrary sequence assume directly
would child scalable graphical hierarchical expressive implementation massive massive hierarchical namely bioinformatics graphical merge concept graph structure age extract noisy make network scalability massive datum massive divide arrange datum influence immediate represent bn representation handle scenario massive represent random represent represent disease city connect disease city assume disease bn outcome compose node probability entry represent massive efficient multi classification probabilistic hierarchical apply domain throughput
usually actually relevant similarity fast amenable usefulness propose feature irrelevant feature good knowledge reduce projection furthermore order provide original present experimental section metric attract lot ten effort towards comprehensive exist focus mahalanobis symmetric semi undesirable dimensional early resort dimension loss interpretability satisfactory limitation restrictive weighting r
coefficient method include bic criterion aic cv minimize stein unbiased sure setting variable location empirically correct demand know reliable popular high demand paper select computationally agnostic regression threshold become variable reliably lasso jointly estimate result benefit sparse property lasso research seek throughout
score merge discrete variant cluster iff well score initialization number operation costly consume solution split suited generalize follow center index centroid heuristic straightforwardly decrease round assignment stage stage neighbor center c lp tc j tc illustrate toy strictly decrease score repeat finite diagram check pdf means mean k centroid pdf k pdf center
quasi give learning construct private proper learner discrete algorithm exhibit private work separate pure approximate case agnostic class easily exploit bound sample predicate private database achieve tight work prove concept examine private differential label therein privacy label private term dimension completely characterize learner constant kind research direction work learner try construct private learner complex private hyperplane would interesting another research try understand learner construction know improve generic pure private learner characterize complexity learner early work another prove separation pure differential privacy differential demonstrate noise pure privacy gap private currently unknown show pure term negligible database shorthand use instead mx mx dx j privacy aim database say preserve record learn datum differentially database omit preserve differential pure case function access differentially mechanism differential concatenation moreover adaptively differentially private permit interaction preserve ensure privacy bind privacy oppose follow permit preserve privacy access ensure label domain predicate mapping example sample unknown successful say hypothesis concept I random coin proper improper empirical q class characterize pac learner behavior cardinality hence pac class theorem give sample dimension concept mc mx dy output concept agree learner pac use
specify format depend require solver interface sdp present transform form trivial say usually cone template function return return empty appear template point point lie use template order form expression every represent atom return otherwise child top apply optimization constraint objective side constraint add sense form argument construct problem variable dual respectively transformation primal intuition note expression
comparison meta moment expand hilbert survey exact hand way arithmetic inequality question whether polynomial ask square considerably polynomial polynomial polynomial polynomial conclusion equation oppose purpose transform inequality easier follow reduce clearly degree polynomial write polynomial see degree write represent write hypercube meaningful real value multilinear polynomial assume loss polynomial verify use gr basis put thing several informally state sketch view equation polynomial need incorporate see system form variable interested system small ideally even parameterize call operate exist system mx fine discretization number carry actual corresponding degree
move add remove single away neighbor reversible sampler bias move probability metropolis ensure balance markov chain maintain desire vector potentially serious scalability issue inclusion proposal evaluate neighbor hence inclusion back fitting scale although score evaluate parallel scalable inclusion interactive framework strategy identify quickly occur good inclusion well current many fit trade neighborhood disjoint configuration pair reversible neighborhood sampler remove remove swap reverse pair neighborhood quick dramatically predictor change proposal allow vary encourage component removal size set decrease unimodal p pair setting remain prohibitive computer component proceed move forward neighborhood reverse move construct reverse accept sampler satisfie detailed mcmc desire inclusion define neighborhood likelihood component intuitively seem sufficiently particular pair add configuration remove neighbor pair opposite direction require hence pair add need factor negligible limit random away appear good pair move markov chain resolve issue
table step scale cg stop else compute k ap h r h h ccccc h r h r r q r r diag h r h diag p n diag p k j kp solve z ta r r k tp observe coefficient table introduction cd cd cd cd kk cd kk kk k preferable cg investigate cg allow cg within generate mutually
dark grey indicate place robot behavior show ability learn avoid try robot fail avoid paper embed quantify fuzzy proposition linguistic multiple extensively environment environment compare mobile significance perform statistically grant support education national plan ap ram part european project cn education notice author version accept publication apply soft correction mechanism reflect document work publication j modification necessary adapt example production linguistic rule example different default use new population l le jj correctly j belong e c classified rule belong incorrectly classified rule take definition accuracy represent individual match mutation generalization individual cover change parameter regression default straight learn rule proposition rule interpretable rule also proposition number c tb proposition confusion learn performance measure show tb straight convex concave concave design stage transform variable second machine high actually
center ik bc expression lemma find membership small constructing problem outline view method approximation offer advantage top eigenvector secondly cluster instead early cc ccc mean cluster sample randomly center denote mean update number sample number initialize column matrix algorithm normalize hypergraph partition consensus maximize mutual partition membership represent assign label partition partition lie perfect maximize consensus represent partition hypergraph regular edge meta partition balanced meta cluster meta association meta
matrix partially fraction vary unnormalized theoretical analysis decay small normalize htb adjoint operator constant v v j c ta follow q bernstein make differential recall particular set sign p p dual theoretical collective requirement meet requirement derive optimum factor requirement program affinity relationship movie explicit represent completion task miss entry potentially core gene protein
x iv tx desirable integer inequality decrease ok ok conclude lagrangian denote derivative yield k ni w ni ni next therefore plug supplementary ni ok report good context contribution rigorously general characterize yield prediction call instability throughout classifier adapt elaborate neighbor one popular literature extensive theoretically justify risk comprehensive reader weight near risk attention find control regret nearest specifically trade regret new methodology minimax rate offer comprehensive near neighbor
consider answer map idea naturally lead item provide rating yield latent space translation rate translation rating rating rating item two rating interesting rating represented nan rate modify qualitative result write item rating describe consist combinatorial combinatorial optimization weight item clarity representation depend user item restrict rewrite loss optimize propose consist apply weight weight particularly fit
mass provide illustration mass interpret support finite mass provide project coordinate distance bar histogram together sample coordinate reduction observation prediction reduce space concrete ccccc observation c c c empirical distance distance selection ccccc mass computer first thesis brief introduction justify justify simply domain lipschitz theory generally study dual classifying optimally use hyperplane banach construct lipschitz lipschitz exist lipschitz margin lipschitz margin denote label equally mass follow criterion distance finitely supremum lipschitz support result tx define inside conversely metric mass respect satisfy measure g q margin draw without regard area consequently summary relate margin imply soft classifier soft margin mass use acceptable embed observation banach life mass extremely compute certain assigning individually keep classifier prediction chapter base new mass chapter chapter explain trained predict label life learn training evaluation set time iterate evaluation parameter receiver operate roc explain technique determine classifier collection confusion count occurrence comparative cm positive negative negative call negative confusion confusion subsection measure confusion quick well classifier predict skew trivial predict even power evaluate e confusion matrix observation actual proportion predict label precision use science especially retrieval disadvantage f account quick example confusion cc actual predict accuracy receiver operate characteristic roc prediction explicitly area measure classifier predict reference regard roc classification
field ci comparison assumption ci ignore fashion ci eq j ci l membership comparison make fully approach disagreement level represent agreement believe field increase mass move take fairly one set take general truncate comparison record disagreement large capture disagreement disagreement disagreement relatively concentrated value close especially amount reasoning specify remain l disagreement construct priori notice believe exclude inclusion level probability record depend use comparison nominal frequent take disagreement could supplementary material gibbs joint subsection brief na present illustrate record believe field specify year month inspire name compose correspond family practice record pairwise record record agree except month day month record notice record day pairwise decision take decision pair record fairly decide record table record could record miss name name whether record believe deal situation record datum report occur year month day date scenario report error name date place form information self
simple radius choose convergence update measure term rate increase long proposition bind suggest guarantee projection great cc factor plot expect blue run algorithm independently consistent bar naive sketch sketch roughly twice verify predict range choice predict blue bar height average marked instance iteration corollary imply high solution confirm bar show naive square roughly apply simplex g portfolio problem prove many matrix completion control user define radius observation observe link nuclear dimension scale exhibit qualitatively compare unconstrained square sketch poor solution exhibit near optimal classifier goal collection correspond individual x since classifier conjunction detail database expression neutral pose example image
improvement integrate element observe efficacy preprocesse layer beneficial concept small large scale visualize dimension time second although proper epoch result contain four field pseudo randomly interval trial last speed accuracy cluster time second example consider virtual agent
boundedness treat ensure tight result agree study cc theorems equal bias show figure obtain unbounded svm especially label contaminate unknown world thus estimate contaminate outli reach contaminate indeed non contaminate give satisfy worst admissible classifier acceptable search validation experiment let property calculate case classifier properly term advance however statistical prove property give sort negative margin statistic linear order property estimator investigate mathematical reference therein detail population eq denote nothing accord express three random omit suppose zero
crowdsource crowdsource design suitable mechanism expand scope model human crowdsource system low appendix proof employ contradiction base axiom q upon axiom eq yield desire contradiction property include choose property decrease lx w lx x axiom towards increase function satisfie axiom investigate convexity hyperplane mean proposition corollary crowdsourcing inference truth optimize function maximize
interest size complexitie theorem bn compare improvement aside logarithmic factor passive active conclude bn bn bn constant passive aside factor suffice active low passive learning aside hand bound ignore satisfie bernstein class gap bound unclear necessary sufficient improvement bound improvement reveal improvement interesting active learning since indicate quite passive logarithmic passive aside factor upper bind passive learn roughly roughly know model finally agnostic gap unclear range passive aside passive replace disagreement section base relate star logarithmic include bound know disagreement split x h substantial literature label complexity various ask bad maximize respective result last improvement express study family active certain quantity quite diverse pair inequality relate attempt relate literature isolate give plug bad behavior relevant star maximize collection definition entirely complexity additionally role proof diverse literature establish devote together represent summary know relevant present discussion measure star minimax complexity implicit literature case compare additionally complexity passive learn loose argue maximize low implication star relation measure star summarize literature thesis disagreement define let let er xy er hx xy bind label complexity learn effective variant reader thorough disagreement b thesis well work characterize disagreement various survey thorough survey disagreement bad disagreement coefficient survey ex ex general date disagreement survey survey survey detailed description well know logarithmic ex ex bn ex ex ex ex general literature advance understanding capability learn section ex ex ex find quantity connection play proof ex ex ex upper offer ex ex simplicity logarithmic ex bn replace ex offer refinement case ex I case replace theorem appendix refine analogous ex desirable bound case near intuition behind achieve rate approximated objective reduce consistent classifier observe far classifier obtain fine grain pair eliminate request worse eliminate classifier inconsistent guarantee eliminate separate arrive space return apply remain allow separate eliminate lead eliminate g hx gx x xy find request capture label request unlabele abundance unlabele potentially increase whether though intuitively however comparison section ignore logarithmic
define define project feasible backtracking project detail also implementation completion variation minimization backtrack search signal formulate unconstraine unconstrained generalize difference descent feasible differentiable solve backtrack implementation graph signal variation signal initialize stop backtrack search cost decomposition matrix discuss general focus whose corresponding minimize nuclear norm study connection f tr tr r cyclic total word rank naturally f frobenius equivalence variation relate nuclear norm nuclear potentially minimize rewrite shift insufficient cause return quantity vector smooth also quantity signal belong subspace span coefficient follow spectral signal belong recovery graph shift cyclic graph shift identity robust shift index matrix anomaly
learn visually especially ba er graph ba evaluate recover position testing average ten three graph competitive rbf ba graph c c ex measure ex precision well understand behavior number gaussian rbf behind behavior uniform entry decrease opposite increase trace tend zero decrease implicitly laplacian b lead ratio dominate fidelity ratio fig learn evaluate edge increase graph intersect reach peak keep drop edge similar graph match combination random rbf graph investigate present respectively different number signal initially deviation signal deviation random temperature measuring show period I month
dictionary middle threshold impose refined digit efficacy model half layer digits layer row furthermore refinement much excellent interpolation result upper digit reconstruction htbp htbp analyze face category deep dictionary infer discuss map fig third layer similar mnist interpolation dictionary max pooling part accurately face detail second layer dictionary deep convolutional testing via project layer deconvolution
provide rate give hoeffding rate regardless empirical risk classical frequently use theory generalization rademacher complexity inequality value domain difference eq show condition generalize domain domain condition furthermore similarly inequality coincide one match inequality manner generalization number vc dimension distribution k tn q show eq characteristic denote recall evident q q another clear result hoeffding range proof ready prove theorem law iterate z respect analyze one representative measure two meanwhile domain adaptation hoeffding
dependence recursive specification symbol note procedure sufficiently long elsewhere include sake seek symbolic sense infinity within string derivative turn reach general state merging process simultaneously symbolic derivative fail already encounter create state find match merge two string crucial seek stre right extension carry split ensure find error call lf observe consist trace set consist string large depth infer identify convex hull mapping qx recursively symbolic q qx terminate necessary connectivity initial run direct symbol read move arc generate normalization recursive synchronization step infer short large approximation upper step approximate arc normalization traversal traversal counting assume row state separate multiple however state algorithm modify two identical discuss carry efficiently use space complexity assume computation involve encode string inspection string identification traversal count normalization arc sum bound symbolic distinguished complete refer probably approximately pac said target output metric language class language efficiently pac learnable establish probabilistic appropriate strongly symbolic denote g define therefore string sequence satisfy triangular define symbolic derivative combination correspond class learnable pac finite long generate estimate runtime initial identify extension occurrence visit right
interesting cognitive posteriori hmm estimate develop problem relate reduction solution determine active reduction examine uncertainty reduction solution inference hide base improve information obtain costly beneficial inference notion active active optimize model principle specifically assess analytical simulation excellent within rest active respectively analytical finding inference follow discuss relationship inference direction realization noisy alphabet write give general overlap average overlap request
pyramid l cnn acc fc ap yes ap fc baseline yes baseline fc fc fc yes pool yes fc yes baseline acc part fc yes fc yes fc yes yes fc yes mit pt l description ft bb cnn fc yes baseline fc yes baseline fc baseline fc yes fc yes fc sp fc yes fc fc yes baseline bb cnn map yes yes ta fc mlp yes fc yes fc art ft plane car cat
produce similarity kernel function look angular project angular nonlinearity instance space similarity representation robustness induce similarity I instance vary discrete normalize anchor representation variation robustness objective function raw distance towards thus dominate objective kind fisher differ compare induce first specify proximity statistical generalize system metric coordinate induce differential map equation differential metric differential metric width difference
encourage logistic technique dc programming concave pointwise compute non zero entry complexity leverage simplex control sparsity heuristic justify perform reduce vector show word whenever enough operation use bounding box positive window box construct window htbp dataset bias htbp c c car cat cover cover multiple design bag
frame music iii iv chemical services environmental school bss use matrix remove word letter default try score full select dataset bss leverage score leverage bss list occurring word supervise bss score five five table five category belong bss document music bss select closely good error selection dataset result fig bss leverage score selection full
explore material describe model nmf nonnegative complexity technique magnitude audio signal simplify take seek approximately decompose factored form index prototype spectra combine activation equivalent maximize eq unobserved observe fill white thick f thick f edge z pt right
dot mathematically tractable dot draw attribute enyi latent process inference author count include operate dynamic temporal stochastic et al temporal extension sbm memberships sbm main author change specifie combination gibbs sampling algorithm demand base dynamic inference procedure hyperparameter estimation investigate approximation priori estimate discuss iid follow scale distribution gaussian observation variance dynamic available noisy evolution linear dynamic transition apply previous gaussian entry process matrix noise unlike construction evolve correlate manner state either estimate system observation define graphical linearity kalman
stage diagram explain e step classify simple actor interact complex actor person interact web page weight assign actor type average secondary scenario transaction activity either entirely count counting case interface interface entity user interface complexity htb complex actor count many actor degree add product multiply weighting add adjust value factor table assign project essential
derivative output obtain complex composition form dag variable argument implicit argument dag dependency variable bipartite acyclic evaluated iteratively sort depend come scalar network interested derivative output dag modify remove make evaluate dag likewise respect recursively accumulate feature layer easy integrate variant visual pre easily implementation cpu gpu version contain require gpu available cnn implementations capable cnn language google convnet project architecture file convnet somewhat individual block matlab gpu convenient toolbox implement cnn seem general purpose framework toolbox several computation library dependency thank
time inferior lstm cope lag transform rnn output sequence via linear analytically train therefore suffer lstm synthetic complex fail dependency optimize attempt hessian adapt rnn allow rnn solve term lag performance number great network training multiplicative rnn multiplicative unit factor way reduce training day graphic filter lag strictly temporal stack
trace penalization determine pair start op ij ij satisfied include solve singular unfortunately particular replace efficient algorithm pca significant relaxation heuristic truncate power psd provide accurate solution rip algorithm case step truncation denote truncation consist tolerance ab ab ab b b ti new active find small termination algorithm algorithm instance add type method maximum suboptimal component reduce thereby select solve matrix small computational bottleneck condition hold evaluation thin rip hold multiplication total cost reach complexity would warm run keep track fast reliable solver discussion go beyond report low rank theoretical assess several pca numerically add mean error dimension numerically take measure square constrain form norm projection noise fast constrained atom confirm summarize dimension trace depend increase confirm capture certainly set correctly solution z kk second set theoretical atom overlap roughly dimension regularizer sum atom bottom right shape curve support atom overlap case consistently outperform suggest linear support factored covariance form add k right make psd form method compare basic presence problem
small entail neighborhood dd state two proceed random norm hold sub lemma exponential gaussian norm I tt jk ik jk jk jk deduce constant obtain e hence last component wise cauchy schwarz derivation yield e argument lipschitz set assume implicitly yet misspecification application investigate leibl principle dimensional asymptotic expansion principle misspecification generalize dimension suggest logarithmic dimensionality complexity establish general misspecification kullback principle rapid advance modern technology throughput set genetic fmri functional datum economic finance frequently make contribute
trait outcome trait parameter estimate rapidly trait brownian along tree trait assess uncertainty space development different along branch thus naturally development area lead model dependent evolve variable aspect explicitly tailor assess trait history assess combine threshold trait unobserve brownian trait spend current interpretation outcome underlie represent effect number genetic factor evolution usually model brownian diffusion trait assess diffusion threshold create continuous development trait evolution trait presence possible importantly trait correlation control trait share evolutionary trait analogy inference association molecular pi less development unobserved tree degree degree independence refer
matrix level remove representation alternate projection see user heavily speed recovery complexity rank e number stage drastically number iteration require convex denote approximation thresholding initial km ts remain result correctness incoherent row respectively row unique sparse e ensure recover recovery result tight constant sparsity sparsity exceed sparsity
near optimality class problem translate alternative cover every around near optimality dimension fact another interesting entire drastically arm many application notably complex position portfolio management switch might formally node bound lem number thm lead exist arm context policy process notice extension observable mdps ergodicity mdp tuple p mapping pair action find long formally receive parameterize policy induce mt relate transition policy initial sequence ergodic reward goal find policy coincide mdp cover sect notably correspond search space mdp special context sect evolve time determine reward state history translate sect sect mdps sect simply advantage work policy finite mdps thm sub regret average transform accumulate readily continuous restrictive size extend comparison use strong reward globally finite
signal dominant font font legend style font system challenge extract analysis conventional perform maximum mutual information hmm clean read news corpus clean multi condition hmm technique know well alignment hmm implementation maxout accordance extract static
construction asymptotically optimal furthermore discount mdps problem keep denote matrix last action dynamic subroutine take action inexact nature near tv ta x state result meet special case assumption rate mean rate reduce study variance question directly ucb concern equation regular solution cost optimality action exist abuse call sometimes necessary guarantee condition continuity state boundedness
imply recursive formula look obtain calculate resort iii treating case ii closed form inequality hold calculate resort case treat proposition microsoft lin minimization concave saddle propose dual dual primal perform accelerate parallel extension weight theoretically arise often regularize minimization convex loss regularization predictor solve problem associate label linear vector machine hinge loss regularize regression obtain linear problem lasso regularize erm book especially interest develop algorithm case evaluating incremental operate component extensive incremental method gradient batch iteration complexity precision quantify complexity
functional independence graph interest definite inner product operator precision element correspond absence indicate convenience abuse notation edge present dependency prior matrix scaling evaluate hold distribution multivariate wishart wishart coincide importantly doubly intractable return implication wish perform gaussian around outline approximate wishart update either clique substantial making
automate start linear association follow learn use observe direction source spectra vary invariant feature spectral variation hundred thousand tf point source aggregate observation formulation lead problem problematic firstly case prohibitive secondly clear regression natural speech common source content frequency source long period devise recently use compose association noise ref inverse extend tf posterior direction gaussian covariance show perform localization separation consume runtime head mechanism collect association position keep position experiment head onto theoretical introduce elegant associated truth marker manually hold front camera place horizontal image head camera setup locate accurate direction quantify propose organized section sound localization mapping onto sound extend input section obtain localization source draw direction head setup record series direction sound sound center frame angle capture single sound static source setup static sound source sound subset possibly sound source general much set acoustic embed
simple classifier possibility classifier classifier use nlp consider sentiment grow small manually label occurrence compound sentiment table test outperform specific purpose specific book see opinion mining review e actor camera lot target challenge sentiment different language specific sentiment analysis english reason english visit internet internet http en wikipedia language complexity country even country widely develop sentiment lack sentiment dataset sentiment end book rating star comprehensive experiment expand set also extract domain sentiment training help space reproduce
accord trial belong sx sx classification discriminate belong source domain experience decode subject subject variability across situation share multiple train underlying idea try capture try instability diversity combine prediction ensemble learn decode set subject partition training one represent diversity train
issue extensive quantile return examine use sample period median box version throughout directional stock stock quantile stock bootstrap confidence replicate quantile lag low lag mean likely positive next half box significant quantile stock median return return lag forecast forecast quantile stock stock return note result first lag cross lag lag less likely half figure box stock return stock previous trend absolute imply quantile increase loss year significant quantile gain next year figure box lag quantile
fact follow hold follow innovation associate latter product appendix value satisfy integer kullback distribution form n duality recursively note nh also bias coin flip probability since measurable conditionally history repeatedly unconditional n concentration size lr kl condition thm fact assume imply rule gain
vc coordinate coordinate coordinate see flip immediately close desire build contain union complete inside contain collection queue comprise collection projection queue collection jump onto swap produce empty complement usual proceed iteratively canonical class consider maximum eventually within check whether project onto contain project retain recall collection height connect sect filter soon clear cube cube construct maximum filter subsequent early vc binary cube vc maximum vc yield simple picture embed compression trivial serve vc class table nd project maximum vc cube argument straightforward complement vc cube union maximum possibility two verify component component must apart diameter check vertex verify symmetry cube
acc rmse acc auc acc acc acc auc acc rmse auc acc rmse pt department university university calibrate calibrate prediction particularly machine present parametric algorithm learn apply step learn make wide select scoring measure second advantage exist calibration convert discriminative probabilistic posterior svm calibration calibration prediction I assumption method nb reduce remainder
minimal candidate program already trace minimal intermediate candidate minimal p either element language return trace result last last p last since element set empty verification engine language element stop yet minimal single progress parse infinitely case long progress intermediate nm mp fp p terminate program synthesis engine verification engine return history synthesis history program amenable long synthesis history synthesis arbitrary history powerful language program language second language language candidate program language form consist
field context aware interaction assume preference interaction interaction way include argue model train recommendation preference modeling perspective item preference focus interact interact additional justification compatibility interaction classic influence interact item dimension user role rank interaction additional context interact context interaction help well context follow scheme early indicate text green layer either preference dimension thing dimension one novel novel context preference divide user reweighte context dependent weight include simultaneously preference dependent interaction strongly affect interaction solely minor item relation context recommendation bias certain bias composite reduce interaction treat reduced way model context omit dimension interaction set reach scope nonetheless cm user bias item bias interaction six five perform traditional case interaction good case intuitively sound perform assumption preference model second interestingly difference heavily noise member much reduce pairwise pairwise
result pass randomly normalised check sure respectively linear give predictor normalise source configuration already prove correlation mixing normalise make stepsize fig define statistic generalize accommodate white bss mixture assume
identically past expect accumulate exploration new arm arm arm know optimal slot high difference accumulate reward good arm profile decide instant accumulate reward regret balance tradeoff show arm accumulate accumulate loss unbounde logarithm ucb mab several arm ucb prove file cache cost algorithm order prove order special number play content place content cache content popularity advance file cache user content cache memory cache user readily cache want rate request cache otherwise directly carry user example background user request either cache observe request store cache cache memory unit file file file divide instantaneous demand file request file period normalise maximum serve demand bound support popularity request cache file consider reward unit file gain user
cm certain transform reaction reaction diffusion j measure cm fractional cm j reaction journal cm calculus fractional calculus transform instability multi medium study apply mathematics growth journal fractional build output mechanism reaction reaction production physical technique describe tail law stochastic
access regard past observe policy execute round distribution nature choose th policy round round maker dm select accord nature dm issue mdp accurate value expect
propose new var call hierarchical selection regularizer provide shift obtain generally nonzero lag motivate goal interpretable model flexible computationally method lag lag procedure early attempt square criterion develop grow lag hold size work information criterion tend whereas tend despite tool lag practitioner approach typically lag reduce economic justification dynamic component forecast false specification add accord predictive inherently offer var specification utilize aic lag selection approach component specification
appropriate suppose allow grow also recall close neighbor index recall close multiple rate want validate rhs rewrite sign close pr pr inclusion rhs q rewrite use substitute expression rhs rhs eq empirical validation bind empirical term multiply term convert linearity apply know length range many term partition subset hoeffding place probability draw use
criterion contaminate outli estimate laplace skew approach noisy spurious cluster contaminate approach asymmetric aforementioned robust cluster model cluster use help point spurious noise discuss contaminate preferable herein contaminate
prior plausible accept parameter abc accept bayes estimator sufficient statistic practice statistic determine rely sufficient low problematic continuous compare summary set p sx generate set accept discard accept I I tolerance evident simulation study implement hasting mh target approximate distribution highly draw upon mh proposal incorporate smc technique propose partial control utilize rejection mutation incorporation mutation improve abc ar
exploration paper aware recommender validate series name compute accord situation recommend desire read interesting current case orient avoid ucb exp ucb mobile paper review involve illustrate conclude point bandit algorithm consider recommendation work dedicated propose base experience preference user preference period association concept profile experience combine describe three complement highly book social among user combine spatio quality recommendation device explicit rating rating proximity natural mobile recommender people read behaviour predict combined preference content provide mobile author
weight available matlab author home denote entry th standard entry remain singular row leverage score leverage replace coherence coherence sampling attain whose th equal row q leverage first two prove row rank leverage weight therefore leverage standard minimization complexity norm nuclear problem nuclear approximate study literature uniform later k th probability exact coherence norm model completion complete whose accord align jj devise paper tb partially matrix initialize perturb let matrix max factorization model alternate poor coordinate
section conditional hidden output face total entry py positive arise reinforcement observable namely state process bellman manifold conditional come feedforward arbitrarily approximate arbitrarily feedforward output sigmoid model arbitrarily feedforward network hide unit feedforward linear threshold example feedforward input output follow boolean function f k km class deterministic distribution arbitrarily hide unit k km paper give description capability conditional boltzmann machine relate restrict boltzmann theoretical rbms trivial study unit imply finitely choice bias quality input unit kullback form universal tight depend attain complexity star improvement worth open analytic integral upper bound expectation draw
lc depict manifold may solve sum one locality coding albeit intrinsic purpose embed manifold aside purpose embed exploit codebook learning explain dictionary analytic locality code represent point surface rank four square b locality code contribute neighbor get manifold initialization dictionary I ip ni lc lc f f dictionary label dictionary tie generate fed base like classification generate code code atom jx dirac efficient utilize residual residual eq class could like preliminary high compare aforementioned alternative solve mean combination represent project close point follow principle completeness closely adjacent intrinsic atom show formulation code solve scope coding efficient learning manifold n elaborate usually employ obtain common depict learn write convexity inspire set break update atom eq minimize r project manifold detail pseudo atom dominant visually informative
choose optimize divide program run incremental hold exception optimize rate code gradient backward optimize couple day smoothed model avoid knowledge internal nonetheless smooth I broad tuple token token distribution number child independently smooth support token manner low log scope test word length replace j word length word replace j j author microsoft research amenable statistical possibility learn become task level potential first code rely primarily integrate offer suggestion massive source public indeed show even improve idea suggest basis code enforce programming language code representation purpose visualization hope learn program programming program task develop tool improve
predict situation category within invariance preserve consistency loss state hard q bootstrap variant term allow predict object unlabele handwritten face train consistency refer describe consistency soft bootstrap reconstruction detailed section mnist handwritten digits degree figure vary label training model mini sgd architecture linear work soft consistency network layer initialize could quickly bootstrappe phase bootstrapping provide significant bootstrap nearly soft
scene affine transformation human mesh smooth generator representation baseline object lot attribute minor lack c illustration run image buffer mesh image close complicated program combine scene generator scene affine scene generator scene configuration induce mid simulator engine formulate interpretation inverting simulator observation widely source numerous graphic simulator drive put rich world contour recent mid
transformation neuron neuron neuron neuron neuron proceed achieve proportion explain compare regression performance rest neural backpropagation compare linear realistic significantly high one compare explain look plot model fig residual fit term besides normal two big curve end plot partial residual effect rest assumption explanatory establish log transformation improve quality still violate verify novel dataset b backpropagation begin analysis dataset find nine attribute plot value substantial drop indicate big
simplify imputation average estimate complete datum variance variation simplify create member
efficiently satisfy require perfectly predict unit intersection homogeneous normal hide incoming weight since input hence intersection hypothesis feed hide single hypothesis normal pac learnable show feed minimize class predictor different weight weight many hence feed capture behavior capture network expressive indeed know network minimize even unit
historical sensor network task without important reason dynamic change rapidly desirable operate environment collect water exploratory initially operate system newly acquire environment researcher accurate meanwhile task prescribe learn low system sensor unable correlation ensure connectivity requirement fulfil sensor hardware resource discover use communication internet introduce motivation support make learn ai limit human intervention drawback consider use technique sensor framework considerable hypothesis consensus trade specifically requirement system employ centralized resource unit speak intend capability bound full control process past decade advanced machine survey machine discuss machine wireless ad hoc network application tree communication specialized survey learn development outli proper action take meanwhile discuss intelligence challenge fusion scheduling intelligence branch focus inspire fuzzy survey decide instead variety strength provide comprehensive roughly unsupervised distinction survey way learn work discuss learn challenge encourage lastly survey classify compare effort provide researcher interested explore research introduce section review effort address localization medium essential determine enhance behavior requirement security specialize difficulty section comparative guide introduction wireless sensor sensor characterize collection create expert recognize rich pattern understand beneficial machine numerous flexibility benefit provide concept adopt context exist intend structure algorithm reinforcement supervise learn learn classify group cluster investigate third include reinforcement agent interact environment online machine characteristic supervise learn hybrid term supervise aim strength category adopt section omit please therein thorough discussion predefine input output represent parameter fact extensively medium security call output
forecast contribute monitoring protocol wind future next author discussion environment project generation wind author analyze introduce analyze load wind physics properly adapt multivariate comprise velocity power fluctuation produce wind langevin drive wind
unimodal operator limit issue adopt alternative generative neural estimator transition alternate main landscape although restrict boltzmann softmax fine specific autoencoder begin present overview include autoencoder autoencoder generative autoencoder autoencoder training autoencoder feed neural aim minimize input
way partition possible modal cluster scenario adopt use wu generate partition either u u last partition identify choice consistent parameter importantly cluster observation close true simple bayesian cart need one ht u show identify interesting deviation mean result count generation randomness reflect quite realistic heterogeneous datum form scheme overlap partition u u u
v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v stroke v v v v v v v v v v v stroke v v v stroke v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v stroke v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v stroke v v v v v v v stroke v v v v stroke v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v stroke v v v v stroke v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v stroke v v v v stroke v stroke lt v v v v v v v v v v v v v v v v v v v stroke v v v v v v v stroke v v v v v v v v v v v v v v v v v v v v v v v v v v v v v stroke v v v v v v stroke v v v v v v v v v v v v v v v v v v v stroke v v v v v v v v v v v v v v v v v v stroke v v stroke v v v v v v v v v v v v v v v stroke v v v v v v v v v v v stroke v v v v v v v v v v v v stroke v v v v v v stroke v v v v v v v stroke v v v v v v v v v stroke v v stroke ltb def exch exch exch def roll exch def exch mul mul sub mul mod ifelse ifelse ifelse ifelse ifelse def constrain lt exch ifelse def copy exch add constrain roll mul exch constrain roll mul exch add constrain roll def exch roll exch roll exch def rgb exch mul exch exch constrain roll mul mul add exch mul roll exch add exch mul exch roll ifelse ifelse def def gidx gidx gidx gidx loop def gidx get sub gidx gidx get def gidx gidx gidx sub add def gidx gidx sub gidx get sub mul gidx gidx get gidx sub mul def gidx sub le gidx gidx ifelse def def def pm mul ifelse mul def def pm exch def stroke constrain exch cf constrain exch def ifelse pm pm exp def ifelse ltb ltb stroke ltb stroke ltb stroke ltb ltb r v stroke ltb stroke ltb stroke stroke ltb stroke ltb v ltb stroke ltb ltb ltb lt v lt v v v v v stroke lt v stroke v v v v v v stroke exch exch exch def mul roll def sub mul mul mul mod def ifelse ifelse ifelse ifelse ifelse def constrain lt exch ifelse def copy mul exch mul constrain roll copy mul mul roll mul exch constrain roll rgb exch roll exch roll exch exch mul exch exch constrain roll copy mul exch mul add exch mul constrain roll mul mul mul roll def eq ifelse ifelse ifelse def gidx gidx gidx def gidx get gidx gidx gidx gidx gidx gidx get gidx sub def gidx gidx get gidx mul add def gidx gidx gidx gidx def ifelse def ifelse pm gamma stroke exch def cf constrain exch cf exch cf constrain ifelse stroke ifelse ltb ltb stroke stroke ltb v stroke ltb v stroke ltb ltb stroke stroke ltb r stroke ltb stroke ltb stroke stroke ltb stroke v ltb r stroke ltb v stroke ltb v stroke ltb v stroke stroke ltb stroke stroke ltb stroke v ltb stroke ltb ltb ltb lt v v v v v v v v stroke v v v v stroke v v v v v v v v v v v v v v v v v v stroke v v v v v v
arguably happen solve spurious fw employ small time consume large sampling convenient cardinality across analyze minimization linear gap
inform learner select exp exploit factor induce feedback direct probability term dominate current computing dominating inform dominating dominate minimal logarithmic yet another proving operate feedback another call internal randomization ifelse b dominate tw tp ti ti ti see algorithm inform run time compute direct induce index since adaptively dominate choice feedback variable cause tuning reason exp slight describe appendix trick direct satisfie trick importantly graph whenever dominate system step exp dominating set regret exp combine bind expect inform set moreover quick comparison reveal feedback advantage inform symmetric factor derive informed direct theorem inform also bound unable bound exp exponentially weight programming action fact program name regret bound utilize e instead use weight action oppose decrease merely affect ifelse generate loss generate simplex w tw w ti p ti I ti ti tw sufficiently notation
reduce step communication ignore last globally ignore involve scalar al lin al demonstrate elimination shrink technique heuristic condition eliminate eliminate multiplier keep optimality eliminate primary behind shrink contribute working variable sample eliminate present condition previously lin shrink overhead condition possible previously eliminate eventually definition conservative approach decide likely essence difficult execute lin shrink reasoning value execute shrink discussion heuristic shrink reconstruction ensure previously eliminate positive step update gradient reconstruction shrinking correspond multiple eliminate gradient sample communication negligible heuristic
scope take largely literature generators generator interpretable different drawing generator last induced generator game payoff dispersion player arm instance match iteration take cpu year upon conduct reward conclusion sample mean confident conclusion way check experiment run experiment confident percent interval fact serve sufficient evidence repeat verify summary statistic experiment take day computer time bootstrappe datum experiment point subsample create form confidence distribution subsampling would diversity moderate distribution would explain bootstrappe summary also check distribution gaussian mean distribution ks vertical vertical analysis unless significant monotonic pair game relationship variable interested study convergence issue appear converge frequency exhibit whether early empirical profile mixed establish nash exact checking multinomial unfortunately situation expensive fail large balanced typically close calibrate incomplete ground truth positive vary roc concept distribution metric quality dominate quantile high quantile maximize dominate value attain probabilistic check probabilistic perform curve everywhere dominate latter higher shift discuss begin fundamental action mix formally record I average reward detail begin accord generator explore examine game across equilibria action consider different raw iteration game reward
kl k ff inequality give summing give knn cauchy note logistic accord convergence depend rate e constant c since k k hence n ns ns argument complete class ahead indicate point significant speed set follow large intuition taylor origin run benefit compare randomly datum generate dimension keep vary median iteration algorithm quite per almost linearly size degree speedup regularization dominate second regularization
book sake occur change book bid ask count resp mid resp order bid move price limit order resp bid move count bid move stress price move correspond occurrence market bid place bid ask distinguish handle consider characterize intensity kernel code intensity bid mid unchanged move intensity estimate book impose bid ask symmetry p estimation estimation scale method retrieve indeed unstable allow retrieve symmetry provide http www book keep year file list limit well ask book micro second precision thus compute ask bid market limit limit mid move shall book depend size asset large small asset asset summarize c simulation
counterpart regression ignore high persistence state encounter scenario simulation ms ms lin viterbi relate ms lin predict regime ms result exploit strength flexible ms strength inferential particular ease interactive formulation selection explore detail along validate aic formulation local class less price outperform need need regime apparent plot regime address fail short day variation interesting explore regime persistent autoregressive capturing
consider infection incorporate period parametric fit able infection epidemic individual remain arguably know epidemic call infect individual state contact individual infect remove infection proportional individual multiply give action individual recover take despite g disease majority assignment parameter advantage manifold avoid result fit often simplicity parametric often tend homogeneity mix well specify non allow
transform shape fill bl dot cm dot left l cm l l cycle leave right node cm l node distance h scale style shape circle line pt inner fill bl dots end dot style shape circle line width cm bl dot h cm r cm h node p rs cm probability distribution bottom top composition originally represent layer lk n lp define conditional feedforward output feedforward bias clearly conditional
bridge target conditionally unbiased diffusion bridge bridge algorithm bridge one simulate intersect simulate bridge go diffusion soon equal independent approximate mcmc nearly approach right calculate expectation produce diffusion mcmc sample simulate euler discretization avoid simulate multi version transform diffusion bridge simulation simulation exact even computational approximate coupling simulate approximate mix bridge simulation paper dimensional generalize bridge method process give wiener bridge simulate diffusion term see standard decide coupling process follow discuss meet manner meet simplify assumption diffusion ignore influence sufficiently assume diffusion dimensional wiener wiener brownian start brownian onto plane orthogonal pass process coupling occur q find alternative brownian motion plane plane
thus condition learner consider instance j learner index training equal tree last equation learner vice versa flip well exponential cut plane cutting constraint solution original start add qp continue program cut major bottleneck lie combinatorial combinatorial solve briefly optimization associate speed instances break maximization restrict order instance optimization problem compute search reader analysis decrease decrease q material initialize learn optimize range else cache access meet weak structured svm cut weight cascade adjust node partial auc cut plane mn problem svm violate learner cut plane weak consist step learn decision fast bin cost total sample call structural cutting cost scale linearly algorithm cost cut plane complexity spend train learner table compare complexity adaboost discuss adaboost next discussion adaboost detector identical adaboost adaboost play adaboost calculate u indicator minimal point subset instance tight instance position
wide cluster smoothness kde make construction powerful algorithm mode dataset laplacian mode mnist acc pt n dataset mnist handwritten digit object vary angle topic semantic category appear keep statistic collect pixel randomly special laplacian search spectral implementation nmf cluster walk mode normalize several graph laplacian graph build let select run respective hyperparameter acc run clear algorithm smooth superior demonstrate importance cluster perform poorly laplacian mode close good criterion find exist range hyperparameter homotopy decrease c
previous retained carry separately central activity pattern among select accord span area eeg signal ec investigate separately outcome perform test feature provide framework preprocesse describe section psd previously psd measure single region extract mahalanobis distance detail carry test condition preliminary report false maps channel psd ec feature j eeg head code axis map psd feature ec already eeg study alpha evidence close brain reflect genetic ec well
knowledge rate convergence strong analysis fail course limit guarantee final bind study believe intuition source improve standard thought weight non paragraph average result average without strong assumption algorithm utilize grow oppose unweighted set nice standard effectively control favorable one problem work deviation interaction algorithm predictor index potential author differ algorithm plausible weight weight justify weight single sparse feature weight magnitude easy sub strictly
gate quantum fix complexity serious advantage perceptron process superposition superposition information extract via quantum quantum discuss introduce variation quantum perceptron weight read consistent digit binary w u x control control parameter additional sign shift iteratively adjust
motivate improve termination beneficial decrease accomplish project vary learner relevance explanatory rely automate constitute methodology specification capability dependencie advantageous explanatory neighbor explanatory accord nearby probability weight intuitively value class different write increase positivity relevance n w experiment strength form weakly stock price indicate substantial furthermore boost procedure ensemble section identify important predictor consideration project demonstrate explanatory purpose visualization particular low toward indicate explanatory high predictive variable vary efficacy stock return summary explanatory b b b great period consideration
neural network without precisely get large boost supervise fine tuning conclusion put optimisation da train low yet classification minima autoencoder manifold though loss autoencoder way primarily stack would improvement diverse representation composite autoencoder proceed local explore dropout comment supplementary material sampling change batch try variant value setup unit first method much da experiment evidence denoise sequence success supervise tuning completeness try encoder layer size
observation spend require size respectively correspond analyze author interest mean unique note enyi I n htbp confidence level obtain interval summarize statistic interval wide confidence day presence see function despite significance ci ci ci ci ci first eq day fact notice day possible day dropping outlier approximate new outlier day comparison n detect lack study examine cc pdf fig
hermitian henceforth bs decode bss overlap sense sample b nonempty bss bs bss bs collect bss full practice bss whose sample helpful intend traffic simplest bs traffic diagonal sec bss overlap bss belong jointly user network architecture proximity site formation may gain pattern costly intra traffic hierarchy mode mcp simplicity exposition sec traffic proportional nonzero mcp favor position inspire advance mmse entry absolute discussion seek f control incur nonconvex norm absolute bs amount traffic site bs user incur even entry decoding
keep triangular nonnegative distribution low triangular loading concern continue prior property row inferential setting comprise factor row associate factor comprise nonnegative
initialize small train phase call train algorithm know item description pass along return weight line perceptron element initialize normal mean perceptron learning rate store yet initialize line train detect fail convergence detect fall specifie improvement else rate differ perceptron instead single held line perform phase perceptron refine together order ic ic backward f ic completeness train stochastic gradient backpropagation individually present conditionally line item
autoencoder validate effectiveness autoencoder useful supervise autoencoder dropout tuning noise large autoencoder purely gaussian noise mnist interaction unsupervise supervised suggest optimize stack type relative autoencoder one layer train input accomplish network object recognition autoencoder map hide encoder yield autoencoder typically square autoencoder autoencoder train reconstruct clean corrupt
support innovation global department contract er office program fa motivate whose b introduce form note equation bandwidth sparsity semi semi equation would variable extend separable rank variable surprisingly length
cut plane cp project subgradient training compare cutting method surrogate involve breast task dataset uci repository training average test use buffer length buffer perform pass c letter c cut plane task vary buffer gradient method accuracy within second even cut plane well trend measure second cut plane could achieve accuracy epoch
dynamic stay step policy content gate last directly properly expect gate regression result respectively conclude neural feedback connection reinforcement selective internal certain rapid shot stack feedforward filter feedback certain filter guess internal maxout selective internal art consider acknowledge include relate convolutional cnn neither try human strategy cnns
n sign lem fx lem immediate result moreover lem let p p proof easily need confirm consequence integrate virtue matrix solution recover furthermore let probably except result keep bound detail quasi
dataset truth actually small overcome ground truth generate type item user type truth rating user netflix yahoo music issue real dataset truth evaluation issue usually solve use reject known restrict choice time rating per meaningful comparison miss rating obviously live people full experiment mind allow update recommender policy need decomposition precisely item current unknown rating evaluation would methodology interest currently evaluate bandit bandit uniform strategy dataset provide yahoo available able rating netflix policy think methodology contribution paper
employ four mini batch sampling batch obtain ratio since tail speed incorporate several explore handle state context portfolio management finance business pt study constrain shortest consider conditional locally employ four importance incorporate procedure along utilize cost objective necessary first second mini spirit monte relate rare importance variance constrain attract lot recently learn unlike previous mostly return measure conditional
flexible analyse market goal market reason market understand mechanism objective market objective market find market aim motivate would analyse whole unlike agent measure market result paper establish strength mathematical giving aim market trading sequential optimisation connection market learn prediction market associate specifically payment unit security pay state require vector subset linearly security space discrete contain th continuous practice trade agent share security portfolio payment essence call
dnn much dnn much select hidden choose size slight frame trend frame suggest network objective theory deep counterpart act difficulty deep deep understanding compare system final component decompose hmm dnn layer dnn dnn largely substitution rate constant small component overall decrease substitution fairly quality confident matching audio component link system vocabulary dictionary capture variation tb analyze group understanding act root leave analyze hide analysis height bar bar break classification base error phone network root three base phone base phone combine set phone classification non dnn correctness spread substantial base base base error fairly base occurrence generally exhibit pattern dnn acoustic nature observe phone find model gradually across large improve expense tb performance level hmm speech decode yet address question architecture achieve describe algorithmic necessary prediction setting offer analysis aim first pattern compute forward dnn non compute fraction example plot sort sense activation help code code researcher learn hide active size unit layer equally share code perfect dispersion would flat sparsity representation axis size deep dnn layer every dnn case activation decrease deeply dnn per deep layer suggest transform compressed dispersion particular dnn fairly flat percentage typically use recognition result relatively dnn previous speech maximum training experiment dnn serve discriminative task dnn generally neural dnn acoustic drive unsupervised yield speech benchmark appear modern dnn neural approach
storing segment calculate high store update primarily still detect section heavily whether examine pruning partitioning apply prune neighbourhood call functional pruning segment neighbourhood prune functional pruning partition version functional pruning efficiency show slightly condition c part depend define segment thus q recursively return last vary firstly problem store update store empty define cost ft update candidate change least leave hand
close z since constant moreover since round therefore programming detail supervise annotation secondly solution annotate sense label annotation correspond every descriptor incorporate modify discriminative explain domain label assignment set subject link center quadratic b constant admissible due avoid class fraction ideally hard proportion problem everywhere except column make operation sec intractable dynamic programming modify constraint minimal avoid trivial objective desire incorporate multipli vector still heavily unbalanced towards deal unbalanced dataset square weight
neighbor popular underlie lsh lsh functions property domain function high formal hash mapping locality sensitive family sensitive satisfy task search query lsh provide mechanism create idea hash function meta lsh lsh need independent meta processing assign hash retrieve element whose query lsh element probability family function one query lsh query preprocesse existence lsh translate sublinear nn note lsh lsh near search dimensionality lsh widely popular present lsh scheme lsh family generate hash eq dx cumulative euclidean distance lsh part lsh
generate question sense properly ss follow condition ns argument model say assign mass strategy posterior vanish assume lemma simplify eq partial convergent since vanish ask posterior concentrate important answer suppose theorem event accord hold former consequently two get let entry min condition say give unnecessary mass together converge bayes start joint fractional joint due conjugacy integrate leave marginal model penalty complexity favor adequate provide rao
sdca minimize sdca importance sampling n option version square loss prox sgd sdca sdca sdca adopt subsection adopt algebra several world aspect dataset dataset dataset fair comparison algorithm adopt experiment prox sgd parameter estimate bound ratio sample sgd verify empirical sampling accelerate duality conduct fix seed learn measure objective gap examine generalization ability finally also gradient importance sdca uniform sgd sdca respectively test variance gradient learn dataset sgd summarize sgd sampling sgd fast rate two adopt propose importance error rate last two indicate sampling effective generalization right proximal prox sgd coordinate
existence cauchy rational adopt locality sensitive lsh lsh question clear answer literature study theoretical similarity similarity basis comparison retrieval lsh inequality framework indexing lsh popular lsh lsh work abundance binary question lsh prefer attempt various aspect example show
I c g r ht pdf monte carlo nest stop criterion ns contribution simulator chain analytical monte visible nest little accurate stop always see meaning estimation heavy bayesian originally bias introduce theoretical framework derive estimator allow law bring modification ideal infinite sum carlo way implement practically totally non parametric conditional draw hasting overcome sample markov pareto carlo heavy tail substantial budget approximately optimal variable far thank university paris energie alternative suggestion comment improve manuscript
protocol communication budget take jointly independent satisfy budget message protocol minimax understand definition instead infimum interactive protocol consequence metric entropy confirm bit problem bind tight problem refine substantially begin low geometric space capture define pack claim family distribution interactive bound q proposition interactive although exploit structure problem mean receive packing entropy distribute minimax yield factor achieve simple machine compute minimax machine result set message serve pre interactive protocol low bound receive unknown low communication budget universal constant section proof centralize sample low machine individually decentralize match ignore factor achievable machine average fusion center average technique family
function great exceed sparse starting point analysis let represent whenever restrictive ty demonstrate whose grouping constant penalize distance converge norm demonstrate sparse signal property recovery also bring solid theoretical justification let value noise identically candidate use plain loss newly behind follow possibility propose produce introduce objective
broad word include formal logic entail task lexical relation recognize pair important lexical semantic word semantic particular semantic relation classification work lexical lexical perhaps lexical concatenation concatenation six vector ease hypothesis tendency learnable context context occur context tend imply cat condition choice would make well possibility weighting choose non weighting word apply lexical semantic relation semantic might seem lexical result past past relational operational definition understand definition attempt lexical intend case exclude non lexical argue agreement lexical decision fit believe inter agreement trade definition lexical section word relation item attribute attribute fulfil item attribute fulfil fulfilled limitation lexical one speech entail noun substitute speech entail situation involve something lexical relational capture act reasonable say act entail noun address limitation one cope part speech noun something pattern could speech limitation definition relation lexical phrase entail phrase corpus successfully lexical lexical useful section preliminary inspection semantic relation systematically label entail instance relation nine lexical asymmetric symmetry apply classify word pair classify symmetric lexical relation capable symmetric asymmetric relation lexical behave behave case car car car see detail researcher apply classification lexical argue
see g rely question dependency necessarily dependencie solely data response theory use propose share analysis however rely estimate solely interpretable factor response ordinal determine several consider learner value response ordinal factor miss ordinal interpretability matrix negativity sparsity ii tag estimation ii tag question oracle concept association concept sparfa jointly association knowledge difficulty extend sparfa response correct incorrect exploitation concept ordinal sparfa estimate concept demonstrate real ordinal sparfa outperform sparfa collaborative predict learner response education interest
factor big generalization planning take way parallel randomize planning generator direction whether plan effectively factor e simulate en expert policy perform factor corrupt establish factor write neighborhood overlap agent constant sample bias pair eq action action expert estimate receive take expert expert e bias turn cause correlation count component payoff policy bias case bias action overlap contribute introduce mixture recover action q joint affect affect maximize joint performance case overlap intersection action profile lie solution joint cause wrong bind explain two omit
anomalous event denote type ii anomalous occur say hypothesis compound nature interval anomalous length small anomalous infinity detection length anomalous become anomalous anomalous successful change asymptotically successful bound distribution bound characterize candidate anomalous scale detector successfully gaussian anomalous distribution differ hypothesis correspondingly interval happen even hypothesis nonparametric distribution unknown arbitrary capture distribution
far tb evaluation consist tweet rank remain tb per evaluation task tweet triple high ir discount gain cut ndcg ndcg performance recommendation tweet predict ideal tweet evaluate formally discount cumulative measure maximum ideal give triple normalize ndcg ndcg user correspond ndcg obtain split baseline rank baseline describe factorization machine fm provide monte deviation
unique infinite number matrix row infinite solution choose similar decay fit flat generalization easy solution see angle original scaling implementation interpolation transform filter much consider horizontal filter capture symmetric invertible integrate tie filter equation transformation mean tie transformation back convenient corresponding filter training propagation distribute transform aggregated filter canonical filter experimental begin achieve follow baseline
exposition maximization expression part problem dirichlet seek maximize interpretation expand objective dirichlet guarantee strictly negligible due set select add maximize q essentially infimum fractional actual simple try unlabele weakly connect label end involve numerous high cost involve account expect sense underlie manifold comment compute eigen iterative specifically method write computation break product typically eigen pair computation iteration iterative filter complexity major atomic operation need store vector moreover structure aforementione suited package eq eigenvectors less
leibler variational change expectation expectation prior compute hold depend rewrite take term obtain nm probability task choose order combine obtain set hold eq nm use weight learner expect vote predictor multiply hand obtain vote predictor variance differ vector distribution learner use return vector computing obtain loss z dt gauss describe obtain next task transfer manner sequential information set q nm order group inside group
recovery deal measurement instance guarantee non noiseless sparse chapter volume author technique much wide ensemble finite value locally partly smooth locally solution concern stability additional partial provide guarantee correct unique guarantee source perfectly location norm matrix govern existence degenerate stable need norm minimal remarkable proposition concerned checking closed form pre define linearize definition note hypothesis set non uniquely similarly hypothesis involve empty affine uniquely affine terminology actually amount solve x appear compute close regularizers instance read svd find exhibit check sharp sufficient condition model case jx measurement rate assumption imply strong uniqueness identification theorem analyze clearly see minimal distinction theorem plain enough fact quite case minimal dominate amplitude entry achieve zero identification manifold conclusion strong guarantee fidelity account loss fidelity quadratic strictly argument prove show theorem sharp almost characterize unique obey correctly cover neither stand affine hull conclude either illustrate finite discretization operator operator make estimate require handle operator row sensitivity perturbation concrete row vector suppose model see g assumption index associate whose generalize previous machine jx state
road turn right road analogy road arrange address dimensional degree reduce must prove select direction suggest analysis order dependency exist remove second insufficient structure solution order representation term another direction require along statistically class algorithm term ica experience help underlie technique I I write section prove transpose matrix tt bi directional require start matrix hope address aspect well principal pca tool graphic parametric extract relevant minimal effort lower reveal simplify intuitive
desire answer probability set representative pick distribute multiply pick occur great claim clear gap monotonically true pick threshold large eq setting thus en inside right dominate order failure slight modification argument let degree set polynomial form iid straightforward estimate roughly speak large magnitude subspace accordingly recurrence theorem let similarly use number canonical
base want able whether importance plausibility depth e scenario calculate r plausible sense depth reverse stress relate set skew generalise family family skew distribution financial close exactly near behaviour structured multivariate briefly half introduce generalize univariate examine computation canonical skewness prove contour skew cauchy elliptical simplification construction investigate angular deviation angular measure explain quality approximation show misclassification interpretable approximation difficult give half hyperplane euclidean affine
benefit also insight approximation scan field even scan section relate power structural formulate approximation power real set describe section paper statistical genomic short read sequence refer successfully fail call alignment allow read often read soft also read produce map close expect detect structural end region genome read pair minus apart expect boundary produce minus read boundary plus read produce end entire produce read close dna dna sequencing read end dna call seq mapping read template detect genome read bind bind shape peak site roughly triangular et jump locate read plausible match kernel scale parameter width equal unknown one maximize statistic ds coverage equivalently process see effect tt ds ratio map length sequence pair map opposite orientation map read length map read variant mean deviation cause cause number variant unknown although segment target map detect consider toy field intensity read alternatively think marked mark starting read plus logic q likelihood index scan genome detect scan statistic simple product function relatively involve compound
approximately score probability contain initialize add matrix row proper size thin singular value singular order orthogonal svd row score fortunately
lemma claim multiplying element inverse q jj mn assumption together prove final note expression positive put sparsity graphical lasso oracle maximum eq q observation give sparsity equivalently suitably zero tail function put theorem estimator subsequently analyze novel minimize penalize determinant bregman
dynamic improve future poorly might reward exist knowledge efficient exploration exploitation cumulative vast majority efficient upon knowledge environment beyond design attain sample state strong difference agent cumulative controller kind regret curse level many practical prior beyond state time step machine direct reduce easily exploit factor
bag instance region region difficult say label generally determine instance formulation ki svm bag different svm instance positive combination linear simplex gradient optimize object take image bag derivative formulation combine respect calculation formulation sparse gradient compare art positive half extract sift densely select sift descriptor visual fig split extract sift codebook main purpose
dimension alone circumstance condition default optimum eigenvector eigenvector without average delay early take exclude term result reveal temporal detection false indicator track event study key movement movement constant spatial subsequently compare three scenario baseline environmental historical baseline strategy scenario compare consider environmental week data environmental day set day search environmental baseline tensor match set environmental setting compare different figure illustrate versus scenario versus result dynamic reason approach unseen environmental setting robust provide reference surveillance alarm art benchmark performance maintain alarm rate considerable extent feature introduce novel decomposition overall eigenvalue change dimension helpful
counter might purpose affect play become notion make sure simulation crucially supremum infimum objective consider objective temporal trivially quantitative vector atomic closure complement multiplication objective quantitative three game expression player determining satisfy condition checking objective extension player contain conjunction atom bound play provably fundamental long quantitative specification boolean union intersection complement strategy single game prove payoff payoff exploit different limit infimum knowledge consider next formal multidimensional payoff reduction counter counter intuition correctness formal
convex assumption convexity equivalently q continuity second strong side result note proof rate convergence generally vanish size discuss finite assume choose large since use induction complete algorithm remark let section restrictive nonsmooth start lemma find nonnegative scalar continuity follow eq summing note ik jensen eq follow boundedness instead generate subsequence large summing inequality sum side eq bound hence subsequence ki take inequality ik satisfy asymptotically condition note immediately
involve attribute often find solution attribute parameter update private setting particular predict depend heuristic come large attribute good attribute like get run range truly know principled handle private area g new mechanism handle order magnitude large continue answer minute dimensional remarkable free accuracy previous approach possible maintain distribution art private set specify subset query record record take like wise answering query answer wise record query answer like marginal query handle
apply table besides r another primary contribution divide find anchor merely low low hyperplane easily handle solver geometry partially probability solver problem plane compute thus learn hundred improve randomization rise multiple subroutine apply hull five learn latent allocation lda nmf subspace comparable generalize separability minimum hull model minimum present divide extremely original convex hull point vertex hull separable hull replace definition cone separability geometrically cone empty generator integer separability assumption cover finitely point cone generator algebraic form x ki give model also negativity allow contain rule hull extreme ray separability uniqueness constitute use actual
maximum ise variational computationally job mean isolate approach towards allow explain variational illustrated ise letter sample letter experimental valuable statistical mechanic drive provide insight behavior organization evolution protein mechanic construction
fidelity order approximate similarly residual approximation residual operator approximation would avoid high dimensional matrix construction computable computable model general residual indicator accuracy reduce order solution change basis increase surrogate dual attribute error even indicator generally frequency depend variable distribute validate surrogate enable perfectly compute inverse stochastic gaussian process generate prediction point denote infer transform reduce error contain joint distribution gp construct via training begin analytically kernel noise covariance kernel geometrically assume generate treat arise prediction variable n derivation expression q likelihood component account gp play crucial account I represent incur employ indicator therefore interpret quantify decrease increase interpret correction due high employ hand reduce employ indicator albeit employ include due lack I pa discretization k kk basis constitute polynomial dependent center radial basis domain employ approach vector hyperparameter affect compute maximum hyperparameter identify remove indicator kernel q graphic axis axis cs cs axis axis axis cs node cs cs axis axis axis block nine input basis reduce section introduce experiment
computationally furthermore modular likelihood choose need spatial choice pareto distribution present advantage choice modeling paper see lower low another choice marginal base extreme likelihood present appeal suited realization extreme year unobserved site observe implement within framework main spatial quantile planning extent simulate km km regular spatial predict method account covariate scientific statistical recent year various model extreme extreme distribution suit quantify physical argue model modeling explore pareto vast framework appear level
half cluster belong belong expect involve four report model requirement mostly force triple appear big might heterogeneity low region indicate possible direction see section region intensity estimate posterior background express reduce considerable contextual sensitivity detail test value much sensitive implement alternative represent association event observe cluster distribute figure value denote number pairwise value association suggest current reduce specie many measure literature interpretable influence section step configuration assess section time run intel processor design random model behaviour project strong prior estimate meaningful context accordance contextual see posterior design cluster complementary applicable context stanford biological context dissimilarity different result specie carefully consider computational aspect consider problem mh
node sum mkl I consist along path call path sum leaf layer consist combination embed atomic path color atomic layer
informative behind many associate permutation describe represent think subspace whole decide fit distribution covariance power associate independent functional gaussian close capture notational drop give graph compute run matrix permutation solve big open graph summary power discriminate effective graph encounter call distribution never think distribution analogy object intuition adjacency gaussians design kernel ensure positive previously satisfy property study semidefinite overall adjacency summarize
ascent rewrite kind objective function note building characterization degenerate completeness selective vector expect occur let lyapunov point lyapunov q negative quick iff negative assume degeneracy probability semi occur selective objective briefly like tensor rule power rule slide threshold neuron slide neuron rewrite define lagrange expansion orthogonal lagrange expansion eigenvector objective attempt degeneracy maxima eigenvector look successively subtract repeat orthogonal local optima tensor correspond optima stage need ensure ascent
sort suffice paper overview prior work really approximate furthermore write suppose dimension sketch column goal work attack splitting implicitly compare adopt make comparison explicit analyze separately section additionally generalize split r kk preserve sketch side constrain suffice side dependent note linearity trace step property trace lemma I bind equation immediately low requirement k f lemma sketch broad error next rank preserve sketch side k k specifically rank handle term starting analogous long eigenvector f f step trace semidefinite schwarz symmetric combine equation recall framework analyze dimensionality start sketch simply onto top rotation truncate vector claim suffice low enough appendix take robust computation often pass limit gain substantial let condition semidefinite q final small
therefore irrelevant relevant htbp evaluating production respective input parameter production column separate pc contribution description generate correlate independently varied input contribution correlate varied independently case independently generate variable specification remain candidate refer set efficiency obtain include input production concern adopt interval simulation try time trial note train full report result htbp parameter follow recommendation al significance comparative use begin full efficiency significance without generality include production
long context propose layer specifically design dependency nonlinearity gradient vanish connected time recurrent pattern suffer rapidly time removing keep constant allow period precisely efficiently recurrent would never vanish gradient variation type memory overview diagonal connection unit differ recurrent achieve retain topic large besides argue learn recurrent cache
error order element finite error accurate accurate inaccurate error table low estimating location generate operate compatible location center project precisely away boundary five away boundary step away mc truncate become increasingly implicit implicit sampling importance generate mode via sample algebraic implicit solving error define norm parameter
individual future mathematical prediction system reliable system may major yet lack indeed obstacle besides collective behavior challenge complex attract lot attention prediction estimate link help biological network protein fact recommendation spurious sound design recommendation irrelevant reliability identify noisy progress largely field accordingly link researcher apply field ref
cycle nucleotide normalization calculate length expand respectively formula separately variance cycle sequence q follow mean cycle increase linearly sequence incorporation extra evaluated value nucleotide incorporation complete cycle derivative vanish cycle nucleotide nucleotide incorporation nucleotide incorporation numerical integer expansion computer continuous calculate evident nucleotide incorporation accurately normal variance
optimization solution I algorithm accuracy output efficiently cope dimensional refer pc algorithm recover set approximate let solution lead solution imply x ta denote modify method reduce hessian method solution line
university california berkeley center education pt berkeley edu interest due streaming energy end use profile principal pca classical however introduce family generalization kl mahalanobis bregman come interested generalize view extend theory property discuss end
cc sgd variety relatively find high dimensional nearly neural perform extensive problem sgd easily likely aim problem easy sgd slow advanced enable could like review thank experiment hyperparameter directly literature specify maxout describe modify maxout network maxout publicly relu dropout intended nearly reproduce relu standard provide relu network dropout precede file
composition private program mechanism oracle find private everything unfold definition suffice previous note tight row amount approach final sensitivity depend private simple throughout weight q instance neighbor objective concrete part add laplace lp exactly private optimal draw perturb q sensitivity objective solve perturb accuracy least laplace happen event perturb lp add bounded perturbed find solution sensitivity trivial exception solve relaxed sense reconstruction attack privacy reconstruct database due completeness restrict entry round round input entry uniformly identical eq uniformly privacy private lp convert database lp solver attack lp likewise say optimal solution lp change bit change right mechanism least feasible lp note guarantee reconstruct impossible differential privacy zero lp zero change coefficient similar lp coefficient amount mechanism private additive lp solution probability solution place index observe private change small constraint want solution constraint mechanism sensitivity private feasible satisfie public probability lp
first realize realize outcome repeat estimate initialize tt give critical underlie imputation realize realize collective mechanism expect typical agent causal issue proceed make behavior prior outcome round consequence good mechanism agent mechanism offer utility strategy behavior agent expect benefit agent adopt game agent vector action softmax behavior agent nash mild condition agent strong high experimental function
lead without additional weak global show exactly eigenvalue exact semi nest dropout autoencoder principal component procedure intrinsic speed quality gain offer procedure na I hamming database semantic within memory associate retrieval complexity grow computationally prohibitive bit address code likely many query locality sensitive seek preserve projection inefficient code impose retrieval decay datum capture coarse allow logarithmic adaptively code large code hundred exist example retrieval dataset entry average per fast order also use continuous degradation compression give rise quality combination small
go tc audio affect notably live device versus combinatorial conventional domain adaptation shot adaptation address calibrate acoustic environment type music speech track categorical variable live lr sr iv live degradation toolbox audio split training overall hold good exploit semantic descriptor descriptor regular recognition environment environment demonstrates effectively covariate see data r origin lr sr avg lr lr tc attribute
special spline use outcome display array stagewise regression noisy curve draw equally knot stagewise figure show stagewise along dotted stagewise computationally update solve exact computed stagewise produce visually reasonable observation thick run stagewise step curve stagewise curve smooth visually reasonable difficulty stagewise set set easy previous iterative invariance around stagewise generalize give term since lasso special case covers choose wherein encourage order piecewise choice dimensional fuse statistic literature incidence correspond encourage component piecewise constant respect framework use trend filtering update linear perspective form switch justify two argument write q conjugate satisfy stagewise path convert primal stagewise differentiable stagewise aside reduce multiplication one fuse trend filter sparse make strategy estimate stagewise update outline compute stationarity th view iterate point simplify considerably eq interpret convention think stagewise strategy compute primal note stagewise estimate generalize lasso stagewise encounter far increase stagewise begin towards opposite usual direction note loss primal initialization update express eq stagewise begin trivial fit along form towards row vector concrete fuse incidence evaluate difference nonzero towards build shrinkage across difference move constant amount towards graph grid graph appendix general various computational three major far present find room comparison tune grid parameter compute recognize fair near computationally superior stagewise competitive capable produce overview path implement group apply accelerate gradient idea complicate backtrack line step refer section description stagewise step coefficient define uncorrelated population independently block predictor correlation group stagewise fit warm run stagewise top row uncorrelated stagewise fit underlie quite competitive fit plot stagewise fit exact top took stagewise fit meanwhile group uncorrelated stagewise take second surprisingly middle computer uncorrelated case stagewise stagewise estimate within criterion frank wolfe mean frank lastly bottom lasso stagewise component path one draw path uncorrelated setup correlate bottom display
relationship pseudo ensemble perturbation space notion robustness focus perturbation novel regularizer make behavior pseudo ensemble generate regularizer match unlike dropout naturally art tensor ensemble world original conclude approximate pseudo child provide pseudo dropout sample activity common minimize consist sampling mask extract use fairly form impose tractable allow variety ensemble formalize follow parent perturbation pseudo approach come broad
quality improve training induce benefit hyper empirically demonstrate improve great parameter optimization supervise generalize hypothesis induce algorithm data set noisy learning associate parameter induce datum hyper improve weighting noisy significantly hyper improve algorithm validation characterize validation induce parameter characterize g optimize training impact induced instance induce even instance induce outlier beneficial instance case instance induce b line represent classification dash line induce boundary
example link uncertainty associate network mining link change uncertain nature analysis need link failure human interaction recently mining investigate uncertain propose algorithm find database support uncertain propose et semantic represent move trajectory configuration edge region connect neighboring main originally cope directly despite number enumeration become intractable overcome simplify leverage simplicity tuple attribute uncertainty uncertainty tuple model database may edge underlying instance consider additional generation rule correlation study combine mine extensively management neighbor context mine frequent mining mining uncertain scenario though entire graph node label collective
end end output dag root guarantee process similarly tree version dag traversal distant sense child child responsible positive choice choice meaningful priori select maximize score available percentile example experimentally choose
part nsf modal mixed cluster datum journal american association shift mode cluster transaction intelligence toward transaction machine intelligence bandwidth selection unsupervise unify self coverage journal recognition research sure screen journal cluster american association journal selection journal american optimality et rkhs
log expression empirical moreover entry give standardized write identify learn especially computationally modern exceed number inference calculate model result take universal infinite limit derive expression selection logistic
multiplication power weight vector lexical noun represent component noun lexical like net tune take give experimental second calculate solution candidate super candidate single pass rank candidate top candidate last super task remove irrelevant candidate tends improve mean impact fortunately affect adjustment greatly evaluation mean candidate top percent candidate candidate super noun super model median composition similarity believe well probability kind see table pseudo context believe pseudo context vector want really learn section suggest red thing separate noun train target super noun noun noun never describe super dataset build table possible rank rank marked candidate build list super rank unfortunately candidate super guess super good member red specie specie table summarize super target evaluation metric top super work composition super super composition mean candidate percent percent top percent c c decomposition baseline vector take significantly super percent fisher test significantly super percent top confidence level candidate median candidate percent top percent top percent percent c compare baseline decompose super decompose although restrict super performance percent versus top versus versus test confidence level near
forward help help chain mix coupling implement mixture gaussian would many sample forward gibbs test code sigma replace subtle never subtle become top soon major drawback unfortunately therefore test pass test general drive development pass outline code method code modular
great q side q q absolutely absolutely inexact kk kk show update inexact saddle
inequality unit vector also inequalitie subspace column draw dimensional correct know pick among label oracle pick maximum success return failure original pick incorrect neighbor pick incorrect near pick among pick incorrect close oracle failure incorrect close point correct fails therefore analyze independent form dependent analyze stochastically dominate random onto isotropic subspace stochastically dominate bind condition least give simultaneously k j inequality since independent fix union choose therefore
online generalize click ad happen currently develop pricing model effort towards click strong assume query click iid let click indicator event position count click capture click click click beta update quantity involve click proposition remark conjecture theorem axiom claim wang click tool leverage feedback click aspect share sequentially click engine search key information important key argue click user experience incorporate search display click relevance search evaluate extensive engine system user query list rank link ad display call search click ad interact search particular ad click attribute ad never search surface user query understand document compare approach crowd employ human
either semantic concept various semantic concept popular show dataset manually annotate large class tag tag depend tag internet verification label noisy contain intra image benchmark recognition high semantic imagenet use imagenet visual recognition k internal imagenet guarantee image consider entire comprise million manually million report recognition manually class tag ground without verification therefore supervision noisy characterize intra class image visually similarity object video content manually video semantic category belong category video size set evaluation protocol binary multi semantic mean map activity publish video sharing release video release development training video learn video first frame phase score network architecture convolution layer convolution architecture compose layer convolution layer fully layer last classification layer resolution image resolution convolution pixel convolution layer max perform layer connect
noise come ignore three decay minibatch fisher gradient size keep govern control see size big big minibatch gradient cause especially close minibatch small per setting digit contain indicate model mnist parameter gamma precision result insensitive generate parameter sampling carry alternate follow step sample weight pa minibatch gamma conditional sgd try give good result use sgd momentum try setting sgd momentum find good performance factorization use movie simplify consider prior however benefit inference sample minibatch rating hyper experiment select try sgd momentum try sgd definition lemma hamiltonian mechanism distant metropolis enable efficient
cascade implementation base round overfitte cascade round team compute count run validation occur iteration additional randomized cascade cascade
statistical exponentially weight forecaster case recently learn property characterize special function space develop rich condition loss understand connection bernstein stochastic fast previous secondly establish slow notion stochastic effective convexity stochastic toward get rate involve er chernoff moment excess moment application er chernoff bind finite vc function describe extend fast notion call weak conclude connection topic theory discussion fx operate compose comprise loss compose frequently throughout
main gps integrate connect explicitly unlike gp corresponding discover employ one convexity lead multiple optima feature optima severe alternative bfgs hessian additionally beneficial extreme easy easy per rmse ard sigmoid number mapping covariance exceed notice undesirable slightly offset actual location ideally describe would intractable gps uncertain input kind although exist strongly appropriate designing covariance g
evaluate serial include follow sampling denote lda conduct yahoo approach wise sampling recently fair strategy three comparison result lda lda three document follow figure speed performance document wise lda indicate discuss document order document fast document compare justify use core distribute parallelization lead new lda huge compare yahoo implementation scale large yahoo lda yahoo server therefore outperform server yahoo disk base implementation assume associated token disk fair run yahoo normal disk yahoo run storage yahoo disk code disk yahoo yahoo parallel machine conduct dataset amazon present figure lda outperform disk yahoo well desire time fast yahoo next
great schema derivation could diagram piece reality expect diagram encode versa throughout deal library student university library offer book article scientific conference publication kind stock book diagram diagram entity author right author author book auto swap translate entity book author relationship book resp child tree schema correspond whereas represent resp option resp option third element attribute book author reflect practical good avoid b author book observe diagram relational incur point diagram orient organization diagram order specification child datum impact match consider map onto list specify concept element phone name phone phone convert negative drawback explore refined level meaningful reduction negative examine highlight semantic properly finding improve schema process user nice schema page extract subsequently exploit refine query page contribution discuss structural link instance schema match source exploit user search web experiment life car site semantic precision web far provide schema primitive build type instance offer integer boolean compatibility type whether violate instance constraint convert soft constraint constraint detect positive schema conclude semantic state reasoning involve primitive solve compatibility table compatibility string integer short second whereas report compatibility coefficient high compatibility type define usually associate management human type simple furthermore allow create type start exist implicitly induce base extend mean new construct intend analogous fashion derive assume type implement relationship implement concept specify sub construct adopt sub appear construct specify semantic exploit management compatibility type principle decide repeatedly simple already complex element match match cardinality specify occurrence element occurrence denote zero indicate occurrence allow analogously compatibility check two schema discard
picture maps assignment assignment constraint partition function equal solution notice abuse col j satisfied represent bipartite include two node denote neighbor factor summation true evaluate clause false circle graph versa refer factor summation ix message typically initialize distribution update estimate loop summarize bp take opposite bp true true true bp influential loop rather instance may converge time bp update message message substitute variable substitution repeatedly message reach point become individual update operator employ sp refer heuristic bias biased fix convergence additional assume iteration sp summarize schedule h return otherwise
complicate pick wang censor covariate usefulness need verify censor datum need lot censor robust slight section simple technique work optimality scenario due apply life efficiency crucially tune choose carefully censor proportion give censor enjoy many property inference censor chen f wang et many assume availability associate target e g medical diagnostic pressure covariate give instead infer response alone interested conditional covariate often study act nuisance component th parametric calculate eq censor distribution also adjust tie parametric censor prove several consistency functional life science know survival censor experience similar disease
hide constrain produce output reality allow criterion htbp hour ahead forecasting present ann ff ann sa forecast svm mae period ff ann sa mae mae rmse ff perform input wind wind previous hour time day presence feed greatly forecast htbp value convert algorithm obtain value train machine mlp neural
long admissible instead combinatorial hold trivially
e te cx z jx jk tw estimate influence derivative q note operate certain redundancy shift constraint component ambiguity scale sample ambiguity lead therefore estimate ambiguity ambiguity cluster similarity value sample neighbor nn efficiently entropy compute ambiguity scalable representation semi supervise spectral operation mm current mix cluster gmm ambiguity entropy approximately evaluate uncertainty qualitative shown notice appearance boundary increasingly model set complexity nonparametric parametric complexity active adopt large scale sample compute scale method
fact throughout expand expand term stationarity last thus sgd var var var ti ti optima capturing randomness tv similar algebra lead desire assumption ml size turn satisfy cluster considerable code framework specialized implementation recent server ps allowing distribute high throughput allow insufficient really ml algorithm output many theoretically computational throughput guarantee convergent exist ps ps communication mechanism implement ps enable ml volume internet activity pressure ml scale beyond single size ml single partition machine machine practitioner turn server ps server
network traditional improve distance representation influence choice label outperform method create label online deep tool analyze build representation evaluate representation task increase sparsity improvement micro outperform even training demonstrate scalability moreover build arrange section formulation classification relate outline relate work conclusion member member edge partially social attribute map label utilize dependence embed achieve superior literature traditional relational inference markov iterative topology label unsupervise structural task
close enable approximate optimize certain distribution good maximize bind overall algorithm initial randomly mask independently update update parameter decrease properly increment go termination r respect replace accord descent assumption intractable summation binary evaluation practice use carlo method last sample well two two think amount
focused datum instead large approach bias try give discrepancy sample feature trick inner equation feature mmd db kb
distribution interest inspection datum rank set arrange arrange arrange arrange constant middle rank rank large rank seven ten set small probability observe extreme discover pattern similar gradually optimal switching minimize mean surprising value choice nine pair nine dot degradation respectively bad w significantly marginally bad significantly accept large
rule student logistic class generalise example transformation distribution list exp control univariate baseline typically reduce poorly limitation consider weakly correlate
write take feasible outside x proper satisfying q subdifferential denote proper hilbert space fr continuous frequently lemma ball lipschitz nonempty briefly apply formulation cover broad input generalization frank programming gradient apply know computational geometry elsewhere appear various signal name boost greedy method orthogonal recent descent discuss reason conditional evident example proximal reference iteration low whereas gradient exhibit rate side step operator section
report encourage positive segment posterior bin interval obtain calibration seem calibrate present simulate detailed use call choose analyse platform replicate datum run pass replicate segment large would distribution uniform namely elsewhere justify simulate cc ccc start ccc length ex pass ideally infer replicate datum happen consistent across replicate infer replicate
extension dependence jensen jensen jensen offer extension control complex value dynamic combination beyond quick estimator cover analytical gram iii modularity modularity computation
gpu set axis run yahoo ranking response goal song audio dataset figure even gpu completely dominate computation massive speedup gpu marker leave fail complete solution recognition datum version match exactly parallelization improve scalability highly besides code burden reduce spend establish parallelization square hinge solve entirely large
hide dnn neuron random forest foreground dnn spatially color transform three regressor choose dnn forest forest scalable dnn forest color dnn fair adapt table foreground obtain slightly dataset forest dnn obtain foreground obtain dnn inspection color forest spatially enhanced image random forest retrieve node retrieve dnn color visual th pt p c method run method mit learn level mapping use experimental work testing mit hence c mit hence train column table capable prediction term predefine mostly concentrated show see enhance close ground could near neighbor slow search percentage contain pixel hand powerful thus method capability exploit near inconsistent color remain testing image th enhance expert enhance enhance effectiveness algorithm naive selection sensor interestingly achieve well gaussian primarily nonlinear deep network rich compare entropy method achieve especially select number mobile filter different color warm rise filter light choose mit enhance training half verify
vary densely achieve error gp I multiple field smallest tend low field usually field step gp e exploit temperature light measure denote colored end vertical horizontal show localization gp field gp small light field small low low field field geodesic e path true road segment road topology fig localization road segment average run cc mobile trajectory light b location achieve error average gp gp clearly scalable I offline incur paper localization whose constant memory filtering theoretically analyze outperform localization algorithm scalability robustness
form alignment gap alphabet denote gray help track protein family position alphabet go cf transform alignment gap one position zero otherwise denote length I cx entry measure protein similarly protein approach approximate value former enforce rely fact input consequence position e site anti work suitable anti correlation model bayesian inference behind density determinant block independence protein sequence read attain constitute inference proper parameter bayesian need introduction require computed accounting prior inverse prior p meaning inverse constant euler wishart integrable px eqs formulae posterior covariance abuse notation shall provide estimate differ attempt protein contact
unnecessary lexical stress keep discard rely cause corpus without lexical evaluation development subset vocabulary decode vocabulary convert window frame form final vector preprocessing alphabet token total recurrent accelerate maximum pass rate pass train gpu implementation decode cross set bt lm lm sort fairly mistake character word
correspond begin discussion scoring distribution framework scoring outcome attention mechanism relationship market semantic implication market behavior vary depend agent market agent accord characterize market reach potential argument market proportional eq though work density turn draw well duality specifically program care specific outcome outcome entropy tend towards always constrain entropy whereas objective let negative solution take form multipli integrate ensure family almost interest consider regular family regular family convex differentiable lie follow property onto inverse statistic statistic log log family relate invertible expect statistic family depend underlie proper scoring consider logarithmic density parametrize proper belief let strict converse score characterization proper strictly px show intuition bregman divergence strictly equality know relate bregman state full lead expectation maximum agent report interpretation principal according agent usual
set extreme point polytope polytope express k solution vi inequality simultaneously polytope reduction formalize denote set put arbitrary vector polytope exist inequality arbitrary inequality dimensional construct qx eq side product satisfie use point derive admit full rank whereby orthogonal eq ks induce orthogonal follow hold cauchy consequently follow unitary similarly since cf likewise definition complete cf theorem obey orthonormal probability depend else implicit
unlike optimize small become high one plausible greedy fast achieve good enough provide approximation gram check number greedy adaptive iteration adaptive subsection ridge ridge near summarize ccccc greedy greedy census ground truth adaptive error sampling since result report two much size case big case adaptive central bounding uniformity phenomenon large art sometimes dnn close reveal accuracy speech match dnn clear improve efficiency kernel high exploit integration technique context include analyze gram author would thank sequence wiener anonymous point helpful program project air laboratory contract fa detailed make term notational lemma claim improve
image hypergraph construction sequentially introduce clique star expansion propose probabilistic hypergraph assign accord centroid pairwise vertex similarity individual pairwise similarity represent designing share context corresponding similarity contextual take account neighborhood vertex corrupt still information corruption context aware hypergraph similarity measure type hypergraph nn hypergraph hypergraph pairwise hypergraph affinity vertex hypergraph hypergraph capture manifold structure modeling contextual combine aware hypergraph intrinsic robustness corruption contribution fold spectral order property build hypergraph encode affinity type end type information hypergraph similarity vertex
detail tackle distribution bag bag bag possible algorithm basis similarity formula finite gaussian heuristic relate parametric gaussians kernel hilbert reference appeal gaussians close product gaussian divergence lack theoretically approach statistical divergence metric non number construction metric kernel overlap concentrated value dispersion type hilbert define guarantee certain domain estimate kernel open plug algorithm similarity distribution index paradigm bag treat instance solve ray set label bag example fit configuration handle shape patch region document web identify link customer characterize record
domain recommendation rating user dataset train remain bc bc em give bc em conduct vs vs model outperform show latent domain common rating aggregate propose art cross domain enhance recommendation dataset give nmf bc nmf vs setting bc vs bc vs
space kernel propose q ensure cluster matrix mn multiplicative overhead greatly require case random partition kernel directly interpret partition sensible machine mean exhaustive example random cluster algorithm couple generate randomness algorithm exist randomized initialization bagging feature return
set separate system system primal build presentation shall familiar na I extend constant purpose like upper unlikely behave shall intersection detail construct element fix set prove system vc verify bound book discrepancy edge lemma vertex drop total drop
control fitting variational paragraph range matching term correspond independent half assign prior find model lp match lp estimate guarantee particular dependent plausible evaluation satisfactory use four model spectra extract image nonlinear pixel accord feature component interaction pixel p b fan bilinear fm generalize bilinear adjust bilinear interaction th pixel stand hadamard admissible set robustness pixel impose cutoff abundance remove drawn version namely two appear
example produce low sometimes hinge loss introduce slack apply result problem select accuracy ht cpu problem c cpu second result solver give relatively matlab get active almost regularization addition parallelization implementation immediately look number plot figure ht technique construct analyze thank smoothing theoretically rate surprisingly analysis enable inexact augment lagrangian expect deep smoothing help adaptive strategy connect european future foundation grant lemma feasibility induction notice assumption nonempty dual saddle inequality lead tt follow outline lemma lemma k definition follow write k obtain also h get subtract inequality definition equality substitute rule third ff substitute final third hand substitute since first inequality lipschitz continuity respect give refinement express kf line easily get prove q g second inequality refine since right hand find proof lipschitz inequality c update c estimate ga update k augment lead obtain combine q prove set obtain inequality imply
orthogonal basis learn distribution pose vector advance determine coin uniformly accord relevant define family underlie prove bind choose key ingredient relation learn coin relation first adversarial ki j identify fraction adversarial specify gb picking assume orthonormal hand attain attain reason observation th coin assumption previously informally coin formalize coin problem hypothesis use mechanism
n constant lin k scan union bind bernstein scan multiplicative chernoff eq several condition suppose fix measure fix suffice positivity q thus tv q inequality variation product measurable subset independent copy variation distance p p eq cauchy prove ball index bin index ball bin fix condition index index need follow association fix independent copy condition vector index vector moreover distinct statement expectation decrease q view therefore q denote bipartite vertex vertex denote distribution match edge proceed clique let clique define conditional
reach activity c room interact activity trajectory model pr robot purpose human environment pr presence available learn preference activity preference discrete preference expert rich encode learn pr robot video thm minus user preference trajectory environment human challenge trajectory interaction environment preferred trajectory new cost system motion segment neutral use preference express environment extensive preference plan preferred trajectory environment preference environment rich object human challenge define good trajectory environment trajectory trajectory preference expert learn video robot segment good neutral parameter preference run environment validate claim planning environment good define vary environment paper object lie type feedback expect user training trajectory currently system argue co active preference feedback non intuitive nevertheless match rate algorithm task
weight define px px property family divergence turn problem yield bregman bregman connect thus family property dp return family density point support include laplacian family densitie family contiguous property intensity grey small distinct grey
update metropolis hasting proposal accept necessity compute normalize acceptance current algorithm keep reasonable acceptance function symmetric current truncate zero proposal draw step package sampler assumption exchange tend auxiliary
solution matrix stack root via identify eigenvector interpret scale direction indicate transformation ica far operation familiar term whiten whitening remove second along term datum whitening demonstrate first eigenvector perform rotation order dependency mathematically transform expectation covariance dimension operation familiar principal pca eigenvector remove reduction remove low figure ensure preferred direction symmetric much sphere whitening simplify ica rotation simplification observe reduce simplified provide additional structure recover highlight likewise consistent whitening recover mixed decomposition whiten rotation statistic variable whiten remove correlation last dependency require correlation ica special term factorial search rotation order instead remove correlation rotation achievable therefore term order estimate
repeat optimisation output discriminative network generative model represent likelihood training use label label bind extension latent bind handle unobserved q entire unseen label contribute relate undesirable classifier ideally variational add also learn label purely discriminative motivating model also variational instead categorical symmetric unified objective optimisation generative optimisation jointly resort
show understand interest discuss detail mention one assume specifically denote let j j linear time follow auxiliary slack solve algorithm convex unconstraine separable separable encounter across rewrite splitting considerable constraint dual eq conjugate fit asynchronous manner describe include least agent allocation therein empirical examine coordinate descent analyze pt experiment convergence rate establish laplacian graph demonstrate topology star run follow decomposable constraint choose evaluate h clique topology topology acceptable long topology sparsity communication require portion essential pair compare star diameter diameter
hyperplane class origin kernel svms introduce derive note lin standard construction derive kernel principle nonetheless euclidean leave new base locality sensitive unless kernel except binomial recognition video comprise individual capture video represent recognition use video capture normal create subspace individual set remain new derive pl outperform new outperform polynomial achieve overall bar outperform performance contain pose image pyramid descriptor acquire pose result use compute
obtain integrate variable term equality equation htp move rate vertical processor plot dark red corner processor code deep color mse green color red color violate achieve significant reduction increase go iff iff next consequence let corollary processor grow mse rate average double equation converge weakly notation covariance interest double averaging convenience non vanish variance asymptotic variance lemma extend dimensional symmetric technique matrix generalize multidimensional go look notation move keep parameter symmetric precise contain scale become block
symbol membership gmm represent level green compete increase explain formulate assume membership although position parameter distribution true covariance figure gmm sign pair monte replicate indeed aside flat prior directly indistinguishable position utilize multivariate gmm method see case position vector distribute sbm note illustrative sbm specifically obtain mean rate see slightly empirical sbm demonstrate robustness sbm final sbm parameterize position case approximately error approximately pair gmm competitive bayes pair analysis
cumulative straightforwardly follow basic additionally unitary length end lie challenge still partially researcher number type regular cube generic examine model
op mat op suggest shall adopt unless drop subscript derive directly perturbation perturbation op singular op op op let top follow k op claim technical analyzing lie non leverage upon theory sharp critical loss signal tensor q large mat mat unfold almost limit subsequence invariance distribution limit surely eq normal since weaker sufficient odd iterate map find result use conjunction available literature amp qualitative asymptotic amp recursion follow establish iteration generic standard satisfied next two initialization ground
abc ic ed gr please integer factor feedback give high high f dt rely definite follow deterministic process mean propose ol ff ff iw start iteratively know feasible dimension spurious parameterization order jointly propose integrate penalty optimize define form follow minimize eigenvector minimize respect respect allow select alternate inner optimize outer update simulation finite estimation notable globally global optimum start follow eigenvalue unobserve span orthogonal possible regressor factor heavily specify exist recommend factor slope know z ik jt correlation time exist panel proportional center lead statistic overcome diagonal exist affect asymptotic term
contrast compressive measurement case justification exploit across become phenomenon matrix achieve concrete projection hope irrespective leverage approximation large usually suffice require achieve observe compared bind highlight average careful similar spirit entry column
guarantee method factorization utilize sound source type mix complex sound autoencoder separation autoencoder successfully speech
control backward induction residual dynamic treatment regime decide treatment assign effect cost attractive follow outcome depend also intervention population equal decision pseudo translate self method optimal setting design regime optimize utility fix manuscript infinite datum collect develop regime treatment datum trajectory remainder manuscript organize follow explain develop gradient descent minimize section conclude remark time trajectory th trajectory decision maximal take summary capital letter variable subscript potential effect treatment
precisely define quantity estimator base weighted mixed weight suitable strong sparsity give strong group assumption review support work work suppose satisfy bind scale corollary work assumption exhibit although dominate interpretation order initial coefficient projection nearly pseudo inverse propose naturally control tucker kkt automatically tx tx j tx closely relate constrain lasso guarantee tn whenever
south east table index header txt txt index txt index header txt table index true txt index header table index header plot index header txt index header minor title xlabel k ylabel ndcg pos east header plot header txt header header plot txt plots txt header header txt index scale minor title xlabel ndcg legend pos east header txt txt header plot index header header plot txt header xlabel ylabel ndcg header txt index header txt table header x plot txt index header plot txt table x header true txt index header plot txt header txt x index true plot minor title xlabel ylabel ndcg index header txt header true plot header txt table index header txt index txt table header txt table index header plot txt index table header scale title xlabel ylabel ndcg header txt
coordinate nearly plot good rank increase rank substitute width xlabel ylabel coordinate coordinate error finally depend find insensitive fit improve htb xlabel ylabel coordinate coordinate coordinate error add wish row continue set estimate new row computation alternate representation new global minimum new new three rank serial share memory implementation fit implementation subproblem implementation date encourage reader package implementation special subproblem program implementation find encourage interested date collection specify store array correspond specify miss correspond object characterize stop alternate procedure fit miss loss store penalty miss list fit automatically add offset scale py correspond iterate solve problem criterion meet regularizer support py quadratic huber hinge ordinal quadratic implement regularizer code model implementation aspect usage date encourage reader specify datum like tuple list loss rx rank code fit loss loss rx x regularize loss rx fit history mark convention regularizers may nonconvex simple globally useful factorization view unified parametrize modeling view pca loss model loss ica thesis hinge divergence regularizer nuclear max norm thesis regularizer factorization constraint literature change regularizer may pca pca svd indexing nonnegative divergence al community induce penalize decomposition pca review focus tool integrate heterogeneous canonical understand eigenvector structure de low data nominal ordinal et label label image text dimensional space recently language processing document computationally generalize np compute weight completion result distinguish way optimization refer matrix factorization present method alternate newton gradient relaxation semidefinite program iteratively entry solve observe method conjunction exploit gram semidefinite intractable optimality semidefinite lead semidefinite factorization
define claim bind enough lem worker belong part use rescale last inequality k k ta rgb op title title proposition definition conjecture definition berkeley berkeley berkeley berkeley significant performance dramatically necessary communication strong well mini batch converge quality quickly batch gain fast theoretically justify sgd approach distinguish communication efficiency optimization assumption cover case case mini though update locally process mini dramatically
near whereas near leibler contain case less ccccc lr tr tr lr tr lr tr tr tr symmetric first mixture number replication ccccc tr lr tr tr tr tr log density value replication respectively beta replication assign label fit two mixture assign label base posterior large wherein next fit component concave record density list size run local run different allow iteration case wherein likelihood flat would em estimate setting flat expect performance outperform two half component dramatically even cutoff point define move usually low difference affect label somewhat totally find log see detail concave percentage datum however uncertainty differ two particularly useful clustering assign less cutoff asymmetric g screen cancer want misclassification case near center great divide center component normality log concavity section new three estimation old national many old analyze explanation people day fit symmetric concave fit variance around bin plot histogram symmetric log mixture et fitting estimate component assume force
use minute matlab clinical intensity function asynchronous medical investigate around investigate make method intensity event increase flexibility infer abstraction event purpose transform raw form standard clinical intensity intensity contact system usually increase increase condition instability probably generate contact severe frequency medical contact similar value
eq partition small mass size capacity energy landscape bins space bin energy bin estimate size energy landscape contain barrier leaf node volume repeatedly smooth landscape volume merge measure desirable difficulty figure landscape look difficulty learn record length color bar leaf true node ii minimum minima curve error vertical like roc operator pattern recognition sliding threshold curve characterize difficulty auc area ii task close impossible problem correspond difficulty measure difficulty experiment move proposal convex involve use first
essential capture topology diagram moreover persistence diagram persistence diagram come persistence diagram create connect maxima merge birth persistence diagram way indeed parametrize persistence diagram fig rectangular particular pixel pixel threshold piecewise triangular mesh heat signature yet commonly crucial aspect persistence diagram respect perturbation infer persistence diagram consider map require diagram natural metric associate persistence diagram speak diagram two diagram persistence diagram infinite distance range persistence
norm complexity dictionary least except contrast yield still heuristic give amenable technique precede write version set possibly uncertainty uncertain corollary uncertain give fix dictionary occur independently signal contrast learning arise previous primarily uncertain implication equivalence empty eq equality homogeneity triangle homogeneity triangle satisfy desire complete attain pair regime course regularization upper problem recover long equivalent unless one discrepancy computable well see satisfie long equality dimensional long strict almost sense gap
circle gray measure order connectivity sequentially maximum c connectivity solid synthetic correlation measure range initial measurement ref hide carefully informative many som nonetheless comparable sir additionally give data specie confident specie infer interaction limit limited number specie output conceptually hard since observe dynamic chemical specie vary case select adaptive single range infer ability condition measurement select able predict chosen range range range fig equation dimensional roughly time measured chemical specie crucially computational complexity sir even hide adaptive sir model exponentially model impossible system many guarantee traditional infer require predict dynamic sir fall process chemical utility say mechanism response unseen qualitatively one necessity feedback importantly analogy model
limitation code hour height width ne u ne ne ne chapter head chapter head subsection head head compatibility corollary conjecture em fc maximize detection fc jointly use investigate adopt theoretic fc theoretic quantization rate address coordinate demonstrate joint approach
update severe systematic limitation hyperplane along filter onto instantaneous dictionary achieve project counterpart algorithm steady yet treat grow paper natural I input present ascent subspace steady mean
error among uncorrelated want make analytic average realization back present show excellent propagation specific task average matrix become eq determine critical determined chosen drop I definition nonlinearity effectively approximately apply row nonlinearity rely analytic need assumption product property vector layer calculation square precisely expand logarithm taylor optimal slope expression low indicate reasonably
cavity similarity indicate methodology near neighbor class annotate phase compare ec annotate ground truth I truth count successive match ec thus ec ec two ec belong conversely similarity ec relevant formally similarity digit ec figure give six correspond ec infer annotated encounter among rank condition query entire ec number annotate ground truth information perform call rank denote couple query database implicit couple conventional optimize together vector minimize set denote truth similarity query contain outer pairwise difference rank loss minimize dual q kronecker feature information feature individual kronecker easily kronecker dual universal indicate access training protein approximation probably kernel yield restriction fp similarity
fall restriction distribution uniformity optimal obtain work restrict broadly fall property decide efficiently property classic problem obvious candidate poisson truly know quite test allow vs contiguous place approach give exploit tolerance identity support effective accuracy barrier complexity answer independent might deduce low however easy member establish dependence tight related distribution
numerical table bootstrap technique substantial gain adjust bias adjust analytically bias ba iteration ba least importantly ba ba reduction expense mse systematically adjust adjusted detailed rule terminate iterative two middle table bias column fall bias record make mse mse conclusion record panel case analytical adjustment record third panel table column fall comparable realization improvement overall summarize confidence size nominal length nominal qualitatively bootstrap ba ba adjust ba ba bootstrap adjust record coverage poor ba produce accurate key point narrow expense inaccurate coverage secondly coverage accuracy yield expense precision negligible interval contrast adjustment analytical adjustment
satisfy latter forest work x notation easily therefore consequently reveal finally eq tend zero assumption satisfied cut recall prove let multivariate univariate know additive implie let tend zero tend assume decrease exist b x x k x since contradiction almost surely necessarily rest x deduce cut k inequality cut perform root case cut direction e cut perform along word calculation besides calculation tail deduce union eq exist q event occur remainder quantity illustration illustration dimension cell case
generate domain graph edge training vs part comparison class need become costly mean use classifying xlabel dimension projection ylabel accuracy plot illustration method art accuracy show generic enough quite author classification validation base feature include gender factor pos patterns report maximum seem acceptable tool gram split reference text vector cosine similarity improve repeatedly set similar accuracy note classified contain token hard reduction discover structure
dr database al match sensitivity specificity dr patients dr report et al automatic must absence clinical detector serve detector prove select modular preprocessing method candidate evaluate value competitive promise component approach support science project develop system office technology contract om om om p inf phone digital issue medical candidate use ensemble would pixel extract
table x index accuracy mention work hard apply task carry krige copula inversion cubic approach inspire informally speak perform primary individually reduce independent multi help yield multiply py x py approximation learn two speedup provide easily numerator advantageous distribution introduce analytical solution precisely predictive copula would obtain reduce gaussian
bind mn first ce ij ij first accurately deterministic propose dependent square recover biased matrix thresholding operator recommend technique provably recover convenience first define rx mn error weight weighted case show constant expect error relate minimizer thresholded order inductive standard inductive want matrix completion special inductive correspond also disease prediction theoretically analysis motivate world example
cr mathematically immediate belong constant rational cr sure avoid channel ps se decode primary queue full corresponding queue empty queue inactive indicate service throughput cr consecutive action maximize secondary service cr queue finite dynamical cr user cr channel cr cr choose feasible action receive immediate take place repeat reinforcement naive implementation discount idea value two
greedy orthogonal omp decode computationally expensive omp collect random random projection sense small sparse context compress sense recovery pursuit nice well page decode count achieve similar accuracy count sketch need
col sep comma sep comma ylabel style west height width axis font axis sep comma expert col comma xlabel ylabel post legend style west major height axis style axis sep comma col comma ylabel post inclusion legend west grid major every font axis sep comma expert table comma inclusion covariate gap size increase eventually low knowledge monotonic averaging repetition yield smoother comprise beyond significant different assess instance datum namely classified range close skewed accuracy instance site classifier return absence information auc area receiver operate insensitive auc auc perfect predictor xlabel ylabel auc legend none legend east grid major height
sampler keep move gradient fairly langevin metropolis reach distant start reverse back phase look origin large move ie change monotonically work well hmc compute minus log ad hoc close reciprocal nd estimate nd independent dominate hmc partial respect coefficient concentrate mode reduce loss fairly update time reduce substantially sum fixed want justify trick important coefficient coefficient update hmc ability hmc random may hard update need update iteration need square gibbs sampling phase sample fix hmc adjustment nd derivative macro pt pt bayesian hyper li high areas sciences example genomic research gene class thousand wish fit date method tail li knowledge fully bayesian appear literature laplace sake attribute restrict dimension laplace hyperparameter recently paper report hyper lasso mcmc
therefore center finite denote integrable kernel linear map gaussians since contain fx function since vanish center expectation xx european european carry support fellowship ex combinatorial locally evaluation method system allow heart allow replace inversion provide blue unbiased noise modern wide medical imaging signal big g recommender system whole property minimize explicit depend explicit compute locality yield tradeoff characterization include explanation circuit combinatorial finding optimality rather natural compressed sense dual sequel field
couple density normalized see smoothed shift bound region define solve poorly component example similarity decrease offset tell semidefinite solve simply couple modification trick show albeit loose bind capture exponential decay gaussian display offset line display straightforward location define solve conclude counter kernel kx calculation compare normalize span top eigenfunction map encode label important level spectral section involve intuition imagine density imagine mathematical bandwidth separate principal vanishe discuss goal quantify equivalently root density measure distance
fast discriminative feed quickly discriminative new task contrast previous learn raw theoretical guarantee contrast construct cross feature mining information provable framework supervise handle pre process score datum framework latent framework scenario input crowdsource different group approach construct score average leverage source challenge scenario section elaborate end supervised framework purely however datum challenge distributional likelihood convex especially involve incorporate generative extract input classification setting consider regression set compute r access understand label vary change locally learn form moment yield high function call order score learn use unlabeled feature apply tensor variate particular setup yield high derivative accurately use feature parametric framework incorporate previous obtain derivative find spectral find notation symmetric decompose hand
weight minimal inside maximal cut edge sample biased increase observation expect look cut inequality cut institute institute spectral computing costly limited notion approximation generalize cluster propose new algorithmic sampling improve considerably learn obtain costly example may metric biology case initially alternative approximation l graph laplacian graph easily value spectral see discrete spectral vs imply use min relevant include nd eigenvector simple example cut theoretical
stable stability implicit td show td td consider assume instability clutter let write assume iterate stay guarantee iterate td performance explanation start argument argument eigenvalue
assume function cover movie show return ranking value table list appeal high list cover diverse list seem cover movie depend need priori hard list diversity section list list property rank item item family x c list item action family maximize item diversity follow solution efficiently behind recommend control consider diversity preference item popularity score predict rating entry utility objective diversity increase typically g address balance increase diversity maintain utility consider primary aim recommend item maximize choice utility recommendation increase diversity diversity weight problem formulate order gain gain choose real instance movie recommend dissimilarity product recommend website diversity diversity item add diversity empty generality particular satisfy general cast maximum argue code work sort decrease utility sg contribute finally recommendation second movie movie list movie place
ip f ip ip ns jj previous markov leave invariant rao together step smooth particle intermediate put present smooth model enable simultaneous smoothing initialize compute notational convenience variable foundation enjoy property certain ergodicity maximizer particle internal together detail empirically
use nlp lasso equivalent equivalent model penalize mm mm decade derivative analysis coefficient popular selector iterate show deal
sx research support nsf grant n research support china cb grant grant dms dms grant w nf name corollary stanford e sparse noisy differential exist correspond sign path discretized setting path piece discretization lead fast sign minimax setting stop rule identify rely development differential time image euler discretization recover signal noisy eq oracle unbiased vanish meet bias remove see multiplying satisfie path time sign thus q motivate differential indeed existence uniqueness good path argue covariate uncorrelated proper early stop sign linearize bregman
convergence selector go zero deterministic convergence rate distance proposal numerical aspect detail incorrectly reject go reject fix ensure cycle must select initial uniformity reject assume enough select uniformity reject close cycle uniformity accept estimator initial nan otherwise hull et al nan uniformity replace definition
asymptotic size many dimensional high situation situation explain use local weight weight asymptotically addition comparable lowest asymptotic among introduce address power addition give asymptotic investigate conclusion summarize proof section average statistic
probability c cccc cccc independent c k bic aic bic exchangeable aic lasso aic bic coverage summarize target procedure set coverage probability independent especially naive interval selector however configuration line lem state vanish selection confidence constant valid moderately probability nominal aic use fail minimal probability selector drastically aic confidence nominal failure interval especially interval reason observation selector stochastic probability small aic bic regressor study regressor make cover former target word situation hold validation implementation lasso course thus depend additional repeat find h df df df f denote last view def establishes follow imply expression like empty beta beta p g prove far l far inspection every every pair satisfy generate construct subsequent abuse shall argument special influence suppose maintain hold satisfy irrespective predictor selection interval model standard target inference post prove
particular discard turn inconsistent example proceed exist consistently first minimize substitute single piecewise equation check small member absolute define interval mean linear zero leave change sign solve q see inconsistent discard accordingly minimize search reflect multiply general technique improve alternate conduct another meaningful mf recommender system descent penalize real mf pr mf construct datum saddle
positive negative class award micro precision fraction positive fraction actual predict harmonic express negative harmonic f translate hold concave accuracy offer discuss actual positive gold actual negative sum fixing linear mention mean assign gold score assign complementary gold see compare negative costly classification domain positive negative considerable cost information mention micro treat prediction score macro f similar accuracy sum monotonically retrieval researcher extensively
implement compare method domain big acknowledgment acknowledge project fellowship fig rgb rgb sections box box intuition appendix chapter integrate couple gps result approximation even cost big
factor radial sphere radial distribute density eq radial density parameter analytically propose reference parameter procedure next likelihood split depend whereby strictly concave whereby long hence attain remarkably possess enough still end maximize optimality definite solution thus rescale constant uniqueness upon existence positive appendix rewrite introduce transformed vector turn solution recover naturally two equivalently ii equivalently positivity preserve also sufficient optimality concave limit column ic prove
position resemble maxout yet filter pool convolution max pool convolution map product follow convolution offset extract patch primitive filter dictionary implement carry library stack convolution layer activation unit subsequent similarly final two imagenet layer supervise architecture experimental c mini cb mini cc pooling normalization mini normalization propose pixel retain response neighborhood pixel apart
c cluster c analogy cluster analogy h mm percentage cycle duration h south mm role theoretical view call learner dependent possibility highlight exploratory behavior simulator qualitative mainly behavior major interest pathway namely possible learn dynamical analysis
enyi entropy generalize gaussian convolution sum describe diversity l remove show entropy low summation core coherent furthermore nn follow distant dictionary equivalent normalize connection approximate become bound increase number grow decrease coherence measure measure online coherence less atom thus diversity within high threshold shannon p p j present kernel write norm span function r n n worth conduct satisfy approximate straightforward theorem lower distant bit turn r enyi entropy case former list include also small
write runtime primary exponent propose complexity algorithm either nature seminal liu make assume generalization decay distinguish require exception family despite informally graphical decay asymptotically know pairwise temperature ise lattice survey article possible efficiently correlation paper convex incoherence explicitly careful provably ise base isometry likely algorithm happen base markov mix closely mix thus class
proposition since proof recursive update stepsize long approximation infinity subsequence solve systematically period horizon horizon move horizon stage programming equation solve identically derivation l j use logic proof update recursively diagonal covariance gradually expand repeat horizon original store potential benefit optimally vary stepsize adapt outlined action problem insight sensitivity consider normally reward discount policy initial approximation furthermore use stepsize secondary stepsize stepsize minimize error scalar secondary stepsize bias stepsize behave early quickly converge limit point behave stepsize tends happen used issue tuning view slightly stepsize rule give yield parameter harmonic stepsize expect convergence million reinforcement issue logarithm stepsize stepsize yield also omit rule harmonic stepsize properly eventually exhibit improvement single performance rule contrast
bad alternative bad precisely statistical dynamical space go equilibrium review dynamical monte utilize method large point correlation partition function evolution auxiliary speed article modify accelerate stem mix equilibrium mathematic chain markov adopt section introduce mathematical definition hasting follow possible configuration markov chain specify chain switching path get independence row length evolution th px evolve ty assume power multiply also define scope
since obtain indeed landscape tell ground though hard away value lie actual growth still point design idea teacher student comparison mnist first half train train network hide layer relu last teacher mistake sgd teacher label second replace tag probability teacher h c error st student student teacher
randomly spread production critical see randomly arrange eq represent dimensional element batch worth discuss modification case lead explicit plan determine lead numerically appear plan study minimization n minimizer put fix minimize numerically denote turn search pass optimizer stage round control simulation conduct insight property procedure design distributional relevance see accurately second mind estimate sample affect operate characteristic first corresponding cf sample production line numerically invert standardize first four employ biased bandwidth gps indirect repetition second stage plan comparable stage gps selector plan situation indirect comparable represent symmetric small whose notable minima
provide prediction note covariance infeasible moderately sized obvious improvement achieve building appropriately scale uniformly replacement possibility detail build great depth produce incorporate estimator become statistic case incomplete situation x I go asymptotic present central without consistency guarantee incomplete order nx nh k n n base approximately root full maintain relatively subsample many computationally equivalent traditional bagging full bootstrap final theorem k though easily tree great assume response bound bound algorithm asymptotically provide load build tree subsample predict prediction final precisely bag use build full estimator may carry distributional tree prediction minimal regularity condition build tree ensemble build method building forest occur additional determine random statistic u
strongly distribute lagrangian connect responsible outline dual primal generally agent propagate c algorithm nc nc highlights sequel covariance diffusion strategy able verify theorem difference relation close desire range step size therefore stable also even dual utilize consensus positive algorithm study solution act enhance dual define always require definite converge prove consensus strategy become unstable agent note unstable growth consensus equation implementation remain show attain term range steady consensus detailed algorithm optimizer since error error vector evolve accord know optimize conclude create resort transformation ignore redundant dual singular decomposition partition diagonal singular non term furthermore
complexity analysis idea label value high q high variance dominant dominant perturbation translate claim rip need proceed thus complexity incoherent whitening since access slice addition perturbation whiten perturbation term whiten whitening differ sense analyze slice moment therefore empirical slice make perturbation new refer whitening term slice different depend detail claim perturbation translate concentration classifier approach except error rip bernstein tensor loose derive tensor z bernstein assume score function convergence perturbation vector suffice otherwise
start environmental dynamic receive environment update posterior bayes action intend maximize expect return criterion tt uncertain world balance exploitation discount play crucial role relative reward opposite illustration exploration exploitation policy index belief decision material find intractable state either potentially unbounded require trivial multinomial rich inference monte fit planning side planning call thompson capable handling involve plan adaptive plan directly albeit cost unclear integrate development approximate scheme perform base sophisticated sampling planning select action solve action current perspective bayes ts computationally proven empirically reach theoretical regret armed bandit handle mcmc generate optimistic
rhs eqs result establish assertion combine bound c c n k go eq rhs q hold unnormalized prior rhs evaluate third term fourth upper bound yield measurable eq mean square sample predictive expect error source time higher two require recommendation comprise rank analogous define singular enable decompose high several naive require
architecture section target choice viewpoint object space correspond classification contribution definition b possibility distance distance similar localization pose require optimize use variant loss network probability test belong prediction pose evaluated extend annotation challenge dataset annotation object annotation introduce joint pose provide accuracy viewpoint standard precision ap computing consider discrete viewpoint associate size threshold set orientation annotation network train imagenet evaluation mark truncate evaluate provide
conclude impossible distinguished expert exponentially tune interesting drawback come uniformity hard one instead bound like get expert regret key quantity instantaneous eq hold develop variant extend propose tuning rate additional factor secondly bring essentially expert report first confidence optimally return excess improvement begin introduction loss rate k cumulative optimize theorem respect desire bind work trick suboptimal quantity new
assume corollary subsection replace theorem corollary simplify treat fix value item give reduce magnitude dropping item however item show continuous simplify monotone enough validation step sized block validation remain drop randomly h nj overlap aside far exclude select candidate matrix base decay power gap separate validation fold validation variability restriction
semantic recent focused building evaluate separate relation focus evaluation closely possible section development parameter section composition surface employ multiclass four first level hyperparameter separately see optimize development follow instance positive published previously publish recent model inform tag predict relation show distributional obtaining score present publish metric expansion attain work vocabulary collect token token vocabulary experimental vs detection distributional semantic many lexical center shift identification relation key parsing task identify seminal task lexical word tag raw syntactic include identify subsequent explored aggregate multiclass relation
r principle many wish analytical available penalty ensures estimate use vector zero goal reduce autoregressive induce eq lasso set coefficient take impose exhibit interesting behave define smooth differentiable nevertheless eq algorithms problem show smooth proximal term purpose follow supremum gradient ss ns accelerate proximal gradient lipschitz
dpp empirical study beyond particularly diverse subset selection compact include quality item coverage supplementary material item sample matrix spherical exhaustive possible noise add drop repeat sample another diverse pair yield training ground precision quantity hamming mle dpp validation testing state various setting dpp thing item among conduct experiment impact fig know ground use outperform number increase method generally improve close oracle specification vary fairly mi quickly subset whereas mle specify parameterization include ground mis specification effectiveness similarity performance ground truth similarity nonetheless mle encouraging parameterization avoid specification outperform track margin similarity panel performance mle plot red color approach blue color along horizontal axis evaluate
step balanced forest draw take split partitioning recursive put choose choose put compare terminal uniform tree forest minimax rate paper performance risk general interpret focus mostly risk tree forest infinity rate tend decrease fast infinite forest estimator provide forest forest contrary empirical infinite bias theoretical illustrate simulation section three model section close rf notation appear line line decomposition purely forest rest build let let independent possibly estimator several every partition convention deal belong partition tree call forest define consider finite finite risk among function q decompose partition integer q proposition average every quadratic risk term rewrite name wise case tree conditionally asymptotically quantity integrate
ij j bb ab aa diagonal square identity let minimizer tucker kkt subgradient evaluate hereafter mean clear review uniqueness fit lemma proof uniqueness kkt unique regard assume column vector augment augment rewrite kkt condition unique uniqueness fix linear determine normal algorithm article basic detail sake understand low go technical utility couple joint subgradient assume mcmc joint sampler simulate error scatter confirm accordingly subgradient autocorrelation surprisingly complicated sampling section approach resample base method tail tail probability challenge accurately weight simulate sample absolutely impossible bootstrappe lasso simulate directly b diverse autocorrelation set writing examine respective space p vector equivalently vice versa equivalent immediately kkt condition rewrite essentially define unique helpful I p solve collection give easy extend nature illustrate map four
network rnn parent child tangent compositional bottom ultimately example sentence little certainly fail correct relation role play entity sentence address representation semantic play entity rather information logical parsing
overlap screening adopt polytope group experiment demonstrate rule dataset efficient rule discard determine represent different hence overlap lasso screening rule dpp dpp form tucker use kkt screen discard solution dpp include rule dpp screening group overlap group screening rule dpp range group derive screening rule dual vector variable dual project
propagation probability small mixture demonstrate topic table individual mixture mixture item htbp topic probability topic topic retrieval item idea raw topic mixture dataset provide nonzero probability column percentile percentile percentile show mostly topic topic propagate mean would investigate topic give iv ij ij j mean fairly separate topic coefficient threshold topic coefficient edge table overlap node network researcher number researcher overlap different pair overlap explain nature movie rating interested category movie friend interesting though overlap nature influence behavior among represent overlap overlap entry row overlap cccc summarize edge overlap coefficient statistic topic fairly area topic dependent topic mostly separate
close observe numerical parameter therefore hand get z z derivation fact limit base slope dependent term become collect supervise extensive find task relation weight hardness analytically binary equilibrium solution explore weight reveal organization isolate explain previously behavior provide analytic simple part physics physics spin tool constraint
definition keep integral dual schema condition basically note set feasibility constraint trivially satisfied element whose naturally termination feasible relaxed set next core high decompose small subproblem mrf computer great success flow estimation object set vertex every master mrf resource adjust maximize relaxation decomposition relaxation affect speed convergence decomposition mrf define consist per subproblem relaxation subgraphs hand lead large subgraph message interestingly consist subgraphs integer replace constraint negativity lp optimal mrf solution exist type subgraph efficiently belief message pass exchange message graph various besides decomposition small mrf even relaxation cut mrf mrfs know mrfs potential mrfs write eq source edge express one mrf see relaxation solve lp furthermore subgradient admm relaxation difference even subgradient long requirement converge alternative smoothed accelerate scheme apply primal area problem long successfully miss pose recover satisfactory need introduce fidelity possibly penalization justify determination posteriori regularization penalty employ impose lead feasibility hybrid
false positive one voxel discovery level active voxel false positive positive hypothesis logistic svm randomize voxel vary select voxel multivariate voxel voxel different restrict probability voxel absolute change change regularization force close show selection show top voxel randomize mostly introduce positive constrain block subsampling bring show work slice brain voxel whole generate show cluster represent index represent cluster spatially I index cluster feature spatially simulate voxel slice multivariate pattern bb eps bb eps bb voxel imaging fmri brain complete probably voxel improve interpretation discover potential voxel space training difficulty attention amount fail use rate probably discriminative voxel randomize superior negative stability selection voxel structural stability randomize block subsampling fmri feature selection recognition brain read kind pattern lead subject brain diagnosis person feature classifier receive attention application diagnosis select voxel candidate feature discover manner accurately mainly construct classifier ignore redundant informative ignore inclusion informative may
memory per partition primitive schedule update schedule decide variable simple possible schedule shall later schedule dynamically fast converging avoid already converge parallel dependency incorrect primitive worker compute write g fraction worker result iterate primitive worker mf application partition key standard array build primitive ensure date variable automatically variety store synchronization parallel asynchronous present worker ap usually risk algorithmic error error guarantee alternative like ap work shall parallelism cover ml
hypothese useful marker marker possibly nest principled technique model fisher kernel propose simple proportional correlation kernel fail give satisfactory setting kernel due setting genetic multiple q jk jj mutually incorporate contribution g jj jk input kernel marker mutually kernel marker effect mainly account sample incorporate via principal marker group denote write j
recovery ensure secondly way deal practical simulation perfectly even constraint early termination affect bound whether perfect recovery lastly approach often retrieval compressive analyze trade adjust aforementioned enable evaluate provide phase retrieval
precision choose namely snr illustrate decay mse algorithm calculate process behave mse fast measurement perform measurement point therefore seems outperform scenario reason behavior change easier narrow need convergence speed multi advantage adapt without manual estimation also complexity iterative weighting exploit figure indicate weight outperform online learn however compare art include mse highlight krige offline scheme entirely krige technique certain measurement despite figure serious computational number fact calculation new measurement apply gaussian krige certain measurement significantly observe base accuracy fashion iteration parametric sample simulation amount point iteration art learn reconstruct addition path measurement mobile smoothly robust low impose operate time predict location exchange point
comparative two public class vs person probe ii vs person available identification pls histogram drive accumulation feature pls first give image rich edge dimensionality space reduce employ person person people group rectangular ratio descriptor rectangular block ratio descriptor foreground construct consider person
hour speech dataset hour corpus model hour speech testing corpus able hour distribute linearly spaced log term window ms evaluate feature frequency critical system particularly hour normalize per feature decode choose hold hide deep speech rnn neuron hour input dnn gmm dnn dnn deep speech baseline al dnn fisher hour corpus system deep speech ht gmm hmm et dnn et et dnn dnn et cnn hmm mlp n n speech testing speech evaluation tv restaurant car drive text
table complex false positive demonstrate effectiveness aggregation false underlie complex figure table quite across distributional score normality instead residual linear df df node score comparison compare simulation log score equivalent posterior package extend dimensional p pa score default non false positive although result total factor least positive perform former considerably still couple mainly bootstrap algorithm drawback practice alternative aforementioned log likelihood operation average average improvement select building good result table couple dag hill dag learn propose structural hamming distance investigate aggregation able greatly compare procedure couple dag remark definition graphical field biology social linguistic direct dag bootstrap
plot expect intercept volume outperform mean versus respectively though respectively inspection criterion seem well addition approximate volume criterion cross euclidean cube behaviour large natural induce topological natural decomposition boundary tb map face blue greatly blue region greatly shrink red wise blue space ideal boundary natural space kind ideal sphere infinity hyperplane spherical rt sf relatively polytope linear inequalities union lastly serve sf ss decreasing make arbitrarily term approximate xx way end section raise possibility volume might generic qr qx integral allow bound neighbourhood sr nc z
j k kk h give contact infect initially adopt direct entail transmission draw transmission transmission time differently edge case infect neighbor continue infect remain infect entire multiple illustrate cascade cascade dimensional infected infected window never infect cascade trivially show likelihood cascade st ft hazard survival infect node cascade hazard infect cascade cascade cascade give st k instance contact associate ji nn cascades cascade direct cascade cast estimation infer edge correspond node subproblem per infer incoming generality problem subproblem cardinality nod node
generally explore space various hyperparameter promise epoch frequently epoch observe hyperparameter tuning drastically model fitting machine partial training curve tend follow decay stationary exponentially decay basis order accurately forecast variety decision dynamically create rapidly find hyperparameter potentially uncertainty exponential problem develop definite develop temporal theoretic framework process experiment several machine
trivial formulation follow efficiency variational technique rank matrix white gaussian unknown completion scenario choose goal measurement reconstruct study convex nuclear norm set regularization advance
alternate diagonal limit fact component tangent account perfectly compute diagonal numerically unstable discrete derivative sensitive grid illustrative figure recover quite straight form field recover component recover alone next two field tangent plane figure show clarity remarkably cccc show top bottom consider sample direction tangent map three dimension longitudinal survey united data consist token identify job pair occur take span node walk number job job diagonal step motivated essence nature phenomenon model largely job notice change affect
predict log class cluster miss unknown assess partition show log aa aa spline regression mixture five spline notice correspond mixture misclassification robust polynomial cc clustering cubic spline knot cluster em seven knot regression spline model variation number well value cluster decrease rapidly regression spline regression discard gradually converge curve objective spline mixture behave similar become cluster value middle experiment dataset cell cycle use effectiveness time course use standardized construct level point analysis cycle curve five cycle figure spline spline provide partition cluster bottom merge middle figure clusters rand index polynomial
train work even second conditioning multi modal boltzmann author generate sentence adversarial net way consist adversarial capture discriminative probability training perceptron generator generator output train adjust adjust min
use non negative factorization square text term topic document sort word word topic pick tweet tweet close consensus tweet connect cluster cluster frequently close tweet cluster tweet way obvious nmf color tweet relate topic tweet slightly create short right see figure topic find
million allow scalable derive qualitatively size entity merging database analysis process many modern database bayes realistic method elaborate merge pose refer merge share unique database unique entity include record identification entity resolution link record computationally
hessian appear show example write gp reconstruct image path result fig poor reconstruction straight go region little mean result reconstruction section synthetic digit motion datum image improve image camera background acquire give total per attain straight reconstruct along clear geodesic avoid region reconstruct image measure distance fig error straight poorly away point geodesic
completely unconstrained bic agreement pick lead group account dependency well performance set covariate respectively directly model well cluster clear numerically however prevent size minimum framework discriminant currently individually advantage response fashion tailed datum distribution multivariate restrictive incorporate mixed acknowledgement work support science author build prove hold almost integrate side g
model describe apply locally lipschitz care possible contraction admm end level epoch epoch cr ir establish epoch establish obtain result contraction break part batch estimate th obtain analysis stochastic I constant require strong impose enable tight strong convexity relate dual note lipschitz prove lemma weak lipschitz length significantly follow short overview establish guarantee build proof since need convenient merge sl couple together two separable part convenient epoch error obtain result play epoch e next add error care proof
estimator suffer connect problem high help overview possible popular access calculate technique joint marginal bias aspect lag involve additional cost slow dimensional naive histogram slow histogram high near would stationarity suffer degree causality measure asymptotic stationary standard framework correspond causality integrate source toolbox causality limited set stationarity augment test schmidt toolbox manual short economic adjustment article stationarity transfer entropy affect highly reliability density bias stationarity slow two result result parameter choice lag lag additionally smoothing understand kernel kx exp correspond infinite express expansion rkh q cross order covariance data parameter choose clearly drawback focus density histogram choice lag choice lag causality big range behave observe causality lag time perform single lag transfer entropy spurious causality case complex observe important general influence smoothness control large conversely overfitte consequence particularly popular convenient size ridge cross expensive determination lag method could increase lag pairwise increase lag degree decrease small lag apart suggest autocorrelation causality testing describe causal instantaneous causality instantaneous coupling simplicity several repeat acceptance approach acceptance loss decrease calculate test reasonable big decide window allow choose window discussion
proportion curve apply conditionally iy additional summarize b bag growth prediction f essentially instances build bag relevant real application political survey bag space instance formally generation let determine pick bag size variable let bag drawing bag p instance bag well cover instance bag bag draw bag hypothesis distribution choose sequence easily translate addition small subset section easily bag size bag hold immediately number bag addition hold bag algorithm version interest useful privacy learn preliminary align implicitly utilize one
implicit gp equation straightforward would thank helpful input differential suggest explicit analogous iteration previously suggest yield standard ode xt differentiable order embed order vast introduction several family implicit speed ode potentially represent value ideal problematic classical solution approximation correctly
ratio observation
tackle discovery formulate privacy preserve decentralize detection multi protocol detection walk preserve discuss future network people recommend people interest topology discovery operate social must interest people may raw towards routine necessary involve current service facebook play role party discovery accomplish node profile social return recommendation complex one level datum successfully
numerically evaluation positively correlate discrimination performance correlation strong value cancer z range cancer implement spectra detailed generate list reference z ratio indicate process spectrum code task outline section vs systematically explore belong three cancer discrimination vs vs cancer restriction force priori number signature signature length inferior computing force signature inferior mrf discrimination spectra outline yielded associate correct decision achieve derive procedure maximize optimize signature reference vs signature retain clique section classify sign evaluate absence see optimize spectra dataset map affine separate cancer display good affine explicit signature discrimination first signature fairly length cancer discrimination task length obviously advantage ratio benchmark discrimination signature take account mrf discrimination reach discrimination h cancer vs vs vs cancer vs vs vs vs cancer performance reach simulated signature discovery performance margin vs vs discrimination vs
consider canonical discriminant literature canonical vector fashion separate estimate come guarantee set express close simulation datum keyword block discriminant feature selection wide application finance biology challenge popular visualization seek maximize variability respect variability pattern novel estimation selection group discriminant goal addition group set propose efficiently generation point suggest natural grid grid carefully choose mostly computational simplify consistent user rest follow discuss insight
back pa ga ti step yield improvement bi prediction accuracy propose primarily way restrict abstract series fine tuning generative learn represent frame across frame video frame common bi machine model whose formally bi image angle pixel space frame hide unit subspace transformation hide encode image rather content
independently detect amount illustrate numerous shift anomaly scenario involve delay describe explicitly specific sign provide detailed setting aggregation maximize anomaly detector knowledge shift quantity early sign normally natural test test coverage case shift compare statistic two possible population length window change case rough maximize several recommendation scale simple ease indeed raw generally interpret easier still therefore indicator correspond rejection give case shift finally point increase specificity detector quantity compare
nonetheless achieve expect operator derive unnormalized semidefinite trace operator ref two semidefinite define triangular vary convenient parametrization q semidefinite vector define triangular take transpose number partial tr b additional relate trace unnormalized operator term cast norm enforce absolute lagrange multiplier enforce infer relation relevant wide give correlation whether influence unless intervention causal generalize conventional complete causal analogue classical alone provide optical optical randomization extract relation quantum correlation follow explain variable causal variable cause act ambiguity system exhibit quantum find surprisingly pair order system top cause common cause acyclic graph direct conventional quantum circuit quantum system box state operation discard circuit gate either top swap circuit dash scheme output state setting outcome passive scheme output measurement projective fixing setup gate notation optical
denote training similarity affinity pairwise relation hash preserve hash closely hamming calculate code mh similar hamming pair multiplication prevent task employ tree hash hash decision tree output hash compare method technique th datum bit auxiliary decompose problem code datum binary way complicated decision tree become task binary bit conditioning th bit equivalently quadratic code bit bit bit
edge argue particular stochastic able even limit entirely satisfactory dimensional number edge term secondly problem partially operator exist extend strong limitation spectral backtracking operator actually bethe hessian physics field show operator directly non backtracking perform stochastic sense community soon standard world benchmark give bethe detail property connection backtrack ise spin spectrum
incorporate tu penalty tu envelope worth positively homogeneous give cf q one zero x direct section ccc limit encode graph tu edge incidence bipartite encoding pairwise function fact envelope ball linearization trick reduce tu cc show objective structural minimizing envelope regularize regularization unit section produce spike value dimension compressive random datum produce datum relative compressive regularize formulation bp tu budget describe lie include constraint point
observation algorithm extreme quite relevant see distribute among fig significant general fail median median regard moderate huge attribute amount miss within result frame q would able unseen two test dr spectra ls rest frame miss spectra definition spectra miss component discuss may missing explain highlighted coverage range illustrate regard absence datum eigenvector show good algorithm ls ls noticed width emission miss consistent come large notice l subset fail converge large correctly reproduce emission
reader question necessity efficient hilbert directly wasserstein latter preferable nonparametric setting establish weak concentration around dirac wasserstein rather posterior respect hellinger universal detail outline yield I assume condition hold eq part independent imply typically slightly bad contraction difference handle bind throughout consider simplicity generalization conclusion obtain condition see address situation statement problematic hellinger wasserstein hard estimate practice next geometric metric distribution hilbert discuss define separable reproduce property therefore l define give hence imply convergence wasserstein hellinger metric translate constant hellinger two distribution q
cycle convenience scalar iteration previous update work task local linearization involve introduce accordance former linearization multiple assimilation justify situation conventional iterative enkf sub optimal instance nonlinearity challenge filter statistically show later consideration account iteration local evaluation jacobian discuss give formula formula root precisely transform observation linear error localization eq addition generalize introduce scalar front optimization algorithm special intuitively approximation estimate tends inverse implementation conventional use assimilation difference regularize method focus residual specify residual residual criterion change inverse useful iteration rule scalar gradually reduce aim prevent drop analytically residual norm sequence satisfied may desirable residual norm prevent either state reach b process outline eqs
aggregation estimation proof need theorem existence asymptotic expansion natural variable probability click valid bandit produce go concentrate static actually issue rather tend make evaluate policy algorithm produce long nothing due chernoff classical especially randomized example argue cost complicated smoothness non justify true introduce experiment compare model linear news news display kind news universal interesting ii news consist contextual approach contextual bandit recommend news evaluating fast
specific identify understood fig topic document topic document generation share topic iteratively current specific word non proportion document likewise fix topic estimate follow computationally jointly perform minimize widely accept comparison logarithm bic balance goodness limit type goodness derive bic modeling dimensionality huge bic aspect type share model topic validation solely corpus minimum topic bic estimation subject introduce prior skewness topic asymmetric prevent word dominate however parsimonious proportion specific fashion spike sparsity use share provide relevant topic sparsity word probability use model small proportion cross combination background argue generalize keyword introduce huge every topic topic proportion probabilistic give proportion topic specify topic lda exhibit salient fashion rest universal
step support common update index row measurement update individual projection w combine refer sc belong weight equation test correct also term notation express matrix form stack training linear assume go solution residual residual residual recognition task single available realistic assumption sequence comprise video frame
exhibit also movie via cluster factorization real rating algorithm simulate movie recommendation recommendation algorithm amongst friend star dense still miss entry rate item rating simulation receive reward mark netflix rating nonzero entry netflix result user u user recommend make movie recommendation movie completion item recommendation item feature rest furthermore reveal dm thresholded rating thresholded rating dm estimate training give compete outperform result netflix similar propose online justify collaborative work key type al neighbor collaborative filter predict unseen study popularity friend examine ability predict rating understand move limit user type type recommendation
grow literature optimisation software package hyper unknown hyper focus deterministic new hence base assume set assumption trivial suppose z combine get convenience know write know f f f c simplify
accuracy hierarchical hold water equivalent cnns feedforward one quick intuitive rigorous necessary van inspire become default report mnist handwritten digit straightforwardly extreme rapid training error implementation individually combination indistinguishable innovation ensure operate patch weight percent hardware iteration backpropagation number unit require suggest challenging dataset type network problem activation occur vector notation convert activation plus value utility combination describe introduce denote vector relevant class mathematically entrie entry class vector activation training vector ideally eq although exact potentially usually solution standard solution express equivalent solution
hard thresholding infinitely penalty exact group divide variable group entry belong gradient detail lb ji g ij matrix matrix cover scad many proper form especially nonconvex penalty enforce parameter easy prohibitive rather q control diagonal loose solve quantile j p successful base matrix set thresholding threshold element zero see thresholde minimum put upper belief network specific usually guarantee expression network seem formula cone difficulty size propose simple asynchronous type denote search subsection convergent satisfied basic line restrict along direction condition satisfied accept carry otherwise try iteration initialize decrease
conjecture thm remark proximal admm collection solver cholesky subproblems intuitively try binary mapping matrix rectangular subproblem briefly
far make preferable gain style algorithm know avoid invert storing form p expect row pick space positive feature ridge regression see main approximation matrix aim style never form update update cost per ridge cost per iteration parameterization latter track via
fig look right look copy make provide reconstruction tumor innovation previous method impact allele frequency contain incorrect inference frequency affect allele integrate datum seminal observation provide automated method reconstruction overlap alone describe infer branching contain lead read base accurately tumor clear extent automate reconstruction describe furthermore demonstrate correction change perform specifically much recover highly copy number state fail reconstruction rely sequencing assume copy occur copy development decompose restricted region copy branch preprocessing population completely detect population detect rely error reconstruction branch equally alternative reconstruction seq even highly achieve reconstruction limited region benchmark stage manuscript new combine reconstruction unlike allele reconstruction
program appeal strong threshold admit expectation polynomial idea polynomial exist make reduce theorem upper jump threshold impossible low formal infinite finite inequality weight finite essentially learn log normal laplace capture everything law main motivating relax concavity end result agnostic technique start concept learnable concave distribution absolutely logarithm multivariate logarithm concave concave laplace natural contain heavy tailed distribution smooth distribution hypercube sphere degree l prove interested
match truth two source calculate statistic tp fp x produce produce carefully entity source true feature feature score entity score randomly entity pair correspond add true entity start value hard vary create level entity source entity evaluate source entity entity pair correct precision harmonic recall employ unconstrained er denote give source score allow entity match entity datum help resolution pair world million scalable operate primarily answer multi match bipartite source statistical total matching movie effectiveness matching add bipartite matching source bipartite obtain movie source six record source result dot result pass plus record color pr threshold specifically regard entity similarity never match entity resolution new range threshold message pass greedy add constraint movie one new source go addition
appear unlike necessarily sometimes see fig subsection update dictionary step simultaneously traffic traffic representation implementation first depend polynomial total parameter practice easy store dictionary learn svd mod forward adjoint case ty vector compute cost fast way exploit ty efficiently compute numerous processing advantage structure dictionary representation omp soft thresholding main chebyshev detail third benefit costly sensor structure learn computed structure make signal fashion parameterize dictionary polynomial give belonging translate vertex polynomial kernel localize fact translate area lead comparable dictionary approximation dictionary robust available size training
latent stochastic update every learn progress markov chain chain close data acceptance rate decrease notable effect dual operator split big previous comparison method score preference analysis mf factored model use ascent loop also run provide mf mf randomly matrix rank mf accord ground rating performance normalise discount cumulative gain truncate ndcg reciprocal sure metric put measure ndcg rank evaluate collaborative netflix challenge apply movie user increment divide
ct ct varied voxel mm ct convert software package ks structure thresholding acquisition partial medical ct ct intensity density interior topological appear thin head cavity bridge extraction segmentation surface surface mesh remove equivalent sphere various topology topological already extract surface topological simplification directly correction surface surface intersect check characteristic automatically equivalent solid definition object image convert slice ct axis topology slice result correct surface topological operation mainly relatively implementation neighboring voxel image concave operation operation instead operation may sequentially binary volume operation operation direction operation operation cavity operation easily disk correction reduce slice unfortunately
could equation py px py compare p xt dt x x indicate optimization whether maximize directly maximize theoretically relationship sm derivation start agreement agreement close maximize maximizing maximize intuition subsection follow subsection clear motivation associate detailed get big bias towards indicate hence py indicate prediction px px py px hence sm predict output x gp maximize sum htb toy example task pose schmidt kernel pose dataset evaluation beyond estimation inverse behind distribution therefore choose start specification toy subsection setting select argument task value uniformly star regression suffer effect indicate space example use input prediction example ht challenge see error total toy
transform autocorrelation autocorrelation obtain order result double bivariate gaussian variance double integral enhance numerical use use eq piece piece wise transform comprised polynomial segment q piece partition domain doubly doubly truncate gaussian piece affect transform therefore estimate interval rather use continuity monotonicity maintain comparative bootstrap large linear accurate estimation adopt multivariate variable pair addition correlation sufficiently consider power frequency difficult achieve case match well spectrum realization decompose four
return alg require nevertheless practice execute likelihood part false p vs p vs p p k p policy alg select sequence pyramid treat alg summarize active htb pyramid likelihood pyramid compute break break compute location uninformative line round part store policy terminate foreground obtain final discriminative label background filter final compare score operation location search possible score alg total turn incorrectly
multivariate skew mm discrimination multivariate skew elliptical mm scalar skew graphical model skew j leibler asymmetric heavy tailed b enyi relate family pre mm e leibl la skew error application profile mm n incomplete j mm ng w skew elliptical beyond volume van discriminant van skew mm univariate york mm population performance discriminant
also layer prevent dropout dropout select select back propagation activation multiply normalization network refer detail heterogeneous collect pose label whole sa collect body body stick left body part head leave sa illustrate definition annotate bound dataset randomly box extract human mirror double augment factor body remove extended pose predict position window equal find initial start dataset initial
file efficient essential various analysis etc testing important book review property gradually new base characterization follow negative identically distribute f equally exponential able validity nearest proved observation continuous f composite family unknown f empirical f observation
equip denote net sup aggregate element give upper net fail metric slow one obtain trivially bound phenomenon concept mention empirical finite even upper bind illustration bound minimax rate couple example linear predictor combination cover bound metric space entropy banach rademacher yield control minimax rademacher enjoy rate specifically regime minimax depend rate depth say tree large tree denote cover follow
define domain make low respectively cross scatter clearly pc look pc rescale unweighted cca pc pc decrease rapidly datum treat input agree relate wrong distance situation bad
dependency template state rating person interest movie production movie classic logic template semantic also concentration inequality example increase interest structure datum citation chemical interaction important field assumption extract many
brevity term mask mask exploiting trick require white background figure public specifically provide angle angle include near color model around keep aspect subtract build cnn value pass whole epoch learn matrix recommend energy never start decrease expect network last use except viewpoint interpolation section reduce time task largely remain restriction interesting supplementary material successfully force row another column transform even transformation quality image basically office fine row
invariance preprocesse mm computation learn individually optima practice free free improve performance freedom need case retain invariance rotation adjust spectrum require instance rbf rbf suffer concentrate fix relatively design e part fourier transform analytic compute inverse fouri readily expansion parametrization provide expression function mode parametrize piecewise moreover piecewise draw occur probability invert quadratic note considerable choice location cumulative location particle yet basis I b feature rescale relevance determination piecewise approximate radial piecewise generate require basis partly gaussian matrix expressive obtain optimize scale property sample chi adjust radial marginal combine procedure input generalize additionally wide class hadamard preserve modify keep multiplication fouri unchanged repository flexible applicable million process beyond secondly hyperparameter involve order e ghz gb ram marginal ard gm grouping rbf ard dataset every pick test report result rmse partitions rbf form kx ard kernel kx ard
fa david decomposition column together row km via randomized algebra approximation input nr r factorization k desirable attractive algorithmic construct optimization k progress span numerical algebra science remain sparsity running question ask question stream topic second run polynomial error column row indicate column row construct section section fast dense contribution subsection subsection idea summary rank later algorithm explain implement etc summarize put context summarize present contribution answer section sparsity select optimal column theorem exist algorithm column form column row column match low ever relative select additionally lower progress accurate approximation non although include matrix input address although deterministic construct
density student distribution pdf density parameterize discuss student identification use variance residual probabilistic review identification set kernel due gaussian realization define scalar impulse response assume joint vector value represent covariance instrumental derive impulse minimum square bayesian jointly efficient marginal obtain integrate adopt paper description briefly previous apply deal represent normal adopt pdf compare thought generate two draw eq model due vector value
assume minimize thus infimum support part li lemma challenge grind applicable attribute overcome challenge propose essential unsupervised meta rule regard unsupervised attribute model cubic curve control hypercube meta rule rank able manifold interpret dataset comparison state art approach list unsupervise multi monotonicity smoothness skeleton htbp viewpoint machine hierarchical able evaluate ground seem challenge difficult issue viewpoint type rank divide multi data representative approach rank variant candidate link attribute attribute object summation scalar assignment give rank first learn object determine propose attack mapping preserve requirement neither vector quantization score attribute nonlinear
statistic rv analogous long move however von freedom sharp note neighborhood p detail repeat monte material sequence covariance apply transformation sequence nk sd base ep rv nk sd ep rv statistic case times ep construct rv standardize
exactly v result conclude root denote extract triangle definite third inequality due unconstrained optimization em qp definite know volume since space challenge instead focus hypothesis measure principal axis hypothese long axis near axis large principal preferable close volume loss fit binary setting naturally include adopt link ellipsoid volume unlabele
player though game rich attempt score depict b stay move take action turn north east west player player player ball toward opponent opponent act simultaneously period attempt opponent reach ball b attempt reach assign randomly correspond ball aim maximize goal subject discount party observer play try play statistically calculate rate h worth would offer help decide b reward brief decide reward opponent conventional opponent part respect specification prior later subsection quality reward negative state ball
n weight lead estimation sgd verify instance parameterization parameterization canonical generalize formal sgd parameterization family log idea carry extensive experiment fisher scoring estimator sgd computational fisher score r versus package implicit sgd package set analyze national air specifically numerical incur small efficiency cost quantify explicit euler approximate differential explicit sgd straightforward attract method large problem sgd datum find diverse area scale online implicit sgd procedure poorly understand arguably least filter implicit robust form usual likelihood normal model attract method mirror implicit proximal function minimize proximal operator establish proven problem present quantity know predictor affect variance appropriate parameterization appear furthermore convenient known function rest glm space hold n variance property devise sgd initially estimate reasonable every covariate vector distribution accord glm observe application definition stochastic generalize iteratively step vector implicit sgd definition concrete omit gradient see factored definition correspond baseline
apply sense bayesian context property estimator absolute loss instead brownian bridge study derive mle investigation interval base performance investigation similar credible
attention database exhibit human tend prominent object occurrence answer time answer capture produce wrong propose inspire measure boundary become fuzzy seek answer naive answer n answer set membership equality whenever variant wu word empirically require answer weight magnitude figure refer weighting correspond plain benchmark architecture require answer score highlight challenge unknown true form advantage world distinguish
next invoke eigenvalue covariance expression operator computationally avoid explicitly list equation nothing bayesian eigenfunction eigenvalue product trick j mathematical physical account mathematical rise pde functional functional determine pde physical distinguish incorporate procedure give form third parametric provide systematic optimally covariance observation posterior saddle admit variational permit formulation flexibility quantify prediction popular inference estimate estimate determined therefore inference powerful quantify prediction assimilation problem incorporate state gaussian
table build another fine precede initialize net minor share layer dramatically mb testing hyperparameter affect number large categorization enable obtain substantial minor much fine cnns mb mb baseline similar cifar convolutional layer cifar double cnn fundamentally average average classify category difference initializations cnn classify cnn average independently train prediction cnn average cnn performance category consistency tune multinomial fine tune coarse consistency method cifar imagenet generality cnn nets cnn significantly test block net ht truth net imagenet top make top fine public building
device box maker cd computer disk head pilot phone machine table gate light house pyramid medium air school daily item play top mask saddle music background unfortunately concept class manually class new newly class group minor fluctuation compare cifar employ interestingly relate similarity visual computer computer computer electrical device codebook help boost codebook might due complexity shot still object miss stage
know precisely characteristic rarely theoretic elegant definition express rich dependency one define tree measure set experience game name player action move causal experiment crucially player express special ability see assumption strictly knowledge player move miss state game indistinguishable player move game analogy people yet go great maintain people draw thing thing novel responsible maintain symbolic rule syntactic perhaps finding analysis appendix firstly interactive order logical state refer appendix consist represent weather pressure assume elementary physics know weather dag subject plausibility compete h depict crucially separate induction causal arc variable two single diagram add diagram dag importantly specify wu inferences htbp contrast situation tree allow causal dependency dynamically weather semantic self indicate probability importantly encode determined path indicate leaf multiply htbp belief encode node finally root causal w consequently distinguish purely decide player consist collection nest causal enough algebra instance suitable modelling state label htbp action observation course game subject keep track probabilistic move depend private subject interaction happen possible move namely experiment transition compatible transition
orientation cluster keep focus difference group connectivity kernel modify affinity geometrically adapt dataset positively influence capability already simulation relevance orientation circular clearly succeed distinguish remain partition plot diffusion constant correctly contour distinguish semi circle partition case affinity case coefficient stimulus wide nan half condition observe affinity object straight retrieve object two element circular contour get unit prevent contour recognize object embed orient segment kernel contour grouping affinity time dataset intrinsic causality reflect carried affect efficiency define cope modify still normalize affinity transition matrix balance reversible eigenvalue obtain modulus perform thresholding far cut approximately transition discuss minimal cut cluster different direct set mark difference symmetric hermitian understand n equivalent eq n rewrite natural double copy sufficient grouping broken speed orthogonal movement segment find maximum local speed make indeed random speed could background lead brain group together rely orientation coherent velocity field smoothly study determinant purpose contour trajectory primary temporal optimally measure stimulus velocity stimulus direction movement
disk bayesian jeffreys proper range ensure choose long tail uninformative shape uniform embed evaluate multiply conditional determinant improve block gamma form pointwise accept accept unchanged choose probability spatio provide huge large conditioning processor gb ram mac os system fourier package mcmc disk shape burn indicate clear unimodal histogram draw minus summarize spatial posterior mean panel draw agree observation converge unconditional unconditional center panel illustrate posterior field plot domain
sampling loo use used obtain laplace toolbox use monte provide result ep approximation hyperparameter weight point grid point usually usually small monte well loo remain loo towards integration quadrature loo loo quadrature quadrature accurate e true quadrature exclude however posterior unstable approximation accuracy tail loo contribution influential correlation strong la simplify problem approximation fail loo case removal datum failure loo la ep several severe failure la ep fact failure marginal ep la ep loo l loo alternative insight improvement write global cavity ep correction increase evaluation small point interpolation sometimes instability produce global result
cite sophisticated meaningful publication evolve centrality centrality fan fan cite author paper table among paper selection primary paper collect author wish influential important area need effort area author community area contain author interpret presentation name group author paper finally network real name beyond community organize centrality discuss contain discuss limitation pattern trend frequently author paper centrality measure centrality centrality closeness centrality reciprocal extent locate centrality conceptually centrality centrality author citation citation paper centrality degree paper table present author identify different centrality centrality largely fan author htb closeness fan fan fan identify centrality measure cite paper selector paper know penalization lasso many wave research devote penalization htb l ccc closeness fan variable fan al li bayes multiple variable li variable selection include covariance empirical www edu list file account number citation count list cite paper adaptive hand important lot broad community google cite
result dim c rbm cd rbm compare algorithm close applicability rate considerably rate along table deviation compare counterpart range layer interestingly autoregressive train model perform well train outperform expressive model rbm cd cd considerably sample compute handle posterior ability autoregressive within dependency considerably one factorial layer autoregressive connection r dim test factorial apply document train generative represent know bag train corpus similar model
kp vertical discussion promise perform improve abc call simulator naturally pseudo several technical gps modeling likely exception simulator noise noise limitation gp assumption covariance major gp abc role phenomena value expensive setting interested insight importantly challenge likelihood partly abc free describe section simulate proxy provide framework progress model inefficient mind simulation maintain surrogate parameter main able posterior accept expensive procedure expensive course model gradually time reject uncertainty insight determine proceed mh simulation
player know probability chain game base hide quite approach fact nash ad prove play nash equilibrium least equilibrium game show equilibrium type nash pure mixed nash equilibrium bayesian games player payoff profile tuple game player strategy
vary depend matrix benefit regularizer smoothness uniform pattern follow uniform user movie independently sparse identically epoch distribution scheme rating set surprisingly suffer uniformity fig value nuclear medium graph uniform report real result aforementioned rating star increment two purpose firstly arbitrary submatrix secondly construction detailed non uniformity sort movie th percentile observe correspond rating movie user movie discard quality important detail
important aspect identification summary model summary kernel statistic poorly information necessity abc summary probably principle identify develop score interest couple choice computational abc classic example imply development efficiency abc convert sophisticated importance generation early likely reject develop computation research state development tractable closed likelihood usage decade could write max stable close population use abc tractable far remain toolbox availability abc nature construction likelihood computationally prohibitive problematic purpose focus framework bayesian computation fit idea behind demonstrate interest modelling multivariate hold mind location due differ non unit use distribution transform multivariate fr generality fr margin preserve monotone eq simplex angular spectral measure determine marginal pz
account consider axis objective reward sufficiently one force pareto front occur return get wide tend expand opposite trend devote metric learn algorithm learn start indicator converge parametrization
search entire average training transform localization response sometimes filter provide localization thereby result localization ht localization inter mean across run bar design without account ccccc leave evaluated design consider window e shape dataset consist category consist decomposition cf filter precision precision object single template learn conclusion paper fundamental formulate frequency explicitly multiplication circular result exist intend function remove circular effect cf show derivation filter design address challenge introduce design eliminate author like acknowledge air force research laboratory resource quickly present bs electrical engineer ms electrical ph university currently future technical gold award student engineering interest filter efficiency processing receive engineering institute ms degree electrical engineering program currently institute research machine institute technology receive conference research air force laboratory sensor interest recognition computer
term derivative formulation replace constraint positivity ik ki convexity constraint verify jk compactly notation ik jk n kx center therefore eq alternative formulation define ki optimization likewise solution define jk k stage output residual dc formulate estimation subject screen consistency variable ac set part first sufficient sparsity stochastic argue probability change line ac dc output use optimization boundedness practice construct relevant satisfie kkt define show optimality index lie contain set th th large entry solution solution program hold regardless impose boundedness set additive convex set guarantee relevant analyze positive restrict analysis standard analogous nonparametric detail probabilistic treat setup positive response support f unique minimizer k independent subset twice differentiable f signal excess risk incur support strictly risk uniqueness play coefficient
length gamma variance notice parametrization gamma scale scale multiply shape limit arrive challenge series process main consideration approximate numerical treatment type search intel ghz implement alg homogeneous laplace allow predefined branch regime free specify branch share trait environmental see biological jj f gamma
sdca execute machine ghz memory gives save compare net compare nonlinear apply top select update resource utilize rate alternative synthetic dataset datum rbf bandwidth choose time median batch justify convergence blue dotted guide iteration could converge seven ridge demonstrate clearly full cost sdca nystr sc vs solve dataset sc sc kernel obtain trick regularization parameter feature go rate achieve sdca illustrate flat region unbalanced rbf bandwidth obtain median trick regularization block sc stop good rbf kernel
publicly specificity disease disease process dr diabetes frequent cause case work population people diabetes far million people develop suffer form people dr quality fm dr screen partially clinical protocol dr reliability medical ensemble applicability breast use classifier ensemble classifier improvement processing system fusion fusion detector efficient consider processing image regard dr
identifying loading cpu run kb ms contexts shoot shot formulate core entity match bipartite graph structure huge appearance variation ambiguity efficiency basis co statistic dataset demonstrate scenario demand storage computational apply real question consider computational calculate wise latent modify learn separable appearance accelerate would optimal affect behavior word occurrence interesting camera add match entity enforce acknowledgment material department science university award ed view represent express imply zhang person identification person identification surveillance maintain entity individual location camera challenge appearance individual illumination maintain entity camera jointly account weight graph base co occurrence occurrence base appearance infer unlikely occurrence appear shoot multi shot computationally person match visual co occurrence shoot surveillance camera deal maintain location overlap camera
canonical correlation minute minute c minute minute canonical correlation variant ghz amount run drastically I store order multiplication allow model cca implicitly randomness random fit available budget contrary art ten parameter autoencoder regularizer view validate grid desire several truncate learner exclusive feature teacher build construct help reveal little achieve leverage randomness key multivariate spectral demonstrate experiment compare art
toolbox frame adequate domain orthonormal synthesis accelerate describe level pre tune keep reconstruction bss al decompose source sum interpretation target word account stand invariant ratio sir result design latter criterion evaluate since account reconstruction provide section train decompose combination negative atom problem negativity w use version laplacian compose performance role level trend smooth enforce method type adapt cope peak suit impose orthonormal expect wavelet enhance separation source retrieve model mixture base tune tuned reconstruction source source particular information reliable later smooth nmf able recover adapt well
art technique therein seminal dictionary learning make k et mod svd denoise behind capable patch two clean image noisy suitable overhead svd fidelity perspective denoise image super resolution suboptimal moreover minimization learning solution
implement forward description dual essentially amount weight span norm solve implement variant achieve well difference likewise set fw description optimally convexity inequality note deduce use bind subtract define q decrease always choice iteration obtain subsequent choice eq substitute eq denominator sequence inductive step assume hand increase quantity upper substitute complete example remark aim atom atomic norm regularizer solve well atomic norm describe call produce reconstruction generally norm truncation exploit nature reduce via experiment signal forward outperform tailor
discover recover portion subspace hypothesis plant sparse embed nonzero entry recover comparison special drive round complexity scalable moderate regime algorithm well perform dictionary vector break barrier literature optimistic applicable dictionary nonconvex also global basis point formulation nonconvex optimize relax necessary force live give
virtual pass token perfectly connect virtual infer great message underlie contact g social anonymous communication dc vast topic accommodate unbounded fully network message model detect origin epidemic recent analogy receive infected act infection adversary contact network infect certain pose centrality centrality measure random increase multiple source source infect allowed recover sir centrality number infect node message platform overcome design protocol fast source focus anonymous underlie contact phone facebook friend system introduce delay delay user message discrete system message immediately protocol delay message inference message difficult pass model adversarial know contact observe snapshot state node receive message strong whole contact state adversary aware receive adversary device device adversary learn snapshot infection give observe adversarial also capture state continuously anonymous protocol call strong adversarial start spread contact edge assume delay user via deterministic
programming express entry package square estimation two way bold impose integrate feature constrain square voxel j like pilot nuisance projection form separable least admit closed form solve respect little singular solve iterative notation remainder j equation q initially pilot generalize least program update use lagrange multiplier compute singular find monotone omit give voxel induce voxel assumption sufficiently j multivariate distribution uncertainty vice result allow glm correspond across brain degree justification rely voxel activate unconstrained easily test pilot define asymptotic inference voxel sensible test activation activate voxel inside identically sized dimension bold fmri stimulus varied systematically duration vary nine create stimulus canonical nonlinear net
vector combination weight vector art locally svms vector local predictive model neighborhood process near neighbor model initialization locally anchor produce construct weight manifold value cluster svms learn svms simultaneously al kernel intermediate respect tree region change weight svms maintain ability art propose exhibit specific predictor meaningful case denote basic notation c index partition input weight th linear partitioning region figure concept dash specify weight whether apply point figure specific partition wise weight
triangular follow corresponding get digit within triangle zero beyond corner triangle converge convention center triangle yet arbitrary follow corner infinite integer rule triangular desirable balanced center symmetric final reasonable third first triangular van discrepancy triangle work unit discrepancy calculation side standard parallel panel decomposition centroid horizontal pointing purpose triangle integer degenerate numerous discrepancy attain hold sign discrepancy maximal exclude van point signed attain discrepancy contain point
relation set relation close range behavior relation lexical embedding linguistic observe rank penalty embedding model certain refine show task embedding benefit quality lexical
sentence translation automatic test sentence term removal refer keep character long appear sentence vocabulary dataset present qualitative reconstruction help aforementioned denote denote pre tune operate understand amount distortion network update much calculate softmax case comparable preserve reconstruction similarity descent backpropagation well assess understand notice half cosine report representation explain
prove time cholesky dimensional brownian motion hold tuple stream detect change independent propagate sensor unknown sensor consider markovian propagation different propagate configuration point max approach consider criterion delay path point objective stop stream observation couple correlate correlation correlation assume work decentralize sensor sensor cumulative binary asynchronous center well centralized distinct alarm alarm actual run uniquely appropriate parameter would delay optimality prove delay recent include sensor couple drift
project project category rank opt previously return outcome project project sort twitter recommend score content recommendation table static feature percentile mm initially project view nothing return list seek financial reward support cause contribute small traditionally fs friend site act act different project project largely project technology game frequent project rely friend family perhaps employ social site reach technology project look active frequent news identify frequent might recommend pool expand pool beneficial twitter project predict potential project twitter activity
predict beyond future history dynamic pass attempt shoot transition player transition influence transition pass include pass pressure shot outcome pass shoot informative yield combine contribution full resolution view full rewrite eq substitute reasonably briefly alternative proof immediately omit chain resolution illustrate conditioning markov resolution short state call transition pass shoot attempt mark shift form basis carry among process well pass
allow action power ratio learn mixed mab define triplet scheme indicate product ucb arm discretize learn discretized value discretization old continuity figure another advantage know horizon horizon stop end every round trick mab ucb completeness arm maximize tu duration play obtain algorithm choose I strategy choose algorithm maximize objective time scheme increase performance get converge decrease indicate separate euclidean cumulative increase learn learn fast converge strategy hence dependent wireless etc wide wireless case knowledge wireless channel bad confidence bound confidence regret arm define set arm optimality
support technology innovation publication reflect ex calculate similarity modality challenge many modal compare similarity realization similarity object link service face infinite
perform performance label object therefore propose eq unlabeled objective supervise lda additional assignment partial assignment class problem form particular labeling toy benchmark dataset train supervise website illustrate consider toy normal center bottom gaussian top decision boundary actually correspond happens draw force fall
fig boltzmann backpropagation deep boltzmann work except likelihood deep boltzmann probabilistic similar would somewhat flexible sigmoid mixture expert one ask happen would line eq ensemble hand expert criterion resemble handwritten digit see middle minibatch rate stop
analytically computational method propose hasting geometric rwm space proposal rwm proposal manifold take qx x g x simplified mala proposal markov chain et conjunction marginal mcmc article simulation highlight geometric mcmc inverse surface ground vary fraction scatter reflect later observation return return times challenge infer medium wave manifold mala figure density incorporate particularly biological nonlinear dynamical comprise six cell system nonlinear eight complicated observation rwm geometric mala report inference size per consider infer jump chemical reaction consider chemical seven rwm simplify mala report accord simplified conceptually simple correlation make al method clinical author highlight difficulty sparsity identifiability discuss describe activity likelihood base approximate spectral evaluate take less five
alternate left figure target elementary conditionally von dispersion sampler accomplish aforementione normalizing estimate partition function classical four node right add left display add node node circle circle draw right right edge edge edge cm main circle edge edge edge topic comparable square mrf relation variable graphical simulate seek distribution four propose one sampler sampler particle sampler arbitrary non gaussian discrete tree tree sampler partially gibbs top ordering give
size stochastic stochastic good step english wang english train uniformly equal understand model english wikipedia embedding english belong create name entity wikipedia second cope annotation extend matching maintain wikipedia article wikipedia language topic category entity region organization organization person map internal entity bias label overcome bias example section language language english identify entity article unfortunately english annotation mention page leave entity phrase english
statistic nonlinearity substitute mixture avoid es probabilistic mixture exploit fact motivated detect align dna knowledge connection two node inside community bernoulli inside community otherwise guess belong community one mixture scenario figure correspond community take advantage enforce community mix community interaction figure
consequently asymptotic probit probit probit analogue factor evaluate fix expectation conditional expectation coincide correction assume small derivation covariate correction probit conditional measure form logistic necessary fitting perform fit covariate omit model covariate normal correlation case probability take skew normal conservative response
nonetheless gradient convergence oracle go proof discuss interpretation relation read fy fx smoothness quadratic fy fy fy assumption precise framework duality clearly always project step size extract lipschitz come back project subgradient obtain multiply yield claim condition improve notation convex smooth descent satisfie obtain x g x conclude one factor complexity step gain reasoning lemma convex convexity smooth prove thus get give claim smoothness prove convex smoothness one x fx x simplify presentation black mapping simplify section make black box span n lipschitz procedure convex strongly oracle ask ask subgradient span necessarily remain compute value minimal prove take low proceed eigenvalue hessian small eq symmetric convex must imply prove remain minimizer span define immediately conclude simplify limit precisely chapter valid choose work clarity exposition let infinite already thank give infinite equation conclusion straightforward computation gap prove respectively gap accelerate published restrict attention case everything constrain describe nesterov accelerate descent context smooth strongly oracle reduce gradient descent improvement quite describe order numerical strong convexity case quite illustration token right token token label token token right token label let strongly strongly induction intuitively fine prove use strong equation understand far answer devote combine fy fx x induction true get fy fy fx fy fx
select section show diabetes datum section range residual htbp penalize score value alone show transform regression display respectively score widely piece wise jump phenomenon detail vertical tc zero multiple qualitatively linear regression notable strongly regression result suggest feature associate htbp black correspond normal diabetes calculate conservative formula value decision pattern correspond display thick black become instance threshold age coefficient well sample thresholding far limit canonical super efficient page lc observe penalize score lasso two variable exact conservative
stand duration overlap classification contiguous segment constitute baseline middle section dataset average successful accuracy frame frame overlap table although yield well yield yield yield well baseline dataset notice see baseline combination good work paper music evaluate contiguous summarize publish mainly music produce song summary way image short version song
possibility flexibility algorithm high work propose genomic gene sequence present genomic measurement complex genomic entropy entropy entropy generate observe respective genomic sequence way network
reflect inter decomposition representation plot tight representation inter good predictor importantly control single alone logarithmic growth break tail result inter high case strong latent classification nn summarize report peak micro macro accuracy yield correspond initialization achieved report show micro accuracy average initialization range dimension peak notice remark worth care
gibbs infer parent parent conjugate constant rate gamma minor elsewhere kronecker delta parameterization k conditional graph gamma weight color theoretically dot indicate show recurrent process feedback lead infinite perturbation mean deterministic rank one equal vector random large come uniformly achieve high weight asymptotic unclear various theoretical indicate dot figure variance example grow large eigenvalue increase greater unstable additional adjacency matrix feedback loop perform link process identity goal event hold link probability interaction compute roc complete set interaction similarly infer
scale property final respect satisfy sbm property node affect definition specify sequence adjacency aside requirement linear unlike stack column top system commonly process equation procedure dynamic sbm utilize central density scale sum bernoulli dependency identically condition asymptotically gaussian q time scale occur occur edge mean increase scale scale identically long apply
concave passive exp concave function several regression portfolio management batch sharp risk space result manner properly individual exp concavity goal help statistical investigate know combinatorial online theory trend characteristic convex condition concavity generalization recent smoothness
represent false error small see correctly nonzero component exactly true quite short algorithm insensitive attractive capacity demonstrate well propose compete multi multi denote th sample weight sample uniform vector sample I response compute recover figure test number advantage plain note estimation rapidly insensitive average estimation error default heuristic formula
solve system nearest search compute pca dense base cost column empirically manifold ambient cost could nearby simultaneously vector field vector field discover geometry however key firstly solve try preserve dimensionality try directly manifold note diffusion geodesic method approximate parallel adopt scalar heat propose learn heat field field gradient obtain try directly heat field note scalar field order direction initial vector manifold manifold idea another parallel tangent ambient another direction employ tangent pca tangent pca noisy method develop heat well partial empirically effectiveness geodesic algorithm representative le unfold mr euclidean distance
gradient stochastic introduce carlo issue mcmc alternative conventional sequential appealing variational gp predictive derivation prediction new induce approximate lp approach induce low expensive compute hyperparameter q summation unbiased segment sequence locally around segment justify time smoothing measurement future variational gp implementation interest
differentiable periodic use exhibit periodic dynamic incorporate prior nonparametric sophisticated modify compound periodic dynamic whereas periodic path draw compound covariance control variance intermediate shift experiment compound level bias dataset apply performance intuitive checking agree knowledge flow consist flow apply variational gp dimension ard initialize initialize neutral dimensional reveal figure visualization keep remarkably visualization visualization sparse run use induce variable variable gp initialize would reduce quantify dominant plotted axis visualization near variational temporal variational follow consider take motion frame time pose jointly paragraph walk learn investigate reconstruct test test test dynamical latent variable additionally test dynamical explore sub model model correlation discover reflect latent assess cumulative per joint scale space model initialize nine mat ern function prior infer dynamical mat retain quadratic dimension ard body ern smooth body model infinitely mat ern one easy motion significantly approach train successfully furthermore dynamical discover prove robust indeed table show gp outperform neighbor gp map worth intuition investigate encode smooth circular shape regime encoding explain force smooth interact supplementary video mat ern use quadratic correspond taylor space cl cb sc sc sc gp
object estimate extreme rank show unbiased parametric learn expert aggregation aggregation base aggregation score available work combine task refer input rank list problem list feature deal difficulty aggregation list geometric normalise rank spirit result build heavily theory copula find available rank describe distance ranking review copula function hypercube unit review bivariate multivariate next definition continuous variable copula distribution univariate effect ordinary extend
optimizer drop negativity constraint feasibility give rise semidefinite careful reader enforce relaxation marginalization widely lp relaxation map q turn redundant feasible solution feasibility intuitively propertie diagonal sdp approach problem employ similar state material despite hard difficulty advantage algorithm advantage property pose solver despite superior accuracy primal regular pc scalable solver propose solve problem negativity produce numerous dual solver restrictive solve pc exceed develop alternate direction multiplier
investigate paired screening deal regularization prevent toolbox lastly show naturally loss generalization able well allow screen dedicate case one corollary base concept extend group group inactive optimum construct contain prevent closed solution center sphere test sphere screening sphere require inequality obtain close rt give dual compute feasible screening test feasible dual feasible safe argument feasible dual dual problem conclude proof construct recall define contain point ellipsoid ellipsoid tangent lemma new sphere q check use convert
expansion fourier entail appear construct law decay gx decay exponent provide first distribution knowledge theorem improve good sbm block sufficiently spectral therein mixed block order additional spectral vanish community exceed universal show consistently condition similar attribute attribute recently show empirical probability apply probability low algorithm correct focus regime e degree edge linearly fundamentally infer distribution result endow label similar enyi isolated neighbor isolate label information
error independent define correspond calculate shift detecting shift shift series line estimate observe bootstrappe figure calculate bootstrap statistic significance observe detect magnitude score extend change user typically twitter decade review amazon book book different book micro google books corpus enable analysis social linguistic extract book language gram gram restrict gram focus span snapshot pos distribution google syntactic amazon review amazon include million start review rating plain text show amazon twitter span period month domain book micro dataset tweet tweet tweet location tweet text change usage time
enhance rest paper organize follow present idea discuss convert raw labeling section systematic conclusion come work lasso well formally training denote output n j j lasso accurate interpretable independently risk regularization
absolute independent complexity parameter well may replace indexing unit ball obvious parameter index result version average latter little heavy tail situation parameter tail typical trying probability fact extend loss explain throughout denote mean empirical minimizer procedure select empirical risk minimization erm context problem natural success erm erm close start know deal distance produce erm sign q exist target heavily target restrictive exclude like arguably independent corrupt since gaussian choice tailed term heavy tail
depend base therefore accumulation always asymptotically consistent prior zero bb test case study test bb percentage instance arrive almost regard surprising important consequence try effect medical well data situation bb always well turn coin guess equal test case rich analyst analyst information collect matlab code available probability let borel borel equip field measure element dirichlet process measure finite measurable dirichlet pa beta beta letter expectation dp reflect prior scale stand normalize define dirichlet atomic probability dirichlet satisfy property sense unnormalized easily posterior hereafter useful dp sequel
result htp result turn selective ol mcp turning kind ng turn turn well method respectively nonzero display htp scad mcp good tendency
deep fact deep alone train competition learning decay mostly visible particle contain feature million heavily regularize arbitrary quantity datum unnecessary neural network lead significantly well manner decide well prior train level variable perform deep suggest derive variable discovery accumulate network classifier
heavily representation huge face computation extreme addition pyramid multi scale face increase discriminative sharing network validate learn introduce face benchmark collect realistic representation present pyramid representation fast compact new social validate hand sift various heuristic descriptor aim design trial also semantic meaning
group many thank lee science foundation dms support apply mathematics activity office advance scientific computing de ac discovery grant david le lee constrain approach come address nontrivial scope handle noisy analysis narrow gap propose response surface expect lagrangian optimization hybrid allow think augment act bottleneck effort yet orient approach surrogate heuristic constrain minor modification motivated word nonparametric design mathematical algorithm provable algorithm optimize handle run computer little constraint modern guarantee statistical approach potential surrogate property objective evaluation expensive monte optima conventional usually local method statistical handle explore perspective consider scalar ba include assume feasible cx nonempty clear distinction difficult numerical allow
guide meaningful analysis may may aware modeling seem reasonable thing prevent try activity research take yet empirical suffer reflect piece finding order likely publication file fact light publication fact among expert thorough also one create ever activity currently account background valid selection selection
ball contain conclude contain fix arbitrarily let coordinate along follow valid along every fix arbitrarily suppose note argument need establish argument part r I follow hence every sequence iterate clearly distinct set still q various step case b generality let b I eq give result play convergence empty subset belong choice distinct follow empty hence set close follow solution identical set linear I lemma I x e r
filter learner advance embed advantage stand advance kernel rely formulation field see spin reconstruct shape bayesian discuss title probabilistic mit artificial title intelligence introduction title author journal volume page gene title gene regression base aic author journal mathematics author journal machine volume page year title bayesian paradigm journal volume page dimension interpretable cpu memory dimensional dataset subset
convolutional weight amongst worker many require synchronization convolutional worker worker particularly near convolutional net inefficient parallelization net dimension extend particular connected neural introduce asynchronous efficacy whenever part neuron activity output worker worker parallelism worker gradient ensure consistent exploit parallelism neither degree scheme informed
validation compare likelihood consider vs f component learn utility visualize specific death indicate high odd vote favorable compute ip support compute align issue capital issue conservative favor toward magnitude actual follow analysis use illustration ct power care topic support topic respectively figure switch notably support average concern support tend focus put concern individual argue fall uncertain analysis predict vote circumstance utility fig result support favor support confident vote favor except interestingly side success favor crucial turn conservative vote favor situation viewpoint
seen form rank due however rank
notion multiclass necessary condition calibration respect later introduce calibration dimension derive calibration dimension subset shorter include main maintain text closely proof completeness calculation multiclass surrogate calibration multiclass loss surrogate throughout denote integer integer e hull iii multiclass describe finite generality training instance goal many space e option notion common predict loss multiclass consider prediction away heavily satisfy ordinal evaluating system predict star hamming label bit bit representation frequently sequence hamming loss prediction reject loss assign incorrect prediction application costly diagnosis uncertain may well request intervention n goal prediction expect loss example draw randomly yx py ideally e difficult risk cd u vector risk achieve model optimal surrogate surrogate
apply unbiased learn membership contradict relation essentially derive hypothesis get optimal latter denote go hamming vertex hamming weight fraction lie middle level matter edge middle ratio go weight amongst setting imply level let study monotonicity giving circuit contain new circuit extend employ fouri analytic tool hardness circuit I boolean boolean study many decade depth survey many
important robust receive real response among datum par principal effect outlier example similar parameter affect close learning optimize require loading time method linearly intrinsic ambient robust linear rapidly computation obstacle big easily recent year rapid fusion machine order negligible communication cost implement distribute communication generic yet preserve reduce evenly machine aggregation operation compatible enhance efficiency factor potentially ability cope big yet machine reduce usage computation communication negligible assign machine step
modularity truth landscape grind truth gibbs phase energy modularity give low free marginal truth bp converge fix find approximate propose phase physics retrieval marginal group energy marginal generative model network contrast group free markov marginal need obtain thus cavity independence exact tree regime block marginal quite free message estimate send probability update message denote neighbor act update derivation bethe energy edge edge network moreover easily converge appear although grow least solution converge chance modularity
cm determine fluctuation level maintain compare versus elliptical variable co fluctuation discussion support order equation solution term confidence maximal interval individual q actual elliptical minimal saddle point
advanced term embed vector inside sub state onto term state likelihood convolution prior compute perform posterior illustrate synthetic real mm keyword model dark galaxy mcmc science understanding aspect behaviour system relevant select aim arise indeed though onto projection know arise science relationship functional situation unknown ill pose condition quantification uncertainty entirely scientific scientific namely datum comprise therefore refer whole learn measurement datum learn set however characteristic system physical biological system simulation consider basic behind learn system simulation useful way appear need add via thus datum achieve quality approach estimate bayesian learn measurement methodology remains learn model process however train formulation training possible form continuous popular covariance length impose smoothness learn
design location origin various transform separable importantly datum avoid however make perform dimensionality mapping feature mu mu j universal much dimension formulate eq hard dimensionality construct utilize related vector also j mu I find transformation introduce reduction particularly substitute equivalent add another specific complexity regularize problem handle corruption robustness term regularize rewrite eq augment lagrange multipli alm subsection review pseudo reduction present preserve utilize pseudo work pseudo transformation easily singular svd method stand within class scatter dimensionality performed view structured counterpart critical preserve less one elsewhere reduction perform especially small compare dimension dictionary size identity matrix pseudo graph one atom neighborhood neighbor among among
skew distribution triplet visible preference classical skew rw examine early point corner capability largely variable gender already appear herein reason extensively illustrative purpose literature skew symmetric proceed consider triplet ari gender reference classify variable analogous fashion clearly distribution comparative one argue package implement restricted skew terminology supplementary result figure dispersion around exhibit many repeat package circle triplet extra identity package attribute skew package
bad optima processing training amount prove art entire take minute model scale modelling present distribute dataset million experiment demonstrate gp amount show inference resource load gps perform big datum open contain extensively derivation equation present supplementary explanation google european fellowship lemma ex pt ex mark computation distribute implementation requirement evenly
overfitte neuron share neuron discrimination optimize descent please propose ns reflect see inference predict sparse decoder sigmoid sparse code force combine oppose include ns classification performance study highlight efficiency framework object categorization carry classification finally comparison subset ar recognition gender classification database consist capture illumination gender testing age database span age leave set finally categorization experiment toolbox level soft pooling different demonstrate face gender categorization absolute mae superiority ns compare control self st mid face database thousand alignment internet vary dictionary basis face record neuron basis unlabele reliable dictionary st amount basis ns level
diag kde library diagram diagram class plot band tb option produce persistence band kernel previous interpret band validity persistence diagrams kde detail option draw place diagram option figure tb c band legend col pt complex consist diameter simplex complex gradually radius create compute persistence diagram user persistence diagram library generate circle
know efficient say algorithm mapping also pairwise entail produce canonical determine whether np show element polytope discuss backward box variational characterization log project substantial iterate without enforce onto section give necessary background conjugate hardness computing partition family specialize duality statement subsection relationship conjugate conjugate partition finite furthermore supremum uniquely entropy express q forward mapping map unique canonical q useful describe hardness
momentum coefficient search like feedforward saddle improvement gradient saddle exceed since hessian plot universit universit universit de stanford universit cifar central challenge many minimize continuous high quasi newton almost often local method minima high global physics theory network evidence point minima interest saddle dramatically local minimum motivated saddle rapidly high saddle descent quasi newton apply recurrent neural provide superior geometric experience geometry surface choose error maxima probability typical random function dimension increasingly saddle point indeed saddle minima minima saddle curvature gradient away saddle curvature
approach efficiency high classifier soft connect soft consider natural connect hard soft rather specific soft classifier novel partial estimation number separate group clinical patient whether disease great setting emphasis individual estimation probability three boundary classifier approach provide assumption entire underlie conditional weight two briefly rejection option introduce third option reject label notably directly prediction probability belong exceed specified require classification view intermediate classification application option include certain medical weighted extend classification accounting difference problem soft remainder paper problem introduce generalize misclassification class surrogate surrogate hinge binary classification theoretical performance rule sub solve piecewise surrogate behavior datum disease database discussion framework
procedure raw word fail word significantly topic topic success correctly topic identifying frequency spread svd toy frequency say ideally rest threshold job extend toy different threshold word also ideally would either achieve condition split part partition thresholde clustered call project namely dimensional svd projection recently also yield start cluster center classic produce cluster dominant approximately identify frequently document nearly approximately document occur pure svd thresholding threshold let high ij b
report publicly indicator prior compare identity link package tuning fold set increase find association country journal association fit repeatedly split data gray bar predictor logistic order gene gene child available measurement cancer available measurement tumor normal available performance spam split training gene set cv percent classifier spam performance spam may successfully model display six
initialize convergence decrease criterion difference projection enforce matrix step pt step svd max z cm c c tuple tuple label relation label adjust four kind public two widely automatically use lexical syntactic name regard feature attribute whereas htb trade size contribution feature formulae assign z
turn plot function despite early surprisingly seem depend characteristic implementation detail often turn correlated direction cause happen causal chance level case discrepancy actual cause assume look accuracy chance benchmark accuracy around explain fact choose appropriately perturbation uniform base drop back add base accuracy slightly well chance become improved slope use estimator bit discretization still chance level amount benchmark accuracy depend entropy estimator due generally behavior method estimator suffer discretization affect perturbation perturbation however base level estimator perform base base make interestingly estimator distribution turn base obviously ratio support explain performance measure benchmark cause know ground benchmark several set bivariate causal discovery base additive geometric inference conclusion robust perturbation include implementation perturbation characteristic causal former conclusion report surprising report time method cut test one correct conservative correction would study method outperform benchmark conservative many result dependent across quantitative nontrivial believe chance continuous discretized digit record find use differential entropy coarse discretization cause entropy entropy robust perturbation include observation contribution extend straightforward distinguishing exploit certain pattern current enable closely originally finally fix bandwidth heuristic modification lead hilbert schmidt definition hilbert schmidt notation original biased propose joint define distribute choice justification stem characteristic special originally intuitively hilbert rkhs rkhs embed marginal give variable detail notion bias tuple center clear typically unbiased bound everywhere negative write kernel follow continuity show lipschitz novel consistency kernel q q start inequality take briefly main expect converge increase regression precisely consist q weakly moment many weakly might go training vanish asymptotically actually result seem kernel assumption bring even distinguish scenario half independence splitting training call use call mean residual vanish asymptotically take split suitable simply reduce scenario weakly consistent suitable state residual use estimate
centroid determine choose observation close uniformly centroid cluster close uniformity display real feature distribute smoothly centroid final less important sequential value perfectly persistence measure start two maximally distant feature add deterministic perfectly persistent avoid get optima appear however suggest optima criterion mean low persistence centroid htbp effectively request simulate much sampling entire replacement choose human
rule rule randomness involve learn seed worker affect success realization weak notion compatibility monotonicity allocation depend realization know ex compatible every irrespective ex rational success realization correspond allocation realization say post monotone mab call see previous need compatibility ns step add agent satisfy cost lead new algorithm monotonicity ensure modify ns select worker satisfy respect algorithm ex monotone compatible ex go analysis stochastically ex post allocation transformation obtain apply allocation stochastically ex ex individually rational success realization parameter allocation induce payment mechanism desirable game theoretic property label task ucb worker initially worker select k n ts tt exploit observe allocation natural apply payment absence worker determine stop compute compatible ex post individual rational post monotone achieve post monotonicity ex post mechanism algorithm post monotone monotonicity fix success brevity worker irrespective
make independent task variational lda synchronization scheduling thing become challenge due small gpu see parallelism subgraphs recent study variational pass scenario outline begin lda collapse big picture lda technical distribute finally section dirichlet allocation bayesian topic low dimensional capture semantic underlie widely amount conceptual overview inference collapse lda vocabulary word document token bayesian lda k intractable approximation gibbs sampling integrate dirichlet yield collapse gibbs algorithm current token token assign document assign topic token exclude th token collapse leave room improvement sub
hold critical case case cover proposition since noise example illustrate situation variation derivative operator turn design vector assume joint fourth canonical obtain say estimator consistent estimator split property implement j proximity forward backward identifie theorem hold close inspection apply generally variable forward assume
piece noise recover figure cm concave minimization whose projection proximal illustrate flexible stepsize affect proximal objective compact see cluster terminate successive instance random standard instance quantity upper usual proximal iteration less observe affect easy solution depend cc e e admm algorithm solve cluster point generate merely choice point generate admm actually convergent condition boundedness proximal admm future direction adapt convex problem case author discussion author also thank anonymous suggestion manuscript cm definition corollary remark partially grant minimize sum nonsmooth latter composition mapping
reweighte property par par image par comparable par trial previous ard objective par practically feasible assess provide convergence par establish addition compute posterior variance variance indicate variance therefore relative reconstruction variance suggest variance boundary subject future use os par split process lead considerable report literature os os thank provide access laboratory experimental acquire like thank mr dr trust newton mr david manuscript provide add penalty positivity consist detect ray ct therefore measurement noise ray medical community class measurement model accord inherent linearity transmission class statistical inspire automatic relevance determination ard allow incorporation positivity poisson underlie apart fidelity avoid parameter determine knowledge ard likelihood mainly ray ct medical method present transmission transmission denote exponential represent ray voxel pi py line source would th detector object standard independent calibration construct entry line source th voxel
minima nonetheless carry rank great explore visualize region nonetheless numerically feasible trivial convenient rank scalar penalty varied feasible minimum characteristic zero canonical finally element varied display nuclear norm produce rank cost display incorrect global appear success depend heavily carefully decay hope initial towards problematic additional spurious explore underlie regard guarantee unchanged iteration function unless satisfied show must bound cluster guarantee converge homotopy regularizer require carefully choose minimum factor perform poorly practice necessity prove formal practice zero promising remain important conventional minimization penalty whether fall nuclear respect consequently cost globally nonetheless distinction lead minima circumstance either construction lead nonetheless improvement covariance column overall kronecker sum q proceed fashion modification alternate upper accommodate use list compare art affine noiseless effectively
introduce change prior construction inform gauss function gauss newton give truncation threshold correspond eigenvalue impact likelihood balance retain informed direction typically threshold extend criterion inform direction global consider use posterior hessian feasible store scale construction global inform suppose sample truncate construct global hessian orthogonal access newton one approximate either construct dominant explore directional landscape project global inform subspace onto self adjoint induce complement factorization key correspond compute orthonormal choose form complete naturally decompose cs transformation illustrate hold prior reformulate r define rewrite reduced complement expectation approximate prior sample posterior importance weight far
worth formula c descent instance accelerate coordinate specialized nice sampling sometimes minimize restrict serial picking coordinate uniquely coordinate never coordinate serial form complexity eq equal time systematic inequality descent method normalize hadamard define already establish different potentially outside randomized c elementary intersection restriction vector define output diagonal hadamard vector index eigenvalue shorthand notation obtain pr day design complexity randomize particular variant refer involve capture way established develop systematic one eigenvalue
fairly relate score lem continuous ki comparing q scaling lemma imply contribution triangle follow trivially k title corollary mit solve massive improve processing time row sample unfortunately leverage score difficult compute row eliminate critical row information instance look sampling examine sampling fraction weak enough observation approximation preserve addition spectral sufficient leverage turn enough preserve key understanding preserve nonetheless increasingly leverage score iterative uniformly row estimate score leverage score reduce aim shoot estimate cut sum estimating
skewness link mobile phone many ratio much fall aim range visualize performance score base metric past example propose elaborate take classic follow number share correspond weighted class calculate whole distant compute weighted link start probability return steady process active connect exact metric expensive approximation use sum keep term predict imbalance distant pair likely pair class imbalance dramatically large distance expensive especially ranking present merging method system rank rank
limitation user priori stage even though large information contrast select design criterion may stage stage implement design pe implement test platform pe window package user generate design use phase phase iii package adaptive adaptive design multiple term outcome available stage stage though stage stage package platform open web little interaction knowledge language interactive web application computation manual application default www www package locally google view option free quick slow encourage heavy locally web run currently com window www window internet line third progress package library package internet library mm calculation directly compute design far application input table table plot see interface generally interface run locally panel output figure design panel describe
period actual need estimate one trial take robot hz try period result control trial minute finish learn limited verify robot controller master controller frequency hz synchronization mechanism controller trial start period e individually fig angle trial sd front front every scenario result combination period robot indicate generic learning anneal determine learn e possible period acceptable permutation acceptance combination period discard previous accept bad condition anneal reduce greedy bar represent permutation green bar greedy annealing situation value depict upon bar figure permutation increase large bad scenario perform period l situation increase combination discard however suitable annealing search bad
expansion recent genetic analysis tumor analysis insight understanding predict response heterogeneity population chinese reconstruction associate merge sampling overview reconstruction remainder consist description series variant illustrative tumor evolution visualization evolution grey expand tumor circle refer share distinguish parent parent mutation highlight
common simplify ray ct application often convergence show term convergent inexact method although term open admm admm convergent admm appropriately concrete mathematically regularize denote measurement image degradation simplify far I clearly algorithm quadratic
right trajectory spend third horizon generate along log plot average length plot natural logarithm horizon plot detail linear dependence natural logarithm linear value dependence measure average increase budget value imply imply depict trajectory regret plot figure average trajectory slope illustrate regret since hold affect particular size top structure describe linear log already depict bottom leave slope estimate bottom figure illustrate support minimax level traditional tight mab quantify price stationarity mathematically capture reward stationary together characterize minimax separate arbitrarily growth quantify price non stationary reward one allow
subscript remain loading remove follow previous estimate hold x give fix solution ty ty ty ty substituting problem maximize convex see split length column become solve back case sufficiently small ty w substitute ty ty back finally solution partly program cb national china education china theorem lemma component aim interpretability dense pca meanwhile exist problem add penalty various objective pca paper whose motivation rotation basis basis approximate three loading bound scale physical unified load loading loading loading lead loading dense loading less important indeed helpful property concern globally global otherwise property type loading exist compute loading one loading scheme former
associate lagrange solve problem minimize problem traditional rectangular significance digit varied digit five digit assume
recent cluster matrix whose relatively zero subspace express member subspace express combination variant ssc ssc bound representation lrr norm penalty regularization guarantee paper object maintain two form linear linear combination propose far page object tensor entry object write oriented weight scalar employ multiplication product product construct introduce scalar font bold matlab indexing treat
alternatively randomize quasi base asymptotic sample simulate different simulate ce random variable discrete base ce longitudinal count propose copula distributional transform early appearance simulate advantage copula continuous margin ce likelihood cl modelling aggregate simulated ce dt cl efficiency calculation ce inefficient lead univariate multivariate cl joint lead study base dt recommend dt previously dependence modelling dt study copula discrete simulate proceed brief overview provide theoretical efficiency discuss estimation two turn surrogate dt could background copula multivariate cdf margin variate cdf margin
credible fourier approach intensity phase distribution approach conventional transform acquire free induction signal quantify approach outperform uncertainty ratio peak induction powerful signal impractical chemical alternative technique chemical relatively alternate approach decay inference principle latent variable conventional transform decay frequency promise typically conventional research tool delay detail chemical quantification robust overlap peak fouri transform large apply resolution less brief conventional fourier transform introduce parameter resolve statistical prior number jointly phase multimodal surface
geometry fundamental reconstruction presence characterization misclassification tractable misclassification bound k k union asymptotic characterization identify absence misclassification perfect absence misclassification lead characterization within boundary bind bound boundary metric offer refined behavior upper decay regime name diversity slope characterize one characterize quantity term classification gaussian distribution ik observation feature pair affect key diversity order side information expansion misclassification draw condition give ik misclassification j ik ik ik ik ik see provide characterization feature description diversity diversity pair classification diversity pairwise dimension linear index dimension side provide subspace possible discriminate among correlation rank conditionally class kp ik r r ik ik ik diversity correspond span decoder discriminate class dimension space span diversity classification consequence characterization misclassification probability access upper zero upper misclassification feature misclassification zero accord ki ik exhibit ik misclassification zero j ik ik r ik ik r ik ik ik cycle node start dot cycle cycle dot ik j ik ik r ik ik r characterization necessary depend whether ik distinct r r r depict tradeoff case bind misclassification correspond index space matrix determine effect correlation transition transition free classification ik importantly ik side ik r ik
derivative derivative increase negative turn complete ex w q define dropout symmetry contribution affect express condition label satisfy constraint minimize define show output divide eq must complete sufficient make informally order term show pay margin error probability adapt analysis provide term actually something somewhat complete proof normal random complete drop complete lemma eq complete ex since show need proof convexity since complete use regularize specialized contribution case affect furthermore leave since fix symmetry identical recall q prove need rough proof inequality constraint x ex moment inequality relate binomial moment directly imply fact constant change enough ex work less rough minimize range give take among
location mean approach sufficiently perform recovery convex measurement greedy however measurement already exactly recover versus left trial figure report accurately estimate recovery noiseless robust regard recovery support plain dash sect start set measurement square magnitude fourier ii compressive signal measurement sparsity reduce seminal phase retrieval support technique signal idea apply linearize measurement minimize
fit main fit model property well observe geodesic distance plot figure summary line reasonably low degree statistic fail actor network high degree geodesic correctly predict actor connect number actor connect variability cause section htbp b recently introduce extend provide analyse relational mixed beyond interact majority actor assign partial membership group interaction interpret accounting heterogeneity model latent incorporate demonstrate variational computational still take single model outline validation burden likelihood variational effective literature group identification care appear treat nuisance observable blockmodel latent actor form
adjacent detect tried compare propose code platform good survey statistic local network filter clique lastly clique one clique another clique process stop clique share cover clique process maximal seed maximize function fc c I c ic ic label discover overlap receive variable evaluate measure extent compare superior superiority exist expand expand process network mix build expand clear smc identical smc identical except expand method smc expand method run smc parameter provide clique default community threshold value network iterate pick performance membership parameter subscript rd figure nd eight rd column first diversity
parameter connect auxiliary interest efficiency present contrast suggest start association variance nuisance correlation fisher regard auxiliary statistic perspective argument easy check assigning inference admissible auxiliary control expression accuracy predict result logical reasoning argument usefulness infer reduced q nuisance auxiliary variable association involve associate allow auxiliary depend nuisance nuisance
simple inequality dt obtain analog next suppose follow following hold optimal minimax us auxiliary function transform function function moreover mx dx furthermore x dx uv e v second statement lemma density density v eq distance mx p dy dy mx identity derive
constitute half boundedness although contribute belong correspond constitute edu optimal approximation remark ex cm asymptotically nonlinear filter stochastically convergent process approximation approximate mmse nonlinear filter property stochastically convergent approximation observable stochastic comprise stochastic state conditionally assumption time stochastically well compactly unity present basis prevent design nonlinear even practical design fashion apply constitute novel mobile research author elsewhere go deal mild technical definition prove later devoted result partially observable relate essential background theoretic measure mode stochastic convergence development
generator extreme negative apply identification show generator author acknowledge non deal separability introduce reduce constraint nmf datum nmf nmf point topic generate
cr il sentiment un cr une un un les phase du attractive clearly show insensitive replacement projection neural translation simple language feedforward list strong mt reliably improve researcher include source combine improve follow incorporated decoder mt system decoder useful improvement baseline sentence although sentence order word work rnn sentence back focus et direct network
incomplete incorporation base complete incorporation relation function principle insight q incorporation probability nucleotide incorporation agree count q elementary nucleotide incorporation reduce incorporation lead result compare check cycle pn f nucleotide incorporation distribution nucleotide incorporation nucleotide use nucleotide incorporation case non nucleotide incorporation complete nucleotide incorporation close expression distribution small module expansion speed technology give distribution associate synthesis practical technology development testing sequencing contain mathematical traditional termination currently available
combine hyperparameter note relevant solution configuration role determine analysis datum combine choose change describe exception cluster obtain way performance level overlap simulated true generation regard value configuration configuration centre setting hyperparameter sensitivity estimate group complete result ccc group hyperparameter default number repetition always recover separate method outcome obtain hyperparameter evaluate suggest rate want combine guarantee obtain well combine solution compare choice hyperparameter overlap final considerably hyperparameter meaningful reasonably return experiment bivariate essentially divide cluster make highlight
business attractive behavior la la htb pt contain author allow simultaneous continuous ordinal nominal logistic adequate theory ordinal use extend result term ordinal logistic ordinal logistic response dimension column produce span row odd obtain multidimensional item geometry computational calculation prediction direction logistic response surface project score odd multidimensional item prediction representation apply study knowledge innovation extract information international jointly carry operation development institute brief discussion concern nominal
accurately dimensionality hardness nearly dimensionality large computationally unbounded adaptively build use hardness result work identify hardness slightly weak nice code code achieve thus hardness environment interactive combinatorial object hardness hardness give interactive interactive code intuitive hardness able code privacy structure analyst know draw analyst answer reconstruct adversary representative adversary object query originally digital interactive code interactive adversary follow game pick specifie response adversary adversary refer interactive contrast code interactive interactive code identify suboptimal achieve technical result boost non interactive code recover code technique interactive independent copy interactive code shorter interactive give suitable extend interactive well interactive achieve still interactive except negligible false security detail section discovery roughly reconstruct answer private reveal hold notion privacy seminal attack use establish interactive combined framework every computationally accurate adaptively
remainder specify framework section dimensional definite summarize sample call precision precision matrix symmetric gaussian undirected node denote maximum
density eq relation step et al hold converge proof vector converge contradict conclusion subsequence equation subsequence subsequence converge since subsequence probability one year formal machine approximation typically possess new state assumption divergence representation generalization review joint large directly important ii learning presence iii active adaptive training statistical environment specify estimate density use paper white
execution decrease try second second average try variant present median reported present combination lie span orthogonal position perturb hardware assumption negligible time stable choose outperform speed correspond implementation hardware extend nature set complement datum q already hull proposition depth hull increase
normality suppose theorem regularity f pn begin expansion around assumption second use regularity second q therefore weakly class kde bound pn pm give bias via conditioning stein estimate respectively recall boundedness argument inside via jensen third precisely schwarz us denote obtain instance occur argument last term take bit difference stein complete alternative separation adapt hellinger divergence index follow satisfied q straightforward modification theorem tf np vs vs eq make complete proof index perturbation use result hellinger zero satisfie exist ball
think independent fix weak risk plausible depend derivative calculus variation yield minimizer however dominant let minimum context existence quite approach optimality dominant estimating make expression converge proof underlie one although successful intuition prove minimizer aspect behavior converge bound explicit result perhaps high start stand rate classic say shrink sense minimax shrink neighborhood say neighborhood achieve rate shrink neighborhood strongly consistent least shrink neighborhood sense
sequence notably compete second compete location performance indicate book average indicate second well precision video book precision heavy object resolve handle especially frame good mainly handle propose extent face object pose variation fail object follow object fail partly variation obtains successfully contrast severe illumination rotation plane rotation track method able track begin fail frame track person
electrical computer engineering interest vision recognition winner microsoft fellowship lin receive ph university key laboratory computer normal university microsoft compute technology chinese interest vision graphic learn associate intelligence international computer vision area extensive iteratively reweighte recent lead broad recognition collaborative filtering popular guarantee practice sparse either frobenius nuclear norm
utilize several index suggest conservative quantitative risk management able distinct fail dependence compare generally behaviour tail slowly correspond motivate search small possible organize idea include admissible maximal example demonstrate index tail extent extreme family copula somewhat discuss section general copula conclude calculation copula idea present
mm water requirement manual university publication pp frequency day consecutive journal engineering apply science mm chi square size rao poisson series modify statistic goodness test theoretical independently assume asymptotic statistic chi square freedom less et concerned potential conduct fit al et
first return label automatic target mostly correct correspond sample recall track seed inspire associated seed maintain detector pool candidate location region close flow object new location negative spatio temporal smoothness object instance negative target easy self successful object conduct detector seed positive negative transfer generic detector oracle seed box approach background applicable stationary move negative difficulty
increase bad variance bring adapt draw rwm proposal carry acceptance rwm estimator variance increase sampling fraction datum long period produce compare posterior rwm data obtain accurate posterior explore extremely explain accurate figure fractional subsample size work regular integration likelihood note lead even large relative gain note numerical integration slowly increase size numerical problem subsample tune dramatically time consume use efficient scheme bind scheme si work many order assign contribution likelihood sampling satisfied conservative evident application adaptive fraction tight avoid compute probit time effect
representation compare additive well present stack generalization noise style stack involve meta score clean db corrupt previous style classifier stack hence superior performance classifier match average improvement multi test difficult practical scenario classification result expect lie match white stack scenario classifier scenario filter speech agnostic rely second employ environment degree match employ accurate filter classifier square star stack exhibit trend generally improvement condition classifier yield well classifier db db similar conclusion comparative performance regime attain improvement across development input meta vector approach flexibility adaptation environment moderate correspond whereas combination difference confusion show suggest error independent classifier individually value combination parameter value empirically previous explicitly decision performance combination regime
fall sample suitably interpretability translate definition look key generally become due interpretable converge interpretable let small b require converge try construct measure exactly rather act distribution essential equitability measure affect hand fall definition equitability statistic measure intend independence rather idea approximate equitability attempt robustness measure utility two rather strict explicitly allow possibility preserve act measure distribution capture truly seek done unfortunately become however statistic reasonably define second explore fully worth requirement converge requirement merely largely subset equitability ideally instance equitability noisy functional relationship relationship equitability hypothesis testing measure equitability define equitability think nan equitability generalization power behind equitability equitability respect concrete noisy consider tailed statistic nan must yield tail hypothesis relationship relationship relationship reduce independence case nan hypothese non property tail tail distinguish particular subject constraint subject result ready equitability call significance statistical give tail distinguish zero value bivariate distribution ordinary distinguish alternative nan case relationship equal power tail nevertheless power besides contain right interpretability analogous tailed test hypothesis uniquely nature reliability need agnostic detecting aspect power reflect uncertain x
transfer knowledge auxiliary misclassification discuss extensively design collection traditionally advance technology digital storage information rapid collection format image audio document popularity attribute record automate expert reveal desire annotation slow development advance computer vision document annotation rely expert annotate utilize automated cost annotation problematic probability differ crowdsource alternative expert annotation annotation annotation annotation carefully design crowd worker collect
neural generalize glm glm flexible activity phenomena activity technique glm network visual devise quantify spike train student ph thesis pt increasingly tractable analyze predict neuron increasingly statistical framework flexible exhibit
ratio maximum peak basis observe region lattice smoothly datum point sample let subgraph enforce connect encoding define flow belong phase connectivity therefore variation triple straight composition phase enforce enforce obtain actually phase peak regular nmf range hour nmf sparsity violate previous soft assignment reflect fact multiple define c reflect amount violate constraint future report visual incorporate constraint exhibit well quickly analyze realistic hundred could hour dataset phase pattern phase chemical identification
notice point view restrict absolutely continuous respect absolutely key use least theorem indeed ignore moment resample order appear instance change trivial sampling section value motivation resample classification highly asymmetric click weight frequent reference towards multiply importance frequent likely impact case easy take gradient favor less performance ultimately drive case schwarz surely q optimal observe gain term large initially negligible optimize stay clear experiment tend
full ix define opt advantage base current select k I linear advantage improve redundant treatment advantage stop incremental advantage selection decision make step let assign quantity outcome randomly selection corresponding covariate advantage q update regime update sequential increment variable stopping criterion iterate sequence incremental advantage characterize variable point advantage newly explain explain currently mean make decision variable selection sequential selection propose select method way score score fit linear optimal penalty penalty simulation cross r study observational study relevant make treatment patient column consider three generate simulation interaction form baseline ty covariate entry
adaptation account train amount detailed adaptation prevent c activation architecture function adapt day l adaptation bold architecture hide layer bold respective shorthand tangent activation percentage corpora line accord topology layer initialize matrix layer backpropagation record select integrate system table lowest keep topology part incremental decrease minor use result configuration corpora clarity tendency end updating incremental compare last improvement part
satisfied strictly sufficiently also use assumption suitably histogram check cluster thick order thick level restrict distribution density recall dimension illustrate nonetheless exist continuous density use easily generalize establish usual assumption satisfied exponent exponent exact exponent fast component separation exponent monotone separation exponent describe illustrate ensure cluster exact separation exponent polynomial somewhat density asymptotically identical assume derivative vanish analogously class density exponent behave saddle separation influence begin let satisfied additionally exponent pick receive exponent replace theorem double bound cluster separation exponent receive separation exponent sufficiently large exponent discuss thus exponent exponent converges consider hence exponent establish
sample go extend framework high simultaneous integrate selection extensively throughput rna restrictive existence uncorrelated gene cluster gene offers automatically integrate generally address dimensional current related computational
disadvantage solver explore hierarchy encode rich second suppose split overlap often posterior posterior posterior explore variational exponential posterior approximately parametric reasonable right mc independent sampler simply aggregate intractable choice group chain single mcmc parallelism straightforward chain strategy adopt slow massive datum category compute processor gpu computation across combine local likelihood unit responsible update space extensive communication specific compute unit explore independence ci hierarchical ci chain posterior communication key combine sample posterior attract attention consensus monte directly combine valid implicit approximate follow build kde represent posterior embed posterior hilbert parallel mcmc posterior combination inaccurate aware efficiently eventually approximate use process multiple likelihood ahead hasting represent binary stochastically reject evaluate core respect core execution ignore communication efficient branch delay acceptance testing method naturally iterate early parallel separate processor extreme parallelism work fast parallelization maintain target correctness gibbs parallel sampler parallel various distribute sampler augmentation logistic normal mixture ci extremely partition stochastic exclusive often lead efficient solution optimization extensively particular sgd share component scheme particularly langevin specialized hardware use accelerate graphic graphic unit self contain device conventional computer distribute core processor graphic easily maintain code dedicated device consumption smc bayesian work demonstrate accelerate collapse lda likelihood eq gpu parallelization smc sampler gpu fast hamiltonian add acceleration e gate literature extensive progress beyond demonstrate word though
combination method solve performance good improvement show cccc theorem sec sec lemma total combine comparison several related report old old theorem preserve old new version additionally performance well well far subset big whole decrease yield new third learn human dependency quite inferior minimize fourth explore many learner slice old htb success lemma lemma name k nn type lemma type k lemma human nn full new mining conservative probably core plausible library much author cut produce small graph predictor add lemma edge produce dependency produce dependency correspond success possibly cut far account small translate allow automate ai immediately formal knowledge computer interactive library however library contain advanced form proof whole development pre exist symbolic reasoning amount library extraction later exhaustive
compute budget learn except fold observe lead precision forest superior value column remain baseline reach drug interaction go attain randomization optimal maximize dependent task relatively remain comparable consider base show trend randomization input build forest drug interaction feature label control forest randomization control accuracy decrease strength low case learn subspace performance behaviour completely phenomenon curve htb reader study appendix material show different time specific combine reduction trend matrix matrix replacement hadamard rademacher subsample space compare rademacher space sub
proportional hazard specify u weakly gaussian score specify weakly p consistently baseline patient independently uniformly covariate survival take extreme correspond odd censor choose achieve censor obvious treatment class regime contain treatment impose x need survival hazard distribution correctly follow compare well smoothed combination assume model term censor replication genetic implement package save presentation report logistic omit table survival follow treatment year plug establish empirical coverage survival probability cp true year misclassification compare regime standard treatment regime logistic treatment regime match relatively bias close one simulated treatment year survival year survival relatively bias estimate survival
present sequence variational sensible state state gradient likely make possibly gradient enable independently develop train expensive
random mean plot epoch act stop beneficial theoretical finding result suggest epoch appendix repository dataset incremental iterative kernel ridge report method inspection approximately remark definition laboratory usa learning
tensor language gpu analogously safe convolutional forward descriptor attribute filter descriptor specify along dimension tensor primitive convolutional neural special convolution list section forward form backpropagation closely relate mini input width output vertical height zero input convolution convolutional filter mini map per image feature map filter output tensor previously image height computing specify convolution mode set matlab
return estimate combine equation rl equation formalize rl learn policy class classification easily rl vc rademacher equation define use classification see go move equation encode reward dynamic analogous crucial relationship rl rl eq n return observable use calculate describe rademacher describe vc unfortunately vc linear indicator function
upon cutoff empirically simulate conditionally identically set burden compute approximate use become recommend large matrix satisfy theorem building random apply projection random expect accordance far small adjust correlation correlation entirely maximize appendix regularize emphasize power upon challenge minimize test simulation draw e haar matrix manner one sketch generality break evenly use project
curse nmf suffer investigate input scheme rule descent scheme multiplicative lee method problem hyperspectral art nonnegative reproduce become prominent interpretable data scope extraction compression recognition processing lee nmf successfully face recognition biology gene nmf consist approximate constraint physical thank part idea issue hyperspectral illustrate hyperspectral reflect typically acquisition reflect cube characteristic contiguous band resolution vary resolution superposition spectra underlie material hyperspectral extract spectra pure material area obvious spectra nonnegative nmf physical interpretation nmf
equal log function expression easily numerical regularity confidence interval confidence significance denote normal sample size base present simulation triplet initial element distribution independent estimator bias n htbp
interested suggest specific nuclear idea gap inside tolerance information arise differ structured duality possible approximate theory example section respectively dynamical system transfer truncate impulse negligible impulse create choose
extract low effect precision decompose regularize ml high rate rating completeness movie three covariance proxy element actual validate intuition span top residual capturing effect heat thresholded expect rank capture precision much stock stock similar low rank heat show global effective much number stock ac lead global sparsity remain conditional effect prove decomposable learning suffice verify condition elaborate taylor loss true perturbation rsc specify tolerance hold rsc sufficient direction convexity restrict imply rsc rsc precision
construct correlation column span approach column actual use top eigenvector matrix approximate span entire weak use span mean still corner separately modify algorithm single linkage take algorithm rigorous time near approach apply mixture spherical pac small close concentrate pac recent additional spherical align gaussians linear exponent spherical gaussians notation result bind give appendix product mix estimate empirical mixture collection output obtain contain class distribution work albeit constant estimate bind gaussians learn spherical gaussian mixture spherical near
develop university ie expression text sentence fail consistency present temporal expression whole text enforce publish challenge extraction text extraction analysis event increasingly challenge natural process nlp application
base h h p see apply bind chain introduce bound ax dx ax kx k dx n tw n kx x error x quantity lemma mapping chain xx follow expression correspond general error express apply lead approximation hold pick n single define I estimate give rewrite yx follow secondary variable statement accuracy union bound first long side obtain rewrite density rewrite scale deal process endow affect synthesis maximize associate property reach avoid reach deal assess horizon trajectory goal avoid express state reach formal relevant air maximize toward avoid model evolve continuous domain discrete reach specification lead rich unbounded focus specification numerically figure probabilistic study scheme quantify error bound
simple end mm measurable support supremum reach hand deduce write bernstein gibbs hypothesis extend note true rest equation bernstein case note infimum two latter specialized measure specification gaussian spike put consequence q obtain previous lift step define similar previously upper quantity soon put
stream price price parent company despite lack price nevertheless like able go period chain adversary utility efficient word bad function make prediction fall framework contextual web budget ad exchange amazon price power poor self category g store crowdsource management reward daily worker select task view price view purpose optimize fall large bad case online target try value bundle additionally despite linearity utility reward entire budget round repeatedly classic preference set restriction utility particular discretized differ amount induce price upper without set infinity
lead combine minimum mse minimum remain satisfied traditional performance angle gap mse comparison use accuracy angle gap rate angle corrupt white average fig
must supremum f ft surely combine left side decompose ft ft subsequence subsequence surely step complete violate k l almost surely contradict k l justify lemma I f condition yield suppose hold rt n simultaneously observe rt rt rt term converge almost surely suffice lemma suffice triangle imply rt ft rt ft rt ft part rt lem simultaneously conservative imply show readily imply imply condition consistency complete follow argument j j j j rt rt rt testing side involve joint x x give calculation yield bivariate section derivation plug explicitly consider plug p dp p dp p f associate co greatly real theorem support grant dms dms context large certain present sim maintain false availability bivariate preliminary analysis
blue ridge mark usa pool mark randomly mark filter hereafter blue lead hereafter assign randomized identification use light difference observer correctly influence example individual mark mark laboratory know individual variation analyze single error match satisfy history mark recorded history record history observer light would analyse blue light examine accurately temporal modify accordingly light million iteration pilot tuning burn start require long run due movement mh sampler diagnostic somewhat blue effective sample blue find median credible table hence reliably light probability
side fit plain outli basically dominate odd meaningful cluster produce side figure cluster high balance birth death cluster medium rather collect low rather despite produce observation look turn substantial disagreement even fix compare classified rate somewhat consist song true class compute discard either compatible discard correlation carry identical list mean mean std clarity feature standardized inspection reveal systematically although overlap extent coincide rand ari value clustering maximum agreement ari interpret third default ari ari ari suggest ari ari particularly high ari classify ari suggest core region run date compare robust shape cluster cluster method comparable formally
gaussian process ic reasoning function expectation function set frobenius norm g eq upper standard thus c bind decomposition metric regard denote kronecker deviation r overall l briefly outline guarantee asymptotic refine presence naive robustness reasoning naive factor completeness straightforwardly show locally dictionary lipschitz gain lipschitz show denote coefficient minimum yield reader notice eq well hence c eventually k eq cp r notice provide frame low frame thus lemma assumption corollary I imply refined result reason finite minima coding noise thus focus noiseless signal combine almost sure around truth energy predict similarly completeness develop realistic spike second dictionary structure dictionary blind calibration improve
contain member repeat macro member double bind heavy contain express protein protein protein pr psd sec contain protein type protein protein protein non domain alpha domain ig tm short member nuclear envelope repeat contain member specific protein sort member htbp bid interact domain death contain b member alpha bind class homology bind interact b theorem corollary rna usage continuous covariate sensitivity specificity method name usage rna seq correspond gene accomplish covariate latter situation compare allele one tumor specificity effect gene rna seq expression differential usage high dna gene often include separate several rna rna multiple may stage produce rna cell great understand functional change genomic disease traditionally microarray provide gene rna rna sequencing rna seq much purpose rna seq rna end sequencing end pair sequence end rna seq rna seq reference genome rna overlap expression th gene measure adjust depth th gene major challenge rna observe rna specifically rna seq compatible
scenario causal approximate rely auxiliary towards income income hide formalism theoretical satisfy physical satisfying exist notion correlation explain correlation structure partially order direct thick anchor east anchor north west space circle thick fill blue minimum mm lb build consideration quantum conjecture classical quantum challenge partial conventional scenario discuss classical correlation analogy network classical model hide notation collect notation usually omit acyclic think mostly rather similarly edge think link propagate exception hide network v v u edge put section keep distinguish carry notation somewhat since clear present intend rather constitute approach generalize arbitrary structure definition achieve adequate thing certain constitute scenario consider problematic approach observe situation statistic repeat assumption trial spirit justify neuron condition encourage conduct system causal structure compatible need causal structure causal structure may whenever event event regard obtain finite relax albeit technical challenge node write collection disjoint event node causal mathematical interpretation confident biology science situation influence causal formalism nevertheless causal whether compatible observation causal formalism come possibly standard commonly link proceed finite sequence shorthand direct absence conventional formalism network propagate indirect intermediate potential indirect happen depend structure long constitute merely environment expect occurrence causal member causal loop g graph sensible length zero figure acyclic graph mean direct
denote curve th observation th dimensional vector know function zero mean process th account correlation repetition directly smoothing procedure derivative obtaining reconstruct ft ft ij dt identically distribute independent ft dt reconstruct right equation case simple derivative inner product global covariance context functional blue
strongly convex make suitable convexity adopt note optimization besides extend constrain please relation component convexity ii I np prove em solution objective assumption k algorithm take condition next equality k convex unbiased estimate appear optimal l lipschitz gradient together fourth due
observation error matrix error mis show report simulate manually variance similarly present fix approach optimize match temperature case pos accordingly rate exhibit large manually optimize smoothing estimation manually variance perform particular lag smoothing panel variance panel variance flow rate flow lag panel brevity experiment term covariance manually variance lag smoothing lag appear low mean rmse pos cf fig assimilation may attribute eqs optimal circumstance one adopt assimilation precise beyond assimilation manual interval lag mean rmse time rmse soft sense salient flow jump ability dynamic auxiliary toward temperature pressure physical sensor flow rate jump implement utilize available temperature pressure
unknown tune standard error assume q assumption theorem see sake smoothed hinge hinge compose gaussian svm tune validation range test classification class amount gaussian dimension norm training close extract elementary bind key idea function idea derive q maximize use bound b concavity logarithm prove differently use vary often proof max w achieve expression elementary
compactly mf atomic compute via programming toeplitz entry presence propose atomic noiseless sdp frequency decomposition computational sort sampling practically since always ff specify definition exactly atomic atomic late analysis observe contaminate noiseless write sdp signal subset noiseless signal per square noise variance eq optimizer part derive appendix generalize complete sample guarantee noiseless show frequency recovery frequency
filter largely parametric signal dimensional infinitely chain hierarchy evolve dirichlet measure speak upon background filtering review hmms evolve mixture parameter projective process finite cox relate conjugacy static mixture exploit project admit computable result exchange filter valid depict former projective respectively obtain duality propagation cox process rectangle x n projection
low embedding additive approximation question lie substitution rank development assumption nuclear would approximate multiplication research framework fp agreement concentration direct concentration dt case use eqn thus become matrix
os k cluster primarily embed closely resemble practical partly heat traditionally within believe wide partitioning setting search dimensional partitioning factor cut lee study higher partition subset partition expansion formulate outer almost linear gap et center combinatorial contrast result bound perspective heat heat use partition current efficient cut matrix multiplicative use embed guarantee present cluster unweighte u ss extensively algebraic adjacency give eq give matrix v fs formal cluster eigenvector normalized characteristic thick
excess risk transform surrogate could binary risk smooth convex surrogate reduce optimization specifically achieve loss optimistic surrogate excess risk question reveal excess examine smooth account excess favorable appropriate smooth excess briefly discuss classification calibrate surrogate relie derive transform elaborate excess derive smoothness binary section omit n I draw dy
front typically focus label relational network network general gain variable emphasis make combination user wish simultaneously word model text permit incorporate attribute label million explain stanford several type high college public desire depend application five label school college city label primary situation people meet become friend school college situation mostly mutually exclusive may share school simplifying explain necessary friend indicate co locate building meet imply individual friend result type formation mutually
input computer simulate j adapt computer calibration couple simulate conventional call thousand replace ideal modern idea calibration local mesh methodology necessity recent trend simulation synthetic variation motivate limitation generally word nonparametric adaptive rapid increase computational power system field work physical phenomenon study induce bias must simulator extent account interested model calibration require describe physics center system develop field evolution input address mathematical small diameter conduct circular configuration aspect ratio describe shape circular field experiment aspect ratio circular exercise shape c design ce ce diameter length ratio ce ce energy input require detail separately super computer exercise third reveal range computer ce ce explore input circular region vary diameter geometry derive disk hold ce computer simulator energy unknown involve heat mesh output insensitive scale explain
hypothesis condition upon non decision domain bf test procedure frequentist relate statistic try bf frequentist hypothesis test propose test normal frequentist fundamental asymptotically regular smooth tend nuisance still lr x previous variance nuisance next end derive relaxed difficult within statistical frame general equality classical invariance arise haar topological notion necessary integral integration accord context give frequentist assumption parameter transformation call family densitie lebesgue group action measure absolutely measure marginal define integral frame avoid technical consequence relaxed random parametrize condition upon whose family sufficient statistic whose sample replace statistic frequentist measure call respectively haar measure induce absolutely lebesgue call marginal posterior frequentist sx
maximal clique clear limitation high property review detail number appear n ai setting particular count also clique mass nearly see proposition differ write term original instead enforce statistic clique adjacent implicitly prior expression expression work supplementary material analogous set vector factor count consistency approximation go derive distribution allow pass multivariate follow directly indicator fix I I
pixel constant vector pixel output figure benefit classifier convergence observe energy current possibility energy software fit rest problem impact future could overall systematically bethe propagation energy alternate parameter smooth algorithm update stochastic lp iterate perform optimization energy making pass slow
write lagrangian exist irrelevant equivalent lemma correspondingly fix application far advance transform constrain already start optimisation stop constant hold issue poisson give alternative involve integral langevin approximate likelihood iid turn classification f integral hand general likelihood availability estimator gradient counterpart langevin apply straightforward adjust see use logistic logistic appear place general call divergence divergence follow odd replace
library optimize generator seed result bias correspond bias slightly bias line measure observe parameter base parameter choose compare size asymptotic regressor fix regressor table uniform deterministic regressor bias deterministic regressor random cause slow r mm estimator score demonstrate admit test generating
q cauchy eq combined union consider q hence uniformly focus union value provide cauchy know cover entropy center cover probability cover conclusion demonstrate furthermore demonstrate consider rather negative introduce bound term perturbation argument consider second trying argue small let pair consider objective mean theorem assume thus able cover ball also cover radius entropy bound fix eq set dominating eq eq elliptical subsection eq r curvature control lemma imply oracle property
characterization large release motivated analysis collection probably learner accord unknown concept generalize example label take requirement preserve differential mean affect particular sample rigorous private survey determine complexity proportional complexity learner exhibit logarithmic class low possibility complexity learn analogy complexity private combinatorial sample learner towards characterization introduce notion concept vc learner computationally sample simplify exposition ignore dependency privacy follow dependency parameter later section private proportional informally concept ignore union argument use exponential mechanism learner show et et function learnable query learn efficiently positive natural computational et al study examine evaluate point al properly imply
impose forced cutoff define reduce support future must serious arbitrarily number bit power believe approach generative dynamical property power material mt code run machine dependence desire number show compare default run implementation mt triangle green circle run numerical square black triangle run square bit equal
offer use object semantic labeling subset derive label cifar organized word average concept human annotated imagenet label million image million around annotate imagenet class apart unlabele unsupervised belong resolution preprocesse see digit digit image significantly hard digits scene house image resolution mnist handwritten digit center intend object contain image generic human object degree pair good algorithm
shape composition tensor substitute perform grouping give sequence kernel intermediate network essentially wise compute flat utilize convolution backpropagation momentum layer fine layer gradient either convolution operation element require number rank comparable take convolution absence bi multiplication theoretical problematic bi tensor hand require rank considerably assume architecture character bigger train devoted layer make
negative distribution consecutive conditioning string conditioning string formally sign iid mention reflect possibly sign marginal without horizon characterize exponential horizon also horizon independent maximize horizon independent finite statistic may regard bernoulli distribution present sign maximize cardinality measure support tt
inexact alternatively factor keep precisely nonnegative square solve decompose independent since nmf descent differ symmetric input nonnegative factorization nmf generate see section update update update step section describe several nmf important tool order component satisfy condition formal explanation nmf naturally either multiplicative modify follow division develop nmf mu everywhere current decrease monotonically mu interpret rescale another intuitive entry derivative decrease derivative partial entry zero modify occur entry zero partial satisfy mu converge several way rescale descent see modify low bind update become scale iii research mu converge relatively slowly theoretical note
variational techniques variational provide scale normal gamma likelihood gamma correlation medical explore non approximate c predictive ts gamma gamma ts drastically analytic burden essential success variance gradient black work improvement library family simply gradient second dynamically carefully distribution g carlo significantly write expectation start derivative integral q simplify supplement rao integrate full family distribution field rao recall
publish real dataset process relative spectral acknowledge acknowledge generally effective illustrate hyperspectral imaging fusion vector variation nonsmooth alternating multiplier admm term spectrum field hyperspectral spectral visible near band narrow offer resolution range spectrum interest source image fusion
place draw converse potential new cluster every potential cluster draw conditional assignment simply conjugate replace cluster rule assign q straightforward density read state generic implementation detail prior heavy tailed distribution recommend transform j nc ic slice tw rest value break exact need author proceed number auxiliary lead slow memory requirement quantity point correlate slow quantitative ess
polynomial estimation estimator achieve time number distribution enyi entropy measure randomness discrete shannon measure diversity quantify activity anomaly estimate shannon give extension near complexity shannon precisely definition enyi mapping sample entropy minimum require estimating q additive great definition obtain confidence typically interested dependence size alphabet regime essential growth notation sufficiently k k eq namely grow linearly enyi order shannon result completeness
theoretical shot major among small portion section programming sdp follow feasible various solving theoretically sdp detect propose new inspire sdp consistently formal deferred cl first viewpoint viewpoint originally going consider ordinary sbm definition sbm event observe adjacency symmetric function rank ordinary give log function choose appropriate maximize entry let constraint check must entry infeasible relax semidefinite relatively requirement become ij hence nuclear penalization contrary impose cone outlier convenience theoretical objective equivalent great introduction penalization sdp recover use solve reveal penalization natural section optimization improve choose definite must ordinary mild detecting among pursuit
feedforward connection feedforward backpropagation gradient formulae eq step feedforward layer use co multiple drop improve dependency preserve tendency stochastic reduction recurrent apply way possibly drop dropout set drop step dropout search find dropout connection evaluate technique use music source classical music divide training testing approximately make necessity make network step long split dataset sized number unit tangent linearity sigmoid rnn entropy step denote note report ce error norm
sequentially correspond power consumption perform maximize measurement eq c k normalization ensure update component whose high error measure apply see else primary secondary om lm theorem observation protocol theorem insight em claim framework sense maximize family signal sense sequential query algorithm model sparse propose signal bring robustness application monitoring sensor call measurement hence gain guide sense big exploit measurement small ambient dimension work shoot measurement much combination entry seminal assumption help sequential adaptive compressed min algorithm restrict performance recognize adaptive sensing offer benefit metric large gain achieve recover signal know sense consider
concern adaptation evolution tailor careful evolution grant display color display display color color green display green display display display color plus paris france institute sciences investigate random size normal well among child I fitness become parent next increase relax normality movement general linear
classifier transform alternative logistic maximum likelihood regression capture decision boundary serve build know marginal transform feature boundary nonlinear transform feature interpretable combine model different marginal contribution boost view original marginal estimate run logistic penalization paradigm step feature augmentation implementation decision boundary separate stand naive wise mixture I pi tw error bar repetition suggest nb boost nb view compare road eventually well large surprising biased nevertheless road model small finally oracle example newly well reasonably suggest
denote occur denote either stop discrete whether martingale denote time failure subtract finally success union markov substitute isolate produce application average understand sampling arise importantly occur independently specifically draw type incoherent eigenvector incoherence problem incoherent condition neither rectangular recovery something convert constructing eigenvector dominant sampling satisfy entry bind rectangular parameter singular vary big time hold common uniformly problem handle flow sample uniformly parameter vary size convergence time let uniformly spherical distribution furthermore compute analysis sampling arise subspace projection consist column give select independent problem motivate satisfie parameter separate entry case additive independent increase
deal focused decrease chain carlo common condition reveal viewpoint abc nearest neighbor knn transform calibration accord replicate frequent highlight via knn mild knn nonparametric infinity return greatly differ provide necessary consistency include tend address shift approximation return solely knn classifier model obvious whose already minimization sake clarity validation simulate reference evaluate misclassification difference knn abc validation knn evidence indeed know purpose analyse irrelevant classifier prior indicator know whose apply predict give datum misclassification pair distribution loss fy fm suggest misclassification valuable posterior nevertheless simulate solely summary explain practically limited know function subtle basic actually train subset reference axis propose abc
intel gb ram os v depend retrieve record dataset negligible specify linearly time datum short value geometric scheme test natural dimensionality suggest converge minute imputation collection datum take suggestion test iteration remarkably iterate require weather conventional statistical cope invoke
definition consequently directly base boundary towards basis boundary cifar class play exist work regard dissimilarity measure fix set kernel base feature fast comprise image fold class fold fold th fold experiment neighboring cell properly effect paper consistency normalization possible ignore feature center normalization kernel primal objective square hinge loss k number fold show vector fold except pixel folds benefit general consider similarity cell measure give z rx hx rx allow q z cell cell cell pixel significant improvement probably
average fp skeleton fdr mcp fp dag skeleton fdr mcp fp dag skeleton fdr comparison sensitivity closely concave highlight conclusion literature surprising tie analysis method robust small dimension roughly mcp observe base reliable fact hc omit dimensional confirm expectation regularization dimension concave comparison provide display supplementary material fast approach particularly score hc single estimate respectively compare mcp method fast roughly magnitude pc translate runtime compute furthermore mcp runtime dimension limit large test fast mcp algorithm require versus pc term difference pc pc significantly fast material dataset claim dimensional subsection efficiently loss previous assessment number mcp purpose increase dependent relationship sense perform realistic scenario run result depend crucially take dags edge threshold six second per second node tp fp skeleton comparable notice fdr due increase sample combine confirm efficiently complete improve total runtime five minute purely report successfully runtime minute day internal internal cache standard vector stand optimize yield fast imagine incremental order edge penalize equally false one sensitivity efficiently pc bic note empirical suboptimal graphical behaviour develop tune mcp confirm edge suffer reason already performance pc use significance level run bic also select appear perform bad relative report section run restrict candidate pc material edge qualitative already provide briefly level provide detailed assessment algorithm increase decrease show behaviour observe section material speak dimension mcp improve graph remain estimation nonetheless
essentially point average distance simplify average equivalent formulation index hence euclidean cluster sdp optimal satisfied large put together isotropic radius distance problem occur currently center assign near center ii new assign terminate fail optimum separated isotropic median lp sdp bad center away ball ball create copy group cluster copy copy pick center cluster center initially group center configuration center away two nearby easy never cluster fail high space even separation center section report regard median mean sdp input consist disjoint center ball I ball use successful optimization separate ball respective repeat plot empirical success cccc relaxation color probability ccc high show range range lp achieve fact median seem comparable lp require plant test necessarily interesting diagram failure coincide clustering cluster plant disjoint support
lastly architecture decoder despite encoder suffer curse investigation require addition general translation system propose recursive neural find sentence syntactic acknowledgment author acknowledge cifar universit de university universit machine purely decoder encoder decoder correct paper analyze property neural machine rnn decoder convolutional network perform recurrent apply sigmoid variable distribution matrix result activation new activation logistic gate previous unit reader b activation ht besides
minimize x c example th cluster tend expand use parameter note give point opposite box away corner explain loss operate distance decision boundary boundary boundary namely analogously low analogous calculation boundary begin issue deal exponential numerical multiplying term divide term factor add least avoid multiply omit proposition solution close outside box set boundary effectively description classifier make start r box compute interpretation set boundary ensure box away nearest give subscript
share many also intuition crowd bit attempt distribute representation phrase chen point model context probability
simple crowdsource lead quality dataset crowd worker interest collect human triplet collect triplet human researcher variety set author learn triplet alone annotation specifically create embedding create dimensional axis create embedding triplet collect crowd worker become intractable difficulty collect triplet datum human relationship choose triplet intelligence provide affect collect triplet embed primary concern object way collect comparison crowd traditionally triplet collect triplet grid ask collect triplet whereas triplet traditionally collect triplet
product particular sequence way precise relation separate denote use tm formalism discuss become column rewrite reduce express already obvious analogy completeness concept explicitly stationary found physical machine constrain result apply spectral decomposition burden power expression power scalar also closed reveal type ever stack familiar make eigen understand behavior decompose operator moreover analytic find tm operate z contour integration complex unit eigenvalue large contour contour plane depend tm lie unit guarantee index index must algebraic eq strongly stack finite hmm besides form useful inverse stacking circle
display histogram chart song test title subset apply effect consist track respectively descriptor deviation beyond replace separately set descriptor descriptor subtract rating audio descriptor without intermediate estimate audio perform coefficient denote behave assign incorporate fall year individual track group track consistently combine group potential track slide window descriptor track window track window difference chart entry testing window descriptor datum section average vector centre annotate chart entry absolute error mae root error rmse display exploratory song year descriptor slide window restrict analysis spread year plot examine chart entry trend descriptor fine non day slide window chart entry mean descriptor examine autocorrelation lag correlation yet decay non stationarity
projection mahalanobi yield asymmetric bottom well extract elliptical contain namely quasi shape depth come need extra determinant depth satisfy represent projection depth univariate projection exact construct hyperplane attribute additional coordinate mention relevant final hyperplane element respectively extend select involve respective separate straight origin separate form minimal choose onto straight orthogonal separate one illustrate diabetes www ac tr seven pressure diabetes age represent unit square class direction depth depth space extend depth space small two reduce extended subspace separate straight line choose axis separate point initial similarly hypercube step iterate extend decrease extend stop step variate depth direction project depth depth implement depth question computationally
purpose study medium internet china case increase period long mechanism context suggest difficulty pattern china fairly noise context contain significant incidence change evident context flat capture trend traditional surveillance progress year meaningful could capture internet incidence identify plausible three pattern improvement linear could lead suggest tune illustrate fail wikipedia wikipedia actual observation infection cause become internet trace poor sub disease observe pattern activity united good internet connectivity incidence peak differ day internet even major news signal forecast china failure united states success united snr china success location list success forecast offset offset forecasting summarize model location context test produce disease line successful technique approach sufficiently promise explore related popularity wikipedia relationship article total level social internet key capture disease incidence distinguish trace health modeling broad article article query pair c china united united china china score test pair disease meaningful indicate list pair disease concern establish feasibility
temperature simplicity performance compare svm radial ridge denote cart forest denote package logistic hyperparameter std nf rf svm cart rule inductive form rule nf rule input nf note rule necessarily decrease monotonically rule increase use rule list dataset fold list fold roc cart unclear perform svm validate poor cart poorly relative
curvature arise w gauss instead negative curvature intuitively hessian quadratic offset since curvature arguably long distance curvature curvature manner seem regard method statistical enjoy rich apply approximation section establish ultimately define training correspond density divergence learn distribution learn eq substitute efficiently empirical substituting minimize standard maximum note extend agree kind fit conditional composition output mapping proportion fisher first psd trivially psd product negative w directly difficult come actually compute situation version give
cm step yield cycle iteration inversion kp diagonal matrix gaussian factor implement obtain estimate acceleration illustrate measurement make x estimate percent day temperature capital city area matrix dimensionality factor variability contaminate former package run criterion good bic contaminate analysis prefer quantify choice perform reject usual factor contaminate gaussian
patient store disease disease international code patient dataset dimensionality dataset history diagnostic event disease vary million million event disease patient disease consider disease dimension contain diagnostic testing qualitatively evaluate knowledge code high code code goal font w observe w vary establish scalability regime qualitative hierarchy hierarchy truth rf tree tree baseline agglomerative baseline advantage increase support md team e htbp portion represent code dataset together group disease group know disease clinical disease group latent reflect status disease common capture node dimension per variable figure portion disease
channel search select ad user determine piece optimize study combine learning game learn framework historical response mechanism mechanism learn bid next mechanism maximization predict bid period infinity genetic bid future datum know call adjust response mechanism mechanism learn historical overcome drawback novel combine strong handle second effect first historical mechanism predict instead mainly engine click average short
file visualize visible denote layer visible layer encode spin hide hide spin influence visible break labeling language network inspire statistical consist unit layer success unclear architecture possess architecture understand structured explanation dnn view coarse high level learn increasingly abstract layer feed dnn combine successively apply extraction simultaneously learn supervised set concerned solely training compression make successive coarse tackle physic describe phenomenon short
call let identify graph belong apart distance precisely group assign identify center case depth e look center valid cover look depth double peak pair triangle triplet circle triplet display graph fig large triangle triplet graph circle belong nonparametric near look near among sample assign label majority neighbor setup principal variable project equivalently iteratively element maximal
panel size extraction divide patch pattern pattern inference pattern sub image pattern pattern sparse binary concentrated keyword panel normalize heavily term panel sub weight pattern heavily panel differently visualize project
node unseen u pseudo count approximate main method cf dependency variable break statistical independence variational approximate crp however allow fit fix break model component prior comparable infinite predict node notably likelihood well however even evaluation often interaction link complete dataset c close difference use ibp
unlikely level operation comparison color negligible theoretically achievable speedup color increase achievable cnns vision widely result network parameterize redundancy non convex redundancy exploit technique appropriate tune performance decomposition base decomposition cluster similarity learn contribution generic redundancy inherent deep cnns imagenet show layer connect factor convolution tensor denote dimension value convolutional spatial location
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thus bilinear form eq choice without highly machine theory pattern geometry cloud laplace rough operator connection uniformly cloud laplace operator approximation heat rough laplacian cloud identically almost surely rough almost surely spectral assume normalize recover rough generally recover whole fm smooth tensor iterate approximate iterate wu develop
parameter variable logistic majority time logit normal label label ref expert probabilitie vote figure selection horizontal possible iii plain satisfactory hence become iii detect em behavior agree return bad feature expert ref expert expert probability generate increase scenario minimize risk graph case probably bottom table set red content quality unit appropriate label great
failure use failure know learn take failure step shape q set learn step learn previous step predict failure interval approach failure approach also consider along dependency variable consider known product another incur company part decrease period fail replacement minimize cost gradient specify comparison failure section service discuss section present discuss module learn
give semi supervise task demonstrate dataset establish give pairwise submodular necessary cut novel connection classical promise incorporate exploit markov literature randomized map largely acknowledge ep google award comment presentation clarity appendix cox cox comparison literature theoretically experimentally literature insight cox function product local lebesgue borel eigenvalue eigenfunction lebesgue measure use predictive density density contrast gaussian family cox process superposition appeal model unbounded chinese imagine cox tractable alpha hard compute approximate two seem approximate cycle compare interest
gram study open display power law size gram track linguistic phenomenon language behavior pl language relate originally database word list show test pl gram gram gram gram gram gram aic standard column correspond aic value
shift plan effect reverse top bottom reverse detail task evolution vary sequence dynamic hilbert give insight methodology show possible time vary learn restrict space flexible part european european fp grant token problem future interesting predict step set machine embed reproduce operator illustrate principle formally set set distribution possible distribute approximated main smoothly evaluate
band give operation goal approximation situation small ii power basic differently application paper scientific problem statistic normally price power nan become severe found particularly appealing suggest high side version henceforth design motivation ratio statistic hypothesis alternative change poisson generalize statistic order event th size behave likelihood maximize tail ratio statistic become consideration tail get argument compare completely ratio seem excellent desirable goodness theory include size apply discuss briefly behave organization expression approximation monte demonstrate comparative broad discuss confidence
super source relate drift activate time mixed fmri fmri input eight fmri eight map slice course linearize concatenation row voxel eight spatio map mix linearly eight fmri formulation final simulate fmri point voxel linearize algorithm slice voxel volume datum task comprise relate trial inter red green box appear participant leave presentation box red appear box appear red left trial per nan acquire dedicated image contiguous brain tr te ms flip angle slice voxel high acquire te ti flip angle slice dataset public include brain template head movement datum nine subject fmri thresholded area zero voxel roughly fmri voxel filter specifically fmri acquisition practice voxel enhance voxel mm since voxel resolution neighboring voxel average voxel version smoothed version section detail visual dataset comprise collect task reflect change six subject general ms thick te resolution gray face pattern category select comparison
take involve thus first minimize set compute
conv visualize filter assign convolutional final layer sample intermediate produce varied layer hold collection node fully connect convolutional recent cnns generative aspect show cnn cnn also visualize turn generative learn approximated sampling fundamentally yet architecture latter generative visualization cnn sampling draw give
section assess variant direct discretized benchmark storage management benchmark problem consider relative benchmark run continuous continuous discount discount storage period space horizon optimal policy typically cpu primarily discount consider storage operation management table refer demand ba consider trading storage price variation discretize benchmark true discrete process state discretize average wind divide storage capacity load hour device max storage device device within hour price fit real price load fit load mid transition wind wind resource wind price e demand computationally simplify interval day resource state c wind wind storage full c full ba ba ba ba reasonable instrumental bellman test illustrate policy instrumental bellman
ts randomize idea bandit reward initially observe failure select algorithm update generate select adapt fa aware propose situation ts fa ts let user situation past situation compare situation method document recommend observe user choose situation improve strategy retrieve
classification prediction supervision ranking compatible map ranking simply take full query competitive prediction surrogate less suitable set supervision rank full ranking sec vector play subsequent surrogate truly explain weight relevant induce measure crucial perceptron minimize measured surrogate hinge member end proportional vector member upper measure rank however technique different provide ndcg document sort relevance level bound relevance vector relevance document vector permutation sort score follow hold say dominate relevance map relation thus sense choice upper induced loss surrogate consequence ndcg vector ndcg determine permutation sort however still condition generalization
parallelism naive ij substantial hardware node example node gb memory amount presence parallelism fortunately note access block contribute appear ij use update much still call depth logarithmic dropping iteration clutter cholesky first fast cache memory use less node node associate also part regularizer initialize r j ij ij note enter admm proximal proximal admm implementation loss multinomial logistic hinge proximal operator hinge solution total requirement count amount term start cache requirement term undesirable however column splitting reduce memory
bin close consider simultaneous empty bin color simultaneous empty bin bin total position bin way empty add extra choose space bin bin bin q desire expression bin argue total position two empty simultaneously bin improve bin bin position empty bin bin bin empty bin
try use method gradient descent optimization well certain different add decrease variance advance weight weight spirit interesting understand thank art discussion suggestion research support
novel probabilistic interpretation autoencoder generative autoencoder low decoder furthermore keep information optimal unimodal neural network factorial show yield sample stack autoencoder long low gradually yield interesting reveal picture idea unfold transformation regular factorize help understand respective role reconstruction criterion regularize auto acknowledge cifar first rise autoencoder encoder encode activation autoencoder plausible mean near map autoencoder training example
calculus introduction denote variable extension disease cause cause disease notation intuition causal provide acyclic relationship dag prove tool causality variable dependency read graph asymmetric nature causality suggest dependency find descriptor causality suppose causal link order pair dependency correlation define descriptor call descriptor asymmetric property causality causality asymmetric descriptor define descriptor markov direct make connection two ii asymmetric create cause condition separate asymmetric q I I j j mutual term asymmetric descriptor
sequence library average know I joint probability obtain put piece average sequence calculate eqs calculation position along possibility nucleotide mutation position state tackle directly however unique probability nucleotide
object illumination classify information need need irrelevant simple naive invariance would list realization object vast amount realization generalization useful invariance represent invariant transformation kind represent set generalize list realization whether possible realization discard operation pooling implement way variable also along relevant often simultaneous operation extension detector sensitive type increase number expansion pooling stage visual cat feature implement competition call simple generating alternate various specialized pooling version pool convolutional review hand transformation
therefore put divide side tu sa co let cauchy consider convexity q h summing multiply kk right convexity hence equivalent max h convergence desire
choose explicitly later coordinate thresholding rule nonnegative assumption give start tf penalty theorem provide careful essential infimum thresholding penalty convergence nonconvex mcp bridge penalty p define p infinitely may frobenius nontrivial lead p type regularization screening replace satisfy supervised application replace let r hybrid perform inner enough k algorithm complexity initial may construct initialization multi frequently challenge variable screening group screen oracle predictor factor loose small type problem ps ps p define similarly break multivariate p rt modify inner
combination power efficient allow interpretability focus control domain could text corpus eliminate pre relatively straight implement factor relationship grateful child provide gs topic model powerful tool latent structure finance goal even modern discover interpret interpretability hierarchical topic encode word lda interpretable summary real matching performance probabilistic originally develop discover corpora framework domain latent allocation lda vocabulary contain observation scientific article image action modeling vary topic summarie document situation topic understand topic meanwhile discover publicly
query relative per regression tune datum cifar update map slice generalize classification softmax datum belong softmax log match softmax tight chapter cifar discover autoencoder prior langevin mala choose softmax experiment gave qualitatively tune dramatically outperform mcmc fold likelihood likelihood tail call robust perform robust dataset molecular property consist million
predict negative label predict third scenario multi dataset learner number weight rest follow two multi correct label instance learner produce learner rwm mechanism since rwm learner learner expert parameter output label nx mistake probably occur mistake rwm rwm mistake mistake subsequently show mistake rwm mistake enough mistake mistake far fraction total weight wrong answer trial mistake learner
protein structure form model architecture change connectivity unfold auto improve quality representation acknowledgment thank discussion acknowledge computer center university resource support nsf health gm national institute medical sciences center gm institute predict secondary protein new supervise network predict secondary hierarchical representation deep train deep generative extension learn chain apply protein scale sized protein layer architecture focus inform representation learn label state
leave almost neighborhood describe alg efficient way symbolic say alg statement noisy beneficial want remark classical consistency applicable lead sample limit argue proper notion manifold say intuitively correctness noise alg complement consistent thresholded span imply pass alg seem svd therefore work enable look span look reveal important identify notice mathematically free example take suppose feature vary principal generative case matrix row classic
series methodology goodness conditional distribution precisely let univariate real let sigma family subscript abuse distribution apply finance relevant moment necessary check jointly specify conditional feature quantile know essential likelihood fan conditional link hazard autoregressive duration knowledge assess finance description management serve risk finance specify location distribution unconditional student model conditional two moment cumulative ty dynamic scale process commonly interest examine test nonlinear obtain dynamic scale series necessary location distribution model estimation rely
include recommendation co occur search user goal capture towards correspond recommendation list bag consider enter name exclude bag co specific avoid name exception popular name build occurrence collaborative feedback metric neighborhood similarity metric compute co occur name recommendation biased fill list name user implement language library model table individual
corpus examine spam multi document topic modeling within corpora list document remain branch job corpus evaluate internal corpus adopt except learn concentration parameter note comparative result multi corpora sampler corpus result evaluation word fold corpora five test fold corpus fold exhibit regardless picture range model corpora outperform support topic share respectively overfitte tb ad building post fan post word page day comment write facebook origin follow topic hold likelihood corpus consist job table show job com
capability I classifier appendix explore appendix divide study error score h omit covariate location classifier label vector explore source variation al problem study evaluate many estimate se algorithms seq two four logistic discriminative describe random non diverse recommend understand experimental study density omit primary density weighting defer al provide study continue pool label experiment classification classifier seed seed affect pool seed experimental experimental evaluate method iterate al loss label illustrate b base provide metric performance label complexity suggest substantial agreement ranking reason employ several calculate ranking yield five metric primary metric al assess classifier performance single single base metric metric seven al seven tie
combine two r letter fair algorithm setup perform logistic optimization tackle classification multiclass step th minibatch sizes sgd ss sgd ss use cluster also via degradation fix seed measure training effectively epoch
little gain sampling match satisfie bernoulli useful far strategy bandit check fact together strong introduce appear fact illustrate fix resp budget empirical match strategy approximate universal bandit model arm crucial stop arm strategy analyze paper stop approximation exponent drawback relate coincide close stopping criterion kl exactly particular approach pac exploration goal result bandit model fix budget star circle specify running scale average
slowly value distinct strictly string aspect subtle search context yield future symbol stream lead irrespective initial course know priori least approximate always identifiable string parametric statistic error entropy alphabet px px occurrence base generally entropy express random variable entropy alphabet usual manner px px rule entropy calculation definition I x rate stationary still hold h define entropy stationary distinct literature formalism find include brief sake completeness denote string string string denote string cardinality strictly ergodic stochastic ergodic calculate sufficiently stationary moment formalize assume realization fix use ergodicity long string induce assume stationarity construct assume ergodicity extend via countable sum notational relation string string language consideration extension equivalence induce equivalence string
infinitely least zero consequently physical feature denote two nature pass truncate end side feature distinguish proceed
none undirecte national study health health health behavior record one school student th student friend student four edge indicator connection present student important social tendency statistic tendency display robust geometrically share specification statistic thorough justification curve parameter goodness friend
employ auxiliary show image difficult addition presence pixel sensitive carefully specific condition crucial drawback completion challenge truth extensively randomly pixel low approximation prior relate use rank mp performance image runtime mp improve performance recovery high mp completion significantly obtain comparable tuned image visually demonstrate mixture prior advantage recognition surveillance videos classifier pose illumination arise question condition highly model image introduce synthesis contain image people pose miss either intractable order apply ratio initial completion tune ground truth fig synthesis rank determination tensor rank exist account address cp treatment incorporate
preserve multidimensional tensor denoise flexibility wide prominent inverse imaging image fouri measurement scan desire clinical throughput convenience sense far exploit sparsity measure domain wavelet variation enhance quality learning adapt accurately measure structural similarity slice volume denoise aim
investigate ahead prediction xlabel horizon ylabel legend pos north width plot coordinate coordinate p computing error space dimensional accord penalty term order everything eventually less intuitive suitable initialization section network choose especially example true preferable system time combine identification auto encoder high move horizontal good dynamical possible denoise autoencoder noisy real b convolutional c exploit controller carlo investigate reinforcement prediction reinforcement decision
x ta therefore least appear eq h high piece get finally piece main warm largely apart bit objective initially might clearly regret constraint meet call initial recall decrease objective oracle call complete proof variance epoch variance policy every policy q lemma abstraction reward pair oracle create epoch time per suboptimal algorithm explore achieve logarithmic run trick adapt call oracle previous optimal however prohibitive scale logarithmic factor contextual require coordinate policy epoch policy update epoch trade concentrate burden round call round oracle call round stress total develop variant conduct show baseline
exploit matching matching exploit weak worth explore accurate region rotation sophisticated discuss motivate discriminative patch among work benefit problem grain recognition deal lemma rewrite lemma exist proper simply lead tune scalar derive auxiliary monotonically increase submodular split definition denote v increase derivation q h v h vx eq q auxiliary monotonically submodular monotonically increase proposition gx x end proof appendix monotonically monotonically without simple mean integer increase otherwise equality image turn therefore overall prove proof
conditional structure specie information theoretic train physical review cm inf measurement york cm partially exchangeable development game page institute mathematical cm volume page institute mathematical california cm process mathematics ann communication system technical boltzmann statistics journal physics mechanics york mutual document physical review completeness regression univariate generalize quantify presence cm key word entropy diversity estimation shannon entropy non overlap independent classification measure deviation
extremely group geometrically transformation image take rotation cyclic transformation length one rotation correspond cyclic translation mapping ht red k accuracy approximately descriptor descriptor descriptor transformation sub cluster descriptor texture transformation analogous cyclic horizontal following index tuple rotation cyclic equivalence partitioning class detail six coefficient member discard addition six class give autocorrelation also discard coefficient since member average thereby
become automatic alignment segmentation news speech corpora audio difficulty solution segmentation propose novel technique acoustic stream self similarity apply music retrieval audio audio audio segmentation duration news
speak high favorable stability range one area help control dimensional particle develop assimilation accommodate gaussian efficiently operational sampling posterior mcmc strategy generate specifically hybrid incorporate system new operator overview assimilation widely give hmc algorithm experiment filter enkf conclusion brief overview assimilation da highlight behind research assimilation information knowledge numerical physical background uncertainty model initial begin evolution perturbation state evolve tangent linearize time observation observation assume distribution observation assimilation combine background assimilation gain popularity ensemble belong latter two exist kalman likelihood filter review kf assimilation moment sequential assimilation algorithm proceed forecast equation produce quantify forecast ensemble kalman enkf take monte carlo member forecast ensemble simulate reality error covariance estimate background time due localization product forecast formula observation version kalman add assimilation kalman make linearized observation k kalman gain use version formulation
maximally skew skewness dense design fraction simply nonzero matrix analyze require need follow measurement essentially use measurement slightly use small
well inference greedy model ultimately stochastic capable high modelling language substantially particle carlo probabilistic inference approximate bayesian abc summary statistic compute generating vector f repeatedly interpret generate repeatedly time sample unnormalized compatibility text lambda real program text time variance noise skewness observe level probabilistic write generalization employ establish correspondence variable describe particular implicitly line implicitly return vector skewness draw normal mean text output skew square code highlight
class direct nonlinearity hard mnist parallel svms within toolbox replace point matlab processor run machine processor limit speedup right may model parallel toolbox quite inefficient proxy function classifier optimize use secondly without minimum view little yet obtain true minimum want ideal consist centroid locate corner simplex really filter nearly efficiently simply find optima classification allow role dr ideally dr variation domain nearly belong choice ideal theoretically sufficient limited separation nonlinear dr train work help move data space boundary simpler hold view good manifold dr dr informative something
hypergraph set xx xx h dictionary partition satisfy regard minimize little restriction question position learn conversely indicate section rise follow pure give size additionally nontrivial combinatorial give minimum geometric characteristic p dictionary subspace dictionary fit et satisfie frame certain pure hypergraph incoherent existence locally give minimum guarantee span unknown position question svd etc span span learn unknown illustrative p b section find hypergraph specify combinatorial characterization uniqueness magnitude projective notation mean fit subspace incidence convenience incidence subspace incidence index lie span follow combinatorial input incidence constrain solve variety characterize generic give x
cycle come twice flow nucleotide probability target nucleotide signal signal approach various limit distribution derive paper asymptotically organize seq formula cycle obtain analytical carefully simulation summarize far r length variance distribution incorporated flow nucleotide incorporation represent usual case letter throughout independent row nucleotide successive four cycle number
induction operational semantic basis technique pattern generation worth whether satisfy resort qualitative semantic appear deeply high capture pattern generate disjoint set accuracy rule translate characterize obtain eq formula predicate label write formula path translate interpret else pattern simulate reaction system generate contain negative choose observation steady state reason trajectory reach steady state discard separate step intel processor ghz gb memory first rule root explain translate follow formula
simulation norm converge figure control iteration dot value control approach gain convergent control evaluate machine theoretical incremental name td rl general trace two propose mainly mdp optimal thus promise introduce address optimal control usually bellman expression extensive digital sensor availability information past attract
construction assumption incoherence exist strong incoherence property necessity regular hence serve least square satisfy psd psd singular large requirement give standard psd stress os enyi fairly straightforward schwarz argument inequality second exist approach moreover sample necessarily element observe vector rip strong connection incoherence assume
change easier may increased lastly adapt admit gain parameter often discard update summary promising scalability cm engineering division ridge national laboratory tn com computer science north nc edu recommendation express publication reflect view novel graph address anomaly label stream introduce detection describe coarse build aggregate fine closely relate hierarchical simultaneously deviation technique insight internal detect event user narrow focus anomalous particular subgraph anomaly evaluation anomaly detector base detector baseline label accurately anomalie synthetic subgraph graph level possible anomaly interactive visualization tool identify precision social playing increasingly role yet insight
regularity seek represent regressor guarantee represent observe entire processing accumulate introduce twice regressor implies sequentially asymptotically regressor piecewise regression piecewise region accumulate asymptotically performance regression provide algorithmic detail let regressor fig denote node represent tree leave incremental asymptotically sequentially piecewise particular piecewise leave achieve compare introduce introduce piecewise whose competition base regression regressor unknown length piecewise performance huge high fine region show regressor competition significant upper incremental intermediate introduce present twice differentiable arbitrary regressor twice differentiable fig datum complexity introduce state twice differentiable note
build introduction let consider target state lebesgue partition choice discuss later multimodal word vary lot remove consider probability bias typically dynamic implement principle learn eventually adaptive adaptive daily practitioner computation physics context partition call give measure choice partition reaction coordinate course problem context weight free reaction focus method wang practical viewpoint main wang tune efficiency prove convergence wang spirit estimate technique wang adaptive dynamic efficiency wang numerical term set follow correctness wang argument wang draw conclusion convergence present probability biased density bias observe
change lemma mh mh grant complex network separate region characterize component goal fmri clinical ica covariate ica decomposition inaccurate inefficient hc ica testing ica hc em analytically tractable step develop subspace approximate em computation high test voxel expensive covariance advantage hc ica fmri connectivity relevant brain introduce hc ica preprocessing hc ica algorithms ica center whitening facilitate subsequent ica preprocessing hc fmri subject fmri acquire across voxel vi paradigm ica dimension whiten fmri eigenvector
vector uniform near isotropic position factor volume shrinkage epoch shrinkage cut body drop high third weak later improve affine q find isotropic least affect big notation bring isotropic mix ball independent sense algorithm call oracle upper
change indice entry confusion modify precisely two increase overall hold directly unchanged suffice end six unchanged proof value randomly k near trees forest tree sequential bayes test uci repository mnist digit short description dataset ensemble appear p classify background spam classify spam classify mnist binary mnist york ny usa various raise priori classifier accuracy unsupervise solve independence assumption fact classifier competitive artificial contrast supervise unlabeled label occur consideration train label
coincide nominal bias adjustment contrast severe selection reflect confidence long nominal significantly less essentially potentially severe selection address conceptual allow base reader may randomness carry second analyst control thing analyst carry include concern select select appropriate rule scientific ever use model split special reason choose require model clinical trial hypothesis selective writing state formal mathematical great mistake inherently rather give analyst must many know particular choice sensible analyst specify analyst course misspecification procedure splitting model leave bad design property selective selective constrain want behave give reasonable since condition selective setting converge truncate much bad threshold truncate become reasonable take provide extended regularization regression respectively present theoretical generalizing optimality generalize work view selective recent distinct work notably fix regard estimate apply selection statistic require work entire typically marginal py py selection prior reflect credible interval article present adjust goal inference make control multiple perform enough threshold simultaneous less
second category approach greedy signal construct principle greedy gain include orthogonal pursuit omp sparse signal index main difference omp ol greedy update support omp find strongly signal seek maximally omp computational call multiple orthogonal square multiple optimal ol reliable hence utilize
ranking inferential ranking bic interpret I plausibility validity interpret assertion specific hypercube random uncertainty plausibility ranking find validity sa plausible strategy select single base datum minimal select sa plausibility seem cf become reliable identify beyond classical frequentist bayesian challenge example objective bayes dependent inferential I feature assertion meaningful summary datum assertion summary calibrate easy interpretation summary guarantee frequentist error use random set predict auxiliary variable statement two development case multiple I complex identify shape involve consideration shape assertion simultaneous balance discuss distinguish optimal general hyper set I model validity I present I drive selection procedure base I present outperform several work meaningful summary must technical deferred I development design instead evidence summary property I focus specify efficient valid summary I model eq q predictor fix non singular without column center ignore intercept I refer
incoherent none try motivate ask early without model model raise issue tend ignore risk care worth point nice sound justify assumption sense risk I bl
generate obtained successively construct derive hard fuzzy dr scalar intra first dissimilarity vector dissimilarity w define soft value quantify target class membership function please close general adopt provide exploit extent dr perform might classify consider get membership target cast optimization well maximize entropy modularity ds maximize datum modularity partitioning solution community cluster suitably optimize separate modularity whose find approximation modularity follow calculate enyi sec edge separate component pruning repeat time partition use however modularity terminate greedy edge weight community modularity modularity return code herein describe vertex modularity increase remove partitioning consider
illumination condition cluster come insight given illumination condition terminology correspond would contain image individual minimization multiplier admm matlab reproduce http www collect individual specifically choose row dft time corresponding ssc apply average instance subset realization show theorems ssc ssc outperform run increase
regard whereas suggest contain fix include inference enable subtle detect integrate trace give great empirical increase accordingly extra add generally apart less match empirical high increase thereby barrier associate inter empty poisson increase imply equal account two way observe point inter distance match option match behave ccc alignment protein alphabet letter letter one sequence align measure develop scoring assign pair alignment evolutionary evolutionary would shorter evolutionary derive say substitution give chain evolution substitution mutation entire protein period place substitution via derivation use markov period evolutionary one substitution period denote substitution probability long evolutionary
literature function scale triplet metric modality realize wu modality apart similar pair information ignore algorithm two distance modality space al information graph accelerate gradient nuclear formulation modality learn medium retrieval database give brief firstly distance point feature besides binary call link contain pair learn metric follow
continuously concave monotone know increase increase continuity show lipschitz continuous p g thm penalize aim extract generalize pair involve tackle surrogate develop iteratively surrogate separable regular ascent eigenvector provide systematic arise closed derive experiment eigenvalue eigenvector extremely useful numerous analysis machine tool component pca cca instance eigenvalue interest eigenvector large although surrogate smoothed equivalent maximize stationary admit every iteration remain organize generalized surrogate use give brief review systematic issue arise convergence section numerical conclusion real field strictly positive low case scalar transpose transpose denote element denote diagonal main interest regularize include
subset theorem informative envelope subset affine ss would intersect contradict adapt replace interval obvious straightforward ss paragraph k ptc france sciences france mail fr david fr france mail ec fr china e mail mail edu cn document unknown know impose receive replace efficient solver homotopy minimize inspire homotopy minimization heuristic search backward previously inspire homotopy respect empirically problem involve dictionary usual least square homotopy homotopy least square sparse traditionally address constrain nonzero entry quadratic fidelity quality formulation adapt select contrary although objective indeed deduce square bi axis namely objective integer thus pareto kk classify former envelope point fig nonconvex well support reach objective support solution method answer depend size nonconvex area pareto font lb lb lb lb lb lb lb lb square pareto support notice objective pareto minimizer tt sum formulation constrain define constrain problem many able minimizer motivate strategy
select tend dispersion regression recommend beta whose present table differ precision function use regressor include covariate consider covariate diagnostic measure considerably bootstrap base testing inference must model dispersion beta regression model criterion dispersion must enter dispersion less costly viewpoint carlo criteria finite approach argue pseudo dispersion beta regression former sensitive dispersion application acknowledgement acknowledge anonymous suggestion regression criterion monte dispersion parameter include precision
I description possible condition vector interest optimization voxel brain volume voxel voxel great importance detail base quasi glm voxel one respect kronecker identity rewrite minimize update procedure high cost iteration problem optimize concatenation cast solver numerical gradient whose easily kronecker identity parametric equivalent partial derivative avoid product kronecker identity compute compute model case define concatenation f read play crucial role iterative non prevent glm glm significantly matrix case significantly thin euclidean orthogonal matrix triangular objective function reduce liu limited constraint constraint problem include convex box support solver arbitrarily degenerate case smooth cost since bias force direction search length wolfe optimization converge equality constraint
efficient product marginalization universal variability densely basis competitive possibility potential area real find predict unseen approximate notice conceptually noise source noisy noisy important observable distribution infinite sample least approximate training linear linear linear define factor factored basis I linear factor factor base ideally approximate basis restrict powerful fourier infinitely approximation factor gaussian limit reproduce kernel hilbert hilbert
imply efficiently combination multiple leaf allow thing instance representation though internal become increasingly class combination value root boundary locality different future direction promise would tree tree adaptively autoencoder tree internal node multivariate encoder tree generate leave original autoencoder handwritten
high large motivate procedure source nonzero source exception dot share two scatter plane estimate ball dot match would dotted ball along source correctly toy highlight significantly source entail low norm display black direction crucial source separation generally simultaneously amplitude locate amplitude source sample non discriminant mix likely one complement diversity hold discriminant relevant separation way restrict bss trivially substitute unfortunately extra reasonably directly extra flexible belong problem equivalent sample zero adaptive procedure mix estimate manner equation discriminant diagonal highlight previously bss conversely discriminant take amplitude source suggest discriminant weight th scalar exist vanish generally source choice particularly account sparsity amplitude entail source several high amplitude bss source novel blind adaptive analysis source build weight ability discriminate estimating source transform domain
ny imply ny mn must chain element maximal allow encode iff iff iff translation initially head clause head update correctly clause cell symbol state complete rational rational whenever rational uniquely ks ms immediately q rational gaussian elimination clear less rational rational value linearly subset desire rational rational positive may subset differ set particular alternative justification proof thm enumeration sequence element test signature finite turn model j proposition value integer integer outer loop show e problem rational f identically countable unary relation relation rational f validity finitely resp finitely resp sentence inter reduction result last finite countable order language finitely finitely valid obviously finitely finitely sentence finitely e valid work countable exponential immediately finitely finitely resp binary finitely valid sentence order relation finitely valid case first language valid complete computable validity inter reduction countable first language symbol add infinite unary predicate normally valid via computable normally generally work computable sentence carry restrict attention normally valid validity finitely validity model give finite preserve consider duality sentence sentence follow
benefit scientific field physics especially large internet variety arise network detection community detection definition often regard success depend consequently community unknown yield z j variable approach literature include blockmodel accounting log blockmodel maximum ab ab ab within community often tackle effect z satisfy community derivation allow self edge thus assume edge community assignment likelihood observe adjacency constraint give elaborate bernoulli reason approximate blockmodel computational burden parameter reader comprehensive review find form
fw h fw fw fw fw fw k fw take give relate bind construct inequality follow fw fw v start eq imply establish bound iterate hessian approximation satisfie stochastic sgd method stochastic newton arise supervised learning discuss method well amongst method choose set heuristic make neighborhood choose constant give experiment try decrease increment inferior us clutter introduce elaborate training size computation control frequency limited bfg update every curvature remove limited update sgd stochastic datum bfgs form repeat every guarantee equally form stochastic
dominate appropriately bayes present store answering query term storage memory come rigorous compression ill pose evaluation neighbor cite quantify tradeoff front generalization argue nn fall advantage seem nn consistent work recently say classifier induce opposite label obviously separable
stage validity confidence simultaneous test figure stage sample level shift candidate replacement element confidence let illustrate mean group xt ct involve distinct signal vertical perturbation basis datum modal distinct modal clutter curve spurious shift algorithm curve grey red mean identify modal asymmetric top panel percentile subsample mode bootstrap summarize curve line clutter highlight candidate local shift replication entirely lie leave therefore correspond mode mean shift cluster curve display behavioral experiment thank provide datum reach target virtual environment curve standardized activity potential specific array spike sort distinct curve tend characteristic spike analysis curve cluster curve summarize first curve fit truncate heuristic
program rgb rgb rgb rgb study phenomenon consume study asset management analysis molecular run code paper code contrary code stochastic paper build code output base density propose yet apply two les code es pour
q conclude follow proof immediately definition first follow immediately weight focus q r follow hence easy q recalling follow proof th k knn hamming k slight abuse zero enough eq theorem op satisfy q constant sufficiently suffice note therefore follow high variable order equivalently sample toeplitz optimality estimator convex minimax adaptive norm factor recover exactly convex admit empirical demonstrate effectiveness illustrate practical improve classify sound hierarchical high cholesky weight result sparse may large gap insight contribution high problem convex amount dependent allow logarithmic population proved logarithmic moreover member newly estimator also
tolerance p algorithm expect sample attempt computational although tolerance successful computational mention introduction cost solve failure variable lie article explore prescribed error tolerance
object cut option probably question like teacher mixture language understand detect person person understand gender person detect teacher part cut learn thing co together stand knowledge instance box never window help architecture cut utilize limit search common sense location front
tuning estimator hill approach robust regression bivariate tail univariate observation scale univariate exponential regression erm identically approximation likelihood lack robustness identically develop observation univariate approach homogeneous erm fr pareto marginal estimate estimate equation marginal pareto coincide obtain produce help behavior contamination variate distribution want robust empirical corresponding note distribution belong consider contaminate degenerate contamination space define thus measure
